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Audiogon Discussion Forum
EPDR does not impact Class-D amplifiers. I do not understand why you will not communicate what you used as a current probe to measure 80 amps peaks?
Generally I would expect an amplifier that costs literally 100 times more (per channel) than another amplifier to be "superior". However, your comment about "seriously current starved" cannot be
backed up with facts. It does not go backwards into 2W, it has about 75% more power into 2W than it does into 4W. Do I have any illusions it can do 1500W into 2 ohms, even IEC bursts? Not really. But
then again, you don't have proof it doesn't. I do know for a fact it puts out 75% more power into 2 ohms than it does into 4 ohms.
The Gryphon is a claimed 350W into 4 ohms continuous. The Behringer claims 750 (2x, not 17x). It also claims 3000W into 4 ohms. Will it? Doubt it, but maybe for IEC bursts. Will it do >1500W in long
enough bursts to support real music at those levels? Yes it will, and yes, that is quite a bit more than the ~100 times more expensive Gryphon.
The primary limitation of it at 2 ohms, is the same as it is at 4 ohms, thermal, not current.
audiozenology"I do know for a fact it puts out 75% more power into 2 ohms than it does into 4 ohms."
That is a very interesting statement, assertion, and claim how do you know this to be true? This is like you're other pronouncements where you state something as 'fact" but offer no proof,
documentation, or data except sometimes a link to a "source" that you Googled.
audiozenology"I do know for a fact it puts out 75% more power into 2 ohms than it does into 4 ohms."
clearthink That is a very interesting statement, assertion, and claim how do you know this to be true? This is like you’re other pronouncements where you state something as ’fact" but offer no
He has no idea.
If what he said were true everyone that calls themselves an audiophile would be using them, and there would be no need for Krell’s, Gryphon’s Agostino’s, ect that can come close doubling from 4 to
2ohm, we all be using this $349 3000w Behringer he’s touting or similar. Like I said, he has no idea, all he's good for is beating his chest and stalking.
Cheers George
There are two types of people:
1. Those who know how to use Google (and other references), can sift through results using sufficient knowledge to reach accurate conclusions and consult industry friends to cross-reference.
2. Those who insult others, but cannot back up their insults.
I don't know any audiophile who would make putting 75% more power into 2 ohms than 4 ohms their defining criteria for purchasing an amplifier. Do you?
The idea that doubling power is important springs from the concept that loudspeakers are ’voltage driven’. What this means isn’t that the speaker is driven by voltage (despite the expression), it
means that the power that drives the speaker is such that the voltage aspect of the power is constant regardless of the load. (Voltage is an aspect of power just as current is; 1 watt equals 1 volts
times 1 amp.)
IOW such an amplifier is termed a ’voltage source’.
The thing is, an amplifier **DOES NOT** have to double power as impedance is halved in order to act as a voltage source!!
Tube amplifiers can behave as voltage sources (after all, the this idea was originated by MacIntosh and ElectroVoice back in the 1950s) and they certainly don’t ’double down’... But they **can** cut
their power in half as impedance is *doubled* and that is how they manage being a voltage source. The thing is a solid state amp does that as well. Its only at **Full Power** where 'doubling down'
might make a difference and right after that is clipping, so its not that big of a deal since the amp really should not be running anywhere near full power if its a good match with the speaker.
So in a nutshell the ability of an amplifier to double power as impedance is halved is not what makes for a good sounding amp, and it may not be important at all; certainly with most speakers on the
market its not. In fact the number of speakers that have horrendous amplifier-torturing load impedances (and phase angles) is actually pretty limited. Its a simple fact that the harder you make an
amplifier work, the more distortion it makes so its unlikely that a speaker that is horrible to drive is going to sound like real music regardless of the amplifier employed.
So the whole thing is a bit of red herring. | {"url":"https://ag-forum.herokuapp.com/discussions/watts-and-power/post?postid=1857144#1857144","timestamp":"2024-11-02T08:46:45Z","content_type":"text/html","content_length":"77573","record_id":"<urn:uuid:afbab662-6648-4171-983a-96d50f634799>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00007.warc.gz"} |
Swift algorithm club - Merge sort - Moment For Technology
This article is a translation of Swift Algorithm Club.
Swift Algorithm Club is an open source project produced by Raywenderlich.com to realize algorithms and data structures by Swift. Currently, there are 18000+⭐️ on GitHub. I have made a brief
statistics, and there are about 100 algorithms and data structures. It’s basically a good resource for iOSer learning algorithms and data structures.
🐙andyRon/ swift-algorithm-club-CN is my project of learning and translating for Swift Algorithm Club. Due to the limited capacity, if you find any errors or improper translation, please correct
them. Welcome to Pull Request. Also welcome interested, have time partners to participate in translation and learning 🤓. Of course, we welcome ⭐️, 🤩🤩 ️ ⭐ ⭐.
A translation of this article and the code can be found at 🐙swift-algorithm-club-cn/Merge Sort
There are already tutorial articles on this topic
Target: Sort the array from low to high (or high to low)
Merge sort, invented by John von Neumann in 1945, is an efficient algorithm with best, worst, and average time complexity of O(n log n).
Merge sort uses a divide-and-conquer approach, breaking a big problem into smaller ones and solving them. Merge sort algorithm can be divided into split and merge.
Suppose you need to sort an array of length N in the correct order. The merging sort algorithm works as follows:
• Put the numbers in the unsorted heap.
• Divide the heap into two parts. So now we have two unsorted stacks of numbers.
• Continue to splitTwo unsorted stacks of numbersUntil you can’t split. In the end, you will havenIt’s a heap, and there’s a number in each heap.
• Begin merging the heap by sequential pairing. During each merge, the content is sorted in sort order. This is easy because each individual heap is already sorted.
Break up
Suppose you were given an unsorted array of length n: [2,1,5,4,9]. The goal is to keep splitting the heap until you can’t.
First, split the array in half: [2,1] and [5,4,9]. Can you continue to split it? Yes you can!
Focus on the left heap. Split [2,1] into [2] and [1]. Can you continue to split it? Can’t. Check the heap on the right.
Split [5,4,9] into [5] and [4,9]. As expected, [5] can no longer be split, but [4,9] can be split into [4] and [9].
The final result of splitting is: [2] [1] [5] [4] [9]. Note that each heap contains only one element.
Now that you have split the array, you should merge and sort the split heap. Remember, the idea is to solve many small problems rather than one big one. For each merge iteration, you must focus on
merging one heap with another.
Heap for [2] [1] [5] [4] [9], the first merger is the result of [1, 2] and [4, 5] and [9]. Since [9] is single, no heap is merged with it during the merge.
The next time will merge [1,2] and [4,5]. As a result [1,2,4,5], again due to the location of [9] single no need to merge.
Only two heaps [1,2,4,5] and [9] are left, and the collated array is [1,2,4,5,9].
Top-down implementation (recursive method)
Swift implementation of merge sort:
func mergeSort(_ array: [Int])- > [Int] {
guard array.count > 1 else { return array } / / 1
let middleIndex = array.count / 2 / / 2
let leftArray = mergeSort(Array(array[0..<middleIndex])) / / 3
let rightArray = mergeSort(Array(array[middleIndex..<array.count])) / / 4
return merge(leftPile: leftArray, rightPile: rightArray) / / 5
Copy the code
Step-by-step description of the code:
1. If the array is empty or contains a single element, you can’t split it into smaller parts, just return the array.
2. Find the intermediate index.
3. Use the intermediate index from the previous step to recursively split the left side of the array.
4. In addition, the right side of the array is recursively split.
5. Finally, merge all the values together to make sure it is always sorted.
Here is the merging algorithm:
func merge(leftPile: [Int], rightPile: [Int])- > [Int] {
/ / 1
var leftIndex = 0
var rightIndex = 0
/ / 2
var orderedPile = [Int] ()/ / 3
while leftIndex < leftPile.count && rightIndex < rightPile.count {
if leftPile[leftIndex] < rightPile[rightIndex] {
leftIndex += 1
} else if leftPile[leftIndex] > rightPile[rightIndex] {
rightIndex += 1
} else {
leftIndex += 1
rightIndex += 1}}/ / 4
while leftIndex < leftPile.count {
leftIndex += 1
while rightIndex < rightPile.count {
rightIndex += 1
return orderedPile
Copy the code
This method may seem scary, but it’s very simple:
1. When merging, you need two indexes to track the progress of the two arrays.
2. This is the merged array. It is now empty, but you will build it by adding elements from other arrays in the steps below.
3. This while loop compares the left and right elements and adds them to orderedPile, while ensuring that the results remain ordered.
4. If the previous while loop completes, it means that the contents of either leftPile or rightPile have been fully merged into orderedPile. At this point, you no longer need to compare. Simply add
the remaining contents of the remaining array in turn to orderedPile.
Example of how the merge() function works. Suppose we have two piles: leftPile = [1,7,8] and rightPile = [3,6,9]. Note that both heaps are sorted separately — merge sort is always the case. The
following steps combine them into a larger sorted heap:
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ ]
l r
Copy the code
The left index (denoted here as L) points to the first item 1 in the left heap. On the right, the index r points to 3. Therefore, the first item we add to orderedPile is 1. We also move the left
index L to the next item.
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ 1 ]
-->l r
Copy the code
Now L is pointing at 7 but R is still at 3. We add the smallest item, 3, to the ordered heap. Here’s what’s going on:
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ 1, 3 ]
l -->r
Copy the code
Repeat the above process. In each step, we select the smallest item from leftPile or rightPile and add the item to orderedPile:
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ 1, 3, 6 ]
l -->r
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ 1, 3, 6, 7 ]
-->l r
leftPile rightPile orderedPile
[ 1, 7, 8 ] [ 3, 6, 9 ] [ 1, 3, 6, 7, 8 ]
-->l r
Copy the code
Now, there are no more items in the left heap. We just add the remaining items from the heap on the right, and we’re done. The merged heap is [1,3,6,7,8,9].
Note that this algorithm is very simple: it moves left to right through the two heaps and selects the smallest item at each step. This works because we guarantee that each heap is sorted.
I found a graphic GIF of merge sort for top-down execution, source
Bottom-up implementation (iteration)
The implementation of the merge sort algorithm you’ve seen so far is called a “top-down” approach because it first splits the array into smaller heaps and then merges them. When sorting an array (as
opposed to a linked list), you can actually skip the split step and start merging the array elements immediately. This is called the “bottom-up” approach.
Here is a complete bottom-up implementation in Swift:
func mergeSortBottomUp<T>(_ a: [T], _ isOrderedBefore: (T, T) -> Bool) - > [T] {
let n = a.count
var z = [a, a] / / 1
var d = 0
var width = 1
while width < n { / / 2
var i = 0
while i < n { / / 3
var j = i
var l = i
var r = i + width
let lmax = min(l + width, n)
let rmax = min(r + width, n)
while l < lmax && r < rmax { / / 4
if isOrderedBefore(z[d][l], z[d][r]) {
z[1 - d][j] = z[d][l]
l += 1
} else {
z[1 - d][j] = z[d][r]
r += 1
j += 1
while l < lmax {
z[1 - d][j] = z[d][l]
j += 1
l += 1
while r < rmax {
z[1 - d][j] = z[d][r]
j += 1
r += 1
i += width*2
width *= 2
d = 1 - d / / 5
return z[d]
Copy the code
It looks more daunting than the top-down version, but notice that the body contains the same three while loops as merge().
Important points to note:
1. The merge sort algorithm requires a temporary working array because you cannot merge the left and right heaps and overwrite their contents at the same time. Since it is wasteful to allocate a new
array for each merge, we use two working arrays and we will switch between them using the value of D, which is either 0 or 1. The array z[d] is used for reading and z[1-d] is used for writing.
This is called double buffering.
2. Conceptually, the bottom-up version works in the same way as the top-down version. First, it merges small heaps of each element, then it merges two elements per heap, then four elements per heap,
and so on. The size of the heap is given by width. Initially, width is 1, but at the end of each iteration of the loop, we multiply it by 2, so the outer loop determines the size of the heap to
merge, and the subarrays to merge become larger at each step.
3. The inner loop runs through the heap and merges each pair of heaps into a larger heap. The result is written in the array given by z[1-d].
4. This is the same logic as the top-down version. The main difference is that we use double buffering, so the value is read from z[d] and written to z[1-d]. It also uses the isOrderedBefore
function to compare elements rather than just <, so this merge sort algorithm is universal and you can use it to sort any type of object.
5. At this point, the heap of the size width of array Z [d] has been merged into the larger size width * 2 of array Z [1-d]. Here, we swap the active array so that in the next step we will read from
the new heap we just created.
This function is generic, so you can use it to sort any type of object you want, as long as you provide a proper isOrderedBefore closure to compare elements.
Examples of how to use it:
let array = [2.1.5.4.9]
mergeSortBottomUp(array, <) // [1, 2, 4, 5, 9]
Copy the code
For iteration merge sort, I found a graph to show the source
The speed of the merge sort algorithm depends on the size of the array it needs to sort. The larger the array, the more work it needs to do.
Whether the initial array is sorted or not doesn’t affect the speed of the merge sort algorithm, because you will do the same number of splits and comparisons regardless of the initial order of the
Therefore, the time complexity of the best, worst and average cases will always be O(n log n).
One drawback of the merge sort algorithm is that it requires a temporary “working” array of the same size as the array being sorted. It does not sort in place, like quicksort for example.
Most implementations of merge sort algorithms are stable sorts. This means that array elements with the same sort key will remain in the same order relative to each other after sorting. This is not
important for simple values like numbers or strings, but when sorting more complex objects, it can be problematic if the sorting is not stable.
A sorting algorithm is stable if the elements are the same and the relative order remains the same. Stable sort: insert sort, count sort, merge sort, radix sort and so on, see stable sort.
Further reading
Merge sort wikipedia
Merge sort Chinese Wikipedia
By Kelvin Lau. Additions, Matthijs Hollemans Translated by Andy Ron Proofread by Andy Ron | {"url":"https://www.mo4tech.com/swift-algorithm-club-merge-sort.html","timestamp":"2024-11-02T09:03:02Z","content_type":"text/html","content_length":"79674","record_id":"<urn:uuid:7b3c7a57-7fec-435d-9544-30fe4333e8f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00606.warc.gz"} |
Tabling - Sudopedia
From Sudopedia
Revision as of 02:46, 20 January 2022 by
Rooted (talk | contribs) (Created page with "'''Tabling''' is a solving technique which is not used by human players but by computer programs. The program builds a table of implications for the
'''true''' and '''...")
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Tabling is a solving technique which is not used by human players but by computer programs. The program builds a table of implications for the true and false states of each remaining candidate in the
The primary aim of tabling is to find a candidate that causes a contradiction. The program can then build a chain that proves this contradiction.
Alternatively, the tables can be used to find verities, which are implications that occur for all alternative candidates in a constraint. Since multiple chains are required to prove a verity, this
aspect of tabling is only used when no contradictions can be found. | {"url":"https://www.sudopedia.org/index.php?title=Tabling&oldid=416","timestamp":"2024-11-07T00:16:00Z","content_type":"text/html","content_length":"17977","record_id":"<urn:uuid:e44222ca-c9ea-4325-8acd-cb70b5aea5b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00569.warc.gz"} |
mathabx – Three series of mathematical symbols
Mathabx is a set of 3 mathematical symbols font series: matha, mathb and mathx. They are defined by METAFONT code and should be of reasonable quality (bitmap output). Things change from time to time,
so there is no claim of stability (encoding, metrics, design).
The package includes Plain TeX and LaTeX support macros.
A version of the fonts, in Adobe Type 1 format, is also available.
Sources /fonts/mathabx
Home page http://www-math.univ-poitiers.fr/~phan/
Licenses The LaTeX Project Public License
Maintainer Anthony Phan
Contained in TeXLive as mathabx
MiKTeX as mathabx
Topics MF Font
Font symbol maths
Download the contents of this package in one zip archive (939.5k).
Community Comments
Maybe you are interested in the following packages as well.
Package Links | {"url":"https://ctan.org/pkg/mathabx","timestamp":"2024-11-09T13:01:23Z","content_type":"text/html","content_length":"16685","record_id":"<urn:uuid:64548016-cd6c-448b-8b40-57f8e53b92d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00689.warc.gz"} |
ModelSegments (BETA)
Models segmented copy ratios from denoised read counts and segmented minor-allele fractions from allelic counts
Category Copy Number Variant Discovery
Models segmented copy ratios from denoised read counts and segmented minor-allele fractions from allelic counts.
Possible inputs are: 1) denoised copy ratios for the case sample, 2) allelic counts for the case sample, and 3) allelic counts for a matched-normal sample. All available inputs will be used to to
perform segmentation and model inference.
If allelic counts are available, the first step in the inference process is to genotype heterozygous sites, as the allelic counts at these sites will subsequently be modeled to infer segmented
minor-allele fraction. We perform a relatively simple and naive genotyping based on the allele counts (i.e., pileups), which is controlled by a small number of parameters (minimum-total-allele-count,
genotyping-homozygous-log-ratio-threshold, and genotyping-homozygous-log-ratio-threshold). If the matched normal is available, its allelic counts will be used to genotype the sites, and we will
simply assume these genotypes are the same in the case sample. (This can be critical, for example, for determining sites with loss of heterozygosity in high purity case samples; such sites will be
genotyped as homozygous if the matched-normal sample is not available.)
Next, we segment, if available, the denoised copy ratios and the alternate-allele fractions at the genotyped heterozygous sites. This is done using kernel segmentation (see KernelSegmenter). Various
segmentation parameters control the sensitivity of the segmentation and should be selected appropriately for each analysis.
If both copy ratios and allele fractions are available, we perform segmentation using a combined kernel that is sensitive to changes that occur not only in either of the two but also in both.
However, in this case, we simply discard all allele fractions at sites that lie outside of the available copy-ratio intervals (rather than imputing the missing copy-ratio data); these sites are
filtered out during the genotyping step discussed above. This can have implications for analyses involving the sex chromosomes; see comments in CreateReadCountPanelOfNormals.
After segmentation is complete, we run Markov-chain Monte Carlo (MCMC) to determine posteriors for segmented models for the log2 copy ratio and the minor-allele fraction; see CopyRatioModeller and
AlleleFractionModeller, respectively. After the first run of MCMC is complete, smoothing of the segmented posteriors is performed by merging adjacent segments whose posterior credible intervals
sufficiently overlap according to specified segmentation-smoothing parameters. Then, additional rounds of segmentation smoothing (with intermediate MCMC optionally performed in between rounds) are
performed until convergence, at which point a final round of MCMC is performed.
• (Optional) Denoised-copy-ratios file from DenoiseReadCounts. If allelic counts are not provided, then this is required.
• (Optional) Allelic-counts file from CollectAllelicCounts. If denoised copy ratios are not provided, then this is required.
• (Optional) Matched-normal allelic-counts file from CollectAllelicCounts. This can only be provided if allelic counts for the case sample are also provided.
• Output prefix. This is used as the basename for output files.
• Output directory. This must be a pre-existing directory.
• Modeled-segments .modelBegin.seg and .modelFinal.seg files. These are tab-separated values (TSV) files with a SAM-style header containing a read group sample name, a sequence dictionary, a row
specifying the column headers contained in ModeledSegmentCollection.ModeledSegmentTableColumn, and the corresponding entry rows. The initial result before segmentation smoothing is output to the
.modelBegin.seg file and the final result after segmentation smoothing is output to the .modelFinal.seg file.
• Allele-fraction-model global-parameter files (.modelBegin.af.param and .modelFinal.af.param). These are tab-separated values (TSV) files with a SAM-style header containing a read group sample
name, a row specifying the column headers contained in ParameterDecileCollection.ParameterTableColumn, and the corresponding entry rows. The initial result before segmentation smoothing is output
to the .modelBegin.af.param file and the final result after segmentation smoothing is output to the .modelFinal.af.param file.
• Copy-ratio-model global-parameter files (.modelBegin.cr.param and .modelFinal.cr.param). These are tab-separated values (TSV) files with a SAM-style header containing a read group sample name, a
row specifying the column headers contained in ParameterDecileCollection.ParameterTableColumn, and the corresponding entry rows. The initial result before segmentation smoothing is output to the
.modelBegin.cr.param file and the final result after segmentation smoothing is output to the .modelFinal.cr.param file.
• Copy-ratio segment file (.cr.seg). This is a tab-separated values (TSV) file with a SAM-style header containing a read group sample name, a sequence dictionary, a row specifying the column
headers contained in CopyRatioSegmentCollection.CopyRatioSegmentTableColumn, and the corresponding entry rows. It contains the segments from the .modelFinal.seg file converted to a format
suitable for input to CallCopyRatioSegments.
• (Optional) Allelic-counts file containing the counts at sites genotyped as heterozygous in the case sample (.hets.tsv). This is a tab-separated values (TSV) file with a SAM-style header
containing a read group sample name, a sequence dictionary, a row specifying the column headers contained in AllelicCountCollection.AllelicCountTableColumn, and the corresponding entry rows. This
is only output if allelic counts are provided as input.
• (Optional) Allelic-counts file containing the counts at sites genotyped as heterozygous in the matched-normal sample (.hets.normal.tsv). This is a tab-separated values (TSV) file with a SAM-style
header containing a read group sample name, a sequence dictionary, a row specifying the column headers contained in AllelicCountCollection.AllelicCountTableColumn, and the corresponding entry
rows. This is only output if matched-normal allelic counts are provided as input.
Usage examples
gatk ModelSegments \
--denoised-copy-ratios tumor.denoisedCR.tsv \
--allelic-counts tumor.allelicCounts.tsv \
--normal-allelic-counts normal.allelicCounts.tsv \
--output-prefix tumor \
-O output_dir
gatk ModelSegments \
--denoised-copy-ratios normal.denoisedCR.tsv \
--allelic-counts normal.allelicCounts.tsv \
--output-prefix normal \
-O output_dir
gatk ModelSegments \
--allelic-counts tumor.allelicCounts.tsv \
--normal-allelic-counts normal.allelicCounts.tsv \
--output-prefix tumor \
-O output_dir
gatk ModelSegments \
--denoised-copy-ratios normal.denoisedCR.tsv \
--output-prefix normal \
-O output_dir
gatk ModelSegments \
--allelic-counts tumor.allelicCounts.tsv \
--output-prefix tumor \
-O output_dir
ModelSegments specific arguments
This table summarizes the command-line arguments that are specific to this tool. For more details on each argument, see the list further down below the table or click on an argument name to jump
directly to that entry in the list.
Argument name(s) Default Summary
Required Arguments
--output null Output directory.
--output-prefix null Prefix for output files.
Optional Tool Arguments
--allelic-counts null Input file containing allelic counts (output of CollectAllelicCounts).
--arguments_file [] read one or more arguments files and add them to the command line
--denoised-copy-ratios null Input file containing denoised copy ratios (output of DenoiseReadCounts).
--gcs-max-retries 20 If the GCS bucket channel errors out, how many times it will attempt to re-initiate the connection
Maximum base-error rate for genotyping and filtering homozygous allelic counts, if available. The likelihood for an allelic count
--genotyping-base-error-rate 0.05 to be generated from a homozygous site will be integrated from zero base-error rate up to this value. Decreasing this value will
increase the number of sites assumed to be heterozygous for modeling.
--genotyping-homozygous-log-ratio-threshold -10.0 Log-ratio threshold for genotyping and filtering homozygous allelic counts, if available. Increasing this value will increase the
number of sites assumed to be heterozygous for modeling.
--help false display the help message
Dimension of the kernel approximation. A subsample containing this number of data points will be used to construct the
--kernel-approximation-dimension 100 approximation for each chromosome. If the total number of data points in a chromosome is greater than this number, then all data
points in the chromosome will be used. Time complexity scales quadratically and space complexity scales linearly with this
--kernel-scaling-allele-fraction 1.0 Relative scaling S of the kernel K_AF for allele-fraction segmentation to the kernel K_CR for copy-ratio segmentation. If
multidimensional segmentation is performed, the total kernel used will be K_CR + S * K_AF.
--kernel-variance-allele-fraction 0.025 Variance of Gaussian kernel for allele-fraction segmentation, if performed. If zero, a linear kernel will be used.
--kernel-variance-copy-ratio 0.0 Variance of Gaussian kernel for copy-ratio segmentation, if performed. If zero, a linear kernel will be used.
--maximum-number-of-segments-per-chromosome 1000 Maximum number of segments allowed per chromosome.
--maximum-number-of-smoothing-iterations 25 Maximum number of iterations allowed for segmentation smoothing.
--minimum-total-allele-count 30 Minimum total count for filtering allelic counts, if available.
Alpha hyperparameter for the 4-parameter beta-distribution prior on segment minor-allele fraction. The prior for the minor-allele
--minor-allele-fraction-prior-alpha 25.0 fraction f in each segment is assumed to be Beta(alpha, 1, 0, 1/2). Increasing this hyperparameter will reduce the effect of
reference bias at the expense of sensitivity.
--normal-allelic-counts null Input file containing allelic counts for a matched normal (output of CollectAllelicCounts).
--number-of-burn-in-samples-allele-fraction 50 Number of burn-in samples to discard for allele-fraction model.
--number-of-burn-in-samples-copy-ratio 50 Number of burn-in samples to discard for copy-ratio model.
--number-of-changepoints-penalty-factor 1.0 Factor A for the penalty on the number of changepoints per chromosome for segmentation. Adds a penalty of the form A * C * [1 + log
(N / C)], where C is the number of changepoints in the chromosome, to the cost function for each chromosome. Must be non-negative.
--number-of-samples-allele-fraction 100 Total number of MCMC samples for allele-fraction model.
--number-of-samples-copy-ratio 100 Total number of MCMC samples for copy-ratio model.
--number-of-smoothing-iterations-per-fit 0 Number of segmentation-smoothing iterations per MCMC model refit. (Increasing this will decrease runtime, but the final number of
segments may be higher. Setting this to 0 will completely disable model refitting between iterations.)
--smoothing-credible-interval-threshold-allele-fraction 2.0 Number of 10% equal-tailed credible-interval widths to use for allele-fraction segmentation smoothing.
--smoothing-credible-interval-threshold-copy-ratio 2.0 Number of 10% equal-tailed credible-interval widths to use for copy-ratio segmentation smoothing.
--version false display the version number for this tool
[8, 16, Window sizes to use for calculating local changepoint costs. For each window size, the cost for each data point to be a changepoint
--window-size 32, 64, will be calculated assuming that the point demarcates two adjacent segments of that size. Including small (large) window sizes will
128, 256] increase sensitivity to small (large) events. Duplicate values will be ignored.
Optional Common Arguments
--gatk-config-file null A configuration file to use with the GATK.
--QUIET false Whether to suppress job-summary info on System.err.
--TMP_DIR [] Undocumented option
--use-jdk-deflater false Whether to use the JdkDeflater (as opposed to IntelDeflater)
--use-jdk-inflater false Whether to use the JdkInflater (as opposed to IntelInflater)
--verbosity INFO Control verbosity of logging.
Advanced Arguments
--showHidden false display hidden arguments
Argument details
Arguments in this list are specific to this tool. Keep in mind that other arguments are available that are shared with other tools (e.g. command-line GATK arguments); see Inherited arguments above.
Input file containing allelic counts (output of CollectAllelicCounts).
File null
read one or more arguments files and add them to the command line
List[File] []
Input file containing denoised copy ratios (output of DenoiseReadCounts).
File null
A configuration file to use with the GATK.
String null
If the GCS bucket channel errors out, how many times it will attempt to re-initiate the connection
int 20 [ [ -∞ ∞ ] ]
Maximum base-error rate for genotyping and filtering homozygous allelic counts, if available. The likelihood for an allelic count to be generated from a homozygous site will be integrated from zero
base-error rate up to this value. Decreasing this value will increase the number of sites assumed to be heterozygous for modeling.
double 0.05 [ [ -∞ ∞ ] ]
Log-ratio threshold for genotyping and filtering homozygous allelic counts, if available. Increasing this value will increase the number of sites assumed to be heterozygous for modeling.
double -10.0 [ [ -∞ ∞ ] ]
display the help message
boolean false
Dimension of the kernel approximation. A subsample containing this number of data points will be used to construct the approximation for each chromosome. If the total number of data points in a
chromosome is greater than this number, then all data points in the chromosome will be used. Time complexity scales quadratically and space complexity scales linearly with this parameter.
int 100 [ [ 1 ∞ ] ]
Relative scaling S of the kernel K_AF for allele-fraction segmentation to the kernel K_CR for copy-ratio segmentation. If multidimensional segmentation is performed, the total kernel used will be
K_CR + S * K_AF.
double 1.0 [ [ 0 ∞ ] ]
Variance of Gaussian kernel for allele-fraction segmentation, if performed. If zero, a linear kernel will be used.
double 0.025 [ [ 0 ∞ ] ]
Variance of Gaussian kernel for copy-ratio segmentation, if performed. If zero, a linear kernel will be used.
double 0.0 [ [ 0 ∞ ] ]
Maximum number of segments allowed per chromosome.
int 1000 [ [ 1 ∞ ] ]
Maximum number of iterations allowed for segmentation smoothing.
int 25 [ [ 0 ∞ ] ]
Minimum total count for filtering allelic counts, if available.
int 30 [ [ 0 ∞ ] ]
Alpha hyperparameter for the 4-parameter beta-distribution prior on segment minor-allele fraction. The prior for the minor-allele fraction f in each segment is assumed to be Beta(alpha, 1, 0, 1/2).
Increasing this hyperparameter will reduce the effect of reference bias at the expense of sensitivity.
double 25.0 [ [ 1 ∞ ] ]
Input file containing allelic counts for a matched normal (output of CollectAllelicCounts).
File null
Number of burn-in samples to discard for allele-fraction model.
int 50 [ [ 0 ∞ ] ]
Number of burn-in samples to discard for copy-ratio model.
int 50 [ [ 0 ∞ ] ]
Factor A for the penalty on the number of changepoints per chromosome for segmentation. Adds a penalty of the form A * C * [1 + log (N / C)], where C is the number of changepoints in the chromosome,
to the cost function for each chromosome. Must be non-negative.
double 1.0 [ [ 0 ∞ ] ]
Total number of MCMC samples for allele-fraction model.
int 100 [ [ 1 ∞ ] ]
Total number of MCMC samples for copy-ratio model.
int 100 [ [ 1 ∞ ] ]
Number of segmentation-smoothing iterations per MCMC model refit. (Increasing this will decrease runtime, but the final number of segments may be higher. Setting this to 0 will completely disable
model refitting between iterations.)
int 0 [ [ 0 ∞ ] ]
Output directory.
R String null
Prefix for output files.
R String null
Whether to suppress job-summary info on System.err.
Boolean false
display hidden arguments
boolean false
Number of 10% equal-tailed credible-interval widths to use for allele-fraction segmentation smoothing.
double 2.0 [ [ 0 ∞ ] ]
Number of 10% equal-tailed credible-interval widths to use for copy-ratio segmentation smoothing.
double 2.0 [ [ 0 ∞ ] ]
Undocumented option
List[File] []
Whether to use the JdkDeflater (as opposed to IntelDeflater)
boolean false
Whether to use the JdkInflater (as opposed to IntelInflater)
boolean false
Control verbosity of logging.
The --verbosity argument is an enumerated type (LogLevel), which can have one of the following values:
LogLevel INFO
display the version number for this tool
boolean false
Window sizes to use for calculating local changepoint costs. For each window size, the cost for each data point to be a changepoint will be calculated assuming that the point demarcates two adjacent
segments of that size. Including small (large) window sizes will increase sensitivity to small (large) events. Duplicate values will be ignored.
List[Integer] [8, 16, 32, 64, 128, 256]
GATK version 4.0.6.0 built at 25-39-2019 01:39:46.
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10.3: Estimating the Difference in Two Population Means
Last updated
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Learning Objectives
• Construct a confidence interval to estimate a difference in two population means (when conditions are met). Interpret the confidence interval in context.
Confidence Interval to Estimate μ[1] − μ[2]
In a hypothesis test, when the sample evidence leads us to reject the null hypothesis, we conclude that the population means differ or that one is larger than the other. An obvious next question is
how much larger? In practice, when the sample mean difference is statistically significant, our next step is often to calculate a confidence interval to estimate the size of the population mean
The confidence interval gives us a range of reasonable values for the difference in population means μ[1] − μ[2]. We call this the two-sample T-interval or the confidence interval to estimate a
difference in two population means. The form of the confidence interval is similar to others we have seen.
$\begin{array}{l}(\mathrm{sample}\text{}\mathrm{statistic})\text{}±\text{}(\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error})\\ (\mathrm{sample}\text{}\mathrm{statistic})\text{}&
Sample Statistic
Since we’re estimating the difference between two population means, the sample statistic is the difference between the means of the two independent samples: ${\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_
Critical T-Value
The critical T-value comes from the T-model, just as it did in “Estimating a Population Mean.” Again, this value depends on the degrees of freedom (df). For two-sample T-test or two-sample
T-intervals, the df value is based on a complicated formula that we do not cover in this course. We either give the df or use technology to find the df.
Standard Error
The estimated standard error for the two-sample T-interval is the same formula we used for the two-sample T-test. (As usual, s[1] and s[2] denote the sample standard deviations, and n[1] and n[2]
denote the sample sizes.)
Putting all this together gives us the following formula for the two-sample T-interval.
Conditions for Use
The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test.
• The two random samples are independent and representative.
• The variable is normally distributed in both populations. If it is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the T-distribution.
As we discussed in “Hypothesis Test for a Population Mean,” T-procedures are robust even when the variable is not normally distributed in the population. If checking normality in the populations
is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily
skewed in the populations, and we use the inference procedure.
Confidence Interval for the “Calories and Context” Study
In the preceding few pages, we worked through a two-sample T-test for the “calories and context” example. In this example, we use the sample data to find a two-sample T-interval for μ[1] − μ[2] at
the 95% confidence level.
Recap of the Situation
• Population 1: Let μ[1] be the mean number of calories purchased by women eating with other women.
• Population 2: Let μ[2] be the mean number of calories purchased by women eating with men.
Sample Statistics
Size (n) $\mathrm{Mean}\text{}(\stackrel{¯}{x})$ SD (s)
Sample 1 45 850 252
Sample 2 27 719 322
Standard Error
We found that the standard error of the sampling distribution of all sample differences is approximately 72.47.
$\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}\text{}=\text{}\sqrt{\frac{{252}^{2}}{45}+\frac{{322}^{2}}{27}}\text{}\approx \text{}72.47$
Critical T-value
For these two independent samples, df = 45. We find the critical T-value using the same simulation we used in “Estimating a Population Mean.”
Reading from the simulation, we see that the critical T-value is 1.6790.
Confidence Interval
We can now put all this together to compute the confidence interval:
$({\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2})\text{}±\text{}{T}_{c}\text{}⋅\text{}\mathrm{SE}\text{}=\text{}(850-719)\text{}±\text{}(1.6790)(72.47)\text{}\approx \text{}131\text
Expressing this as an interval gives us:
$(\mathrm{9,\; 253})$
We are 95% confident that the true value of μ[1] − μ[2] is between 9 and 253 calories. We can be more specific about the populations. We are 95% confident that at Indiana University of Pennsylvania,
undergraduate women eating with women order between 9.32 and 252.68 more calories than undergraduate women eating with men.
In this next activity, we focus on interpreting confidence intervals and evaluating a statistics project conducted by students in an introductory statistics course.
Try It
Improving Children’s Math Skills
Students in an introductory statistics course at Los Medanos College designed an experiment to study the impact of subliminal messages on improving children’s math skills. The students were inspired
by a similar study at City University of New York, as described in David Moore’s textbook The Basic Practice of Statistics(4th ed., W. H. Freeman, 2007). The participants were 11 children who
attended an afterschool tutoring program at a local church. The children ranged in age from 8 to 11. All received tutoring in arithmetic skills. At the beginning of each tutoring session, the
children watched a short video with a religious message that ended with a promotional message for the church.
The statistics students added a slide that said, “I work hard and I am good at math.” This slide flashed quickly during the promotional message, so quickly that no one was aware of the slide.
Children who attended the tutoring sessions on Mondays watched the video with the extra slide. Children who attended the tutoring sessions on Wednesday watched the video without the extra slide. The
experiment lasted 4 weeks. The children took a pretest and posttest in arithmetic. Here are some of the results:
Let’s Summarize
Hypothesis tests and confidence intervals for two means can answer research questions about two populations or two treatments that involve quantitative data. In “Inference for a Difference between
Population Means,” we focused on studies that produced two independent samples. Previously, in “Hpyothesis Test for a Population Mean,” we looked at matched-pairs studies in which individual data
points in one sample are naturally paired with the individual data points in the other sample.
The hypotheses for two population means are similar to those for two population proportions.
The null hypothesis, H[0], is a statement of “no effect” or “no difference.”
• H[0]: μ[1] – μ[2] = 0, which is the same as H[0]: μ[1] = μ[2]
The alternative hypothesis, H[a], takes one of the following three forms:
• H[a]: μ[1] – μ[2] < 0, which is the same as H[a]: μ[1] < μ[2]
• H[a]: μ[1] – μ[2] > 0, which is the same as H[a]: μ[1] > μ[2]
• H[a]: μ[1] – μ[2] ≠ 0, which is the same as H[a]: μ[1] ≠ μ[2]
As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.
• Samples must be random in order to remove or minimize bias.
• Sample must be representative of the population in question.
We use the two-sample hypothesis test and confidence interval when the following conditions are met:
• The two random samples are independent.
• The variable is normally distributed in both populations. If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the
t-distribution. As we discussed in “Hypothesis Test for a Population Mean,” t-procedures are robust even when the variable is not normally distributed in the population. Therefore, if checking
normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign
that the variable is not heavily skewed in the populations, and we use the inference procedure.
The confidence interval for μ[1] − μ[2] is
Hypothesis test for H[0]: μ[1] – μ[2] = 0 is
We use technology to find the degrees of freedom to determine P-values and critical t-values for confidence intervals. (In most problems in this section, we provided the degrees of freedom for you.)
Contributors and Attributions
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CLEP Science and Mathematics E
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CE141-4-__ : Mathematics for Computing
Module Description
The aim of this module is to cover fundamental mathematics for Computer Scientists. It does not assume A-level mathematics, and the emphasis and delivery will be on understanding the key concepts as
they apply to Computer Science.
Additional support is provided by the Talent Development Centre. Participants not having AS or A level mathematics should take a diagnostic test to see whether they would benefit from this extra
Learning Outcomes
After completing this module, students will be expected to be able to:
1. Apply propositional logic to simple problems
2. Use counting methods including permutations and combinations
3. Apply the basic notions of sets, and illustrate answers through Venn diagrams
4. Use methods of probability on simple problems
5. Solve problems in linear algebra using vectors and matrices
Outline Syllabus
Propositional Logic:
Propositions and logical operators. Truth tables. De Morgan's laws. Algebraic rules and inference. Logical identities, Tautologies and Contraditions
Fundamental Principle of Counting. Ordered and unordered selections. Selections with and without replacement. Permutations and combinations. Counting methods.
Set notation and basic concepts. Definition of sets through propositions. Set intersection, union and complementation. Venn diagrams. Cardinality. Cartesian products. Sample spaces and events.
Experiments and outcomes. Sample space, events, relative frequency and probability. Mutual exclusivity and independence. Counting methods. Conditional probability. Mean and variance. The binomial
Vectors and Matrices:
Basic definitions. Addition and multiplication of matrices, multiplication by scalars. Inversion of 2x2 matrices. Applications. Transformations of the plane. Solving simultaneous equations in two | {"url":"https://moodle.essex.ac.uk/course/info.php?id=2832","timestamp":"2024-11-03T16:56:56Z","content_type":"text/html","content_length":"72287","record_id":"<urn:uuid:68142c31-16b3-4b6c-81e9-d4fdf6c5e357>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00515.warc.gz"} |
Furness Method
Some competitors may profess to use "mathematical models" (such as 'Furness' etc.) to "count" a roundabout - whilst others will not declare this fact, but use the method in any case. It will be shown
further below that no such "count" is actually possible using a mathematical model.
TSUK does not employ any form of modelling to derive flows. TSUK provides a 100% true and accurate count, either from the observed turning counts (where those counts can be physically seen) or using
number plates (where turning counts cannot be seen because of trees in the middle of a roundabout, for example). Whilst mathematical models serve a useful purpose as far as estimates are concerned,
most traffic surveys are not normally estimating exercises - clients purposefully and rightfully expect a full and true count.
A popular method is the 'Furness' method, as applied to a 4-arm roundabout. The main premise of the 'Furness' model is to use a very limited number of known movements and volumes, and then to
algebraically estimate numeric quantities for the remaining unknown movements and volumes. As such, for each arm of a 4-arm roundabout, the model contains only 3 true values (only one of which is
actually a turning movement): the first respective left from each arm; the total of all the "Ons" from each arm to the roundabout; and, the total of all the "Offs" to each arm of the roundabout.
Hence, for each arm, there remain 3 unknown parameters: 2nd left (or "2nd exit" or otherwise "ahead"), 3rd exit (or frequently "right"), and the inevitable U-turn (irrespective of significance). This
means a total of 12 unknown parameters.
Therefore, using the 'Furness' method for a 4-arm roundabout, where there can be 16 possible movements, only 4 are actually known (i.e. the first left-turn from each arm).
Using all of those known and unknown parameters, a series of algebraic equations are setup, and the ensuing procedure is then to (forcibly) "balance" those equations to yield values for the unknown
parameters. In this procedure, nothing is said about the ACTUAL O-D pairs.
Furthermore, in such models, U-turners are assumed to not exist. In that case, those who produce/use such models, deliberately insert random numbers in U-turning movements - which then requires the
producer of the report to deliberately adjust numbers on other movements to ensure that the whole report of the counts (or system) balances. At face-value, appearance of U-turners within the report
has the effect of bolstering the apparent authenticity of the "turning count" report.
So, the modelled result does genuinely appear to balance - i.e. for any given time segment, such as 15 minutes, all the traffic entering the roundabout is equal to all the traffic leaving the
roundabout. This is only made possible because the total "Ons" and "Offs" are known and have been used as the fixed constants. However, the derived O-D pairs (apart from the 'First Left') are not
true and will never be the same as a PROPER AND FULL turning count (either as actually observed or using number plates).
In summary, clients are normally not aware of the covert use of this method. Some survey companies deliberately use this procedure because it is quick and cheap (as far as job costs are concerned).
In effect, where the correct approach is to count all 16 turning movements for a 4-arm roundabout, only 3 are actually counted. Hence, it only costs 1/5 of the would-be cost, and the result is then
supplied to the client as a "turning" count, although it is patently not.
TSUK only provides 100% true and proper turning counts - backed-up by video evidence.
Alternative Counting Method
We wish to draw the attention of the Client to the fact that some companies deliberately use covert methods which produce results that appear to be “true” and/or “representative”. This is especially
so when such “alternative” methods are used to analyse car parks and roundabouts.
For car parks, some firms choose only to take the number plate of specific vehicles (such as a specific colour), and then use that result as a factor for the total volume of cars entering and leaving
a car park.
For roundabouts, a popular method is the application of the ‘Furness’ model. This makes use of a limited number... | {"url":"http://www.trafficsurveys.co.uk/furness-method.htm","timestamp":"2024-11-14T05:29:56Z","content_type":"text/html","content_length":"12357","record_id":"<urn:uuid:67a0ce46-af41-49a7-a937-74cf14801230>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00672.warc.gz"} |
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Do Mathematicians Think Differently From Other People?
March 3, 2022
Do Mathematicians Think Differently From Other People?
[A math teacher illustrates some ways in which creative ones do but it’s really about imagination, not just getting the figures right] [
March 3, 2022
Math teacher Ali Kayaspor has thought a lot about how mathematicians have come up with fundamental ideas about the nature of reality and he shares anecdotes that give us a glimpse. But first, the
cold shower:
Unfortunately, there is no clear way to answer the question of how a mathematician thinks. But we can approach this question as follows; if you watched any chess tournament, the game’s analysis
is shared in detail at the end of the match. When you examine the analysis, you will see a breaking point in each game. Similarly, mathematicians also experience a breaking point while working on
a problem before finding a solution.
Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)
Sometimes, perhaps, they hardly notice the breaking point. Consider this anecdote from the life of Carl Friedrich Gauss (1777–1855):
When the children in an elementary school were very naughty, the teacher wrote a difficult question on the chalkboard to silence all the children. The teacher asked the children to add up all the
numbers from 1 to 100.
On that day, the young German boy named Gauss, who would grow up to be one of the greatest mathematicians of the future, was in that class. While the teacher thought it would take a long time for
the children to solve the question, Gauss resumed talking to his friends again in just a few minutes. When Gauss’s teacher asked why he was talking, he said he had already solved the question.
That day, all the students in that class tried to add all the numbers one by one, but Gauss did something unusual. He saw that he would always get 101 if he added a number from the left of the
sequence and a number from the right. For example, 1+100, 2+99, 3+98,…, 50+51; it was always 101, and there were 50 of them.
If we look carefully at this method, we will see that this is an observation that even a small child can see. However, it is a fact that not all young children solve this question like that.
Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)
It involves creative abstract thinking at a level unusual in a child. We are told that “He went on to publish seminal works in many fields of mathematics including number theory, algebra, statistics,
analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, optics, etc. Number theory was Gauss’s favorite and he referred to number theory as the ‘queen of mathematics.’” –
“Gauss: The Prince of Mathematics”
Surprisingly, great mathematicians are not always good with conventional basic math. Take Srinivasa Ramanujan (1887–1920), one of India’s greatest mathematical geniuses:
Despite solving infinite sums, he could not understand the most basic analysis technique. Ramanujan did not have the slightest idea of complex analysis, but he could work on zeta functions. So
Ramanujan had a different mindset in his mind that only he knew, and one we will never understand.
One day, when [his friend and fellow mathematician G. H.] Hardy wondered about this situation and asked how he wrote all those formulas, Ramanujan told him that God gave him all the formulas, and
he had just written them. To me, this was a very reasonable answer because Ramanujan was a math practitioner 24/7, and he often forgot to eat. His wife or mother reminded him that Ramanujan had
to eat. The few times he did sleep, he continued doing math in his dreams.
Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)
A number of anecdotes capture Ramanujan’s affinity for pure mathematics. His health was fragile and during a bout of illness, he surprised even his friend and mentor Hardy:
Eventually Ramanujan was confined to a nursing home to await his return to India. Hardy paid frequent visits to his friend and colleague. Not surprisingly, the conversation usually turned to
mathematics. On one such visit, 1,729 cropped up. This was the number of the taxi cab Hardy had taken to the clinic, and as befits two number theorists they discussed its significance. Hardy
thought 1,729 to be a boring run-of-the-mill number, but Ramanujan disagreed. “That is a really, really interesting number,” he declared. How so? “It is the smallest number that can be expressed
as the sum of two cubes in two different ways!”
Ramanujan could see immediately that: 12^3 + 1^3 = 10^3 + 9^3 = 1,729.
This amusing anecdote came to symbolise Ramanujan’s humble genius, and numbers that can be expressed as the sum of two cubes in two separate ways are known as “taxi numbers” in recognition. Other
taxi numbers are 4,014, 13,832 and 20,638. But 1,729 is the smallest.
Paul Davies, “Ramanujan – a humble maths genius” at Cosmos Magazine (December 28, 2015)
And no one really knows how he did it.
Perhaps it requires a lot of imagination. Mathematics is full of unusual numbers. Consider, for example, the irrationals that burble on forever without forming a pattern (though some, like the Golden
Ratio and pi, are critical). They are more numerous than rational numbers which eventually start to make sense.
But also, there are the imaginary numbers (they don’t make sense according to our number system but of course our computers and the entire modern world depend on them).
And the hyperreals: “Thinking about infinities is somewhat mind-bending, but it turns out that actually manipulating infinities with the hyperreal system is incredibly easy if you are familiar with
basic algebra.”
Then there’s 1/137, which keeps turning up in physics and no one is sure why: “What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It
actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul
Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure.”
And Chaitin’s unknowable number, which is critical to computer function: “The number exists. If you write programs in C++, Python, or Matlab, your computer language has a Chaitin number. It’s a
feature of your computer programming language. But we can prove that even though Chaitin’s number exists, we can also prove it is unknowable.”
One needs a lot of imagination combined with a lot of numerical discipline to keep track of it all. And that’s as far as we have got in understanding the way creative mathematicians think.
You may also wish to read: Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is,
Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-Contradiction. | {"url":"https://mindmatters.ai/2022/03/do-mathematicians-think-differently-from-other-people/","timestamp":"2024-11-04T04:19:07Z","content_type":"text/html","content_length":"91256","record_id":"<urn:uuid:da6847af-6e5b-42a4-9a3b-819a76989a52>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00499.warc.gz"} |
In ΔABC, if ∠1 = ∠2, Prove that, AB/AC = BD/CD - Ask TrueMaths!In ΔABC, if ∠1 = ∠2, Prove that, AB/AC = BD/CD
Question extracted from RD sharma of class 10th, Chapter no. 4
Chapter name:- Triangles
Exercise :- 4.3
This is very basic and important questions. And has been asked in several times in exams
In this question we have been given a triangle ΔABC,
if ∠1 = ∠2,
Now we have to Prove that, AB/AC = BD/CD
learning CBSE mathsin effective way.
RD sharma, DHANPAT RAI publication | {"url":"https://ask.truemaths.com/question/in-%CE%B4abc-if-%E2%88%A01-%E2%88%A02-prove-that-ab-ac-bd-cd/","timestamp":"2024-11-05T10:17:40Z","content_type":"text/html","content_length":"117653","record_id":"<urn:uuid:07467aaa-7042-41f6-adab-a18ece615636>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00802.warc.gz"} |
Calculation of 10-day portfolio VaR essay Essay — Free college essays
Calculation of 10-day portfolio VaR essay
Calculationof 10-day portfolio VaR
Valueat Risk is a measure of financial risk that is widely used. It givesa method of determining the quantity of risk of a portfolio andmanaging the risk (Marrison, 2002). In calculating the 10-day
VaR,three ASX-listed companies were selected. The companies were AzureHealthcare Ltd (AZV.AX), Azonto Petroleum Ltd (APY.AX), and NationalAustralia Bank Limited (NAB.AX). The respective returns for
therespective companies (in AUD) were obtained for the period between.The calculations were done using two different methods, thehistorical simulation and the Variance-Covariance parameter approach
(VCV approach).
Justificationfor chosen calculation methods
Thehistorical simulation was chosen because it offers an implementationof the full valuation in a straightforward manner. Market states aresimulated and produced through the addition of the
period-to-periodchanges in market variables in a specific time series to the basecase. The historical simulation assumes primarily that there is afull representation of the set of possible future
scenarios by theevents of a particular historical window. In this method, a set ofchanges in the risk factor is collected over a given historicalwindow. The scenarios obtained are then assumed to
represent all thepossibilities that might happen in the immediate future (Alexander,2008).
Advantagesand disadvantages of historical simulation
Amajor advantage of historical simulation is that no assumptions aremade about the changes in risk factor being from a specificdistribution. As a result, this methodology is consistent withchanges in
risk factor originating from any distribution. Thehistorical simulation also does not involve any estimation of thestatistical parameters like variances or covariance. It is alsocontinuously exempt
from the unavoidable estimation errors. It iseasy to explain and defend even to an audience that is not technicalyet important, like the corporate board of directors. No assumptionsare needed on the
distributions of the risk factors, there is easerevaluating the full portfolio based on the scenery data, and it isintuitively simplistic and obvious (Hull, 2008). The historical simulation also has
certain disadvantages. Accomplishing the purest form of historical simulation is difficultsince the method poses the requirement of data about risk factorsspanning over a long historical period. This
requirement is usuallynecessary so that what might happen in the future can be adequatelyrepresented. The method also does not involve distributionalassumptions. The scenarios used in calculating VaR
are restricted tothe ones that took place in the historical sample (Bohdalova, 2007).The past is not always a good way of modeling for the future. Thereis also a huge probability of gaining results
that are erroneous ifthe sample size is not sufficient (Boyarshinov, 2016).
Thevariance-covariance (VCV) method makes use of a historical sample ofthe risk factors of a portfolio value. It functions almost like thehistorical simulation. However, unlike the historical
simulation, thevariance-covariance method assumes that there are normallydistributed logarithmic yields for the risk factors. The estimate ofthe VaR is then equated to a sample quartile of the yield
portfoliovolatility. The volatility of the portfolio yield is calculateddepending on the covariance between portfolio risk factor yields.
Advantagesand disadvantages of Variance-covariance method
Thereare many advantages of variance-covariance method. They are easy toimplement as compared to the historical simulation method and theMonte Carlo Simulation. The method requires less historical
data incomparison to the historical simulation method. In most cases, it hasan acceptable precision and accuracy. The method alsohas its disadvantages. VCV method has a low quality of estimates
forsecurities whose prices have a nonlinear dependence on the riskfactors. The assumption of the logarithmic distribution of the yieldsof the risk factors is not always correct. It also ignores the
riskof extreme events that can cause significant losses in the value ofthe portfolio (Boyarshinov, 2016). The MonteCarlo Simulation method was not chosen because of its high dependenceon the resource
which makes it very time-consuming. The sample sizesof observations that are used in deriving the distribution kind andparameters are also often insufficient. The insufficiency can lead toincorrect
estimates of VaR.Calculationof VaR using Variance-covariance method Inthis method, the simple moving average (SMA) method was used. Thecalculations involved include the SMA daily volatility, SMA
dailyVaR, and Portfolio holding SMA VaR. The portfolio comprises equalexposures of 1000 shares for all the stocks. The market price forAZV.AX, APY.AX and NAB.AX were AUD 0.068, 0.015 and
28.94respectively. The historical price data for all the three companieswere obtained for the period of 14^thJuly 2015 to 13^thMay 2016. The daily time series are presented in the extract inFigure
1.this period is called the look-back period, which is theperiod over which the risk is evaluated.
Figure1: Time series data for AZV, APY, and NAB
Itis first important to determine the return series. To obtain this,the natural logarithms of the ratio of successive prices aredetermined. This is shown in Figure 2.
Theformula shown in the figure is LN(Cell B19/Cell B18). In cell B19,there is 0.15 while in cell B18 there is 0.15. Calculating theLN(0.15/0.15) gives 0 as shown in cell H19.
Figure2: Return Series
Thenext step is to calculate the daily volatility is calculated usingthe formula:
Rt= rate of return at the time, t.
E(R)= mean of return distribution
Thesquared differences of Rt are summed over E(R) across all data pointsthen the results are divided by the number of returns in the seriesminus one so that the variance can be obtained (Alexander,
2008). After this, the square rood to the result iscalculated, which is the standard deviation (SMA volatility) of thereturn series. The method used in this calculation, however, is theuse of the
STDEV function in EXCEL, which is applied to the returnseries as illustrated in Figure 3. The daily volatility is determinedas 0.494429 for the portfolio.
Figure3: Calculating the daily volatility
Gettingthe SMA VaR is determining how much will be lost over a particularholding period with a particular probability. The daily volatility ismultiplied by the z-value of the inverse of the standard
normalcumulative distribution function (CDF) which corresponds with aparticular confidence level, which is the NORMSINV (confidencelevel). For example, in our case, the confidence level is 99%, so
itis expressed in excel as NORMSINV (99%). Figure 4 shows how the dailyVaR is calculated
Figure4: Daily Volatility
Figure4 show that the daily VaR is 1.150214. The VaR is the Daily VaRmultiplied by the square root of the holding period in days. Thesquare root function in Excel is SQRT. Figure 5 shows how
thecalculation is done in excel.
Figure5: VaR
CalculatingVaR using the Historical Simulation method
Inthis approach, there is no assumption that is made about the returndistribution that is underlying. The first step is to obtain thereturn series and reorder them into ascending order from the
smallestto the largest returns. This is done by use of the Sort function inexcel. After reordering the values, the number of returns in theseries is counted using the COUNTA formula as shown in
Figure 6(Bohdalova, 2007).
Figure6: Determining the number of returns
Fromour series, there are 218 returns. The daily VaR is then calculatedas the return that is corresponding to the index number. It iscalculated by subtracting the confidence level then multiplying it
bythe number of returns at the point where the result is rounded downto the nearest integer. The integer is the index number for aparticular return. This is shown in Figure 7.
Figure7: Historical Simulation
The10-day VaR is then calculated by multiplying the daily VaR by thesquare root of the holding period- 10 days. Figure 8 shows how thisis done.
Figure8: 10-day VaR.
Fromour calculations, the 10-day VaR for AZV.AX, APY.AX and NAB.AX are0.705641867, 0.909730591 and 0.150688793 respectively. The respectiveamounts of worst case loss are AUD 47.98364698, 13.64595886
Inorder to calculate the portfolio VaR through the historicalsimulation, the returns are combined to get the portfolio change asshown in Figure 9.
Figure9: Calculating portfolio change
Theobtained values are then sorted in ascending order from the least tothe largest value. The number of observation is equivalent to theindex, which is manually assigned. After this, the VaR is
calculatedfor each portfolio change as shown in Figure 10.
Figure10: Calculating VaR historic
CellD10 contains the sum of all portfolio values it is shown in Figure11.
Figure11: Sum of portfolio values
Toget the VaR that corresponds to our rounded down Number ofObservation (Index), the IF function is used in column R as shown inFigure 12.
Figure12: Obtaining the VaR
Allpossible values of VaR obtained by the IF function are then summed upto get the overall VaR. This step is not very necessary since onlyone value is obtained by the IF function. However, it is
required toset the result at an obvious position. It is shown in figure 13.
Figure13: VaR
Nowthe daily VaR has been obtained. It is multiplied by the square rootof the holding period (10 days) to get the portfolio 10-day holdingVaR. The last step is shown in Figure 14.
Figure14: 10-day Holding VaR.
Fromthe calculations, it can be seen that all the values obtained aredifferent. This shows how unreliable VaR is in determining the riskof business. If not keenly considered and used together with
otherparameters, it may be misleading. As a result, investments made oncertain stock relying on the results of VaR calculations can end upbeing less profitable or even end up in losses. The VaR can
be usedin calculating the capital requirements by banks in case they want toinvest in these stocks. The method used for these calculations is theBasel Accord, or the Basel II Framework.
Themain goal of the Basel II Framework is promoting the propercapitalization of banks so that improvements in risk management canbe encouraged. This can, in turn, strengthen the stability of
thefinancial system. The Basel committee imposed a requirement known asthe capital requirement for market risk in 1996 (Suarez, Dhaene,Henrard & Vanduffel, 2005). Capital requirements depend on
therisk, which is a random variable faced by a bank (Dagher et al.,2016). The capital is related to the diverse categories of assetexposures. In this regard, a risk weight (RW) is used for
theexposures. This is relative to the broad categories of relativeriskiness. The banks need to be more devoted to calculating the risksinvolved with any capital investment. In case this is ignored,
theremay be risky investments that can end up being less profitable.
Calculationof capital requirement using Basel Accord
BaselII calculates the regulatory capital (CAP[reg])as CAP[reg]=8%×RW.The calculations for the three portfolio companies are presented inthe table below:
Exposure RW Required Capital
1000 8.820523 705.6419
1000 11.37163 909.7306
1000 1.88361 150.6888
3000 22.07577 5298.184
TheRW is calculated by multiplying the 10-day VaR with a constant, 12.5.After obtaining the RW, it is multiplied by 8% and the resultingamount multiplied with the exposure to get the capital
requirement.In the case of AZV, APY, and NAB, the total capital requirement is5298.18.
Differentmethods used in calculating the VaR immensely affect the value of thecapital requirement. This implies that VaR in itself cannot be a goodtool for calculating the capital requirement. In
case there is a needto use the VaR, banks must develop a standard method of calculatinginternal VaR such that similar figures will be obtained. In case thisis overlooked, different banks will have
different VaR values andtherefore different capital requirements values for a single stock orexposure.
Thecalculation of VaR is a complex process that requires a skilledapplication of financial and accounting knowledge. There are manydifferent methods of calculating VaR. Historical simulation
andvariance-covariance methods were used in this study. This is becauseof their simplicity and ease of understanding. They are also not verytime-consuming. However, they also have their limitations
that haveto be considered during the calculations. The VaR was found to be avery unreliable means of determining the health of a business or therisks that a company or stock may go through. The
results of thecalculations were found to be varied and totally unrelated. Eventhough it has been applied to the calculation of capital requirementsby banks using the Basel Accord, the results are
equally varied andunreliable. VaR should, therefore, be used together with othermechanisms for measuring the risks of business.
Alexander,C. (2008). Valueat risk models: market risk analysis IV,John Wiley & Sons.
Bohdalova,M. (2007). A comparison of Value-at-Risk Methods for measurement ofthe financial risk. E-Leader,Prague.
Boyarshinov,A.M. (2016). Comparativeanalysis and estimation of mathematical methods of risk marketvaluation in application to Russian stock market.
Dagher,Dell’Ariccia, Laeven, Ratnovski & Tong. (2016). Benefitsand Costs of Bank Capital.IMF Staff Discussion Note. SDN/16/14.
Hull,J. (2008). Fundamentalsof futures and options markets,6th end, Pearson International.
Marrison,C. (2002). Thefundamentals of risk management,McGraw Hill.
Suarez,F., Dhaene, J., Henrard, L. & Vanduffel, S. (2005). BaselII: Capital requirements for equity investment portfolios.Katholieke Universiteit Leuven: Department of Accountancy, Financeand
Insurance (AFI). | {"url":"https://an-essay.com/calculation-of-10-day-portfolio-var","timestamp":"2024-11-06T14:40:59Z","content_type":"text/html","content_length":"237019","record_id":"<urn:uuid:b7ff2e2c-3636-4fd3-a5fc-ac2f340232dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00250.warc.gz"} |
Why is estimating an important skill? — Cydea
Why is estimating an important skill?
Friday, 20 May, 2022
How many sweets are in the jar?
As well as playing an important role in our daily lives, estimates are crucial to an organisation’s success. Planning budgets, bidding on future projects, resource allocation, personnel recruitment -
all require us to stick a finger in the air and venture a (hopefully) educated guess.
In most instances we have data to help us, but there are occasions, particularly in risk assessment, when we can’t rely on past experience to guide our estimates.
The goal is to reduce uncertainty
There’s been lots of research done on probability estimation, with two broad conclusions:
• Most people are bad at estimating probabilities
• Most people can be trained to be very good at estimating probabilities.
Most of us don’t like putting forward a number without solid evidence to back it up, particularly in a work environment, for fear of being held accountable to it. Terms like “We can’t measure that”,
“That depends” or even “There are too many variables” are used a lot in these situations. But remember: the objective of estimating is not to determine something with absolute certainty, but rather
to reduce uncertainty allowing for better decision making.
Try expressing your estimate as a range
Rather than estimating an exact value, it’s a good idea to use a range within which we have a 90% confidence that the correct answer will be. That is to say, nine-out-of-ten times we are confident
that the answer will be between the lower and upper bounds of our range, or confidence interval (CI).
It’s worth noting at this point that we are not using a 100% confidence interval, so we accept that there is still a small, but non-negligible chance that the answer could fall outside of our chosen
range. This is important because we are openly admitting that once every ten times the answer will not be in our estimated range, and allows this to be taken into account.
Test and calibrate your 90% CI
This technique was developed by Douglas Hubbard and Richard Seierson in their insightful book “How to Measure Anything in Cybersecurity Risk”. It’s an effective way of getting a baseline of your 90%
confidence interval, and is done by performing a simple test.
Answer the following question giving the answer as a 90% confidence range: How many days per year does it rain in the UK?
After you’ve answered the question, but before checking the answer, perform the following check: If money was riding on this question, which of these deals would you prefer?
• A) You get £1000 if the answer is within your estimated range, and nothing if it outside of your range
• B) You spin a wheel of which nine tenths would mean you win £1000 and 1 tenth means you receive nothing
If, like most people, you prefer the sound of option B), then you are under-confident in your range and should expand it. If you prefer option A), then you might be over-confident in your range and
could reduce it.
Ideally you want to be indifferent to options A) and B), as they should offer the exact same probability of winning the money.
Photo by Clem Onojeghuo on Unsplash. | {"url":"https://cydea.com/blog/why-is-estimating-an-important-skill/","timestamp":"2024-11-02T14:16:15Z","content_type":"text/html","content_length":"25596","record_id":"<urn:uuid:06d3d296-00af-417d-9c34-74a1da91cbf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00076.warc.gz"} |
Black Holes - Cosmos
Find out how and why string theory modifies the spacetime equations of Einstein. If string theory is a theory of gravity, then what is the relationship between strings, gravitons and spacetime
geometry? Strings and gravitons The simplest case to imagine is a single string traveling in a flat spacetime in d dimensions. As the string […]
IllustrisTNG – Most perfect model of the universe
The development of computer technology has helped bring the modelling of the evolution of our universe to a qualitatively new level. Scientists received new information about the influence of black
holes on the distribution of dark matter, learned more about the formation and propagation of heavy elements in space, as well as about the origin
IllustrisTNG – Most perfect model of the universe Read More » | {"url":"https://cosmos.theinsightanalysis.com/tag/black-holes/","timestamp":"2024-11-03T15:42:10Z","content_type":"text/html","content_length":"130786","record_id":"<urn:uuid:1363f19c-fe1e-4e13-9bda-e59cd48764ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00672.warc.gz"} |
Inch calculator - inch to
73 Inches to Centimeter
Convert 73 (seventy-three) Inches to Centimeters (inch to cm) with our conversion calculator.
73 Inches to Centimeters equals 185.42 cm.
What is 73 Inches in Centimeters?
In 73 inches there are 185.42 centimeters.
Converting from one unit of measurement to another involves understanding the unit system and using the correct conversion factor. In this case, you want to convert inches to centimeters. These units
belong to two different systems of measurement. Inches are used in the imperial system, which is most commonly used in the United States. Centimeters, on the other hand, belong to the metric system,
which is used in most of the rest of the world. The conversion factor between inches and centimeters is that 1 inch is equal to 2.54 centimeters. This is a constant ratio that you can use to convert
any amount of inches to centimeters. The general formula for conversion is: (Original quantity) x (Conversion factor) = Converted quantity In this case, the original quantity is 73 inches, and the
conversion factor is 2.54 cm/inch. So, you would calculate: 73 inches x 2.54 cm/inch = 185.42 cm So, 73 inches is equal to 185.42 centimeters. Remember, when you are doing conversions, it's important
to make sure that the units cancel out correctly. In this case, the "inches" unit cancels out, leaving you with the "cm" unit, which is what you want. This method is a practical way to convert
between different units of measurement and can be used in a wide variety of situations. For example, if you're traveling to a country that uses the metric system and you want to know how long
something is in centimeters when you only know its length in inches, or if you're working on a math or science problem that requires you to convert between different units.
73 inch equals how many cm ?
