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https://www.repository.cam.ac.uk/browse?type=author&value=Cabeleira%2C+Manuel | Now showing items 1-6 of 6
• An association between ICP-derived data and outcome in TBI patients: The role of sample size
(2016)
BACKGROUND: Many demographic and physiological variables have been associated with TBI outcomes. However, with small sample sizes, making spurious inferences is possible. This paper explores the effect of sample sizes on ...
• An association between ICP-derived data and outcome in TBI patients: The role of sample size
(Springer, 2016)
${\bf Background:}$ Many demographic and physiological variables have been associated with outcome after TBI. However, with small sample sizes, making spurious inferences is possible. This paper explores the effect of ...
• Cerebrovascular pressure reactivity monitoring using wavelet analysis in traumatic brain injury patients: A retrospective study.
(Public Library of Science (PLoS), 2017-07-25)
Background After traumatic brain injury (TBI), the ability of cerebral vessels to appropriately react to changes in arterial blood pressure (pressure reactivity) is impaired, leaving patients vulnerable to cerebral hypo- ...
• Monitoring of Optimal Cerebral Perfusion Pressure in Traumatic Brain Injured Patients Using a Multi-Window Weighting Algorithm.
(J Neurotrauma, 2017-11)
Methods to identify an autoregulation guided 'optimal' cerebral perfusion pressure (CPPopt) for traumatic brain injury patients (TBI) have been reported through several studies. An important drawback of existing methodology ...
• Prospective study on non-invasive assessment of ICP in head injured patients: comparison of four methods
(2015-09-28)
Elevation of intracranial pressure (ICP) may occur in many diseases, and therefore the ability to measure it noninvasively would be useful. Flow velocity signals from transcranial Doppler (TCD) have been used to estimate ...
• Prospective study on non-invasive assessment of ICP in head injured patients: comparison of four methods
(Mary Ann Liebert, 2015-09-28)
Elevation of intracranial pressure (ICP) may occur in many diseases and therefore the ability to measure it non-invasively would be useful. Flow velocity signals from Transcranial Doppler (TCD) have been used to estimate ... | 2018-04-22 00:32:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3788720369338989, "perplexity": 12406.543206941973}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945484.57/warc/CC-MAIN-20180422002521-20180422022521-00165.warc.gz"} |
https://lmcs.episciences.org/volume/view/id/269 | # Special Issue for the Conferences on Computer Science Logic and Logic in Computer Sciences (CSL-LICS) 2014
Editors: Thomas A Henzinger, Dale Miller
The Vienna Summer of Logic in July 2014 attracted a great number of researchers working in logic to Vienna. To take advantage of this unprecedented confluence of logicians, especially those working in computational logic, the organizers of the CSL and LICS series of meetings decided to merge the 2014 editions of these meetings into one meeting titled the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). This joint meeting had one program committee, one program, and one proceedings.
A small number of papers from the proceedings were selected and their authors were invited to submit a full version of their paper to this special issue. All submissions were refereed according to the usual standards of LMCS with each paper being reviewed by three experts in the field. We are grateful to the authors for their excellent submissions and to the reviewers for their efforts to evaluate and improve these papers.
Thomas A. Henzinger and Dale Miller
CSL-LICS 2014, Program Committee Chairs
### 1. Preservation and decomposition theorems for bounded degree structures
We provide elementary algorithms for two preservation theorems for first-order sentences (FO) on the class âd of all finite structures of degree at most d: For each FO-sentence that is preserved under extensions (homomorphisms) on âd, a âd-equivalent existential (existential-positive) FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is complemented by lower bounds showing that a 3-fold exponential blow-up of the computed existential (existential-positive) sentence is unavoidable. Both algorithms can be extended (while maintaining the upper and lower bounds on their time complexity) to input first-order sentences with modulo m counting quantifiers (FO+MODm). Furthermore, we show that for an input FO-formula, a âd-equivalent Feferman-Vaught decomposition can be computed in 3-fold exponential time. We also provide a matching lower bound.
### 2. Proof equivalence in MLL is PSPACE-complete
MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linear logic are related by a series of inference permutations. It is also known as the word problem for star-autonomous categories. Previous work has shown the problem to be equivalent to a rewiring problem on proof nets, which are not canonical for full MLL due to the presence of the two units. Drawing from recent work on reconfiguration problems, in this paper it is shown that MLL proof equivalence is PSPACE-complete, using a reduction from Nondeterministic Constraint Logic. An important consequence of the result is that the existence of a satisfactory notion of proof nets for MLL with units is ruled out (under current complexity assumptions). The PSPACE-hardness result extends to equivalence of normal forms in MELL without units, where the weakening rule for the exponentials induces a similar rewiring problem.
### 3. (Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the […]
### 4. On the characterization of models of H*: The semantical aspect
We give a characterization, with respect to a large class of models of untyped lambda-calculus, of those models that are fully abstract for head-normalization, i.e., whose equational theory is H* (observations for head normalization). An extensional K-model $D$ is fully abstract if and only if it is hyperimmune, {\em i.e.}, not well founded chains of elements of D cannot be captured by any recursive function. This article, together with its companion paper, form the long version of [Bre14]. It is a standalone paper that presents a purely semantical proof of the result as opposed to its companion paper that presents an independent and purely syntactical proof of the same result. | 2021-08-04 06:36:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7090038657188416, "perplexity": 760.9775877142298}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00223.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=829127 | # Math Question
by Edwin
Tags: math
P: 167 How would you solve the following system of simultaneous equations for t and b? sin(pi*t) = 0 sin(pi*(t^2 + 35)/(2*t)) = 0 (t^2 + 35)/(2*t) - t/2 - b/2 = 0 t^2/35 +35/t^2 -t/b - b/t = 0 t*b = 35 inquisitively, Edwin G. Schasteen
Sci Advisor HW Helper P: 3,149 For starters, you should recognize the first equation tells you t is an integer. Likewise, the second tells you $$\frac {t^2 + 35}{2t}$$ is also an integer.
P: 167 That is true. But how do you solve for t algebraically? Is it even possible to solve these systems of equations without using a graphing calculator? Is it possible using numerical methods? If so, which methods? Inquisitively, Edwin
Mentor
P: 7,292
## Math Question
Simple observation and common sense go along ways in this sort of problem. I do not know of any numerical method which will work well. The problem comes when you are restricted to the integers. This is not the natural domain of numerical methods which are planted firmly in the real number line.
As Tide pointed out your first equations restricts you to the integers, the second further restricts you to a small set of integers.
$$2n = t + \frac {35} t$$ | 2014-04-21 07:26:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5018443465232849, "perplexity": 802.6994719673403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00335-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://energy.ihed.ras.ru/en/arhive/article/7300 | # Article
Methods of Experimental Investigation and Measurements
1977. V. 15. № 6. P. 1076–1081
Vishnevetskaya I.A., Petrov V.A.
Method of investigation and experimental device for measuring the coefficient of thermal conductivity of refractory compounds
Annotation
A new axial method is proposed to determine the thermal conductivity coefficient of solids at temperatures above $1000^{\circ}$ C with the use of internal heating of specimens by an electric current and with experimental determination of the heat flux from the side surface of the working section of the specimen. This method can be used to investigate the thermal conductivity of materials with unknown or unstable radiative surface characteristics, and also when it is necessary to conduct experiments not only in a vacuum but also in various gaseous environments. The experimental device is described.