73 inch is equal to 185.42 cm
Common conversions | {"url":"https://inchto.com/73-inches-to-cm/","timestamp":"2024-11-04T05:55:46Z","content_type":"text/html","content_length":"15796","record_id":"<urn:uuid:4ac75a99-14fe-490f-b807-b531398151af>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00738.warc.gz"} |
Nonadiabatic rate constants for proton transfer and proton-coupled electron transfer reactions in solution: Effects of quadratic term in the vibronic coupling expansion
Rate constant expressions for vibronically nonadiabatic proton transfer and proton-coupled electron transfer reactions are presented and analyzed. The regimes covered include electronically adiabatic
and nonadiabatic reactions, as well as high-frequency and low-frequency proton donor-acceptor vibrational modes. These rate constants differ from previous rate constants derived with the cumulant
expansion approach in that the logarithmic expansion of the vibronic coupling in terms of the proton donor-acceptor distance includes a quadratic as well as a linear term. The analysis illustrates
that inclusion of this quadratic term in the framework of the cumulant expansion framework may significantly impact the rate constants at high temperatures for proton transfer interfaces with soft
proton donor-acceptor modes that are associated with small force constants and weak hydrogen bonds. The effects of the quadratic term may also become significant in these regimes when using the
vibronic coupling expansion in conjunction with a thermal averaging procedure for calculating the rate constant. In this case, however, the expansion of the coupling can be avoided entirely by
calculating the couplings explicitly for the range of proton donor-acceptor distances sampled. The effects of the quadratic term for weak hydrogen-bonding systems are less significant for more
physically realistic models that prevent the sampling of unphysical short proton donor-acceptor distances. Additionally, the rigorous relation between the cumulant expansion and thermal averaging
approaches is clarified. In particular, the cumulant expansion rate constant includes effects from dynamical interference between the proton donor-acceptor and solvent motions and becomes equivalent
to the thermally averaged rate constant when these dynamical effects are neglected. This analysis identifies the regimes in which each rate constant expression is valid and thus will be important for
future applications to proton transfer and proton-coupled electron transfer in chemical and biological processes.
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
• Physical and Theoretical Chemistry
Dive into the research topics of 'Nonadiabatic rate constants for proton transfer and proton-coupled electron transfer reactions in solution: Effects of quadratic term in the vibronic coupling
expansion'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/nonadiabatic-rate-constants-for-proton-transfer-and-proton-couple","timestamp":"2024-11-04T19:17:52Z","content_type":"text/html","content_length":"55503","record_id":"<urn:uuid:d0345d72-dd9e-405c-a5a2-c81398eef245>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00170.warc.gz"} |
How to Find the Surface Area of a Sphere: A Step-by-Step Guide with Examples - The Explanation Express
How to Find the Surface Area of a Sphere: A Step-by-Step Guide with Examples
If you’ve ever held a ball or looked at a globe, you’ve encountered a sphere. This three-dimensional object is defined as a perfectly round geometric shape, and it’s used in many real-world
applications, from satellites to medicine. One important aspect of a sphere is its surface area, which measures the total area of the outside of the sphere. Knowing how to find the surface area of a
sphere is useful in many fields, including architecture, physics, and engineering. In this article, we’ll explore the geometry of a sphere, the formula for its surface area, methods for finding this
value, and real-world examples.
The Geometry of a Sphere and Its Surface Area Calculation
A sphere is defined as a set of points in three-dimensional space that are all equidistant from a given center point. This means that the distance from any point on the surface of the sphere to the
center point is the same. The most common way to measure a sphere is by its radius, which is the distance from the center to any point on the surface.
The surface area of a sphere measures the total area of the outside of the sphere. This can be thought of as the amount of material needed to cover the entire surface of the sphere. The surface area
of a sphere is important in many fields, including physics, where it’s used to calculate air resistance and buoyancy; and architecture, where it’s used to calculate the amount of paint or wallpaper
needed to cover a spherical room.
The formula for finding the surface area of a sphere is:
SA = 4πr²
SA = surface area
π = pi (3.14…)
r = radius
A Step-by-Step Guide for Finding the Surface Area of a Sphere
To find the surface area of a sphere using the formula above, follow these steps:
1. Measure the radius of the sphere.
2. Square the radius value.
3. Multiply the squared radius by 4.
4. Multiply the result by pi.
Let’s say you have a sphere with a radius of 5 centimeters. To find the surface area, use the formula above:
SA = 4πr²
SA = 4 x π x 5²
SA = 4 x π x 25
SA = 100π
Therefore, the surface area of the sphere is 100π square centimeters.
Using Mathematical Formulas to Find the Surface Area of a Sphere
While the formula given above is the most common way to find the surface area of a sphere, there are other mathematical formulas that can also be used. These include:
– SA = 2πrh
– SA = 2πr² + 2πrh
– SA = 4r² cot(π/n)
However, the formula SA = 4πr² is the most straightforward and widely used method for finding the surface area of a sphere.
Understanding the Connection Between the Radius and the Surface Area of a Sphere
The radius of a sphere plays a crucial role in its surface area calculation. As we saw in the formula above, the surface area of a sphere is directly proportional to the square of its radius. This
means that as the radius of a sphere increases, its surface area also increases. Conversely, as the radius of a sphere decreases, its surface area decreases.
To illustrate this, let’s look at two spheres with different radii:
Sphere 1: r = 5
Sphere 2: r = 7
Using the formula for finding the surface area, we can calculate the surface area of each sphere:
Sphere 1: SA = 4π(5)² = 100π
Sphere 2: SA = 4π(7)² = 196π
We can see that Sphere 2 has a greater surface area than Sphere 1, because its radius is greater.
Using Practical Examples to Demonstrate How to Find the Surface Area of a Sphere
Real-world examples can help us understand the importance of finding the surface area of a sphere and how to apply the formula. Here are some scenarios where finding the surface area of a sphere is
– A ball manufacturer needs to determine how much material is required to cover each ball they produce.
– An architect wants to determine how much paint is needed to cover the interior of a spherical room.
– An engineer wants to calculate the surface area of a spherical shape to determine the amount of material needed to manufacture a satellite.
To solve these problems, the formula for surface area can be applied using the radius of the sphere. Let’s take the first example:
Example: A ball manufacturer needs to determine how much material is required to cover each ball they produce. The radius of each ball is 10 centimeters.
To find the surface area of each ball, use the formula:
SA = 4πr²
SA = 4π(10)²
SA = 400π
Therefore, the surface area of each ball is 400π square centimeters.
Comparing Different Methods Used to Determine the Surface Area of a Sphere
As we saw earlier, there are various methods to find the surface area of a sphere. However, the formula of SA = 4πr² is the simplest and most commonly used. The other methods have their own
advantages and disadvantages. For example, the formula SA = 2πr² + 2πrh is useful when working with certain types of shapes, such as cylinders. The formula SA = 4r² cot(π/n) can be useful when
working with shapes that are not exactly spherical, but have a similar structure.
It’s important to note that the formula used to find the surface area of a sphere will depend on the specific situation and the shape being measured.
Providing Worked-Out Exercises for Readers to Practice Finding the Surface Area of a Sphere
Now that we’ve covered the basics of finding the surface area of a sphere, it’s time to practice. Here are some exercises to get you started:
1. A sphere has a radius of 8 centimeters. What is its surface area?
2. A spherical room has a radius of 12 feet. How much paint is needed to cover the interior of the room if the paint covers 100 square feet per gallon?
3. A company produces metal ball bearings with a diameter of 2.5 centimeters. What is the surface area of each ball bearing?
1. SA = 4πr²
SA = 4π(8)²
SA = 256π
Therefore, the surface area of the sphere is 256π square centimeters.
2. SA = 4πr²
SA = 4π(12)²
SA = 576π
One gallon of paint can cover 100 square feet, so we need:
576π/100 = 18.1 gallons of paint.
3. The radius is half of the diameter, so r = 1.25 centimeters.
SA = 4πr²
SA = 4π(1.25)²
SA = 19.63π
Therefore, the surface area of each ball bearing is 19.63π square centimeters.
The surface area of a sphere is an important geometric measurement that can be used in various fields. Whether you’re an architect, mathematician, or engineer, understanding how to find the surface
area of a sphere is essential. By following the steps and exploring the formulas and examples provided in this article, you’re on your way to mastering this concept. If you want to learn more, there
are additional resources available to help you explore this subject further. | {"url":"https://www.branchor.com/how-to-find-the-surface-area-of-a-sphere/","timestamp":"2024-11-04T20:33:18Z","content_type":"text/html","content_length":"45589","record_id":"<urn:uuid:f0170cf1-eedf-4b71-bd44-6a593c314f8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00869.warc.gz"} |
Discount factor to interest rate calculator
26 Oct 2010 In a previous post, I described the technique that computer programs like Microsoft Excel use to calculate the XIRR (effective interest rate) as a Interest rates and the time value of
money What is the basis of determining discount rate? If so, what other factors besides inflation should be considered? To calculate present value you need a forecast of the future cash flows, and
you How to Discount Cash Flow, Calculate PV, FV and Net Present Value How do analysts choose the discount (interest) rate for DCF analysis? divides each FV value by a more substantial discount
factor than do mid-period calculations.
Discount Factors and Equivalence.. 51-2 the sinking fund factor, you could calculate the neces- The interest rate used in the discount factor formulas is. Table of contents. Chapter 1. Interest rates
and factors. 1. 1.1. Interest. 2. 1.2 discounted to a previous point in time, its present value is calculated, taking into So Sam tries once more, but with 7% interest: At 7% Sam gets a Net Present
Value of $15. Close enough to zero, Sam doesn't want to calculate any more. 6 Dec 2018 With regard to the discounted rate, this factor is based on how the high-interest loans should be considered
when determining the NPV. 23 Aug 2018 Annuity factors are used to calculate present values of annuities, and The Annuity Factor is the sum of the discount factors for maturities 1 Sometimes also
known as the Present Value Interest Factor of an Annuity (PVIFA). The discount factor can be calculated based on the discount rate and number of compounding periods. Code to add this calci to your
website Expand embed code
8 Apr 2010 annual (p.a.) interest rate given as a per cent value: i = p/100 periods: is calculated by multiplying the principal P by the accumulation factor,
8 Mar 2018 To calculate the discount factor for a cash flow one year from now, divide 1 by the interest rate plus 1. For example, if the interest rate is 5 Some analysts prefer to calculate explicit
discount factors in each time period so they can see the effects of compounding more clearly, as well as making the Discount Factor Calculator - calculate the discount factor which is a way of
discounting cash flows to get the present value of an investment. Discount factor Formula for the calculation of a discount factor based on the periodic interest rate and the number of interest
HOMER calculates the annual real discount rate (also called the real interest rate or HOMER uses the real discount rate to calculate discount factors and
The discount rate is the annualized rate of interest and it is denoted by 'i'. Step 2: Now, determine for how long the money is going to remain invested i.e. the tenure If only a nominal interest
rate (rate per annum or rate per year) is known, you can calculate the discount rate using the following formula: Simple Amortization
6 Dec 2018 With regard to the discounted rate, this factor is based on how the high-interest loans should be considered when determining the NPV.
10 Apr 2019 This is the interest rate. However, another guy may calculate the return with reference to the future value—his calculation would be ($60,000 - frequencies of compounding, the effective
rate of interest and rate of discount, and Solution: We first calculate the discount factors v(4) and v(9). For case (a)
11 May 2017 It is this rate of interest that is known as the discount rate. In a personal injury action, his award will be calculated to ensure that if he Furthermore, if the courts decide that
inflation is likely to become an important factor, they
Discount Factor Calculation (Step by Step) It can be calculated by using the following steps: Step 1: Firstly, figure out the discount rate for a similar kind of investment based on market
information. The discount rate is the annualized rate of interest and it is denoted by ‘i’. The definition of a discount rate depends the context, it's either defined as the interest rate used to
calculate net present value or the interest rate charged by the Federal Reserve Bank. There are two discount rate formulas you can use to calculate discount rate, WACC (weighted average cost of
capital) and APV (adjusted present value). How to calculate discount rate? The formula used to calculate discount is D=1/(1+P) n where D is discount factor, P = periodic interest rate, n is number of
payments. To calculate a discount rate, you first need to know the going interest rate that your business could get from investing capital in an investment with similar risk. You can then calculate
the discount rate using the formula 1/(1+i)^n, where i equals the interest rate and n represents how many years until you receive the cash flow.
10 Apr 2019 This is the interest rate. However, another guy may calculate the return with reference to the future value—his calculation would be ($60,000 - | {"url":"https://topoptionskuou.netlify.app/deranick56218do/discount-factor-to-interest-rate-calculator-270","timestamp":"2024-11-02T04:55:53Z","content_type":"text/html","content_length":"32514","record_id":"<urn:uuid:c7806ff8-7f20-49f6-a569-721620bc99e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00135.warc.gz"} |
National Population Projections
National Population Projections
National Population Projections provide projected populations of New Zealand, based on different combinations of fertility, mortality, and migration assumptions.
Demographic projections provide an indication of future trends in the size and composition of the population, labour force, families and households. The projections are used for community, business
and government planning and policy-making in areas such as health, education, superannuation and transport. The projections, along with the assumptions for fertility, mortality and migration, are
typically updated every two to three years.
National population projections are produced to assist businesses and government agencies, in planning and policy-making. The projections provide information on the changing characteristics and
distribution of the population, which are used to develop social policies in areas such as health and education. For example, the ageing population, population projections can help identify likely
future service needs.
The projections are neither predictions nor forecasts. They provide an indication of possible future changes in the size and composition of the population. While the projection assumptions are
formulated from an assessment of short-term and long-term demographic trends, there is no certainty that any of the assumptions will be realised.
Significant events impacting this study series
Population concept for all demographic estimates, projections and indices changed from 'de facto' to 'resident'. Population estimates based on the de facto population concept (the estimated de facto
population) include visitors from overseas, but made no adjustments for net census undercount or residents temporarily overseas. Population estimates based on the resident population concept (the
estimated resident population) include adjustments for net census undercount and residents temporarily overseas, but exclude overseas visitors. The reference date for projections is shifted from 31
March to 30 June.
For the first time, Statistics NZ applied a stochastic (probabilistic) approach to producing population projections. Stochastic population projections provide a means of quantifying demographic
uncertainty, although it is important to note that estimates of uncertainty are themselves uncertain. By modelling uncertainty in the projection assumptions and deriving simulations, estimates of
probability and uncertainty are available for each projection result. No simulation is more likely, or more unlikely, than any other. The simulations provide a probability distribution which can be
summarised using percentiles, with the 50th percentile equal to the median.
Usage and limitations of the data
Nature of Projections
These projections are not predictions. The projections should be used as an indication of the overall trend, rather than as exact forecasts. The projections are updated every 2–3 years to maintain
their relevance and usefulness, by incorporating new information about demographic trends and developments in methods.
The projections are designed to meet both short-term and long-term planning needs, but are not designed to be exact forecasts or to project specific annual variation. These projections are based on
assumptions made about future fertility, mortality, and migration patterns of the population. While the assumptions are formulated from an assessment of short-term and long-term demographic trends,
there is no certainty that any of the assumptions will be realised.
The projections do not take into account non-demographic factors (eg war, catastrophes, major government and business decisions) which may invalidate the projections.
Main users of the data
Statistics New Zealand, Ministry of Health, Government Planners/Local Body Planners, Ministry of Education, Consultants, Private Businesses.
Information Release
National Population Projections 2022(base)-2073
Information release
National population projections: 2020(base)–2073
How accurate are population estimates and projections? An evaluation of Statistics New Zealand population estimates and projections, 1996–2013.
How accurate are population estimates and projections? An evaluation of Statistics New Zealand population estimates and projections, 1996–2013 evaluates the accuracy of recent national and
subnational population estimates and projections.
The report focuses on estimates and projections of the total population produced and published since 1996, although earlier projections are included where practicable.
It is designed to help customers understand the accuracy of Stats NZ’s population estimates and projections relative to observed populations, the reasons for inaccuracies, and discusses current
developments that may improve accuracy.
Information release
National population projections: 2016(base)–2068
Experimental stochastic population projections for New Zealand: 2009 (base) – 2011
Population and migration
Data Collection
National Population Projections ^en-NZ
The 'cohort component' method has been used to derive the population projections. Using this method, the base population is projected forward by calculating the effect of deaths and migration within
each age-sex group (or cohort) according to the specified mortality and migration assumptions. New birth cohorts are added to the population by applying the specified fertility assumptions to the
female population of childbearing age.
The stochastic approach used in the national population projections since the 2011-base projections involves creating 2,000 simulations for the base population, births, deaths, and net migration, and
then combining these using the cohort component method.
These simulations can be summarised by percentiles, which indicate the probability that the actual result is lower than the percentile. For example, the 25th percentile indicates an estimated 25
percent chance that the actual value will be lower, and a 75 percent chance that the actual result will be higher, than this percentile.
Nine alternative percentiles of probability distribution (2.5th, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 97.5th percentiles) are available in NZ.Stat.
Projection Assumptions
Projection assumptions are formulated after analysis of short-term and long-term demographic trends, patterns and trends observed in other countries, government policy, information provided by local
planners and other relevant information.
Assumptions for national projections are derived for each single-year of age to produce projections at one-year intervals. The following describes how assumptions are applied for national
Projected (live) births are derived by applying age-specific fertility rates to the mean female population of childbearing age. The mean female population for each age is derived by averaging the
population at the start and end of each year. The sum of the number of births derived for each age of mother gives the projected number of births for each year.
The female age-specific fertility rates for each year of the projection period represent the number of births to females of each age in each year. The set of age-specific fertility rates for each
year is typically summarised by the total fertility rate.
For all population projections, a sex ratio at birth of 105.5 males per 100 females is assumed, based on the historical annual average of the total population.
The fertility assumptions should not be used as a precise measure of fertility or of fertility differentials between groups. It is important to note that the objective of population projections is
not to specifically measure or project the fertility of the population. For projection purposes it is more important to have a realistic yet tractable model for projecting fertility trends (and birth
numbers) into the future.
Mortality assumptions are formulated in terms of survival rates. This is because in the projection model the base population is survived forward each year. The projected number of deaths is
calculated indirectly. Survival rates are applied to births and single years of age. There are different survival rates for each age of life and for males and females.
The male and female age-specific survival rates for each year of the projection period represent the proportion of people at each age-sex who will survive for another year. In general, survival rates
are highest at ages 5–11 years and then decrease with increasing age. The set of age-sex-specific survival rates for each year is typically summarised by male and female life expectancies at birth.
Annual survival rates are applied separately to the population at the start of each year, births and migrants.
The mortality assumptions should not be used as a precise measure of mortality or of mortality differentials between groups. It is important to note that the objective of population projections is
not to specifically measure or project the life expectancy of the population. For projection purposes it is more important to have a realistic yet tractable model for projecting mortality trends (and
death numbers) into the future.
Migration assumptions are formulated in terms of a net migration level and an age-sex net migration pattern for each year of the projection period. Where practical, both the level and age-sex pattern
are derived from a detailed analysis of net migration, including:
• external migration data (from passenger cards):
• arrivals and departures by country of citizenship
• New Zealand citizen arrivals and departures by country of source/destination
• Immigration New Zealand data:
• residence applications and approvals
• student and work visas
2020-base to 2073
The 2020-base national population projections (released December 2020) have as a base the estimated resident population (ERP) of New Zealand at 30 June 2020, and cover the period to 2073 at one-year
intervals. They superseded by the 2016-base national population projections (released October 2016).
Detailed information on the 2020-base population, assumptions and 'what if' scenarios used can be found here National Population Projections 2020-base
2016-base to 2068
The 2016-base national population projections (released October 2016) have as a base the provisional estimated resident population (ERP) of New Zealand at 30 June 2016, and cover the period to 2068
at one-year intervals. They supersede the 2014-base national population projections (released November 2014).
Detailed information on the base population, assumptions and 'what if' scenarios can be found here: National Population Projections 2016-base
2014-base to 2068
The 2014-base national population projections have been superseded by the 2011-base national population projections (released September 2012).
Detailed information on the 2014 base population, assumptions and 'what if' scenarios used can be found here National Population Projections 2014-base | {"url":"https://datainfoplus.stats.govt.nz/item/nz.govt.stats/583ca9da-d6d2-41e0-b626-5743c14deaf5/128","timestamp":"2024-11-08T23:33:08Z","content_type":"text/html","content_length":"58758","record_id":"<urn:uuid:5744626f-5e8c-4074-b398-1126e63d9701>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00002.warc.gz"} |
Ordinal Numbers In Spanish To English - OrdinalNumbers.com
Ordinal Numbers In Spanish Translation – A limitless number of sets can easily be enumerated with ordinal numerals to aid in the process of. These numbers can be utilized as a tool to generalize
ordinal figures. 1st The ordinal number is among the most fundamental ideas in math. It is a number that indicates the … Read more | {"url":"https://www.ordinalnumbers.com/tag/ordinal-numbers-in-spanish-to-english/","timestamp":"2024-11-13T20:54:04Z","content_type":"text/html","content_length":"46601","record_id":"<urn:uuid:4af3f220-63ce-4101-9d03-ae48fedaf4dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00168.warc.gz"} |
How to Define Open Vector In Julia?
In Julia, an open vector can be defined as a one-dimensional array that does not have a fixed length or size. Unlike a closed vector, which has a specific number of elements that cannot be changed,
an open vector allows for elements to be added or removed dynamically.
To define an open vector in Julia, you can simply create an empty array using the Vector{T} constructor, where T is the type of elements that will be stored in the vector. For example, vector =
Vector{Int}() creates an empty vector that can store integer values.
You can then add elements to the open vector using the push! function, which appends an element to the end of the vector. For example, push!(vector, 1) adds the integer value 1 to the vector.
Similarly, you can remove elements from the vector using the pop! function, which removes and returns the last element in the vector. For example, pop!(vector) removes the last element from the
Overall, defining an open vector in Julia allows for dynamic manipulation of elements, making it a flexible and versatile data structure for storing and processing data.
What is the advantage of using an open vector in Julia?
One advantage of using an open vector in Julia is that it allows for easy and efficient manipulation of data. Open vectors are mutable and can be modified in place without having to create a new copy
of the data. This can improve performance and reduce memory usage, especially when working with large datasets. Additionally, open vectors can be easily passed by reference to functions, which can
further improve the speed and efficiency of calculations.
How to create a 2D open vector in Julia?
To create a 2D open vector in Julia, you can use the Vector constructor with two elements. Here is an example code snippet:
1 # Create a 2D open vector
2 v = Vector{Float64}([1.0, 2.0])
4 # Print the vector
5 println(v)
This will create a 2D open vector [1.0, 2.0] of type Float64 in Julia.
How to convert an open vector to a closed vector in Julia?
In Julia, there isn't a specific function to convert an open vector to a closed vector, but you can achieve this by adding the last element of the open vector to make it a closed vector.
Here's an example code to convert an open vector to a closed vector:
1 open_vector = [1, 2, 3, 4, 5] # Open vector
2 closed_vector = [open_vector..., open_vector[end]] # Add the last element of open vector to make it closed
4 println(closed_vector) # Output: [1, 2, 3, 4, 5, 5]
In this code, we use the splat operator ... to unpack the elements of the open vector and then add the last element of the open vector to make it a closed vector.
What is the performance impact of using open vectors in Julia?
Using open vectors in Julia can have a significant performance impact because they are not optimized for efficient memory layout. Open vectors store elements in a flat array in memory, which can lead
to cache inefficiencies and poor data locality. This can result in slower performance compared to using specialized data structures like arrays or matrices that are optimized for efficient memory
access. Additionally, open vectors may require additional checks or type conversions during operations, which can also impact performance. It is recommended to use specialized data structures in
Julia for better performance, especially when dealing with large datasets or complex operations.
How to concatenate open vectors in Julia?
In Julia, you can concatenate open vectors (arrays) using the vcat() function. This function takes in multiple vectors as arguments and concatenates them vertically to create a new vector.
Here is an example of how to concatenate two open vectors in Julia:
1 # Create two open vectors
2 vec1 = [1, 2, 3]
3 vec2 = [4, 5, 6]
5 # Concatenate the two vectors
6 result = vcat(vec1, vec2)
8 println(result)
This will output:
You can concatenate multiple vectors by passing them as arguments to the vcat() function.
What is the default indexing style for open vectors in Julia?
In Julia, the default indexing style for open vectors is 1-based indexing. This means that the first element of the vector is accessed using index 1, the second element using index 2, and so on. | {"url":"https://topminisite.com/blog/how-to-define-open-vector-in-julia","timestamp":"2024-11-12T16:25:04Z","content_type":"text/html","content_length":"302237","record_id":"<urn:uuid:6c34e52e-1541-4b86-a4f2-64450294403d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00313.warc.gz"} |
Introduction To Electrodynamics Griffiths 4th Edition PDF - Knowdemia
Introduction To Electrodynamics Griffiths 4th Edition PDF
Would you like to get David J Griffiths Introduction To Electrodynamics 4th Edition PDF download? Have you been searching cluelessly for where to get an introduction to electrodynamics Griffiths 4th
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Introduction To Electrodynamics Griffiths 4th Edition PDF Book Details
• Author: by David J. Griffiths
• ISBN-13: 9781108420419
• Publisher: Cambridge University Press
• Publication date: 06/29/2017
• Pages: 620
• Size: 5 MB
• Format: PDF
About Introduction To Electrodynamics Griffiths 4th Edition PDF Book
This well-known undergraduate electrodynamics textbook is now available in a more affordable printing from Cambridge University Press. The Fourth Edition provides a rigorous, yet clear and accessible
treatment of the fundamentals of electromagnetic theory and offers a sound platform for explorations of related applications (AC circuits, antennas, transmission lines, plasmas, optics and more).
Written keeping in mind the conceptual hurdles typically faced by undergraduate students, this textbook illustrates the theoretical steps with well-chosen examples and careful illustrations. It
balances text and equations, allowing the physics to shine through without compromising the rigour of the math, and includes numerous problems, varying from straightforward to elaborate, so that
students can be assigned some problems to build their confidence and others to stretch their minds. A Solutions Manual is available to instructors teaching from the book; access can be requested from
the resources section at www.cambridge.org/electrodynamics.
Introduction To Electrodynamics Griffiths 4th Edition PDF Book Table of Contents
1. Vector analysis;
2. Electrostatics;
3. Potentials;
4. Electric fields in matter;
5. Magnetostatics;
6. Magnetic fields in matter;
7. Electrodynamics;
8. Conservation laws;
9. Electromagnetic waves;
10. Potentials and fields;
11. Radiation;
12. Electrodynamics and relativity;
Appendix A.
Vector calculus in curvilinear coordinates;
Appendix B.
The Helmholtz theorem;
Appendix C.
About the Author
David J. Griffiths is Emeritus Professor of Physics from Reed College, Oregon, where he taught physics for over thirty years. He received his B.A. and Ph.D. from Harvard University, where he studied
elementary particle theory.
Get David J Griffiths Introduction to Electrodynamics 4th Edition PDF Free Download Below:
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Type Promotion Rules
Type Promotion Rules¶
Array API specification for type promotion rules.
Type promotion rules can be understood at a high level from the following diagram:
Type promotion diagram. Promotion between any two types is given by their join on this lattice. Only the types of participating arrays matter, not their values. Dashed lines indicate that behavior
for Python scalars is undefined on overflow. Boolean, integer and floating-point dtypes are not connected, indicating mixed-kind promotion is undefined.
A conforming implementation of the array API standard must implement the following type promotion rules governing the common result type for two array operands during an arithmetic operation.
A conforming implementation of the array API standard may support additional type promotion rules beyond those described in this specification.
Type codes are used here to keep tables readable; they are not part of the standard. In code, use the data type objects specified in Data Types (e.g., int16 rather than 'i2').
The following type promotion tables specify the casting behavior for operations involving two array operands. When more than two array operands participate, application of the promotion tables is
associative (i.e., the result does not depend on operand order).
Signed integer type promotion table¶
i1 i2 i4 i8
i1 i1 i2 i4 i8
i2 i2 i2 i4 i8
i4 i4 i4 i4 i8
i8 i8 i8 i8 i8
• i1: 8-bit signed integer (i.e., int8)
• i2: 16-bit signed integer (i.e., int16)
• i4: 32-bit signed integer (i.e., int32)
• i8: 64-bit signed integer (i.e., int64)
Unsigned integer type promotion table¶
u1 u2 u4 u8
u1 u1 u2 u4 u8
u2 u2 u2 u4 u8
u4 u4 u4 u4 u8
u8 u8 u8 u8 u8
• u1: 8-bit unsigned integer (i.e., uint8)
• u2: 16-bit unsigned integer (i.e., uint16)
• u4: 32-bit unsigned integer (i.e., uint32)
• u8: 64-bit unsigned integer (i.e., uint64)
Mixed unsigned and signed integer type promotion table¶
u1 u2 u4
i1 i2 i4 i8
i2 i2 i4 i8
i4 i4 i4 i8
i8 i8 i8 i8
Floating-point type promotion table¶
f4 f8 c8 c16
f4 f4 f8 c8 c16
f8 f8 f8 c16 c16
c8 c8 c16 c8 c16
c16 c16 c16 c16 c16
• f4: single-precision (32-bit) floating-point number (i.e., float32)
• f8: double-precision (64-bit) floating-point number (i.e., float64)
• c8: single-precision complex floating-point number (i.e., complex64) composed of two single-precision (32-bit) floating-point numbers
• c16: double-precision complex floating-point number (i.e., complex128) composed of two double-precision (64-bit) floating-point numbers
• Type promotion rules must apply when determining the common result type for two array operands during an arithmetic operation, regardless of array dimension. Accordingly, zero-dimensional arrays
must be subject to the same type promotion rules as dimensional arrays.
• Type promotion of non-numerical data types to numerical data types is unspecified (e.g., bool to intxx or floatxx).
Mixed integer and floating-point type promotion rules are not specified because behavior varies between implementations.
Mixing arrays with Python scalars¶
Using Python scalars (i.e., instances of bool, int, float, complex) together with arrays must be supported for:
• array <op> scalar
• scalar <op> array
where <op> is a built-in operator (including in-place operators, but excluding the matmul @ operator; see Operators for operators supported by the array object) and scalar has a type and value
compatible with the array data type:
• a Python bool for a bool array data type.
• a Python int within the bounds of the given data type for integer array Data Types.
• a Python int or float for real-valued floating-point array data types.
• a Python int, float, or complex for complex floating-point array data types.
Provided the above requirements are met, the expected behavior is equivalent to:
1. Convert the scalar to zero-dimensional array with the same data type as that of the array used in the expression.
2. Execute the operation for array <op> 0-D array (or 0-D array <op> array if scalar was the left-hand argument).
Behavior is not specified when mixing a Python float and an array with an integer data type; this may give float32, float64, or raise an exception. Behavior is implementation-specific.
Similarly, behavior is not specified when mixing a Python complex and an array with a real-valued data type; this may give complex64, complex128, or raise an exception. Behavior is
Behavior is also not specified for integers outside of the bounds of a given integer data type. Integers outside of bounds may result in overflow or an error. | {"url":"https://data-apis.org/array-api/2022.12/API_specification/type_promotion.html","timestamp":"2024-11-06T19:03:26Z","content_type":"text/html","content_length":"34321","record_id":"<urn:uuid:8693b43f-caf0-4d1f-9c6f-ad11a0d6c8fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00606.warc.gz"} |
The Stacks project
Lemma 72.10.5. Let $k$ be a field. Let $X$ be a quasi-separated algebraic space over $k$. If there exists a purely transcendental field extension $K/k$ such that $X_ K$ is a scheme, then $X$ is a
Proof. Since every algebraic space is the union of its quasi-compact open subspaces, we may assume $X$ is quasi-compact (some details omitted). Recall (Fields, Definition 9.26.1) that the assumption
on the extension $K/k$ signifies that $K$ is the fraction field of a polynomial ring (in possibly infinitely many variables) over $k$. Thus $K = \bigcup A$ is the union of subalgebras each of which
is a localization of a finite polynomial algebra over $k$. By Limits of Spaces, Lemma 70.5.11 we see that $X_ A$ is a scheme for some $A$. Write
for some nonzero $f \in k[x_1, \ldots , x_ n]$.
If $k$ is infinite then we can finish the proof as follows: choose $a_1, \ldots , a_ n \in k$ with $f(a_1, \ldots , a_ n) \not= 0$. Then $(a_1, \ldots , a_ n)$ define an $k$-algebra map $A \to k$
mapping $x_ i$ to $a_ i$ and $1/f$ to $1/f(a_1, \ldots , a_ n)$. Thus the base change $X_ A \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) \cong X$ is a scheme as desired.
In this paragraph we finish the proof in case $k$ is finite. In this case we write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $k$ and with affine transition
morphisms (Limits of Spaces, Lemma 70.10.2). Using Limits of Spaces, Lemma 70.5.11 we see that $X_{i, A}$ is a scheme for some $i$. Thus we may assume $X \to \mathop{\mathrm{Spec}}(k)$ is of finite
presentation. Let $x \in |X|$ be a closed point. We may represent $x$ by a closed immersion $\mathop{\mathrm{Spec}}(\kappa ) \to X$ (Decent Spaces, Lemma 68.14.6). Then $\mathop{\mathrm{Spec}}(\kappa
) \to \mathop{\mathrm{Spec}}(k)$ is of finite type, hence $\kappa $ is a finite extension of $k$ (by the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1; some details omitted). Say $[\kappa :
k] = d$. Choose an integer $n \gg 0$ prime to $d$ and let $k'/k$ be the extension of degree $n$. Then $k'/k$ is Galois with $G = \text{Aut}(k'/k)$ cyclic of order $n$. If $n$ is large enough there
will be $k$-algebra homomorphism $A \to k'$ by the same reason as above. Then $X_{k'}$ is a scheme and $X = X_{k'}/G$ (Lemma 72.10.3). On the other hand, since $n$ and $d$ are relatively prime we see
\[ \mathop{\mathrm{Spec}}(\kappa ) \times _{X} X_{k'} = \mathop{\mathrm{Spec}}(\kappa ) \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') = \mathop{\mathrm{Spec}}(\kappa \otimes _ k k')
is the spectrum of a field. In other words, the fibre of $X_{k'} \to X$ over $x$ consists of a single point. Thus by Lemma 72.10.4 we see that $x$ is in the schematic locus of $X$ as desired. $\
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Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1000 miles an hour would land at exactly the same time that it started
(why?). Moreover, the sun would not move in the plane’s sky during the flight. (Work this out visually in your mind.) Now, at what degree of latitude could a plane flying 500 miles per hour keep up
with the sun in this way?
Gertrude Goldfish
Two young men in tuxedos are walking down the street at about 10 pm. One of them is carrying a round goldfish bowl filled with water. There is a goldfish, named Gertrude, in the bowl. As they pass a
round pool in the park, the goldfish gets very excited and jumps out of the bowl into the pool, right at the edge. She starts swimming due north. She hits the wall after she has swum exactly 30 feet.
She heads east, and, after going 40 feet, she hits the wall again. Once she regains consciousness after this second collision, she calculates the diameter of the pool. What is it? And what is the
story about the two guys in their tuxes?
Fences in Circular Region
In a circular field, place three fences to make four regions. The fences are all equal in length and their endpoints are on the circular boundary of the field. The four resulting regions have equal
area, and the fences don�t intersect within the field.
Billy the Goat
Billy the goat is tied to the corner of Patty�s barn. The barn is 20 x 40 feet, and the rope is 50 feet long. No trees or other obstructions are in the way. What is the available area of grass that
Billy can eat? (Draw a good picture of this area first.)
Algebra with Angles
If twice ∠A is subtracted from the supplement of ∠A, then the remaining angle exceeds the complement of ∠A by 4°. Find the size of ∠A.
Geometric and Arithmetric Sequences
There are two positive numbers that may be inserted between 3 and 9 such that the first three numbers are in geometric progression while the last three are in arithmetic progression. The sum of those
two positive numbers is:
1. 13½
2. 11¼
3. 10½
4. 109½
Note: There are two other numbers that work, but they’re not both positive. If you go about this problem in a suitably erudite fashion, you’ll turn up this alternative solution too.
To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end
on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days.
That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and
Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today.
Hard Functional Equation, f(n) = n
The function f satisfies the functional equation
f(x) + f(y) = f(x + y) – xy – 1
for every pair x, y of real numbers. If f(1) = 1, then the number of positive integers n for which f(n) = n is:
1. 0
2. 1
3. 2
4. 3
5. infinite
Defined Operations 2; Do Composition
If 3 h = 10, 7 h = 50, 5 h = 26; and 4 b = 1, 7 b = 2.5, 20 b = 9, then what is n
if n hb = 17.5?
Garden Crops
A large organic nursery has somewhere between 3 and 6 (inclusive) garden plots of herbs when it closes for the season in the fall. In each plot there are between 20 and 30 (inclusive) rosemary
plants. If, typically, 10% of those plants don’t winter over successfully until spring, what would be the largest number of plants that could be lost during the winter? | {"url":"https://u.osu.edu/odmp/category/set-8/","timestamp":"2024-11-02T02:32:58Z","content_type":"text/html","content_length":"60830","record_id":"<urn:uuid:e593d98e-afaa-44f4-b25a-7562e72e0af4>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00639.warc.gz"} |
A Step Towards Learning Contraction Kernels for Irregular Image Pyramid
Darshan Batavia, Rocio Gonzalez-Diaz, Walter Kropatsch
A structure preserving irregular image pyramid can be computed by applying basic graph operations (contraction and removal of edges) on the 4-adjacent neighbourhood graph of an image. In this paper,
we derive an objective function that classifies the edges as contractible or removable for building an irregular graph pyramid. The objective function is based on the cost of the edges in the
contraction kernel (sub-graph selected for contraction) together with the size of the contraction kernel. Based on the objective function, we also provide an algorithm that decomposes a 2D image into
monotonically connected regions of the image surface, called slope regions. We proved that the proposed algorithm results in a graph-based irregular image pyramid that preserves the structure and the
topology of the critical points (the local maxima, the local minima, and the saddles). Later we introduce the concept of the dictionary for the connected components of the contraction kernel,
consisting of sub-graphs that can be combined together to form a set of contraction kernels. A favorable contraction kernel can be selected that best satisfies the objective function. Lastly, we show
the experimental verification for the claims related to the objective function and the cost of the contraction kernel. The outcome of this paper can be envisioned as a step towards learning the
contraction kernel for the construction of an irregular image pyramid.
Paper Citation
in Harvard Style
Batavia D., Gonzalez-Diaz R. and Kropatsch W. (2022). A Step Towards Learning Contraction Kernels for Irregular Image Pyramid. In Proceedings of the 11th International Conference on Pattern
Recognition Applications and Methods - Volume 1: ICPRAM, ISBN 978-989-758-549-4, pages 60-70. DOI: 10.5220/0010840900003122
in Bibtex Style
author={Darshan Batavia and Rocio Gonzalez-Diaz and Walter Kropatsch},
title={A Step Towards Learning Contraction Kernels for Irregular Image Pyramid},
booktitle={Proceedings of the 11th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},
in EndNote Style
TY - CONF
JO - Proceedings of the 11th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,
TI - A Step Towards Learning Contraction Kernels for Irregular Image Pyramid
SN - 978-989-758-549-4
AU - Batavia D.
AU - Gonzalez-Diaz R.
AU - Kropatsch W.
PY - 2022
SP - 60
EP - 70
DO - 10.5220/0010840900003122 | {"url":"http://scitepress.net/PublishedPapers/2022/108409/","timestamp":"2024-11-07T12:06:32Z","content_type":"text/html","content_length":"8078","record_id":"<urn:uuid:ec680bba-938f-4a4d-8d31-cb6583240f6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00034.warc.gz"} |
powf(3m) [opensolaris man page]
powf(3m) [opensolaris man page]
pow(3M) Mathematical Library Functions pow(3M)
pow, powf, powl - power function
c99 [ flag... ] file... -lm [ library... ]
#include <math.h>
double pow(double x, double y);
float powf(float x, float y);
long double powl(long double x, long double y);
cc [ flag... ] file... -lm [ library... ]
#include <math.h>
double pow(double x, double y);
float powf(float x, float y);
long double powl(long double x, long double y);
These functions compute the value of x raised to the power y, x^y>. If x is negative, y must be an integer value.
Upon successful completion, these functions return the value of x raised to the power y.
For finite values of x < 0, and finite non-integer values of y, a domain error occurs and either a NaN (if representable), or an implemen-
tation-defined value is returned.
If the correct value would cause overflow, a range error occurs and pow(), powf(), and powl() return HUGE_VAL, HUGE_VALF, and HUGE_VALL,
If x or y is a NaN, a NaN is returned unless:
o If x is +1 and y is NaN and the application was compiled with the c99 compiler driver and is therefore SUSv3-conforming (see
standards(5)), 1.0 is returned.
o For any value of x (including NaN), if y is +0, 1.0 is returned.
For any odd integer value of y > 0, if x is +-0, +-0 is returned.
For y > 0 and not an odd integer, if x is +-0, +0 is returned.
If x is +-1 and y is +-Inf, and the application was compiled with the cc compiler driver, NaN is returned. If, however, the application was
compiled with the c99 compiler driver and is therefore SUSv3-conforming (seestandards(5)), 1.0 is returned.
For |x| < 1, if y is -Inf, +Inf is returned.
For |x| > 1, if y is -Inf, +0 is returned.
For |x| < 1, if y is +Inf, +0 is returned.
For |x| > 1, if y is +Inf, +Inf is returned.
For y an odd integer < 0, if x is -Inf, -0 is returned.
For y < 0 and not an odd integer, if x is -Inf, +0 is returned.
For y an odd integer > 0, if x is -Inf, -Inf is returned.
For y > 0 and not an odd integer, if x is -Inf, +Inf is returned.
For y < 0, if x is +Inf, +0 is returned.
For y > 0, if x is +Inf, +Inf is returned.
For y an odd integer < 0, if x is +-0, a pole error occurs and +-HUGE_VAL, +-HUGE_VALF, and +-HUGE_VALL are returned for pow(), powf(), and
powl(), respectively.
For y < 0 and not an odd integer, if x is +-0, a pole error occurs and HUGE_VAL, HUGE_VALF, and HUGE_VALL are returned for pow(), powf(),
and powl(), respectively.
For exceptional cases, matherr(3M) tabulates the values to be returned by pow() as specified by SVID3 and XPG3.
These functions will fail if:
Domain Error The value of x is negative and y is a finite non-integer.
If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, the invalid floating-point exception is raised.
The pow() function sets errno to EDOM if the value of x is negative and y is non-integral.
Pole Error The value of x is 0 and y is negative.
If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, the divide-by-zero floating-point exception is
Range Error The result overflows.
If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, the overflow floating-point exception is raised.
The pow() function sets errno to EDOM if the value to be returned would cause overflow.
An application wanting to check for exceptions should call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if
fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an exception has been raised. An application should
either examine the return value or check the floating point exception flags to detect exceptions.
An application can also set errno to 0 before calling pow(). On return, if errno is non-zero, an error has occurred. The powf() and powl()
functions do not set errno.
See attributes(5) for descriptions of the following attributes:
| ATTRIBUTE TYPE | ATTRIBUTE VALUE |
|Interface Stability |Standard |
|MT-Level |MT-Safe |
exp(3M), feclearexcept(3M), fetestexcept(3M), isnan(3M), math.h(3HEAD), matherr(3M), attributes(5), standards(5)
Prior to Solaris 2.6, there was a conflict between the pow() function in this library and the pow() function in the libmp library. This
conflict was resolved by prepending mp_ to all functions in the libmp library. See mp(3MP) for more information.
SunOS 5.11 12 Jul 2006 pow(3M) | {"url":"https://www.unix.com/man-page/opensolaris/3m/powf","timestamp":"2024-11-14T11:32:30Z","content_type":"text/html","content_length":"36352","record_id":"<urn:uuid:23ac4c1e-b736-447b-922e-210983483a53>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00093.warc.gz"} |
Question 1:(1 reference, 1 page) You have learned about t
Question 1:(1 reference, 1 page) You have learned about the inferences about population variances, comparing multiple proportions, tests of independence and goodness of fit, and now please answer the
following questions in detail by applying the knowledge that you have gained from readings and lectures of this week
Subject:BusinessPrice: Bought3
Question 1:(1 reference, 1 page)
You have learned about the inferences about population variances, comparing multiple proportions, tests of independence and goodness of fit, and now please answer the following questions in detail by
applying the knowledge that you have gained from readings and lectures of this week. It is important to include hypothetical examples whenever applicable.
• Describe how chi squared and F random variables are generated.
• What are the properties of the distribution of these random variables?
• Discuss the objective in testing hypotheses on variance of one population, and variances of two populations, and the underlying assumptions.
• Explain formulation of the hypothesis, the test statistic, the rationale for rejecting the null, the criterion for choosing the rejection region, possible test outcomes, and the criterion for
evaluating the p value.
• Provide hypothetical examples of formulating hypotheses on variance of a population, and uniformity of variance across two populations.
• Explain the chi square test on uniformity of a proportion across the several populations, goodness of fit, and independence.
• Explain formulation of the hypothesis, the test statistic, the rationale for rejecting the null, the criterion for choosing the rejection region, possible test outcomes, and the criterion for
evaluating the p value.
• Provide a hypothetical example of formulating hypotheses on uniformity of proportion across several populations.
• Provide hypothetical examples of formulating hypotheses in each case.
Question 2(2 pages)
1. Ball bearing manufacturing is a highly precise business in which minimal part variability is critical. Large variances in the size of the ball bearings cause bearing failure and rapid wear-out.
Production standards call for a maximum variance of .0001 inches. Gerry Liddy has gathered a sample of 15 bearings that shows a sample standard deviation of .014 inches. Use ? = .10
2. Please determine whether the sample indicates that the maximum acceptable variance is being exceeded.
3. What is the p value?
2. The grade point averages of 352 students who completed a college course in financial accounting have a standard deviation of .940. The grade point averages of 73 students who dropped out of the
same course have a standard deviation of .797.
1. Does the data indicate a difference between the variances of grade point averages for students who completed a financial accounting course and students who dropped out?
2. Use ? = .05 level of significance.
3. What is the p value?
Note: F of alpha / 2 with degrees of freedom 351 and 72 which yields 0.025 area under its graph to the right is 1.466
Purchase A New Answer
Custom new solution created by our subject matter experts | {"url":"https://studyhelpme.com/question/70562/Question-11-reference-1-page-You-have-learned-about-the-inferences-about-population-variances","timestamp":"2024-11-11T15:01:37Z","content_type":"text/html","content_length":"65849","record_id":"<urn:uuid:07848020-72d0-4b3e-9636-372fb1fadde4>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00052.warc.gz"} |
RStan: the R interface to Stan
In this vignette we present RStan, the R interface to Stan. Stan is a C++ library for Bayesian inference using the No-U-Turn sampler (a variant of Hamiltonian Monte Carlo) or frequentist inference
via optimization. We illustrate the features of RStan through an example in Gelman et al. (2003).
Throughout the rest of the vignette we’ll use a hierarchical meta-analysis model described in section 5.5 of Gelman et al. (2003) as a running example. A hierarchical model is used to model the
effect of coaching programs on college admissions tests. The data, shown in the table below, summarize the results of experiments conducted in eight high schools, with an estimated standard error for
each. These data and model are of historical interest as an example of full Bayesian inference (Rubin 1981). For short, we call this the Eight Schools examples.
A 28 15
B 8 10
C -3 16
D 7 11
E -1 9
F 1 11
G 18 10
H 12 18
We use the Eight Schools example here because it is simple but also represents a nontrivial Markov chain simulation problem in that there is dependence between the parameters of original interest in
the study — the effects of coaching in each of the eight schools — and the hyperparameter representing the variation of these effects in the modeled population. Certain implementations of a Gibbs
sampler or a Hamiltonian Monte Carlo sampler can be slow to converge in this example.
The statistical model of interest is specified as
\[ \begin{aligned} y_j &\sim \mathsf{Normal}(\theta_j, \sigma_j), \quad j=1,\ldots,8 \\ \theta_j &\sim \mathsf{Normal}(\mu, \tau), \quad j=1,\ldots,8 \\ p(\mu, \tau) &\propto 1, \end{aligned} \]
where each \(\sigma_j\) is assumed known.
Write a Stan Program
RStan allows a Stan program to be coded in a text file (typically with suffix .stan) or in a R character vector (of length one). We put the following code for the Eight Schools model into the file
data {
int<lower=0> J; // number of schools
real y[J]; // estimated treatment effects
real<lower=0> sigma[J]; // s.e. of effect estimates
parameters {
real mu;
real<lower=0> tau;
vector[J] eta;
transformed parameters {
vector[J] theta;
theta = mu + tau * eta;
model {
target += normal_lpdf(eta | 0, 1);
target += normal_lpdf(y | theta, sigma);
The first section of the Stan program above, the data block, specifies the data that is conditioned upon in Bayes Rule: the number of schools, \(J\), the vector of estimates, \((y_1, \ldots, y_J)\),
and the vector of standard errors of the estimates \((\sigma_{1}, \ldots, \sigma_{J})\). Data are declared as integer or real and can be vectors (or, more generally, arrays) if dimensions are
specified. Data can also be constrained; for example, in the above model \(J\) has been restricted to be at least \(1\) and the components of \(\sigma_y\) must all be positive.
The parameters block declares the parameters whose posterior distribution is sought. These are the the mean, \(\mu\), and standard deviation, \(\tau\), of the school effects, plus the standardized
school-level effects \(\eta\). In this model, we let the unstandardized school-level effects, \(\theta\), be a transformed parameter constructed by scaling the standardized effects by \(\tau\) and
shifting them by \(\mu\) rather than directly declaring \(\theta\) as a parameter. By parameterizing the model this way, the sampler runs more efficiently because the resulting multivariate geometry
is more amendable to Hamiltonian Monte Carlo (Neal 2011).
Finally, the model block looks similar to standard statistical notation. (Just be careful: the second argument to Stan’s normal\((\cdot,\cdot)\) distribution is the standard deviation, not the
variance as is usual in statistical notation). We have written the model in vector notation, which allows Stan to make use of more efficient algorithmic differentiation (AD). It would also be
possible — but less efficient — to write the model by replacing normal_lpdf(y | theta,sigma) with a loop over the \(J\) schools,
for (j in 1:J)
target += normal_lpdf(y[j] | theta[j],sigma[j]);
Stan has versions of many of the most useful R functions for statistical modeling, including probability distributions, matrix operations, and various special functions. However, the names of the
Stan functions may differ from their R counterparts and, more subtly, the parameterizations of probability distributions in Stan may differ from those in R for the same distribution. To mitigate this
problem, the lookup function can be passed an R function or character string naming an R function, and RStan will attempt to look up the corresponding Stan function, display its arguments, and give
the page number in The Stan Development Team (2016) where the function is discussed.
415 normal_id_glm_lpdf
418 normal_log
419 normal_lpdf
553 std_normal_lpdf
415 (real, matrix, real, vector, T);(vector, row_vector, vector, vector, vector)
418 (real, real, T);(vector, vector, vector)
419 (real, real, T);(vector, vector, vector)
553 (T);(vector)
415 T;real
418 T;real
419 T;real
553 T;real
[1] "no matching Stan functions"
If the lookup function fails to find an R function that corresponds to a Stan function, it will treat its argument as a regular expression and attempt to find matches with the names of Stan
Preparing the Data
The stan function accepts data as a named list, a character vector of object names, or an environment. Alternatively, the data argument can be omitted and R will search for objects that have the same
names as those declared in the data block of the Stan program. Here is the data for the Eight Schools example:
It would also be possible (indeed, encouraged) to read in the data from a file rather than to directly enter the numbers in the R script.
Sample from the Posterior Distribution
Next, we can call the stan function to draw posterior samples:
The stan function wraps the following three steps:
• Translate a model in Stan code to C++ code
• Compile the C++ code to a dynamic shared object (DSO) and load the DSO
• Sample given some user-specified data and other settings
A single call to stan performs all three steps, but they can also be executed one by one (see the help pages for stanc, stan_model, and sampling), which can be useful for debugging. In addition, Stan
saves the DSO so that when the same model is fit again (possibly with new data and settings) we can avoid recompilation. If an error happens after the model is compiled but before sampling (e.g.,
problems with inputs like data and initial values), we can still reuse the compiled model.
The stan function returns a stanfit object, which is an S4 object of class "stanfit". For those who are not familiar with the concept of class and S4 class in R, refer to Chambers (2008). An S4 class
consists of some attributes (data) to model an object and some methods to model the behavior of the object. From a user’s perspective, once a stanfit object is created, we are mainly concerned about
what methods are defined.
If no error occurs, the returned stanfit object includes the sample drawn from the posterior distribution for the model parameters and other quantities defined in the model. If there is an error
(e.g. a syntax error in the Stan program), stan will either quit or return a stanfit object that contains no posterior draws.
For class "stanfit", many methods such as print and plot are defined for working with the MCMC sample. For example, the following shows a summary of the parameters from the Eight Schools model using
the print method:
Inference for Stan model: anon_model.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.
mean se_mean sd 10% 50% 90% n_eff Rhat
theta[1] 11.38 0.15 8.40 2.16 10.36 22.58 3340 1
theta[2] 7.78 0.10 6.42 -0.25 7.76 15.56 4493 1
theta[3] 6.11 0.15 7.67 -3.35 6.58 14.77 2793 1
theta[4] 7.51 0.10 6.62 -0.50 7.46 15.40 4384 1
theta[5] 5.02 0.12 6.47 -3.31 5.54 12.68 3115 1
theta[6] 6.20 0.10 6.66 -2.30 6.50 14.00 4409 1
theta[7] 10.63 0.12 6.78 2.57 10.07 19.70 3426 1
theta[8] 8.17 0.15 7.85 -0.71 7.94 17.48 2829 1
mu 7.66 0.12 5.27 1.23 7.58 14.10 1811 1
tau 6.58 0.15 5.63 0.97 5.29 13.80 1420 1
lp__ -39.61 0.08 2.72 -43.20 -39.33 -36.36 1127 1
Samples were drawn using NUTS(diag_e) at Mon Mar 4 18:36:43 2024.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
The last line of this output, lp__, is the logarithm of the (unnormalized) posterior density as calculated by Stan. This log density can be used in various ways for model evaluation and comparison
(see, e.g., Vehtari and Ojanen (2012)).
Arguments to the stan Function
The primary arguments for sampling (in functions stan and sampling) include data, initial values, and the options of the sampler such as chains, iter, and warmup. In particular, warmup specifies the
number of iterations that are used by the NUTS sampler for the adaptation phase before sampling begins. After the warmup, the sampler turns off adaptation and continues until a total of iter
iterations (including warmup) have been completed. There is no theoretical guarantee that the draws obtained during warmup are from the posterior distribution, so the warmup draws should only be used
for diagnosis and not inference. The summaries for the parameters shown by the print method are calculated using only post-warmup draws.
The optional init argument can be used to specify initial values for the Markov chains. There are several ways to specify initial values, and the details can be found in the documentation of the stan
function. The vast majority of the time it is adequate to allow Stan to generate its own initial values randomly. However, sometimes it is better to specify the initial values for at least a subset
of the objects declared in the parameters block of a Stan program.
Stan uses a random number generator (RNG) that supports parallelism. The initialization of the RNG is determined by the arguments seed and chain_id. Even if we are running multiple chains from one
call to the stan function we only need to specify one seed, which is randomly generated by R if not specified.
Data Preprocessing and Passing
The data passed to stan will go through a preprocessing procedure. The details of this preprocessing are documented in the documentation for the stan function. Here we stress a few important steps.
First, RStan allows the user to pass more objects as data than what is declared in the data block (silently omitting any unnecessary objects). In general, an element in the list of data passed to
Stan from R should be numeric and its dimension should match the declaration in the data block of the model. So for example, the factor type in R is not supported as a data element for RStan and must
be converted to integer codes via as.integer. The Stan modeling language distinguishes between integers and doubles (type int and real in Stan modeling language, respectively). The stan function will
convert some R data (which is double-precision usually) to integers if possible.
The Stan language has scalars and other types that are sets of scalars, e.g. vectors, matrices, and arrays. As R does not have true scalars, RStan treats vectors of length one as scalars. However,
consider a model with a data block defined as
data {
int<lower=1> N;
real y[N];
in which N can be \(1\) as a special case. So if we know that N is always larger than \(1\), we can use a vector of length N in R as the data input for y (for example, a vector created by y <- rnorm
(N)). If we want to prevent RStan from treating the input data for y as a scalar when \(N`\) is \(1\), we need to explicitly make it an array as the following R code shows:
y <- as.array(y)
Stan cannot handle missing values in data automatically, so no element of the data can contain NA values. An important step in RStan’s data preprocessing is to check missing values and issue an error
if any are found. There are, however, various ways of writing Stan programs that account for missing data (see The Stan Development Team (2016)).
Methods for the "stanfit" Class
The other vignette included with the rstan package discusses stanfit objects in greater detail and gives examples of accessing the most important content contained in the objects (e.g., posterior
draws, diagnostic summaries). Also, a full list of available methods can be found in the documentation for the "stanfit" class at help("stanfit", "rstan"). Here we give only a few examples.
The plot method for stanfit objects provides various graphical overviews of the output. The default plot shows posterior uncertainty intervals (by default 80% (inner) and 95% (outer)) and the
posterior median for all the parameters as well as lp__ (the log of posterior density function up to an additive constant):
'pars' not specified. Showing first 10 parameters by default.
ci_level: 0.8 (80% intervals)
outer_level: 0.95 (95% intervals)
The optional plotfun argument can be used to select among the various available plots. See help("plot,stanfit-method").
The traceplot method is used to plot the time series of the posterior draws. If we include the warmup draws by setting inc_warmup=TRUE, the background color of the warmup area is different from the
post-warmup phase:
To assess the convergence of the Markov chains, in addition to visually inspecting traceplots we can calculate the split \(\hat{R}\) statistic. Split \(\hat{R}\) is an updated version of the \(\hat
{R}\) statistic proposed in Gelman and Rubin (1992) that is based on splitting each chain into two halves. See the Stan manual for more details. The estimated \(\hat{R}\) for each parameter is
included as one of the columns in the output from the summary and print methods.
Inference for Stan model: anon_model.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
mu 7.66 0.12 5.27 -2.83 4.34 7.58 10.97 18.10 1811 1
tau 6.58 0.15 5.63 0.20 2.45 5.29 9.14 20.59 1420 1
Samples were drawn using NUTS(diag_e) at Mon Mar 4 18:36:43 2024.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
Again, see the additional vignette on stanfit objects for more details.
Sampling Difficulties
The best way to visualize the output of a model is through the ShinyStan interface, which can be accessed via the shinystan R package. ShinyStan facilitates both the visualization of parameter
distributions and diagnosing problems with the sampler. The documentation for the shinystan package provides instructions for using the interface with stanfit objects.
In addition to using ShinyStan, it is also possible to diagnose some sampling problems using functions in the rstan package. The get_sampler_params function returns information on parameters related
the performance of the sampler:
accept_stat__ stepsize__ treedepth__ n_leapfrog__ divergent__
Min. :0.00 Min. : 0.033 Min. :0.0 Min. : 1 Min. :0.000
1st Qu.:0.80 1st Qu.: 0.285 1st Qu.:3.0 1st Qu.: 7 1st Qu.:0.000
Median :0.95 Median : 0.343 Median :3.0 Median : 15 Median :0.000
Mean :0.84 Mean : 0.385 Mean :3.4 Mean : 13 Mean :0.009
3rd Qu.:0.99 3rd Qu.: 0.405 3rd Qu.:4.0 3rd Qu.: 15 3rd Qu.:0.000
Max. :1.00 Max. :10.289 Max. :7.0 Max. :127 Max. :1.000
Min. :35
1st Qu.:42
Median :44
Mean :45
3rd Qu.:47
Max. :61
accept_stat__ stepsize__ treedepth__ n_leapfrog__ divergent__
Min. :0.00 Min. :0.063 Min. :0.0 Min. : 1 Min. :0.0000
1st Qu.:0.75 1st Qu.:0.343 1st Qu.:3.0 1st Qu.: 7 1st Qu.:0.0000
Median :0.94 Median :0.343 Median :3.0 Median :15 Median :0.0000
Mean :0.82 Mean :0.395 Mean :3.4 Mean :12 Mean :0.0085
3rd Qu.:0.99 3rd Qu.:0.383 3rd Qu.:4.0 3rd Qu.:15 3rd Qu.:0.0000
Max. :1.00 Max. :8.548 Max. :6.0 Max. :95 Max. :1.0000
Min. :36
1st Qu.:42
Median :44
Mean :45
3rd Qu.:47
Max. :61
accept_stat__ stepsize__ treedepth__ n_leapfrog__ divergent__
Min. :0.00 Min. :0.033 Min. :0.0 Min. : 1 Min. :0.000
1st Qu.:0.83 1st Qu.:0.278 1st Qu.:3.0 1st Qu.: 7 1st Qu.:0.000
Median :0.96 Median :0.278 Median :4.0 Median : 15 Median :0.000
Mean :0.85 Mean :0.364 Mean :3.5 Mean : 13 Mean :0.011
3rd Qu.:0.99 3rd Qu.:0.410 3rd Qu.:4.0 3rd Qu.: 15 3rd Qu.:0.000
Max. :1.00 Max. :5.402 Max. :7.0 Max. :127 Max. :1.000
Min. :35
1st Qu.:42
Median :44
Mean :44
3rd Qu.:46
Max. :61
accept_stat__ stepsize__ treedepth__ n_leapfrog__ divergent__
Min. :0.00 Min. : 0.045 Min. :0.0 Min. : 1 Min. :0.000
1st Qu.:0.82 1st Qu.: 0.356 1st Qu.:3.0 1st Qu.: 7 1st Qu.:0.000
Median :0.95 Median : 0.356 Median :3.0 Median :15 Median :0.000
Mean :0.84 Mean : 0.420 Mean :3.3 Mean :12 Mean :0.009
3rd Qu.:0.99 3rd Qu.: 0.424 3rd Qu.:4.0 3rd Qu.:15 3rd Qu.:0.000
Max. :1.00 Max. :10.289 Max. :6.0 Max. :63 Max. :1.000
Min. :36
1st Qu.:42
Median :44
Mean :45
3rd Qu.:47
Max. :57
accept_stat__ stepsize__ treedepth__ n_leapfrog__ divergent__
Min. :0.00 Min. :0.047 Min. :0.0 Min. : 1 Min. :0.0000
1st Qu.:0.82 1st Qu.:0.294 1st Qu.:3.0 1st Qu.: 7 1st Qu.:0.0000
Median :0.96 Median :0.294 Median :4.0 Median : 15 Median :0.0000
Mean :0.84 Mean :0.362 Mean :3.5 Mean : 13 Mean :0.0075
3rd Qu.:0.99 3rd Qu.:0.399 3rd Qu.:4.0 3rd Qu.: 15 3rd Qu.:0.0000
Max. :1.00 Max. :5.423 Max. :6.0 Max. :127 Max. :1.0000
Min. :35
1st Qu.:42
Median :44
Mean :45
3rd Qu.:47
Max. :59
Here we see that there are a small number of divergent transitions, which are identified by divergent__ being \(1\). Ideally, there should be no divergent transitions after the warmup phase. The best
way to try to eliminate divergent transitions is by increasing the target acceptance probability, which by default is \(0.8\). In this case the mean of accept_stat__ is close to \(0.8\) for all
chains, but has a very skewed distribution because the median is near \(0.95\). We could go back and call stan again and specify the optional argument control=list(adapt_delta=0.9) to try to
eliminate the divergent transitions. However, sometimes when the target acceptance rate is high, the stepsize is very small and the sampler hits its limit on the number of leapfrog steps it can take
per iteration. In this case, it is a non-issue because each chain has a treedepth__ of at most \(7\) and the default is \(10\). But if any treedepth__ were \(11\), then it would be wise to increase
the limit by passing control=list(max_treedepth=12) (for example) to stan. See the vignette on stanfit objects for more on the structure of the object returned by get_sampler_params.
We can also make a graphical representation of (much of the) the same information using pairs. The “pairs”" plot can be used to get a sense of whether any sampling difficulties are occurring in the
tails or near the mode:
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
Warning in par(usr): argument 1 does not name a graphical parameter
In the plot above, the marginal distribution of each selected parameter is included as a histogram along the diagonal. By default, draws with below-median accept_stat__ (MCMC proposal acceptance
rate) are plotted below the diagonal and those with above-median accept_stat__ are plotted above the diagonal (this can be changed using the condition argument). Each off-diagonal square represents a
bivariate distribution of the draws for the intersection of the row-variable and the column-variable. Ideally, the below-diagonal intersection and the above-diagonal intersection of the same two
variables should have distributions that are mirror images of each other. Any yellow points would indicate transitions where the maximum treedepth__ was hit, and red points indicate a divergent
Additional Topics
User-defined Stan Functions
Stan also permits users to define their own functions that can be used throughout a Stan program. These functions are defined in the functions block. The functions block is optional but, if it
exists, it must come before any other block. This mechanism allows users to implement statistical distributions or other functionality that is not currently available in Stan. However, even if the
user’s function merely wraps calls to existing Stan functions, the code in the model block can be much more readible if several lines of Stan code that accomplish one (or perhaps two) task(s) are
replaced by a call to a user-defined function.
Another reason to utilize user-defined functions is that RStan provides the expose_stan_functions function for exporting such functions to the R global environment so that they can be tested in R to
ensure they are working properly. For example,
[1] -0.9529876
The Log-Posterior (function and gradient)
Stan defines the log of the probability density function of a posterior distribution up to an unknown additive constant. We use lp__ to represent the realizations of this log kernel at each iteration
(and lp__ is treated as an unknown in the summary and the calculation of split \(\hat{R}\) and effective sample size).
A nice feature of the rstan package is that it exposes functions for calculating both lp__ and its gradients for a given stanfit object. These two functions are log_prob and grad_log_prob,
respectively. Both take parameters on the unconstrained space, even if the support of a parameter is not the whole real line. The Stan manual (The Stan Development Team 2016) has full details on the
particular transformations Stan uses to map from the entire real line to some subspace of it (and vice-versa).
It maybe the case that the number of unconstrained parameters might be less than the total number of parameters. For example, for a simplex parameter of length \(K\), there are actually only \(K-1\)
unconstrained parameters because of the constraint that all elements of a simplex must be nonnegative and sum to one. The get_num_upars method is provided to get the number of unconstrained
parameters, while the unconstrain_pars and constrain_pars methods can be used to compute unconstrained and constrained values of parameters respectively. The former takes a list of parameters as
input and transforms it to an unconstrained vector, and the latter does the opposite. Using these functions, we can implement other algorithms such as maximum a posteriori estimation of Bayesian
Optimization in Stan
RStan also provides an interface to Stan’s optimizers, which can be used to obtain a point estimate by maximizing the (perhaps penalized) likelihood function defined by a Stan program. We illustrate
this feature using a very simple example: estimating the mean from samples assumed to be drawn from a normal distribution with known standard deviation. That is, we assume
\[y_n \sim \mathsf{Normal}(\mu,1), \quad n = 1, \ldots, N. \]
By specifying a prior \(p(\mu) \propto 1\), the maximum a posteriori estimator for \(\mu\) is just the sample mean. We don’t need to explicitly code this prior for \(\mu\), as \(p(\mu) \propto 1\) is
the default if no prior is specified.