Article reference:
Vishnevetskaya I.A., Petrov V.A. Method of investigation and experimental device for measuring the coefficient of thermal conductivity of refractory compounds, High Temp., 1977. V. 15. № 6. P. 1076 | 2020-08-15 00:25:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6910485029220581, "perplexity": 671.5720117984762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740343.48/warc/CC-MAIN-20200814215931-20200815005931-00260.warc.gz"} |
https://nadre.ethernet.edu.et/record/4493/export/csl | Thesis Open Access
# ETHIOPIAN CAR LICENSE PLATE RECOGNITION USING DEEP LEARNING
ERDEY SYOUM
### Citation Style Language JSON Export
{
"abstract": "<p>The focus of this research is to develop a system that assist humans in reading car<br>\nlicense plate. Such a study is important as the number of traffic on roads becomes<br>\nincreasing constantly, the manual process in car license plate recognition becomes a<br>\nserious problem for traffic management system which not only detect and track a<br>\nvehicle but also identify it. Initially a dataset that contains 930 car images was prepared<br>\nfor model comparison purpose. Two object detection algorithms (Faster R-CNN and<br>\nSSD) were trained and tested on the same dataset using the same model to select the<br>\nbest candidate. The metrics for the comparison were accuracy, average prediction time,<br>\nand total training time taken. It was found that Faster R-CNN gives high accuracy, short<br>\naverage prediction time, and short total training time. After that additional car and<br>\ncropped license plate images were added to the prepared dataset and based on this, two<br>\nobject detection networks were trained using Faster R-CNN one for plate detection and<br>\nanother for character recognition on the detected plate. The proposed approach has been<br>\ntested on test set and later collected images of national license plate of Ethiopia. Both<br>\nthe trained models were achieved a high accuracy which is 99 and 98.89 mAP over 0.5<br>\nIoU for plate detection and character recognition respectively and takes on average 12s<br>\nto complete the recognition of a license plate. The study could be further investigated<br>\non other countries.</p>",
"author": [
{
"family": "ERDEY SYOUM"
}
],
"id": "4493",
"issued": {
"date-parts": [
[
2020,
1,
16
]
]
},
"language": "eng",
"title": "ETHIOPIAN CAR LICENSE PLATE RECOGNITION USING DEEP LEARNING",
"type": "thesis"
}
105
26
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https://www.physicsforums.com/threads/max-power-delivered-to-variable-resistor.238174/ | # Max power delivered to variable resistor
1. Jun 1, 2008
### enian
1. The problem statement, all variables and given/known data
http://img126.imageshack.us/img126/6400/picex8.jpg
Determine the maximum power that can be delievered to the varaible resistor R in the circuit of Fig 4.139.
2. Relevant equations
3. The attempt at a solution
I am not sure how to handle this because the resistor is in that diamond structure. I need to find V thev and R thev Any advice or a hint to get started?
2. Jun 1, 2008
3. Jun 1, 2008
### Defennder
To find $$R_{th}$$, short the voltage source. That means drawing a vertical line connecting the blue nodes in your 2nd diagram. Then you must find the equivalent resistance between the red nodes. You have to redraw the circuit in order to solve it easily.
Next to find $$V_{th}$$, from the original diagram find the open circuit potential difference across the red nodes with the voltage source added back in. That would be the thevenin voltage. You can use nodal analysis to solve this. I got 30V for this. Once you got both, you can use the formulae for the maximum power theorem to get the answer. | 2017-03-29 22:54:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5399559140205383, "perplexity": 594.5676220320089}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218191405.12/warc/CC-MAIN-20170322212951-00373-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.bradthiessen.com/html5/stats/m300/activity2.html | require(mosaic)
#### Examples from Activity #2
##### 5. Karl Pearson, a famous statistician, once tossed a coin 24,000 times and observed 12,012 heads. Using his results, we’d estimate the probability of tossing heads to be 0.5005.
Here’s the code to create the graph showing the running proportion of heads in 50,000 flips:
# First, specify how many coin tosses we want. We'll start with 1000
N = 1000
# Flip a coin N times and store the results in **coin**
coin <- do(N) * rflip()
coin <- data.frame(coin,
sum = cumsum(coin$prop)) # Add a column: *toss* = cumulative number of coin tosses coin <- data.frame(coin, toss =1:N) # Add a column: *runprop* = running proportion of heads coin <- data.frame(coin, runprop = coin$sum/coin$toss) Let’s look at the first several rows of our data frame, coin to see all the columns we added: head(coin) ## n heads tails prop sum toss runprop ## 1 1 0 1 0 0 1 0.0000 ## 2 1 1 0 1 1 2 0.5000 ## 3 1 1 0 1 2 3 0.6667 ## 4 1 0 1 0 2 4 0.5000 ## 5 1 1 0 1 3 5 0.6000 ## 6 1 1 0 1 4 6 0.6667 From this, you can see the final column contains our running proportion of heads at each toss. Let’s graph these results: # Graph the results xyplot(runprop~toss, data=coin, #Plot runprop as a function of toss col="steelblue", alpha=0.7, type="o", #Sets the color & transparency xlim=c(1,N), ylim=c(0.0,1.0), #Sets limits for axes xlab="Toss", ylab="Proportion of Heads", #Labels axes main=paste("Final Proportion of Heads = ", coin$runprop[N])) #Creates title
# Add a horizontal line at 0.50
##### 5c) If we roll two dice and calculate their sum, what’s the probability that the sum is 7? To estimate this probability, I had a computer simulate 5,000 rolls of two dice.
Rolling a die 5,000 times is like choosing 5,000 numbers between 1-6 (with replacement, of course). We can use the sample() syntax:
roll1 <- sample(1:6, size=5000, replace=TRUE) #Rolls one die 5000 times
roll2 <- sample(1:6, size=5000, replace=TRUE) #Rolls another die 5000 times
We can now find the sum of the two dice in each of those 5,000 rolls:
sum <- roll1+roll2 #Finds the sum of the two dice
tally( ~sum) #Tallies the sum from each of the 5,000 rolls
##
## 2 3 4 5 6 7 8 9 10 11 12
## 128 296 415 522 701 791 716 569 409 299 154
histogram( ~ sum, width=1, main="Sums from 5,000 Rolls of Two Dice") #Creates a histogram
Finally, let’s count the number (and proportion) of those sums that are equal to 7:
tally( ~(sum==7)) #Tallies the sums that are equal to 7
##
## TRUE FALSE
## 791 4209
prop( ~(sum==7)) #Proportion of sums that are equal to 7
## TRUE
## 0.1582
That proportion represents our estimate of getting a sum of 7.
##### 7) Again, suppose we toss a coin 3 times, but we’re only interested in the number of heads we observe. List the sample space. Is each outcome in the sample space equally likely to occur? What’s the probability that we observe at least 2 HEADS?.
We’ll simulate tossing 3 coins 10,000 times:
ThreeCoins <- do(10000) * rflip(3) #Flips 3 coins 10,000 times
histogram( ~ heads, data=ThreeCoins, v=1.5, width=1) #Graphs outcomes
##
## 0 1 2 3
## 1212 3772 3775 1241
prop( ~(heads >= 2), data=ThreeCoins) #Counts proportion with at least 2 heads
## TRUE
## 0.5016
##### 11) Suppose you forget to study and you randomly guess on a 10-question true/false quiz. What’s the probability that you get a perfect score? What’s the probability you get at least one question correct?
Let’s simulate 25,000 students randomly guessing on this true/false quiz. With random guessing, each question is like flipping a coin (heads = correct answer).
Here’s the estimated probability of getting at least one question correct:
Quiz <- do(25000) * rflip(10) #Flips 10 coins 25,000 times
histogram( ~ heads, data=Quiz, width=1, col="SteelBlue") #Graphs outcomes
##
## 0 1 2 3 4 5 6 7 8 9 10
## 17 252 1138 2960 5033 6122 5151 2978 1088 236 25
prop( ~(heads >=1), data=Quiz) #Counts proportion with at least 1 correct (head)
## TRUE
## 0.9993
Here’s the estimated probability of getting a perfect score:
## TRUE
## 0.001
##### 14) My two older brothers and I all have the same initials, “BAT.” If my parents chose a name at random from 104 commonly used (English) boy names beginning with the letter “B” and 157 names beginning with “A,” what’s the probability I would have gotten the name Bradley Adam?
There is a 1/104 = 0.0096 chance that they would have chosen “Bradley” at random. That’s like flipping a coin that has a 0.0096 chance of coming up heads. Likewise, choosing a middle name of Adam is like flipping a coin that has a 1/157 chance of coming up heads:
FirstName <- do(50000) * rflip(1, prob=1/104) #Flips an unfair coin 50,000 times
MiddleName <- do(50000) * rflip(1, prob=1/157) #Flips another unfair coin 50,000 times
Remember that, for the computer, HEADS=1 and TAILS=0. We can add the results from the two coins on each of the 10,000 trials. If the sum is 2, it means both coins came up heads (and the correct name was chosen):
FullName<- FirstName$heads + MiddleName$heads #Sums the coins
tally(~FullName) #Shows the number of trials with 0, 1, and 2 heads
##
## 0 1 2
## 49259 736 5
prop( ~(FullName ==2)) #Counts proportion with 2 heads (correct name)
## TRUE
## 1e-04
##### 15) In how many ways could we arrange the letters ABCD? In this calculation, are we assuming that we are sampling with or without replacement?