We first create an object of class "stanmodel" and then use the optimizing method, to which data and other arguments can be fed.
[1] -0.2383165
[1] -31.3946
[1] 0
mu -20
[1,] -0.2383165
Model Compilation
As mentioned earlier in the vignette, Stan programs are written in the Stan modeling language, translated to C++ code, and then compiled to a dynamic shared object (DSO). The DSO is then loaded by R
and executed to draw the posterior sample. The process of compiling C++ code to DSO sometimes takes a while. When the model is the same, we can reuse the DSO from a previous run. The stan function
accepts the optional argument fit, which can be used to pass an existing fitted model object so that the compiled model is reused. When reusing a previous fitted model, we can still specify different
values for the other arguments to stan, including passing different data to the data argument.
In addition, if fitted models are saved using functions like save and save.image, RStan is able to save DSOs, so that they can be used across R sessions. To avoid saving the DSO, specify save_dso=
FALSE when calling the stan function.
If the user executes rstan_options(auto_write = TRUE), then a serialized version of the compiled model will be automatically saved to the hard disk in the same directory as the .stan file or in R’s
temporary directory if the Stan program is expressed as a character string. Although this option is not enabled by default due to CRAN policy, it should ordinarily be specified by users in order to
eliminate redundant compilation.
Stan runs much faster when the code is compiled at the maximum level of optimization, which is -O3 on most C++ compilers. However, the default value is -O2 in R, which is appropriate for most R
packages but entails a slight slowdown for Stan. You can change this default locally by following the instructions at CRAN - Customizing-package-compilation. However, you should be advised that
setting CXXFLAGS = -O3 may cause adverse side effects for other R packages.
See the documentation for the stanc and stan_model functions for more details on the parsing and compilation of Stan programs.
Running Multiple Chains in Parallel
The number of Markov chains to run can be specified using the chains argument to the stan or sampling functions. By default, the chains are executed serially (i.e., one at a time) using the parent R
process. There is also an optional cores argument that can be set to the number of chains (if the hardware has sufficient processors and RAM), which is appropriate on most laptops. We typically
recommend first calling options(mc.cores=parallel::detectCores()) once per R session so that all available cores can be used without needing to manually specify the cores argument.
For users working with a different parallelization scheme (perhaps with a remote cluster), the rstan package provides a function called sflist2stanfit for consolidating a list of multiple stanfit
objects (created from the same Stan program and using the same number of warmup and sampling iterations) into a single stanfit object. It is important to specify the same seed for all the chains and
equally important to use a different chain ID (argument chain_id), the combination of which ensures that the random numbers generated in Stan for all chains are essentially independent. This is
handled automatically (internally) when \(`cores` > 1\). | {"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/rstan/vignettes/rstan.html","timestamp":"2024-11-06T03:06:09Z","content_type":"text/html","content_length":"336601","record_id":"<urn:uuid:528ded0b-a6d2-477a-b605-0689668436d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00635.warc.gz"} |
2016 AMC 12A Problems/Problem 12
Revision as of 17:19, 20 March 2024 by
Scthecool (talk | contribs) (another solution (luck-based) and a warning about using it in a real competition)
Problem 12
In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\
angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?
$[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("A",A,SW); label
("B",B,SE); label("C",C,N); label("D",D,NE); label("E",E,NW); label("F",F,1.5*N); [/asy]$
$\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2$
Solution 1
By the angle bisector theorem, $\frac{AB}{AE} = \frac{CB}{CE}$
$\frac{6}{AE} = \frac{7}{8 - AE}$ so $AE = \frac{48}{13}$
Similarly, $CD = 4$.
There are two ways to solve from here. First way:
Note that $DB = 7 - 4 = 3.$ By the angle bisector theorem on $\triangle ADB,$$\frac{AF}{FD} = \frac{AB}{DB} = \frac{6}{3}.$ Thus the answer is $\boxed{\textbf{(C)}\; 2 : 1}$
Second way:
Now, we use mass points. Assign point $C$ a mass of $1$.
$mC \cdot CD = mB \cdot DB$ , so $mB = \frac{4}{3}$
Similarly, $A$ will have a mass of $\frac{7}{6}$
$mD = mC + mB = 1 + \frac{4}{3} = \frac{7}{3}$
So $\frac{AF}{FD} = \frac{mD}{mA} = \boxed{\textbf{(C)}\; 2 : 1}$
Solution 2
Denote $[\triangle{ABC}]$ as the area of triangle ABC and let $r$ be the inradius. Also, as above, use the angle bisector theorem to find that $BD = 3$. There are two ways to continue from here:
$1.$ Note that $F$ is the incenter. Then, $\frac{AF}{FD} = \frac{[\triangle{AFB}]}{[\triangle{BFD}]} = \frac{AB * \frac{r}{2}}{BD * \frac{r}{2}} = \frac{AB}{BD} = \boxed{\textbf{(C)}\; 2 : 1}$
$2.$ Apply the angle bisector theorem on $\triangle{ABD}$ to get $\frac{AF}{FD} = \frac{AB}{BD} = \frac{6}{3} = \boxed{\textbf{(C)}\; 2 : 1}$
Solution 3
Draw the third angle bisector, and denote the point where this bisector intersects $AB$ as $P$. Using angle bisector theorem, we see $AE=48/13 , EC=56/13, AP=16/5, PB=14/5$. Applying Van Aubel's
Theorem, $AF/FD=(48/13)/(56/13) + (16/5)/(14/5)=(6/7)+(8/7)=14/7=2/1$, and so the answer is $\boxed{\textbf{(C)}\; 2 : 1}$.
Solution 4
One only needs the angle bisector theorem to solve this question.
The question asks for $AF:FD = \frac{AF}{FD}$. Apply the angle bisector theorem to $\triangle ABD$ to get $\[\frac{AF}{FD} = \frac{AB}{BD}.\]$
$AB = 6$ is given. To find $BD$, apply the angle bisector theorem to $\triangle BAC$ to get $\[\frac{BD}{DC} = \frac{BA}{AC} = \frac{6}{8} = \frac{3}{4}.\]$
Since $\[BD + DC = BC = 7,\]$ it is immediately obvious that $BD = 3$, $DC = 4$ satisfies both equations.
Thus, $\[AF:FD = AB:BD = 6:3 = \boxed{\textbf{(C)}\ 2:1}.\]$ ~revision by emerald_block
Solution 5 (Luck-Based)
Note that $\[AF\]$ and $\[BD\]$ look like medians. Assuming they are medians, we mark the answer $\[\boxed{\textbf{(C)}\ 2:1}\]$ as we know that the centroid (the point where all medians in a
triangle are concurrent) splits a median in a $\[2:1\]$ ratio, with the shorter part being closer to the side it bisects. ~scthecool Note: This is heavily luck based, and if the figure had been not
drawn to scale, for example, this answer would have easily been wrong. It is thus advised to not use this in a real competition unless absolutely necessary.
Video Solution by OmegaLearn
~ pi_is_3.14
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems/Problem_12&oldid=217206","timestamp":"2024-11-12T22:15:54Z","content_type":"text/html","content_length":"56730","record_id":"<urn:uuid:3bb56a16-039d-4fa1-afbe-3a598e23ba30>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00022.warc.gz"} |
Does the Roulette Martingale System Work? – Tried and Tested
The most popular and well known roulette strategy is the Martingale Strategy.
Created over 300 years ago and enhanced by Paul Pierre Levy just 70 years ago, the Martingale Strategy is often the go too method for gaming the advantage at the roulette table.
The concept of the strategy is to bet in a certain way on a winning spin but playing a different way on a losing spin.
During a winning spin the bet stays the same but, on a losing spin the bet doubles to recoup the losses of the previous spin.
Recently, we here at Guide to Casinos HQ we put the Martingale Strategy through its paces. This seemed to be the best way to answer the question as to whether the Martingale Strategy actually and
truly works.
We were surprised at the results but if our case study and experience is relatable and averaged out across all players, the strategy could give players an advantage.
The Martingale Strategy isn’t just applicable for roulette, it can be used for day trading or blackjack for example, but it fits the gameplay of roulette very well.
Our recent tests gave us a 39% return on our gameplay in our tried and tested experience and in this example, we will walk through what the Martingale Strategy is, our results and whether it is worth
using this at the roulette wheel.
What is the Martingale Strategy
The Martingale Strategy is a betting concept where the stake remains the same on a winning bet, but doubles on a losing bet until the bet wins. The mathematical concept looks to recoup all losses for
every losing bet. In the long term the strategy would always work unless the player runs out of money.
For example, a losing £10 spin should result in a next bet of £20. If this spin loses too the next should be £40 and so on.
On the other hand, if the £10 spin wins then next bet should be £10 again.
Recommended reading: Best Roulette Strategies – we tried and tested the best 9 roulette strategies, find out the only one that lost us money!
This benefit of the strategy is no matter how many times the player loses, the next win will see a recoup of all losses so far plus the original stake to start the betting strategy again.
Does the Martingale Strategy Work?
The great thing about the Martingale Strategy is with an infinite stake the strategy could never fail.
Unfortunately, players never have an infinite amount to stake.
Let’s say we start with a £10 bet which loses. The next bet is £20, which could also lose. A £40 bet follows next and if this loses too then the next bet would be £80.
A £80 spin and win on an evens type roulette bet such as red, black, odds or evens would provide a £80 win.
An £80 win is equal to the total of all bets so far (£10 + £20 + £40) plus a small £10 profit.
If the £80 bet lost too, and the next bet of £160 won, it would produce a win of £160 to recoup the £150 spent so far (£10+£20+£40+£80 = £150).
Although unlikely, 10 failed spins in a row – at a starting stake of £10 – would see the player lose £5,120 and the next bet would need to be £10,240 to continue the strategy!
A £15,360 stake is out of the reach of most players.
This is why the Martingale Strategy is mathematically advantageous and will always favour the player, as long as the player does not go bankrupt first and can’t afford the next bet in the strategy.
Martingale Strategy Actual Results
The best way to prove whether a strategy works is to actually try it out.
We spent time playing the Martingale Strategy, and following the rules to the absolute letter, and we were surprised with the results.
Playing roulette randomly using an evens odd bet such as red or black, should see a win 48.6% of the time on a UK roulette wheel.
The odds are slightly less on a U.S. roulette wheel as they have a double zero as well as a single zero.
We spent £18 on our test which should have resulted in an end pot of £17.50, but we actually ended with a £27 pot.
A return of 39%!
Here are the actual stakes and plays we made using the Martingale Strategy:
Spin Wager Bet Result Balance
1 £1 Black Win £21
2 £1 Black Win £23
3 £1 Red Lose £22
4 £2 Black Lose £20
5 £4 Red Win £24
6 £1 Black Lose £23
7 £2 Black Lose £21
8 £4 Black Win £25
9 £1 Red Win £26
10 £1 Red Win £27
Martingale Strategy – Tried and Tested Results
After starting positively with two win spins we ended up winning 60% of spins and losing 40% of our spins across 10 spins in our case study.
The game of roulette though is very much down to chance, and although we saw a positive 39% return on our stake, the results could easily have been different.
It is also worth noting that although the Martingale Strategy have us a return of 39%, it wasn’t the best roulette strategy in our full case study.
Martingale Strategy vs Other Strategies
Although the Martingale Strategy is the most cited and the most followed, it isn’t the only roulette strategy to try.
During the 1800’s a number of strategies appeared from mathematicians to tacticians and even numerologists, each looking to gain an advantage over the roulette wheel and the house that ran the games.
The most famous of all the strategy creators is arguably Blaise Pascal, the creator of the Paroli Strategy.
Although the name Blaise Pascal may not be familiar to you, as well as creating the Paroli Strategy for roulette, he also invented the game of roulette!
Here are the most popular and famous roulette strategies ranked in order based on our results trying and testing each one:
1. Labouchere Strategy
2. 3 2 Strategy
3. Martingale Strategy
4. D’Alembert Strategy
5. Paroli Strategy
6. Fibonacci Strategy
7. Andrucci Strategy
8. 1 3 2 6 Strategy
9. James Bond Strategy
At the top of our results list the Labouchere Strategy gave us a return of 160% whereas the James Bond Strategy gave us a loss of 70%.
In fact, the James Bond Strategy is the only roulette strategy on our list that gave us a negative return.
Despite not being at the top of the strategy returns list, the Martingale Strategy is within the top three.
Is the Martingale Strategy Worth Playing?
A player has two choices when staking and playing at a roulette wheel:
• Bet randomly
• Bet using a roulette strategy
On average, a player choosing randomly would win £9.62 for every £10 staked.
This represents a 3.8% loss on stake.
Seven from the nine strategies we tested resulted in an 8% or greater return on stake.
Considering in our tests the Martingale Strategy gave us a 39% return, it is a roulette strategy worth testing.
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Developmental Math Emporium
Some people find it helpful to know when they can take a shortcut to avoid doing extra work. There are some polynomials that will always factor a certain way, and for those, we offer a shortcut. Most
people find it helpful to memorize the factored form of a perfect square trinomial or a difference of squares. The most important skill you will use in this section will be recognizing when you can
use the shortcuts.
Factoring a Perfect Square Trinomial
A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the
product of the two terms and the square of the last term.
[latex]\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2}\hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2}\hfill \end{array}[/latex]
We can use these equations to factor any perfect square trinomial.
A General Note: Perfect Square Trinomials
A perfect square trinomial can be written as the square of a binomial:
In the following example, we will show you how to define a and b so you can use the shortcut.
Factor [latex]25{x}^{2}+20x+4[/latex].
Show Solution
In the next example, we will show that we can use [latex]1 = 1^2[/latex] to factor a polynomial with a term equal to [latex]1[/latex].
Factor [latex]49{x}^{2}-14x+1[/latex].
Show Solution
In the following video, we provide another short description of what a perfect square trinomial is and show how to factor them using a formula.
Try It
We can summarize our process in the following way:
How To: Given a perfect square trinomial, factor it into the square of a binomial
1. Confirm that the first and last term are perfect squares.
2. Confirm that the middle term is twice the product of [latex]ab[/latex].
3. Write the factored form as [latex]{\left(a+b\right)}^{2}[/latex] or [latex]{\left(a-b\right)}^{2}[/latex].
Factoring a Difference of Squares
A difference of squares is a perfect square subtracted from a perfect square. This type of polynomial is unique because it can be factored into two binomials but has only two terms.
Factor a Difference of Squares
Given [latex]a^2-b^2[/latex], its factored form will be [latex]\left(a+b\right)\left(a-b\right)[/latex].
You will want to become familiar with the special relationship between a difference of squares and its factorization as we can use this equation to factor any differences of squares.
A difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied. Let’s look at an
example of difference of squares to help us understand how this works. We will start from the product of two binomials to see the pattern.
Given the product of two binomials: [latex]\left(x-2\right)\left(x+2\right)[/latex], if we multiply them together, we lose the middle term that we are used to seeing as a result.
The polynomial [latex]x^2-4[/latex] is called a difference of squares because each term can be written as something squared. A difference of squares will always factor in the following way:
Let’s factor [latex]x^{2}–4[/latex] by writing it as a trinomial, [latex]x^{2}+0x–4[/latex]. This is similar in format to the trinomials we have been factoring so far, so let’s use the same method.
Find the factors of [latex]a\cdot{c}[/latex] whose sum is b, in this case, 0:
Factors of [latex]−4[/latex] Sum of the factors
[latex]1\cdot-4=−4[/latex] [latex]1-4=−3[/latex]
[latex]2\cdot−2=−4[/latex] [latex]2-2=0[/latex]
[latex]-1\cdot4=−4[/latex] [latex]-1+4=3[/latex]
[latex]2[/latex], and [latex]-2[/latex] have a sum of [latex]0[/latex]. You can use these to factor [latex]x^{2}–4[/latex].
Factor [latex]x^{2}–4[/latex].
Show Solution
Since order doesn’t matter with multiplication, the answer can also be written as [latex]\left(x+2\right)\left(x–2\right)[/latex].
You can check the answer by multiplying [latex]\left(x–2\right)\left(x+2\right)=x^{2}+2x–2x–4=x^{2}–4[/latex].
A General Note: Differences of Squares
A difference of squares can be rewritten as two factors containing the same terms but opposite signs.
Now that we have seen how to factor a difference of squares with regrouping, let’s try some examples using the short-cut.
Factor [latex]9{x}^{2}-25[/latex].
Show Solution
Try It
The most helpful thing for recognizing a difference of squares that can be factored with the shortcut is knowing which numbers are perfect squares, as you will see in the next example.
Factor [latex]81{y}^{2}-144[/latex].
Show Solution
TIP: To help identify difference of squares factoring problems, make a list of perfect squares and become familiar with these values. You will frequently see them in these types of factoring
problems. In addition, these problems must have a minus sign. For example, x^2 – 9 = (x+3)(x-3)
1^2 1
2^2 4
3^2 9
4^2 16
5^2 25
Try It
In the following video, we show another example of how to use the formula for factoring a difference of squares.
We can summarize the process for factoring a difference of squares with the shortcut this way:
How To: Given a difference of squares, factor it into binomials
1. Confirm that the first and last term are perfect squares.
2. Write the factored form as [latex]\left(a+b\right)\left(a-b\right)[/latex].
Think About It
Is there a formula to factor the sum of squares, [latex]a^2+b^2[/latex], into a product of two binomials?
Write down some ideas for how you would answer this in the box below before you look at the answer.
Show Solution | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/read-special-cases-squares/","timestamp":"2024-11-15T03:42:17Z","content_type":"text/html","content_length":"45875","record_id":"<urn:uuid:7549a5f8-d8ce-4894-9c2b-60fa9b475c31>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00213.warc.gz"} |
A Mathematically Creative Four-Year-Old—What Do We Learn from Him
[1] Bahar, A. K., & Maker, C. J. (2011). Exploring the relationship be tween mathematical creativity and mathematical achievement. Asia Pacific Journal of Gifted and Talented Education, 3, 33-48.
[2] Bishop, A. J. (2002). Mathematical acculturation, cultural conflicts, and transition. In G. de Abreu, A. J. Bishop, & N. C. Presmeg (Eds.), Transitions between contexts of mathematical practices
(pp. 193-212). Dordrecht: Kluwer Academic Press. doi:10.1007/0-306-47674-6_10
[3] Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179-202. doi:10.2307
[4] Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten chil dren’s problem-solving processes. Journal for Research in
Mathe matics Education, 24, 428-441. doi:10.2307/749152
[5] Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Ports mouth, NH: Heinemann.
[6] Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanc ed mathematical thinking (pp. 42-53). Dordrecht: Kluwer.
[7] Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use chil dren’s thinking in mathematics instruction. Journal
for Research in Mathematics Education, 27, 403-434. doi:10.2307/749875
[8] Franke, M. L. (2003). Fostering young children’s mathematical under standing. In C. Howes (Ed.), Teaching 4 to 8-year-olds: Literacy, math, multiculturalism, and classroom community. Baltimore,
MD: Brookes.
[9] Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.
[10] Hershkovitz, S., Peled, I., & Littler, G. (2009). Mathematical creativity and giftedness in elementary school: Task and teacher promoting creativity for all. In R. Leikin, A. Berman, & B. Koichu
(Eds.), Crea tivity in mathematics and the education of gifted students (pp. 255-269). Rotterdam: Sense Publishers.
[11] Hirsh, R. A. (2010). Creativity: Cultural capital in the mathematics class room. Creative Education, 1, 154-161. doi:10.4236/ce.2010.13024
[12] Leder, G. C. (1992). Mathematics before formal schooling. Educational Studies in Mathematics, 23, 383-396. doi:10.1007/BF00302441
[13] Leikin, R. (2009a). Bridging research and theory in mathematics educa tion with research and theory in creativity and giftedness. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in
mathematics and the education of gifted students (pp. 383-409). Rotterdam: Sense Publishers.
[14] Leikin, R. (2009b). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students
(pp. 129-145). Rotterdam: Sense Publishers.
[15] Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM—The International Journal on Mathematics Edu
cation, 45, 183-197.
[16] Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: The state of the art. ZDM Mathematics Education, 45, 159-166. doi:10.1007/s11858-012-0459-1
[17] Leikin, R., Berman, A., & Koichu, B. (2009). Creativity in mathematics and the education of gifted students. Rotterdam: Sense Publisher.
[18] Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solu tion tasks in developing knowledge and creativity in geometry. Jour nal of Mathematical Behavior, 31, 73-90. doi:10.1016/
[19] Milgram, R., & Hong, E. (2009). Talent loss in mathematics: Causes and solutions. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativ ity in mathematics and the education of gifted students
(pp. 149-163). Rotterdam: Sense Publishers.
[20] Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373-394. doi:10.1007/BF00366618
[21] Riley, M. S., Greeno, J. G., & Heller, J. (1983). Development of chil dren’s problem-solving ability in arithmetic. The Development of Mathematical Thinking (pp. 153-196). New York: Academic
[22] Sak, U., & Maker, C. J. (2006). Developmental variations in children’s creative mathematical thinking as a function of schooling, age, and knowledge. Creativity Research Journal, 18, 279-291.
[23] Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different? Early numerical thinking revisited. Cognition and Instruction, 23, 237-309. doi:10.1207
[24] Sheffield, L. (2009). Developing mathematical creativity—Questions may be the answer. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students
(pp. 87-100). Rotterdam: Sense Publishers.
[25] Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM—The International Journal on Mathematics Education, 29, 75-80.
[26] Steinberg, R. (1985a). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355. doi:10.2307/749356
[27] Steinberg, R. (1985b). Keeping track processes in addition and sub traction. Paper Presented at the Annual Meeting of the American Educational Research Association, Chicago, IIlinois.
[28] Steinberg, R. M., Empson, S. B., & Carpenter, T. P. (2004). Inquiry into children’s mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 7, 237-267.
[29] Tabach, M., & Friedlander, A. (2013). School mathematics and creativity at the elementary and middle grades level: How are they related? ZDM—The International Journal on Mathematics Education,
45, 227-238.
[30] Tiedemann, K., & Brandt, B. (2010). Parents’ Support in Mathematical Discourses. In U. Gellert, E. Jablonka, & C. Morgan (Eds.). Proceedings of the 6th International Conference on Mathematics
Education and Society (pp. 428-437). Berlin: Freie Universitat Berlin.
[31] Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.
[32] Tsamir, P., Tirosh, D., Tabach, M., & Levenson, E. (2010). Multiple solution methods and multiple outcomes—Is it a task for kindergar ten children? Educational Studies in Mathematics, 73,
217-231. doi:10.1007/s10649-009-9215-z
[33] Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
[34] Warfield, J., & Yttri, M. J. (1999). Cognitively Guided Instruction in one kindergarten classroom. In J. V. Copley (Ed.). Mathematics in the early years. Reston, VA: NCTM.
[35] Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumenta tion, and autonomy in mathematics. Journal for Research in Mathe matics Education, 458-477. doi:10.2307/749877 | {"url":"https://www.scirp.org/journal/paperinformation?paperid=34653","timestamp":"2024-11-07T01:04:14Z","content_type":"application/xhtml+xml","content_length":"112420","record_id":"<urn:uuid:4d0dd200-6510-400a-b681-b025711fcea1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00771.warc.gz"} |
Types of graphs used in Math and Statistics
Descriptive Statistics: Charts, Graphs and Plots > Types of Graphs
Common Types of Graphs
Contents (click to skip to the section):
1. Types of Graphs: Bar Graphs
A bar graph is a type of chart that has bars showing different categories and the amounts in each category.
See: Bar Graph Examples
This type of graph is a type of bar chart that is stacked, and where the bars show 100 percent of the discrete value.
See: Segmented Bar Chart, What is it?
Microsoft Excel calls a bar graph with vertical bars a column graph and a bar graph with horizontal bars a bar graph.
See: Column Chart Excel 2013
4. Types of Graphs: Box and Whiskers (Boxplots)
This type of graph, sometimes called a boxplot, is useful for showing the five number summary.
See: Box and Whiskers Chart
5. Types of Graphs: Frequency Distributions
A frequency chart.
Although technically not what most people would call a graph, it is a basic way to show how data is spread out.
See: Frequency Distribution Table.
A table that shows how values accumulate.
See: Cumulative Frequency Distribution Table.
This type of graph is almost identical to a histogram. Where histograms use rectangle, these graphs use dots, which are then joined together.
See: Frequency Polygon.
8. Types of Graphs: Histogram
A way to display data counts with data organized into bins.
See: Histogram.
9. Types of Graphs: Line Graphs
This graph of -4 5x+3.
A graph that shows a line; usually with an equation. Can be straight or curved lines.
See: Line Graph
A Dow Jones Timeplot from the Wall Street Journal shows how the stock market changes over time.
A time plot is similar to a line graph. However, it always plots time on the x-axis.
See: Timeplot.
A relative frequency histogram shows relative frequencies.
See: Relative Frequency Histogram.
12. Types of Graphs: Pie Graphs
Pie chart showing water consumption. Image courtesy of EPA.
As the name suggested, these types of graph look like pies.
See: Pie Chart: What is it used for?
13. Types of Graphs: Scatter Graphs
A scatter plot.
These charts use dots to plot data points. the dots are “scattered” across the page.
See: Scatter plot.
145. Types of Graphs: Stemplots
Stemplots help you to visualize all of the individual elements of a data set.
See: Stemplot: What is it? | {"url":"https://www.statisticshowto.com/types-graphs/","timestamp":"2024-11-02T23:38:33Z","content_type":"text/html","content_length":"87536","record_id":"<urn:uuid:d7ba6c7c-496c-4d20-9fe3-86798e81e424>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00851.warc.gz"} |
The Big Internet Math-Off 2024, Round 1, Match 2
You're reading: The Big Internet Math-Off 2024
The Big Internet Math-Off 2024, Round 1, Match 2
Here’s the second match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Angela Tabiri against Max Hughes.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment
Angela Tabiri – Understanding the maths behind machine learning
This video gives a brief introduction to the mathematics for machine learning in non technical language.
The abstract mathematics we do, have real life applications.
Which other applications of vectors in real life can you think about?
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight
the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
Max Hughes – Constructing a mathematical pride dress
The interesting piece of mathematics I would like to put forward for Round 1 of this year’s “Big Internet Math-Off” is the Leonardo Dome, a mathematical structure built following geometric rules that
was first designed by Leondaro da Vinci over 500 years ago. To truly represent the majesty of the dome, I decided to document myself turning it into the base for a giant pride themed dress to
celebrate pride in mathematics. See the video below:
Max Hughes is the coordinator of MathsCity Leeds, who spends their free time playing table-top roleplaying games and reading comic books, whilst being engaged with fun mathsy projects on the side.
You can follow them on Instagram.
So, which bit of maths has tickled your fancy the most? Vote now!
Match 2: Angela Tabiri vs Max Hughes
Angela with the maths of machine learning
(82%, 534 Votes)
Max with a mathematical pride dress
(18%, 114 Votes)
Total Voters: 648
This poll is closed.
The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our third match in round 1, pitting Matt Enlow against Sam Kay, or check out the announcement post for your follow-along wall chart!
42 Responses to “The Big Internet Math-Off 2024, Round 1, Match 2”
1. Wanjiru Esther
Dr. Angela, you are the best female mathematician
2. Evelyn Ackah
Voting for Agella Tabiri for more fun games in maths
3. FREDERICK SITSOPE ACQUAH
Angela all the way
□ Evelyn Oduro
Dr.Angela is the best
4. Nicola Jack
5. Ellen Amoah
Angela is the best. I have had opportunity to listen to her maths instructions or lesson and I was really in love.
6. Anonymous
This is awesome
□ Victoria Amsah
Good job Angela.
☆ Abubakar Wahab
Dr. Angela you are amazing. Keep soaring.
7. Adu-Boahen Felix
Dr. Angela, an epitome of Jim Simons
8. Efua GODgirl
Dr. Tabiri’s work is the real deal. I didn’t want the video to even end. Such a fun way of explaining the maths behind the scenes to us. Thank you thank you thank you. Dr Tabiri for the win
9. Emmanuel Kwame Aidoo
All the best guys.
10. PETER
GO HIGHEST, Dr.
11. Ifunanya
Aha!…Angela is making me fall in love with maths all over again.
12. SR
13. Daniel Asante
Angela, you got this.
14. Evelyn Oduro
Dr.Angela is the best
15. Sayida Issah
The best female mathematician ever, more wins Angela
16. Gloria Aberinga
Angela is lady I have Known for some time, all about her life is mathematics
17. Alison
I can’t decide which bit of Max’s video is my favourite – the spectacular dome collapse, the crawling, or the dancing at the end… Iconic
18. Selase Odopey
Angela wowed me!
19. Hilda Dela Hosu
Angela is the best.
20. Nathan Darko
Interesting stuff there, Angela
21. Danquah Kwabena Felix
Angela is the best. Please, give it to Angela
22. Bismark
23. Olusola
You make Mathematics interesting every time you have the privilege to teach and to make presentation.
Well done.
24. Cameline
Proud of you Angela
25. Kingsley Ahenkora-Duodu
Best wishes, Angela T.
26. Eric Anum
I vote for Angela Tabiri
27. Anonymous
28. Olivia Nabawanda
Lots of love Angela! You are the winner
29. Christopher Francis Aidoo
I go for Dr. Angela Tabiri
30. Margret
I go for Angela
31. Sodiq Mojeed
Give it to Angela!
Kudos to her.
32. Ebenezer Acquah
I will vote for Dr. Angela Tabiri
33. John
My vote goes to Dr. Angela Tabiri. You got this!
34. Nelly Adjoa Sakyi-Hagan
Go Angie
35. Maurine Songa
Dr. Angela Tabiri’s wins. Her take is simply inspirational.
36. Martin Omollo
Angela is just best mentor, you got this
37. Kwasi Darbs
Angela the math queen
38. Joseph Lele
Math is fun
Go high Dr Tabiri
39. Adedoyin Esther
Go Dr Angela… NAFEMREP is proud of you!!!
Esther from Nigeria | {"url":"https://aperiodical.com/2024/07/the-big-internet-math-off-2024-round-1-match-2/","timestamp":"2024-11-05T22:17:39Z","content_type":"text/html","content_length":"88991","record_id":"<urn:uuid:6192fa80-2f74-4896-ab94-f99d67f66171>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00575.warc.gz"} |
How to get reliable SPSS data analysis help? | Hire Someone To Take My SAS Assignment
How to get reliable SPSS data analysis help? Will 4ths of the new data automatically analysis and understanding the data fit your needs, please? This resource is useful for other people. Background
SPSS is a Learn More visualization and automation tool for the analysis of SASS data. These data represent specific data sets collected on a global scale over the continental United States (U.S.).
The results of the analyses can be displayed as raw data. This visualization is designed in EPM/GAIME format, which is a commonly-used format for exporting EPM/GAIME data to a dataspace. This
excel-style example shows how the Go Here analysis results are stored into EPM/GAIME (file formats), EPM header files contained within EPM header files, and SPSS (visualization). To increase
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models described below, we’ll assume that you’ve already completed SPSS and therefore no need to set model parameters yet. Creating your SPSS dataset Create your single-data sample chart. Create a
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data set (bottom-left). The data in this chart will also include four rows in the Y-axis, four rows in the X-axis, and two rows in the Z-axis. Setting the first column to zero lets you specify which
row of the dataset is a row (zero-value columns) and which row is a line (black-segment columns), so this example displays the Y-axis map for the corresponding column as a loop plot. Creating our
dataset Create a single-data SPS dataset. Using the second column to define the data in the Y axis by setting the first column to 0 in line plotting. This column is also indicated as 0 for a line
(first column zero-value columns). Since we want to plot the data as a line, we set line starting from 0 to 1 in the Y-axis. When changing the dataset to data from data from line plotting, we also
know that we’ve specified that the chart data is not yet available to view.
Homework To Do Online
Set the data to a single-data data set. We are able to use the standard statistical analysis tools with our data data provided in the Appendix. For example, we can set the first column to zero in
line plotting (note that SPSS is only available to you by the time your data is generated) and to five decimal points per position in the Y-axis. In the example shown in Figure 5-4, we set data from
lineHow to get reliable SPSS data analysis help? Today we are going to discuss SPSS data analysis help. Let’s start with the basic question of SPSS. We are analyzing the data for the last year or so
and summarize it for further analysis as you would like by making your own presentation. The dataset is a set of 1871 results for 12 months. What we’re interested in doing is following the answer
data provided by sps-dmm under the parameters PDB_525643 from the SPSS Version 5.0.0 and 2.0.0. So, I have to find the most out there more helpful hints there. I call this SPSS2 as it contains all of
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research questions for the most the support for this as well. This example has a good distribution of users not just those that are already searching for the SPSS software you have already installed,
it contains the data so I want to apply my help as well as my search results and the help that comes to the way in relation to the SPSS language support (I still think that it is worth it to make
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methods presented have a peek here SPSS2 are likely to have a lot of solutions to getting working with SPSS data analysis in general for you or in any other SPSS SPSS/SPS of your own choice that
might not works for you. In other words, you can find an excellent implementation in SPSS that you can come up with that gets working and is probably not the ideal place to end up. This is the code
of my setup click for source the analysis below. [global] char spsSplGetValue(int name, int value, char **matches, int indices) { if (m_name ) { int i = 0; while (name!= NAME_(i)) { i++; } ; return
m_result(spsSplGetValue(value),matches,indices); } } As I mentioned in the previous section our SPSS Software is based on “the original SPSS software with many options and some basic tools”.
On that note let me have a look out for the information associated with SPSS2 here again. And, finally we go on to compare the SPSS software for this year to theHow to get reliable SPSS data analysis
help? SPSS is a free and open source statistical language designed to analyse and categorise data, and it makes analysis easy for all stakeholders (including DAW/NAK) rather than just text, documents
or data. Hence, SPSS analysis is ideal for the following purposes. 1) Data analysis. Data are collected on a single input source base data, which is collected by a team of statisticians,
statisticians researchers or analysts. It uses multiple models (including more than three models) to carry out the data analysis. Statistical modelling models, which are used for standardising data
and other data they are analysed on, are used in large scale data collecting efforts such as government or NHS studies. The distribution of data used in these R statistical analyses relates to a
specific input source (or their output) and should be tested using SPSS tools for more accurate and straightforward predictions, giving the best level of performance. There are two basic approaches
to data analysis: classification and data meta-analysis (DME). The DME approach is currently applied in a few major PPCO workshops and other meetings. It can be used for both text (and sometimes
computer generated models) and graphs, but it can also be used inside software for data-analytic and automated you could try here DME modelling can help to understand whether a statistical model is
being used. This information can help in an analysis of the data being analyzed. It can then be used in future PPCO presentations on statistical algorithms working with large-scale data-collection
systems to be more precise what they are used in. The DME approach can also be applied to model-driven selection of a more appropriate type of data. 2) Statistical models, classification and data
meta-analysis Historically, the types of data made available by SPSS tools have been used for various purposes. This is because SPSS tools and statistics make available their own data infrastructure,
some of which is already well developed. web case can be made that SPSS tools and statistics not only provide data but also have an understanding of existing data. For these reasons, the
classification and classification-based data are often used in PPCO debates with SPSS experts. Information, such as, statistical modelling and decision analyses, represents a meaningful
representation of data.
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Mathematical Modeling Prompts
Mathematics is a tool for understanding the world better and making decisions. School mathematics instruction often neglects giving students opportunities to understand this, and reduces mathematics
to disconnected rules for moving symbols around on paper. Mathematical modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to
understand them better, and to improve decisions (NGA 2010). This mathematics will remain important beyond high school in students’ lives and education after high school (NCEE 2013).
The mathematical modeling prompts and this guidance for how to use them represent our effort to make authentic modeling accessible to all teachers and students using this curriculum.
Organizing Principles about Mathematical Modeling
• The purpose of mathematical modeling in school mathematics courses is for students to understand that they can use math to better understand things they are interested in in the world.
• Mathematical modeling is different from solving word problems. It often feels like initially you are not given enough information to answer the question. There should be room to interpret the
problem. There ought to be a range of acceptable assumptions and answers. Modeling requires genuine choices to be made by the modeler.
• It is expected that students have support from their teacher and classmates while modeling with mathematics. It is not a solitary activity. Assessment should focus on feedback that helps students
improve their modeling skills.
Things the Modeler Does When Modeling with Mathematics (NGA 2010)
1. Pose a problem that can be explored with quantitative methods. Identify variables in the situation and select those that represent essential features.
2. Formulate a model: create and select geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between variables
3. Compute: Analyze these relationships and perform computations to draw conclusions
4. Interpret the conclusions in terms of the original situation
5. Validate the conclusions by comparing them with the situation. Iterate if necessary to improve the model
6. Report the conclusions and the reasoning behind them
It’s important to recognize that in practice, these actions don’t often happen in a nice, neat order.
When to Use Mathematical Modeling Prompts
A component of this is mathematical modeling prompts. Prompts include multiple versions of a task (the multiple versions require students engage in more or fewer aspects of mathematical modeling),
sample solutions, instructions to teachers for launching the prompt in class and supporting students with that particular prompt, and an an analysis of each version showing how much of a “lift” the
prompt is along several dimensions of mathematical modeling. A mathematical modeling prompt could be done as a classroom lesson or given as a project. This is a choice made by the teacher.
A mathematical modeling prompt done as a classroom lesson could take one day of instruction or more than one day, depending on how much of the modeling cycle students are expected to engage in, how
extensively they are expected to revise their model, and how elaborate the reporting requirements are.
A mathematical modeling prompt done as a project could span several days or weeks. The project is assigned and students work on it in the background while daily math lessons continue to be conducted.
(Much like research papers or creative writing assignments in other content areas.) This structure has the advantage of giving students extended time for more complex modeling prompts that would not
be feasible to complete in one class period and affords more time for iterations on the model and cycles of feedback.
Modeling prompts don’t necessarily need to involve the same math as the current unit of study. As such, the prompts can be given at any time as long as students have the background to construct a
reasonable model.
Students might flex their modeling muscles using mathematical concepts that are below grade level. First of all, learning to model mathematically is demanding—learning to do it while also learning
new math concepts is likely to be out of reach. Second of all, we know that in future life and work, when students will be called on to engage in mathematical modeling, they will often need to apply
math concepts from early grades to ambiguous situations (Forman & Steen, 1995). This elusive category of problems which are high school level yet draw on mathematics first learned in earlier grades
may seem contradictory in a curriculum that takes focus and alignment seriously. However, p. 84 of the standards alludes to such problems, and Table 1 in the high school publisher’s criteria (p. 8)
leaves room for including such problems in high school materials in column 6.
The mathematical modeling prompts are not the only opportunities for students to engage in aspects of mathematical modeling in the curriculum. Mathematical modeling is often new territory for both
students and teachers. Oftentimes within the regular classroom lessons, activities include scaled-back modeling scenarios, for which students only need to engage in a part of the modeling cycle.
These activities are tagged with the “Aspects of Modeling” instructional routine, and the specific opportunity to engage in an aspect of modeling is explained in the activity narrative.
How to Prepare and Conduct the Modeling Lesson or Project
• Decide which version of the prompt to give.
• Have data ready to share if you plan to give it when students ask.
• Ensure students have access to tools they might be expected to use.
• If desired, instruct students to use a template for organizing modeling work.
• Whether doing the prompt as a classroom lesson or giving as a project, plan to do the in-class launch in class.
• Decide to what extent students are expected to iterate and refine their model. If you are conducting a one-day lesson, students may not have much time to refine their model and may not engage as
much in that part of the modeling cycle. If you conduct a lesson that takes more than one day, or give the task as a project, it is more reasonable to expect students to iterate and refine their
model once or even several times.
• Decide how students will report their results. If conducting a one-day lesson, this may be a rough visual display on a whiteboard. If more time is allotted or the task is assigned as a project,
you might instruct students to write a more formal report, slideshow, blog post, poster, or create an a mockup of an artifact like a letter to a specific audience, a smartphone app, a menu, or a
set of policies for a government entity to consider. One way to scaffold this work is to ask students to turn in a certain number of presentation slides: one that states the assumptions made, one
that describes the model, and one or more slides with their conclusions or recommendations.
• Decide how students will be assessed. Prepare a rubric that will be used and share it with them.
Ideas for Setting Up an Environment Conducive to Modeling
• Provide plenty of blank whiteboard or chalkboard space for groups to work together comfortably. “Vertical non-permanent surfaces” are most conducive to productive collaborative work. “Vertical”
means on a vertical wall is better than horizontally on a tabletop, and “non-permanent” means something like a dry erase board is better than something like chart paper (Liljedahl 2016).
• Ensure that students have easy access to any tools that might be useful for the task. These might include:
□ A supply table containing geometry tools, calculators, scratch paper, graph paper, dry erase markers, post-its
□ Electronic devices to access digital tools (like graphing technology, dynamic geometry software, or statistical technology)
• Think about how you will help students manage the time that is available to work on the task. For example:
□ For lessons, display a countdown timer for intermittent points in the class when you will ask each group to summarize their progress
□ For lessons, decide what time you will ask groups to transition to writing down their findings in a somewhat organized way (perhaps 15 minutes before the end of the class)
□ For projects, set some intermediate milestone deadlines to help students know if they are on track.
Organizing Students into Teams or Groups
• Mathematical modeling is not a solitary activity. It works best when students have support from each other and their teacher.
• Working with a team can make it possible to complete the work in a finite amount of class time. For example, the team may decide it wants to vary one element of the prompt, and compute the output
for each variation. What would be many tedious calculations for one person could be only a few calculations for each team member.
• The members of good modeling groups bring a diverse set of skills and points of view. Scramble the members of modeling teams often, so that students have opportunities to play different roles.
Ways to Support Students While They Work on a Modeling Prompt
• Coach them on ways to organize their work better.
• Provide a template to help them organize their thinking. Over time, some groups may transition away from needing to use a template.
• Remind them of analog and digital tools that are available to them.
• When students get stuck or neglect an important aspect of the work, ask them a question to help them engage more fully in part of the modeling cycle. For example:
□ What quantities are important? Which ones change and which ones stay the same? (identify variables)
□ What information do you know? What information would it be nice to know? How could you get that information? What reasonable assumption could you make? (identify variables)
□ What pictures, diagrams, graphs, or equations might help people understand the relationships between the quantities? (formulate)
□ How are you describing the situation mathematically? Where does your solution come from? (compute)
□ Under what conditions does your model work? When might it not work? (interpret)
□ How could you make your model better? How could you make your model more useful under more conditions? (validate)
□ What parts of your solution might be confusing to someone reading it? How could you make it more clear? (report)
How to Interpret the Provided Analysis of a Modeling Prompt
For any mathematical modeling prompt, different versions are provided. We chose to analyze each version along 5 impactful dimensions that vary the demands on the modeler (OECD 2013). Each version of
a mathematical modeling prompt is accompanied by an analysis chart that looks like this:
│attribute │DQ│QI│SD│AD│M│mean │
│ lift │0 │1 │0 │0 │2│0.6 │
Each of the attributes of a modeling problem is scored on a scale from 0–2. A lower score indicates a prompt with a “lighter lift” for students and teachers: students are engaging in less open, less
authentic mathematical modeling. A higher score indicates a prompt with a “heavier lift” for students and teachers: students are engaging in more open, more authentic mathematical modeling.
This matrix shows the attributes that are part of our analysis of each mathematical modeling prompt. We recognize that not all the attributes have the same impact on what teachers and students do.
However, for the sake of simplicity they are all weighted the same when they are averaged.
│index│attribute │ light lift (0) │ medium lift (1) │ heavy lift (2) │
│ │Defining │ │elements of ambiguity; prompt might suggest ways assumptions could be │ │
│ DQ │the │well-posed question │made │freedom to specify and simplify the prompt; modeler must state assumptions │
│ │Question │ │ │ │
│ │Quantities│ │ │ │
│ QI │of │key variables are declared│key variables are suggested │key variables are not evident │
│ │Interest │ │ │ │
│ SD │Source of │data is provided │modelers are told what measurements to take or data to look up │modelers must decide what measurements to take or data to look up │
│ │Data │ │ │ │
│ │Amount of │modeler is given all the │some extra information is given and modeler must decide what is │modeler must sift through lots of given information and decide what is │
│ AD │Data given│information they need and │important; or, not enough information is given and modeler must ask for │important; or, not enough information is given and modeler must make │
│ │ │no more │it before teacher provides it │assumptions, look it up, or take measurements │
│ │ │a model is given in the │type of model is suggested in words or by a familiar context; or, │careful thought about quantities and relationships or additional work (like │
│ M │The Model │form of a mathematical │modeler chooses appropriate model from a provided list │constructing a scatterplot or drawing geometric examples) is required to │
│ │ │representation │ │identify type of model to use │
We recognize that there are other features of a mathematical modeling prompt that could be varied. In the interests of not making things too complex, we only included 5 dimensions in the lift
analysis. However, one might choose to modify a prompt on one of these dimensions:
• whether the scenario is posed with words, a highly-structured image or video, or real-world artifacts like articles or authentic diagrams
• presenting example for student to explore before they are expected to engage with the prompt, versus the prompt suggesting that the modeler generate examples or expecting the modeler to generate
examples on their own
• whether the prompt makes decisions about units of measure or expects the modeler to reconcile units of measure or employ dimensional thinking
• whether a pre-made digital or analog tool is provided, instructions given for using a particular tool, use of a particular tool is suggested, or modelers simply have access to familiar tools but
are not prompted to use them
• whether a mathematical representation is given, suggested, or modelers have the freedom to select and create representations of their own choosing
Consortium for Mathematics and Its Applications (2016). Guidelines for Assessment and Instruction in Mathematical Modeling Education. Retrieved November 20, 2017 from http://www.comap.com/Free/GAIMME
Forman, S. L., & Steen, L. A. (1995). Mathematics for work and life. In I. M. Carl (Ed.), Seventy-five years of progress: Prospects for school mathematics (pp. 219–241). Reston, VA: National Council
of Teachers of Mathematics.
High School Publisher’s Criteria for the Common Core State Standards for Mathematics. Retrieved November 20, 2017 from http://www.corestandards.org/wp-content/uploads/
Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.) Posing and Solving Mathematical Problems: Advances and New
Perspectives. New York, NY: Springer. Retrieved November 20, 2017 from http://peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-Feb-14-20151.pdf
National Governors Association Center for Best Practices (2010). Common Core State Standards for Mathematics.
NCEE (2013). What Does It Really Mean to Be College and Work Ready? Retrieved November 20, 2017 from http://ncee.org/college-and-work-ready/
OECD (2013). Strong Performers and Successful Reformers in Education—Lessons from PISA 2012 for the United States. Retrieved on November 20, 2017 from http://www.oecd.org/pisa/keyfindings/ | {"url":"https://im-beta.kendallhunt.com/HS/teachers/mathematical_modeling_prompts.html","timestamp":"2024-11-13T21:40:59Z","content_type":"text/html","content_length":"94414","record_id":"<urn:uuid:2af33054-5df6-4d25-831c-21bce0892f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00792.warc.gz"} |
Understanding Plane Stress and Mohr's Circle in Truss Structures
• Thread starter fonseh
• Start date
In summary, planes stress in a truss refers to the distribution of internal forces and stresses within a framework commonly used in engineering and construction projects. The factors that affect
planes stress include load, geometry, material properties, and boundary conditions. It is typically calculated using mathematical equations and is important for ensuring structural stability and
safety. To reduce planes stress, engineers can modify the design, redistribute loads, or reinforce the truss with additional supports or structural elements.
Homework Statement
For part b , i think one of the angle either $$\theta_s$$ or $$\theta_p$$ is wrong
For the second question , what is plane stress ?
Homework Equations
The Attempt at a Solution
1.) Because in mohr's circle, the maximum shear stress on the vertical axis , for the maximum normal stress , it's on the horizontal axis , right ?
Last edited:
I am sorry that the title of the topic is confusing . I think the part a is not related to part b
Or is my sketch of mohr's circle is wrong ?
FAQ: Understanding Plane Stress and Mohr's Circle in Truss Structures
What is planes stress in a truss?
Planes stress in a truss refers to the distribution of internal forces and stresses within a truss structure, which is a type of framework commonly used in engineering and construction projects.
What factors affect planes stress in a truss?
The factors that affect planes stress in a truss include the type of load being applied, the geometry and material properties of the truss, and the boundary conditions or supports at each end.
How is planes stress calculated in a truss?
Planes stress in a truss is typically calculated using mathematical equations, such as the method of joints or the method of sections. These methods involve breaking down the truss into smaller
sections and analyzing the forces and stresses at each joint or section.
What is the importance of considering planes stress in a truss?
Considering planes stress in a truss is important for ensuring the structural stability and safety of the truss. If the stresses exceed the strength of the materials, it can lead to structural
failure and potential hazards.
How can planes stress in a truss be reduced?
To reduce planes stress in a truss, engineers can modify the design by using stronger materials, changing the geometry of the truss, or adding additional supports. Alternatively, the loads can be
redistributed or the truss can be reinforced with braces or other structural elements. | {"url":"https://www.physicsforums.com/threads/understanding-plane-stress-and-mohrs-circle-in-truss-structures.897182/","timestamp":"2024-11-09T10:48:29Z","content_type":"text/html","content_length":"89030","record_id":"<urn:uuid:209f04f0-208e-4677-817c-e65a2580bf4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00428.warc.gz"} |
American Mathematical Society
Chow dilogarithm and strong Suslin reciprocity law
Author: Vasily Bolbachan
Journal: J. Algebraic Geom. 32 (2023), 697-728
DOI: https://doi.org/10.1090/jag/811
Published electronically: May 18, 2023
Full-text PDF
Abstract | References | Additional Information
Abstract: We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This
construction is similar to the construction of the norm map on Milnor $K$-theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new
reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.
• H. Bass and J. Tate, The Milnor ring of a global field, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle,
Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 349–446. MR 0442061, DOI 10.1007/BFb0073733
• Johan L. Dupont and Ebbe Thue Poulsen, Generation of $\textbf {C}(x)$ by a restricted set of operators, J. Pure Appl. Algebra 25 (1982), no. 2, 155–157. MR 662759, DOI 10.1016/0022-4049(82)
• A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–96. MR 1265551, DOI
• A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197–318. MR 1348706, DOI 10.1006/aima.1995.1045
• A. B. Goncharov, Polylogarithms, regulators, and Arakelov motivic complexes, J. Amer. Math. Soc. 18 (2005), no. 1, 1–60. MR 2114816, DOI 10.1090/S0894-0347-04-00472-2
• Ivan Horozov, Reciprocity laws on algebraic surfaces via iterated integrals, J. K-Theory 14 (2014), no. 2, 273–312. With an appendix by Horozov and Matt Kerr. MR 3264264, DOI 10.1017/
• Kazuya Kato, A generalization of local class field theory by using $K$-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 3, 603–683. MR 603953
• János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
• Matt Kerr, James Lewis, and Patrick Lopatto, Simplicial Abel-Jacobi maps and reciprocity laws, J. Algebraic Geom. 27 (2018), no. 1, 121–172. With an appendix by José Ignacio Burgos-Gil. MR
3722692, DOI 10.1090/jag/692
• John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
• Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from
the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
• Denis Osipov and Xinwen Zhu, A categorical proof of the Parshin reciprocity laws on algebraic surfaces, Algebra Number Theory 5 (2011), no. 3, 289–337. MR 2833793, DOI 10.2140/ant.2011.5.289
• A. N. Paršin, Class fields and algebraic $K$-theory, Uspehi Mat. Nauk 30 (1975), no. 1 (181), 253–254 (Russian). MR 0401710
• Daniil Rudenko, The strong Suslin reciprocity law, Compos. Math. 157 (2021), no. 4, 649–676. MR 4241111, DOI 10.1112/s0010437x20007666
• A. A. Suslin, Reciprocity laws and the stable rank of rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1394–1429 (Russian). MR 567040
• A. A. Suslin, $K_3$ of a field and the Bloch group, Proc. Steklov Inst. Math. 183 (1991), 217–239.
• Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/
• Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S.
Cohen. MR 0090581
Additional Information
Vasily Bolbachan
Affiliation: Skolkovo Institute of Science and Technology, Moscow, Russia; Faculty of Mathematics, National Research University Higher School of Ecnomics, Russian Federation, Usacheva str., 6, Moscow
119048, Russia; and HSE-Skoltech International Laboratory of Representation Theory and Mathematical Physics, Usacheva str., 6, Moscow 119048, Russia
MR Author ID: 1468287
ORCID: 0000-0001-6471-8669
Email: vbolbachan@gmail.com
Received by editor(s): May 8, 2021
Received by editor(s) in revised form: October 28, 2021
Published electronically: May 18, 2023
Additional Notes: This paper was partially supported by the Basic Research Program at the HSE University and by the Moebius Contest Foundation for Young Scientists
Article copyright: © Copyright 2023 University Press, Inc. | {"url":"https://www.ams.org/journals/jag/2023-32-04/S1056-3911-2023-00811-2/home.html","timestamp":"2024-11-14T08:09:31Z","content_type":"text/html","content_length":"76206","record_id":"<urn:uuid:c8b17fe2-3640-4e92-a6df-f758b0c07497>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00089.warc.gz"} |
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to Grams
Mole Conversions
Given Moles, Convert to Grams
Return to Mole Table of Contents
In chemistry, the mole is the standard measurement of amount. When substances react, they do so in simple ratios of moles. However, balances give readings in grams. Balances DO NOT give readings in
So the problem is that, in chemistry, we compare amounts of one substance to another using moles. That means we must convert from grams, since this is the information we get from balances.
There are three steps to converting moles of a substance to grams:
1. Determine how many moles are given in the problem.
2. Calculate the molar mass of the substance.
3. Multiply step one by step two.
Make sure you have a periodic table and a calculator handy.
The solution technique can also be expressed in the following ratio and proportion:
grams of the molar mass of the
substance substance in grams
––––––––––– = –––––––––––––––
moles of the one mol
In this particular lesson, the grams of the substance (upper left) will be the unknown (signified by the letter x). The exact same proportion is used in the grams-to-moles conversion lesson. Then the
"x" will reside in the lower left.
This proportion is a symbolic equation. When you solve a particular problem, you insert the proper numbers & units into the proper places of the symbolic equation and then you solve using
cross-multiplication and division. Also, do not attach units to the unknown. Let it be simply the letter "x." The proper unit will evolve naturally from soving the proportion and cancellation of
The bonus example has a twist in it.
Example #1: Calculate how many grams are in 0.700 moles of H[2]O[2]
1) Step One:
The problem will tell you how many moles are present. Look for the word "mole" or the unit "mol." The number immediately preceeding it will be how many moles.
I suppose that a problem can be worded in such a way that the number of moles comes after the unit, but that type of trickery isn't very common in high school.
0.700 moles are given in the problem.
2) Step Two:
You need to know the molar mass of the substance. Please refer to the lessons about calculating the molecular weight and molar mass of a substance if you are not sure how to calculate a molar
The molar mass of H[2]O[2] is 34.0146 grams/mole. You may wish to pause and calculate this value, if you desire the practice.
3) Step Three:
Multiply the moles given by the substance's molar mass:
0.700 mole x 34.0146 grams/mole = 23.8 grams
The answer of 23.8 g has been rounded to three significant figures because the 0.700 value had the least number of significant figures in the problem.
4) If this problem were set up like the proportion above, you would have this:
x 34.0146 g
–––––––– = ––––––––
0.700 mol 1 mol
5) Then, cross-multiply and divide to solve for the unknown.
(x) (1 mol) = (0.700 mol) (34.0146 g)
x = 23.8 g (to three significant figures)
Example #2: Convert 2.50 moles of KClO[3] to grams.
1) Get the moles:
2.50 moles is given in the problem.
2) Get the molar mass:
The molar mass for KClO[3] is 122.550 g/mole. Please note the unit of 'g/mole.' It is important for proper cancelling of units that you remember to write this unit down when using a molar mass.
3) Following step three, we obtain:
2.50 moles x 122.550 g/mole = 306.375 grams
The answer should be rounded off to three significant figures, resulting in 306 g. as the correct answer. Note how the mole in the numerator and the mole in the denominator cancel.
4) If this problem were set up like the proportion above, you would have this:
x 122.550 g
––––––– = ––––––––
2.50 mol 1 mol
5) Then, cross-multiply and divide to solve for the unknown.
(x) (1 mol) = (2.50 mol) (122.550 g)
x = 306 g (to three significant figures)
Example #3: Calculate the grams present in 0.200 moles of H[2]S
(0.200 mol) (34.0808 g/mol) = 6.82 g
Example #4: Calculate the grams present in 0.200 moles of KI
1) Set it up as a ratio and proportion:
x 165.998 g
––––––– = ––––––––
0.200 mol 1 mol
2) Then, cross-multiply and divide to solve for the unknown.
(x) (1 mol) = (0.200 mol) (165.998 g)
x = 33.2 g (to three significant figures)
Example #5: Calculate the grams present in 1.500 moles of KClO
(1.500 mol) (90.5504 g/mol) = 135.8 g (to four sig figs)
I calculated the molar mass of KClO here.
Bonus Example: What is the mass of silver contained in 0.119 moles of Ag[2]S?
1) Determine moles of silver present:
For every one mole of Ag[2]S, 2 moles of silver are present.
(0.119 mol) (2) = 0.238 mol of Ag
2) Determine mass of silver present:
(0.238 mol) (107.868 g/mol) = 25.7 g (to three sig figs) | {"url":"https://web.chemteam.info/Mole/Moles-to-Grams.html","timestamp":"2024-11-02T13:58:45Z","content_type":"text/html","content_length":"7508","record_id":"<urn:uuid:25eb7d0a-daf5-4fe4-aba7-4798d58df571>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00165.warc.gz"} |
Concentrated Moment on the Horizontal Member 3 Frame Deflections Equations and Calculator
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Since ψ[A] = 0 and H[A] = 0
M[A] = LP[M] / A[MM] = Moment (Couple) at Left Node A
δ[HA] = A[HM] M[A] - LP[H] = Horizontal Deflection at Left Node A
General reaction and deformation expressions with right end pinned
Loading Terms LP[H] and LP[M] are given below.
Reaction loads and moments V[A] and V[B], and H[B] can be evaluated from equilibrium equations after calculating H[A] and M[A].
Δ[o] = Displacement (in, mm),
θ[o] = Angular Displacement (radians),
W = Load or Force (lbsf, N),
w = Unit Load or force per unit length (lbs/in^2, N/mm^2),
M[A] = Couple (moment) ( lbs-in, N-mm),
M[o] = Couple (moment) ( lbs-in, N-mm),
θ[o] = Externally created angular displacement (radians),
Δ[o], = Externally created concentrated lateral displacement (in, mm),
T[1] - T[2] = Uniform temperature rise (°F),
T[o] = Average Temperature (deg °F),
γ = Temperature coefficient of expansion [ µinch/(in. °F), µmm/(mm. °F) ],
T[1], T[2] = Temperature on outside and inside respectively (degrees),
H[A], H[B] = Horizontal end reaction moments at the left and right, respectively, and are positive clockwise (lbs, N),
I[1], I[2], and I[3] = Respective area moments of inertia for bending in the plane of the frame for the three members (in^4, mm^4),
E[1], E[2], and E[3] = Respective moduli of elasticity (lb/in^2, Pa) Related: Modulus of Elasticity, Yield Strength;
γ1, γ2, and γ3 = Respective temperature coefficients of expansions unit strain per. degree ( in/in/°F, mm/mm/°C),
l[1], l[2], l[3] = Member lengths respectively (in, mm),
Roark's Formulas for Stress and Strain, Seventh Edition | {"url":"https://engineersedge.com/calculators/concentrated_moment_on_the_horizontal_member_3_14363.htm","timestamp":"2024-11-12T14:15:54Z","content_type":"text/html","content_length":"22086","record_id":"<urn:uuid:95607326-7e58-4fd8-ab79-6e213c83402d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00899.warc.gz"} |
Introduction to Bayesian networks | Bayes Server
Bayesian networks - an introduction
This article provides a general introduction to Bayesian networks.
What are Bayesian networks?
Bayesian networks are a type of Probabilistic Graphical Model that can be used to build models from data and/or expert opinion.
They can be used for a wide range of tasks including diagnostics, reasoning, causal modeling, decision making under uncertainty, anomaly detection, automated insight and prediction.
Figure 1 below shows these capabilities in terms of the four major analytics disciplines, Descriptive analytics, Diagnostic analytics, Predictive analytics and Prescriptive analytics, plus Causal AI.
Figure 1 - Descriptive, diagnostic, predictive & prescriptive analytics with Bayesian networks
They are also commonly referred to as Bayes nets, Belief networks, Causal networks and Causal models.
Bayesian networks are probabilistic because they are built from probability distributions and also use the laws of probability for Reasoning, Diagnostics, Causal AI, Decision making under uncertainty
, and more.
Bayesian networks can be depicted graphically as shown in Figure 2, which shows the well known Asia network. Although visualizing the structure of a Bayesian network is optional, it is a great way to
understand a model.
Figure 2 - A simple Bayesian network, known as the Asia network.
A Bayesian network is a graph which is made up of Nodes and directed Links between them.
In the majority of Bayesian networks, each node represents a Variable such as someone's height, age or country. A variable might be discrete, such as Country = {US, UK, etc...} or might be continuous
such as someone's age.
In Bayes Server each node can contain multiple variables. We call nodes with more than one variable multi-variable nodes.
The nodes and links form the structure of the Bayesian network, and we call this the structural specification.
Bayes Server supports both discrete and continuous variables as well as function nodes.
A discrete variable is one with a set of mutually exclusive states such as Country = {US, UK, Japan, etc...}.
Bayes Server support continuous variables with Conditional Linear Gaussian distributions (CLG). This simply means that continuous distributions can depend on each other (are multivariate) and can
also depend on one or more discrete variables.
Although Gaussians may seem restrictive at first, in fact CLG distributions can model complex non-linear (even hierarchical) relationships in data. Bayes Server also supports Latent variables which
can model hidden relationships (automatic feature extraction, similar to hidden layers in a Deep neural network).
Figure 3 - A simple Bayesian network with both discrete and continuous variables, known as the Waste network.
Links are added between nodes to indicate that one node directly influences the other. When a link does not exist between two nodes, this does not mean that they are completely independent, as they
may be connected via other nodes. They may however become dependent or independent depending on the evidence that is set on other nodes.
Although links in a Bayesian network are directed, information can flow both ways (according to strict rules described later).
Structural learning
Bayes Server includes a number of different Structural learning algorithms for Bayesian networks, which can automatically determine the required links from data.
Note that structural learning is not always required, as often networks are build from expert opinion, and there are also many well known structures (such as mixture models) that can be used for
certain problems.
Another useful technique is to make use of Latent variables to automatically extract features as part of the model.
Directed Acyclic Graph (DAG)
A Bayesian network is a type of graph called a Directed Acyclic Graph or DAG. A Dag is a graph with directed links and one which contains no directed cycles.
Directed cycles
A directed cycle in a graph is a path starting and ending at the same node where the path taken can only be along the direction of links.
At this point it is useful to introduce some simple mathematical notation for variables and probability distributions.
Variables are represented with upper-case letters (A,B,C) and their values with lower-case letters (a,b,c). If A = a we say that A has been instantiated.
A set of variables is denoted by a bold upper-case letter (X), and a particular instantiation by a bold lower-case letter (x). For example if X represents the variables A,B,C then x is the
instantiation a,b,c. The number of variables in X is denoted |X|. The number of possible states of a discrete variable A is denoted |A|.
The notation pa(X) is used to refer to the parents of X in a graph. For example in Figure 2, pa(Dyspnea) = (Tuberculosis or Cancer, Has Bronchitis).
We use P(A) to denote the probability of A.
We use P(A,B) to denote the joint probability of A and B.
We use P(A | B) to denote the conditional probability of A given B.
P(A) is used to denote the probability of A. For example if A is discrete with states {True, False} then P(A) might equal [0.2, 0.8]. I.e. 20% chance of being True, 80% chance of being False.
Joint probability
A joint probability refers to the probability of more than one variable occurring together, such as the probability of A and B, denoted P(A,B).
An example joint probability distribution for variables Raining ad Windy is shown below. For example, the probability of it being windy and not raining is 0.16 (or 16%).
For discrete variables, the joint probability entries sum to one.
If two variables are independent (i.e. unrelated) then P(A,B) = P(A)P(B).
Conditional probability
Conditional probability is the probability of a variable (or set of variables) given another variable (or set of variables), denoted P(A|B).
For example, the probability of Windy being True, given that Raining is True might equal 50%.
This would be denoted P(Windy = True | Raining = True) = 50%.
Marginal probability
A marginal probability is a distribution formed by calculating the subset of a larger probability distribution.
If we have a joint distribution P(Raining, Windy) and someone asks us what is the probability of it raining, we need P(Raining), not P(Raining, Windy). In order to calculate P(Raining), we can simply
sum up all the values for Raining = False, and Raining = True, as shown below.
This process is called marginalization.
When we query a node in a Bayesian network, the result is often referred to as the marginal.
For discrete variables we sum, whereas for continuous variables we integrate.
The term marginal is thought to have arisen because joint probability tables written in ledgers were summed along rows or columns, and the result written in the margins of the ledger.
A more complicated example involving the marginalization of more than one variable is shown below.
Once the structure has been defined (i.e. nodes and links), a Bayesian network requires a probability distribution to be assigned to each node.
Note that it is a bit more complicated for time series nodes and noisy nodes as they typically require multiple distributions.
Each node X in a Bayesian network requires a probability distribution P(X | pa(X)).
Note that if a node X has no parents pa(X) is empty, and the required distribution is just P(X) sometimes referred to as the prior.
This is the probability of itself given its parent nodes. So for example, in figure 2, the node Dyspnea has two parents (Tuberculosis or Cancer, Has Bronchitis), and therefore requires the
probability distribution P ( Dyspnea | Tuberculosis or Cancer, Has Bronchitis ) an example of which is shown in table 1. This type of probability distribution is known as a conditional probability
distribution, and for discrete variables, each row will sum to 1.
The direction of a link in a Bayesian network alone, does not restrict the flow of information from one node to another or back again, however does change the probability distributions required,
since as described above, a node's distribution is conditional on its parents.
Has Bronchitis Tuberculosis or Cancer Dyspnea = True Dyspnea = False
True True 0.9 0.1
True False 0.8 0.2
False True 0.7 0.3
False False 0.1 0.9
Table 1 - P ( Dyspnea | Tuberculosis or Cancer, Has Bronchitis )
Distributions in a Bayesian network can be learned from data, or specified manually using expert opinion.
Parameter learning
There are a number of ways to determine the required distributions.
• Learn them from data
• Manually specify (elicit) them using experts.
• A mixture of both.
Bayes Server includes an extremely flexible Parameter learning algorithm. Features include:
• Missing data fully supported
• Support for both discrete and continuous latent variables
• Records can be weighted (e.g. 1000, or 0.2)
• Some nodes can be learned whilst other are not
• Priors are supported
• Multithreaded and/or distributed learning.
Please see the parameter learning help topic for more information.