If we sample with replacement, we’re simply shuffling the letters ABCD. Let’s have the computer do this 10,000 times:
# Create a vector with the characters A, B, C, D
ABCD <-c ("A","B","C","D")
# Shuffle the letters 25,000 times and store the results in "Words"
Words <- do(25000) * shuffle(ABCD, replace=FALSE)
# This stores the first letter as V1, the second letter as V2, etc.
If we wanted to estimate the probability of getting the letters “ABCD” in order, we would use the following:
prop( ~(V1=="A" & V2=="B" & V3=="C" & V4=="D"), data=Words)
## TRUE
## 0.04176
If we want a list of all the different 4-letter “words,” we’d need to tally
# Combine the V1-V4 variables into a single 4-character string
Words2<- paste(Words$V1,Words$V2,Words$V3,Words$V4, sep="")
# Tally the number of different words
tally(~ Words2)
##
## 1044 1012 1038 1023 1038 1085 1067 1071 1021 1155 1015 958 1045 994 1049
## CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA
## 1090 1043 1026 1007 995 1052 990 1080 1102
# Make the table easier to view as columns
cbind(duration.freq = table(Words2))
## duration.freq
## ABCD 1044
## ABDC 1012
## ACBD 1038
## ACDB 1023
## BACD 1067
## BCDA 1155
## BDAC 1015
## BDCA 958
## CABD 1045
## CBDA 1090
## CDAB 1043
## CDBA 1026
## DABC 1007
## DACB 995
## DBAC 1052
## DBCA 990
## DCAB 1080
## DCBA 1102
From that table, we can count 24 different outcomes.
##### 16-25) Now might be a good time to demonstrate how to calculate combinations and permutations in R.
Let’s start with permutations. We can create a function in R that calculates the number of permutations given n = number of objects and x = number of objects selected:
perm = function(n, x) {
return(factorial(n) / factorial(n-x))
}
We can then call that function. As an example, suppose we want to replicate our answer to #14 (arranging the letters ABCD). We would set n=4 and x=4:
perm(n=4,x=4)
## [1] 24
To replicate #20 (choosing president, VP, and secretary from 6 people), we would set n=6 and x=3:
perm(n=6,x=3)
## [1] 120
Combinations are just as simple. First, we create a function with inputs n and x:
comb = function(n, x) {
return(factorial(n) / (factorial(x) * factorial(n-x)))
}
To replicate #23 (5-card hands from 52 cards), we would set n=52 and x=5:
comb(n=52,x=5)
## [1] 2598960
To replicate #24 (dividing 8 subjects into two equal-sized groups), we would set n=8 and x=4:
comb(n=8,x=4)
## [1] 70
##### 31) Suppose we have 10 IE majors and 10 other majors in this class. I need to choose 4 students at random to fail this course (thus, satisfying my ego). In how many ways could I choose 4 students out of 20? In how many ways could I choose 4 students out of the 10 IE majors? Using the results from these two questions, what is the probability that I randomly select 4 IE majors to fail?
We can use the combo function that we just wrote. First, let’s calculate the number of ways to choose 4 students out of 20:
comb(n=20,x=4)
## [1] 4845
Next, let’s calculate the number of ways to choose 4 students from 10 IE majors:
comb(n=10,x=4)
## [1] 210
Finally, we divide the two results to get our probability:
comb(n=10,x=4)/comb(n=20,x=4)
## [1] 0.04334
As we’ll learn in Activity #8, we could use the hypergeometric distribution to get this probability:
The function is: phyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)
where:
q = number of type 1 objects chosen
m = number of type 1 objects
n = number of type 2 objects
k = number of objects selected
To get our probability, we’d need:
phyper(4, 10, 10, 4, lower.tail = TRUE, log.p = FALSE)-phyper(3, 10, 10, 4, lower.tail = TRUE, log.p = FALSE)
## [1] 0.04334
One final way to estimate this probability would be via simulation. In this scenario, we have 20 students. Let’s create 10 engineering majors (who all happen to have the name “E”) and 10 other majors (who all have the name “X”).
Students <-c ("E","E","E","E","E","E","E","E","E","E","X","X","X","X","X","X","X","X","X","X")
Students
## [1] "E" "E" "E" "E" "E" "E" "E" "E" "E" "E" "X" "X" "X" "X" "X" "X" "X"
## [18] "X" "X" "X"
To choose 4 students, we simply sample 4 of these letters. We’ll do this 100,000 times:
Failures <- do(100000) * sample(Students, 4, replace=FALSE)
# This stores the first student as V1, the second student as V2, etc.
## V1 V2 V3 V4
## 1 E X X X
## 2 X E E X
## 3 X X E E
## 4 E X X X
## 5 X E X X
## 6 E X X X
If we wanted to estimate the probability of choosing 4 engineering majors, we’d simply find the proportion of our 100,000 trials that yielded “E, E, E, E.”
prop( ~(V1=="E" & V2=="E" & V3=="E" & V4=="E"), data=Failures)
## TRUE
## 0.04229 | 2018-01-17 13:12:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.667871356010437, "perplexity": 3431.0222743271493}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886939.10/warc/CC-MAIN-20180117122304-20180117142304-00418.warc.gz"} |
https://tug.org/pipermail/tex4ht/2018q3/002054.html | [tex4ht] problem with new dvisvgm 2.6, all svg fies have horizontal lines with no math showing.
Nasser M. Abbasi nma at 12000.org
Mon Sep 10 07:37:12 CEST 2018
Hello;
Just finished compile of one large latex file.
This is the one I started yesterday. It took about 32 hrs.
(it seems slower this time than before).
But there is a big problem. This is using the new dvisvgm 2.6,
compiled from sources on Ubuntu with TL 2018.1. Had to use 2.6
due to size limitation of 2.5.
When I open any web page, all the math is just thin horizontal lines.
No actual math shows up. Here is screen shot
https://www.12000.org/tmp/091018/screen_shot.jpg
The SVG files are all generated. Hashing seems to work well,
as all the svg file names have the long hash name. But no math shows.
Tried on firefox and Chrome. Cleaned the cache, no help.
Could someone please have a look?
I put a zip file of the complete folder which contains
everything, and also dvi, idv. The main HTML is called index.htm,
in this folder:
https://www.12000.org/tmp/091018/
There seems to be something got corrupted somewhere.
May be it is due to large size of the IDV or DVI. I do not
know.
Here is an example of one SVG file:
---------------------------
<?xml version='1.0' encoding='UTF-8'?>
<!-- This file was generated by dvisvgm 2.6 -->
<defs/>
<g id='page27' transform='matrix(1.15 0 0 1.15 0 0)'>
</g>
</svg>
-----------------
The zip file in the above folder is large, about 250 MB.
I also put the DVI and IDV files outside the zip file
in the above folder in case you just need to look at
The command I used to compile the index.tex is
make4ht -ulm default -e ./new.mk4 -c ./nma.cfg
-f html5+dvisvgm_hashes index.tex
"htm,3,pic-align,notoc*,p-width,svg"
So any one can try the same thing.
And I also put dvisvgm 2.6 I build as binary in the above
folder, just in case. It did build correctly, but who knows,
may be it using wrong font library. It was complicated to
build from sources, but that was only way to get 2.6 without
having to wait one year for TL 2019.
This is very frustrating. I have not been to convert this latex
to HTML for more than 2 weeks now. Nothing seems to work for
some reason. Tried mathjax with make4ht, but that also did not
work for this file.
Any suggestions what to try?
>which dvisvgm
/usr/local/bin/dvisvgm
>dvisvgm --version
dvisvgm 2.6
Thanks for any help.
TL 2018 on Ubuntu
--Nasser | 2020-04-01 09:13:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6879902482032776, "perplexity": 6685.631191809768}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370505550.17/warc/CC-MAIN-20200401065031-20200401095031-00451.warc.gz"} |
https://brainvis.wustl.edu/wiki/index.php?title=Caret:Documentation:Statistics&oldid=744 | # WARNING
THIS DOCUMENT IS IN DEVELOPMENT AND DESCRIBES FUTURE VERSIONS OF CARET
# Descriptive Statistics
Descriptive statistics provide information about the data such as the mean (average), median (middle value), mode (most common value), standard deviation, and variance. When computing the standard deviation, one must know if the data values represent the entire population in which case division is by N (number of items) or the data values are a subsample of the population in which case division is by N - 1.