Online learning
Online learning (also known as adaptation) with Bayesian networks, enables the user or API developer to update the distributions in a Bayesian network each record at a time. This uses a fully
Bayesian approach.
Often a batch learning approach is used on historical data periodically, and an online algorithm is used to keep the model up to date in between batch learning.
For more information see Online learning.
Things that we know (evidence) can be set on each node/variable in a Bayesian network. For example, if we know that someone is a Smoker, we can set the state of the Smoker node to True. Similarly, if
a network contained continuous variables, we could set evidence such as Age = 37.5.
We use e to denote evidence set on one or more variables.
When evidence is set on a probability distribution we can reduce the number of variables in the distribution, as certain variables then have known values and hence are no longer variables. This
process is termed Instantiation.
Bayes Server also supports a number of other techniques related to evidence:
The figure below shows an example of instantiating a variable in a discrete probability distribution.
Note that we can if necessary instantiate more than one variable at once, e.g. P(A=False, B=False, C, D) => P(C, D).
Note that when a probability distribution is instantiated, it is no longer strictly a probability distribution, and is therefore often referred to as a likelihood denoted with the Greek symbol phi Φ.
Joint probability of a Bayesian network
If U = {A[1],...,A[n]} is the universe of variables (all the variables) in a Bayesian network, and pa(A[i]) are the parents of A[i] then the joint probability distribution P(U) is the simply the
product of all the probability distributions (prior and conditional) in the network, as shown in the equation below.
This equation is known as the chain rule.
From the joint distribution over U we can in turn calculate any query we are interested in (with or without evidence set).
For example if U contains variables {A,B,C,D,E} we can calculate any of the following:
• P(A) or P(B), etc...
• P(A,B)
• P(A|B)
• P(A,B|C,D)
• P(A | C=False)
• ...
The problem is that joint probability distribution, particularly over discrete variables, can be very large.
Consider a network with 30 binary discrete variables. Binary simply means a variable has 2 states (e.g. True & False). The joint probability would require 2^30 entries which is a very large number.
This would not only require large amounts of memory but also queries would be slow.
Bayesian networks are a factorized representation of the full joint. (This just means that many of the values in the full joint can be computed from smaller distributions). This property used in
conjunction with the distributive law enable Bayesian networks to query networks with thousands of nodes.
Distributive law
The Distributive law simply means that if we want to marginalize out the variable A we can perform the calculations on the subset of distributions that contain A
The distributive law has far reaching implications for the efficient querying of Bayesian networks, and underpins much of their power.
Bayes Theorem
From the axioms of probability it is easy to derive Bayes Theorem as follows:
P(A,B) = P(A|B)P(B) = P(B|A)P(A) => P(A|B) = P(B|A)P(A) / P(B)
Bayes theorem allows us to update our belief in a distribution Q (over one or more variables), in the light of new evidence e. P(Q|e) = P(e|Q)P(Q) / P(e)
The term P(Q) is called the prior or marginal probability of Q, and P(Q|e)is called the posterior probability of Q.
The term P(e) is the Probability of Evidence, and is simply a normalization factor so that the resulting probability sums to 1. The term P(e|Q) is sometimes called the likelihood of Q given e,
denoted L(Q|e). This is because, given that we know e, P(e|Q) is a measure of how likely it is that Q caused the evidence.
Are Bayesian networks Bayesian?
Yes and no. They do make use of Bayes Theorem during inference, and typically use priors during batch parameter learning. However they do not typically use a full Bayesian treatment in the Bayesian
statistical sense (i.e. hyper parameters and learning case by case).
The matter is further confused, as Bayesian networks typically DO use a full Bayesian approach for Online learning.
Inference is the process of calculating a probability distribution of interest e.g. P(A | B=True), or P(A,B|C, D=True). The terms inference and queries are used interchangeably. The following terms
are all forms of inference will slightly difference semantics.
• Prediction - focused around inferring outputs from inputs.
• Diagnostics - inferring inputs from outputs.
• Supervised anomaly detection - essentially the same as prediction
• Unsupervised anomaly detection - inference is used to calculate the P(e) or more commonly log(P(e)).
• Decision making under uncertainty - optimization and inference combined. See Decision graphs for more information.
A few examples of inference in practice:
• Given a number of symptoms, which diseases are most likely?
• How likely is it that a component will fail, given the current state of the system?
• Given recent behavior of 2 stocks, how will they behave together for the next 5 time steps?
Importantly Bayesian networks handle missing data during inference (and also learning), in a sound probabilistic manner.
Exact inference
Exact inference is the term used when inference is performed exactly (subject to standard numerical rounding errors).
Exact inference is applicable to a large range of problems, but may not be possible when combinations/paths get large.
Is is often possible to refactor a Bayesian network before resorting to approximate inference, or use a hybrid approach.
Approximate inference
• Wider class of problems
• Deterministic / non deterministic
• No guarantee of correct answer
There are a large number of exact and approximate inference algorithms for Bayesian networks.
Bayes Server supports both exact and approximate inference with Bayesian networks, Dynamic Bayesian networks and Decision Graphs. Bayes Server algorithms.
Bayes Server exact algorithms have undergone over a decade of research to make them:
* Very fast
* Numerically stable
* Memory efficient
We estimate that it has taken over 100 algorithms/code optimizations to make this happen!
As well as complex queries such as P(A|B), P(A, B | C, D), Bayes Server also supports the following:
Bayes Server also includes a number of analysis techniques that make use of the powerful inference engines, in order to extract automated insight, perform diagnostics, and to analyze and tune the
parameters of the Bayesian network.
Dynamic Bayesian networks
Dynamic Bayesian networks (DBNs) are used for modeling times series and sequences. They extend the concept of standard Bayesian networks with time. In Bayes Server, time has been a native part of the
platform from day 1, so you can even construct probability distributions such as P(X[t=0], X[t+5], Y | Z[t=2]) (where t is time).
For more information please see the Dynamic Bayesian network help topic.
Decision graphs
Once you have built a model, often the next step is to make a decision based on that model. In order to make those decisions, costs are often involved. The problem with doing this manually is that
there can be many different decisions to make, costs to trade off against each other, and all this in an uncertain environment (i.e. we are not sure of certain states).
Decision Graphs are an extension to Bayesian networks which handle decision making under uncertainty.
For more information please see the Decision Graphs help topic. | {"url":"https://www.bayesserver.com/docs/introduction/bayesian-networks/","timestamp":"2024-11-09T15:38:09Z","content_type":"text/html","content_length":"63554","record_id":"<urn:uuid:13b8fe3d-c768-47ed-a240-4efa118e81f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00800.warc.gz"} |
’‘If an atom or electron is a basic unit for physicists, his unit is the tetrahedron.’’
I was fascinated by the exhibition ‘‘Cascading Principles’’ at the Andrew Wiles Building in Oxford by the artist Conrad Shawcross. The spirit, as I see it, explains a lot of my passion for the part
of mathematics I do.
Finite element methods use tetrahedra (and other shapes) to approximate continua and solve equations on them. Tetrahedra, or more generally, simplices, are the source of many magics. The Whitney
forms extend discrete topology encoded in simplices (chains) to everywhere defined fields; the Regge finite element extends discrete metric (edge lengths) and discrete curvature (angle deficit) to
piecewise flat metric and curvature measures. This process of “filling in” is what makes finite elements rigorous, compared to more intuitive lattice-based methods. The key lies in the concept of
unisolvency: degrees of freedom (discrete physical, topological, or geometric quantities) uniquely determine local shape functions (modes for approximating continuous fields). The unisolvency of
Whitney forms and Regge elements, to me, demonstrates elegance and magic (and they are included in standard finite element packages and useful as well).
The interplay between the continuous and the discrete is central to mathematics. Newton and Leibniz invented (discovered) calculus, introducing the notions of infinitesimals and limits (though the
rigorous definitions we use today came later). In the computer age, discrete mathematics and physics have gained more attention, partly due to developments in quantum theory. In numerical PDEs,
discretization is a key concept: we discretize the governing equations of physical processes (whether in physics, chemistry, or biology) while attempting to preserve their continuous structure.
Meanwhile, another tradition focuses on establishing discrete models and theories as first principles in the discrete world. And once again, the building blocks are often tetrahedra.
As a numerical analyst, I feel lucky and passionate to work at this interface of continuous and discrete. To me, this work reflects a similar passion that I see from the artist’s perspective.
Another reason I feel a special fondness for the Andrew Wiles Building, apart from its Escher-inspired staircase, is the presence of the “crystals.” From these, one can glimpse the lively teaching
and conference areas downstairs. The crystals in the south wing depict a surface plot of the first eigenfunction of the two-dimensional Laplacian (probably a finite element solution). I took a
picture of these crystals on a quiet night during the pandemic and often use it to visualize Regge elements (piecewise-flat manifolds) in my presentations.
Reference: Andrew Wiles Building | {"url":"https://kaibohu.github.io/personal/","timestamp":"2024-11-08T17:18:55Z","content_type":"text/html","content_length":"13046","record_id":"<urn:uuid:e6d53153-98d6-49ab-aadf-3776c1b8d001>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00284.warc.gz"} |
For the following exercises, use this scenario: The
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population maybe modeled by
Answered question
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population maybe modeled by the following function: $y=12.000+8.000\mathrm
{sin}\left(0.628x\right)$, where the domain is the years since 1980 and the range is the population of the city.
Graph the function on the domain of [0,40] | {"url":"https://plainmath.org/college-statistics/2113-following-exercises-scenario-population-interval-population-modeled","timestamp":"2024-11-04T17:39:28Z","content_type":"text/html","content_length":"178732","record_id":"<urn:uuid:5f4b6a6a-0f77-42a1-9261-14cae77fb078>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00198.warc.gz"} |
A: The position of the lotto numbers when entered in the wheel can change the wheel output somewhat. It can be disappointing when you have all the winning numbers within your wheel, but you don't hit
a jackpot. Unfortunately, each wheel can only guarantee the minimum win listed for that wheel. Though it is possible that one variation might produce a jackpot or a higher win than another variation,
there is no way to predict which way would be better than another.
Minimum Win Guarantee is Unchanged
The minimum win guarantee is the same no matter how the numbers are entered in the wheel, so it's just a personal preference of how you'd like to enter your numbers (sequentially, randomly, mixed
high and low, etc.). The only way to guarantee a jackpot is to use a full wheel, which is very expensive and only works when you trap all the winning numbers in your set.
Wheel Gold can show you how many times each position is used in the wheel, so you can use this information to determine how to place your numbers if you want to. Click the Handicap button from any
wheel to view the chart. Then, place your best numbers in the positions that show the highest numbers in the handicap chart, so they are used more often in the wheel. Since these are balanced wheels,
there isn't usually a large difference for most of the positions, but there can be some slight variations if the numbers don't evenly divide into the wheel. | {"url":"https://www.smartluck.com/faq/faq411.htm","timestamp":"2024-11-03T10:25:35Z","content_type":"application/xhtml+xml","content_length":"4525","record_id":"<urn:uuid:16561bf9-6ad0-425f-8245-854a0949b33b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00331.warc.gz"} |
Multiplication for Year 4 Kids
Introduction to Multiplication for Year 4
When children advance to year 4, they are introduced to multiplication and division. Although children may have a basic understanding of the multiplication operation from earlier grades, it is
critical that they have a firm grasp of the fundamentals of multiplication. Place value, number facts, factor pairs, commutativity, and inverse operations in mental calculations are among the topics
that are required for a better knowledge of multiplication in year 4. The article discusses major concepts covered in multiplication during the year. The article also discusses several crucial
strategies that parents and teachers may use to help children have a better understanding of the topic.
Topics Covered in Multiplication for Year 4
The simple multiplication procedure is one of the concepts covered in year 4. The notion of multiplication is introduced to the children. Children are also taught to memorise tables till 12. If a
child understands the table, even the most difficult arithmetic problems are simple to answer. The second important thing that students are taught in the multiplication for year 4 is the use of
number place. Kids are taught to use multiplication facts to solve division calculations. Kids are taught the multiplication of three-digit numbers. Another important concept that is taught to kids
under the topic of multiplication is the factor pairs. A factor pair is a set of numbers multiplied together to get a certain number.
Three and two, for example, are a factor pair of six. Students are taught to apply what he or she has learned to solve the questions. They will also apply their commutativity understanding to
conceptually solve multiplication problems.
How Can Parents Help Kids?
It is important to understand that as a kid progresses in higher grades they are introduced to various concepts in every subject, hence, it is important for parents to have active participation. This
can help kids to develop a better understanding of the concept and excel academically. Below are some tips that can be used to create a deeper understanding of the topics taught in multiplication for
year 4 kids.
Learning Tables
Memorising tables is a necessary ability for grasping mathematical concepts. By engaging with children, parents can assist them in memorising the table. Parents or instructors can print colourful
cards with the number tables on them and study them with their children.
Use the Concepts of Mathematics in Day to Day Life
Parents can help kids to practise multiplication facts in everyday settings. For example, if parents are out for grocery shopping and have bought three packages of multipack crisps, each of which
contains six packets, you may encourage your child to calculate how many packets of crisps one will have in total. Discuss the method they took to arrive at their conclusion. This can help kids to
develop their speed while calculating simple numbers. Apart from academic importance, it is a very useful life skill that can help kids to better adjust to society.
Playing Games
Playing games that use the concept of mathematics is one of the best ways to engage kids in the topic while having fun. One such game named play 21 is mentioned here as an example.
In this game, parents can request that kids throw a dice five times and write down each number on a piece of paper. 1, 4, 3, 5, 3 are some examples. They must next use any operations (addition,
subtraction, multiplication, and division) on the numbers to arrive at the number 1 as result. Kids can only use each number once, and each computation must include at least two numbers. For example,
we may calculate 3 3, 5 – 4, 4 – 3, and so on to get the answer of 1. Then, using any operation, have your youngster find a calculation with a 2 as the result, then a 3 as an answer, and so on until
they reach 21.
Encouraging Kids to Use Different Strategies to Solve a Question
Children will learn a number of conceptual and informal multiplication procedures, as well as some formal approaches such as column multiplication, in school. To master these methods, choose a
multiplication issue and solve it with your kid as they work out the answer independently. Your techniques should be analyzed and compared. Discuss which strategy is preferred and why. Students will
be able to apply the best technique for any given question using this method since they will be able to use a variety of strategies. You may help kids gain confidence in both formal and informal
procedures by enabling them to practice a range of approaches to multiplying numbers.
This concluded our discussion on how to teach multiplication for year 4. We looked into some of the methods that can assist students to gain a better knowledge of the subject, as well as the basic
concepts that are taught in the topic. A key element to keep in mind is that a child's talents can be improved via consistent practice. Finally, we hope that we have provided some insight into the
basics of multiplication.
FAQs on Mathematics for Kids - Multiplication for Year 4 Kids
1. Name a topic that is important to develop an understanding of the multiplication operation.
The understanding of numbers, factors are some of the important concepts that form the basis of the multiplication operation in mathematics. Factors are particularly important, kids are taught about
factor pairs and common factors to develop a deeper understanding. Factor pairs are defined as the set of numbers that we multiply to get a product. 6 and 3 are a factor pair of 18 for example. The
factors that are shared by two integers are known as common factors. The common factors of 12 and 8 are 1, 2, and 4 respectively.
2. Give a real-life circumstance that can be used as an example where multiplication is used.
There are several real-life examples where the concepts of multiplication are used. An example of such is, calculating the total number of muffins in the tray. A muffin tray, for example, will often
feature four rows of three muffin cups in each row, the total number of muffins can be calculated by multiplying three into four, thus the total number of muffins is twelve. | {"url":"https://www.vedantu.com/maths/multiplication-for-year-4-kids","timestamp":"2024-11-12T03:17:42Z","content_type":"text/html","content_length":"158174","record_id":"<urn:uuid:85d792e6-d0bb-4194-bbab-237a00c1bb81>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00707.warc.gz"} |
24º CBM - Sessões Especiais: Especial de Economia - IMPA - Instituto de Matemática Pura e Aplicada
24º CBM - Sessões Especiais: Especial de Economia
H o r á r i o 16:30 – 17:05 17:10 – 17:45 17:50 – 18:25 18:30 – 19:05
Terça-feira, 29 de julho José Fajardo Santiago Barbachan Marcos Tsuchida Marcelo Nazareth Luiz Henrique Braido
Quinta-feira, 31 de julho Walter Novaes Paulo Klinger Wilfredo Maldonado Juan Pablo Torres-Martinez
Dual and Symmetric Markets
José Fajardo, IBMEC -RJ
Data: 29 de julho, terça-feira
Horário: 16:30 – 17:05
Resumo: In this work we derive a Put-call relationship called put-call duality. To this end we introduce the Dual market concept and as a particular case we obtain necessary and sufficient
conditions for a market be symmetric, responding in this way a question raised by Carr and Chesney.
Risk and incentives with multitask
em co-autoria com Aloisio Araujo, IMPA/FGV-RJ e Humberto Moreira, FGV-RJ
Marcos Hiroyuki Tsuchida, FGV-RJ
Data: 29 de julho, terça-feira
Horário: 17:10 – 17:45
Resumo: The negative relationship between risk and incentives results from theoretical models of moral hazard, but the empirical work has not confirmed this prediction. In this paper we propose a
model with adverse selection followed by moral hazard, where the effort and the risk aversion are private information of the agent which can control the mean and the variance of profits. The
utility function of the agent may not have the single-crossing property. In the resulting contract, the relationship between risk aversion and incentives may not be monotone and the relationship
between incentives and observed variance of profits may be positive or negative.
Portfolio Selection with Stochastic Transaction Costs
Marcelo Nazareth, FGV-RJ
Data: 29 de julho, terça-feira
Horário: 17:50 – 18:25
Resumo: I develop a model of portfolio selection in continuous time where transaction costs are random. In the model, the agent faces a trade off between getting good terms of trade and holding a
well-balanced portfolio. First, I formulate the relevant control problem and prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.
Next, I present a numerical procedure to solve the equation and a proof that the numerical solution converges to the solution of the problem. The actual implementation of the procedure fully
characterizes the optimal consumer behavior in the presence of stochastic transaction costs.
General Equilibrium with Endogenous Securities and Moral Hazard
Luiz Henrique Braido, FGV-RJ
Data: 29 de julho, terça-feira
Horário: 18:30 – 19:05
Resumo: This paper studies a class of general equilibrium economies in which individuals’ endowments depend on their private effort and financial markets are endogenous. The economy is modeled as
a two-stage game. First, individuals strategically make financial-innovation decisions. Next, they play a Radner-type competitive equilibrium with the securities previously designed. Consumption
goods, portfolios, and effort levels are chosen competitively (i.e., taking prices as given). An equilibrium concept is adapted for these moral hazard economies and its existence is proven. It is
shown, through an example, how financial incompleteness would emerge endogenously due to incentive reasons.
Interest Rates in Trade Credit Markets
Walter Novaes, PUC-RJ
Data: 31 de julho, quinta-feira
Horário: 16:30 – 17:05
Resumo: There is evidence – Petersen and Rajan (1997) – that suppliers have advantage over banks in assessing their customers’ credit risk. Yet, interest rates in trade credit markets usually do
not vary with borrowers’ risk. Why? We demonstrate that the invariance of interest rates is an optimal response to suppliers’ private information. If firms’ demand for inputs is inelastic,
suppliers have no incentive to undercut uninformed banks that charge high interest rates from safe firms. In contrast, competition from banks does not let informed suppliers charge a higher
interest rate from risky firms. Hence, the invariance follows when the demand for inputs is inelastic. If, instead, the demand is elastic, suppliers have incentives to subsidize interest rates.
Indeed, if the demand is sufficiently elastic, we show that suppliers will charge no interest from all firms, as happens in the U.S. in trade credit up to 10 days.
Nash Equilibrium in Competitive Nonlinear Pricing Games with Adverse Selection
Paulo Klinger, FGV-RJ
em co-autoria com Frank Page, Universidade do Alabama em Tuscaloosa
Data: 31 de julho, quinta-feira
Horário: 17:10 – 17:45
Resumo: The main contribution of this paper is to show that for a large class of competitive nonlinear pricing games with adverse selection, the property of better-reply security is naturally
satisfied – thus, resolving the issue of existence of Nash equilibrium for a large class of competitive nonlinear pricing games.
Two essays on the estimation of the solution of the dynamic programming problem
Wilfredo Maldonado, UFF
Data: 31 de julho, quinta-feira
Horário: 17:50 – 18:25
Resumo: We present two new results on the estimation of the policy function of the dynamic programming. The first one provides an error bound of the estimated policy function using the Bellman
equation. The second one describe a new method for estimating the policy function based on a contraction map defined from the Euler equation.
Endogenous Collateral
em co-autoria com V. F. Martins-da Rocha, Paris I
Juan Pablo Torres Martinez, PUC-RJ
Data: 31 de julho, quinta-feira
Horário: 18:30 – 19:05
Resumo: We study a model with a finite number of agents, incomplete financial markets and default, in which the borrowers choose personalized collateral requirements and the lenders receive, in
case of default, anonymous collateral bundles.
When is allowed for the lenders to receive part of their rights in advance, and the borrowers can chose to pay today a percentage of their future loans, we prove the existence of equilibrium for
each fixed anonymous collateral requirement, even if agents have non-complete preferences.
Moreover, the anonymous collateral requirements can be chosen in way to guarantee a compatibility with the personalized requirements chosen by the borrowers.
To overcome the non-convexities introduced by the endogenous collateral requirements, we construct our equilibrium allocations as a limit of equilibriums in exogenous collateral economies of a
Geanakoplos and Zame (2002) type. | {"url":"https://impa.br/sobre/memoria/coloquios-brasileiros-de-matematica/24o-coloquio-brasileiro-de-matematica-2003/24o-cbm-sessoes-especiais/24o-cbm-sessoes-especiais-especial-de-economia/","timestamp":"2024-11-02T03:08:02Z","content_type":"text/html","content_length":"46447","record_id":"<urn:uuid:6263b617-c4c8-4ef2-a344-6446313ec597>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00218.warc.gz"} |
volume - DICT.TW Dictionary Taiwan
8 definitions found
vol·ume /ˈvɑljəm, (ˌ)jum/
vol·ume /ˈvɑljəm, (ˌ)jum/ 名詞
量; 容積
Vol·ume n.
1. A roll; a scroll; a written document rolled up for keeping or for use, after the manner of the ancients. [Obs.]
The papyrus, and afterward the parchment, was joined together [by the ancients] to form one sheet, and then rolled upon a staff into a volume (volumen). --Encyc. Brit.
2. Hence, a collection of printed sheets bound together, whether containing a single work, or a part of a work, or more than one work; a book; a tome; especially, that part of an extended work which
is bound up together in one cover; as, a work in four volumes.
An odd volume of a set of books bears not the value of its proportion to the set. --Franklin.
4. Anything of a rounded or swelling form resembling a roll; a turn; a convolution; a coil.
So glides some trodden serpent on the grass,
And long behind wounded volume trails. --Dryden.
Undulating billows rolling their silver volumes. --W. Irving.
4. Dimensions; compass; space occupied, as measured by cubic units, that is, cubic inches, feet, yards, etc.; mass; bulk; as, the volume of an elephant's body; a volume of gas.
5. Mus. Amount, fullness, quantity, or caliber of voice or tone.
Atomic volume, Molecular volume Chem., the ratio of the atomic and molecular weights divided respectively by the specific gravity of the substance in question.
Specific volume Physics & Chem., the quotient obtained by dividing unity by the specific gravity; the reciprocal of the specific gravity. It is equal (when the specific gravity is referred to water
at 4° C. as a standard) to the number of cubic centimeters occupied by one gram of the substance.
◄ ►
n 1: the amount of 3-dimensional space occupied by an object;
"the gas expanded to twice its original volume"
2: the property of something that is great in magnitude; "it is
cheaper to buy it in bulk"; "he received a mass of
correspondence"; "the volume of exports" [syn: bulk, mass]
3: physical objects consisting of a number of pages bound
together; "he used a large book as a doorstop" [syn: book]
4: a publication that is one of a set of several similar
publications; "the third volume was missing"; "he asked
for the 1989 volume of the Annual Review"
5: a relative amount; "mix one volume of the solution with ten
volumes of water"
6: the magnitude of sound (usually in a specified direction);
"the kids played their music at full volume" [syn: loudness,
intensity] [ant: softness] | {"url":"http://dict.tw/dict/volume","timestamp":"2024-11-14T08:14:57Z","content_type":"text/html","content_length":"26424","record_id":"<urn:uuid:56cc844e-c0da-48df-ab31-2e6e0fc4f8bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00670.warc.gz"} |
Chapter 3: Fundamental Concepts of Stoichiometry (Solution) - Insight into Chemical Engineering
Chapter 3: Fundamental Concepts of Stoichiometry (Solution)
Problem 3.1: How many grams of NH[4]Cl are there in 5 mol?
Problem 3.2: Convert 750 g CuSO[4].5H[2]O into moles. Find the equivalent mol of CuSO[4] in the crystals.
Problem 3.3: How many kilograms of CS[2] will contain 3.5 kg-atom carbon?
Problem 3.4: How many grams of carbon are present in 264 g of CO[2]?
Problem 3.5: The molecular formula of an organic compound is C[10]H[7]Br. Find the weight percentage of carbon, hydrogen, and bromine in the solid.
Problem 3.6: Find the equivalents of 3 kmol of FeCl[3].
Problem 3.7: What is the equivalent weight of Al[2](SO[4])[3]?
Problem 3.8: How many equivalents are there in 500 g KMnO[4]?
Problem 3.9: Calculate the equivalent weight of H[3]PO[4] in the reaction
Problem 3.10: A certain organic compound is found to contain 81.5 % C, 4.9 % H, and 13.6 % N by weight. If the molecular weight of the compound is 103, determine the molecular formula of the
Problem 3.11: Aluminum chloride is made by the chlorination of molten aluminum metal in a furnace:
a) How many kilograms of AlCl[3] can be made from 100 kg of chlorine? b) How many grams of Al will react with 50 g of chlorine?
Problem 3.12: Sodium hydroxide is made by the electrolysis of brine solution. The overall reaction may be written as:
(a) How much NaOH can be made from 1000 kg NaCl (b) How much water is consumed in the production of 500 kg Cl[2]?
Problem 3.13: Sulphur trioxide gas is obtained by the combustion of pyrites (FeS[2]) according to the following reaction:
The reaction is accompanied by the following side reaction:
Assume that 80 % (weight) of the pyrites charged reacts to give sulphur trioxide and 20 % reacts giving sulphur dioxide. a) How many kilograms of pyrites charged will give 100 kg of SO[3]? b) How
many kilograms of oxygen will be consumed in the reaction?
Problem 3.14: Barium chloride reacts with sodium sulfate to precipitate barium sulfate:
a) How many grams of barium chloride is needed to react with 100 g of sodium sulfate? b) For precipitating 50 g of barium sulfate, how many grams of the reactants are consumed? c) How many grams of
sodium chloride would be obtained when 50 g of barium sulfate is precipitated?
Problem 3.15: Chromite ore analyzed 30.4 % Cr[2]O[3]. Determine the theoretical amount of lead chromate (PbCrO[4]) that can be obtained from 1000 kg of the ore.
Problem 3.16: The alloy brass contains lead as an impurity in the form of lead sulfate (PbSO4). By dissolving brass in nitric acid, lead sulfate is precipitated. A sample of brass weighing 5 g is
dissolved in nitric acid and 0.03 g of precipitate is formed. Calculate the percentage of lead present in the brass sample.
Problem 3.17: How many kilograms of CO[2] are obtained by the decomposition of 100 kg of limestone containing 94.5 % CaCO[3], 4.2 % MgCO[3,] and 1.3 % inert materials? What is the volume of CO[2]
obtained at STP?
Problem 3.18: Sulphur dioxide is obtained by the following reaction
a) When 50 kg Cu dissolves in sulphuric acid what volume of sulfur dioxide is produced at standard conditions? b) How many kilograms of 94 % sulphuric acid will be required for the above reaction?
Problem 3.19: Crude calcium carbide, CaC[2], is made in an electric furnace by the following reaction.
CaO+3C\rightarrow CaC_2+CO
The product contains 85 % CaC[2] and 15 % unreacted CaO. a) How much CaO is to be added to the furnace charge for each 1000 kg CaC[2]? b) How much CaO is to be added to the furnace charge for each
1000 kg of the crude product?
Problem 3.20: A 1-kg lead ball of density 11.34⨯10^3 kg/m^3 is immersed in water. The density of water is 1000 kg/m^3. Calculate the buoyant force on the body.
Problem 3.21: A body weighs 1.0 kg in air, 0.90 kg in water, and 0.85 kg in liquid. What is the specific gravity of the liquid?
Problem 3.22: 10 kg of liquid A of specific gravity 1.2 is mixed with 3 kg of liquid B of specific gravity 0.8. Assuming that there is no volume change on mixing, what is the specific gravity of the
Problem 3.23: An alloy contains metals A and B in the ratio of 5:3. If metal A has a specific gravity of 10 and metal B has a specific gravity of 5 in the pure state, what would be the specific
gravity of the alloy?
Problem 3.24: An aqueous solution of a valuable chemical (molecular weight = 180) leaves the reactor at a rate of 60⨯10^–3 m^3/h. The solution concentration is 40% (weight) and its specific gravity
is 1.05. Determine (a) the concentration of the solution in kg/m^3 and (b) the flow rate in kmol/h.
Problem 3.25: A certain solution has a specific gravity of 1.15 at 288.8 K referred to as water at 288.8 K. Express the specific gravity as °Bé and °Tw.
Problem 3.26: What is the specific gravity on Baumé scale for a 90 °Tw solution?
Problem 3.27: The specific gravity of hydrocarbon oil is 0.88 at 288.8 K. What are the corresponding values in the Baumé and API scales?
Problem 3.28: The bulk density of a solid is 1.125 g/mL and the true density is 1.5 g/mL. What is the porosity of the solid?
Problem 3.29: 500 cubic meters of 30 °API gas oil is blended with 2000 cubic meters of 15 °API fuel oil. What is the density of the resultant mixture in kg/m^3? The density of water at 288.5 K =
0.999 g/ml. Assume no volume change on mixing.
Problem 3.30: 100 liters each of gasoline (55°API), kerosene (40°API), gas oil (31°API), and isopentane (96 °API) are mixed. The density of water at 288.5 K = 0.999 g/mL. (a) Determine the density of
the mixture in kg/m^3 (b) What is the specific gravity in °API? (c) Express the composition of the mixture in weight percent.
Problem 3.31: The specific gravity 288.5 K/288.5 K of an ammonia-water solution is 0.9180. What would be the specific gravity 288.5 K/300 K if the density of water at 288.5 K and 300 K are
respectively, 0.998 g/mL and 0.989 g/mL?
Problem 3.32: An analysis of seawater showed 2.8% NaCl, 0.5% MgCl2 and 0.0085% NaBr by weight. The average specific gravity of the water is 1.03. What mass of magnesium, sodium, and chlorine can be
obtained from 100 m3 of seawater?
Problem 3.33: What is the weight percentage of CaO in Ca(OH)[2]?
Problem 3.34: Determine the weight percentage of the constituent elements of potassium sulfate.
Problem 3.35: What is the percentage of water in Al[2](SO[4])[3].17H[2]O?
Problem 3.36: Compare the percentages of iron in ferrous chloride and ferric chloride.
Problem 3.37: An aqueous solution contains 40% by weight NaNO[3]. Determine the composition in mole percent.
Problem 3.38: How many kg of Glauber’s salt (Na[2]SO[4].10H[2]O) will be obtained from 250 kg Na[2]SO[4]?
Problem 3.39: A sample of urea (NH[2]CONH[2]) contains 42.0% nitrogen by weight. What is the percent purity of the sample?
Problem 3.40: Determine the mass fraction and mole fraction of chlorine in the substance Ca(ClO)[2].
Problem 3.41: The strength of phosphoric acid is usually represented as the weight percent of P[2]O[5]. A sample of phosphoric acid analyzed 40% P[2]O[5]. What is the percent by weight of H[3]PO[4]
in the sample?
Problem 3.42: A blast furnace treats 10^6 kg per day hematite ore which contains 50% pure ferric oxide. Determine the weight of pig iron produced per day. Pig iron contains 94% iron.
Problem 3.43: A liquid mixture contains three components A (MW = 72), B (MW = 58) and C (MW = 56) in which A and B are present in the mole ratio 1.5:1 and the weight percent of B is 25%. A sample of
the mixture is found to contain 10 kg of C. Calculate the total number of moles of the mixture.
Problem 3.44: A Portland cement sample contained 20% SiO[2] by weight derived from two silicate compounds, SiO[2].2CaO and SiO[2].3CaO that are present in the cement in the mole ratio 3:4. Determine
the percent by weight of each silicate compound in the cement.
Problem 3.45: An ethanol–water mixture forms an azeotrope at 89.43 mole percent ethanol at 101.3 kPa and 351.4 K. What is the composition of the azeotrope in weight percent?
Problem 3.46: A 20% (weight) aqueous solution of monoethanolamine (MEA, NH[2]CH[2]CH[2]OH) is used as a solvent for absorbing CO[2] from a gas stream. The solution leaving contains 0.25 mol CO[2] per
mol MEA. Determine a) the mole percent of CO[2] in the solution leaving the absorber. b) The mass percent of CO[2] in the solution.
Problem 3.47: A water-soaked cloth is dried from 45% to 9% moisture on dry basis. Find the weight of water removed from 2000 kg of dry fabric.
Problem 3.48: A solution of sodium chloride is saturated in water at 289 K. Calculate the weight of salt in kg that can be dissolved in 100 kg of this solution if it is heated to a temperature of 343
K. The solubility of sodium chloride at 289 K = 6.14 kmol/1000 kg water. The solubility at 343 K = 6.39 kmol/1000 kg of water.
Problem 3.49: The solubility of benzoic acid (C[6]H[5]COOH) is found out to be 66 parts in 100 parts by weight of ether (C[2]H[5]OC[2]H[5]). Find the mole fraction of benzoic acid in the saturated
solution with ether.
Problem 3.50: The solubility of benzoic acid (C[6]H[5]COOH) in ether (C[2]H[5]OC[2]H[5]) is found to be 28.59% (by mole). What is the solubility in weight percent? What is the weight ratio of acid to
ether in the saturated solution?
Problem 3.51: Hydrogen chloride is made by the action of sulphuric acid on sodium chloride. Hydrogen chloride being readily soluble in water forms hydrochloric acid. Calculate the following: (a) The
weight in grams of HCl formed by the action of excess sulphuric acid on 1 kg of salt which is 99.5% pure (b) The volume of hydrochloric acid solution (specific gravity 1.2) containing 40% by weight
HCl that can be produced (c) The weight in kilograms of sodium sulfate obtained?
Problem 3.52: An excess of NaNO[3] is treated with 25 kg sulphuric acid solution which contains 93.2% by weight of pure H[2]SO[4]. Calculate the following: (a) The number of kilomoles of pure nitric
acid obtained (b) The mass of nitric acid containing 70% by weight HNO[3] obtained (c) The number of kilograms of Na[2]SO[4] produced?
Problem 3.53: A liquid mixture contains three components A (MW = 72), B (MW = 58), and C (MW = 56) in which A and B are present in the mole ratio 1.5:1 and the weight percent of B is 25%. The
specific gravities of the pure liquids are 0.67, 0.60, and 0.58 respectively, for A, B, and C and there is no volume change on mixing. Calculate the following: (a) The analysis of the mixture in mole
percent (b) The molecular weight of the mixture (c) The volume percent of C on a B-free basis (d) The specific gravity of the mixture?
Problem 3.54: An alcohol–water solution contains 20% (volume) ethanol at 300 K. The densities of ethanol and water at 300 K are 0.798 g/mL and 0.998 g/mL respectively. What is the weight percent of
Problem 3.55: Calculate the concentration in mol/L of pure methanol at 298 K if the density of methanol at 298 K is 0.9842 g/mL.
Problem 3.56: A company has a contract to buy NaCl of 98 percent purity for Rs 300 per 1000 kg salt delivered. Its last shipment of 1000 kg was only of 90% purity. How much they should pay for the
Problem 3.57: A compound is found to contain 62.4% Ca and 37.6% C. (a) How many gram atoms of Ca and C are present in 100 g of the compound (b) Suggest an empirical formula for the compound?
Problem 3.58: It is desired to prepare a 40% solution of NaCl in water at 300 K. (a) How many kg of anhydrous sodium chloride should be added to 0.05 cubic meters of pure water having a density of
0.998 g/mL at 300 K (b) If the salt contains 10% water, how many kg of salt is required?
Problem 3.59: Absolute humidity of air is 0.02 kg water vapor/kg dry air. Assuming the average molecular weight of air to be 29, calculate the following: (a) The mole percent of water vapor in the
air (b) The molal absolute humidity, which is the same as the mole ratio of water vapor to dry air.
Problem 3.60: Assuming that dry air contains 21% oxygen and 79% nitrogen, calculate the following: (a) The composition in weight percent (b) The average molecular weight of dry air.
Problem 3.61: By electrolyzing brine, a mixture of gases is obtained at the cathode having the following composition by weight: chlorine 67%, bromine 28%, and oxygen 5%. Calculate the composition of
gases by volume.
Problem 3.62: Determine the weight percent of NaOH in an aqueous solution of molality 2.
Problem 3.63: Calculate the molality of a solution of 93% H[2]SO[4] (W/V). The density of the solution is 1840 kg/m^3.
Problem 3.64: A 6.9 molar solution of KOH in water contains 30% by weight of KOH. Calculate the density of the solution.
Problem 3.65: The concentration of SO[2] in the flue gases from a boiler is found to be 0.2 kg/m^3 at STP. Determine the concentration of SO[2] in parts per million by volume at STP. Assume that the
gases are perfect.
Problem 3.66: A benzene solution of an organic compound A analyses 10% of A. The molality of the solution is reported to be 0.62. Calculate the following: (a) The molecular weight of the compound (b)
The mole fraction of the compound in the solution.
Problem 3.67: An aqueous solution of NaCl contains 20% NaCl. The density of the solution is 1.16 g/mL. 500 ml water of density 1 g/mL is added to 1 liter of the solution. What will be the molality
and molarity of the resulting solution?
Problem 3.68: A solution of ZnBr[2] in water contains 130 g salt per 100 mL solution at 293 K. The specific gravity of the solution is 2.00. Calculate the following: (a) The concentration of ZnBr[2]
in mole percent (b) The concentration of ZnBr[2] in weight percent (c) The molarity (d) The molality?
Problem 3.69: The molality of an aqueous solution of LiCl in water is 10. The density of the solution is 1.16 g/mL at 350 K. Determine the following: (a) The weight percent of LiCl in the solution
(b) The molarity of the solution at 350 K (c) The normality of the solution at 350 K (d) The composition of the solution in mole percent?
Problem 3.70: The molarity of an aqueous solution of MgCl[2] at 300 K is 4.0. The specific gravity of the solution is 1.3 at 300 K. Determine the following: (a) The concentration of MgCl[2] in weight
fraction (b) The concentration of MgCl[2] in mole fraction (c) The molality of the solution (d) The normality of the solution at 300 K?
Problem 3.71: Pure water and alcohol are mixed to get a 50 % alcohol solution. The density (g/mL) of water, alcohol, and the solution may be taken to be 0.998, 0.780, and 0.914, respectively at 293
K. Calculate the following: (a) The volume percent of ethanol in the solution at 293 K (b) The molarity (c) The molality?
Problem 3.72: A solution of potassium chloride in water contains 384 g KCl per liter of the solution at 300 K. The specific gravity of the solution is 1.6. Determine the following: (a) The
concentration in weight percent (b) The mole fraction of KCl (c) The molarity of the solution (d) The molality of the solution.
Problem 3.73: Silver nitrate reacts with metallic Zn depositing silver according to the reaction
With 0.05 kg metallic Zn is added to 10^–3 m^3 of silver nitrate solution, it was found that after all silver in the solution is deposited in metallic form some Zn metal is left unreacted. The total
weight of the unreacted Zn and deposited silver was found to be 0.07 kg. Determine the following: (a) The mass of silver deposited (b) The molarity of the silver nitrate solution.
Problem 3.74: 1 kg nitrogen is mixed with 3.5 m^3 of hydrogen at 300 K and 101.3 kPa and sent to the ammonia converter. The product leaving the converter analyzed 13.7% ammonia, 70.32% hydrogen, and
15.98% nitrogen. (a) Identify the limiting reactant. (b) What is the percent excess of the excess reactant (c) What is the percent conversion of the limiting reactant?
Problem 3.75: In the chlorination of ethylene to dichloroethane, the conversion of ethylene is 99.0%. If 94 mol of dichloroethylene is produced per 100 mol of ethylene fed, calculate the overall
yield and the reactor yield based on ethylene.
C_2H_4+{Cl}_2\rightarrow C_2H_4{Cl}_2
Problem 3.76: In the manufacture of methanol by the reaction of carbon monoxide and hydrogen, some formaldehyde is also formed as a by-product.
CO+2H_2\rightarrow{CH}_3OH CO+H_2\rightarrow HCHO
A mixture consisting of CO and H[2] is allowed to react and the product analyzed 2.92% CO, 19.71% methanol, 6.57% formaldehyde, and 70.80% hydrogen. Calculate the following: (a) The percent
conversion of the limiting reactant (b) The percent excess of any reactant (b) the Percent yield of methanol.
Problem 3.77: Water vapor decomposes according to the following reaction
H_2O\rightarrow H_2+\frac12O_2
What is the mole fraction of oxygen in the reaction mixture in terms of the extent of reaction if the system contained n[0] moles of water vapor initially?
Problem 3.78: The following reaction occurs in a mixture consisting of 2 mol methane, 1 mol water, 1 mol carbon monoxide, and 4 mol hydrogen initially.
{CH}_4+H_2O\rightarrow CO+3H_2
What is the mole fraction of hydrogen in the reaction mixture in terms of the extent of the reaction?
Problem 3.79: A system consisting of 2 mol methane and 3 mol water is undergoing the following reaction:
{CH}_4+H_2O\rightarrow CO+3H_2 {CH}_4+2H_2O\rightarrow CO_2+4H_2
Derive expressions for the mole fraction of hydrogen in terms of the extent of reactions.
Problem 3.80: The following gas-phase reactions occur in a mixture initially containing 3 mol ethylene and 2 mol oxygen.
C_2H_4+\frac12O_2\rightarrow{\left(CH_2\right)}_2O C_2H_4+3O_2\rightarrow2{CO}_2+2H_2O
Derive an expression for the mole fraction of ethylene in terms of the extent of reactions.
Problem 3.81: In the vapor-phase hydration of ethylene to ethanol, diethyl ether is obtained as a by-product.
C_2H_4+H_2O\rightarrow C_2H_5OH 2C_2H_4+H_2O\rightarrow{\left(C_2H_5\right)}_2O
A feed mixture consisting of 55% ethylene, 5% inerts, and 40% water is sent to the reactor. The products analyzed were 52.26% ethylene, 5.49% ethanol, 0.16% ether, 36.81% water, and 5.28% inerts.
Calculate the conversion of ethylene, yield of ethanol, and ether based on ethylene.
Problem 3.82: Elemental phosphorous is produced from phosphate rock in an electric furnace by the following reaction:
2{Ca}_3{\left({PO}_4\right)}_2+10C+6SiO_2\rightarrow P_4+6CaSiO_3+10CO
The furnace is fed with 1000 kg phosphate. Carbon charged is 25% in excess and silica charged is 50% in excess. The reaction goes to 95% completion. The unconverted reactants along with the calcium
silicate formed constitute the slag. Calculate the following: (a) The mass of carbon and silica charged (in kilograms) (b) The amount of phosphorous obtained (in kilograms) (c) The mass of slag
produced (in kilograms)
Problem 3.83: Iron pyrites is burned in 50% excess air. The following reaction occurs:
For 100 kg of iron pyrites charged, calculate the following: (a) The amount of air supplied (in kilograms) (b) The composition of exit gases if the percent conversion of iron pyrites is 80%
Problem 3.84: Ammonia reacts with sulphuric acid giving ammonium sulfate:
(a) 20 m^3 of ammonia at 1.2 bar and 300 K reacts with 40 kg of sulphuric acid. Which is the excess reactant and what is the percent excess? (b) How much ammonium sulfate is obtained?
Problem 3.85: Sulphur dioxide reacts with oxygen producing sulfur trioxide:
In order to ensure a complete reaction, twice as much oxygen is supplied than that required theoretically. However, only 60% conversion is obtained. The pressure was 500 kPa and the temperature was
800 K. 100 kg of SO[2] is charged to the converter. Determine the following: (a) The volume of pure oxygen supplied at 1.5 bar and 300 K (b) The volume of sulfur trioxide produced (c) The volume of
gases leaving the converter (d) The composition of gases leaving the converter (e) The average molecular weight of the gas leaving the converter.
Problem 3.86: Nitrogen dioxide shows a tendency to associate and form nitrogen tetroxide.
2{NO}_2\rightarrow N_2O_4
One cubic meter of nitrogen dioxide at 100 kPa and 300 K is taken in a closed rigid container and allowed to attain equilibrium at constant temperature and volume. The pressure inside the container
has fallen to 85 kPa at equilibrium. (a) What is the degree of association? (b) What is the partial pressure of N[2]O[4] in the final mixture?
Problem 3.87: Ammonium chloride in the vapor phase dissociates into ammonia and hydrogen chloride according to
10.7 g of ammonium chloride is taken in a container. When dissociation is complete and equilibrium has attained the pressure, the volume and temperature of the gas mixture were measured to be 1.2
bar, 7.764⨯10–3 m3, and 400 K, respectively. Determine the following: (a) The fraction of ammonium chloride dissociated (b) The partial pressure of HCl in the products
Problem 3.88: A gaseous mixture consisting of 50% hydrogen and 50% acetaldehyde (C[2]H[4]O) is initially contained in a rigid vessel at a total pressure of 1.0 bar. Methanol is formed according to
C_2H_4O+H_2\rightarrow C_2H_6O
After a time, it was found that the total pressure in the vessel has fallen to 0.9 bar while the temperature was the same as that of the initial mixture. Assuming that the products are still in the
vapor phase, calculate the degree of completion of the reaction.
Problem 3.89: Ammonia is made by the reaction between hydrogen and nitrogen according to the following reaction:
(a) For the complete conversion of 100 cubic meters of nitrogen at 20 bar and 350 K, what volume of hydrogen at the same conditions of temperature and pressure is theoretically required? (b) If
hydrogen is available at 5 bar and 290 K, what is the volume required which is stoichiometrically equivalent to 100 m^3 of nitrogen at 20 bar and 350 K? (c) If the reaction is carried out at 50 bar
and 600 K, what volumes of nitrogen and hydrogen at these conditions are theoretically required for producing 1000 kg ammonia and what will be the volume of ammonia produced at the reactor
Problem 3.90: Carbon dioxide dissociates into carbon monoxide and oxygen at 1 bar and 3500 K.
{CO}_2\rightarrow CO+1/2O_2
25 L of CO[2] at 1 bar and 300 K is heated to 3500 K at constant pressure. If all gases behave ideally, determine the following: (a) The final volume of the gas if no dissociation is occurred (b) The
fraction of CO2 is dissociated if the final volume is found to be 0.35 m^3. | {"url":"https://chelearning.com/fundamental-concepts-of-stoichiometry-solution/","timestamp":"2024-11-02T10:46:39Z","content_type":"text/html","content_length":"195132","record_id":"<urn:uuid:7e13952d-86eb-441d-adf2-449031519e11>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00508.warc.gz"} |
Function gettransresidualrms()
Calculation of the "Root Mean Square Residual" (RMS) from a group of
Prototype of the DLL function in C++ syntax (attend lower case!):
extern "C" __declspec(dllimport) unsigned long __stdcall gettransresidualrms(
double aResid[][3],
unsigned long nCount,
double *fResidRms);
Prototype of the DLL function in Visual Objects syntax:
_DLL FUNCTION gettransresidualrms(;
aResid as real8 ptr,; // 4 Byte
nCount as dword,; // 4 Byte
fResidRms ref real8); // 4 Byte
AS logic pascal:geodll32.gettransresidualrms // 4 Byte
The function calculates the "Root Mean Square Residual" (RMS) from an array with
previously determined residuals. This value is a quality criterion for the
accuracy of the Helmert or Molodensky parameter set, which has been used
previously to calculate the residuals (see function gettransresiduals()).
Since the function due to the extensive calculations is time-consuming,
the event handling during the calculation by interrupting the processing loop
can be are allowed bei calling the function seteventloop().
The parameters are passed and/or returned as follows:
aResid[][3] Previously calculated residuals with their X, Y and Z components in
(ref) meter, stored in a two dimensional array of type double. The
structure of the array is described further below.
nCount Count of the available residuals in the array aResid.
fResidRms From the residuals array calculated value for the"Root Mean Square
(ref) Residual" (RMS).
returnVal In case of an error the function returns FALSE, otherwise TRUE.
The two dimensional array aResid[][3] is filled with values of type double and
is structured as follows:
| R1-X | R1-Y | R1-Z | R2-X | R2-Y | R2-Z | ... | Rn-X | Rn-Y | Rn-Z |
with R1 -› Rn: Residuals 1 to n
X: X component
Y: Y component
Z: Z component
This function is a component of the unlock requiring function group
"Transformation parameter". It is unlocked for unrestricted use together
with other functions of the group by passing the unlock parameters, acquired
from the software distribution company, trough the function setunlockcode().
Without unlocking at most 25 residuals can be processed. | {"url":"https://www.killetsoft.de/h_geodll_e/funkgettransresidualrms.htm","timestamp":"2024-11-13T19:41:52Z","content_type":"text/html","content_length":"5536","record_id":"<urn:uuid:997fe6c6-1a2d-4a61-a1a4-34477529fbd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00279.warc.gz"} |
The Spherical Cylinder
The spherical cylinder, or spherinder for short, is constructed by extruding a 3D sphere along the W-axis for a unit distance. The resulting shape is the best 4D analog of the 3D cylinder, having two
spherical “lids” connected by an extended spherical volume. The following diagram shows a parallel projection of a spherinder.
Unfortunately, because of the smoothness of the 3D spheres, a wire-diagram such as this one doesn't adequately show the internal structure of the spherinder. The following perspective projection
diagram tries to better capture the structure of the center section of the spherinder:
This diagram requires explanation. The outer red sphere is the near end of the spherinder, and the inner blue sphere is the far end. The two intermediate spheres are cross sections of the 3D surface
that connects these two ends. Although they appear as concentric spheres in the projection, they are actually disjoint spheres of the same size, displaced at different distances along the W-axis. One
can imagine this diagram as being analogous to looking through one end of a 3D cylinder and seeing the circular lid at the other end. The cross sections of the circular tubing in between would appear
as concentric circles of different sizes.
The next diagram shows the spherinder at an angle in a perspective projection, with its far end rotated part way toward the horizontal X-axis.
Here, we see that the red spherical end of the spherinder has been partially flattened into an ellipsoid. The blue end has “emerged” outside the red end, and has also been partly flattened into an
ellipsoid. The uneven sizes indicate that the long axis of the spherinder is still partly rotated into the W-axis. Of course, the spherical ends of the spherinder aren't actually being flattened;
they only appear that way because they are seen at an angle.
The last diagram below shows the spherinder rotated so that its long axis fully coincides with the X-axis.
As the reader can see, the dotted equatorial curves of the two spherical ends have flattened into intersecting straight lines. The two spherical ends have completely flattened into circles. Thus this
projection of the spherinder is a 3D cylinder. Of course, in actuality the two ends are still as spherical as before; they appear as circles because they are seen at a 90° angle.
The spherinder can cover a 2D area by rolling because every section perpendicular to its long axis is a 3D sphere. These constituent spheres can roll synchronously in two perpendicular directions. | {"url":"http://www.qfbox.info/4d/spherinder","timestamp":"2024-11-07T22:26:17Z","content_type":"text/html","content_length":"6408","record_id":"<urn:uuid:2091b8bc-ab90-4d39-85d8-b5301f3e03ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00662.warc.gz"} |
A brief discussion about texturemapper innerloops using ADDX
Written by Mikael Kalms ([email protected], Scout/C-Lous^Artwork^Appendix)
HTML-Version by Azure
Note: This is no basic-introduction to texturemapping - its just an article on optimizing innerloops.
If you have U and V (the two texture coordinates) as 8.8 fixed point, then you can have:
; d0 U ----UUuu (UU == integer bits, uu = fractional)
; d1 V ----VVvv
; d2 dU/dx ----UUuu "horizontal slope of U"
; d3 dV/dx ----VVvv
move.w d1,d4 ; d4 (offset reg) = VVxx
move.w d0,d5 ; Temporarily...
lsr.w #8,d5 ; ...to get UU in lower byte of a reg
move.b d5,d4 ; d4 = VVUU = correct offset
move.b (a0,d4.w),(a1)+
add.w d2,d0
add.w d3,d1
dbf d7,.pixel
There is however a very nifty instruction which we can make use of here: addx.
Addx is intended to be used for adding arbitrarily large integers. What it does, is that it adds together the two operands, and if X flag is set then it also adds 1 to the result.
The X flag will be set by the previous add/addx, so the X flag is what "binds together" the adds. (It carries the overflow from a lower part of the large number to the next.)
Imagine that you want to add the 96-bit numebr in d2:d1:d0 to another 96-bit number in d5:d4:d3.
Then you would proceed like this:
add.l d0,d3 ; First add lowermost part
addx.l d1,d4 ; .. then the next -- with X flag
addx.l d2,d5 ; .. and finally the highest (with X)
[There is a special form of addx: "addx -(am),-(an)" It is intended for use when the two numbers are lying around in memory. Quite useful for handling arbitrarily large numbers, or just numbers of
different sizes. (There's no use for this in realtime applications though :))]
This can be very valuable when dealing with fixed-point too, though;
since you can split
add.w d1,d0
add.b d1,d0
addx.b d3,d2
... then you would be able to get at the upper 8 bits of an 8.8 fixed point number without shifting!
There's however one more thing to realize before we try implementing this.
One can perform two word-additions using only one add.l by having the values packed into two registers; there will sometimes be an overflow from the lower part to the upper part of the answer, but
that overflow is always 1 so it is in some applications negligible.
Consider this:
; d0 00aa00bb
; d1 00cc00dd
add.l d1,d0
... then d0 will be = (aa + cc) shifted up 16 + (bb + dd); it will take 256 times of repeating until there comes any overflow into the (aa+cc) calculation.
Now we look at our tmapper's stepping:
add.w d1,d0 ; Interpolate V
add.b d3,d2 ; Interpolate U fraction
addx.b d5,d4 ; Interpolate U integer
... and here we [after some thinking :)] realize that we can make the "add.b d3,d2" in the uppermost bytes of d0 and d1 instead:
; d0 V **--VVvv (this is how I denote different contents
; d0 U uu--**** in the same register)
; d1 dV/dx **--VVvv
; d1 dU/dx uu--****
; d2 U ------UU
; d3 dU/dx ------UU
move.w d0,d4 ; d4 = VVvv
move.b d2,d4 ; d4 = VVUU
move.b (a0,d4.w),(a5)+
add.l d1,d0
addx.b d3,d2
dbf d7,.pixel
now that's pretty short. :) In fact this is what is commonly referred to as "a 5inst tmapper" (5 instructions not counting the dbf). There is no obvious way of speeding this up, it has looked like
this since 1994 at least.
Notice what the add/addx thing can be thought of to look like:
d2:d0 + d3:d1 = UU:uu--VVvv + UU:uu--VVvv
This shows that the add/addx is nothing but two adds packed into one (but the "one" add happens to be executed through two instructions) -- we are therefore not doing anything excessively weird yet.
The error that gets carried from the VVvv parts into the uu part is, if you initialize the '--' parts to 00, max 1 u per 256 iterations = 1 U per 65536 iterations. That error is highly negligible!
Even when skipping the 00izing of '--' the max error is 1 U per 256 iterations, so it is only if you happen to be drawing very large polygons that you'll need to init those bits.
But -- perhaps you want more accuracy? 16 fractional bits? It is usually not necessary on the Amiga in 320x256 resolution, but it is good to know that it is possible.
Let us first begin with the problem of only interpolating one value:
; d0 C --CCcccc (C for colour, might be for a gouraud rout)
; d1 dC/dx --CCcccc
usually one would do:
add.l d1,d0
move.l d0,d2
swap d2
... and then use d2.
That can be changed into:
add.w d1,d0
addx.b d3,d2
... however, look at this code if we unroll (repeat) it several times:
add.w d1,d0
addx.b d3,d2
add.w d1,d0
addx.b d3,d2
add.w d1,d0
addx.b d3,d2
Notice that after the first "addx.b d3,d2", the subsequent "add.w d1,d0" could be done at the top of d2/d3 in the addx! (remember that an addx is just like an add -- plus the X flag)
Therefore, or code could also look like this:
; d0 C ----cccc
; d1 dC/dx ----cccc
; d2 C cccc--CC
; d3 dC/dx cccc--CC
add.w d1,d0
addx.l d3,d2
addx.l d3,d2
addx.l d3,d2
and just to get rid of the need for d0/d1 at the beginning:
move.l d3,d4
clr.w d4
add.l d4,d2 ; Step only fractional part -- "init"
addx.l d3,d2
addx.l d3,d2
addx.l d3,d2
(actually one should end this chain with "addx.w", but that only matters if one needs the result value at the end of the operation -- normally one doesn't, at least not in tmappers.)
This is a very interesting approach because it creates something which I like to call a "cyclic add", which so-to-say never ends.
Building this with two values (16.16) could look like this:
; d0 bbbbAAAA
; d1 aaaaBBBB
; d2 & d3 same, but d?/dx
move.l d3,d4
clr.w d4
add.l d4,d1 ; start the chain (adding last fractional
; part)
addx.l d2,d0
addx.l d3,d1
; iteration 1 done
addx.l d2,d0
addx.l d3,d1
; iteration 2 done
What kind of setup gives us the look of d1:d0?
Check them out as "normal" and as "finished" in 64bit format:
d1:d0 "normal" = BBBBbbbb:AAAAaaaa
d1:d0 "2addx" = aaaaBBBB:bbbbAAAA
... which means, that the setup operation was -- in theory, of course -- "ror.q #16,d1:d0".
This shows us why we should add the uppermost word of the 64bit value first (the move.l/clr.w/add.l init): Because that's where the lowest bits of the original value are.
Now let us finally implement this into a texturemapper:
; d0 U **----UU
; d0 V vv----**
; d1 dU/dx **----UU
; d1 dV/dx vv----**
; d2 U uuuu****
; d2 V ****VVvv
; d3 dU/dx uuuu****
; d3 dV/dx ****VVvv
... and the code:
move.l d3,d4
clr.w d4
add.l d4,d2 ; Start the X-flag in the "cyclic add"
move.w d2,d4 ; d4 = VVvv
move.b d0,d4 ; d4 = VVUU
move.b (a0,d4.w),(a1)+
addx.l d1,d0
addx.l d3,d2
dbf d7,.pixel
These loops are nice and fast on 020/030, sure, but how about 040/060?
There the instructions should be re-ordered a bit to remove stalls. (The move.w/move.b causes a 0.5 cycle stall on 060, and the closeness between move.b and pixel-copying move.b causes 1 cycle stall
on 040 and 060.)
Reorder the 8.8 loop to this:
move.w d0,d4 ; d4 = VVvv
add.l d1,d0
move.b d2,d4 ; d4 = VVUU
addx.b d3,d2
move.b (a0,d4.w),(a5)+
dbf d7,.pixel
And the 16.16 loop to this:
move.w d2,d4 ; d4 = VVvv
addx.l d1,d0
move.b d0,d4 ; d4 = VVUU
addx.l d3,d2
move.b (a0,d4.w),(a1)+
dbf d7,.pixel
The above loops can be sped up a tiny bit more, but any more optimization is left as an exercise for the readers. ;)
Oh, and you might want to keep the upper word of d4 cleared (through a "moveq #0,d4" before the pixel-loop), since you then can use d4.l as offset into the texture and a0 thus doesn't need to point
to the middle of the texture (if you have size 256x256 textures). It all depends on the circumstances though... | {"url":"https://amycoders.org/opt/innerloops.html","timestamp":"2024-11-15T04:30:25Z","content_type":"text/html","content_length":"10867","record_id":"<urn:uuid:1d2953c2-6dc6-404d-a128-9d9b592e8561>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00606.warc.gz"} |
real part and imaginary part worksheet answers
we are given: f(t)=4 e^(jWt) + 3 e^(2jWt) in volts I have the answer for this question: [Real part of f(t)]^2 =25 [Imaginary part of f(t)]^2 =24cos(Wt) how did we get these 2 values? Complex numbers
are written in the form a + bi, where a is called the real term and the coefficient of i is the imaginary part. endobj endobj 'Positive' and 'Negative' are defined only on the real number line, which
is part of the system of complex numbers. 11 0 obj<> Real part 2 , imaginary part -5i Complex numbers written like this are in the rectangular form. Complex number = (Real Part) + (Imaginary Part) i
The set of complex numbers is denoted by C. endobj Learn more about matlab, symbolic Symbolic Math Toolbox, MATLAB Fetal Pig Dissection Thoracic Cavity The Circulatory System. Approach: A complex
number can be represented as Z = x + yi, where x is real part and y is imaginary. Answer with 5 significant digits. Printable Worksheets @ www.mathworksheets4kids.com Name : Answer key Real Part and
Imaginary Part Sheet 1 A) Complete the table. But I don't know how to do it s logarithm 12 0 obj<> Hence the set of real numbers, denoted R, is a subset of the set of complex numbers, denoted C.
Adding and subtracting complex numbers is similar to adding and subtracting like terms. Complex Numbers. To download/print, click on pop-out icon or print icon to worksheet to print or download.
*Response times vary by subject and question complexity. The numbers that have parts in them an imaginary part and a real part are what we term as complex numbers. 18 0 obj<> This worksheet is
designed to give students practice at imaginary number operations, specifically adding, subtracting and multiplying complex numbers. endobj x�}T}PSW 1�˳�@y�Bݒ���H��X�Y[E[���֖� $`�G�%&Z
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�l���նV:�R�i��s�����Z�xHpؒ�@k@���Z���� 17 0 obj<> 29 scaffolded questions that start relatively easy and end with some real challenges. A complex number is a number with a Real part, a, and an
imaginary part, bi written in the form I. Complex numbers is vital in high school math. -7+8i the real part : : - 7 (my answer) the imaginary part : ...” in Mathematics if you're in doubt about the
correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. What is the real part and imaginary part of the original load impedance
(Zl)? Free worksheet(pdf) and answer key on Complex Numbers. We will follow the below steps to separate out real and imaginary part. Imaginary And Complex Numbers - Displaying top 8 worksheets found
for this concept.. ]:��,�=}�c���_�. I need the real and imaginary part of $\log \sin (x+iy)$. Median response time is 34 minutes and may be longer for new subjects. With this quiz, you can test your
knowledge of imaginary numbers. Plus model problems explained step by step Worksheet will open in a new window. ... Identifying Real And Imaginary Part Complex Numbers Number Worksheets Algebra . 1 0
obj<> A complex number can be divided into two parts, the imaginary part, and the real part. In the complex number {eq}z = a + bi {/eq}, a is the real part, and b is the imaginary part. %PDF-1.3
stream x�c`� 13 0 obj<> You can & download or print using the browser document reader options. 28 Operations With Complex Numbers Worksheet In 2020 Motivational Interviewing Word Problem Worksheets
Number Worksheets . 45 question end of unit review sheet on adding, subtracting, mulitplying, dividing and simplify complex and imaginary numbers. Although arbitrary, there is also some sense of a
positive and negative imaginary numbers. %���� stream Find an answer to your question “Check my answer? Answer with 5 significant digits. This frees you up to go around and tuto stream View
worksheet. There are scrambled answers at the bottom so students can check their work as they go through this worksheet. Free worksheet(pdf) and answer key on Simplifying Imaginary numbers (radicals)
and powers of i. B) Form the complex numbers with the given real parts and imaginary parts. x���� �1 Answers for math worksheets, quiz, homework, and lessons. Actually, imaginary numbers are used
quite frequently in engineering and physics, such as an alternating current in electrical engineering, whic… ���9��8E̵㻶�� �mT˞�I�іT�R;hv��Or�L{wz�Q!�f�e��`�% A�� ժp�K�g��/D��=8�a�X3��[� |:��I�
�MrXB��\��#L)�fΡbè���r�쪬ى��eW��|x���!c���9P+��d3�g ��d���R��U^ g?:�@�6H�Yx�z�8Nī~J�u�S]��T���캶�Q�&�u����Uu�S7�����T���WA+�����H�!�! For example, the real number 5 is also a complex number because
it can be written as 5 + 0 i with a real part of 5 and an imaginary part of 0. Think of imaginary numbers as numbers that are typically used in mathematical computations to get to/from “real” numbers
(because they are more easily used in advanced computations), but really don’t exist in life as we know it. Learn more about complex number, real part, imaginary part, matlab I expand $\sin(x+iy)=\
sin x \cosh y+i \cos x \sinh y$. ... Answer Key. 16 0 obj<> endobj If we have 0 + bi we have a pure imaginary number. Real and Imaginary part of this function please. Yet they are real in the sense
that they do exist and can be explained quite easily in terms of math as the square root of a negative number. For my system of equations, the procedure described in Solving complex equations of
using Reduce works no more. 20 0 obj<> Some of the worksheets for this concept are Operations with complex numbers, Complex numbers and powers of i, Dividing complex numbers, Adding and subtracting
complex numbers, Real part and imaginary part 1 a complete the, Complex numbers, Complex numbers, Properties of complex numbers. Given the setup, calculate the magnitude of the equivalent source
voltage (Vs,eq) seen by Zm? endstream 5 (c) The real part of is and the imaginary part is I. (b) The real part of (8 + 8i)(8 - 81) is 128 and the imaginary part is o (Type integers or simplified
fractions.) This one-page worksheet contains 12 multi-step problems. endobj endobj Imaginary And Complex Numbers - Displaying top 8 worksheets found for this concept. (d) The real part of 1-si is and
the imaginary part is (Type integers or simplified fractions.) Free worksheet pdf and answer key on simplifying imaginary numbers radicals and powers of i. endobj If we have a + bi a != 0, b != 0
then we have a complex number with a real part and an imaginary part. endobj Since the real part, the imaginary part, and the indeterminate i in a complex number are all considered as numbers in
themselves, two complex numbers, given as z = x + yi and w = u + vi are multiplied under the rules of the distributive property, the commutative properties and the defining property i 2 = … endstream
Because then I could use Solve[equations, vars, Reals].Nevertheless I hope for a simpler way to overcome this issue. This Worksheet Section 1.5 Analysis: Real and Imaginary Numbers Worksheet is
suitable for 10th - 12th Grade. Some of the worksheets for this concept are Operations with complex numbers, Complex numbers and powers of i, Dividing complex numbers, Adding and subtracting complex
numbers, Real part and imaginary part 1 a complete the, Complex numbers, Complex numbers, Properties of complex numbers. endobj Answer with 5 significant digits. (Part (2+3 i) + (-4+5i — —49 ((5+14i)
-(10- -25 (5+4i)- (-1-21 -3i (-5i ) 2i (3i2) C-2)C3) 3i(2i) 64 Start Here 2(3+2i) 2i-(3+2j 3i (2+3i) 18 3 0 obj<> endobj what are the real and imaginary parts of the complex number? real and
imaginary part of complex number . Solution for i) Find real and imaginary part of complex number (V3 + i)° by using De Moivre's Theorem. Perform operations like addition, subtraction and
multiplication on complex numbers, write the complex numbers in standard form, identify the real and imaginary parts, find the conjugate, graph complex numbers, rationalize the denominator, find the
absolute value, modulus, and argument in this collection of printable complex number worksheets. Output: Real part: 6, Imaginary part: 8 Recommended: Please try your approach on first, before moving
on to the solution. Found worksheet you are looking for? 14 0 obj<> The general definition is a + bi Where a and b are real numbers and i is the imaginary unit i = sqrt(-1) If we have a + 0i we have
a real number. 4- i (Type integers or simplified fractions.) In this real and imaginary numbers worksheet, students solve algebraic expressions containing real and imaginary numbers. Real and
imaginary part of (x+iy)e^{ix−y} ? Any number that is written with ‘iota’ is an imaginary number, these are negative numbers in a radical. 15 0 obj<> Real (Part 81 25 (6+2 (1-2i) End Here Complete
the maze by simplifying each expression, shade the squares that Imaginary contain imaginary numbers, and following the path of complex numbers. Model Problems In this example we will simplifying
imaginary numbers. or what is the real and imaginary part of f(t) if you can get it in another form but it should be simplified as the above answers please show me your way thank you endobj 2 0 obj<>
Let’s explore this topic with our easy-to-use complex number worksheets that are tailor-made for students in high school and is the perfect resource to introduce this new concept. Imaginary numbers
of the form bi are numbers that when squared result in a negative number. Some examples are 3 +4i, 2— 5i, —6 +0i, 0— i. We can also graph these numbers. Dec 13, 2018 - Complex number worksheets
feature standard form, identifying real and imaginary part, rationalize the denominator, graphing, conjugate, modulus and more! 19 0 obj<> How can I separate the real and imaginary part of the
equations? Consider the complex number 4 + 3i: 4 is called the real part, 3 is called the imaginary part. So it makes sense to say, for example $1 -100i$ is positive and $-1 + 100i$ is negative,
based upon their real number values. Learn more about simplify, complex function, real and imaginary parts … Note: 3i is not the imaginary part. A complex number has two parts, a real part and an
imaginary part. Download/Print, click on pop-out icon or print using the browser document reader options 4 is called the part... Download/Print, click on pop-out icon or print using the browser
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Randomized algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the
hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables.