## Population Descriptive Statistics
• Population Mean $\mu = \frac{\sum_{i=1}^N x_i}{N}$
• Population Standard Deviation $\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}$ OR $\sigma = \sqrt{\frac{\sum_{i=1}^N x_i^2 - \frac{(\sum_{i=1}^N x_i)^2}{N}}{N}}$
• Population Variance = σ2
• Standard Deviation of the Mean $SD_{\overline{x}} = \frac{\sigma}{\sqrt{N}}$
## Sample Descriptive Statistics
• Sample Mean $M = \frac{\sum_{i=1}^N x_i}{N}$
• Sample Standard Deviation $S = \sqrt{\frac{\sum_{i=1}^N (x_i - M)^2}{N-1}}$ OR $S = \sqrt{\frac{\sum_{i=1}^N x_i^2 - \frac{(\sum_{i=1}^N x_i)^2}{N}}{N-1}}$
• Sample Variance = S2
• Standard Error of the Mean $SE_{\overline{x}} = \frac{S}{\sqrt{N}}$
## Miscellaneous Descriptive Statistics
• Z-Score $Z = \frac{x_i - \mu}{\sigma}$
# Inferential Statistic Tests
## Parametric Inferential Tests
For parametric tests, the data is assumed to be in a specific probability distribution, typically the normal (gaussian) distribution.
### ANOVA (Analysis of Variance), One Way
A one-way ANOVA determines if the mean values at each node for two or more groups of subjects are statistically different. The groups being compared are allowed to have a different number of subjects.
K = Number of Groups
N = Total Number of Subjects
Ni = Number of Subjects in Group "i"
dfTotal = N − 1
$df_{Error} = \sum_{i=1}^{K} (N_i - 1) = N - K$
dfTreatment = K − 1
Xij = Measurement for subject "j" in group "i"
Mean of group i, $\bar{X_i} = \frac{\sum_{j=1}^{N_i} x_{ij}} {N_i}$
Grand Mean, $\bar{X_{..}} = \frac{\sum_{i=1}^{K} \sum_{j=1}^{N_i} X_{ij}}{N}$
$SS_{Total} = \sum_{i=1}^{K} \sum_{j=1}^{N_i} (X_{ij} - \bar{X_{..}})^2$
$SS_{Error} = \sum_{i=1}^{K} \sum_{j=1}^{N_i} (X_{ij} - \bar{X_i})^2$
$SS_{Treatment} = \sum_{i=1}^{K} N_i (\bar{X_i} - \bar{X_{..}})^2$
SSTotal = SSWithin + SSTreatment
$MS_{Treatment} = \frac{SS_{Treatment}} {df_{Treatment}}$
$MS_{Error} = \frac{SS_{Error}} {df_{Error}}$
$F = \frac{MS_{Treatment}} {MS_{Error}}$
If the ANOVA is run with two groups of data, the F-statistic is equivalent to the square of the T-Statistic produced by a Two-Sample T-Test.
### T-Test, One-Sample (Single Sample)
A one-sample T-Test determines if the mean value at each node is statistically different than a specified value, often zero.
t = $\frac{\mathrm{M} - \mu}{\sqrt{\frac{\mathrm{s}^2}{N}}}$
df = N − 1
### T-Test, Paired (Dependent Means)
A paired T-Test determines if mean at each node is statistically different for two measurements (X and Y) on one group of subjects.
$\overline{D} = \frac{\sum_{i=1}^N (x_i - y_i)}{N}$
t = $\frac{\overline{D} - \mu}{\sqrt{\frac{\mathrm{s}^2}{N}}}$
df = N − 1
### T-Test, Two-Sample (Independent Means)
A two-sample T-Test determines if the means at each node for two groups of subjects are statistically different. The groups being compared are allowed to have a different number of subjects.
#### Equal (Pooled) Variances
$S^2 = \frac{ \sum_{i=1}^{N_1} (x_i - \overline{x}_1)^2 + \sum_{j=1}^{N_2} (x_j - \overline{x}_2)^2} {N_1 + N_2 - 2}$
$t = \frac{\overline{x}_1 - \overline{x}_2} { \sqrt{S^2(\frac{1}{N_1} + \frac{1}{N_2})} }$
df = N1 + N2 − 2
#### Unequal (Unpooled) Variances
$S_1^2 = \frac{\sum_{i=1}^{N_1} (x_i - \overline{x}_1)^2} {N_1 - 1}$
$S_2^2 = \frac{\sum_{j=1}^{N_2} (x_j - \overline{x}_2)^2} {N_2 - 1}$
$t = \frac{\overline{x}_1 - \overline{x}_2} {\sqrt{\frac{S_1^2}{N_1} + \frac{S_2^2}{N_2} }}$
$d\mathit{f} = \frac{(\frac{S_1^2}{N_1} + \frac{S_2^2}{N_2})^2} {\frac{(\frac{S_1^2}{N_1})^2}{N_1 - 1} + \frac{(\frac{S_2^2}{N_2})^2}{N_2 - 1} }$
## Non-Parametric (Distribution Free) Inferential Statistic Tests
For non-parametric tests, no assumptions are made about the distribution of the data.