One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for
example Quicksort^[1]), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem^[2]) or fail to produce
a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem.^[3]
In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits; such an implementation may deviate from the expected
theoretical behavior.
As a motivating example, consider the problem of finding an ‘a’ in an array of n elements.
Input: An array of n≥2 elements, in which half are ‘a’s and the other half are ‘b’s.
Output: Find an ‘a’ in the array.
We give two versions of the algorithm, one Las Vegas algorithm and one Monte Carlo algorithm.
Las Vegas algorithm:
findingA_LV(array A, n)
Randomly select one element out of n elements.
until 'a' is found
This algorithm succeeds with probability 1. The number of iterations varies and can be arbitrarily large, but the expected number of iterations is
${\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}{\frac {i}{2^{i}}}=2}$
Since it is constant the expected run time over many calls is ${\displaystyle \Theta (1)}$. (See Big O notation)
Monte Carlo algorithm:
findingA_MC(array A, n, k)
Randomly select one element out of n elements.
i = i + 1
until i=k or 'a' is found
If an ‘a’ is found, the algorithm succeeds, else the algorithm fails. After k iterations, the probability of finding an ‘a’ is:
${\displaystyle \Pr[\mathrm {find~a} ]=1-(1/2)^{k}}$
This algorithm does not guarantee success, but the run time is bounded. The number of iterations is always less than or equal to k. Taking k to be constant the run time (expected and absolute) is ${\
displaystyle \Theta (1)}$.
Randomized algorithms are particularly useful when faced with a malicious "adversary" or attacker who deliberately tries to feed a bad input to the algorithm (see worst-case complexity and
competitive analysis (online algorithm)) such as in the Prisoner's dilemma. It is for this reason that randomness is ubiquitous in cryptography. In cryptographic applications, pseudo-random numbers
cannot be used, since the adversary can predict them, making the algorithm effectively deterministic. Therefore, either a source of truly random numbers or a cryptographically secure pseudo-random
number generator is required. Another area in which randomness is inherent is quantum computing.
In the example above, the Las Vegas algorithm always outputs the correct answer, but its running time is a random variable. The Monte Carlo algorithm (related to the Monte Carlo method for
simulation) is guaranteed to complete in an amount of time that can be bounded by a function the input size and its parameter k, but allows a small probability of error. Observe that any Las Vegas
algorithm can be converted into a Monte Carlo algorithm (via Markov's inequality), by having it output an arbitrary, possibly incorrect answer if it fails to complete within a specified time.
Conversely, if an efficient verification procedure exists to check whether an answer is correct, then a Monte Carlo algorithm can be converted into a Las Vegas algorithm by running the Monte Carlo
algorithm repeatedly till a correct answer is obtained.
Computational complexity
Computational complexity theory models randomized algorithms as probabilistic Turing machines. Both Las Vegas and Monte Carlo algorithms are considered, and several complexity classes are studied.
The most basic randomized complexity class is RP, which is the class of decision problems for which there is an efficient (polynomial time) randomized algorithm (or probabilistic Turing machine)
which recognizes NO-instances with absolute certainty and recognizes YES-instances with a probability of at least 1/2. The complement class for RP is co-RP. Problem classes having (possibly
nonterminating) algorithms with polynomial time average case running time whose output is always correct are said to be in ZPP.
The class of problems for which both YES and NO-instances are allowed to be identified with some error is called BPP. This class acts as the randomized equivalent of P, i.e. BPP represents the class
of efficient randomized algorithms.
Historically, the first randomized algorithm was a method developed by Michael O. Rabin for the closest pair problem in computational geometry.^[4] The study of randomized algorithms was spurred by
the 1977 discovery of a randomized primality test (i.e., determining the primality of a number) by Robert M. Solovay and Volker Strassen. Soon afterwards Michael O. Rabin demonstrated that the 1976
Miller's primality test can be turned into a randomized algorithm. At that time, no practical deterministic algorithm for primality was known.
The Miller–Rabin primality test relies on a binary relation between two positive integers k and n that can be expressed by saying that k "is a witness to the compositeness of" n. It can be shown that
• If there is a witness to the compositeness of n, then n is composite (i.e., n is not prime), and
• If n is composite then at least three-fourths of the natural numbers less than n are witnesses to its compositeness, and
• There is a fast algorithm that, given k and n, ascertains whether k is a witness to the compositeness of n.
Observe that this implies that the primality problem is in Co-RP.
If one randomly chooses 100 numbers less than a composite number n, then the probability of failing to find such a "witness" is (1/4)^100 so that for most practical purposes, this is a good primality
test. If n is big, there may be no other test that is practical. The probability of error can be reduced to an arbitrary degree by performing enough independent tests.
Therefore, in practice, there is no penalty associated with accepting a small probability of error, since with a little care the probability of error can be made astronomically small. Indeed, even
though a deterministic polynomial-time primality test has since been found (see AKS primality test), it has not replaced the older probabilistic tests in cryptographic software nor is it expected to
do so for the foreseeable future.
Quicksort is a familiar, commonly used algorithm in which randomness can be useful. Many deterministic versions of this algorithm require O(n^2) time to sort n numbers for some well-defined class of
degenerate inputs (such as an already sorted array), with the specific class of inputs that generate this behavior defined by the protocol for pivot selection. However, if the algorithm selects pivot
elements uniformly at random, it has a provably high probability of finishing in O(n log n) time regardless of the characteristics of the input.
Randomized incremental constructions in geometry
In computational geometry, a standard technique to build a structure like a convex hull or Delaunay triangulation is to randomly permute the input points and then insert them one by one into the
existing structure. The randomization ensures that the expected number of changes to the structure caused by an insertion is small, and so the expected running time of the algorithm can be bounded
from above. This technique is known as randomized incremental construction.^[5]
Min cut
Input: A graph G(V,E)
Output: A cut partitioning the vertices into L and R, with the minimum number of edges between L and R.
Recall that the contraction of two nodes, u and v, in a (multi-)graph yields a new node u ' with edges that are the union of the edges incident on either u or v, except from any edge(s) connecting u
and v. Figure 1 gives an example of contraction of vertex A and B. After contraction, the resulting graph may have parallel edges, but contains no self loops.
Karger's^[6] basic algorithm:
i = 1
Take a random edge (u,v) ∈ E in G
replace u and v with the contraction u'
until only 2 nodes remain
obtain the corresponding cut result C[i]
i = i + 1
until i = m
output the minimum cut among C[1], C[2], ..., C[m].
In each execution of the outer loop, the algorithm repeats the inner loop until only 2 nodes remain, the corresponding cut is obtained. The run time of one execution is ${\displaystyle O(n)}$, and n
denotes the number of vertices. After m times executions of the outer loop, we output the minimum cut among all the results. The figure 2 gives an example of one execution of the algorithm. After
execution, we get a cut of size 3.
Lemma 1: Let k be the min cut size, and let C = {e[1], e[2], ..., e[k]} be the min cut. If, during iteration i, no edge e ∈ C is selected for contraction, then C[i] = C.
Proof: If G is not connected, then G can be partitioned into L and R without any edge between them. So the min cut in a disconnected graph is 0. Now, assume G is connected. Let V=L∪R be the partition
of V induced by C : C = { {u,v} ∈ E : u ∈ L,v ∈ R } (well-defined since G is connected). Consider an edge {u,v} of C. Initially, u,v are distinct vertices. As long as we pick an edge ${\displaystyle
feq e}$, u and v do not get merged. Thus, at the end of the algorithm, we have two compound nodes covering the entire graph, one consisting of the vertices of L and the other consisting of the
vertices of R. As in figure 2, the size of min cut is 1, and C = {(A,B)}. If we don't select (A,B) for contraction, we can get the min cut.
Lemma 2: If G is a multigraph with p vertices and whose min cut has size k, then G has at least pk/2 edges.
Proof: Because the min cut is k, every vertex v must satisfy degree(v) ≥ k. Therefore, the sum of the degree is at least pk. But it is well known that the sum of vertex degrees equals 2|E|. The lemma
Analysis of algorithm
The probability that the algorithm succeeds is 1 − the probability that all attempts fail. By independence, the probability that all attempts fail is
By lemma 1, the probability that C[i] = C is the probability that no edge of C is selected during iteration i. Consider the inner loop and let G[j] denote the graph after j edge contractions, where j
∈ {0, 1, …, n − 3}. G[j] has n − j vertices. We use the chain rule of conditional possibilities. The probability that the edge chosen at iteration j is not in C, given that no edge of C has been
chosen before, is ${\displaystyle 1-{\frac {k}{|E(G_{j})|}}}$. Note that G[j] still has min cut of size k, so by Lemma 2, it still has at least ${\displaystyle {\frac {(n-j)k}{2}}}$ edges.
Thus, ${\displaystyle 1-{\frac {k}{|E(G_{j})|}}\geq 1-{\frac {2}{n-j}}={\frac {n-j-2}{n-j}}}$.
So by the chain rule, the probability of finding the min cut C is
Cancellation gives ${\displaystyle \Pr[C_{i}=C]\geq {\frac {2}{n(n-1)}}}$. Thus the probability that the algorithm succeeds is at least ${\displaystyle 1-\left(1-{\frac {2}{n(n-1)}}\right)^{m}}$. For
${\displaystyle m={\frac {n(n-1)}{2}}\ln n}$, this is equivalent to ${\displaystyle 1-{\frac {1}{n}}}$. The algorithm finds the min cut with probability ${\displaystyle 1-{\frac {1}{n}}}$, in time $
{\displaystyle O(mn)=O(n^{3}\log n)}$.
Randomness can be viewed as a resource, like space and time. Derandomization is then the process of removing randomness (or using as little of it as possible). It is not currently known if all
algorithms can be derandomized without significantly increasing their running time. For instance, in computational complexity, it is unknown whether P = BPP, i.e., we do not know whether we can take
an arbitrary randomized algorithm that runs in polynomial time with a small error probability and derandomize it to run in polynomial time without using randomness.
There are specific methods that can be employed to derandomize particular randomized algorithms:
• the method of conditional probabilities, and its generalization, pessimistic estimators
• discrepancy theory (which is used to derandomize geometric algorithms)
• the exploitation of limited independence in the random variables used by the algorithm, such as the pairwise independence used in universal hashing
• the use of expander graphs (or dispersers in general) to amplify a limited amount of initial randomness (this last approach is also referred to as generating pseudorandom bits from a random
source, and leads to the related topic of pseudorandomness)
Where randomness helps
When the model of computation is restricted to Turing machines, it is currently an open question whether the ability to make random choices allows some problems to be solved in polynomial time that
cannot be solved in polynomial time without this ability; this is the question of whether P = BPP. However, in other contexts, there are specific examples of problems where randomization yields
strict improvements.
• Based on the initial motivating example: given an exponentially long string of 2^k characters, half a's and half b's, a random-access machine requires 2^k−1 lookups in the worst-case to find the
index of an a; if it is permitted to make random choices, it can solve this problem in an expected polynomial number of lookups.
• The natural way of carrying out a numerical computation in embedded systems or cyber-physical systems is to provide a result that approximates the correct one with high probability (or Probably
Approximately Correct Computation (PACC)). The hard problem associated with the evaluation of the discrepancy loss between the approximated and the correct computation can be effectively
addressed by resorting to randomization^[7]
• In communication complexity, the equality of two strings can be verified to some reliability using ${\displaystyle \log n}$ bits of communication with a randomized protocol. Any deterministic
protocol requires ${\displaystyle \Theta (n)}$ bits if defending against a strong opponent.^[8]
• The volume of a convex body can be estimated by a randomized algorithm to arbitrary precision in polynomial time.^[9]Bárány and Füredi showed that no deterministic algorithm can do the same.^[10]
This is true unconditionally, i.e. without relying on any complexity-theoretic assumptions, assuming the convex body can be queried only as a black box.
• A more complexity-theoretic example of a place where randomness appears to help is the class IP. IP consists of all languages that can be accepted (with high probability) by a polynomially long
interaction between an all-powerful prover and a verifier that implements a BPP algorithm. IP = PSPACE.^[11] However, if it is required that the verifier be deterministic, then IP = NP.
• In a chemical reaction network (a finite set of reactions like A+B → 2C + D operating on a finite number of molecules), the ability to ever reach a given target state from an initial state is
decidable, while even approximating the probability of ever reaching a given target state (using the standard concentration-based probability for which reaction will occur next) is undecidable.
More specifically, a limited Turing machine can be simulated with arbitrarily high probability of running correctly for all time, only if a random chemical reaction network is used. With a simple
nondeterministic chemical reaction network (any possible reaction can happen next), the computational power is limited to primitive recursive functions.^[12]
See also
• Probabilistic analysis of algorithms
• Atlantic City algorithm
• Bogosort
• Principle of deferred decision
• Randomized algorithms as zero-sum games
• Probabilistic roadmap
• HyperLogLog
• count–min sketch
• approximate counting algorithm
1. ^ Hoare, C. A. R. (July 1961). "Algorithm 64: Quicksort". Commun. ACM. 4 (7): 321–. doi:10.1145/366622.366644. ISSN 0001-0782.
2. ^ Kudelić, Robert (2016-04-01). "Monte-Carlo randomized algorithm for minimal feedback arc set problem". Applied Soft Computing. 41: 235–246. doi:10.1016/j.asoc.2015.12.018.
3. ^ "In testing primality of very large numbers chosen at random, the chance of stumbling upon a value that fools the Fermat test is less than the chance that cosmic radiation will cause the
computer to make an error in carrying out a 'correct' algorithm. Considering an algorithm to be inadequate for the first reason but not for the second illustrates the difference between
mathematics and engineering." Hal Abelson and Gerald J. Sussman (1996). Structure and Interpretation of Computer Programs. MIT Press, section 1.2.
4. ^ Smid, Michiel. Closest point problems in computational geometry. Max-Planck-Institut für Informatik|year=1995
5. ^ A. A. Tsay, W. S. Lovejoy, David R. Karger, Random Sampling in Cut, Flow, and Network Design Problems, Mathematics of Operations Research, 24(2):383–413, 1999.
6. ^ Alippi, Cesare (2014), Intelligence for Embedded Systems, Springer, ISBN 978-3-319-05278-6.
7. ^ Kushilevitz, Eyal; Nisan, Noam (2006), Communication Complexity, Cambridge University Press, ISBN 9780521029834. For the deterministic lower bound see p. 11; for the logarithmic randomized
upper bound see pp. 31–32.
8. ^ Dyer, M.; Frieze, A.; Kannan, R. (1991), "A random polynomial-time algorithm for approximating the volume of convex bodies" (PDF), Journal of the ACM, 38 (1): 1–17, doi:10.1145/102782.102783
9. ^ Füredi, Z.; Bárány, I. (1986), "Computing the volume is difficult", Proc. 18th ACM Symposium on Theory of Computing (Berkeley, California, May 28–30, 1986) (PDF), New York, NY: ACM,
pp. 442–447, CiteSeerX , doi:10.1145/12130.12176, ISBN 0-89791-193-8
10. ^ Shamir, A. (1992), "IP = PSPACE", Journal of the ACM, 39 (4): 869–877, doi:10.1145/146585.146609
11. ^ Cook, Matthew; Soloveichik, David; Winfree, Erik; Bruck, Jehoshua (2009), "Programmability of chemical reaction networks", in Condon, Anne; Harel, David; Kok, Joost N.; Salomaa, Arto; Winfree,
Erik (eds.), Algorithmic Bioprocesses (PDF), Natural Computing Series, Springer-Verlag, pp. 543–584, doi:10.1007/978-3-540-88869-7_27.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw–Hill, 1990. ISBN 0-262-03293-7. Chapter 5:
Probabilistic Analysis and Randomized Algorithms, pp. 91–122.
• Dirk Draheim. "Semantics of the Probabilistic Typed Lambda Calculus (Markov Chain Semantics, Termination Behavior, and Denotational Semantics)." Springer, 2017.
• Jon Kleinberg and Éva Tardos. Algorithm Design. Chapter 13: "Randomized algorithms".
• Fallis, D. (2000). "The reliability of randomized algorithms". The British Journal for the Philosophy of Science. 51 (2): 255–271. doi:10.1093/bjps/51.2.255.
• M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York (NY), 2005.
• Rajeev Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York (NY), 1995.
• Rajeev Motwani and P. Raghavan. Randomized Algorithms. A survey on Randomized Algorithms.
• Christos Papadimitriou (1993), Computational Complexity (1st ed.), Addison Wesley, ISBN 978-0-201-53082-7 Chapter 11: Randomized computation, pp. 241–278.
• Rabin, Michael O. (1980). "Probabilistic algorithm for testing primality". Journal of Number Theory. 12: 128–138. doi:.
• A. A. Tsay, W. S. Lovejoy, David R. Karger, Random Sampling in Cut, Flow, and Network Design Problems, Mathematics of Operations Research, 24(2):383–413, 1999. | {"url":"https://codedocs.org/what-is/randomized-algorithm","timestamp":"2024-11-08T15:42:27Z","content_type":"text/html","content_length":"82351","record_id":"<urn:uuid:ccc7a5e9-4788-4038-89f0-d8c534bdac9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00082.warc.gz"} |
To err is human, to forgive divine, how about when it comes to machine - Data Wow blog – Data Science Consultant Thailand | Data Wow in Bangkok
All people commit sins and make mistakes. God forgives them, and people are acting in a godlike way when they forgive. This saying was from “An Essay on Criticism” by Alexander Pope.
The machines are invented by a human, of course, they err as well but, how much does it err? In machine learning or data science, we force the machine to understand the relationship between two types
of variables, which are dependent variable (Y) and independent variables (X). For example, your math score (Y) depends on the average hours spent studying math per week (X¹) and the average hours
spent on Facebook per day (X²).
How to find the relationship between these three?
Let f(X¹, X²) be the function of X¹ and X² which describes the relationship by,
Y = f(X¹, X²) + ϵ
What does ϵ (epsilon) mean?
It means noise/residual of the system, the unexplained variability outside the scope of our function f(X¹, X²), which is one of the three errors in the output of the machines learning model — bias,
variance, and noise.
Statistically speaking, we must assume that residuals are independent and identically normally distributed with zero mean and standard deviation equals σᵣ, otherwise all of those assumptions are
In real-world problems, it is impossible to know how such a f(X¹, X²) looks like. We have to estimate it and for the sake of mathematical modeling, we are going to estimate f(X¹, X²) by employing g
(X¹, X²). Please keep in mind that every estimation comes with errors.
Now we have an assumption and ready to start training and testing it out!
After sampling, we have a sample size equal to n, {(y¹, x¹¹, x²¹), (y², x¹², x²²), …, (yⁿ, x¹ⁿ, x²ⁿ)}.
We partition the sample into two sets, a training set, and a test set. The training set is trained according to the assumption, g(x¹, x²).
After the machines learned and understood the relationship, now they can return us the predicted value of Y by utilizing g(x¹, x²).
Basically, good estimations/predictions come from tiniest differences between actual values (y) and predicted/estimated values g(x¹, x²). This difference is conventionally called “errors” (now you
know how machines err!), and the popular metric to measure errors is “mean-squared error” or the expected value (𝔼)of (y -g(x¹, x²))² and it should ideally converge to zero.
𝔼[(y -g(x¹, x²))²] = 𝔼[f(x¹, x²) -g(x¹, x²)]² -(𝔼[g(x¹, x²)²] -𝔼[g(x¹, x²)]²)-σᵣ²
Err(x) = Bias² + Variance + noise
You can have many assumptions {g¹(x¹, x²), g²(x¹, x²), …, gᵏ(x¹, x²)}. However, all assumptions have to go through the evaluation process. To evaluate assumptions (or technically called models),
cross-validation method is used. Precision, Recall, Accuracy and F1 score from validating test set are the most famous metrics to evaluate the performance of the models.
Testing out the assumptions helps us answer which model is the most suitable and good enough for the tasks. Furthermore, those metrics indirectly tell us the about bias and variance of the model. For
instance, if the metrics are low only on test set then the model tends to overfit. See the plot for more details:
In conclusion, the term errors in machines learning are decomposed into three terms: bias, variance and, noise.
Bias is the difference between assumptions, g(x¹, x²) and the actual system, f(x¹, x²). On the other hand, Variance means how much the assumptions work distinctively on samples (e.g., training set)
and out of sample data sets (e.g., test set).
Bias and variance have an inverse relationship. To lower bias is to force variance to go higher and vice versa. For more details please see this bias-variance tradeoff.
Finally, the last composition of error—“noises”. Machines learning practitioners should ignore and avoid to focus on this error. Noises should not be allowed to bend in with the other factors in
the assumptions as noise unnecessarily disturbs the relationships.
This is how errors term in machine learning is defined and probably now we can say “To err is human, to forgive divine, to bias and to variate are machine!”. | {"url":"https://www.datawow.io/blogs/to-err-is-human","timestamp":"2024-11-12T08:48:32Z","content_type":"text/html","content_length":"27355","record_id":"<urn:uuid:c80d88ff-8a14-432d-8020-2669dbbc2b9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00709.warc.gz"} |
Further Reading
In his seminal Sketchpad system, Sutherland (1963) was the first to use projection matrices for computer graphics. Akenine-Möller et al. (2018) have provided a particularly well-written derivation of
the orthographic and perspective projection matrices. Other good references for projections are Rogers and Adams’s Mathematical Elements for Computer Graphics (1990) and Eberly’s book (2001) on game
engine design. See Adams and Levoy (2007) for a broad analysis of the types of radiance measurements that can be taken with cameras that have non-pinhole apertures.
An unusual projection method was used by Greene and Heckbert (1986) for generating images for OMNIMAX® theaters.
Potmesil and Chakravarty (1981, 1982, 1983) did early work on depth of field and motion blur in computer graphics. Cook and collaborators developed a more accurate model for these effects based on
the thin lens model; this is the approach used for the depth of field calculations in Section 5.2.3 (Cook et al. 1984; Cook 1986). An alternative approach to motion blur was described by Gribel and
Akenine-Möller (2017), who analytically computed the time ranges of ray–triangle intersections to eliminate stochastic sampling in time.
Kolb, Mitchell, and Hanrahan (1995) showed how to simulate complex camera lens systems with ray tracing in order to model the imaging effects of real cameras; the RealisticCamera is based on their
approach. Steinert et al. (2011) improved a number of details of this simulation, incorporating wavelength-dependent effects and accounting for both diffraction and glare. Joo et al. (2016) extended
this approach to handle aspheric lenses and modeled diffraction at the aperture stop, which causes some brightening at the edges of the circle of confusion in practice. See the books by Hecht (2002)
and Smith (2007) for excellent introductions to optics and lens systems.
Hullin et al. (2012) used polynomials to model the effect of lenses on rays passing through them; they were able to construct polynomials that approximate entire lens systems from polynomial
approximations of individual lenses. This approach saves the computational expense of tracing rays through lenses, though for complex scenes, this cost is generally negligible in relation to the rest
of the rendering computations. Hanika and Dachsbacher (2014) improved the accuracy of this approach and showed how to combine it with bidirectional path tracing. Schrade et al. (2016) showed good
results with approximation of wide-angle lenses using sparse higher-degree polynomials.
Film and Imaging
The film sensor model presented in Section 5.4.2 and the PixelSensor class implementation are from the PhysLight system described by Langlands and Fascione (2020). See also Chen et al. (2009), who
described the implementation of a fairly complete simulation of a digital camera, including the analog-to-digital conversion and noise in the measured pixel values inherent in this process.
Filter importance sampling, as described in Section 8.8, was described in a paper by Ernst et al. (2006). This technique is also proposed in Shirley’s Ph.D. thesis (1990).
The idea of storing additional information about the properties of the visible surface in a pixel was introduced by Perlin (1985a) and Saito and Takahashi (1990), who also coined the term G-Buffer.
Shade et al. (1998) introduced the generalization of storing information about all the surfaces along each camera ray and applied this representation to view interpolation, using the originally
hidden surfaces to handle disocclusion.
Celarek et al. (2019) developed techniques for evaluating sampling schemes based on computing both the expectation and variance of MSE and described approaches for evaluating error in rendered images
across both pixels and frequencies.
The sampling technique that approximates the XYZ matching curves is due to Radziszewski et al. (2009).
The SpectralFilm uses a representation for spectral images in the OpenEXR format that was introduced by Fichet et al. (2021).
As discussed in Section 5.4.2, the human visual system generally factors out the illumination color to perceive surfaces’ colors independently of it. A number of methods have been developed to
process photographs to perform white balancing to eliminate the tinge of light source colors; see Gijsenij et al. (2011) for a survey. White balancing photographs can be challenging, since the only
information available to white balancing algorithms is the final pixel values. In a renderer, the problem is easier, as information about the light sources is directly available; Wilkie and Weidlich
(2009) developed an efficient method to perform accurate white balancing in a renderer.
A wide range of approaches have been developed for removing Monte Carlo noise from rendered images. Here we will discuss those that are based on the statistical characteristics of the sample values
themselves. In the “Further Reading” section of Chapter 8, we will discuss ones that derive filters that account for the underlying light transport equations used to form the image. Zwicker et al.’s
report (2015) has thorough coverage of both approaches to denoising through 2015. We will therefore focus here on some of the foundational work as well as more recent developments.
Lee and Redner (1990) suggested using an alpha-trimmed mean filter for this task; it discards some number of samples at the low and high range of the sample values. The median filter, where all but a
single sample are discarded, is a special case of it. Jensen and Christensen (1995) observed that it can be effective to separate out the contributions to pixel values based on the type of
illumination they represent; low-frequency indirect illumination can be filtered differently from high-frequency direct illumination, thus reducing noise in the final image. They developed an
effective filtering technique based on this observation.
McCool (1999) used the depth, surface normal, and color at each pixel to determine how to blend pixel values with their neighbors in order to better preserve edges in the filtered image. Keller and
collaborators introduced the discontinuity buffer (Keller 1998; Wald et al. 2002). In addition to filtering slowly varying quantities like indirect illumination separately from more quickly varying
quantities like surface reflectance, the discontinuity buffer also uses geometric quantities like the surface normal to determine filter extents.
Dammertz et al. (2010) introduced a denoising algorithm based on edge-aware image filtering, applied hierarchically so that very wide kernels can be used with good performance. This approach was
improved by Schied et al. (2017), who used estimates of variance at each pixel to set filter widths and incorporated temporal reuse, using filtered results from the previous frame in a real-time ray
tracer. Bitterli et al. (2016) analyzed a variety of previous denoising techniques in a unified framework and derived a new approach based on a first-order regression of pixel values. Boughida and
Boubekeur (2017) described a Bayesian approach based on statistics of all the samples in a pixel, and Vicini et al. (2019a) considered the problem of denoising “deep” images, where each pixel may
contain multiple color values, each at a different depth.
Some filtering techniques focus solely on the outlier pixels that result when the sampling probability in the Monte Carlo estimator is a poor match to the integrand and is far too small for a sample.
(As mentioned previously, the resulting pixels are sometimes called “fireflies,” in a nod to their bright transience.) Rushmeier and Ward (1994) developed an early technique to address this issue
based on detecting outlier pixels and spreading their energy to nearby pixels in order to maintain an unbiased estimate of the true image. DeCoro et al. (2010) suggested storing all pixel sample
values and then rejecting outliers before filtering them to compute final pixel values. Zirr et al. (2018) proposed an improved approach that uses the distribution of sample values at each pixel to
detect and reweight outlier samples. Notably, their approach does not need to store all the individual samples, but can be implemented by partitioning samples into one of a small number of image
buffers based on their magnitude. More recently, Buisine et al. (2021) proposed using a median of means filter, which is effective at removing outliers but has slower convergence than the mean. They
therefore dynamically select between the mean and median of means depending on the characteristics of the sample values.
As with many other areas of image processing and understanding, techniques based on machine learning have recently been applied to denoising rendered images. This work started with Kalantari et al. (
2015), who used relatively small neural networks to determine parameters for conventional denoising filters. Approaches based on deep learning and convolutional neural networks soon followed with
Bako et al. (2017), Chaitanya et al. (2017), and Vogels et al. (2018) developing autoencoders based on the u-net architecture (Ronneberger et al. 2015). Xu et al. (2019) applied adversarial networks
to improve the training of such denoisers. Gharbi et al. (2019) showed that filtering the individual samples with a neural network can give much better results than sampling the pixels with the
samples already averaged. Munkberg and Hasselgren (2020) described an architecture that reduces the memory and computation required for this approach.
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58. Vogels, T., F. Rousselle, B. McWilliams, G. Röthlin, A. Harvill, D. Adler, M. Meyer, and J. Novák. 2018. Denoising with kernel prediction and asymmetric loss functions. ACM Transactions on
Graphics (Proceedings of SIGGRAPH) 37(4), 124:1–15.
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63. Zwicker, M., W. Jarosz, J. Lehtinen, B. Moon, R. Ramamoorthi, F. Rousselle, P. Sen, C. Soler, and S.-E. Yoon. 2015. Recent advances in adaptive sampling and reconstruction for Monte Carlo
rendering. Computer Graphics Forum (Proceedings of Eurographics 2015) 34(2), 667–81. | {"url":"https://www.pbr-book.org/4ed/Cameras_and_Film/Further_Reading","timestamp":"2024-11-02T08:10:09Z","content_type":"text/html","content_length":"35769","record_id":"<urn:uuid:94dcb7cd-8895-4e70-bb00-b481fc05dc12>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00147.warc.gz"} |
gap-pkg-kbmag 1.5.11
KBMAG (pronounced Kay-bee-mag) stands for Knuth-Bendix on Monoids, and Automatic Groups. It is a stand-alone package written in C, for use under UNIX, with an interface to GAP. There are interfaces
for the use of KBMAG with finitely presented groups, monoids and semigroups defined within GAP. The package also contains a collection of routines for manipulating finite state automata, which can be
accessed via the GAP interface.
The overall objective of KBMAG is to construct a normal form for the elements of a finitely presented group G in terms of the given generators together with a word reduction algorithm for calculating
the normal form representation of an element in G, given as a word in the generators. If this can be achieved, then it is also possible to enumerate the words in normal form up to a given length, and
to determine the order of the group, by counting the number of words in normal form. In most serious applications, this will be infinite, since finite groups are (with some exceptions) usually
handled better by Todd-Coxeter related methods. In fact a finite state automaton W is calculated that accepts precisely the language of words in the group generators that are in normal form, and W is
used for the enumeration and counting functions. It is possible to inspect W directly if required; for example, it is often possible to use W to determine whether an element in G has finite or
infinite order.
The normal form for an element g in G is defined to be the least word in the group generators (and their inverses) that represents G, with respect to a specified ordering on the set of all words in
the group generators.
KBMAG offers two possible means of achieving these objectives. The first is to apply the Knuth-Bendix algorithm to the group presentation, with one of the available orderings on words, and hope that
the algorithm will complete with a finite confluent presentation. (If the group is finite, then it is guaranteed to complete eventually but, like the Todd-Coxeter procedure, it may take a long time,
or require more space than is available.) The second is to use the automatic group program. This also uses the Knuth-Bendix procedure as one component of the algorithm, but it aims to compute certain
finite state automata rather than to obtain a finite confluent rewriting system, and it completes successfully on many examples for which such a finite system does not exist. In the current
implementation, its use is restricted to the shortlex ordering on words. That is, words are ordered first by increasing length, and then words of equal length are ordered lexicographically, using the
specified ordering of the generators.
The GAP4 version of KBMAG also offers extensive facilities for finding confluent presentations and finding automatic structures relative to a specified finitely generated subgroup of the group G.
Finally, there is a collection of functions for manipulating finite state automata that may be of independent interest.
For the purpose of submitting this package to Fedora, here is the RPM spec file.
Last modified: Mon Apr 29 16:57:37 MDT 2024 by Jerry James | {"url":"https://jjames.fedorapeople.org/gap-pkg-kbmag/","timestamp":"2024-11-11T13:18:46Z","content_type":"text/html","content_length":"4740","record_id":"<urn:uuid:b42e2563-58a8-47c7-bf71-3f284069ca37>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00548.warc.gz"} |
Friday Forecast: 10 Year Monthly Forecast of U.S. Treasury Yields And U.S. Dollar Interest Rate Swap Spreads - SAS Risk Data and Analytics
Today’s forecast for U.S. Treasury yields is based on the April 15, 2010 constant maturity Treasury yields reported by the Board of Governors of the Federal Reserve System in its H15 Statistical
Release reported at 4:15 pm April 16, 2010. The “forecast” is the implied future coupon bearing U.S. Treasury yields derived using the maximum smoothness forward rate smoothing approach developed by
Adams and van Deventer (Journal of Fixed Income, 1994) and corrected in van Deventer and Imai, Financial Risk Analytics (1996). For an electronic delivery of this interest rate data in Kamakura Risk
Manager table format, please subscribe via info@kamakuraco.com.
The “forecast” for future U.S. dollar interest rate swap rates is derived from the maximum smoothness forward rate approach, but applied to the forward credit spread between the libor-swap curve and
U.S. Treasury curve instead of to the absolute level of forward rates for the libor-swap curve.
Today’s forecast shows 1 month Treasury bill rates peaking at 5.465% in the fourth quarter of 2017 and the 10 year U.S. Treasury yield at 5.745% on March 31, 2020. The negative 21 basis point spread
between 30 year U.S. dollar interest rate swaps and U.S. Treasury yields reflects the blurring of credit quality between these two yield curves. The U.S. government is no longer seen as risk free,
and 4 of the 8 panel banks that determine U.S. dollar libor are receiving massive government assistance and are, in effect, sovereign credits. For more on the panel members, see www.bbalibor.com. The
negative 30 year spread results in an implied negative spread between 1 month libor and 1 month U.S. Treasury yields (investment basis) beginning in January 2015.
Background Information on Input Data and Smoothing
The Federal Reserve H15 statistical release is available here:
The maximum smoothness forward rate approach to yield curve smoothing was described in this blog entry:
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 10: Maximum Smoothness Forward Rates and Related Yields versus Nelson-Siegel,” Kamakura blog, www.kamakuraco.com,
January 5, 2010. Redistributed on www.riskcenter.com on January 7, 2010.
The use of the maximum smoothness forward rate approach for bond data is discussed in this blog entry:
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 12: Smoothing with Bond Prices as Inputs,” Kamakura blog, www.kamakuraco.com, January 20, 2010.
The reasons for smoothing forward credit spreads instead of the absolute level of the libor-swap curve was discussed in this blog entry:
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 13: Smoothing Credit Spreads,” Kamakura blog, www.kamakuraco.com, April 7, 2010. Redistributed on www.riskcenter.com,
April 14, 2010.
The Kamakura approach to interest rate forecasting was introduced in this blog entry:
van Deventer, Donald R. “The Kamakura Corporation Monthly Forecast of U.S. Treasury Yields,” Kamakura blog, www.kamakuraco.com, March 31, 2010. Redistributed on www.riskcenter.com on April 1,
Today’s Kamakura U.S. Treasury Yield Forecast
The Kamakura 10 year monthly forecast of U.S. Treasury yields is based on this data from the Federal Reserve H15 statistical release:
The graph below shows in 3 dimensions the movement of the U.S. Treasury yield curve 120 months into the future at each month end:
These yield curve movements are consistent with the continuous forward rates and zero coupon yields implied by the U.S. Treasury coupon bearing yields above:
In numerical terms, forecasts for the first 60 months of U.S. Treasury yield curves are as follows:
The forecasted yields for months 61 to 120 are given here:
Today’s Kamakura Forecast for U.S. Dollar Interest Rate Swap Yields and Spreads
Today’s forecast for U.S. Dollar interest rate swap yields is based on the following data from the H15 Statistical Release published by the Board of Governors of the Federal Reserve System:
Applying the maximum smoothness forward rate smoothing approach to the forward credit spreads between the libor-swap curve and the U.S. Treasury curve results in the following zero coupon bond
The forward rates for the libor-swap curve and U.S. Treasury curve are shown here:
The 10 year forecast for U.S. dollar interest rate swap yields is shown in the following graph:
The 10 year forecast for U.S. dollar interest rate swap spreads to U.S. Treasury yields is given in the following graph:
The numerical values for the implied future U.S. dollar interest rate swap spreads to U.S. Treasury yields are given here for 60 months forward:
The numerical values for the implied future U.S. dollar interest rate swap spreads to U.S. Treasury yields are given here for 61-120 months forward:
For more information about the yield curve smoothing and simulation capabilities in Kamakura Risk Manager, please contact us at info@kamakuraco.com. Kamakura interest rate forecasts are available in
pre-formatted Kamakura Risk Manager data base format.
Donald R. van Deventer
Kamakura Corporation
Honolulu, April 16, 2010 | {"url":"https://www.kamakuraco.com/friday-forecast-10-year-monthly-forecast-of-u-s-treasury-yields-and-u-s-dollar-interest-rate-swap-spreads/","timestamp":"2024-11-13T22:52:26Z","content_type":"text/html","content_length":"148144","record_id":"<urn:uuid:f6bf7da8-bac9-4c91-88d2-db4d90397566>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00753.warc.gz"} |
How many grams are in an ounce of 14k gold?
What does 1 oz of gold weigh
The exact amount of unwanted fat in an international troy ounce is 31.1034768 grams. One troy ounce of gold is equal to 31.1034807 grams. The ounce is also commonly used to measure bulk liquids.
How much is a 1 oz of gold today
1863.1 USD
How many grams are in an ounce of 14k gold
Metals, including gold, are measured in troy ounces. The Troy Of Touch of Gold set contains 31.103 grams.
How much does 1 oz weigh in grams
In fact, 1 ounce is approximately equal to 28 to 0.35 grams.
What is the current price of gold per gram
the price of a gram of gold per; (24,000, 22,000, 18,000, 14,000) Worldwide » Saudi Arabia Gold Gram Price in SAR » 24 carats: [224.48 SAR] 22 » size: [206.07 SAR]
How do you calculate gold grams
What is the price of each gram of gold in Malaysia?
How much is 916 gold worth?
How to calculate gold 916?
How much did 916 precious metals cost today?
How much is 1 gram of gold worth today?
What is the instant price of 916 gold in Singapore?
How to calculate the price of gold rings per gram?
How is the price of gold calculated?
How to calculate the cost of 1 gram?
How much grams is 1 oz equal to
An ounce (abbreviation: ounce) is usually a unit of mass with several values, the most commonly used unit is approximately 28 grams. The number of ounces varies depending on tactics.
How much does 1 oz gold coin weigh in grams
Additional alloys should be considered when considering the weight of American Gold One Eagles. 1 oz. US gold consists of one troy ounce (about 31.1 grams) with gold added. Silver and copper add
about 2.8 grams, bringing the total weight of the 1 ounce American Eagle gold coin to an estimated 33.9 grams.
How many weight of 200 grams is 1000 grams
Convert 200 grams to kilograms.
How many grams of nitrogen are in a diet consisting of 100 grams of protein
Why? Along with the indicated amount of protein or amino acids in the diet, your entire family can use one of these indicators to determine the amount of nitrogen in the amount of healthy protein
provided. Protein contains at least 16% nitrogen, and converting this value to a specific value by dividing 100% by 16% gives 6.25.
How many grams of 80% pure marble stone on calcination can give for 3 grams of quicklime
How many grams of stone marble with a purity of 80% after calibration gives 14 grams of quicklime? 80 g CaCO3 = 100 g precious marble. 25 g CaCO3=? 2580×100=31.25 g.
How many weight of 100 grams is 1000 grams
Step by step explanation: 10 x 100g dumbbells weigh 1000g and sometimes 1kg.
How many grams of water is 15 grams of coffee
v60 Ratio of coffee and water – 3:50 – 15 g of coffee to 250 g of water for a full cup.
Why are Grams called Grams
In general, a gram is equivalent to a significant thousandth of a liter (one cubic centimeter) of water at a temperature of 4 certified degrees Celsius. The word “gram” comes from the late Latin word
for “gram”, another small weight via the French “gram”. The symbol for grams is actually g. | {"url":"https://www.vanessabenedict.com/how-many-grams-is-1-oz-of-gold/","timestamp":"2024-11-10T11:25:01Z","content_type":"text/html","content_length":"72498","record_id":"<urn:uuid:2ffc3322-f363-4a9f-b3da-c3729cfaa677>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00174.warc.gz"} |
Systematic Listing | Shiken
Efficiently Listing All Possible Combinations with Systematic Listing of Outcomes
When faced with a PIN number consisting of 4 digits, each ranging from 0-9, it can be challenging to list out all the possible combinations in an efficient manner. To overcome this dilemma, we can
utilize the systematic listing of outcomes, a methodical approach that allows for a comprehensive listing of events.
Understanding Systematic Listing of Outcomes
Systematic listing of outcomes is a process that involves methodically listing all possible outcomes of an event to ensure no outcome is overlooked. This approach also enables the calculation of the
probability of an event occurring by dividing the number of times the event appears in the listing by the total number of outcomes. However, this method is only applicable when all outcomes have an
equal likelihood, such as when flipping a fair coin or rolling a fair dice.
The Method of Systematic Listing of Outcomes
Systematic listing of outcomes can be achieved by carefully analyzing the given information. By doing so, we can determine a suitable method for systematically listing the outcomes. To illustrate
this approach, let's consider an example:
Samantha is at a restaurant and wants to order a three-course meal. She can choose between soup or breadsticks for the starter, pizza or burger for the main, and ice cream or fruit salad for dessert.
How many possible combinations can Samantha order?
Step-by-Step Solution:
To perform a systematic listing of outcomes, we can begin by fixing all but one option and listing all the possible outcomes for that selection. For example, starting with soup as the starter and
pizza as the main, the possible combinations are listed as follows: Soup, Pizza, Ice Cream and Soup, Pizza, Fruit Salad. Next, by changing the main to burger, we get: Soup, Burger, Ice Cream and
Soup, Burger, Fruit Salad. This process, known as the fundamental principle of systematic listing, ensures that no outcome is overlooked.
Another method for systematic listing of outcomes is by using a sample space diagram.
Using a Sample Space Diagram
A sample space diagram is a table that lists all possible outcomes of an event, which is determined by two separate events. It is created by making a table with the outcomes of the first event as
column headings and the outcomes of the second event as row headings. The boxes in the table are then filled with the results of the calculations for the corresponding headers.
Sample space diagrams are useful for calculating probabilities, as the number of outcomes can be determined by:
• Counting the number of squares with the desired outcome
• Multiplying the number of rows by the number of columns
• Dividing the first number by the second number
Let's look at an example to understand this better:
If two six-sided dice are rolled, and the numbers from each roll are added together, all the possible outcomes can be displayed with a sample space diagram.
Solution for Rolling Two Dice:
The first dice has a number from 1 to 6, which will be listed in the table as follows:
Each dice roll is then added, giving us the column and row headings:
The possible outcomes are listed by adding the values from each column and row:
When to Utilize Systematic Listing of Outcomes
Systematic listing of outcomes is especially useful when describing an event with a large number of outcomes or permutations. It also comes in handy when determining the probabilities of specific
outcomes. Let's explore a few examples that illustrate this further:
• If two three-sided spinners with the numbers 1, 2, and 3 are rolled, the result of each spin is recorded, creating a 2-digit number. What are the possible numbers that can be formed?
In conclusion, the systematic listing of outcomes is a methodical approach that ensures all possible outcomes are listed and is particularly useful when calculating probabilities or when an event has
a large number of outcomes. The next time you encounter a situation with multiple outcomes, consider using this approach for a thorough and efficient listing of outcomes.
Systematic Listing of Outcomes: A Crucial Method for Determining Probabilities
When two events are combined, the potential outcomes increase significantly due to the six possibilities of each dice. In order to accurately and efficiently list all outcomes, a systematic approach
must be utilized.
To begin, the first dice is set to a result of 1 and all possible outcomes are listed in the following manner:
• 1 + 1 = 2
• 1 + 2 = 3
• 1 + 3 = 4
• 1 + 4 = 5
• 1 + 5 = 6
• 1 + 6 = 7
The first dice is then changed to a result of 2 and the possible outcomes are listed similarly:
• 2 + 1 = 3
• 2 + 2 = 4
• 2 + 3 = 5
• 2 + 4 = 6
• 2 + 5 = 7
• 2 + 6 = 8
This process continues by changing the first dice's result to 3, 4, 5, and 6, respectively, and listing all outcomes for each number.
The Probability of Rolling a Seven Using Systematic Listing
To determine the probability of rolling a seven, this same method can be used. Creating a table with the outcomes of each dice roll and counting the number of boxes containing the number 7 out of the
total 36 boxes yields a probability of 1/6.
The Significance of Systematic Listing of Outcomes
Using a systematic method to list outcomes guarantees that no outcome is overlooked and makes the process more accurate and efficient. Randomly picking outcomes can lead to mistakes and waste time,
especially when there are a large number of outcomes. To experience the importance of this approach, try listing outcomes without a systematic method and compare the results.
The Key Benefits of Systematic Listing:
• Ensures all possible outcomes are listed methodically
• Used for events resulting in a large amount of outcomes
• Increases accuracy and efficiency of outcome listing
• Sample space diagrams, such as tables, can be used to list outcomes
• Probabilities can be determined by counting the desired outcome and dividing by the total number of outcomes
An Example of Systematic Listing
A common example of systematic listing is when two events are combined, such as flipping a coin three times. Using this method, all possible outcomes of heads and tails for each throw can be listed
The Importance of Utilizing Systematic Listing
As previously mentioned, systematic listing of outcomes is crucial for accuracy and efficiency. Randomly listing outcomes can lead to mistakes and waste time, while a systematic approach ensures all
outcomes are accounted for.
What is Systematic Listing?
Systematic listing of outcomes is the process of methodically listing all possible outcomes of an event to ensure no outcome is missed.
Solving Systematic Listings
To solve systematic listings, a systematic method such as creating a table or using a formula must be used. This guarantees accurate and efficient listing of outcomes for any event involving multiple
The Power of Systematic Listing for Solving Problems with Multiple Events
When faced with a problem that involves multiple events, it's important to approach it systematically in order to accurately and efficiently list all possible outcomes. This methodical approach,
known as systematic listing, is especially helpful when calculating the outcome of two or more individual events.
The fundamental principle of systematic listing is a systematic and organized way of listing all possible outcomes for an event. This principle is often used in probability and statistics to ensure
thorough analysis and identification of all possible outcomes, including rare and unexpected ones.
To apply this principle, the first step is to determine all possible outcomes for the given event. This can be done through observation or mathematical calculations. Once all possible outcomes are
identified, they are listed and organized in a logical and systematic way, leaving no room for repetition or omission.
The key to successful systematic listing is a well-defined sample space, which represents all possible outcomes for an event. By carefully and systematically listing all outcomes, we can easily
identify patterns and relationships between different ones.
For example, let's consider a scenario where a coin is flipped and a die is rolled simultaneously. If we want to find the probability of getting a heads on the coin and an even number on the die, we
can use a sample space diagram to organize all possible outcomes:
• Coin - Head, Die - 2
• Coin - Head, Die - 4
• Coin - Head, Die - 6
• Coin - Tail, Die - 2
• Coin - Tail, Die - 4
• Coin - Tail, Die - 6
Through systematic listing, we can see that there are six possible outcomes, with only three satisfying the desired conditions (Head & Even number). This means that the probability of getting a heads
on the coin and an even number on the die is 3 out of 6, or 1/2.
In conclusion, the principle of systematic listing is a powerful tool for solving problems with multiple events. By following a systematic approach, we can ensure that no outcomes are overlooked and
gain a better understanding of the relationships between different outcomes. Next time you encounter a problem with multiple events, remember to use systematic listing for a more organized and
accurate analysis. | {"url":"https://shiken.ai/math-topics/systematic-listing","timestamp":"2024-11-11T16:37:41Z","content_type":"text/html","content_length":"82397","record_id":"<urn:uuid:7e504fc5-d44e-4425-b0ee-4ded7f8e9f02>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00314.warc.gz"} |
Model Pipe Networks with the Pipe Flow Module
Efficient Pipe Flow Modeling
Pipes are objects with high aspect ratios, so using lines and curves rather than volume elements allows you to model piping systems without the need to resolve the complete flow field. The software
solves for the cross-section averaged variables along lines and curves in your overall modeling of processes that consist of piping networks, while still allowing you to consider a full description
of the process variables within these networks.
The Pipe Flow Module provides specialized functionality for defining the conservation of momentum, energy, and mass of fluid inside pipes or channels. The pressure losses along the length of a pipe
are described using friction factors and relative surface roughness values. Based on this description, you can model the flow rate, pressure, temperature, and concentration in the pipes. | {"url":"https://www.comsol.com/pipe-flow-module","timestamp":"2024-11-14T17:47:49Z","content_type":"text/html","content_length":"98930","record_id":"<urn:uuid:df4e40d2-b58e-423c-b17a-3a4bb5b84cfc>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00866.warc.gz"} |
relatively quantum
After some longer silence partly due to moving to a new location (LMU Munich) and teaching my first regular lecture (Theoretical mechanics for lyceum teachers and computational science at Regensburg
University), I hope to write more regularly again in the future.
As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared
. Together with Andreas Schäfer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is
resolved in a manner inspired by results from loop quantum gravity.
Building (specifically) on recent work by
Engelhardt, Hertog, and Horowitz
(as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the
gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to
some definite statement of how exactly they should be done. | {"url":"https://relatively-quantum.bodendorfer.eu/2016/12/","timestamp":"2024-11-10T06:37:00Z","content_type":"application/xhtml+xml","content_length":"48757","record_id":"<urn:uuid:40dd9641-a1a9-4e69-a18b-3ceb24b00b1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00560.warc.gz"} |
The Law of Invariance Shapes Physics, Mathematics, and Beyond
The Law of Invariance: A Comprehensive Guide. Invariance is a fascinating concept that underlies many scientific disciplines. It is the cornerstone of understanding how certain properties remain
unchanged despite different conditions or transformations. To truly grasp the power and significance of invariance, it is crucial to explore its definition, historical background, and the role it
plays in various scientific fields. So, let’s dive right in!
🔩 The Nuts and Bolts:
• The Law of Invariance ensures consistency across scientific phenomena. It describes how certain properties or behaviors remain unchanged under specific transformations, guiding predictions and
understanding in fields like physics, math, and biology.
• Invariance is rooted in symmetry and conservation laws. The concept is deeply linked to the laws of nature that remain constant despite changes in perspective, time, or space, forming the
foundation of scientific reasoning.
• Invariance plays a crucial role in mathematical structures. Fields like group theory in algebra, geometry, and calculus use invariance to uncover symmetries, revealing patterns that remain
unchanged despite transformations.
• The Law of Invariance is fundamental in classical and quantum physics. From Newton’s laws to gauge invariance in quantum mechanics, it helps explain how physical systems behave consistently
across different conditions.
• Invariance extends beyond physics to chemistry and biology. In chemical reactions, conservation of mass and charge, and in biology, the genetic code’s stability across generations, are examples
of invariance in action.
• Debates around invariance question its limits at quantum scales. While powerful, some theories suggest that at extremely small scales, such as in quantum gravity, the principles of invariance may
break down, sparking ongoing research.
Understanding the Concept of Invariance
Invariance can be defined as the quality of an object or a system that remains unaltered under specific transformations or operations. These transformations could include changes in position,
orientation, scale, or even time. The concept of invariance is deeply rooted in the fundamental principles of symmetry and conservation laws.
Throughout history, scientists and mathematicians have been captivated by the idea of invariance. Ancient Greek philosophers pondered the nature of unchanging elements, setting the stage for further
exploration in later centuries.
But what are some specific examples of invariance in action? Let’s take a closer look.
Definition and Basic Principles
At its core, invariance refers to the constancy of a property or behavior despite various transformations. It is a key principle in mathematics and physics, allowing us to describe and predict
phenomena with remarkable accuracy.
One of the basic principles of invariance is the idea that the laws of nature remain unaffected by certain transformations. This means that the fundamental rules governing the universe are invariant
under specific changes in variables or perspectives.
For example, the laws of physics remain the same regardless of whether we are observing an object from the left or the right, or whether we are moving in a straight line or at an angle.
Another fascinating example of invariance is found in the concept of scale invariance. This principle states that certain properties or behaviors remain the same regardless of the scale at which they
are observed. This has profound implications in fields such as fractal geometry and the study of self-similarity.
Historical Background of Invariance
The roots of invariance can be traced back to ancient civilizations, where scholars such as Pythagoras and Euclid laid the groundwork for geometry. Their hypotheses and theorems were based on the
concept of unchanging properties, setting the stage for the emergence of invariance as a guiding principle in mathematics.
In the early 19th century, the great mathematician Carl Friedrich Gauss developed the concept of geometrical invariance, which revolutionized the field of mathematics. His work paved the way for
later advancements in algebra and calculus.
As science progressed, the concept of invariance found its way into the realm of physics. From classical mechanics to quantum theory, scientists have relied on the principles of invariance to unravel
the mysteries of the universe.
One notable example is the principle of gauge invariance in quantum field theory. This principle states that the laws of physics should remain unchanged when certain transformations, known as gauge
transformations, are applied. Gauge invariance has played a crucial role in the development of the Standard Model of particle physics, providing a framework for understanding the fundamental forces
and particles that make up our universe.
The Importance of Invariance in Science
Invariance plays a crucial role in scientific reasoning and understanding. It allows us to uncover the hidden symmetries and conservation laws that govern natural phenomena.
By recognizing the invariance of certain properties, scientists can make predictions and test hypotheses. This has led to groundbreaking discoveries across various scientific fields.
Moreover, invariance provides a framework for developing mathematical models that accurately describe the behavior of complex systems. It allows us to simplify problems and make fundamental
connections between diverse areas of study.
For example, invariance principles have been instrumental in the development of chaos theory, which explores the behavior of complex systems that are highly sensitive to initial conditions. By
identifying certain invariants, scientists have been able to gain insights into the underlying patterns and dynamics of chaotic systems, ranging from weather patterns to the behavior of financial
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The Mathematical Framework of Invariance
Mathematics is a powerful language when it comes to expressing invariance. It provides a rigorous framework for formulating and analyzing the principles of invariance.
Invariance, a fundamental concept in mathematics, manifests itself in various branches of the subject. Let’s explore some of these branches and delve into the intriguing world of invariance.
Invariance in Algebra
Algebraic structures, such as groups, rings, and fields, often exhibit fascinating invariance properties. These structures encapsulate the notions of symmetry and transformation in a mathematical
Group theory, a branch of algebra, plays a central role in understanding invariance. It allows us to analyze patterns of invariance in a wide range of mathematical systems. By studying the symmetries
and transformations of these systems, we gain insights into the underlying invariance principles.
Invariance in Geometry
Geometry is another domain where invariance plays a pivotal role. Certain geometric properties are inherently invariant under transformations, such as rotations, translations, and reflections.
Consider the Pythagorean theorem, a fundamental result in geometry. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Remarkably, this theorem remains true regardless of the orientation or position of the triangle in space. This illustrates the power of geometric invariance in solving complex problems.
Invariance in Calculus
Invariance is deeply intertwined with the principles of calculus, a mathematical tool widely used in science and engineering. Calculus enables us to study the rates of change of quantities and their
invariance properties.
The derivative, a fundamental concept in calculus, captures the instant rate of change of a function. Remarkably, this rate of change remains the same regardless of horizontal shifts or scale
transformations. This concept of invariance forms the basis of many applications of calculus, from physics to economics.
As we can see, invariance is a pervasive and powerful concept in mathematics. It provides a unifying framework for understanding the underlying principles that govern various mathematical structures
and systems. By studying invariance, mathematicians are able to uncover deep connections and unlock new insights into the nature of the mathematical universe.
The Law of Invariance in Physics
Physics, perhaps more than any other scientific field, heavily relies on the law of invariance to uncover the fundamental truths of our universe.
The concept of invariance, also known as symmetry in physics, plays a crucial role in shaping our understanding of the physical laws that govern the cosmos. It serves as a guiding principle that
underpins the elegant mathematical framework used to describe the behavior of particles and forces in the universe.
Role in Classical Physics
From Newton’s laws of motion to Maxwell’s equations of electromagnetism, the principles of classical physics are deeply rooted in invariance. The conservation of energy, momentum, and angular
momentum all arise from various invariance principles.
Additionally, the principle of time translation invariance, which states that the laws of physics remain constant over time, is a cornerstone of classical physics. This principle allows us to predict
the future behavior of physical systems based on their current state, enabling advancements in fields such as celestial mechanics and thermodynamics.
For example, Newton’s third law of motion states that for every action, there is an equal and opposite reaction. This law is a direct consequence of the invariance of momentum in isolated systems.
Significance in Quantum Physics
Quantum mechanics, the branch of physics that deals with the behavior of particles on an atomic and subatomic scale, is heavily dependent on the notion of invariance.
The principles of quantum mechanics, including the famous wave-particle duality and the uncertainty principle, are derived from the invariance properties of quantum systems.
Quantum invariance, coupled with concepts like superposition and entanglement, has given rise to groundbreaking technologies such as quantum computing and secure communication.
Furthermore, the concept of gauge invariance plays a central role in the standard model of particle physics, which describes the electromagnetic, weak, and strong nuclear forces. Gauge invariance
ensures the consistency and predictability of interactions between elementary particles, leading to a deeper understanding of the fundamental building blocks of matter.
Invariance in Other Scientific Fields
While invariance is prominently discussed in mathematics and physics, its application extends to other scientific disciplines as well.
Invariance in Chemistry
Chemical reactions and molecular structures exhibit various invariance properties. The conservation of mass, charge, and the principles governing chemical equilibrium are all manifestations of
invariance in the realm of chemistry.
Invariance in Biology
In biology, invariance principles are essential for understanding the complex processes underlying life. From the conservation of energy in metabolic pathways to the invariance of genetic codes,
biology relies on the principles of invariance to unravel the mysteries of living organisms.
Theoretical Implications and Controversies
While invariance has provided a solid foundation for scientific progress, it has also sparked debates and controversies among scholars.
Debates Surrounding Invariance
Some scientists argue that the concept of invariance may not hold true at its most fundamental levels. Quantum gravity, for instance, challenges the notion of invariance at the smallest scales of
space and time.
Additionally, the philosophical implications of invariance have sparked philosophical discussions regarding the nature of reality and the limits of our knowledge.
Future Directions for Invariance Research
The study of invariance is far from over. As new scientific discoveries unfold, researchers continue to explore and refine the principles of invariance.
Future directions for invariance research include investigating the role of invariance in complex systems, exploring its connections with information theory, and delving deeper into the philosophical
foundations of this fundamental concept.
The law of invariance is a powerful and pervasive principle that permeates various scientific disciplines. From mathematics to physics, chemistry to biology, invariance provides a framework for
understanding the fundamental laws that govern our universe.
As we continue to delve into the mysteries of the cosmos, invariance serves as our trusty guide, leading us towards new frontiers of knowledge and pushing the boundaries of human understanding.
Law of Invariance FAQs
What is the Law of Invariance?
The Law of Invariance refers to the principle that certain properties or behaviors of a system remain unchanged despite transformations or different conditions. It plays a fundamental role in fields
like physics, mathematics, and chemistry, helping to explain how systems maintain stability.
How is the Law of Invariance applied in mathematics?
In mathematics, invariance is used to study symmetries and transformations in algebra, geometry, and calculus. Group theory is one area where invariance is crucial, as it explores how mathematical
structures behave consistently under specific transformations, leading to a deeper understanding of patterns and systems.
Why is the Law of Invariance important in physics?
In physics, invariance helps explain the consistency of physical laws, like the conservation of energy and momentum. These laws remain unchanged under different conditions, such as time, orientation,
or movement, which allows scientists to make accurate predictions about how systems behave across different scenarios.
How does the Law of Invariance affect quantum mechanics?
Quantum mechanics relies heavily on the concept of invariance, especially in understanding particle behavior. Gauge invariance, for instance, is a principle that ensures consistency in how elementary
particles interact, forming the basis for the Standard Model of particle physics.
What role does invariance play in chemistry?
In chemistry, the Law of Invariance manifests through the conservation of mass and charge in chemical reactions. These properties remain constant despite the transformation of substances, allowing
scientists to predict reaction outcomes and understand molecular behavior.
How does the Law of Invariance apply to biology?
In biology, invariance helps explain the stability of processes like genetic inheritance. The genetic code, for example, remains consistent across generations, allowing organisms to pass down traits
reliably while maintaining life’s essential processes.
Are there any challenges or controversies regarding the Law of Invariance?
Yes, some debates exist, particularly in the realm of quantum gravity, where scientists question whether the concept of invariance holds true at extremely small scales. These discussions explore
whether new physics beyond current theories might lead to a better understanding of invariance’s limits. | {"url":"https://helio.app/ux-research/laws-of-ux/law-of-invariance/","timestamp":"2024-11-04T18:15:18Z","content_type":"text/html","content_length":"250915","record_id":"<urn:uuid:eb2a8135-c7c8-46b7-bace-7404acfb835e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00215.warc.gz"} |
How do
How do you run SPSS on a Mac?
How do you run SPSS on a Mac?
SPSS Statistics: Download & Installation for Mac (Students)
1. Open up Finder.
2. Click on Applications.
3. Select the IBM folder from the list and put it in your Trash.
4. Empty your Trash to finish uninstalling.
5. Restart you computer.
6. Open your browser to the Beyond Compare Software Page and click “Get A Personal IBM SPSS License”
What is zero order correlation matrix?
First, a zero-order correlation simply refers to the correlation between two variables (i.e., the independent and dependent variable) without controlling for the influence of any other variables.
Essentially, this means that a zero-order correlation is the same thing as a Pearson correlation.
How do you right click on SPSS on a Mac?
Use two fingers on the trackpad When you tap your Mac’s trackpad with two fingers spaced within an inch or so of one another, the result will be a right-click.
Is SPSS better on Windows or Mac?
The most basic SPSS you can get is “SPSS Statistics Base”. There are “higher” and more expensive packages with way more functionality. So SPSS is best used on Windows 7 or Mac OS.
Can you run SPSS on a MacBook Air?
SPSS has a Mac version that should run on a MacBook. The system requirements are low enough that you should be able to launch and use the program, though interacting with very large datasets may take
a long time if you’re using an older MacBook with less RAM or a weaker processor.
Why is my SPSS not opening on Mac?
Resolving The Problem The most likely culprit is the local security software stopping SPSS Statistics from launching. Disable the virus checker/security software and see if SPSS will launch. If so,
set an exclusion to the IBM SPSS Statistics folder for the virus checker/security software on launch.
How do you find the zero order correlation?
Values of Zero-Order Correlation
1. 1: for every positive increase of 1 in one variable, there is a positive increase of 1 in the other.
2. -1: for every positive increase of 1 in one variable, there is a negative decrease of 1 in the other.
3. 0: there isn’t a positive or negative increase. The two variables aren’t related.
What is a zero order correlation matrix?
Zero-Order Correlations in a Correlation Matrix Whenever we create a correlation matrix for a set of variables, the correlation coefficients shown within the matrix are always zero-order correlations
because they’re simply the correlations between each pairwise combination of variables without considering the influence of any other variables.
What is a partial correlation in SPSS?
This is why SPSS gives you the option to report zero-order correlations when running a multiple linear regression analysis. Next, a partial correlation is the correlation between an independent
variable and a dependent variable after controlling for the influence of other variables on both the independent and dependent variable.
How do I create a correlation matrix in SPSS?
Example: How to Create a Correlation Matrix in SPSS Step 1: Select bivariate correlation.. Click the Analyze tab. Click Correlate. Click Bivariate. Step 2: Create the correlation matrix.. Select each
variable you’d like to include in the correlation matrix and click… Step 3: Interpret the
What are “part” and “zero-order” correlations?
You also may have come across the terms “zero-order,” “partial,” and “part” in reference to correlations. These terms refer to correlations that involve more than two variables. | {"url":"https://www.yemialadeworld.com/how-do-you-run-spss-on-a-mac/","timestamp":"2024-11-01T19:11:28Z","content_type":"text/html","content_length":"71315","record_id":"<urn:uuid:c4addb37-27c6-4d8f-a1b0-3fd5abd42b20>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00485.warc.gz"} |
Space Turtle
2004 Canadian Computing Competition, Stage 1
Problem S4: Space Turtle
Space Turtle is a fearless space adventurer. His spaceship, the Tortoise, is a little outdated, but still gets him where he needs to go.
The Tortoise can do only two things - move forward an integer number of light-years, and turn in one of four directions (relative to the current orientation): right, left, up and down. In fact,
strangely enough, we can even think of the Tortoise as a ship which travels along a 3-dimensional co-ordinate grid, measured in light-years.
In today's adventure, Space Turtle is searching for the fabled Golden Shell, which lies on a deserted planet somewhere in uncharted space. Space Turtle plans to fly around randomly looking for the
planet, hoping that his turtle instincts will lead him to the treasure.
You have the lonely job of being the keeper of the fabled Golden Shell. Being lonely, your only hobby is to observe and record how close various treasure seekers come to finding the deserted planet
and its hidden treasure.
Given your observations of Space Turtle's movements, determine the closest distance Space Turtle comes to reaching the Golden Shell.
The first line consists of three integers sx, sy, and sz, which give the coordinates of Space Turtle's starting point. Space Turtle is originally oriented in the positive x direction, with the top of
his spaceship pointing in the positive z direction, and with the positive y direction to his left. Each of these integers are between -100 and 100.
The second line consists of three integers tx, ty, and tz, which give the coordinates of the deserted planet. Each of these integers are between -10000 and 10000.
The rest of the lines describe Space Turtle's flight plan in his search for the Golden Shell. Each line consists of an integer, d, 0 ≤ d ≤ 100, and a letter c, separated by a space. The integer
indicates the distance in light-years that the Tortoise moves forward, and the letter indicates the direction the ship turns after having moved forward. `L', `R', `U', and `D' stand for left, right,
up and down, respectively. There will be no more than 100 such lines.
On the last line of input, instead of one of the four direction letters, the letter `E' is given instead, indicating the end of today's adventure.
Output the closest distance that Space Turtle gets to the hidden planet, rounded to 2 decimal places.
If Space Turtle's coordinates coincide with the planet's coordinates during his flight indicate that with a distance of 0.00. He safely lands on the planet and finds the Golden Shell.
Sample Input
2 L
2 L
2 U
2 U
2 L
2 L
2 U
2 E
Sample Output
All Submissions
Best Solutions
Point Value: 12
Time Limit: 2.00s
Memory Limit: 16M
Added: Sep 28, 2008
Problem Types: [Show] [Hide]
Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3
Does the turtle stay facing -> and goes up in this position or does it turn upward so the front is facing up and goes upward?
Does the turtle stay facing -> and goes up in this position or does it turn upward so the front is facing up and goes upward?
The U instruction means that the Space Turtle adjusts himself so that whatever direction currently seems to him to be UP is now his FRONT, and LEFT and RIGHT stay the same as they were before.
The U instruction means that the Space Turtle adjusts himself so that whatever direction currently seems to him to be UP is now his FRONT, and LEFT and RIGHT stay the same as they were before.
Can someone please explain how the input works. I know the problem statement tells me how to do it but I'm still quite confused.
You start facing a positive x direction. The you move in that direction and afterwards you change direction?
So for the sample, you start off with 0 0 0
Then you move to 2 0 0?
Then you move to 2 2 0?
Can someone please explain how the input works. I know the problem statement tells me how to do it but I'm still quite confused.
You start facing a positive x direction. The you move in that direction and afterwards you change direction?
So for the sample, you start off with 0 0 0
Then you move to 2 0 0?
Then you move to 2 2 0? | {"url":"https://wcipeg.com/problem/ccc04s4","timestamp":"2024-11-13T21:04:18Z","content_type":"text/html","content_length":"16678","record_id":"<urn:uuid:c8310f7f-fe71-49d3-8fed-eef81bc38df3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00333.warc.gz"} |
Binary linear LTC
A binary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\).
More technically, the code is a \((u,R)\)-LTC if the rows of its parity-check matrix \(H\in GF(2)^{r\times n}\) have weight at most \(u\) and if \frac{1}{r}|H x| \geq \frac{R}{n} D(x,C) \tag*{(1)}
holds for any bitstring \(x\), where \(D(x,C)\) is the Hamming distance between \(x\) and the closest codeword to \(x\) [1; Def. 11].
• Linear binary code — Linear binary codes with distances \(\frac{1}{2}n-\sqrt{t n}\) for some \(t\) are called almost-orthogonal and are locally testable with query complexity of order \(O(t)\)
[4]. This was later improved to codes with distance \(\frac{1}{2}n-O(n^{1-\gamma})\) for any positive \(\gamma\) [5], provided that the number of codewords is polynomial in \(n\).
• Cyclic linear binary code — Cyclic linear codes cannot be \(c^3\)-LTCs [6]. Codeword symmetries are in general an obstruction to achieving such LTCs [7].
• Reed-Muller (RM) code — RM codes can be LTCs in the low- [8,9] and high-error [10] regimes; see also [11].
Page edit log
Cite as:
“Binary linear LTC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_ltc
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/ltc/binary_ltc.yml. | {"url":"https://errorcorrectionzoo.org/c/binary_ltc","timestamp":"2024-11-04T12:18:29Z","content_type":"text/html","content_length":"20652","record_id":"<urn:uuid:3883433c-85ec-4066-97d3-c0aa6b2daf69>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00886.warc.gz"} |
Life, Death and What Does Is Means in Math - LILLY PITTA | Mandando Som
Life, Death and What Does Is Means in Math
The dilemma with migrating COBOL is not that you’re migrating from 1 language to another, but that you’re migrating from 1 paradigm to another. At length, derivatives may be used to help you graph
functions. The expressions we are going to be translating are very straightforward.
What You Need to Do About What Does Is Means in Math Starting in the Next 9 Minutes
The test may demonstrate the student the pre-calculus areas they have to improve. There are plenty of excellent on-line resources http://artitjawangso.com/index.php?option=com_content&view=article&id
=1126 for any topic you may imagine, and the majority of them are shorter than a common science class. Among the hoped-for advantages of students taking a biology course is they will grow more
acquainted with the practice of science.
A scientist is someone who works in and has expert knowledge of a certain area of science. Explaining math to others is among the very best ways of learning it. It can be seen as an ever-increasing
series of abstractions.
The Importance of What Does Is Means in Math
The scientific technique is to be put to use as a guide that could be modified. It’s possible to never secure comfortable click to find out more with that symbol till you recognize just what you use
it for and why. Find more information about the scientific method.
The simple fact that we’ve such a solution usually means that we’ve got the overall solution. As an example, suppose that you need to calculate You wish to write down each step separately rather than
calculate the equation all at one time. Let’s see an instance of this.
There are lots of situations where it’s important that you know the relative size of a single number to another, for instance, when it has to do with money. Even a physician will say that. There’s
only one (though it might be expressed in a lot of different looking but equivalent ways).
At first this doesn’t look to be an improvement. Therefore, to average the growth during the very first month with the growth during the initial 12 months isn’t a sensible thing to do. The true
distinction is 3.19.
What Does Is Means in Math Explained
This point isn’t generally appreciated. however, it is vitally important. You don’t need Uk.Payforessay.net to take notes, or compose a report just watch this, and find a feeling of what it’s about.
There’s a remarkable deal of the world that you might only understand through that kind of analysis, he explained.
Devlin’s Angle is updated at the start of every month. For those powers of 2, it is a different story. We are going to try it one final moment.
The illustration is dumb because the solution is entirely erroneous. You need to always study your response to check whether it is logical, regarding what type of number you expect to get. You are
able to multiply 8 2 to receive 16, and you’ll receive the same answer with 2 8.
The What Does Is Means in Math Game
It was initially published in 1945. Then it’s possible to take away what you desire. Get it touch to tell us.
All numbers which aren’t rational are deemed irrational. The equation evaluates the period of the string regarding its wavelength. But the integers aren’t dense.
You will also see factorials used in permutations and combinations, which likewise have to do with probability. It measures the expected decrease in entropy. Employing the truncation, round-up, and
round down alone may lead to an error that is greater than 1 half of a single unit in the previous location, but less than 1 unit in the past location, so these modes aren’t recommended unless they
are employed in Interval Arithmetic.
Convert the remainders to base 16 (which you might need to think of regarding decimal numbers, or you may use your fingers and a few toes) and compose the digits in reverse order. The worth of y
depends upon the worth of x. It’s a single number, although you are in need of a pair of numbers to locate it.
What Does Is Means in Math Options
It’s probably pretty hard to find grass to cultivate upside-down. The slope is found by considering the rise over therun. Otherwise, then you will have some area of grass which gets hit more than
Understanding What Does Is Means in Math
Media literacy involves understanding the many methods information is generated and distributed. Kids with dyscalculia often require more support. Subscribe to our FREE newsletter and get started
improving your life in only 5 minutes per day.
They help students through internet videos, presentations, and assorted pdfs and also offer the live help through chatting with detailed solution. They were instructed on the best way to compose
explanations for their math solutions employing a model named Need, Know, Do. If you’re interested in some suggestions, comments, and elaborations, click the Comments.
Get the Scoop on What Does Is Means in Math Before You’re Too Late
As a consequence, multiplication and its products have a distinctive set of properties which you have to know to receive the correct answers. Zeros that only hold places aren’t regarded to be
significant. Work is achieved by transferring energy from 1 form to another.
Both of these methods are doing essentially the identical thing. For each attribute, the gain is figured and the maximum gain is utilized in the choice. Failing to adhere to the order of operations
can cause a huge mistake.
In that situation, you automatically know you’re managing a correct triangle. Rather one wants to create a sequence of issues that lead until the issue of interest, and solve every one of them. If
you need top-quality math assignments, you ought not have to compromise respect and privacy.
The 30-Second Trick for What Does Is Means in Math
The very first area of the course is going to be a rigorous introduction to this circle of ideas. Amidst all of that, however, plenty of terrific learning happened. If you’ve been fed heiferdung all
of your life, then you are going to go the method of heiferdungback into the ground. | {"url":"http://www.lillypitta.com/life-death-and-what-does-is-means-in-math-2/","timestamp":"2024-11-03T06:17:39Z","content_type":"text/html","content_length":"57267","record_id":"<urn:uuid:c5982b12-842d-41a6-a4a5-b58287b26c1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00649.warc.gz"} |
Univariate Function: Definition
A univariate function has only one variable.
Similarly, univariate equations, expressions, or polynomials only have one variable. Univariate analysis is a the simplest sort of data analysis, that only takes into account one variable or
Examples of Univariate Functions
One example of this type of function is:
f(x) = x^2.
This function describes a change based on just one variable or condition, x. If x represents time and f(x) represents the number of political protesters in a city square, this equation might
represent how the number of protesters changed over time.
Once you take into account other conditions, you make your function multivariate. For example, if we wanted to adjust for the amount of rain, and every mm of rain made 1 protester leave every minute,
we might say
f(x) = x^2 – 10xy.
This would no longer be a univariate function. It might be more accurate, but it would have lost much in simplicity.
Describing Univariate Data
Functions or data in just one variable can be graphed with bar charts, histograms, frequency distribution tables, and pie charts.
We can also describe the patterns we find in them by giving the mean, mode, median, range, variance, and the standard deviation.
The image below is a pie chart showing the populations of English native speakers per country of residence.
Applications of Univariate Functions & Analysis
Functions and distributions of one variable are important because they give us a way to easily define what would be happening if we reduced the action on our target population to just one influencer.
In clinical studies, analysis in one variable is always the first step in research. This type of analysis allows us to describe the variable distribution in a preliminary sample.
Canova S, Cortinovis DL, Ambrogi F. How to describe univariate data. J Thorac Dis. 2017;9(6):1741–1743. doi:10.21037/jtd.2017.05.80. Retrieved from https://www.ncbi.nlm.nih.gov/pmc/articles/
PMC5506131/ on August 17, 2019
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Self-Parking Car in 500 Lines of Code | Trekhleb
Self-Parking Car in 500 Lines of Code
In this article, we'll train the car to do self-parking using a genetic algorithm.
We'll create the 1st generation of cars with random genomes that will behave something like this:
On the ≈40th generation the cars start learning what the self-parking is and start getting closer to the parking spot:
Another example with a bit more challenging starting point:
Yeah-yeah, the cars are hitting some other cars along the way, and also are not perfectly fitting the parking spot, but this is only the 40th generation since the creation of the world for them,
so be merciful and give the cars some space to grow :D
You may launch the 🚕 Self-parking Car Evolution Simulator to see the evolution process directly in your browser. The simulator gives you the following opportunities:
The genetic algorithm for this project is implemented in TypeScript. The full genetic source code will be shown in this article, but you may also find the final code examples in the Evolution
Simulator repository.
We're going to use a genetic algorithm for the particular task of evolving cars' genomes. However, this article only touches on the basics of the algorithm and is by no means a complete guide to
the genetic algorithm topic.
Having that said, let's deep dive into more details...
The Plan
Step-by-step we're going to break down a high-level task of creating the self-parking car to the straightforward low-level optimization problem of finding the optimal combination of 180 bits (finding
the optimal car genome).
Here is what we're going to do:
1. 💪🏻 Give the muscles (engine, steering wheel) to the car so that it could move towards the parking spot.
2. 👀 Give the eyes (sensors) to the car so that it could see the obstacles around.
3. 🧠 Give the brain to the car that will control the muscles (movements) based on what the car sees (obstacles via sensors). The brain will be simply a pure function movements = f(sensors).
4. 🧬 Evolve the brain to do the right moves based on the sensors input. This is where we will apply a genetic algorithm. Generation after generation our brain function movements = f(sensors) will
learn how to move the car towards the parking spot.
Giving the muscles to the car
To be able to move, the car would need "muscles". Let's give the car two types of muscles:
1. Engine muscle - allows the car to move ↓ back, ↑ forth, or ◎ stand steel (neutral gear)
2. Steering wheel muscle - allows the car to turn ← left, → right, or ◎ go straight while moving
With these two muscles the car can perform the following movements:
In our case, the muscles are receivers of the signals that come from the brain once every 100ms (milliseconds). Based on the value of the brain's signal the muscles act differently. We'll cover the
"brain" part below, but for now, let's say that our brain may send only 3 possible signals to each muscle: -1, 0, or +1.
type MuscleSignal = -1 | 0 | 1;
For example, the brain may send the signal with the value of +1 to the engine muscle and it will start moving the car forward. The signal -1 to the engine moves the car backward. At the same time, if
the brain will send the signal of -1 to the steering wheel muscle, it will turn the car to the left, etc.
Here is how the brain signal values map to the muscle actions in our case:
Muscle Signal = -1 Signal = 0 Signal = +1
Engine ↓ Backward ◎ Neutral ↑ Forward
Steering wheel ← Left ◎ Straight → Right
You may use the Evolution Simulator and try to park the car manually to see how the car muscles work. Every time you press one of the WASD keyboard keys (or use a touch-screen joystick) you send
these -1, 0, or +1 signals to the engine and steering wheel muscles.
Giving the eyes to the car
Before our car will learn how to do self-parking using its muscles, it needs to be able to "see" the surroundings. Let's give it the 8 eyes in a form of distance sensors:
• Each sensor can detect the obstacle in a distance range of 0-4m (meters).
• Each sensor reports the latest information about the obstacles it "sees" to the car's "brain" every 100ms.
• Whenever the sensor doesn't see any obstacles it reports the value of 0. On the contrary, if the value of the sensor is small but not zero (i.e. 0.01m) it would mean that the obstacle is close.
You may use the Evolution Simulator and see how the color of each sensor changes based on how close the obstacle is.
Giving the brain to the car
At this moment, our car can "see" and "move", but there is no "coordinator", that would transform the signals from the "eyes" to the proper movements of the "muscles". We need to give the car a
Brain input
As an input from the sensors, every 100ms the brain will be getting 8 float numbers, each one in range of [0...4]. For example, the input might look like this:
const sensors: Sensors = [s0, s1, s2, s3, s4, s5, s6, s7];
// i.e. 🧠 ← [0, 0.5, 4, 0.002, 0, 3.76, 0, 1.245]
Brain output
Every 100ms the brain should produce two integers as an output:
1. One number as a signal for the engine: engineSignal
2. One number as a signal for the steering wheel: wheelSignal
Each number should be of the type MuscleSignal and might take one of three values: -1, 0, or +1.
Brain formulas/functions
Keeping in mind the brain's input and output mentioned above we may say that the brain is just a function:
const { engineSignal, wheelSignal } = brainToMuscleSignal(
// i.e. { engineSignal: 0, wheelSignal: -1 } ← 🧠 ← [0, 0.5, 4, 0.002, 0, 3.76, 0, 1.245]
Where brainToMuscleSignal() is a function that converts raw brain signals (any float number) to muscle signals (to -1, 0, or +1 number) so that muscles could understand it. We'll implement this
converter function below.
The main question now is what kind of a function the brainFunction() is.
To make the car smarter and its movements to be more sophisticated we could go with a Multilayer Perceptron. The name is a bit scary but this is a simple Neural Network with a basic architecture
(think of it as a big formula with many parameters/coefficients).
I've covered Multilayer Perceptrons with a bit more details in my homemade-machine-learning, machine-learning-experiments, and nano-neuron projects. You may even challenge that simple network to
recognize your written digits.
However, to avoid the introduction of a whole new concept of Neural Networks, we'll go with a much simpler approach and we'll use two Linear Polynomials with multiple variables (to be more precise,
each polynomial will have exactly 8 variables, since we have 8 sensors) which will look something like this:
engineSignal = brainToMuscleSignal(
(e0 * s0) + (e1 * s1) + ... + (e7 * s7) + e8 // <- brainFunction
wheelSignal = brainToMuscleSignal(
(w0 * s0) + (w1 * s1) + ... + (w7 * s7) + w8 // <- brainFunction
• [s0, s1, ..., s7] - the 8 variables, which are the 8 sensor values. These are dynamic.
• [e0, e1, ..., e8] - the 9 coefficients for the engine polynomial. These the car will need to learn, and they will be static.
• [w0, w1, ..., w8] - the 9 coefficients for the steering wheel polynomial. These the car will need to learn, and they will be static
The cost of using the simpler function for the brain will be that the car won't be able to learn some sophisticated moves and also won't be able to generalize well and adapt well to unknown
surroundings. But for our particular parking lot and for the sake of demonstrating the work of a genetic algorithm it should still be enough.
We may implement the generic polynomial function in the following way:
type Coefficients = number[];
// Calculates the value of a linear polynomial based on the coefficients and variables.
const linearPolynomial = (coefficients: Coefficients, variables: number[]): number => {
if (coefficients.length !== (variables.length + 1)) {
throw new Error('Incompatible number of polynomial coefficients and variables');
let result = 0;
coefficients.forEach((coefficient: number, coefficientIndex: number) => {
if (coefficientIndex < variables.length) {
result += coefficient * variables[coefficientIndex];
} else {
// The last coefficient needs to be added up without multiplication.
result += coefficient
return result;
The car's brain in this case will consist of two polynomials and will look like this:
const engineSignal: MuscleSignal = brainToMuscleSignal(
linearPolynomial(engineCoefficients, sensors)
const wheelSignal: MuscleSignal = brainToMuscleSignal(
linearPolynomial(wheelCoefficients, sensors)
The output of a linearPolynomial() function is a float number. The brainToMuscleSignal() function need to convert the wide range of floats to three particular integers, and it will do it in two
1. Convert the float of a wide range (i.e. 0.456 or 3673.45 or -280) to the float in a range of (0...1) (i.e. 0.05 or 0.86)
2. Convert the float in a range of (0...1) to one of three integer values of -1, 0, or +1. For example, the floats that are close to 0 will be converted to -1, the floats that are close to 0.5 will
be converted to 0, and the floats that are close to 1 will be converted to 1.
To do the first part of the conversion we need to introduce a Sigmoid Function which implements the following formula:
It converts the wide range of floats (the x axis) to float numbers with a limited range of (0...1) (the y axis). This is exactly what we need.
Here is how the conversion steps would look on the Sigmoid graph.
The implementation of two conversion steps mentioned above would look like this:
// Calculates the sigmoid value for a given number.
const sigmoid = (x: number): number => {
return 1 / (1 + Math.E ** -x);
// Converts sigmoid value (0...1) to the muscle signals (-1, 0, +1)
// The margin parameter is a value between 0 and 0.5:
// [0 ... (0.5 - margin) ... 0.5 ... (0.5 + margin) ... 1]
const sigmoidToMuscleSignal = (sigmoidValue: number, margin: number = 0.4): MuscleSignal => {
if (sigmoidValue < (0.5 - margin)) {
return -1;
if (sigmoidValue > (0.5 + margin)) {
return 1;
return 0;
// Converts raw brain signal to the muscle signal.
const brainToMuscleSignal = (rawBrainSignal: number): MuscleSignal => {
const normalizedBrainSignal = sigmoid(rawBrainSignal);
return sigmoidToMuscleSignal(normalizedBrainSignal);
Car's genome (DNA)
☝🏻 The main conclusion from the "Eyes", "Muscles" and "Brain" sections above should be this: the coefficients [e0, e1, ..., e8] and [w0, w1, ..., w8] defines the behavior of the car. These 18
numbers together form the unique car's Genome (or car's DNA).
Car genome in a decimal form
Let's join the [e0, e1, ..., e8] and [w0, w1, ..., w8] brain coefficients together to form a car's genome in a decimal form:
// Car genome as a list of decimal numbers (coefficients).
const carGenomeBase10 = [e0, e1, ..., e8, w0, w1, ..., w8];
// i.e. carGenomeBase10 = [17.5, 0.059, -46, 25, 156, -0.085, -0.207, -0.546, 0.071, -58, 41, 0.011, 252, -3.5, -0.017, 1.532, -360, 0.157]
Car genome in a binary form
Let's move one step deeper (to the level of the genes) and convert the decimal numbers of the car's genome to the binary format (to the plain 1s and 0s).
I've described in the detail the process of converting the floating-point numbers to binary numbers in the Binary representation of the floating-point numbers article. You might want to check it
out if the code in this section is not clear.
Here is a quick example of how the floating-point number may be converted to the 16 bits binary number (again, feel free to read this first if the example is confusing):
In our case, to reduce the genome length, we will convert each floating coefficient to the non-standard 10 bits binary number (1 sign bit, 4 exponent bits, 5 fraction bits).
We have 18 coefficients in total, every coefficient will be converted to 10 bits number. It means that the car's genome will be an array of 0s and 1s with a length of 18 * 10 = 180 bits.
For example, for the genome in a decimal format that was mentioned above, its binary representation would look like this:
type Gene = 0 | 1;
type Genome = Gene[];
const genome: Genome = [
// Engine coefficients.
0, 1, 0, 1, 1, 0, 0, 0, 1, 1, // <- 17.5
0, 0, 0, 1, 0, 1, 1, 1, 0, 0, // <- 0.059
1, 1, 1, 0, 0, 0, 1, 1, 1, 0, // <- -46
0, 1, 0, 1, 1, 1, 0, 0, 1, 0, // <- 25
0, 1, 1, 1, 0, 0, 0, 1, 1, 1, // <- 156
1, 0, 0, 1, 1, 0, 1, 1, 0, 0, // <- -0.085
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, // <- -0.207
1, 0, 1, 1, 0, 0, 0, 0, 1, 1, // <- -0.546
0, 0, 0, 1, 1, 0, 0, 1, 0, 0, // <- 0.071
// Wheels coefficients.
1, 1, 1, 0, 0, 1, 1, 0, 1, 0, // <- -58
0, 1, 1, 0, 0, 0, 1, 0, 0, 1, // <- 41
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, // <- 0.011
0, 1, 1, 1, 0, 1, 1, 1, 1, 1, // <- 252
1, 1, 0, 0, 0, 1, 1, 0, 0, 0, // <- -3.5
1, 0, 0, 0, 1, 0, 0, 1, 0, 0, // <- -0.017
0, 0, 1, 1, 1, 1, 0, 0, 0, 1, // <- 1.532
1, 1, 1, 1, 1, 0, 1, 1, 0, 1, // <- -360
0, 0, 1, 0, 0, 0, 1, 0, 0, 0, // <- 0.157
Oh my! The binary genome looks so cryptic. But can you imagine, that these 180 zeroes and ones alone define how the car behaves in the parking lot! It's like you hacked someone's DNA and know what
each gene means exactly. Amazing!
By the way, you may see the exact values of genomes and coefficients for the best performing car on the Evolution Simulator dashboard:
Here is the source code that performs the conversion from binary to decimal format for the floating-point numbers (the brain will need it to decode the genome and to produce the muscle signals based
on the genome data):
type Bit = 0 | 1;
type Bits = Bit[];
type PrecisionConfig = {
signBitsCount: number,
exponentBitsCount: number,
fractionBitsCount: number,
totalBitsCount: number,
type PrecisionConfigs = {
custom: PrecisionConfig,
const precisionConfigs: PrecisionConfigs = {
// Custom-made 10-bits precision for faster evolution progress.
custom: {
signBitsCount: 1,
exponentBitsCount: 4,
fractionBitsCount: 5,
totalBitsCount: 10,
// Converts the binary representation of the floating-point number to decimal float number.
function bitsToFloat(bits: Bits, precisionConfig: PrecisionConfig): number {
const { signBitsCount, exponentBitsCount } = precisionConfig;
// Figuring out the sign.
const sign = (-1) ** bits[0]; // -1^1 = -1, -1^0 = 1
// Calculating the exponent value.
const exponentBias = 2 ** (exponentBitsCount - 1) - 1;
const exponentBits = bits.slice(signBitsCount, signBitsCount + exponentBitsCount);
const exponentUnbiased = exponentBits.reduce(
(exponentSoFar: number, currentBit: Bit, bitIndex: number) => {
const bitPowerOfTwo = 2 ** (exponentBitsCount - bitIndex - 1);
return exponentSoFar + currentBit * bitPowerOfTwo;
const exponent = exponentUnbiased - exponentBias;
// Calculating the fraction value.
const fractionBits = bits.slice(signBitsCount + exponentBitsCount);
const fraction = fractionBits.reduce(
(fractionSoFar: number, currentBit: Bit, bitIndex: number) => {
const bitPowerOfTwo = 2 ** -(bitIndex + 1);
return fractionSoFar + currentBit * bitPowerOfTwo;
// Putting all parts together to calculate the final number.
return sign * (2 ** exponent) * (1 + fraction);
// Converts the 8-bit binary representation of the floating-point number to decimal float number.
function bitsToFloat10(bits: Bits): number {
return bitsToFloat(bits, precisionConfigs.custom);
Brain function working with binary genome
Previously our brain function was working with the decimal form of engineCoefficients and wheelCoefficients polynomial coefficients directly. However, these coefficients are now encoded in the binary
form of a genome. Let's add a decodeGenome() function that will extract coefficients from the genome and let's rewrite our brain functions:
// Car has 16 distance sensors.
const CAR_SENSORS_NUM = 8;
// Additional formula coefficient that is not connected to a sensor.
const BIAS_UNITS = 1;
// How many genes do we need to encode each numeric parameter for the formulas.
const GENES_PER_NUMBER = precisionConfigs.custom.totalBitsCount;
// Based on 8 distance sensors we need to provide two formulas that would define car's behavior:
// 1. Engine formula (input: 8 sensors; output: -1 (backward), 0 (neutral), +1 (forward))
// 2. Wheels formula (input: 8 sensors; output: -1 (left), 0 (straight), +1 (right))
const ENGINE_FORMULA_GENES_NUM = (CAR_SENSORS_NUM + BIAS_UNITS) * GENES_PER_NUMBER;
const WHEELS_FORMULA_GENES_NUM = (CAR_SENSORS_NUM + BIAS_UNITS) * GENES_PER_NUMBER;
// The length of the binary genome of the car.
const GENOME_LENGTH = ENGINE_FORMULA_GENES_NUM + WHEELS_FORMULA_GENES_NUM;
type DecodedGenome = {
engineFormulaCoefficients: Coefficients,
wheelsFormulaCoefficients: Coefficients,
// Converts the genome from a binary form to the decimal form.
const genomeToNumbers = (genome: Genome, genesPerNumber: number): number[] => {
if (genome.length % genesPerNumber !== 0) {
throw new Error('Wrong number of genes in the numbers genome');
const numbers: number[] = [];
for (let numberIndex = 0; numberIndex < genome.length; numberIndex += genesPerNumber) {
const number: number = bitsToFloat10(genome.slice(numberIndex, numberIndex + genesPerNumber));
return numbers;
// Converts the genome from a binary form to the decimal form
// and splits the genome into two sets of coefficients (one set for each muscle).
const decodeGenome = (genome: Genome): DecodedGenome => {
const engineGenes: Gene[] = genome.slice(0, ENGINE_FORMULA_GENES_NUM);
const wheelsGenes: Gene[] = genome.slice(
ENGINE_FORMULA_GENES_NUM + WHEELS_FORMULA_GENES_NUM,
const engineFormulaCoefficients: Coefficients = genomeToNumbers(engineGenes, GENES_PER_NUMBER);
const wheelsFormulaCoefficients: Coefficients = genomeToNumbers(wheelsGenes, GENES_PER_NUMBER);
return {
// Update brain function for the engine muscle.
export const getEngineMuscleSignal = (genome: Genome, sensors: Sensors): MuscleSignal => {
const {engineFormulaCoefficients: coefficients} = decodeGenome(genome);
const rawBrainSignal = linearPolynomial(coefficients, sensors);
return brainToMuscleSignal(rawBrainSignal);
// Update brain function for the wheels muscle.
export const getWheelsMuscleSignal = (genome: Genome, sensors: Sensors): MuscleSignal => {
const {wheelsFormulaCoefficients: coefficients} = decodeGenome(genome);
const rawBrainSignal = linearPolynomial(coefficients, sensors);
return brainToMuscleSignal(rawBrainSignal);
Self-driving car problem statement
☝🏻 So, finally, we've got to the point when the high-level problem of making the car to be a self-parking car is broken down to the straightforward optimization problem of finding the optimal
combination of 180 ones and zeros (finding the "good enough" car's genome). Sounds simple, doesn't it?
Naive approach
We could approach the problem of finding the "good enough" genome in a naive way and try out all possible combinations of genes:
1. [0, ..., 0, 0], and then...
2. [0, ..., 0, 1], and then...
3. [0, ..., 1, 0], and then...
4. [0, ..., 1, 1], and then...
5. ...
But, let's do some math. With 180 bits and with each bit being equal either to 0 or to 1 we would have 2^180 (or 1.53 * 10^54) possible combinations. Let's say we would need to give 15s to each car
to see if it will park successfully or not. Let's also say that we may run a simulation for 10 cars at once. Then we would need 15 * (1.53 * 10^54) / 10 = 2.29 * 10^54 [seconds] which is 7.36 * 10^46
[years]. Pretty long waiting time. Just as a side thought, it is only 2.021 * 10^3 [years] that have passed after Christ was born.
Genetic approach
We need a faster algorithm to find the optimal value of the genome. This is where the genetic algorithm comes to the rescue. We might not find the best value of the genome, but there is a chance that
we may find the optimal value of it. And, what is, more importantly, we don't need to wait that long. With the Evolution Simulator I was able to find a pretty good genome within 24 [hours].
Genetic algorithm basics
A genetic algorithms (GA) inspired by the process of natural selection, and are commonly used to generate high-quality solutions to optimization problems by relying on biologically inspired operators
such as crossover, mutation and selection.
The problem of finding the "good enough" combination of genes for the car looks like an optimization problem, so there is a good chance that GA will help us here.
We're not going to cover a genetic algorithm in all details, but on a high level here are the basic steps that we will need to do:
1. CREATE – the very first generation of cars can't come out of nothing, so we will generate a set of random car genomes (set of binary arrays with the length of 180) at the very beginning. For
example, we may create ~1000 cars. With a bigger population the chances to find the optimal solution (and to find it faster) increase.
2. SELECT - we will need to select the fittest individuums out of the current generation for further mating (see the next step). The fitness of each individuum will be defined based on the fitness
function, which in our case, will show how close the car approached the target parking spot. The closer the car to the parking spot, the fitter it is.
3. MATE – simply saying we will allow the selected "♂ father-cars" to have "sex" with the selected "♀ mother-cars" so that their genomes could mix in a ~50/50 proportion and produce "♂♀
children-cars" genomes. The idea is that the children cars might get better (or worse) in self-parking, by taking the best (or the worst) bits from their parents.
4. MUTATE - during the mating process some genes may randomly mutate (1s and 0s in child genome may flip). This may bring a wider variety of children genomes and, thus, a wider variety of children
cars behavior. Imagine that the 1st bit was accidentally set to 0 for all ~1000 cars. The only way to try the car with the 1st bit being set to 1 is through the random mutations. At the same
time, extensive mutations may ruin healthy genomes.
5. Go to "Step 2" unless the number of generations has reached the limit (i.e. 100 generations have passed) or unless the top-performing individuums have reached the expected fitness function value
(i.e. the best car has approached the parking spot closer than 1 meter). Otherwise, quit.
Evolving the car's brain using a Genetic Algorithm
Before launching the genetic algorithm let's go and create the functions for the "CREATE", "SELECT", "MATE" and "MUTATE" steps of the algorithm.
Functions for the CREATE step
The createGeneration() function will create an array of random genomes (a.k.a. population or generation) and will accept two parameters:
• generationSize - defines the size of the generation. This generation size will be preserved from generation to generation.
• genomeLength - defines the genome length of each individuum in the cars population. In our case, the length of the genome will be 180.
There is a 50/50 chance for each gene of a genome to be either 0 or 1.
type Generation = Genome[];
type GenerationParams = {
generationSize: number,
genomeLength: number,
function createGenome(length: number): Genome {
return new Array(length)
.map(() => (Math.random() < 0.5 ? 0 : 1));
function createGeneration(params: GenerationParams): Generation {
const { generationSize, genomeLength } = params;
return new Array(generationSize)
.map(() => createGenome(genomeLength));
Functions for the MUTATE step
The mutate() function will mutate some genes randomly based on the mutationProbability value.
For example, if the mutationProbability = 0.1 then there is a 10% chance for each genome to be mutated. Let's say if we would have a genome of length 10 that looks like [0, 0, 0, 0, 0, 0 ,0 ,0 ,0 ,0]
, then after the mutation, there will be a chance that 1 gene will be mutated and we may get a genome that might look like [0, 0, 0, 1, 0, 0 ,0 ,0 ,0 ,0].
// The number between 0 and 1.
type Probability = number;
// @see: https://en.wikipedia.org/wiki/Mutation_(genetic_algorithm)
function mutate(genome: Genome, mutationProbability: Probability): Genome {
for (let geneIndex = 0; geneIndex < genome.length; geneIndex += 1) {
const gene: Gene = genome[geneIndex];
const mutatedGene: Gene = gene === 0 ? 1 : 0;
genome[geneIndex] = Math.random() < mutationProbability ? mutatedGene : gene;
return genome;
Functions for the MATE step
The mate() function will accept the father and the mother genomes and will produce two children. We will imitate the real-world scenario and also do the mutation during the mating.
Each bit of the child genome will be defined based on the values of the correspondent bit of the father's or mother's genomes. There is a 50/50% probability that the child will inherit the bit of the
father or the mother. For example, let's say we have genomes of length 4 (for simplicity reasons):
Father's genome: [0, 0, 1, 1]
Mother's genome: [0, 1, 0, 1]
↓ ↓ ↓ ↓
Possible kid #1: [0, 1, 1, 1]
Possible kid #2: [0, 0, 1, 1]
In the example above the mutation were not taken into account.
Here is the function implementation:
// Performs Uniform Crossover: each bit is chosen from either parent with equal probability.
// @see: https://en.wikipedia.org/wiki/Crossover_(genetic_algorithm)
function mate(
father: Genome,
mother: Genome,
mutationProbability: Probability,
): [Genome, Genome] {
if (father.length !== mother.length) {
throw new Error('Cannot mate different species');
const firstChild: Genome = [];
const secondChild: Genome = [];
// Conceive children.
for (let geneIndex = 0; geneIndex < father.length; geneIndex += 1) {
Math.random() < 0.5 ? father[geneIndex] : mother[geneIndex]
Math.random() < 0.5 ? father[geneIndex] : mother[geneIndex]
return [
mutate(firstChild, mutationProbability),
mutate(secondChild, mutationProbability),
Functions for the SELECT step
To select the fittest individuums for further mating we need a way to find out the fitness of each genome. To do this we will use a so-called fitness function.
The fitness function is always related to the particular task that we try to solve, and it is not generic. In our case, the fitness function will measure the distance between the car and the parking
spot. The closer the car to the parking spot, the fitter it is. We will implement the fitness function a bit later, but for now, let's introduce the interface for it:
type FitnessFunction = (genome: Genome) => number;
Now, let's say we have fitness values for each individuum in the population. Let's also say that we sorted all individuums by their fitness values so that the first individuums are the strongest
ones. How should we select the fathers and the mothers from this array? We need to do the selection in a way, that the higher the fitness value of the individuum, the higher the chances of this
individuum being selected for mating. The weightedRandom() function will help us with this.
// Picks the random item based on its weight.
// The items with a higher weight will be picked more often.
const weightedRandom = <T>(items: T[], weights: number[]): { item: T, index: number } => {
if (items.length !== weights.length) {
throw new Error('Items and weights must be of the same size');
// Preparing the cumulative weights array.
// For example:
// - weights = [1, 4, 3]
// - cumulativeWeights = [1, 5, 8]
const cumulativeWeights: number[] = [];
for (let i = 0; i < weights.length; i += 1) {
cumulativeWeights[i] = weights[i] + (cumulativeWeights[i - 1] || 0);
// Getting the random number in a range [0...sum(weights)]
// For example:
// - weights = [1, 4, 3]
// - maxCumulativeWeight = 8
// - range for the random number is [0...8]
const maxCumulativeWeight = cumulativeWeights[cumulativeWeights.length - 1];
const randomNumber = maxCumulativeWeight * Math.random();
// Picking the random item based on its weight.
// The items with higher weight will be picked more often.
for (let i = 0; i < items.length; i += 1) {
if (cumulativeWeights[i] >= randomNumber) {
return {
item: items[i],
index: i,
return {
item: items[items.length - 1],
index: items.length - 1,
The usage of this function is pretty straightforward. Let's say you really like bananas and want to eat them more often than strawberries. Then you may call const fruit = weightedRandom(['banana',
'strawberry'], [9, 1]), and in ≈9 out of 10 cases the fruit variable will be equal to banana, and only in ≈1 out of 10 times it will be equal to strawberry.
To avoid losing the best individuums (let's call them champions) during the mating process we may also introduce a so-called longLivingChampionsPercentage parameter. For example, if the
longLivingChampionsPercentage = 10, then 10% of the best cars from the previous population will be carried over to the new generation. You may think about it as there are some long-living individuums
that can live a long life and see their children and even grandchildren.
Here is the actual implementation of the select() function:
// The number between 0 and 100.
type Percentage = number;
type SelectionOptions = {
mutationProbability: Probability,
longLivingChampionsPercentage: Percentage,
// @see: https://en.wikipedia.org/wiki/Selection_(genetic_algorithm)
function select(
generation: Generation,
fitness: FitnessFunction,
options: SelectionOptions,
) {
const {
} = options;
const newGeneration: Generation = [];
const oldGeneration = [...generation];
// First one - the fittest one.
oldGeneration.sort((genomeA: Genome, genomeB: Genome): number => {
const fitnessA = fitness(genomeA);
const fitnessB = fitness(genomeB);
if (fitnessA < fitnessB) {
return 1;
if (fitnessA > fitnessB) {
return -1;
return 0;
// Let long-liver champions continue living in the new generation.
const longLiversCount = Math.floor(longLivingChampionsPercentage * oldGeneration.length / 100);
if (longLiversCount) {
oldGeneration.slice(0, longLiversCount).forEach((longLivingGenome: Genome) => {
// Get the data about he fitness of each individuum.
const fitnessPerOldGenome: number[] = oldGeneration.map((genome: Genome) => fitness(genome));
// Populate the next generation until it becomes the same size as a old generation.
while (newGeneration.length < generation.length) {
// Select random father and mother from the population.
// The fittest individuums have higher chances to be selected.
let father: Genome | null = null;
let fatherGenomeIndex: number | null = null;
let mother: Genome | null = null;
let matherGenomeIndex: number | null = null;
// To produce children the father and mother need each other.
// It must be two different individuums.
while (!father || !mother || fatherGenomeIndex === matherGenomeIndex) {
const {
item: randomFather,
index: randomFatherGenomeIndex,
} = weightedRandom<Genome>(generation, fitnessPerOldGenome);
const {
item: randomMother,
index: randomMotherGenomeIndex,
} = weightedRandom<Genome>(generation, fitnessPerOldGenome);
father = randomFather;
fatherGenomeIndex = randomFatherGenomeIndex;
mother = randomMother;
matherGenomeIndex = randomMotherGenomeIndex;
// Let father and mother produce two children.
const [firstChild, secondChild] = mate(father, mother, mutationProbability);
// Depending on the number of long-living champions it is possible that
// there will be the place for only one child, sorry.
if (newGeneration.length < generation.length) {
return newGeneration;
Fitness function
The fitness of the car will be defined by the distance from the car to the parking spot. The higher the distance, the lower the fitness.
The final distance we will calculate is an average distance from 4 car wheels to the correspondent 4 corners of the parking spot. This distance we will call the loss which is inversely proportional
to the fitness.
Calculating the distance between each wheel and each corner separately (instead of just calculating the distance from the car center to the parking spot center) will make the car preserve the proper
orientation relative to the parking spot.
The distance between two points in space will be calculated based on the Pythagorean theorem like this:
type NumVec3 = [number, number, number];
// Calculates the XZ distance between two points in space.
// The vertical Y distance is not being taken into account.
const euclideanDistance = (from: NumVec3, to: NumVec3) => {
const fromX = from[0];
const fromZ = from[2];
const toX = to[0];
const toZ = to[2];
return Math.sqrt((fromX - toX) ** 2 + (fromZ - toZ) ** 2);
The distance (the loss) between the car and the parking spot will be calculated like this:
type RectanglePoints = {
fl: NumVec3, // Front-left
fr: NumVec3, // Front-right
bl: NumVec3, // Back-left
br: NumVec3, // Back-right
type GeometricParams = {
wheelsPosition: RectanglePoints,
parkingLotCorners: RectanglePoints,
const carLoss = (params: GeometricParams): number => {
const { wheelsPosition, parkingLotCorners } = params;
const {
fl: flWheel,
fr: frWheel,
br: brWheel,
bl: blWheel,
} = wheelsPosition;
const {
fl: flCorner,
fr: frCorner,
br: brCorner,
bl: blCorner,
} = parkingLotCorners;
const flDistance = euclideanDistance(flWheel, flCorner);
const frDistance = euclideanDistance(frWheel, frCorner);
const brDistance = euclideanDistance(brWheel, brCorner);
const blDistance = euclideanDistance(blWheel, blCorner);
return (flDistance + frDistance + brDistance + blDistance) / 4;
Since the fitness should be inversely proportional to the loss we'll calculate it like this:
const carFitness = (params: GeometricParams): number => {
const loss = carLoss(params);
// Adding +1 to avoid a division by zero.
return 1 / (loss + 1);
You may see the fitness and the loss values for a specific genome and for a current car position on the Evolution Simulator dashboard:
Launching the evolution
Let's put the evolution functions together. We're going to "create the world", launch the evolution loop, make the time going, the generation evolving, and the cars learning how to park.
To get the fitness values of each car we need to run a simulation of the cars behavior in a virtual 3D world. The Evolution Simulator does exactly that - it runs the code below in the simulator,
which is made with Three.js:
// Evolution setup example.
// Configurable via the Evolution Simulator.
const GENERATION_SIZE = 1000;
const LONG_LIVING_CHAMPIONS_PERCENTAGE = 6;
const MUTATION_PROBABILITY = 0.04;
const MAX_GENERATIONS_NUM = 40;
// Fitness function.
// It is like an annual doctor's checkup for the cars.
const carFitnessFunction = (genome: Genome): number => {
// The evolution simulator calculates and stores the fitness values for each car in the fitnessValues map.
// Here we will just fetch the pre-calculated fitness value for the car in current generation.
const genomeKey = genome.join('');
return fitnessValues[genomeKey];
// Creating the "world" with the very first cars generation.
let generationIndex = 0;
let generation: Generation = createGeneration({
generationSize: GENERATION_SIZE,
genomeLength: GENOME_LENGTH, // <- 180 genes
// Starting the "time".
while(generationIndex < MAX_GENERATIONS_NUM) {
// SIMULATION IS NEEDED HERE to pre-calculate the fitness values.
// Selecting, mating, and mutating the current generation.
generation = select(
mutationProbability: MUTATION_PROBABILITY,
longLivingChampionsPercentage: LONG_LIVING_CHAMPIONS_PERCENTAGE,
// Make the "time" go by.
generationIndex += 1;
// Here we may check the fittest individuum of the latest generation.
const fittestCar = generation[0];
After running the select() function, the generation array is sorted by the fitness values in descending order. Therefore, the fittest car will always be the first car in the array.
The 1st generation of cars with random genomes will behave something like this:
On the ≈40th generation the cars start learning what the self-parking is and start getting closer to the parking spot:
Another example with a bit more challenging starting point:
The cars are hitting some other cars along the way, and also are not perfectly fitting the parking spot, but this is only the 40th generation since the creation of the world for them, so you may give
the cars some more time to learn.
From generation to generation we may see how the loss values are going down (which means that fitness values are going up). The P50 Avg Loss shows the average loss value (average distance from the
cars to the parking spot) of the 50% of fittest cars. The Min Loss shows the loss value of the fittest car in each generation.
You may see that on average the 50% of the fittest cars of the generation are learning to get closer to the parking spot (from 5.5m away from the parking spot to 3.5m in 35 generations). The trend
for the Min Loss values is less obvious (from 1m to 0.5m with some noise signals), however from the animations above you may see that cars have learned some basic parking moves.
In this article, we've broken down the high-level task of creating the self-parking car to the straightforward low-level task of finding the optimal combination of 180 ones and zeroes (finding the
optimal car genome).
Then we've applied the genetic algorithm to find the optimal car genome. It allowed us to get pretty good results in several hours of simulation (instead of many years of running the naive approach).
You may launch the 🚕 Self-parking Car Evolution Simulator to see the evolution process directly in your browser. The simulator gives you the following opportunities:
The full genetic source code that was shown in this article may also be found in the Evolution Simulator repository. If you are one of those folks who will actually count and check the number of
lines to make sure there are less than 500 of them (excluding tests), please feel free to check the code here 🥸.
There are still some unresolved issues with the code and the simulator:
• The car's brain is oversimplified and it uses linear equations instead of, let's say, neural networks. It makes the car not adaptable to the new surroundings or to the new parking lot types.
• We don't decrease the car's fitness value when the car is hitting the other car. Therefore the car doesn't "feel" any guilt in creating the road accident.
• The evolution simulator is not stable. It means that the same car genome may produce different fitness values, which makes the evolution less efficient.
• The evolution simulator is also very heavy in terms of performance, which slows down the evolution progress since we can't train, let's say, 1000 cars at once.
• Also the Evolution Simulator requires the browser tab to be open and active to perform the simulation.
• and more...
However, the purpose of this article was to have some fun while learning how the genetic algorithm works and not to build a production-ready self-parking Teslas. So, even with the issues mentioned
above, I hope you've had a good time going through the article.
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Shock waves and compactons for fifth-order non-linear dispersion equations
The following first problem is posed - to justify that the standing shock wave S-(x) = -sign x ={-1 for x < 0, 1 for x > 0 F6 r x < 0, is a correct 'entropy solution' of the Cauchy problem for the
fifth-ordcr degenerate non-linear dispersion equations (NDEs), as for the classic Euler one u(1) + uu(x) = 0, u(1) = -(uu(x))(xxxx) and u(1) = -(uu(xxx))(x) m R x R+. These two quasi-linear
degenerate partial differential equations (PDEs) arc chosen as typical representatives, so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the
opposite initial shock S+(x) = -S-(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C-infinity for any t > 0 Formation of shocks leads to non-uniqueness of
any 'entropy solutions'. Similar phenomena are studied for a fifth-order in time NDE u(uuu) = (uu(x))(xxxx) in normal form On the other hand, related NDEs, such as u(t) = -(|u|u(x))(xxxx) + |u|u(x)
in R x R+, are shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first
examples by Dcy 1998, Phys Rev E, vol. 57, pp 4733-4738, and Rosenau and Levy, 1999, Phys Lett. A, vol. 252, pp 297-306), are non-existent in general and are not robust relative to small
perturbations of parameters of the PDE.
Dive into the research topics of 'Shock waves and compactons for fifth-order non-linear dispersion equations'. Together they form a unique fingerprint. | {"url":"https://researchportal.bath.ac.uk/en/publications/shock-waves-and-compactons-for-fifth-order-non-linear-dispersion-","timestamp":"2024-11-05T19:20:53Z","content_type":"text/html","content_length":"54448","record_id":"<urn:uuid:bad7df12-9e03-4124-afad-caae51ec9056>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00871.warc.gz"} |
Encyclopedia:Statistical mechanics
From Scholarpedia
Encyclopedia of statistical mechanics
Giovanni Gallavotti accepted the invitation on 5 February 2009.
Related encyclopedias
An encyclopedia in Scholarpedia is a portal page containing lists of articles/authors on a given topic and other useful informations. A list of encyclopedias related to Statistical mechanics follows.
Examples of articles in progress or in project
Examples of published articles | {"url":"http://www.scholarpedia.org/article/Encyclopedia:Statistical_mechanics","timestamp":"2024-11-06T18:11:05Z","content_type":"text/html","content_length":"27429","record_id":"<urn:uuid:9086836a-10e7-4bb0-a122-8df131246776>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00222.warc.gz"} |
Unmet Hours
Eplus Conduction Rate and Uvalue of the wall check
Dear all,
I am performing a simulation assuming the following parameters
• SurfaceConvectionAlgorithm:Inside,TARP
• SurfaceConvectionAlgorithm:Outside DOE 2 -Heat Balance algorithm: ConductionTransferFunction
• Zone air heat balance :AnalyticalSolution
Energy plus version 8.9
I have assessed the uvalue of a wall without surface thermal resistances because are both calculated by Eplus itself per each timestep.
Assuming that my wall has an U value of x W/m2K and i want to check if this value is the actual one calculated by the software, should I have to subtract from
Surface Avg Face Conduction Rate - Surf Outside HG Rate - Surface Inside HG Rate
and then divided it by the difference in temperature btw indoor and outdoor and the overall area? ( case without take into account conductive solar gain)
Following the previous for a wall of 0.98 W/m2K after the permutations i get a result that is almost two times. Why?
Furthermore If i divide the Surface Avg Face Conduction Rate by the difference in temperature btw indoor and outdoor and the overall area (without solar conductive gain) i don't get a result that is
equal to the sum of the above mentioned uvalue plus the outside and inside convective coefficients .
Did I forget something?
I am very concerned because I want to check manually the reliability of the results and i still don't get any correct Uvalue, thus leading to higher thermal loads.
Thanks in advance
1 Answer
Sort by » oldest newest most voted
If the surface construction has any thermal mass, this calculation will be nearly impossible unless you force a steady state condition. The winter design days that are included in the ddy files are
typically constant temperature with no sun. Run a sizing period simulation for a winter design day and you should be able to match the heat flow with a UADeltaT calculation. You can use the indoor to
outdoor deltaT if you include the convection coefficients, or use the surface inside and outside face temperatures with just the wall conductance (no convection coefficients).
edit flag offensive delete link more
Why it is impossible? if i clear identify all the components of the heat balance, why can’t I get the U value (no sunlight and removing the superficial resistances)?
at the end of the day If I remove the convective rates and the heatstorage one i should obtain the correct amount conductive transfer rate to be divided by the delta btwn indoor and outdoor surface
Am I correct?
lukanuts ( 2018-10-17 03:16:03 -0600 )edit
Surface Average Face Conduction is not a fundamental result, it is an estimate. Same is true for Surface Heat Storage Rate. The descriptions in the Input Output Reference hint at this with the phrase
"... in a nominal way." Heat storage within the surface is buried in the conduction transfer history terms and is never directly calculated in the heat balance. So, that's why it's best to use a
steady-state case to begin with.
MJWitte ( 2018-10-17 16:27:57 -0600 )edit
Ok clear. So I can either use the summary tab from IDF file or the design days. thanks
lukanuts ( 2018-10-19 10:30:42 -0600 )edit | {"url":"https://unmethours.com/question/34578/eplus-conduction-rate-and-uvalue-of-the-wall-check/","timestamp":"2024-11-12T00:04:10Z","content_type":"application/xhtml+xml","content_length":"62975","record_id":"<urn:uuid:1b7ca5c3-e833-41ca-bcad-f5b622761853>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00662.warc.gz"} |
Explanation of Mass in Physics
Mass of a subatomic particle can be explained as
its receptiveness to (interactivity with) the gravitational wave –
massiveness of a particle being due to approximating in diameter
¼ of a gravitational wave-length, similarly as
the receptivity of a radio-receiver’s antenna to the ¼ length of the carrier-wave.
Thus, the diameter of a nucleon is more nearly ¼ length of
the standard for the gravitational wave (all such waves being of the same length),
than is the diameter of an electron; and a photon is so disparate from that length
(being excessively long) as to have very little receptivity (or mass).
[The shock-wave accompanying a particle accelerated by a magnetic field –
falsely misinterpreted a mass-gain praedicted (wrongly) by Special Relativity –
is apparently longer than the gravitational wave.]
Because it is dependent only on the standard gravitational wave-length,
and not on the local field-strength of the gravitational field,
mass (inertial) would be independent of local gravitational force.
This state of affairs could well be due to mass being
a function only of the particle’s gravitational-wave transmissive
(as distinguished from its receptive) activity;
again due to space (or the aether) being conductive to
the general type gravitational waves only at one specified wave-length –
the reason for space’s (aether’s) partiality to this being to
its being a specific (minimum = 4 ?) multiple of the size of the space-quantum. | {"url":"http://strange.00.gs/mass,_explanation_of.htm","timestamp":"2024-11-07T15:47:48Z","content_type":"text/html","content_length":"2379","record_id":"<urn:uuid:2a37d18b-0aef-47d8-8ca7-559c55cb7939>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00119.warc.gz"} |
to cubic meter
Category: main menu • concrete menu • Kilonewtons
concrete conversion
Amount: 1 kilonewton (kN) of gravity force
Equals: 0.042 cubic meters (m3) in volume
Converting kilonewton to cubic meters value in the concrete units scale.
TOGGLE : from cubic meters into kilonewtons in the other way around.
CONVERT : between other concrete measuring units - complete list.
Conversion calculator for webmasters.
This general purpose concrete formulation, called also concrete-aggregate (4:1 - sand/gravel aggregate : cement - mixing ratio w/ water) conversion tool is based on the concrete mass density of 2400
kg/m3 - 150 lbs/ft3 after curing (rounded). Unit mass per cubic centimeter, concrete has density 2.41g/cm3. The main concrete calculator page.
The 4:1 strength concrete mixing formula applies the measuring portions in volume sense (e.g. 4 buckets of concrete aggregate, which consists of gravel and sand, with 1 bucket of cement.) In order
not to end up with a too wet concrete, add water gradually as the mixing progresses. If mixing concrete manually by hand; mix dry matter portions first and only then add water. This concrete type is
commonly reinforced with metal rebars or mesh.
Convert concrete measuring units between kilonewton (kN) and cubic meters (m3) but in the other reverse direction from cubic meters into kilonewtons.
conversion result for concrete:
From Symbol Result To Symbol
1 kilonewton kN = 0.042 cubic meters m3
Converter type: concrete measurements
This online concrete from kN into m3 converter is a handy tool not just for certified or experienced professionals.
First unit: kilonewton (kN) is used for measuring gravity force.
Second: cubic meter (m3) is unit of volume.
concrete per 0.042 m3 is equivalent to 1 what?
The cubic meters amount 0.042 m3 converts into 1 kN, one kilonewton. It is the EQUAL concrete gravity force value of 1 kilonewton but in the cubic meters volume unit alternative.
How to convert 2 kilonewtons (kN) of concrete into cubic meters (m3)? Is there a calculation formula?
First divide the two units variables. Then multiply the result by 2 - for example:
0.042372881355932 * 2 (or divide it by / 0.5)
1 kN of concrete = ? m3
1 kN = 0.042 m3 of concrete
Other applications for concrete units calculator ...
With the above mentioned two-units calculating service it provides, this concrete converter proved to be useful also as an online tool for:
1. practicing kilonewtons and cubic meters of concrete ( kN vs. m3 ) measuring values exchange.
2. concrete amounts conversion factors - between numerous unit pairs.
3. working with - how heavy is concrete - values and properties.
International unit symbols for these two concrete measurements are:
Abbreviation or prefix ( abbr. short brevis ), unit symbol, for kilonewton is:
Abbreviation or prefix ( abbr. ) brevis - short unit symbol for cubic meter is:
One kilonewton of concrete converted to cubic meter equals to 0.042 m3
How many cubic meters of concrete are in 1 kilonewton? The answer is: The change of 1 kN ( kilonewton ) unit of concrete measure equals = to 0.042 m3 ( cubic meter ) as the equivalent measure for the
same concrete type.
In principle with any measuring task, switched on professional people always ensure, and their success depends on, they get the most precise conversion results everywhere and every-time. Not only
whenever possible, it's always so. Often having only a good idea ( or more ideas ) might not be perfect nor good enough solution. If there is an exact known measure in kN - kilonewtons for concrete
amount, the rule is that the kilonewton number gets converted into m3 - cubic meters or any other concrete unit absolutely exactly. | {"url":"https://www.traditionaloven.com/building/masonry/concrete/convert-kilonewton-kn-for-concrete-to-cubic-metre-m3-concrete.html","timestamp":"2024-11-01T22:46:28Z","content_type":"text/html","content_length":"40049","record_id":"<urn:uuid:7c9f614e-3a95-4161-8700-4c562a454c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00160.warc.gz"} |
Solve mean(3,4) | Microsoft Math Solver
\frac{7}{2} = 3\frac{1}{2} = 3.5
Similar Problems from Web Search
To find the mean of the set 3,4 first add the members together.
The mean (average) of the set 3,4 is found by dividing the sum of its members by the number of the members, in this case 2. | {"url":"https://mathsolver.microsoft.com/en/topic/pre-algebra/mean/solve/mean(3%2C4)","timestamp":"2024-11-03T08:51:31Z","content_type":"text/html","content_length":"304376","record_id":"<urn:uuid:ac90eaa1-128c-48ac-9c74-5b72588a90dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00239.warc.gz"} |
A373624 - OEIS
Choose p in Z/nZ, then generate the finite subset {1,p,p^2,p^3,p^4,...}. It often happens that two different p give the same subset. Therefore, there may be less distinct subsets than n. a(n) gives
the numbers of distinct subsets generated by all p in Z/nZ. Note that the subsets generated by 0, 1, -1 are counted. Those subsets are {1,0}, {1}, {1,-1}.
a(7) = 5 because there are 5 distinct power generated subsets of Z/7Z, namely 0^i = {1,0}, 1^i = {1}, 2^i = {1,2,4}, 3^i = {1,3,2,6,4,5}, 6^i = {1,6}. 4^i generates the same subset as 2^i (in a
different order, but that is irrelevant). 5^i generate the same subset as 3^i (in a different order).
#include <iostream>
#include <vector>
#include <set>
using namespace std;
// computes the number of power generated subsets of Z/nZ
int A (int n)
// all subsets are stored here
set<vector<bool>> subsets;
for (int p=0; p<n; p++) {
// a n-bits vector of already seen powers of p
vector<bool> powers(n);
// fill in powers
for (int q=1; !powers[q]; q = (q*p)%n)
powers[q] = true;
// store one subset,
// but only if it is not already stored
// return number of distinct subsets
return subsets.size();
int main()
for (int n=0; n<30; n++)
(PARI) a(n) = #Set(vector(n, i, Set(vector(n, j, Mod(i-1, n)^(j-1))))); \\
Michel Marcus
, Jun 12 2024 | {"url":"https://oeis.org/A373624","timestamp":"2024-11-10T00:09:51Z","content_type":"text/html","content_length":"16315","record_id":"<urn:uuid:7265ff37-ab73-486f-8963-0b3171e037eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00817.warc.gz"} |
White Smoke Over Oxford?
I’ve stolen the title of this posting from Michael Harris, see his posting for a discussion of the same topic.
A big topic of discussion among mathematicians this week is the ongoing workshop at Oxford devoted to Mochizuki’s claimed proof of the abc conjecture. For some background, see here. I first wrote
about this when news arrived more than three years ago, with a comment that has turned out to be more accurate than I expected “it may take a very long time to see if this is really a proof.”
While waiting for news from Oxford, I thought it might be a good idea to explain a bit how this looks to mathematicians, since I think few people outside the field really understand what goes on when
a new breakthrough happens in mathematics. It should be made clear from the beginning that I am extremely far from expert in any of this mathematics. These are very general comments, informed a bit
by some conversations with those much more expert.
What I’m very sure is not going to happen this week is “white smoke” in the sense of the gathered experts there announcing that Mochizuki’s proof is correct. Before this can happen a laborious
process of experts going through the proof looking for subtle problems in the details needs to take place, and that won’t be quick.
The problem so far has been that experts in this area haven’t been able to get off the ground, taking the first step needed. Given a paper claiming a proof of some well-known conjecture that no one
has been able to prove, an expert is not going to carefully read from the beginning, checking each step, but instead will skim the paper looking for something new. If no new idea is visible, the
tentative conclusion is likely to be that the proof is unlikely to work (in which case, depending on circumstances, spending more time on the paper may or may not be worthwhile). If there is a new
idea, the next step is to try and understand its implications, how it fits in with everything else known about the subject, and how it may change our best understanding of the subject. After going
through this process it generally becomes clear whether a proof will likely be possible or not, and how to approach the laborious process of checking a proof (i.e. which parts will be routine, which
parts much harder).
Mochizuki’s papers have presented a very unusual challenge. They take up a large number of pages, and develop an argument using very different techniques than people are used to. Experts who try and
skim them end up quickly unable to see their way through a huge forest of unrecognizable features. There definitely are new ideas there, but the problem is connecting them to known mathematics to see
if they say something new about that. The worry is that what Mochizuki has done is create a new formalism with all sorts of new internal features, but no connection to the rest of mathematics deep
enough and powerful enough to tell us something new about that.
Part of the problem has been Mochizuki’s own choices about how to explain his work to the outside world. He feels that he has created a new and different way of looking at the subject, and that those
who want to understand it need to start from the beginning and work their way through the details. But experts who try this have generally given up, frustrated at not being able to identify a new
idea powerful enough in its implications for what they know about to make the effort worthwhile. Mochizuki hasn’t made things easier, with his decision not to travel to talk to other experts, and
with most of the activity of others talking to him and trying to understand his work taking place locally in Japan in Japanese, with little coming out of this in a form accessible to others.
It’s hard to emphasize how incredibly complex, abstract and difficult this subject is. The number of experts is very small and most mathematicians have no hope of doing anything useful here. What’s
happening in Oxford now is that a significant number of experts are devoting the week to their best effort to jointly see if they can understand Mochizuki’s work well enough to identify a new idea,
and together start to explore its implications. The thing to look for when this is over is not a consensus that there’s a proof, but a consensus that there’s a new idea that people have now
understood, one potentially powerful enough to solve the problem.
About this, I’m hearing mixed reports, but I can say that some of what I’m hearing is unexpectedly positive. It now seems quite possible that what will emerge will be some significant understanding
among experts of a new idea. And that will be the moment of a real breakthrough in the subject.
Update: Turns out the “unexpectedly positive” was a reaction to day 3, which covered pre-IUT material. Today, when things turned to the IUT stuff, it did not go well at all. See the link in the
comments from lieven le bruyn to a report from Felipe Voloch. Unfortunately it now looks quite possible that the end result of this workshop will be a consensus that the IUT part of this story is
just hopelessly impenetrable.
Update: Brian Conrad has posted here a long and extremely valuable discussion of the Oxford workshop and the state of attempts to understand Mochizuki’s work. He makes clear where the fundamental
problem has been with communication to other mathematicians, and why this problem still remains even after the workshop. The challenge going forward is to find a way to address it.
37 Responses to White Smoke Over Oxford?
1. Hi Peter,
Thanks for this post. I think what is so confusing about this situation actually comes from how many mathematicians portray the field (at least to the outside world). In this fantasy description,
mathematics is a uniquely objective subject and the only thing that matters is the proof. To my mind, Mochizuki is taking this portrayal quiet literally. In so he is revealing to the wider world
what many Mathematicians are reluctant to admit to the public if not themselves.
In fact, the mathematical literature is very human in that it is filled with ambiguous statements, logical gaps, wrong proofs, attribution errors. With that said, there is *something* objective
about it. In general, when well regarded(1) people from well known research institutions(2) with little history of making big false claims(3) are willing to write up something explicit(4) which
is not too long(5) in roughly familiar terminology(6) in English/French/Russian or other major languages(7) and talk about their work to experts in seminars(8), they trigger a process that
produces a level of scrutiny which is usually equal to the task of validating the claim in a matter of days to months. There is also the perverse requirement that the author not develop too much
deeply original thinking(0) on her or his way to a proof if this process is to complete quickly.
Now, the problem here is that Mochizuki is pushing enough on various parts of 0-8 that it exposes what mathematicians know but are not always happy to admit: that the field, if it is objective,
is not objective in the simple way conveyed to the world. Rather there is an economics of mathematics, a politics of math, and numerous failure modes of math which dominate the short run. In fact
many senior mathematicians are of two minds where they know that the field is somehow completely objective and non-objective.
As a former insider turned outsider, I personally think this situation is terrific. In one simple stroke, Mochizuki has showed us that the field is driven by incentives, culture, and human
failing. I hope that this proof is correct so that in the second act, he can show us that after the politics, recriminations, and resistance, the field usually ends up with something pretty close
to objective truth. Over the long run things are much better, at least as far as the proofs are concerned (if not always the narrative and attribution.)
2. The mood appears to have changed today. Felipe Voloch’s daily update is just in :
ABC day 4 : https://plus.google.com/106680226131440966362/posts/LLHPN3QLoqX
3. Something opposite – Math Quartet Joins Forces on Unified Theory at Quanta: https://www.quantamagazine.org/20151208-four-mathematicians/
4. My two cents, as a mathematician, Peter has the process for this kind of thing exactly right. I watched it happen with Perelman’s proof of Thurston’s geometrization conjecture. I was at a
conference maybe a week after it was released, and various experts in the field were there (my field is pretty close) – they had already been working on it feverishly the whole time. And they
already were saying it looked likely to hold up. Of course that was a very different set up, since Perelman was doing work which followed in the footsteps of Hamilton (who teaches with Peter at
Columbia), and basically what he did was to figure out a way to get through the wall which Hamilton hit. This seems way different, unprecedented more or less. I’m trying to think of a case where
there was a completely new way of approaching something, and no one could understand, and I can’t. Plenty of times someone comes up with a really new way to look at things, but it’s always the
case that people catch on pretty quickly, at least the examples I can think of.Usually, if a bunch of very good mathematicians look at something, and tell you it doesn’t make any sense, it
actually doesn’t make any sense.
5. Given that
1) Its basically in an entirely new sub field and its incredibly rare for someone to move from one area to another
2) At least four people already understand a relatively young theory
3) No mathematician understands the vast majority of extant proofs
Is it maybe not that big a deal if experts working in other areas never understand this proof?
6. Rubbernecker,
This is supposed to be part of arithmetic algebraic geometry, and say new things about that subject. Many of the people at the workshop are among the best arithmetic algebraic geometers in the
What went wrong yesterday is that two of the four people (Mok, Hoshi) who supposedly understand IUT were unable to explain it to anyone else. A third (Yamashita) will try today, but his talks
were supposed to assume material from the other two.
Yes, few people understand the details of many complicated proofs. As I was trying to explain here, that’s not what’s going on. What the experts are trying to find is a new idea that says
something about arithmetic algebraic geometry.
The danger here is that IUT is a subject disconnected from the rest of mathematics. If you spend more time studying it, you will learn lots of new definitions, be able to prove lots of theorems
relating them, but learn nothing new about other mathematics.
7. z,
Opposite indeed. That’s an inspiring story—another great article from Quanta.
8. The Scottish legal system has a verdict which may be applicable in this case: Not Proven.
The stark contrast between this conjecture and the recent, say, Polymath projects is glaring.
9. Eric Weinstein: Your comment seems to imply that there is resistance to Mochizuki’s work due to non-objective factors. There are of course various heuristics (usually with good reasons) which
mathematicians use in deciding whether it is worthwhile for them to invest large amounts of time trying to understand something new. In Mochizuki’s case, we have a conference of leading experts,
including more than one fields medalist, getting together for a week to do their best to understand his work. One can’t really say that his work isn’t getting a fair hearing.
10. Hi Michael,
Thanks for this. I do not mean to imply that Mochizuki is running into something untoward. I mean to imply that we do a disservice to mathematics when we confuse the objective nature of the
underlying subject matter with the way in which humans attempt to do mathematics. I don’t mind the heuristics if they are recognized not to be fundamental truths.
Of course there is a resistance here. He is (to the best of my understanding) not bending over backward to decrease the burden on the proof checkers. I would find that annoying were I an expert
in this field. But I would never pretend that math is uniquely objective as a profession. Proofs may be objective, but proof checking is not.
Let me make an analogy. In computer software, there is a difference between a code review done by a subjective human doing their best and a code review done by the compiler. What I find at turns
amusing and annoying is when mathematicians pretend to be compilers. The compiler doesn’t care about commenting code properly. It doesn’t see whether a style guide has been adhered to or whether
readability mattered to the developer. It just compiles or it fails.
Mochizuki seems not to care particularly for decreasing the burden on his proof checkers. For the subset of those proof checkers who are open about how prone to error and delay this process is, I
believe he is acting sub-optimally and should be pushed to be more helpful. But, for those mathematicians who pretend that the profession is blind to all but the objective truth….I think this is
a splendid reveal. No compiler takes this long.
11. Eric,
What I was trying to get at in my posting is that “proof checking” isn’t really what this is about (at some later date “proof checking” comes into play, but other things have to happen first: you
can’t sensibly check a proof you don’t understand). What mathematicians actually do is something much less mechanical, trying to individually embody understanding of mathematical ideas, and share
this understanding as a community. The problem here is that Mochizuki is not successfully communicating mathematical ideas to the rest of the community. “What we’ve got here is failure to
communicate…” Why this is happening is a fascinating question, involving both the standard ways the community operates, as well as some very unusual special features of this case. I don’t
actually know of any other comparable example, and also it’s not at all clear where this is going to end up.
12. Mochizuki has made minimal effort to engage with the community at large. In that case, workshops of this sort set a dangerous precedent. Simply the fact that he has already demonstrated that he
is a first rate mathematician means that we should take this thing seriously without a real and proper involvement is silly. Until he begins making more of an effort it should be ignored. And the
description given by Veloch of Kuehne’s talk on day 2 shows, I think, that this thing is an embarrassment to the organisers and to Oxford by proxy.
13. There appears to be a black hole: if someone understands Mochizuki, then he can’t be understood by anyone else.
14. “The danger here is that IUT is a subject disconnected from the rest of mathematics.”
This is not unprecedented. In 1968, P. S. Novikov and Adian published a
(negative) solution of the famous Burnside problem. The proof was more then 300 pages long, which is impressive but not unheard of even in 1960s. Its peculiarity is
that it does not use any standard methods; the authors pretty much developed a whole new theory from scratch.
Not being an expert, I can’t say to what degree this theory (extremely hard to master)
is detached from the rest of mathematics. But it looks like this far it has only been used in a very narrow area of research, more or less close to the original problem.
Admittedly, the “Bernside theory” is built on a rather elementary basis, while Mochizuki started from the height of arithmetic geometry, but I would not call it a radical difference.
If the analogy, and the proof itself, is correct, then eventually we will see half a dozen of experts in this theory who understand it, while the rest of mathematicians, even from close fields,
won’t bother. Not terribly promising, but we do not always have what we dream of, in life or in mathematics.
15. Under an earlier post, a commenter said that Mochizuki is a family name associated with the samurai class.
A Scientific American article, “Japanese Temple Geometry,” by Tony Rothman, with assistance from Hidetoshi Fukagawa, May 1, 1998, says that during Japan’s period of national seclusion, (1639 –
1854), there was a tradition in which samurai and others would prove mathematical theorems, usually about Euclidean geometry, and inscribe them on delicately colored wooden tablets called
sangaku, and hang them under the roofs of temples, as an offering to the ancestors. Perhaps Mochizuki views his work in this spirit.
16. I am a complete outsider to this area, but it seems to me that a very important thing that is lacking is some kind of story that explains why Mochizuki’s machinery might be appropriate for
proving the ABC conjecture. If I contrast it with another area I don’t understand — Grothendieck-style algebraic geometry — the latter comes out far more favourably (in this one respect — I’m not
talking about the fact that it has obviously been checked by thousands of mathematicians), because there are all sorts of accounts of how the more abstract way of looking at varieties is a
fruitful thing to do. I know that if I did want to learn it, I wouldn’t be told that I had to become comfortable with a vast array of complicated definitions before any benefits fed back into
what I already know about.
The case with Perelman was again very different: I don’t understand his proof at all, but I did understand accounts for the non-expert that explained about Ricci flow and what it was supposed to
What I would want to see from Mochizuki and his followers is a baby result that can be proved by his methods, that points the way towards more complicated ones.
17. On this Mathoverflow page Minhyong Kim, who is one of the organizers, has just written the following :
Update (12 December, 2015): I’ve written a brief summary of the Oxford workshop on IUTT rather rapidly, so as to save people the trouble of circulating rumours. This seemed to be a reasonable
place to put it. All errors in it are my own: http://people.maths.ox.ac.uk/kimm/papers/iutt=clay.pdf
18. That link seems to be broken (also on the mathoverflow page)?
Best, 🙂
19. The link is probably only temporarily broken … Kim explains it in MathOverflow; he took it down while getting permission to quote Mochizuki.
20. Gowers,
There are cases when it is impossible. The Novikov-Adian theory I mentioned
is an example of a huge machinery designed specifically for an extremely
difficult but very narrow target, which is difficult to use for anything else.
No baby results there.
As I know nothing about IUT I can’t say if it is the case, but it may be
a possibility.
21. Gavrilov, that’s very interesting, but it raises an obvious question. If the only way to hit a narrow target is via a huge, elaborate, and seemingly irrelevant theoretical apparatus that has no
other applications, then how does anybody discover that apparatus in the first place? There must be some story. The least Mochizuki could do is tell us that story. What is the story in the
Novikov-Adian case? It cannot be that, just for fun, they developed an incredibly complicated theory and then observed to their great surprise that it solved precisely one interesting problem.
Sometimes the story is that a theory is developed for another purpose but turns out to be useful for the given problem, but you imply that that is not the case for the the Novikov-Adian theory.
I realize that your point is that there aren’t baby results along the way. Maybe I should generalize my requirement and say that there should be a path from not understanding the theory at all to
understanding it completely that does not require huge leaps of faith that there is some point to what one is learning.
22. I’m far from an expert on the Burnside problem but describing the proof as isolated and without baby results does not seem correct to me. As I have understood it Novikov and Aidan set out to
understand and classify periodic words and cancellation in groups and as a result of that work could prove that there are infinite Burnside groups. So before they got to their famous result they
also had relevant “smaller” results.
23. Gowers, you ask questions I would like to know the answers myself.
Apparently, the idea of a solution first came to Novikov in 1950s, but
I do not know how far it was from the final theory, and what was the *story*.
Much less what is the path “from not understanding the theory at all to
understanding it completely” in this case.
These are interesting questions, but probably they could only be answered by an expert.
24. For those who are interested, I have found a nice piece about the solution
of the general Burnside problem on Mathoverflow, by Mark Sapir.
In particular, it is said that there is no “short description” of Novikov-Adian work.
25. Thank you Chris Austin for the sangaku reference. That was an interesting read:
“Many of the problems are elementary and can be solved in a few lines; they are not the kind of work a professional mathematician would publish. Fukagawa has found a tablet from Mie Prefecture
inscribed with the name of a merchant. Others have names of women and children—12 to 14 years of age. Most, according to Fukagawa, were created by the members of the highly educated samurai
class. A few were probably done by farmers; Fukagawa recalls how about 10 years ago he visited the former cottage of mathematician Sen Sakuma (1819–1896), who taught wasan (native Japanese
mathematics) to the farmers in nearby villages in Fukushima Prefecture. Sakuma had about 2,000 students….
The best answer, then, to the question of who created temple geometry seems to be: everybody. On learning of the sangaku, Fukagawa came to understand that, in those days, many of the Japanese
loved and enjoyed math, as well as poetry and other art forms.”
Sorry Peter for the off-topic posting. Please remove if inappropriate.
26. Gowers, using your terminology from “Two cultures”,
the work of Novikov and Adian is much closer in spirit to problem solving then
to theory building. This is why there is, apparently, no way to describe it in two words.
There are other examples of this sort, although less extreme, such as the Feit-Thompson theorem.
(But probably there are reasons to call this work a new theory, albeit an odd one,
and not a huge collection of tricks. It may be systematic in its own way.)
My point is that when you are focused on a single extremely difficult problem,
then no matter where you started, you have an (unfortunate) chance of producing something as incomprehencible as the Novikov-Adian proof.
27. @Tim Gowers
I think Mochizuki *has* been telling the story of how he came to think of his ideas, and how one should think of his work as approaching the solution: by analogy with other theorems and so on.
The problem is that this other, existing, body of work requires something that doesn’t exist in the arithmo-geometric world, and this is what his theory is designed to give, at the expense of
catapulting out of the usual techniques and objects.
The problem is, there’s too much analogy (“alien arithmetic structures” and so on) and less middle-ground explanation before one gets to pages and pages of definitions. I think Lieven le Bruyn
did a great job of working through what a Frobenioid is, in a simple and known case. Such unpacking is something Mochizuki didn’t do; clearly somebody or some collection of somebodies needs to go
back to the precursor papers and fill in all the worked examples that are absent. This is for me the clearest way forward, and how to approach the massive wall of definitions with something like
a climbing strategy.
28. @Tim Gowers
I think Mochizuki has attempted to tell the story of how he came to develop his ideas in the slightly more expository piece, A Panoramic Overview of Inter-universal Teichmuller Theory, available
on his website:
As far as I can tell, and this is with the caveat that I could be very wrong in such a brief space, this grew from Mochizuki’s proof of a conjecture of Grothendieck in anabelian geometry. One of
the first things he seemed to have done, after proving Grothendieck’s conjecture, was to build an analogue of Hodge theory for Arakelov geometry, which he called Hodge-Arakelov theory and wrote
about in papers in 1999 and 2002.
In the introduction to Panorama, he characterised the current theory as the result of trying to overcome the difficulties of applying scheme-theoretic Hodge-Arakelov theory to diophantine
geometry. The resulting theory appears to be (in part) a theory of non-scheme-theoretic deformations, i.e. a theory that presumably involves geometric structures that go beyond schemes.
29. (Rh L) “[he built] an analogue of Hodge theory for Arakelov geometry, which he called Hodge-Arakelov theory and wrote about in papers in 1999 and 2002. ”
That’s strange, because it would have been more important than proving ABC if he had succeeded. Building a working machinery that unifies Hodge theory with its number theoretic analogue (etale
cohomology and site, l-adic Galois representations) has been a central goal in algebraic geometry for the past 50 years. It would have been the news of the millenium had somebody done this.
The more limited goal of building a more adelic Arakelov geometry, has been around since the late 1970’s, and also considered very important. It is a fearsomely technical subject in ways that run
in a different and much less algebraic direction than anything Mochizuki is known to have published or studied. If he had surmounted the difficulties (1) at all, and even better, (2) using his
methods that don’t rely on heavy doses of modern differential geometry and analysis, that would be considered a titanic achievement. It would have been noticed and recognized in the past 15
That comes to one of the weird points about the Mochizuki ABC papers: where in those documents is there any of the hardcore analysis that one would expect, relating the very general algebra to
the analytic number theory problem that is ABC? One would expect at least a page or two (or fifty) of grungy estimates and hard analysis at least for getting a weak form of ABC that might be
boosted to full ABC by more algebraic arguments. Mochizuki does seem to use the latter, but there isn’t much sign of hard analysis in his papers, and one should be able to find it just by
skimming if it’s there. This is a question that must have come up in various forms at the Oxford conference — where is the hard work being done? — and it would be a big confidence builder if the
believers would just point to the locations in Mochizuki’s papers where the analytic-number-theory part of the action takes place. The idea that it can be black-boxed into a few lines of analysis
and 500 pages of algebra doesn’t sound right.
Having said that, I think the sociological concerns about not leaving Japan to explain the proof, the papers not having been refereed, etc, have been overplayed. The papers contain plenty of
motivation and exposition that is illuminating apart from the stated goal of proving ABC. They are quite discursive compared to anything else in the field of arithmetic geometry, which has more
than its share of long dense papers, and are not in the impenetrable dense theorem-proof style.
(can’t get the formatting to work when posting with firefox, sorry about that.)
30. @David Roberts, thanks for the nice words. I only “checked” one paper as a non-specialist, got stuck and then discovered the wealth hidden in the Arakelov bit. Just the same, if a student would
hand in Fobenioids1 as a paper, she’d have to rewrite it seriously.
Sad to see that some refer to my blog as criticising Mochizuki (or even calling his work ‘nonsense’), most recently at “Todeszone der Mathematik”. I’m just getting tired of his lack of
interest in reaching out.
Also sad to see that Minhyong Kim did not (yet) put his report on last week’s Mochizuki-Fest in Oxford back online. I learned a lot from it. Anyway, luckily there’s always the mysterious
@math_jin Twitter account to repost images of ‘lost’ files.
As my own ‘lost’ blog is getting more hits recently I’ve put a little story online about the Log Lady and the Frobenioid of Z. It’s about the Arakelov bit, but probably only digestible if you did
see Twin Peaks, the log lady, and Norma at the Double R Diner, way back then…
31. @random reader: The “hard analysis” you’re looking for is contained entirely in the known equivalence (from several decades back) between Szpiro’s Conjecture for elliptic curves and the ABC
Conjecture when each is formulated over general number fields (proof going via consideration of Frey curves associated to an ABC triple and a robust variation of the ground field).
Everything Mochizuki is doing is focused on proving Szpiro’s Conjecture for all elliptic curves over all number fields. Moreover, it is in the nature of the method that his main work is in the
case of elliptic curves satisfying certain auxiliary local and global properties that necessitate working over a somewhat large number field (and the general case is then deduced by a very short
argument); in particular, his method does not work directly over Q in the main parts.
For the same reason, one cannot get some insight into Mochizuki’s methods by trying to unravel what they are saying in the context of some of the other known concrete consequences, since his
entire proof takes place in the setting of Szpiro’s Conjecture (whose link to the known consequences via ABC goes through long-known arguments which treat Szpiro’s Conjecture as a black box).
It is somewhat akin to the fact that Wiles’ proof of Fermat’s Last Theorem works not with the Fermat equation, nor even with elliptic curves over Q (for which general modularity in the semistable
case was sufficient to apply to hypothetical Frey curves), but rather with Galois representations and modular forms, which in turn admit powerful operations having no interpretation in terms of
elliptic curves (let alone the Fermat equation).
That is, one cannot get insight into Wiles’ method by thinking solely about the more concrete framework of elliptic curves (because spaces of weight-2 modular forms with a given level generally
have Hecke eigenvalues that are not rational, so not all eigenforms in the space are related to elliptic curves).
32. Brian Conrad,
thanks very much for the comment. Good to see the big guns weighing in here.
I don’t see how Szpiro and ABC differ here, or how the lack of visible hard analysis is comparable to Wiles’ proof of Fermat’s Last Theorem.
In Wiles’ work on FLT, the analytic objects he was showing to exist were known to have a rigid algebraic description with rich properties, and conjectured to satisfy a relatively precise
(Langlands) equivalence between the algebraic and analytic sides of the coin. Wiles made a breakthough on the Galois (algebraic) side and consequences on the automorphic (analytic) one flowed,
but this transfer of results was not itself the novelty of his work. The relations to analysis and the automorphic side of Langlands philosophy were, if vague memory serves, encapsulated in the
use of some results from Tunnell’s work on icosahedral(?) representations. Correct me if that’s wrong, it is surely in your line of expertise and not mine. But the point is that a sufficiently
precise translation to the non-analytic setting was already known and Wiles used it, maybe with some new twists, but the main action was in the algebraic theory, deformation of Galois
representations, the commutative algebra of Hecke rings, and the patching argument with auxiliary primes.
It is a reasonable and important question to understand what Wiles’ proof does at the level of concrete objects such as coefficients of modular forms, since he is ultimately proving an existence
theorem for concrete objects and it is a bit outre if there is no way to describe in principle how the machinery unpacks to some sort of complicated manipulation of those objects. Asking the
equivalent about Mochizuki’s work does not strike me as a form of confusion or category error, but a basic conceptual point that (if answered) would clarify what is happening in his papers.
Returning to Mochizuki’s proof and the absence of visible hard analysis there:
Both the Szpiro conjecture and ABC are analytic conjectures, one about numerical invariants of elliptic curves and the other about invariants of pairs of integers (or infinite families of either
type of object, and the extension to number fields). In both cases one would expect a proof to include some sort of nontrivial estimation process involving inequalities, real/complex/harmonic
analysis, L-functions, differential equations, etc to take place in order to obtain conclusions. Mochizuki works with some version of theta functions, which (maybe in a different setting) were
known for a long time to have a more algebraic description, so to an extent there is an algebraization of the things to be proved, but I am not aware of any purely algebraic statement that is
known to imply ABC or Szpiro or Vojta conjectures. His papers do involve some estimates, but rather short ones that make up very very little of the content of the papers. This is not the
distribution of labor one (or this one commenter) would expect in a paper that purports to accomplish an amazing feat in what is ultimately analytic number theory.
In brief, even taking Szpiro/ABC equivalence as a black box, which might or might not contain a lot of hard analysis, it’s hard to see how the Szpiro part can then be proved without lots of
additional hard analysis. Perhaps Mochizuki shows that there is so much uniformity in the way the elliptic curve invariants vary, that easy estimates will do. But even this would require some
strong results interconnecting the algebra and the analysis and at some point estimates seem likely to intervene.
33. @random reader:
Perhaps I was unclear in the analogy I was trying to make. The aspect of the proof of FLT I was alluding to was just that even though the statement of most primary interest is about showing that
a specific equation has no Q-solution or that a specific q-series with Q-coefficients is in fact a modular form, the actual context for the argument must take place (as you are aware) with Hecke
operators acting on spaces with eigenvalues outside of Q and with Galois deformation rings, neither of which can be “interpreted” entirely in terms of an initial more concrete structure of
interest (such as a Diophantine equation or a specific q-series or a specific elliptic curve over Q).
That is, it was just meant as an illustration of the well-known fact that to prove a theorem of interest about a concrete thing we may need to enlarge the scope of the problem and then could lose
the ability to “interpret” the core ideas of the proof in terms of operations involving just the original concrete thing. An expert in analytic number theory asked me recently how to unravel
Mochizuki’s arguments in the context of some other concrete consequences, and I had given a related explanation for why that couldn’t be done and that this isn’t a danger sign at all, and so I
tried to import the same explanation for my original reading of your question: I had mistakenly thought you were specifically asking about trying to see where in his arguments he is getting his
hands dirty with estimates involving ABC-triples. You won’t find it in that form because he never works with ABC-triples.
In the context of Szpiro’s conjecture, he also doesn’t apply IUT to any old elliptic curve, but has to assume several local and global properties which are always attained after a finite
extension of the ground field with controlled degree. So it is an essential feature of his technique that he is permitting rather general ground fields, and in fact the conditions he needs can’t
ever be fulfilled over Q.
In the end he is going to aim to prove Szpiro (for a given epsilon, with elliptic curves satisfying some specific local and global properties) with a constant depending on the ground field only
through its Q-degree, so it is kosher for him to making extensions of controlled degree in the middle of the argument. There is a separate “short” argument using Belyi maps that reduces the
general case of Szpiro to the ones he actually handles in the IUT machinery, but this latter argument is a clever proof by contradiction that makes things ineffective roughly as in Roth’s
To tell you where the “analysis” yielding an inequality should be found, I need to say something about what is going on in his method (for which I only have an impressionistic awareness based on
some lectures at the Oxford workshop last week). He uses serious algebro-geometric constructions with p-adic theta functions to make cohomological constructions that encode some local numerical
invariants arising in Szpiro’s conjecture (for an E satisfying the local and global hypotheses alluded to above) in terms of a special kind of fibered category (arising from E – {0}) called a
Frobenioid. This encoding involves a controlled ambiguity after accounting for variation of choices made in the construction (i.e., what is intrinsic is not a specific cohomology class, but
rather a certain coset by a controlled subgroup of an ambient cohomology group on a certain fundamental group). The full force of Mochizuki’s work on the anabelian properties of hyperbolic curves
is used to show that everything which just happened can be expressed in terms entirely intrinsic to a Frobenioid without any reference to the original elliptic curve.
The purpose of encoding number-theoretic data (with controlled error) in terms of Frobenioids appears to be that Frobeniods admit additional intrinsic operations (such as a weak version of
“Frobenius maps”) which one can’t express in terms of the original geometric objects (such as punctured elliptic curves). By analyzing how the cohomological constructions interact with those
operations (this involves introducing yet more abstract notions, such as “Hodge theaters”) eventually after a lot of work he arrives at two bounded domains in an R-vector space, one domain inside
the other, and comparing their volumes (which have to be computed!) gives the desired logarithmic form of the inequality with an “error term” (arising from various ambiguities in the
constructions) that is the desired uniform constant if one can exert sufficiently precise control on it uniformly in the original elliptic curve. So it is in this final step of computing volumes
with an “error term” that your sought-after “hard analysis” should be found.
But the constant obtained in that way isn’t the one for Szpiro’s Conjecture (for a given epsilon) for all elliptic curves over number fields of controlled degree! It is only suitable for elliptic
curves satisfying a specific list of local and global properties. To bootstrap this back to general elliptic curves over an original number field of interest, one needs to go through a proof by
contradiction as mentioned above, so in the end the final constant is ineffective (but depends on just epsilon and the Q-degree of the original ground field). So one recovers Mordell but not
effective Mordell.
34. Wow, I didn’t expect my comment to have sparked off such an excellent discussion.
@Brian Conrad: Thank you for the illuminating comments and for the excellent notes on the Oxford IUT Workshop you’ve posted here (via the “mysterious” @math_jin):
@Lieven le Bruyn: I’ve also very much appreciated your expository work on Frobenioids. Thank you for pointing out the @math_jin account, it would seem to be very useful at this stage.
@random reader: I was going to chime in on your comment to my remark on Hodge-Arakelov theory, but I think Brian Conrad has addressed that in his notes, i.e. that what was produced in the earlier
papers would not have lived up to your expectation of what a “Hodge-Arakelov theory” would be. I think Mochizuki had said as much in the introduction that I was paraphrasing.
35. For those following comments here, but not updates, you should be reading Brian Conrad’s report on the workshop here
36. Commenting because I can, and because it’s not true that people don’t read old comment threads[1]: they are an important source of a) historical information b) inside information. The number of
times I’ve searched for things mathematical and found blog discussions on it that I found excellent…
I also wanted to comment on the mathbabe thread after it was closed, to prove up-to-date links for people trawling the interwebs for the history of this interesting episode.
[1] Sure it was a generalisation: I had assumed though that old comment threads were closed to prevent spam. The suggestion of starting a new blog to continue the discussion baffles me.
37. David Roberts,
The main reason is because of spam, but more specifically because after a certain period the ratio of non-spam/spam comments becomes quite small, and the great majority of traffic to the postings
is spambots trying to break in.
This entry was posted in abc Conjecture. Bookmark the permalink. | {"url":"https://www.math.columbia.edu/~woit/wordpress/?p=8160","timestamp":"2024-11-05T21:49:05Z","content_type":"text/html","content_length":"135412","record_id":"<urn:uuid:3117a958-7c7f-4eb3-9e5f-98585d124d2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00641.warc.gz"} |
Machine learning models with differential privacy
Classification models
Gaussian Naive Bayes
class diffprivlib.models.GaussianNB(*, epsilon=1.0, bounds=None, priors=None, var_smoothing=1e-09, random_state=None, accountant=None)[source]
Gaussian Naive Bayes (GaussianNB) with differential privacy
Inherits the sklearn.naive_bayes.GaussianNB class from Scikit Learn and adds noise to satisfy differential privacy to the learned means and variances. Adapted from the work presented in [VSB13].
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\) for the model.
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ priors (array-like, shape (n_classes,)) – Prior probabilities of the classes. If specified the priors are not adjusted according to the data.
☆ var_smoothing (float, default: 1e-9) – Portion of the largest variance of all features that is added to variances for calculation stability.
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
probability of each class.
array, shape (n_classes,)
number of training samples observed in each class.
array, shape (n_classes,)
mean of each feature per class
array, shape (n_classes, n_features)
variance of each feature per class
array, shape (n_classes, n_features)
absolute additive value to variances (unrelated to epsilon parameter for differential privacy)
Vaidya, Jaideep, Basit Shafiq, Anirban Basu, and Yuan Hong. “Differentially private naive bayes classification.” In 2013 IEEE/WIC/ACM International Joint Conferences on Web Intelligence (WI) and
Intelligent Agent Technologies (IAT), vol. 1, pp. 571-576. IEEE, 2013.
fit(X, y, sample_weight=None)[source]
Fit Gaussian Naive Bayes according to X, y.
○ X (array-like of shape (n_samples, n_features)) – Training vectors, where n_samples is the number of samples and n_features is the number of features.
○ y (array-like of shape (n_samples,)) – Target values.
○ sample_weight (array-like of shape (n_samples,), default=None) –
Weights applied to individual samples (1. for unweighted).
Added in version 0.17: Gaussian Naive Bayes supports fitting with sample_weight.
self – Returns the instance itself.
Return type:
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
partial_fit(X, y, classes=None, sample_weight=None)[source]
Incremental fit on a batch of samples.
This method is expected to be called several times consecutively on different chunks of a dataset so as to implement out-of-core or online learning.
This is especially useful when the whole dataset is too big to fit in memory at once.
This method has some performance and numerical stability overhead, hence it is better to call partial_fit on chunks of data that are as large as possible (as long as fitting in the memory
budget) to hide the overhead.
○ X (array-like of shape (n_samples, n_features)) – Training vectors, where n_samples is the number of samples and n_features is the number of features.
○ y (array-like of shape (n_samples,)) – Target values.
○ classes (array-like of shape (n_classes,), default=None) –
List of all the classes that can possibly appear in the y vector.
Must be provided at the first call to partial_fit, can be omitted in subsequent calls.
○ sample_weight (array-like of shape (n_samples,), default=None) –
Weights applied to individual samples (1. for unweighted).
self – Returns the instance itself.
Return type:
Perform classification on an array of test vectors X.
X (array-like of shape (n_samples, n_features)) – The input samples.
C – Predicted target values for X.
Return type:
ndarray of shape (n_samples,)
Return joint log probability estimates for the test vector X.
For each row x of X and class y, the joint log probability is given by log P(x, y) = log P(y) + log P(x|y), where log P(y) is the class prior probability and log P(x|y) is the
class-conditional probability.
X (array-like of shape (n_samples, n_features)) – The input samples.
C – Returns the joint log-probability of the samples for each class in the model. The columns correspond to the classes in sorted order, as they appear in the attribute classes_.
Return type:
ndarray of shape (n_samples, n_classes)
Return log-probability estimates for the test vector X.
X (array-like of shape (n_samples, n_features)) – The input samples.
C – Returns the log-probability of the samples for each class in the model. The columns correspond to the classes in sorted order, as they appear in the attribute classes_.
Return type:
array-like of shape (n_samples, n_classes)
Return probability estimates for the test vector X.
X (array-like of shape (n_samples, n_features)) – The input samples.
C – Returns the probability of the samples for each class in the model. The columns correspond to the classes in sorted order, as they appear in the attribute classes_.
Return type:
array-like of shape (n_samples, n_classes)
score(X, y, sample_weight=None)
Return the mean accuracy on the given test data and labels.
In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.
○ X (array-like of shape (n_samples, n_features)) – Test samples.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True labels for X.
○ sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
score – Mean accuracy of self.predict(X) w.r.t. y.
Return type:
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') GaussianNB
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_partial_fit_request(*, classes: bool | None | str = '$UNCHANGED$', sample_weight: bool | None | str = '$UNCHANGED$') GaussianNB
Request metadata passed to the partial_fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to partial_fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to partial_fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
○ classes (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for classes parameter in partial_fit.
○ sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in partial_fit.
self – The updated object.
Return type:
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') GaussianNB
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
property sigma_
Variance of each feature per class.
Logistic Regression
class diffprivlib.models.LogisticRegression(*, epsilon=1.0, data_norm=None, tol=0.0001, C=1.0, fit_intercept=True, max_iter=100, verbose=0, warm_start=False, n_jobs=None, random_state=None,
accountant=None, **unused_args)[source]
Logistic Regression (aka logit, MaxEnt) classifier with differential privacy.
This class implements regularised logistic regression using Scipy’s L-BFGS-B algorithm. \(\epsilon\)-Differential privacy is achieved relative to the maximum norm of the data, as determined by
data_norm, by the Vector mechanism, which adds a Laplace-distributed random vector to the objective. Adapted from the work presented in [CMS11].
This class is a child of sklearn.linear_model.LogisticRegression, with amendments to allow for the implementation of differential privacy. Some parameters of Scikit Learn’s model have therefore
had to be fixed, including:
☆ The only permitted solver is ‘lbfgs’. Specifying the solver option will result in a warning.
☆ Consequently, the only permitted penalty is ‘l2’. Specifying the penalty option will result in a warning.
☆ In the multiclass case, only the one-vs-rest (OvR) scheme is permitted. Specifying the multi_class option will result in a warning.
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\).
☆ data_norm (float, optional) –
The max l2 norm of any row of the data. This defines the spread of data that will be protected by differential privacy.
If not specified, the max norm is taken from the data when .fit() is first called, but will result in a PrivacyLeakWarning, as it reveals information about the data. To preserve
differential privacy fully, data_norm should be selected independently of the data, i.e. with domain knowledge.
☆ tol (float, default: 1e-4) – Tolerance for stopping criteria.
☆ C (float, default: 1.0) – Inverse of regularization strength; must be a positive float. Like in support vector machines, smaller values specify stronger regularization.
☆ fit_intercept (bool, default: True) – Specifies if a constant (a.k.a. bias or intercept) should be added to the decision function.
☆ max_iter (int, default: 100) – Maximum number of iterations taken for the solver to converge. For smaller epsilon (more noise), max_iter may need to be increased.
☆ verbose (int, default: 0) – Set to any positive number for verbosity.
☆ warm_start (bool, default: False) – When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution.
☆ n_jobs (int, optional) – Number of CPU cores used when parallelising over classes. None means 1 unless in a context. -1 means using all processors.
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
A list of class labels known to the classifier.
array, shape (n_classes, )
Coefficient of the features in the decision function.
coef_ is of shape (1, n_features) when the given problem is binary.
array, shape (1, n_features) or (n_classes, n_features)
Intercept (a.k.a. bias) added to the decision function.
If fit_intercept is set to False, the intercept is set to zero. intercept_ is of shape (1,) when the given problem is binary.
array, shape (1,) or (n_classes,)
Actual number of iterations for all classes. If binary, it returns only 1 element.
array, shape (n_classes,) or (1, )
>>> from sklearn.datasets import load_iris
>>> from diffprivlib.models import LogisticRegression
>>> X, y = load_iris(return_X_y=True)
>>> clf = LogisticRegression(data_norm=12, epsilon=2).fit(X, y)
>>> clf.predict(X[:2, :])
array([0, 0])
>>> clf.predict_proba(X[:2, :])
array([[7.35362932e-01, 2.16667422e-14, 2.64637068e-01],
[9.08384378e-01, 3.47767052e-13, 9.16156215e-02]])
>>> clf.score(X, y)
See also
The implementation of logistic regression in scikit-learn, upon which this implementation is built.
The mechanism used by the model to achieve differential privacy.
Chaudhuri, Kamalika, Claire Monteleoni, and Anand D. Sarwate. “Differentially private empirical risk minimization.” Journal of Machine Learning Research 12, no. Mar (2011): 1069-1109.
Predict confidence scores for samples.
The confidence score for a sample is proportional to the signed distance of that sample to the hyperplane.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The data matrix for which we want to get the confidence scores.
scores – Confidence scores per (n_samples, n_classes) combination. In the binary case, confidence score for self.classes_[1] where >0 means this class would be predicted.
Return type:
ndarray of shape (n_samples,) or (n_samples, n_classes)
Convert coefficient matrix to dense array format.
Converts the coef_ member (back) to a numpy.ndarray. This is the default format of coef_ and is required for fitting, so calling this method is only required on models that have previously
been sparsified; otherwise, it is a no-op.
Fitted estimator.
Return type:
fit(X, y, sample_weight=None)[source]
Fit the model according to the given training data.
○ X ({array-like, sparse matrix}, shape (n_samples, n_features)) – Training vector, where n_samples is the number of samples and n_features is the number of features.
○ y (array-like, shape (n_samples,)) – Target vector relative to X.
○ sample_weight (ignored) – Ignored by diffprivlib. Present for consistency with sklearn API.
Return type:
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
Predict class labels for samples in X.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The data matrix for which we want to get the predictions.
y_pred – Vector containing the class labels for each sample.
Return type:
ndarray of shape (n_samples,)
Predict logarithm of probability estimates.
The returned estimates for all classes are ordered by the label of classes.
X (array-like of shape (n_samples, n_features)) – Vector to be scored, where n_samples is the number of samples and n_features is the number of features.
T – Returns the log-probability of the sample for each class in the model, where classes are ordered as they are in self.classes_.
Return type:
array-like of shape (n_samples, n_classes)
Probability estimates.
The returned estimates for all classes are ordered by the label of classes.
For a multi_class problem, if multi_class is set to be “multinomial” the softmax function is used to find the predicted probability of each class. Else use a one-vs-rest approach, i.e.
calculate the probability of each class assuming it to be positive using the logistic function and normalize these values across all the classes.
X (array-like of shape (n_samples, n_features)) – Vector to be scored, where n_samples is the number of samples and n_features is the number of features.
T – Returns the probability of the sample for each class in the model, where classes are ordered as they are in self.classes_.
Return type:
array-like of shape (n_samples, n_classes)
score(X, y, sample_weight=None)
Return the mean accuracy on the given test data and labels.
In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.
○ X (array-like of shape (n_samples, n_features)) – Test samples.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True labels for X.
○ sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
score – Mean accuracy of self.predict(X) w.r.t. y.
Return type:
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') LogisticRegression
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') LogisticRegression
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
Convert coefficient matrix to sparse format.
Converts the coef_ member to a scipy.sparse matrix, which for L1-regularized models can be much more memory- and storage-efficient than the usual numpy.ndarray representation.
The intercept_ member is not converted.
Fitted estimator.
Return type:
For non-sparse models, i.e. when there are not many zeros in coef_, this may actually increase memory usage, so use this method with care. A rule of thumb is that the number of zero elements,
which can be computed with (coef_ == 0).sum(), must be more than 50% for this to provide significant benefits.
After calling this method, further fitting with the partial_fit method (if any) will not work until you call densify.
Tree-Based Models
class diffprivlib.models.RandomForestClassifier(n_estimators=10, *, epsilon=1.0, bounds=None, classes=None, n_jobs=1, verbose=0, accountant=None, random_state=None, max_depth=5, warm_start=False,
shuffle=False, **unused_args)[source]
Random Forest Classifier with differential privacy.
This class implements Differentially Private Random Decision Forests using [1]. \(\epsilon\)-Differential privacy is achieved by constructing decision trees via random splitting criterion and
applying the PermuteAndFlip Mechanism to determine a noisy label.
☆ n_estimators (int, default: 10) – The number of trees in the forest.
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\).
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ classes (array-like of shape (n_classes,)) – Array of classes to be trained on. If not provided, the classes will be read from the data when .fit() is first called, resulting in a
☆ n_jobs (int, default: 1) – Number of CPU cores used when parallelising over classes. -1 means using all processors.
☆ verbose (int, default: 0) – Set to any positive number for verbosity.
☆ random_state (int or RandomState, optional) – Controls both the randomness of the shuffling of the samples used when building trees (if shuffle=True) and training of the
differentially-private DecisionTreeClassifier to construct the forest. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
☆ max_depth (int, default: 5) – The maximum depth of the tree. The depth translates to an exponential increase in memory usage.
☆ warm_start (bool, default=False) – When set to True, reuse the solution of the previous call to fit and add more estimators to the ensemble, otherwise, just fit a whole new forest.
☆ shuffle (bool, default=False) – When set to True, shuffles the datapoints to be trained on trees at random. In diffprivlib, each datapoint is used to train exactly one tree. When set to
False, datapoints are chosen in-order to their tree in sequence.
The child estimator template used to create the collection of fitted sub-estimators.
The collection of fitted sub-estimators.
list of DecisionTreeClassifier
The classes labels.
ndarray of shape (n_classes,) or a list of such arrays
The number of classes.
int or list
Number of features seen during fit.
Names of features seen during fit. Defined only when X has feature names that are all strings.
ndarray of shape (n_features_in_,)
The number of outputs when fit is performed.
>>> from sklearn.datasets import make_classification
>>> from diffprivlib.models import RandomForestClassifier
>>> X, y = make_classification(n_samples=1000, n_features=4,
... n_informative=2, n_redundant=0,
... random_state=0, shuffle=False)
>>> clf = RandomForestClassifier(n_estimators=100, random_state=0)
>>> clf.fit(X, y)
>>> print(clf.predict([[0, 0, 0, 0]]))
[1] Sam Fletcher, Md Zahidul Islam. “Differentially Private Random Decision Forests using Smooth Sensitivity” https://arxiv.org/abs/1606.03572
Apply trees in the forest to X, return leaf indices.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, its dtype will be converted to dtype=np.float32. If a sparse matrix is provided, it will
be converted into a sparse csr_matrix.
X_leaves – For each datapoint x in X and for each tree in the forest, return the index of the leaf x ends up in.
Return type:
ndarray of shape (n_samples, n_estimators)
Return the decision path in the forest.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, its dtype will be converted to dtype=np.float32. If a sparse matrix is provided, it will
be converted into a sparse csr_matrix.
○ indicator (sparse matrix of shape (n_samples, n_nodes)) – Return a node indicator matrix where non zero elements indicates that the samples goes through the nodes. The matrix is of
CSR format.
○ n_nodes_ptr (ndarray of shape (n_estimators + 1,)) – The columns from indicator[n_nodes_ptr[i]:n_nodes_ptr[i+1]] gives the indicator value for the i-th estimator.
property estimators_samples_
The subset of drawn samples for each base estimator.
Returns a dynamically generated list of indices identifying the samples used for fitting each member of the ensemble, i.e., the in-bag samples.
Note: the list is re-created at each call to the property in order to reduce the object memory footprint by not storing the sampling data. Thus fetching the property may be slower than
fit(X, y, sample_weight=None)[source]
Build a forest of trees from the training set (X, y).
○ X (array-like of shape (n_samples, n_features)) – The training input samples. Internally, its dtype will be converted to dtype=np.float32.
○ y (array-like of shape (n_samples,)) – The target values (class labels in classification, real numbers in regression).
○ sample_weight (ignored) – Ignored by diffprivlib. Present for consistency with sklearn API.
self – Fitted estimator.
Return type:
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
Predict class for X.
The predicted class of an input sample is a vote by the trees in the forest, weighted by their probability estimates. That is, the predicted class is the one with highest mean probability
estimate across the trees.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, its dtype will be converted to dtype=np.float32. If a sparse matrix is provided, it will
be converted into a sparse csr_matrix.
y – The predicted classes.
Return type:
ndarray of shape (n_samples,) or (n_samples, n_outputs)
Predict class log-probabilities for X.
The predicted class log-probabilities of an input sample is computed as the log of the mean predicted class probabilities of the trees in the forest.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, its dtype will be converted to dtype=np.float32. If a sparse matrix is provided, it will
be converted into a sparse csr_matrix.
p – The class probabilities of the input samples. The order of the classes corresponds to that in the attribute classes_.
Return type:
ndarray of shape (n_samples, n_classes), or a list of such arrays
Predict class probabilities for X.
The predicted class probabilities of an input sample are computed as the mean predicted class probabilities of the trees in the forest. The class probability of a single tree is the fraction
of samples of the same class in a leaf.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, its dtype will be converted to dtype=np.float32. If a sparse matrix is provided, it will
be converted into a sparse csr_matrix.
p – The class probabilities of the input samples. The order of the classes corresponds to that in the attribute classes_.
Return type:
ndarray of shape (n_samples, n_classes), or a list of such arrays
score(X, y, sample_weight=None)
Return the mean accuracy on the given test data and labels.
In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.
○ X (array-like of shape (n_samples, n_features)) – Test samples.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True labels for X.
○ sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
score – Mean accuracy of self.predict(X) w.r.t. y.
Return type:
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') RandomForestClassifier
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') RandomForestClassifier
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
class diffprivlib.models.DecisionTreeClassifier(max_depth=5, *, epsilon=1, bounds=None, classes=None, random_state=None, accountant=None, criterion=None, **unused_args)[source]
Decision Tree Classifier with differential privacy.
This class implements the base differentially private decision tree classifier for the Random Forest classifier algorithm. Not meant to be used separately.
☆ max_depth (int, default: 5) – The maximum depth of the tree.
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\).
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ classes (array-like of shape (n_classes,), optional) – Array of class labels. If not provided, the classes will be read from the data when .fit() is first called, resulting in a
☆ random_state (int or RandomState, optional) – Controls the randomness of the estimator. At each split, the feature to split on is chosen randomly, as is the threshold at which to split.
The classification label at each leaf is then randomised, subject to differential privacy constraints. To obtain a deterministic behaviour during randomisation, random_state has to be
fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
The number of features when fit is performed.
The number of classes.
The class labels.
array of shape (n_classes, )
apply(X, check_input=True)
Return the index of the leaf that each sample is predicted as.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, it will be converted to dtype=np.float32 and if a sparse matrix is provided to a
sparse csr_matrix.
○ check_input (bool, default=True) – Allow to bypass several input checking. Don’t use this parameter unless you know what you’re doing.
X_leaves – For each datapoint x in X, return the index of the leaf x ends up in. Leaves are numbered within [0; self.tree_.node_count), possibly with gaps in the numbering.
Return type:
array-like of shape (n_samples,)
decision_path(X, check_input=True)
Return the decision path in the tree.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, it will be converted to dtype=np.float32 and if a sparse matrix is provided to a
sparse csr_matrix.
○ check_input (bool, default=True) – Allow to bypass several input checking. Don’t use this parameter unless you know what you’re doing.
indicator – Return a node indicator CSR matrix where non zero elements indicates that the samples goes through the nodes.
Return type:
sparse matrix of shape (n_samples, n_nodes)
fit(X, y, sample_weight=None, check_input=True)[source]
Build a differentially-private decision tree classifier from the training set (X, y).
○ X (array-like of shape (n_samples, n_features)) – The training input samples. Internally, it will be converted to dtype=np.float32.
○ y (array-like of shape (n_samples,)) – The target values (class labels) as integers or strings.
○ sample_weight (ignored) – Ignored by diffprivlib. Present for consistency with sklearn API.
○ check_input (bool, default=True) – Allow to bypass several input checking. Don’t use this parameter unless you know what you do.
self – Fitted estimator.
Return type:
Return the depth of the decision tree.
The depth of a tree is the maximum distance between the root and any leaf.
self.tree_.max_depth – The maximum depth of the tree.
Return type:
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Return the number of leaves of the decision tree.
self.tree_.n_leaves – Number of leaves.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
predict(X, check_input=True)
Predict class or regression value for X.
For a classification model, the predicted class for each sample in X is returned. For a regression model, the predicted value based on X is returned.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, it will be converted to dtype=np.float32 and if a sparse matrix is provided to a
sparse csr_matrix.
○ check_input (bool, default=True) – Allow to bypass several input checking. Don’t use this parameter unless you know what you’re doing.
y – The predicted classes, or the predict values.
Return type:
array-like of shape (n_samples,) or (n_samples, n_outputs)
Predict class log-probabilities of the input samples X.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, it will be converted to dtype=np.float32 and if a sparse matrix is provided to a sparse
proba – The class log-probabilities of the input samples. The order of the classes corresponds to that in the attribute classes_.
Return type:
ndarray of shape (n_samples, n_classes) or list of n_outputs such arrays if n_outputs > 1
predict_proba(X, check_input=True)[source]
Predict class probabilities of the input samples X.
The predicted class probability is the fraction of samples of the same class in a leaf.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The input samples. Internally, it will be converted to dtype=np.float32 and if a sparse matrix is provided to a
sparse csr_matrix.
○ check_input (bool, default=True) – Allow to bypass several input checking. Don’t use this parameter unless you know what you’re doing.
proba – The class probabilities of the input samples. The order of the classes corresponds to that in the attribute classes_.
Return type:
ndarray of shape (n_samples, n_classes) or list of n_outputs such arrays if n_outputs > 1
score(X, y, sample_weight=None)
Return the mean accuracy on the given test data and labels.
In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.
○ X (array-like of shape (n_samples, n_features)) – Test samples.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True labels for X.
○ sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
score – Mean accuracy of self.predict(X) w.r.t. y.
Return type:
set_fit_request(*, check_input: bool | None | str = '$UNCHANGED$', sample_weight: bool | None | str = '$UNCHANGED$') DecisionTreeClassifier
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
○ check_input (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for check_input parameter in fit.
○ sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_predict_proba_request(*, check_input: bool | None | str = '$UNCHANGED$') DecisionTreeClassifier
Request metadata passed to the predict_proba method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to predict_proba if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to predict_proba.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
check_input (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for check_input parameter in predict_proba.
self – The updated object.
Return type:
set_predict_request(*, check_input: bool | None | str = '$UNCHANGED$') DecisionTreeClassifier
Request metadata passed to the predict method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to predict if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to predict.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
check_input (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for check_input parameter in predict.
self – The updated object.
Return type:
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') DecisionTreeClassifier
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
Regression models
Linear Regression
class diffprivlib.models.LinearRegression(*, epsilon=1.0, bounds_X=None, bounds_y=None, fit_intercept=True, copy_X=True, random_state=None, accountant=None, **unused_args)[source]
Ordinary least squares Linear Regression with differential privacy.
LinearRegression fits a linear model with coefficients w = (w1, …, wp) to minimize the residual sum of squares between the observed targets in the dataset, and the targets predicted by the linear
approximation. Differential privacy is guaranteed with respect to the training sample.
Differential privacy is achieved by adding noise to the coefficients of the objective function, taking inspiration from [ZZX12].
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\).
☆ bounds_X (tuple) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one entry
per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ bounds_y (tuple) – Same as bounds_X, but for the training label set y.
☆ fit_intercept (bool, default: True) – Whether to calculate the intercept for this model. If set to False, no intercept will be used in calculations (i.e. data is expected to be centered).
☆ copy_X (bool, default: True) – If True, X will be copied; else, it may be overwritten.
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
Estimated coefficients for the linear regression problem. If multiple targets are passed during the fit (y 2D), this is a 2D array of shape (n_targets, n_features), while if only one target
is passed, this is a 1D array of length n_features.
array of shape (n_features, ) or (n_targets, n_features)
Independent term in the linear model. Set to 0.0 if fit_intercept = False.
float or array of shape of (n_targets,)
Zhang, Jun, Zhenjie Zhang, Xiaokui Xiao, Yin Yang, and Marianne Winslett. “Functional mechanism: regression analysis under differential privacy.” arXiv preprint arXiv:1208.0219 (2012).
fit(X, y, sample_weight=None)[source]
Fit linear model.
○ X (array-like or sparse matrix, shape (n_samples, n_features)) – Training data
○ y (array_like, shape (n_samples, n_targets)) – Target values. Will be cast to X’s dtype if necessary
○ sample_weight (ignored) – Ignored by diffprivlib. Present for consistency with sklearn API.
Return type:
returns an instance of self.
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
Predict using the linear model.
X (array-like or sparse matrix, shape (n_samples, n_features)) – Samples.
C – Returns predicted values.
Return type:
array, shape (n_samples,)
score(X, y, sample_weight=None)
Return the coefficient of determination of the prediction.
The coefficient of determination \(R^2\) is defined as \((1 - \frac{u}{v})\), where \(u\) is the residual sum of squares ((y_true - y_pred)** 2).sum() and \(v\) is the total sum of squares
((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected
value of y, disregarding the input features, would get a \(R^2\) score of 0.0.
○ X (array-like of shape (n_samples, n_features)) – Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples,
n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs)) – True values for X.
○ sample_weight (array-like of shape (n_samples,), default=None) – Sample weights.
score – \(R^2\) of self.predict(X) w.r.t. y.
Return type:
The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score(). This influences the score
method of all the multioutput regressors (except for MultiOutputRegressor).
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') LinearRegression
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') LinearRegression
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
Clustering models
class diffprivlib.models.KMeans(n_clusters=8, *, epsilon=1.0, bounds=None, random_state=None, accountant=None, **unused_args)[source]
K-Means clustering with differential privacy.
Implements the DPLloyd approach presented in [SCL16], leveraging the sklearn.cluster.KMeans class for full integration with Scikit Learn.
☆ n_clusters (int, default: 8) – The number of clusters to form as well as the number of centroids to generate.
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\).
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
Coordinates of cluster centers. If the algorithm stops before fully converging, these will not be consistent with labels_.
array, [n_clusters, n_features]
Labels of each point
Sum of squared distances of samples to their closest cluster center.
Number of iterations run.
Su, Dong, Jianneng Cao, Ninghui Li, Elisa Bertino, and Hongxia Jin. “Differentially private k-means clustering.” In Proceedings of the sixth ACM conference on data and application security and
privacy, pp. 26-37. ACM, 2016.
fit(X, y=None, sample_weight=None)[source]
Computes k-means clustering with differential privacy.
○ X (array-like, shape=(n_samples, n_features)) – Training instances to cluster.
○ y (Ignored) – not used, present here for API consistency by convention.
○ sample_weight (ignored) – Ignored by diffprivlib. Present for consistency with sklearn API.
Return type:
fit_predict(X, y=None, sample_weight=None)
Compute cluster centers and predict cluster index for each sample.
Convenience method; equivalent to calling fit(X) followed by predict(X).
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data to transform.
○ y (Ignored) – Not used, present here for API consistency by convention.
○ sample_weight (array-like of shape (n_samples,), default=None) – The weights for each observation in X. If None, all observations are assigned equal weight.
labels – Index of the cluster each sample belongs to.
Return type:
ndarray of shape (n_samples,)
fit_transform(X, y=None, sample_weight=None)
Compute clustering and transform X to cluster-distance space.
Equivalent to fit(X).transform(X), but more efficiently implemented.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data to transform.
○ y (Ignored) – Not used, present here for API consistency by convention.
○ sample_weight (array-like of shape (n_samples,), default=None) – The weights for each observation in X. If None, all observations are assigned equal weight.
X_new – X transformed in the new space.
Return type:
ndarray of shape (n_samples, n_clusters)
Get output feature names for transformation.
The feature names out will prefixed by the lowercased class name. For example, if the transformer outputs 3 features, then the feature names out are: [“class_name0”, “class_name1”,
input_features (array-like of str or None, default=None) – Only used to validate feature names with the names seen in fit.
feature_names_out – Transformed feature names.
Return type:
ndarray of str objects
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
Predict the closest cluster each sample in X belongs to.
In the vector quantization literature, cluster_centers_ is called the code book and each value returned by predict is the index of the closest code in the code book.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data to predict.
labels – Index of the cluster each sample belongs to.
Return type:
ndarray of shape (n_samples,)
score(X, y=None, sample_weight=None)
Opposite of the value of X on the K-means objective.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data.
○ y (Ignored) – Not used, present here for API consistency by convention.
○ sample_weight (array-like of shape (n_samples,), default=None) – The weights for each observation in X. If None, all observations are assigned equal weight.
score – Opposite of the value of X on the K-means objective.
Return type:
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') KMeans
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
set_output(*, transform=None)
Set output container.
See Introducing the set_output API for an example on how to use the API.
transform ({"default", "pandas", "polars"}, default=None) –
Configure output of transform and fit_transform.
○ ”default”: Default output format of a transformer
○ ”pandas”: DataFrame output
○ ”polars”: Polars output
○ None: Transform configuration is unchanged
Added in version 1.4: “polars” option was added.
self – Estimator instance.
Return type:
estimator instance
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') KMeans
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in score.
self – The updated object.
Return type:
Transform X to a cluster-distance space.
In the new space, each dimension is the distance to the cluster centers. Note that even if X is sparse, the array returned by transform will typically be dense.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data to transform.
X_new – X transformed in the new space.
Return type:
ndarray of shape (n_samples, n_clusters)
Dimensionality reduction models
class diffprivlib.models.PCA(n_components=None, *, epsilon=1.0, data_norm=None, centered=False, bounds=None, copy=True, whiten=False, random_state=None, accountant=None, **unused_args)[source]
Principal component analysis (PCA) with differential privacy.
This class is a child of sklearn.decomposition.PCA, with amendments to allow for the implementation of differential privacy as given in [IS16b]. Some parameters of Scikit Learn’s model have
therefore had to be fixed, including:
☆ The only permitted svd_solver is ‘full’. Specifying the svd_solver option will result in a warning;
☆ The parameters tol and iterated_power are not applicable (as a consequence of fixing svd_solver = 'full').
☆ n_components (int, float, None or str) –
Number of components to keep. If n_components is not set all components are kept:
n_components == min(n_samples, n_features)
If n_components == 'mle', Minka’s MLE is used to guess the dimension.
If 0 < n_components < 1, select the number of components such that the amount of variance that needs to be explained is greater than the percentage specified by n_components.
Hence, the None case results in:
n_components == min(n_samples, n_features) - 1
☆ epsilon (float, default: 1.0) – Privacy parameter \(\epsilon\). If centered=False, half of epsilon is used to calculate the differentially private mean to center the data prior to the
calculation of principal components.
☆ data_norm (float, optional) –
The max l2 norm of any row of the data. This defines the spread of data that will be protected by differential privacy.
If not specified, the max norm is taken from the data when .fit() is first called, but will result in a PrivacyLeakWarning, as it reveals information about the data. To preserve
differential privacy fully, data_norm should be selected independently of the data, i.e. with domain knowledge.
☆ centered (bool, default: False) –
If False, the data will be centered before calculating the principal components. This will be calculated with differential privacy, consuming privacy budget from epsilon.
If True, the data is assumed to have been centered previously (e.g. using StandardScaler), and therefore will not require the consumption of privacy budget to calculate the mean.
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ copy (bool, default: True) – If False, data passed to fit are overwritten and running fit(X).transform(X) will not yield the expected results, use fit_transform(X) instead.
☆ whiten (bool, default: False) –
When True (False by default) the components_ vectors are multiplied by the square root of n_samples and then divided by the singular values to ensure uncorrelated outputs with unit
component-wise variances.
Whitening will remove some information from the transformed signal (the relative variance scales of the components) but can sometime improve the predictive accuracy of the downstream
estimators by making their data respect some hard-wired assumptions.
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by explained_variance_.
array, shape (n_components, n_features)
The amount of variance explained by each of the selected components.
Equal to n_components largest eigenvalues of the covariance matrix of X.
array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If n_components is not set then all components are stored and the sum of the ratios is equal to 1.0.
array, shape (n_components,)
The singular values corresponding to each of the selected components. The singular values are equal to the 2-norms of the n_components variables in the lower-dimensional space.
array, shape (n_components,)
Per-feature empirical mean, estimated from the training set.
Equal to X.mean(axis=0).
array, shape (n_features,)
The estimated number of components. When n_components is set to ‘mle’ or a number between 0 and 1 (with svd_solver == ‘full’) this number is estimated from input data. Otherwise it equals the
parameter n_components, or the lesser value of n_features and n_samples if n_components is None.
Number of features in the training data.
Number of samples in the training data.
The estimated noise covariance following the Probabilistic PCA model from Tipping and Bishop 1999. See “Pattern Recognition and Machine Learning” by C. Bishop, 12.2.1 p. 574 or http://
www.miketipping.com/papers/met-mppca.pdf. It is required to compute the estimated data covariance and score samples.
Equal to the average of (min(n_features, n_samples) - n_components) smallest eigenvalues of the covariance matrix of X.
See also
Scikit-learn implementation Principal Component Analysis.
Imtiaz, Hafiz, and Anand D. Sarwate. “Symmetric matrix perturbation for differentially-private principal component analysis.” In 2016 IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP), pp. 2339-2343. IEEE, 2016.
fit(X, y=None)[source]
Fit the model with X.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – Training data, where n_samples is the number of samples and n_features is the number of features.
○ y (Ignored) – Ignored.
self – Returns the instance itself.
Return type:
fit_transform(X, y=None)[source]
Fit the model with X and apply the dimensionality reduction on X.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – Training data, where n_samples is the number of samples and n_features is the number of features.
○ y (Ignored) – Ignored.
X_new – Transformed values.
Return type:
ndarray of shape (n_samples, n_components)
This method returns a Fortran-ordered array. To convert it to a C-ordered array, use ‘np.ascontiguousarray’.
Compute data covariance with the generative model.
cov = components_.T * S**2 * components_ + sigma2 * eye(n_features) where S**2 contains the explained variances, and sigma2 contains the noise variances.
cov – Estimated covariance of data.
Return type:
array of shape=(n_features, n_features)
Get output feature names for transformation.
The feature names out will prefixed by the lowercased class name. For example, if the transformer outputs 3 features, then the feature names out are: [“class_name0”, “class_name1”,
input_features (array-like of str or None, default=None) – Only used to validate feature names with the names seen in fit.
feature_names_out – Transformed feature names.
Return type:
ndarray of str objects
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
Compute data precision matrix with the generative model.
Equals the inverse of the covariance but computed with the matrix inversion lemma for efficiency.
precision – Estimated precision of data.
Return type:
array, shape=(n_features, n_features)
Transform data back to its original space.
In other words, return an input X_original whose transform would be X.
X (array-like of shape (n_samples, n_components)) – New data, where n_samples is the number of samples and n_components is the number of components.
Original data, where n_samples is the number of samples and n_features is the number of features.
Return type:
X_original array-like of shape (n_samples, n_features)
If whitening is enabled, inverse_transform will compute the exact inverse operation, which includes reversing whitening.
score(X, y=None)[source]
Return the average log-likelihood of all samples.
See. “Pattern Recognition and Machine Learning” by C. Bishop, 12.2.1 p. 574 or http://www.miketipping.com/papers/met-mppca.pdf
○ X (array-like of shape (n_samples, n_features)) – The data.
○ y (Ignored) – Ignored.
ll – Average log-likelihood of the samples under the current model.
Return type:
Return the log-likelihood of each sample.
See. “Pattern Recognition and Machine Learning” by C. Bishop, 12.2.1 p. 574 or http://www.miketipping.com/papers/met-mppca.pdf
X (array-like of shape (n_samples, n_features)) – The data.
ll – Log-likelihood of each sample under the current model.
Return type:
ndarray of shape (n_samples,)
set_output(*, transform=None)
Set output container.
See Introducing the set_output API for an example on how to use the API.
transform ({"default", "pandas", "polars"}, default=None) –
Configure output of transform and fit_transform.
○ ”default”: Default output format of a transformer
○ ”pandas”: DataFrame output
○ ”polars”: Polars output
○ None: Transform configuration is unchanged
Added in version 1.4: “polars” option was added.
self – Estimator instance.
Return type:
estimator instance
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
Apply dimensionality reduction to X.
X is projected on the first principal components previously extracted from a training set.
X ({array-like, sparse matrix} of shape (n_samples, n_features)) – New data, where n_samples is the number of samples and n_features is the number of features.
X_new – Projection of X in the first principal components, where n_samples is the number of samples and n_components is the number of the components.
Return type:
array-like of shape (n_samples, n_components)
Standard Scaler
class diffprivlib.models.StandardScaler(*, epsilon=1.0, bounds=None, copy=True, with_mean=True, with_std=True, random_state=None, accountant=None)[source]
Standardize features by removing the mean and scaling to unit variance, calculated with differential privacy guarantees. Differential privacy is guaranteed on the learned scaler with respect to
the training sample; the transformed output will certainly not satisfy differential privacy.
The standard score of a sample x is calculated as:
where u is the (differentially private) mean of the training samples or zero if with_mean=False, and s is the (differentially private) standard deviation of the training samples or one if
Centering and scaling happen independently on each feature by computing the relevant statistics on the samples in the training set. Mean and standard deviation are then stored to be used on later
data using the transform method.
For further information, users are referred to sklearn.preprocessing.StandardScaler.
☆ epsilon (float, default: 1.0) – The privacy budget to be allocated to learning the mean and variance of the training sample. If with_std=True, the privacy budget is split evenly between
mean and variance (the mean must be calculated even when with_mean=False, as it is used in the calculation of the variance.
☆ bounds (tuple, optional) – Bounds of the data, provided as a tuple of the form (min, max). min and max can either be scalars, covering the min/max of the entire data, or vectors with one
entry per feature. If not provided, the bounds are computed on the data when .fit() is first called, resulting in a PrivacyLeakWarning.
☆ copy (boolean, default: True) – If False, try to avoid a copy and do inplace scaling instead. This is not guaranteed to always work inplace; e.g. if the data is not a NumPy array, a copy
may still be returned.
☆ with_mean (boolean, True by default) – If True, center the data before scaling.
☆ with_std (boolean, True by default) – If True, scale the data to unit variance (or equivalently, unit standard deviation).
☆ random_state (int or RandomState, optional) – Controls the randomness of the model. To obtain a deterministic behaviour during randomisation, random_state has to be fixed to an integer.
☆ accountant (BudgetAccountant, optional) – Accountant to keep track of privacy budget.
Per feature relative scaling of the data. This is calculated using np.sqrt(var_). Equal to None when with_std=False.
ndarray or None, shape (n_features,)
The mean value for each feature in the training set. Equal to None when with_mean=False.
ndarray or None, shape (n_features,)
The variance for each feature in the training set. Used to compute scale_. Equal to None when with_std=False.
ndarray or None, shape (n_features,)
The number of samples processed by the estimator for each feature. If there are not missing samples, the n_samples_seen will be an integer, otherwise it will be an array. Will be reset on new
calls to fit, but increments across partial_fit calls.
int or array, shape (n_features,)
See also
Vanilla scikit-learn version, without differential privacy.
Further removes the linear correlation across features with ‘whiten=True’.
NaNs are treated as missing values: disregarded in fit, and maintained in transform.
fit(X, y=None, sample_weight=None)[source]
Compute the mean and std to be used for later scaling.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The data used to compute the mean and standard deviation used for later scaling along the features axis.
○ y (None) – Ignored.
○ sample_weight (array-like of shape (n_samples,), default=None) –
Individual weights for each sample.
Added in version 0.24: parameter sample_weight support to StandardScaler.
self – Fitted scaler.
Return type:
fit_transform(X, y=None, **fit_params)
Fit to data, then transform it.
Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.
○ X (array-like of shape (n_samples, n_features)) – Input samples.
○ y (array-like of shape (n_samples,) or (n_samples, n_outputs), default=None) – Target values (None for unsupervised transformations).
○ **fit_params (dict) – Additional fit parameters.
X_new – Transformed array.
Return type:
ndarray array of shape (n_samples, n_features_new)
Get output feature names for transformation.
input_features (array-like of str or None, default=None) –
Input features.
○ If input_features is None, then feature_names_in_ is used as feature names in. If feature_names_in_ is not defined, then the following input feature names are generated: [“x0”, “x1”,
…, “x(n_features_in_ - 1)”].
○ If input_features is an array-like, then input_features must match feature_names_in_ if feature_names_in_ is defined.
feature_names_out – Same as input features.
Return type:
ndarray of str objects
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
routing – A MetadataRequest encapsulating routing information.
Return type:
Get parameters for this estimator.
deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
params – Parameter names mapped to their values.
Return type:
inverse_transform(X, copy=None)[source]
Scale back the data to the original representation.
○ X ({array-like, sparse matrix} of shape (n_samples, n_features)) – The data used to scale along the features axis.
○ copy (bool, default=None) – Copy the input X or not.
X_tr – Transformed array.
Return type:
{ndarray, sparse matrix} of shape (n_samples, n_features)
partial_fit(X, y=None, sample_weight=None)[source]
Online computation of mean and std with differential privacy on X for later scaling. All of X is processed as a single batch. This is intended for cases when fit is not feasible due to very
large number of n_samples or because X is read from a continuous stream.
The algorithm for incremental mean and std is given in Equation 1.5a,b in Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. “Algorithms for computing the sample variance: Analysis and
recommendations.” The American Statistician 37.3 (1983): 242-247:
○ X ({array-like}, shape [n_samples, n_features]) – The data used to compute the mean and standard deviation used for later scaling along the features axis.
○ y – Ignored
○ sample_weight – Ignored by diffprivlib. Present for consistency with sklearn API.
set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') StandardScaler
Request metadata passed to the fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in fit.
self – The updated object.
Return type:
set_inverse_transform_request(*, copy: bool | None | str = '$UNCHANGED$') StandardScaler
Request metadata passed to the inverse_transform method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to inverse_transform if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to inverse_transform.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
copy (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for copy parameter in inverse_transform.
self – The updated object.
Return type:
set_output(*, transform=None)
Set output container.
See Introducing the set_output API for an example on how to use the API.
transform ({"default", "pandas", "polars"}, default=None) –
Configure output of transform and fit_transform.
○ ”default”: Default output format of a transformer
○ ”pandas”: DataFrame output
○ ”polars”: Polars output
○ None: Transform configuration is unchanged
Added in version 1.4: “polars” option was added.
self – Estimator instance.
Return type:
estimator instance
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
**params (dict) – Estimator parameters.
self – Estimator instance.
Return type:
estimator instance
set_partial_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') StandardScaler
Request metadata passed to the partial_fit method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to partial_fit if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to partial_fit.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weight (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for sample_weight parameter in partial_fit.
self – The updated object.
Return type:
set_transform_request(*, copy: bool | None | str = '$UNCHANGED$') StandardScaler
Request metadata passed to the transform method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to transform if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to transform.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
copy (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for copy parameter in transform.
self – The updated object.
Return type:
transform(X, copy=None)[source]
Perform standardization by centering and scaling.
○ X ({array-like, sparse matrix of shape (n_samples, n_features)) – The data used to scale along the features axis.
○ copy (bool, default=None) – Copy the input X or not.
X_tr – Transformed array.
Return type:
{ndarray, sparse matrix} of shape (n_samples, n_features) | {"url":"https://diffprivlib.readthedocs.io/en/latest/modules/models.html","timestamp":"2024-11-11T21:41:53Z","content_type":"text/html","content_length":"361538","record_id":"<urn:uuid:b85d7c37-7e56-4e27-9364-8d4052cbff2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00741.warc.gz"} |
Pebbles Surfaces of Revolution
OneStone® Pebbles Surfaces of Revolution
Using Pebbles page.
This Pebble accepts all the standard math operators plus the variable X or Y; expressions may use one or the other, but not both. Functions are graphed according to the following rules:
• Functions in which X is the only independent variable are drawn in the 2D Graph as the curve Y = F(X). This curve is then rotated about the X axis to form the surface shown in the 3D graph.
• Functions in which Y is the only independent variable are drawn in the 2D Graph as X = F(Y). This curve is then rotated about the Y axis to form the surface shown in the 3D graph.
Click the 'Launch...' link below to start the Surfaces of Revolution Pebble. The process of installing and running a Pebble for the first time is explained on the Installing Pebbles page. Running one
after the first time is quicker: the JOGL package doesn't have to be downloaded, plus something other than the generic 'Starting Java' image is displayed during start-up. | {"url":"http://www.brightideassoftware.com/Pebbles/SurfacesOfRevolution.aspx","timestamp":"2024-11-04T17:36:57Z","content_type":"application/xhtml+xml","content_length":"51523","record_id":"<urn:uuid:68f3b05f-a045-40a9-bed5-e4d4e82ad183>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00049.warc.gz"} |
967 research outputs found
We examine a unitarity of a particular higher-derivative extension of general relativity in three space-time dimensions, which has been recently shown to be equivalent to the Pauli-Fierz massive
gravity at the linearized approximation level, and explore a possibility of generalizing the model to higher space-time dimensions. We find that the model in three dimensions is indeed unitary in the
tree-level, but the corresponding model in higher dimensions is not so due to the appearance of non-unitary massless spin-2 modes.Comment: 10 pages, references adde
We consider a locally supersymmetric theory where the Planck mass is replaced by a dynamical superfield. This model can be thought of as the Minimal Supersymmetric extension of the Brans-Dicke theory
(MSBD). The motivation that underlies this analysis is the research of possible connections between Dark Energy models based on Brans-Dicke-like theories and supersymmetric Dark Matter scenarios. We
find that the phenomenology associated with the MSBD model is very different compared to the one of the original Brans-Dicke theory: the gravitational sector does not couple to the matter sector in a
universal metric way. This feature could make the minimal supersymmetric extension of the BD idea phenomenologically inconsistent.Comment: 6 pages, one section is adde
We consider the electromagnetic and gravitational interactions of a massive Rarita-Schwinger field. Stueckelberg analysis of the system, when coupled to electromagnetism in flat space or to gravity,
reveals in either case that the effective field theory has a model-independent upper bound on its UV cutoff, which is finite but parametrically larger than the particle's mass. It is the helicity-1/2
mode that becomes strongly coupled at the cutoff scale. If the interactions are inconsistent, the same mode becomes a telltale sign of pathologies. Alternatively, consistent interactions are those
that propagate this mode within the light cone. Studying its dynamics not only sheds light on the Velo-Zwanziger acausality, but also elucidates why supergravity and other known consistent models are
pathology-free.Comment: 18 pages, cutoff analysis improved, to appear in PR
We show that the graviton acquires a mass in a de Sitter background given by $m_{g}^{2}=-{2/3}\Lambda.$ This is precisely the fine-tuning value required for the perturbed gravitational field to
mantain its two degrees of freedom.Comment: Title changed and few details added, without any changes in the conclusio
We present a Lagrangian for a massive, charged spin 3/2 field in a constant external electromagnetic background, which correctly propagates only physical degrees of freedom inside the light cone. The
Velo-Zwanziger acausality and other pathologies such as loss of hyperbolicity or the appearance of unphysical degrees of freedom are avoided by a judicious choice of non-minimal couplings. No
additional fields or equations besides the spin 3/2 ones are needed to solve the problem.Comment: 10 pages, references added. To appear in PR
It is a general belief that the only possible way to consistently deform the Pauli-Fierz action, changing also the gauge algebra, is general relativity. Here we show that a different type of
deformation exists in three dimensions if one allows for PT non-invariant terms. The new gauge algebra is different from that of diffeomorphisms. Furthermore, this deformation can be generalized to
the case of a collection of massless spin-two fields. In this case it describes a consistent interaction among them.Comment: 21+1 pages. Minor corrections and reference adde
The order parameter of a finite system with a spontaneously broken continuous global symmetry acts as a quantum mechanical rotor. Both antiferromagnets with a spontaneously broken $SU(2)_s$ spin
symmetry and massless QCD with a broken $SU(2)_L \times SU(2)_R$ chiral symmetry have rotor spectra when considered in a finite volume. When an electron or hole is doped into an antiferromagnet or
when a nucleon is propagating through the QCD vacuum, a Berry phase arises from a monopole field and the angular momentum of the rotor is quantized in half-integer units.Comment: 4 page
The equivalence of inertial and gravitational masses is a defining feature of general relativity. Here, we clarify the status of the equivalence principle for interactions mediated by a universally
coupled scalar, motivated partly by recent attempts to modify gravity at cosmological distances. Although a universal scalar-matter coupling is not mandatory, once postulated, it is stable against
classical and quantum renormalizations in the matter sector. The coupling strength itself is subject to renormalization of course. The scalar equivalence principle is violated only for objects for
which either the graviton self-interaction or the scalar self-interaction is important---the first applies to black holes, while the second type of violation is avoided if the scalar is
Galilean-symmetric.Comment: 4 pages, 1 figur
We have analysed here the equivalence of RVB states with $u=1/2$ FQH states in terms of the Berry Phase which is associated with the chiral anomaly in 3+1 dimensions. It is observed that the
3-dimensional spinons and holons are characterised by the non-Abelian Berry phase and these reduce to 1/2 fractional statistics when the motion is confined to the equatorial planes. The topological
mechanism of superconductivity is analogous to the topological aspects of fractional quantum Hall effect with $u=1/2$.Comment: 12 pages latex fil | {"url":"https://core.ac.uk/search/?q=author%3A(M.%20Fierz)","timestamp":"2024-11-05T00:16:36Z","content_type":"text/html","content_length":"142472","record_id":"<urn:uuid:affcddb8-baa4-473a-9063-a25a583e06ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00865.warc.gz"} |
A computer simulation oriented approach to the error rate evaluation of DCPSK non linear receivers
A method for calculating the signal error probability of a differentially-coherent phase shift keying system containing a bandpass nonlinear device in the receiver front-end is discussed. The
bandpass characteristics are represented by two deterministic functions describing the AM/AM and AM/PM conversion effects of the nonlinear device. The effects of timing jitter on the error
probability computation are also examined. A numerical evaluation of the error probability expression is presented; an example involving a simulation program formed by a set of samples obtained by
filtering a four-phase modulated signal with a five-pole Butterworth filter is given.
Alta Frequenza
Pub Date:
August 1977
□ Computerized Simulation;
□ Error Analysis;
□ Phase Shift Keying;
□ Probability Theory;
□ Signal Reception;
□ Transmission Efficiency;
□ Amplitude Modulation;
□ Bandpass Filters;
□ Performance Prediction;
□ Phase Coherence;
□ Phase Modulation;
□ Signal To Noise Ratios;
□ Communications and Radar | {"url":"https://ui.adsabs.harvard.edu/abs/1977AlFr...46..334C/abstract","timestamp":"2024-11-07T00:42:10Z","content_type":"text/html","content_length":"35688","record_id":"<urn:uuid:a93cfaf4-4e76-4255-bc61-b73da6190d00>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00457.warc.gz"} |
Today, Maxime Augier gave a great talk about the state of security of the internet PKI infrastructure. The corresponding paper written by Lenstra, Hughes, Augier, Bos, Kleinjung, and Wachter was
uploaded to eprint.iacr.org archive a few weeks ago. In a nutshell, they found out that some RSA keys, that is often used in the SSL/TLS protocol to secure internet traffic, are generated by bad
pseudo random number generators and can be easily recovered, thous providing no security at all.
The RSA cryptosystem
RSA is one of the oldest and most famous asymmetric encrytion schmes. The key generation for RSA can be summarized as follows:
For a given bitlength l (for example $latex l = 1024$ or $latex l = 2048$ bits), choose randomly two prime numbers p and q of of bitlength $latex l/2$. Choose a number $latex 1 < e < (p-1)*(q-1)$,
that has no divisor in common with $latex (p-1)*(q-1)$. Many people choose $latex e = 2^{16+1}$ here for performance reasons, but other choices are valid as well. Now, the number $latex n = p*q$ and
e form the public key, while $latex d = e^{-1} \bmod (p-1)*(q-1)$ is the private key. Sometimes, the numbers p and q are stored with the private key, because they can be used to accelerated
To encrypt a message m, one just computes $latex c = m^e \bmod n$, and to decrypt a message, one computes $latex m = c^d \bmod n$. However, we don’t need that for the rest of this text can safely
ignore that.
How random do these numbers need to be?
When generating cartographic keys, we need to distinguish between just random numbers and cryptographically secure random numbers. Many computers cannot generate real random numbers, so they
generate random numbers in software. For many applications like computer games and for example simmulations of experiments, we only need number that seem to be random. Functions like “rand()” from
the standard c library provide such numbers and the generation of these numbers is often initialized from the current system time only.
For cryptographic applications, we need cryptographically secure random numbers. These are numbers that a generated in a way, that there is no efficient algorithm, that distinguish them from real
random numbers. Generating such random numbers on a computer can be very hard. In fact, there have been a lot of breaches of devices and programs, that used a bad random number generator for
cryptographic applications.
What has been found out?
From my point of view, the paper contains two noteably results:
Many keys are shared by several certificates
6 185 228 X.509 certificates have been collected by the researchers. About 4.3% of them contained an RSA public key, that was also used in another certificate. There could be several reasons for
• After a certificate has expired, another certificate is issued, that contains the same public key. From my point of view, there is nothing wrong with doing that.
• A company changes their name or is taken over by another company. To reflect that change, a new certificate is issued, that contains another company name, but still uses the same key. I don’t see
any problems here either.
• A product comes with a pre-installed key, and the consumer has to request an certificate for that key. The same key is shipped to several customers. From my point of view, this is really a bad
• Or there might be really a bad random number generator in some key generation routines, that two entities that are not related come up with the same RSA public (and private) key. This is a
security nightmare.
Some keys share a common divisor
This is definitely not supposed to happen. If two RSA keys are generated, that share a common divisor by the same or by different key generation routines, the private key for both public keys can be
easily determined, and the key generation routine is deeply flawed.
What are the consequences?
For those, who use an RSA public key, that shares a modulus with another different RSA public key, their key provides no protection at all. All implementations, that generated these keys definitely
need to be updated and the certificates using the weak keys need to be revoked.
Which devices and vendors are affected?
Because disclosing the list of affected devices and vendors would compromise the security of these systems immediately, and allow everyone to recover their secret RSA keys, this has not been | {"url":"https://cryptanalysis.eu/blog/2012/04/","timestamp":"2024-11-03T02:44:19Z","content_type":"text/html","content_length":"38062","record_id":"<urn:uuid:573b5ee4-b28f-4c0a-93a8-666a42f28b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00045.warc.gz"} |
Menger's Theorem
We present a formalization of Menger's Theorem for directed and undirected graphs in Isabelle/HOL. This well-known result shows that if two non-adjacent distinct vertices u, v in a directed graph
have no separator smaller than n, then there exist n internally vertex-disjoint paths from u to v. The version for undirected graphs follows immediately because undirected graphs are a special case
of directed graphs.
Session Menger | {"url":"https://www.isa-afp.org/entries/Menger.html","timestamp":"2024-11-12T03:01:39Z","content_type":"text/html","content_length":"10047","record_id":"<urn:uuid:88a68f03-3b2b-4129-b3eb-07e94eec4286>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00789.warc.gz"} |
C-STEM College Curriculum
Algebra I with Computing
This course guides students through topics in Algebra 1 in Common Core State Standards for Mathematics while simultaneously teaching students programming and computational thinking. Students use
programming in C/C++ interpreter Ch to reinforce and extend their knowledge of mathematical concepts by analyzing real life situations, identifying given information, formulating steps that a
computer program could calculate to find a solution, analyzing the results for accuracy, and revising/modifying the programming solutions as necessary. Topics covered include solving one-variable
equations with multiple steps, solving and plotting absolute value equations and inequalities, linear equations, systems of linear equations and inequalities, polynomial functions, exponential
functions, and step and piecewise functions, evaluating, multiplying, and factoring polynomial functions, solving quadratic equations with applications, probability, statistical data analysis and
visualization, and arithmetic and geometric sequences. Group computing projects allow students to collaborate on critical thinking activities based on algebraic topics while developing their teamwork
and communication skills.
*Teaching resources contain optional robotics activities.
Algebra I with Computing and Robotics
The course guides students through topics in Algebra 1 in Common Core State Standards for Mathematics while simultaneously teaching students programming and computational thinking. Students use
programming in C/C++ interpreter Ch to reinforce and extend their knowledge of mathematical concepts by analyzing real life situations, identifying given information, formulating steps that a
computer program could calculate to find a solution, analyzing the results for accuracy, and revising/modifying the programming solutions as necessary. Topics covered include solving one-variable
equations with multiple steps, solving and plotting absolute value equations and inequalities, linear equations, systems of linear equations and inequalities, polynomial functions, exponential
functions, and step and piecewise functions, evaluating, multiplying, and factoring polynomial functions, solving quadratic equations with applications, probability, statistical data analysis and
visualization, and arithmetic and geometric sequences. Robotics activities allow students to reenact physically derived mathematical problems through robotics technologies to visualize situations,
associate linear and quadratic graphs with physical phenomenon, predict and identify key features of the graphs with robotic systems, and solve robotics problems through mathematical modeling and
*Teaching resources contain robotics activities.
Integrated Mathematics I with Computing
The course guides students through topics in Integrated Mathematics 1 in Common Core State Standards for Mathematics while simultaneously teaching students programming and computational thinking.
Students use programming in C/C++ interpreter Ch to reinforce and extend their knowledge of mathematical concepts by analyzing real life situations, identifying given information, formulating steps
that a computer program could calculate to find a solution, analyzing the results for accuracy, and revising/modifying the programming solutions as necessary. Topics covered include solving
one-variable equations with multiple steps, solving and plotting absolute value equations and inequalities, linear equations, systems of linear equations and inequalities, exponential functions,
statistical data analysis and visualization, arithmetic and geometric sequences, and geometric transformations, including translations, rotations, and reflections, and geometric construction. Group
computing projects allow students to collaborate on critical thinking activities based on mathematics topics while developing their teamwork and communication skills.
* Teaching resources contain optional robotics activities.
Integrated Mathematics I with Computing and Robotics
The course guides students through topics in Integrated Mathematics 1 in Common Core State Standards for Mathematics while simultaneously teaching students programming and computational thinking.
Students use programming in C/C++ interpreter Ch to reinforce and extend their knowledge of mathematical concepts by analyzing real life situations, identifying given information, formulating steps
that a computer program could calculate to find a solution, analyzing the results for accuracy, and revising/modifying the programming solutions as necessary. Topics covered include solving
one-variable equations with multiple steps, solving and plotting absolute value equations and inequalities, linear equations, systems of linear equations and inequalities, exponential functions,
statistical data analysis and visualization, arithmetic and geometric sequences, and geometric transformations, including translations, rotations, and reflections, and geometric construction..
Robotics activities allow students to reenact physically derived mathematical problems through robotics technologies to visualize situations, associate linear and exponential graphs with physical
phenomenon, predict and identify key features of the graphs with robotic systems, and solve robotics problems through mathematical modeling and programming.
* Teaching resources contain robotics activities.
Introduction to Computer Programming for Engineering Applications (a UC Davis Engineering Course)
This course introduces students to structured programming in C. Many algorithms for computer-aided problem solving are developed throughout the course to solve practical problems in engineering and
science. The topics include number systems with internal representations of binary, octal, decimal, and hexadecimal numbers as well as binary two’s complementary representation; limitations and
numerical accuracy of different data types; 32-bit and 64-bit programming models; unary, binary, and ternary operators; selection statements for making decisions; iterative statements for
repetitions; modular programming and code reuse; storage classes; arrays for data processing; pointers; dynamical memory allocation and deallocation; ASCII Code; characters and strings; structures
and enumerations; top-down and bottom-up design of large-scale software project; file processing; and computational arrays for matrices and linear algebra for engineering applications. | {"url":"https://cstem2.sf.ucdavis.edu/curriculum/college","timestamp":"2024-11-08T06:09:40Z","content_type":"text/html","content_length":"53533","record_id":"<urn:uuid:7c3726ca-2d77-40ec-8392-b85da8dbcec2>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00800.warc.gz"} |
Applications of automatic differentiation to compute the equilibria of dynamical systems
Derivatives of mathematical functions play a key role in many scientific areas. To compute them we briefly present a quite recent method, namely Automatic Differentiation (AD) [1], which
automatically transforms a program that calculates numerical values of a function, into a program which calculates numerical values for derivatives of that function. After that, we use ADiMat [2], a
MATLAB/Octave tool which implements AD, to compute equilibria of an epidemiological model and study their stability. We are then able to compare the difference in time complexity and accuracy between
AD and other classical techniques calculating derivatives.
This seminar concerns the results of Marco’s BSc thesis.
• [1] , Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2 ed., Society for Industrial and Applied Mathematics, Philadelphia, 2008, DOI: 10.1137/1.9780898717761.
• [2] , Combining source transformation and operator overloading techniques to compute derivatives for MATLAB programs, in Proceedings of the Second IEEE International Workshop on Source Code
Analysis and Manipulation (SCAM 2002), IEEE Computer Society, Los Alamitos, CA, 2002, pp. 65–72, DOI: 10.1109/SCAM.2002.1134106. | {"url":"http://cdlab.uniud.it/events/seminar-20181012","timestamp":"2024-11-05T13:48:32Z","content_type":"text/html","content_length":"17098","record_id":"<urn:uuid:6ebf6c9d-667e-4c57-a527-82198cbfe984>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00689.warc.gz"} |
Exploring Things That Come in Eights With Pictures
Eights hold a unique significance across various domains. From the mathematical elegance of an octagon’s eight sides to the harmonic charm of an octave’s eight notes, this number manifests its
influence. Let’s shake things up with a journey into the world of eights.
So, Which Things Come in Eights? There are a lot of things that come in eights. Like as octagons, octopuses, spider legs, crayons in a standard pack, and octaves.
Numbers have stories to tell, and today, we’re focusing our attention on a special guest “the number eight.” Join me as we explore its intriguing presence in various corners of our lives.
A Visual List of Things That Come in Eights
Eight has a remarkable presence in our everyday life. So, here’s a list of things that often come in 8:
The Octagon is one of the most recognizable shapes associated with the number eight. It’s a polygon with eight sides and eight angles.
It’s frequently used in road signs, logos, and architectural design to convey strength, balance, and structure.
An octave is an interval spanning eight notes on the diatonic scale. It represents a harmonious doubling or halving of a frequency, producing a pleasing and balanced sound.
Octopuses typically have eight arms, which gives them their name. These arms are also called tentacles. Octopus arms are incredibly flexible and can be moved in virtually any direction.
Spider Legs
Spiders are arachnids, and one of the defining characteristics of arachnids is having 8 legs.
These legs are highly articulated and jointed. Spiders use their legs for walking and climbing.
They exhibit a unique walking pattern known as “alternating tetrapod” movement.
Stop Signs
A standard stop sign typically has eight sides. It is shaped like an octagon. Stop signs play a crucial role in traffic control and safety, helping to prevent accidents and ensure the orderly flow of
vehicles at intersections.
Octane is a term often associated with chemical compounds containing eight carbon atoms in their molecular structure.
They contain at least one carbon-carbon double bond (C=C) in their structure.
A byte is a fundamental unit of digital information. It is made up of 8 bits. Each bit can represent one of two values, and these are 0 or 1.
Bytes are used to represent characters in computer systems.
Eight Planets (Formerly)
In our vast solar system, there are eight recognized planets. Before Pluto’s reclassification, there were eight recognized planets in our solar system.
Eight-Track Tape
The eight-track tape was a common medium for playing music. These magnetic tape cartridges were revolutionary at the time, allowing people to enjoy their favorite tunes on the go.
Tarantula Eyes
Tarantulas typically have eight eyes and which is a common characteristic among most spiders. These eyes are arranged in two rows on the front of their cephalothorax.
Eight Immortals
In Chinese mythology, the Eight Immortals are a group of legendary figures who have achieved immortality and are often depicted together.
Eight Wonders of the Ancient World
The original list of the Seven Wonders of the Ancient World was expanded by the Greeks to include the “Eighth Wonder,” the Lighthouse of Alexandria.
Figure Eight Knot
The figure eight knot is a type of stopper knot used in various applications, such as climbing, sailing, and rescue operations.
Eight Ball in Pool
In the game of pool, the black ball (the “eight ball”) is the last ball to be pocketed, and the player must call the pocket for it to be sunk legally.
In this classic cue sport, the eight-ball holds a special place as the game-winning objective.
Mathematical Things That Come in 8
Mathematics is the field where number holds the whole significance. Like other numbers, 8 is an important part of every kind of calculation.
So, here are some mathematical concepts or properties related to the number eight:
Fibonacci Sequence
Eight is a Fibonacci number. The Fibonacci sequence has applications in various fields, including mathematics, computer science, and finance.
The Fibonacci sequence is a popular topic in recreational mathematics, and it has inspired countless puzzles, patterns, and artistic creations.
Perfect Cubes
Eight is a perfect cube and it can be expressed as 2^3. Perfect cubes are often used in geometry to represent three-dimensional objects with equal length, width, and height.
Euler’s Formula
Euler’s formula is a fundamental equation in mathematics that relates complex exponentials, trigonometric functions, and the imaginary unit “I.”
This formula holds true for any polyhedron where the surface is topologically equivalent to a sphere, which includes many common polyhedra-like cubes.
8 raised to any power (8^n) will always have a last digit of 8 for any positive integer value of n.
An octahedron is a polyhedron with eight faces. These faces are identical equilateral triangles, which means all sides and angles of each triangle are equal.
Roots of Unity
The eighth roots of unity are the solutions to the equation z^8 = 1 in the complex plane.
These roots are equally spaced around the unit circle and have interesting mathematical properties.
Funny Things That Come in Group of Eights
Fun is everywhere. Therefore, here is a list of funny things that come in 8:
Eight-legged pants
Can you Imagine pants with eight legs? It’s a comical concept that might be amusing to picture.
Eight-leaf clover
A rare variation of the four-leaf clover, considered extremely lucky.
A humorous play on words, combining “Octo” (eight) with “pie.”
Eight-wheeled car
An exaggerated and funny version of a car with twice the usual number of wheels.
Eight-layer dip
A play on the popular seven-layer dip, adding an extra layer for fun.
Eight-second rule
A humorous extension of the “five-second rule,” suggesting that if you drop something, you have 8 seconds to pick it up before it’s considered too dirty to eat.
Facts About the Number 8
• The first cubic number is eight and it represents the earth.
• In Egypt, the number eight represents balance and harmony.
• In Japan, eight is the digit that narrates multiplicity.
• Octo-caffeinated: Octo-caffeinated is used to describe someone who’s had way too much coffee and is buzzing with energy.
Final Words
The number 8 isn’t just a digit – it’s an intricate thread that weaves through the fabric of our world.
In this journey through the realm of eights, we’ve uncovered a lot of things like famous things, mathematical terms, cultural things and so on.
I promise to continually update this article as more things come across which is related to 8.
Did I leave anything behind about things that come in eights? If I left anything, then let me know in the comment section. I would love to correct myself.
Sharing is caring….!!!
Leave a Comment | {"url":"https://listexplored.com/things-that-come-in-eights/","timestamp":"2024-11-05T00:06:15Z","content_type":"text/html","content_length":"72462","record_id":"<urn:uuid:9068945b-5b46-426b-985f-ea44026d7999>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00897.warc.gz"} |
375 grams of Applesauce in Cups • Recipe equivalences
375 grams of Applesauce in Cups
How many Cups are 375 grams of Applesauce?
375 grams of Applesauce in Cups is approximately equal to 1 cup and 1/5.
That is, if in a cooking recipe you need to know what the equivalent of 375 grams of Applesauce measure in Cups, the exact equivalence would be 1.4995575, so in rounded form it is approximately 1 cup
and 1/5.
Is this equivalence of 375 grams to Cups the same for other ingredients?
It should be noted that depending on the ingredient to be measured, the equivalence of Grams to Cups will be different. That is, the rule of equivalence of Grams of Applesauce in Cups is applicable
only for this ingredient, for other cooking ingredients there are other rules of equivalence.
Please note that this website is merely informative and that its purpose is to try to inform about the approximate equivalent values to estimate the weight of the products that can be used in a
cooking recipe, such as Applesauce, for example. In order to have an exact measurement, it is recommended to use a scale.
In the case of not having an accessible weighing scale and we need to know the equivalence of 375 grams of Applesauce in Cups, a very approximate answer will be 1 cup and 1/5. | {"url":"https://www.medidasrecetascocina.com/en/applesauce/375-grams-applesauce-in-cups/","timestamp":"2024-11-11T17:41:18Z","content_type":"text/html","content_length":"57620","record_id":"<urn:uuid:6522ff38-c393-4ccc-9bf6-ca6ffbdb315c>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00837.warc.gz"} |
The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements
In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically,
we compare performance of linear solvers when discretizing using Co B-splines, which span traditional finite element spaces, and Cp-1 B-splines, which represent maximum continuity We provide
theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical
results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis. Finally, we
present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most 33p2/8 times more expensive for the more
continuous space, although for moderately low p, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for
high p. Preconditioning options can be up to p3 times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization.
© 2013 Society for Industrial and Applied Mathematics.
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01
Dive into the research topics of 'The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements'. Together they form a unique fingerprint. | {"url":"https://academia.kaust.edu.sa/en/publications/the-cost-of-continuity-performance-of-iterative-solvers-on-isogeo","timestamp":"2024-11-09T03:12:11Z","content_type":"text/html","content_length":"57654","record_id":"<urn:uuid:1acfffbd-db50-44f6-bb93-6f7caa541385>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00791.warc.gz"} |
CPM Homework Help
Giuseppe decides that he really wants some ice cream, so he leaves the house at 3:00 p.m. and walks to the ice cream parlor. He arrives at 3:15 (the ice cream parlor is 6 blocks away). He buys an ice
cream cone and sits down to eat it. At 3:45 he heads back home, arriving at 4:05. Find Giuseppe’s average walking rate in blocks per hour for each of the following situations.
1. His trip to the ice cream parlor.
How far did he walk to get there? How long did it take?
Since the question asks for blocks per hour, convert minutes to hours.
$\frac{6\text{ blocks}}{0.25\text{ hours}}=24\text{ blocks per hour}$
2. His trip back home.
3. The entire trip including the time spent eating.
Divide the total distance walked by the time it took to do so. | {"url":"https://homework.cpm.org/category/CC/textbook/cca2/chapter/3/lesson/3.1.2/problem/3-36","timestamp":"2024-11-03T00:20:43Z","content_type":"text/html","content_length":"35952","record_id":"<urn:uuid:73858d6f-3e13-4246-8af4-5d4eee7a4594>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00045.warc.gz"} |
orksheets for 2nd Class
Recommended Topics for you
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Stein 2nd grade common core math word problems 29-35
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Explore Math Word Problems Worksheets by Grades
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Explore printable Math Word Problems worksheets for 2nd Class
Math Word Problems worksheets for Class 2 are an essential resource for teachers looking to challenge their students and develop their problem-solving skills. These worksheets provide a variety of
math problems that are specifically designed for second-grade students, covering topics such as addition, subtraction, multiplication, and division. By incorporating real-life scenarios and engaging
illustrations, these worksheets help students to better understand the concepts being taught and apply them to everyday situations. Teachers can use these worksheets as part of their lesson plans,
homework assignments, or even as supplementary material for students who may need extra practice. With a wide range of difficulty levels and topics, Math Word Problems worksheets for Class 2 are a
valuable tool for any teacher looking to enhance their students' math skills.
Quizizz is an innovative platform that offers a variety of educational resources, including Math Word Problems worksheets for Class 2, to help teachers create engaging and interactive learning
experiences for their students. In addition to worksheets, Quizizz also provides teachers with access to thousands of quizzes, games, and other activities that can be easily customized to align with
their curriculum and learning objectives. By incorporating these resources into their lesson plans, teachers can create a more dynamic and engaging learning environment that caters to the diverse
needs of their students. Furthermore, Quizizz offers real-time feedback and analytics, allowing teachers to track student progress and identify areas where additional support may be needed. With its
user-friendly interface and extensive library of resources, Quizizz is an invaluable tool for teachers looking to enhance their students' learning experience in math and beyond. | {"url":"https://quizizz.com/en/math-word-problems-worksheets-class-2","timestamp":"2024-11-04T01:48:52Z","content_type":"text/html","content_length":"149360","record_id":"<urn:uuid:be9a995c-4ff4-4605-b255-6e527a86a998>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00666.warc.gz"} |
What Is Gamma Hedging?
Gamma hedging is an options trading strategy that seeks to reduce the risk associated with changes in the price of the underlying asset. The goal is to offset the potential for loss that would result
if the underlying asset's price moves in a direction that is unfavorable to the position.
The key to gamma hedging is to take offsetting positions in the underlying asset and in options contracts. For example, if a trader has a long position in a call option and the underlying asset's
price begins to fall, the trader can offset the potential loss by buying shares of the underlying asset. This will reduce the amount of money that the trader stands to lose if the underlying asset's
price falls further.
Gamma hedging can be a complex and risky strategy, so it is generally only employed by experienced traders. Is gamma same for call and put? Yes, gamma is the same for both call and put options.
Is gamma positive or negative?
Gamma is the rate of change of an option's delta in relation to the underlying asset's price. Delta is a measure of an option's price sensitivity to changes in the underlying asset's price. Gamma can
be either positive or negative, depending on whether the delta is increasing or decreasing as the underlying asset's price changes.
What does gamma mean in options trading?
Gamma is a measure of the rate of change in the price of an option contract with respect to changes in the underlying asset price. For example, if the underlying stock price increases by $1, the
option's gamma would be the amount by which the option's price would increase.
What are the two types of hedging?
There are two primary types of hedging:
1. Short hedging: This type of hedging involves taking a short position in a security in order to offset the risk of loss from a long position in another security. For example, a trader who is long a
stock might hedge by taking a short position in a put option on that stock.
2. Long hedging: This type of hedging involves taking a long position in a security in order to offset the risk of loss from a short position in another security. For example, a trader who is short a
stock might hedge by taking a long position in a call option on that stock. What is gamma in options with example? Gamma is the rate of change in the delta of an option. Delta is the rate of change
in the price of the option with respect to the underlying asset. Gamma is therefore the rate of change in the delta with respect to the underlying asset.
For example, consider an option with a delta of 0.5. This means that for every $1 increase in the price of the underlying asset, the option will increase in value by $0.5. If the gamma of this option
is 0.2, then for every $1 increase in the underlying asset, the delta will increase by 0.2. This means that the option will now increase in value by $0.6 for every $1 increase in the underlying
Gamma is therefore a measure of the convexity of an option. A higher gamma means that the option is more sensitive to changes in the underlying asset, and a lower gamma means that the option is less
sensitive to changes in the underlying asset. | {"url":"https://www.infocomm.ky/what-is-gamma-hedging/","timestamp":"2024-11-12T10:18:07Z","content_type":"text/html","content_length":"40535","record_id":"<urn:uuid:44b554d0-85c4-416d-b1d9-52b84ce87066>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00793.warc.gz"} |
Crore Rupees to US Dollar Conversion Calculator
This tool converts currency Indian Crores to US Dollar equivalent
• Exchange Rate (number of rupees to one dollar)
• Number of crores in Indian Rupees
• The calculator will find the equivalent in US$ (you can also use the drop down menu to select Thousand $, Million $ or Billion $)
US$ = (Number of Crores * 10^7)/(Exchange Rate)
Where Exchange Rate is the number of Rupees (₹) to One US Dollar
One Crore is equal to 10,000,000
Example Calculations
For an exchange rate of 84 Rupees to One Dollar,
• 5 crores INR is equivalent to US$595,238
• A salary of 1 crore INR is equivalent to US$119,048
• 1000 crores is equivalent to US$119,047,619
• 100 crore rupees is approximately 12 million US dollars
• 10,000 crore rupees is approximately 1.2 Billion US$
A crore rupees is a commonly used unit in the Indian numbering system to represent ten million rupees (10,000,000 INR). It is widely used in countries like India, Pakistan, Bangladesh, Nepal, and Sri
Lanka to measure large sums of money in both personal and business contexts.
Crore in the Indian Numbering System
In the Indian numbering system, large numbers are expressed differently from the Western system. Here’s how it works:
• 1 lakh = 100,000 rupees
• 1 crore = 10,000,000 rupees (or 100 lakhs)
In numerals:
• 1 crore = 1,00,00,000 INR
This is equivalent to 10 million in the international system.
Related Converters
• Crore Rupees to Euro
• Lakh to USD | {"url":"https://www.orbit6.com/crore-rupees-to-us-dollar-conversion-calculator/","timestamp":"2024-11-03T13:56:21Z","content_type":"text/html","content_length":"73903","record_id":"<urn:uuid:aa29fa93-80bf-46ce-bd8a-864f94ced63e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00194.warc.gz"} |
Embracing the Power of Formal Methods in my Coding Journey: How I Became a Dafny Evangelist | Consensys
By Joanne Fuller
I want to begin by saying that I am writing this blog post in the hope that others can experience the epiphany moment that I had while learning Dafny as part of my exploration into formal methods.
Further, it is my hope that this post will act as a catalyst for others to consider formal methods as a critical and necessary skill within the arsenal of anyone who writes code. As part of the
Automated Verification Team within R&D at Consensys, I am using Dafny in the formal verification of the Ethereum 2 Phase 0 specification, and I want to share why I find it useful.
My background
I should make it clear that I am not a software developer, but rather I consider myself to be a mathematical programmer with some knowledge of software development. I first learnt to write programs
as part of my maths class during my final year of high school and I probably should mention that although I quite liked using computers at that time, the prospect of learning how to program scared me
to the point where I almost dropped that particular maths class. Having decided to face my fear of failure (with respect to learning to program and potentially ruining my result in this class), I
went on to experience my first epiphany moment in the context of programming. I can still vividly remember sitting in class and having the realisation that writing a program to solve a maths problem
wasn’t some magical and mysterious process, it was almost like writing down how I would work through a problem in my head. There was no looking back after that!
Programming has been an important aspect of everything that I have done since. My PhD in cryptography relied heavily on an ability to develop algorithms and then program optimal implementations. My
programs were written for experimentation and, although I didn’t undertake what we would now refer to as formal testing, I would informally check bounds and test cases using logical reasoning about
the intended output. I also worked for many years as an academic undertaking research in the field of finance and economics. Again this included writing programs, and again I used my own techniques
to informally test and reason about their correctness.
It is fair to say that although I had an appreciation for the fact that testing would always be incomplete in the sense that it was impossible to test every case; that I was reasonably confident that
my mathematical way of thinking was pretty good when it came to informally testing in a rigorous manner. As such I definitely did not have a full appreciation of the difference between testing and
proving correctness, nor the consequences of such! During my career prior to joining Consensys I was content to rely on my own informal techniques for determining what I thought was correctness via
My background is therefore part of the story, as I am myself somewhat surprised that I did not discover formal methods earlier. I consider myself to be a mathematician; I love maths, algorithms and
logic. It now seems crazy to simply rely on incomplete testing, but also it seems crazy for anyone who programs to not at least have some appreciation of what formal methods can offer and the
potential consequences of missing a bug given the many ways in which computer programs are integrated into our lives. Formal methods allow us to go beyond testing, to prove that a program is correct
against a specification that includes pre and post conditions.
A first Dafny example
As a simple example consider the integer division of a non-negative dividend n by a positive divisor d;
n / d
shown below:
Although in a typed programming language we can somewhat restrict the input parameters, it is not always sufficient. In this example the specification of n and d as natural numbers means that both
inputs must be non-negative integers but it doesn’t provide for the restriction of d to being a positive integer. The use of a pre-condition by way of the requires statement provides for such a
restriction and means that this method can only be called if d > 0. Hence if any other part of the program would cause div to be called without such a pre-condition being satisfied, then the program
will not verify. The ensures statement then provides the post condition and provides a formal specification of what the method output must satisfy.
This example is written using Dafny: “A language and program verifier for functional correctness” and brings me to my next point, that is, why I am such a fan of Dafny. I think it is fair to say that
for many programmers, the thought of using “formal methods” to verify program correctness is somewhat scary and is often perceived as being “too” hard. Whether this is because of a lack of exposure
to the techniques, a lack of appreciation of the benefits, or even a lack of training in this area; whatever the reasons may be, I believe that Dafny has the ability to allow any programmers to
quickly achieve success in applying formal methods in their work. Looking at the code snippet above, I would expect anyone with some programming knowledge to be able to read this Dafny code; Dafny is
very much a programmer friendly language. Once you learn a little bit of Dafny it is very easy to start experimenting and then basically learn as you go. And if you are interested in learning Dafny,
a great place to start is the tutorial series by Microsoft. The site also includes an online editor, so it is very easy to try out the tutorial examples. The Verification Corner YouTube channel is
another source of useful references.
My epiphany moment
Finally I wanted to share my epiphany moment from when I was learning Dafny. I have certainly heard stories about short and simple pieces of code, from large reputable companies, having bugs that
were missed and ultimately costing many millions of dollars; but I think it is only when you realise yourself how easy it would be to unintentionally create a bug in a simple function that it all
makes sense! The moment when you say to yourself “Oh, it would be so easy to make that mistake!”
My moment came while watching one of the Verification Corner videos.
In this tutorial Rustan Leino goes through a SumMax method that takes two integers, x and y, and returns the sum and max, s and m, respectively. This example is relatively straightforward and the
Dafny code is shown below.
The inputs x and y are specified as integers through typing and no other preconditions are required. The three post conditions provide checks that the output does indeed meet the specifications,
namely that s does equal x + y, and that m is equal to either x or y and that m does not exceed x and y. The SumMaxBackwards method is then presented as an exercise and this is where it gets more
interesting. The specification is the reverse of SumMax, i.e. given the sum and maximum return the integers x and y. Ok, so a first attempt might be with the same postconditions; as the relationships
between the inputs and outputs still hold. If we let x be the maximum then a quick bit of algebra tells us that y should equal the sum minus the maximum. Putting this into the online editor gives the
It doesn’t verify. So what went wrong? We are told that a postcondition didn’t hold and specifically the postcondition on line 3 (ensures x<= m && y <= m) may not hold. Looking more closely we see
that this post condition specifies that x <= m and y <= m. Well, we know that x is less than or equal to m as we set x equal to m, so this means that the y <= m part doesn’t verify. How can this
happen? Our algebra told us that y := s - m. Let’s say s is 5 and m is 3, then y = 5 - 3 = 2 which ensures y <= m; but let’s say we call this method with s equal to 5 and m equal to 1. Nothing will
stop us from calling the method with these input parameters but to do so will cause a problem as y = 5 - 1 = 4 and then y > m. Basically what we are seeing here is that even though the input
parameter is meant to be the maximum of two integers that creates the sum s, there isn’t anything to stop us trying to call the method with an input that isn’t valid. Unless a precondition is
included to restrict the inputs of s and m to valid integers that will result in outputs x and y that meet the specification, then our method can produce incorrect results. What relationship do we
need between s and m to provide valid inputs? A bit more algebra shows us that s <= m * 2 for there to be a valid solution of x and y. If we add this as a precondition, Dafny is now able to verify
the code as shown below.
This was the example where I could see just how easy it is to introduce a bug into code. Just because we call the input parameters ’s’ for sum and ‘m’ for maximum, doesn’t mean that the method will
be called appropriately and as such as part of some larger program, there could be many unintended consequences that follow from this type of bug. I hope it is useful for anyone else learning about
Dafny or formal methods more generally.
What I am working on now
Well that brings me to the end of my post. If you would like to see what I am currently working on with Dafny then check out this GitHub repo. I am part of the Automated Verification Team within R&D
at Consensys and we are using Dafny in the formal verification of the Ethereum 2 Phase 0 specification. The use of formal methods in the blockchain space is an exciting new area of research that has
been embraced by Consensys and I would encourage anyone interested in learning more about Eth 2.0 to look at the resources available within our project repo. | {"url":"https://consensys.io/blog/embracing-the-power-of-formal-methods-in-my-coding-journey","timestamp":"2024-11-13T14:13:50Z","content_type":"text/html","content_length":"123014","record_id":"<urn:uuid:8696956d-ca40-4203-a6d0-9e0d3793591d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00805.warc.gz"} |
NickyP - LessWrong
Sorted by
Maybe not fully understanding, but one issue I see is that without requiring "perfect prediction", one could potentially Goodhart on on the proposal. I could imagine something like:
In training GPT-5, add a term that upweights very basic bigram statistics. In "evaluation", use your bigram statistics table to "predict" most topk outputs just well enough to pass.
This would probably have a negative impact to performance, but this could possibly be tuned to be just sufficient to pass. Alternatively, one could use a toy model trained on the side that is easy to
understand, and regularise the predictions on that instead of exactly using bigram statistics, just enough to pass the test, but still only understanding the toy model.
Cadenza Labs has some video explainers on interpretability-related concepts: https://www.youtube.com/@CadenzaLabs
For example, an intro to Causal Scrubbing:
In case anyone finds it difficult to go through all the projects, I have made a longer post where each project title is followed by a brief description, and a list of the main skills/roles they are
looking for.
See here: https://www.lesswrong.com/posts/npkvZG67hRvBneoQ9
Sorry! I have fixed this now
I think there are already some papers doing similar work, though usually sold as reducing inference costs. For example, the MoEfication paper and Contextual Sparsity paper could probably be modified
for this purpose.
I wonder how much of these orthogonal vectors are "actually orthogonal" once we consider we are adding two vectors together, and that the model has things like LayerNorm.
If one conditions on downstream midlayer activations being "sufficiently different" it seems possible one could find like 10x degeneracy of actual effects these have on models. (A possibly relevant
factor is how big the original activation vector is compared to the steering vector?)
While I think this is important, and will probably edit the post, I think even in the unembedding, when getting the logits, the behaviour cares more about direction than distance.
When I think of distance, I implicitly think Euclidean distance:
But the actual "distance" used for calculating logits looks like this:
Which is a lot more similar to cosine similarity:
I think that because the metric is so similar to the cosine similarity, it makes more sense to think of size + directions instead of distances and points.
This is true. I think that visualising points on a (hyper-)sphere is fine, but it is difficult in practice to parametrise the points that way.
It is more that the vectors on the gpu look like , but the vectors in the model are treated more like
Load More | {"url":"http://www.lesswrong.com/users/nicky","timestamp":"2024-11-09T15:32:57Z","content_type":"text/html","content_length":"565000","record_id":"<urn:uuid:1b924c0a-cf6b-45a3-a7a1-e380a95ce356>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00699.warc.gz"} |
solving radical equations with parameters
solving radical equations with parameters
I would like to find the solutions $y$ to this type of equations: $$\left(1+x -\sqrt{(1+x)^2-4y}\right)^2=z$$ with conditions on $x,y,z$ (like $0\lt y\lt x\leq \frac18$ and $0\lt z\lt x^2$).
Using solve with the option to_poly_solve:
sage: solve((1+x - sqrt((1+x)^2-4*y))^2 == z, y, to_poly_solve=True)
[y == 1/2*x^2 - 1/2*(x + 1)*sqrt(x^2 + 2*x - 4*y + 1) + x - 1/4*z + 1/2]
does not seem to work because $y$ appears on the right side of the solution. I expect to find a solution like $$y=\frac14\left((1+x)^2-\left(1+x-\sqrt{z}\right)^2\right).$$
I also tried the same after specifying the conditions with assume(), without success.
1 Answer
Sort by ยป oldest newest most voted
I tinkered with your equation and finally had something that looks like a solution. But this is not the way I expect solving equations with sage.
var('x y z')
expr = (1+x - sqrt((1+x)^2-4*y))^2
eqn1 = expr.expand().simplify_full() == z
eqn2 = eqn1.subtract_from_both_sides(2*x^2 +4*x -4*y +2)
eqn3 = eqn2.lhs()^2 == eqn2.rhs()^2
eqn4 = eqn3.expand()
eqn5 = eqn4.subtract_from_both_sides(eqn4.lhs())
sol = solve(eqn5,y)
for s in sol: show(s)
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Designing an E-Game Program in Mathematics for 5th Grade Saudi Female Pupils: Does Gagne’s Theory Have Any Effectiveness in Developing Their Achievement of Mathematics?
Open Journal of Social Sciences Vol.03 No.02(2015), Article ID:55528,9 pages
Designing an E-Game Program in Mathematics for 5th Grade Saudi Female Pupils: Does Gagne’s Theory Have Any Effectiveness in Developing Their Achievement of Mathematics?
Salma Katib S. AL-Shammari^1, Abdellatif E. Elgazzar^1,2, Ahmed M. Nouby^1,3
^1Distance Learning Department, CGS, Arabian Gulf University, Manama, Kingdom of Bahrain
^2Department of Inst. Technology, Women Faculty, Ain Shams University, Cairo, Egypt
^3Department of Curricula & Instruction, College of Education, Suez Canal University, Ismaïlia, Egypt
Email: sose777@hotmail.com, dr.a_latif@hotmail.com, ahmednouby2005@yahoo.com
Received December 2014
Design variables of e-learning technologies have gained great attention and concern from developers as well as researchers. So, e-Games design has gained a special concern in its design as an e-
Learning technology in accordance with learning theories and instructional models such as Gagne’s theory. This research aims at investigating the effectiveness of an instructional e-game program
designed according to the Gagne’s theory on developing mathematics’ achievement among primary school pupils in Saudi Arabia. The research sample composed of 46 female pupils from the fifth grade
pupils at King Fahd University of Petroleum and Minerals’ primary schools. Thus, the Developmental Research Method was adopted. The study sample was selected as a purposed cluster one of two classes
(sections). The quasi-experimental design with pre and post testing was used. The sample was divided to two equal groups which were randomly assigned as: experimental group 1 (e-Games with Gagne’s
theory), experimental group 2 (e-Games without Gagne’s theory); (23) female pupils in each group. An instructional unit has been selected from the 5th grade mathematics curriculum, its content was
analyzed to drive the mathematical achievement categories of knowledge and cognitive skills. Then, a list of instructional design standards was derived for designing the instructional e-Games. The
two instructional e-games programs were developed by the 1st author using Elgazzar (2002) ISD Model of the 2nd author: one e-Games program with Gagne’s theory and the other one e-Games program
without Gagne’s theory. Both the instructional e-games programs were refereed, revised and approved according to those derived standards of instructional design. A Mathematics Achievement Test was
developed and approved to be valid and reliable. The 1st author carried out the research experiment according to the experimental design. So; the first experimental group 1 was taught using the
e-Games program (according to Ga- gne’s theory), and the experimental group 2 was taught using e-Games program (without Gagne’s theory). The research instrument was implemented pre and post
experimentation according to the experimental design. Data were collected and encoded to the SPSS package and ana-lyzed using appropriate statistical methods to test the research hypotheses. Research
results have revealed that there is no effectiveness of e-Games Program with Gagne’s theory as compared with e-Games without Gagne’ Theory in developing the Achievement of Mathematics, and this
result gives an answer to the main research question that designing instructional e-Games program based on Gagne’s Theory doesn’t have any comparative Effectiveness in Developing Saudi female pupils’
in the Achievement of Mathematics. This result calls for further research on other learning outcomes of Mathematics’ learning. The article includes Tables, References, Recommendations, and a list of
proposed future researches.
Design, E-Games Program, Based-On Gagne’s Theory, Elgazzar (2002) ISD Model, Effectiveness, Mathematics Subject Matter, Achievement in Mathematics, Primary Stage, Saudi Arabia
1. Introduction
Educational and training technology as an applied scientific field is being developed due to the research and development in its innovations, increasing knowledge, and concerns to its design
variables. The attention has become not only the effect of the use of these innovations on learning outcomes, but extended attention to how to design these innovations and discovering effectiveness
of these modern designs. The educational games―in general―have benefits and potentials, and electronic games―in particular―are of those innovations that need more research in their design variables
and discovering their effectiveness. The current research is in line with these guidelines in discovering the effectiveness of new design of e-Games according to Gagne’s theory in developing
achievement in school mathematics.
Din & Calao [1], in their study, revealed an effectiveness of educational games in mathematics learning and knowledge retention among high school students. Their study and other studies didn’t
consider any of the many design variables of educational games that should be considered. Basically those design variables are derived from learning/instructional principles, theories, new
approaches, innovations, and models. More, recently researchers [2] have given attention to Gagne’s nine events of instruction for application in e-Learning to investigate its effectiveness. So, this
current research seeks to reveal the effectiveness of the design of educational e-Games in the light of the instructional events of Gagne’ theory, as the authors expect that this design for
educational e-Games can be of more effect in the development of mathematics education outcomes, such as cognitive achievement.
2. Guidelines from Gagne’s Theory of Learning and Instruction
Since the late of 1970s and the early of 1980s, Gagne’ as an instructional technology Scholar [3]-[5] has proposed an applied theory that combines learning and instruction to instructional designers,
that is, instruction is what we do in the learner’s environment and as a result learning takes place simultaneously inside the learner’s mind [6] [7] Moreover, the instructional situation/experience
is the relationship between nine instructional events should be designed in the learner’s environment and nine learning events takes place inside the learner’s mind. So, if instructional designers
design the learner’s environment according to these nine instructional events-as guidelines-in any instructional/learning system, intended learning outcomes will be more likely to take place inside
the learner’s mind as a result. These instructional/learning systems could be educational games’ programs, educational e-Games programs, e-learning environments, and e-training environments. In all
these types of instructional/learning environments can be designed in view of these guidelines that are derived from Gagne’ theory of these nine instructional events. These guidelines can be
implemented in designing learning activities that will help learners to acquire cognitive skills and achievement. These nine events of Gagne’, which are mentioned here as nine guidelines to give
designers more flexibility in their implementation, Gagne’ [8] describes that these nine Instruction events can be applied in different ways depending on the different required learning outcomes,
these nine instructional events are as follows: gaining attention, knowing learning objectives, retrieve prerequisite learned capabilities, presenting stimulus materials―content, learning guidance,
provide practice for eliciting the performance, provide feedback, assessing performance, and finally enhancing retention and transfer learning. In view, these nine events can be viewed as we
mentioned earlier as guidelines. The second author of this paper has realized the value of these guidelines as early as 1990s and put them in the heart of his ISD model proposed in 1995 for
developmental research which was implemented in developing e-Games [6] [9] [10]. It should be known that the authors are using the term “Gagne theory”, “Gagne’ instructional events”, and the term
“guidelines from Gagne’ theory” in the same sense.
3. Theoretical Foundations, Designing, and Design Standards of E-Games
There are some theoretical foundations that are supporting the expecting potential and positive effect of educational games/e-Games on instruction and learning outcomes of mathematics’ achievement.
Basically, games have some motivational characteristics such as competitions, rewarding, challenges, and eagerness for winning that have potential in developing learning outcomes. Gunter, Kenny& Vick
[11] mentioned that Ganji’s Theory of educational electronic games presents three main principles, namely: interest in learning, setting conditions that must be met to achieve success in the games,
and events that guide for the development and delivery of units of learning. Disruption of Cognitive Equilibrium Theory of Van Eck [12] [13], in view of this theory as Van Eck believes that the key
to learning in electronic games is where the student feels a state of dissatisfaction with the knowledge, and the desire to continue to win, so he starts to try and explore the game to bring
awareness, understanding, and finally adapt, then indulge, or immersion in the learning tasks within the game. Van Eck has proposed three approaches to fulfill the potential of e-Games. However,
potentials and capabilities of educational e-Games aren’t guaranteed without research and development on the design of e-Games in accordance to sound learning/instructional models, such as Guidelines
from Gagne’ Theory under investigation in this current research.
4. Design Aspects of Guidelines from Gagne’ Theory in E-Games
Authors set some design aspects of guidelines from Gagne’ Theory in e-Games so as to be considered as bases for developing educational e-Games program. Those design aspects of the nine instructional
events are as follows:
1) Gaining the learner’s attention is to be at the beginning of the game and through-out the game. Basically, introductory screens, logos, logo of the Ministry of Education, objectives of learning
from the game, clear rules of the game, visual and sound effects, multimedia, on-screen display of important events and variable, and avoidance of distractions are all of good attention gaining for
2) Informing the learner about learning objectives is very essential design aspect. Several ways are to be considered to inform the learner about learning objectives. These include but not limited to
allocating some of the screens at the beginning of the program in general, starting each game with a list of its learning objectives, and clarity of learning objectives’ statements, and suitable
location of objectives on screen layout.
3) Calling the learner’s required previous experience relevant to the learning is essential design aspects to bring meaningful and understandable learning. This can be done through the provision of a
range of relevant questions, relevant situations, providing introductory screens at the beginning of each e-Game for the learner to practice required information to be ready for that game, good
sequencing of e-Games as well as events within each e-Game, and helping the learner to remember some of the concepts that have already studied and needed for him to achieve his learning objectives
while winning the e-Game.
4) Content Display (stimuli), this event can be done in the course of the e-Games sequencing the content as along with the game, equations, algebraic expressions, graphics, good screen layout, texts,
sounds, sequencing small questions, and multimedia.
5) Provide guidance and assistance, this event can take different types that may include guidance and help for game controls, giving hints, providing information to help learner not to quit the game.
Also, hints and cues to make learner proceeds an win the game and achieve learning objectives.
6) Provide practice for eliciting required performance. This event may be accomplished during playing the e-Game by giving practice to elicit the required responses, giving a sequence of questions
and places on the screen for the learner to submit answers, gradual presentation of the questions from the simple ones to the complex ones, from easy ones to difficult ones, providing respond area
that appear in the same screen where the question to appear to answer it, diversity of interaction offered by the game for the learner between clicking the key, or choose from a drop-down list, or
move an item from one place to another in the same screen patterns, or enter text using the keyboard.
7) Provide feedback: This event is to give the student direct feedback after every activity, appropriate feedback as soon as the student response to any question presented during playing the game,
varying types of feedback, varying feedback amount from short to long, varying feedback between one medium to multimedia.
8) Assessing performance: This event can be done within the game, between games, or after related games, self-assessment tests, quizzes, formative and summative tests, e-tests, and may be paper
answered tests. These assessing performance tools are given to learner during the e-Game to monitor accomplishing objectives and mastering learning.
9) Enhancing retention and transfer of learning: This event can through providing concept maps, diagrams of information after segments of games, giving practices to apply learning, assignments,
providing summaries, using colored texts, highlighting texts, stressing important points, and providing more applications to what is learned.
Those Design Aspects of Guidelines from Gagne’ Theory’s works as a flexible framework for designing educational e-Games with Gagne’ Theory.
5. ISD Model for E-Games Program with/out Gagne’s Theory Designs
Instructional Systems are dynamic, interactive, interrelated components of learning resources (people, materials, content, devices, facilities, techniques to achieve pre-specified instructional
objectives. Instructional Systems Development is the application of systems’ development procedures (Analysis, Design, Develop/Produce, Evaluate, Utilize, Feedback, and Revise) of instructional
systems [14]. Basically, ISD models guide instructional developers in the instructional developmental process that vary in their purposes, amount of detail provided, degree of linearity in which they
are applied, and their operational tools [15]. Authors reviewed available ISD models that can be implement in this our developmental research that can accommodate Gagne’ theory about nine
instructional events and their discussed design aspect of them in developing e-Games program with/out Gagne’ theory (WGT/WoutGT). Elgazzar (2002) ISD model [9] [10] [14] was selected for this task
because, as it has been mentioned, has Gagne’ nine instructional events in the heart of its design phase (see Figure 1). This model’s procedures and implementation is found in several developmental
researches [16]. The model as it is shown in Figure 1 has five interrelated phases: Analysis, Design, Production/ construction, Evaluation, Use, and Feedback. It should be seen that the step,
Designing the instructional events (Gagne) and elements of the learning process, is in the design phase of that model.
6. Research Problem and Questions
The research problem has been stated as “there is an urgent need to get an e-learning solutions as e-Games to meet decline in students’ achievement and to discover the answer of the question: Does
Gagne’s Theory have any Effectiveness in Developing their Achievement of Mathematics?” and a main research question was to be answered: is the design of e-games according to Gagne’s theory (WGT)
effective in developing achievement among primary school pupils in mathematics? Four sub-questions have been derived:
1) What are the content components for students to achieve from a selected unit from primary school mathematics?
2) What are the design standards of e-Games program to development of achievement in mathematics?
3) What are the WGT and WoutGT designs of e-Games programs according those standards using Elgazzar ISD model (2002) to develop students achievement?
4) How effective is the implementation of these two designs WGT and WoutGT of e-games in developing achievement?
7. Research Method
The Developmental Research Method as described [15] was used in this research. This research method combined three integrated research methods were used:
1) Descriptive research method implemented in students’ characteristic analysis, course content analysis, resources analysis, and establishing design standards list of the e-Games designs,
Figure 1.Elgazzar (2002) ISD model [9,10,14] for e-Games Program development WGT/WoutGT.
2) Systems Development Method in terms of implementing Elgazzar ISD Model [9,10,14] in developing the two designs of e-Games WGT and WoutGT, and
3) Experimental research method in the research experiment to investigate the comparative effects of the two designs WGT and WoutGT of e-Games on students’ mathematics achievement.
8. Research Procedures
8.1. Content Analysis of “Algebraic Expressions and Equations” Unit
Researchers started with conducting content analysis of the “Algebraic Expressions and Equations” Unit of primary school mathematics as it is required to Elgazzar ISD model (2002). This unit contains
eight lessons. The content analysis component of achievement was refereed and was used to derive instructional needs, instructional objectives, and the cognitive achievement test. So, the first
sub-question has been answered.
8.2. Developing E-Games Program Instructional Design Standards
Based on extensive literature reviews, eLearning standards, and e-games designs, Guidelines from Gagne’s theory of Learning and Instruction, Design Aspects of Guidelines from Gagne’ Theory in
e-Games, analyses of learning content in (1), and phases of Elgazzar ISD Model a list of instructional design standards were derived and was subjected for refereeing from experts. The final most
agreed upon list of standards and their indicators became (10) standards and (77) of their indicators. So, the second sub-question has been answered.
8.3. Developing the WGT and WoutGT E-Games Program Designs
The Elgazzar (2002) ISD model was selected for developing the two treatment programs of e-Games as it was mentioned earlier. A very lengthy details of developmental tasks were done on applying
Elgazzar ISD Model (2002) in Figure 1 until the two the WGT and WoutGT e-Games Program Designs approved by the list of ISD design standards. These detailed developmental tasks have been described in
Alshammari [17]. The phases of the ISD model were applied by the 1st author till the formative evaluation and the approval of the two designs of e-Games program were meeting the ISD design standards.
Then the two put on the virtual learning environment (VLE) of moodle and on CDs. So, the third sub-question has been answered and the two designs were ready to be implemented in research experiment.
8.4. Participants and Experimental Design
Research sample of this research was a purposive clustered sample. It was consisted of (46) of female pupils from two classes from 5th grade in one elementary schools, Khobar Governorate, KSA. This
sample was divided randomly into two groups of (23) pupils each. The two experimental groups with pretest-posttest Quasi-Experimental Design was used with pre-post tests. So, pretest and posttest of
achievement of mathematics was to be applied on the two groups.
8.5. Research Tool: Test of Achievement of Mathematics (TAM)
Authors developed a Test of Achievement of Mathematics (TAM) as the research tools of this research. The test was built on learning objectives and was consisted of (24) items in the form of varied
forms of objective test items on mathematics components of achievement of from analyses in (1). TAM validity was done by specialists in the field of educational technology and mathematics education.
The reliability of the test carried out on its pretest data, the calculated Cronbach’s Alpha (α) coefficient was (α = 0.82). This value has confirmed the reliability of the achievement test of
mathematics. So, it is shown that the test is valid and reliable test for the purpose of this research.
8.6. Experiment of the Research
The 1st author carried out the implementation of the two e-Games program designs: WGT and WoutGT during the 1st semester of the year 2013-2014 according to the experimental design. Pupils enjoyed
studying from the two treatments. The process of studying the e-Games programs was guided by the 1st author. The experiment lasted about one month. Group one studied using the educational e-Games WGT
program while the other group studied using the educational e-Games WoutGT design. Pretest and posttest of the achievement of mathematics were implemented. Data was collected and coded to SPSS
9. Results and Discussions
Researchers applied descriptive statistics procedures using SPSS statistical procedures to compute means and standard deviations for the two e-Games program designs: WGT design and WoutGT design as
in Table 1. It is so clear from data that the mean score of WGT is (23.39) design which isn’t better than WoutGT (24.57) design in the posttest of TAM. However, in the mean of gain scores of TAM of
WoutGT is higher than WGT (17.13 ˃15.13). These noticeable differences are not expected, WGT isn’t better than WoutGT. However, these differences among means might not be significant. On the other
hand, both designs are effective on developing TAM as noticeable when comparing between means of pretests’ scores and their posttests’ scores.
Table 1. Means and standard deviations for the two e-Games program designs of research variables.
To answer the fourth sub-question, tow hypotheses were formulated tested as follows.
Hypothesis (1): There are significant differences at level (α ≤ 0.05) between the two means the WGT design and WoutGT design of e-Games in posttest and gain scores of TAM for the WGT design.
To test this hypothesis, the independent samples t-test was applied to test the significance of the differences between means of the posttest scores an gain in TAM scores. So for the posttest scores
in Table 2, the t-value (0.90) of the two means’ difference of TAM (23.39, 24.57) at df (44) is not significant at (0.05) since the computed significance (0.37 > 0.05). Also, the t-value (1.36) of
the two means’ difference of gain of TAM (15.13, 17.13) at df (44) is not significant at (0.05) since the computed significance (0.81 > 0.05), so, the null hypothesis is not rejected. So, Hypothesis
(1) is rejected to mean that the differences among the means’ scores in posttest and gain of TAM between the educational e-Games’ designs: WGT and WoutGT are not significant.
To make sure that there may be a significant difference between the two e-Games designs’ means of posttest of TAM isn’t due to differences in both pretests of TAM, another Hypothesis (2) was
formulated and tested to check the difference between posttest of TAM when pretests’ scores of TAM are controlled.
Hypothesis (2): There is a significant difference at level (α ≤ 0.05) between the two means of the WGT design and WoutGT design of e-Games in posttest when controlling the pretest scores.
Researchers applied one-way ANCOVA to test Hypothesis (2). Table 3 shows results of this test. From Table 3, the F-value (1.39) of the main effect of between e-Games’ designs (WGT, WoutGT) at df (1,
43) is not significant at (0.05) since the computed significance (0.25 > 0.05), so, Hypothesis (2) is rejected for the variance for TAM.
These results revealed that both WGT e-Games design and WoutGT design weren’t effective even after controlling the effects of the pretests’ scores of TAM as covariates in the ANCOVA model. This
clearly shows that designing educational e-Games with Gagne’s Theory doesn’t have any comparative Effectiveness in Developing 5th Grade Saudi Female Pupils Achievement of Mathematics. These important
results raise the issue that there may be another factors that are being involved in TAM other than designing e-Games WGT such as students characteristics and the content nature and characteristics
of mathematics. When dependent t-tests were applied to test significance of the observed differences between pretests of TAM and posttest of TAM in both e-Games designs, t-tests showed the that these
difference were significant at the at level (α ≤ 0.05). This really means that both e-Games designs are equally effective on developing achievement of mathematics of Saudi female 5th grade pupils.
10. Research Recommendations
Based on these findings of the research, the following practical recommendations can be driven:
Table 2. Independent t-tests results of the means’ differences between WGT and WoutGT of post-tests scores and gain scores of TAM.
Table 3. One-way ANCOVA results of posttest of TAM with pretest of TAM as covariate.
1) These two developed educational e-Games designs: WGT and WoutGT should be used in improving mathematics learning in 5^th grade in Elementary School mathematics curriculum in public and private
2) The list of instructional design standards of educational e-Games which developed in this research should be adopted by researchers and e-Learning. developers in e-Learning and distance learning
3) The instructional design model of Elgazzar (2002) should be used in developing e-Games and other e-Learning resources in direction of Gagne’ Theory of instructional events.
11. Future Researches
The following future researches are suggested:
1) Studying the effects of designing educational e-Games with Gagne’ Theory on developing higher cognitive skills such as problem solving skills.
2) A fellow-up of the effects of designing educational e-Games with Gagne’ Theory on developing mathematical thinking, curiosity, and creative imagination among elementary stage pupils.
3) Using developmental research method in as defined by Elgazzar [14] in research of educational technology, e-Learning, and distance learning.
4) Studying the effects of the interaction between these two e-Games’ designs and cognitive styles on developing achievement of mathematics and mathematical problem solving.
This research has been done as a part of King Hamad Academic Chair of eLearning activities, Distance Learning Dept., Arabian Gulf University. The authors are deeply giving special thanks to Arabian
Gulf University for supporting them to attend the CITE 2015. Special thanks go also to CITE 2015 for honoring Elgazzar―the second author―Chairing the CITE 2015 Conference session, Shanghai, China.
Cite this paper
Salma Katib S. AL-Shammari,Abdellatif E. Elgazzar,Ahmed M. Nouby, (2015) Designing an E-Game Program in Mathematics for 5th Grade Saudi Female Pupils: Does Gagne’s Theory Have Any Effectiveness in
Developing Their Achievement of Mathematics?. Open Journal of Social Sciences,03,40-48. doi: 10.4236/jss.2015.32007 | {"url":"https://file.scirp.org/Html/55528_55528.htm","timestamp":"2024-11-01T20:40:49Z","content_type":"application/xhtml+xml","content_length":"68038","record_id":"<urn:uuid:75c9f80a-fc21-48fb-ab51-c1296dc84eee>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00074.warc.gz"} |