# caret_stats
caret_stats is a command line program that performs statistical operations on GIFTI surface data files. The first parameter indicates the operation that will be performed. Run the command with just the operation for help information.
The program is written in Java and requires the Java SE Development Kit (JDK) for optimal execution. If you are using a Mac, Java is already installed and you can skip this step. If you are running Linux or Windows, you must download the Java JDK. The Java Development Kit is downloaded from http://java.sun.com/javase/downloads/index.jsp. Download and install the Java SE Development Kit (JDK). You must set the "path" environment variable to the Java installation's "bin" directory so that "java" can be run from the command line.
Note: Do not use the Java Runtime Environment. It does not support Java's "-server" option which reduces the runtime of caret_stats by fifty percent. If you get the error message "No Server JVM" you are using JRE, not JDK.
After Java is installed, download the caret6 distribution. Install in the desired location such as "Program Files" on Windows, "/Applications" on a Mac, or "/usr/local" on Linux. When the distribution is unzipped, it will create the subdirectory "caret6". Located in the caret6 directory are several directories whose names being with "bin". You must update your PATH environment variable to point to the appropriate "bin" directory so that "caret_stats" can be run from the command line. In addition, Windows users will need to set the environment variable CARET6_HOME to the full path of the caret6 directory (eg: C:\caret6).
## Descriptive Statistical Operations
• -descriptive Mean, standard deviation, etc.
## Inferential Statistical Operations
The purpose of the inferential statistic is to take the input files, perform a statistical test at each node, and create a new file containing one or more statistical measurements (F, T, Z, etc) at each node.
## Performing Inferential Statistical Tests in Caret
Inferential statistical tests in Caret are performed on metric or surface shape files. All of the data (metric or shape files) must be on a co-registered surface so that all data files have the same number of nodes and each node number i is "in register" across subjects (i.e., all subjects' surfaces have undergone surface-based registration using Caret, Freesurfer, CIVET, or other software).
The goal is to find clusters (regions) that are statistically different between the groups of input data. That is, one can reject the null hypothesis which states that the metric/shape values at each node are essentially the same.
The steps in Caret are:
1. Run the input files through an inferential statistical test to produce the statistic file and the randomized statistic file.
2. Perform a significance test to assign P-Values to the statistic file.
Each of the inferential tests in Caret produces two files. The statistic file contains the results of the statistical test performed on the input data. The randomized statistic file contains columns with the same statistical test performed on randomly assigned groups of the input data. This randomized file is used during significance testing.
### One Sample T-Test
-inferential-t-test-one-sample
### Paired T-Test
-inferential-t-test-paired
### Two-Sample T-Test
-inferential-t-test-two-sample Two sample T-Test with or without pooled variance.
### Interhemispheric Clusters
-inferential-interhemispheric
The interhemispheric clusters test is used to determine asymmetry (and symmetry) between the left and right hemispheres of two groups of subjects. All subjects left and right hemispheres must be co-registered to an atlas, typically the PALS atlas.
Inputs:
• AL is group A, left hemispheres.
• AR is group A, right hemispheres.
• BL is group B, left hemispheres.
• BR is group B, right hemispheres.
• ITER_LEFT_RIGHT is the number of iterations for T-Statistics of random combinations of left or right subjects.
• ITERATIONS is the number of iterations for the randomized T-Statistic file.
Algorithm:
• Create TL, a T-Statistic metric file comparing the left hemispheres of the two groups, TL = T-Statistic(AL, BL).
• Create TR, a T-Statistic metric file comparing the right hemispheres of the two groups, TR = T-Statistic(AR, BR).
• Create TP, a metric file containing the product of the left and right T-Statistic, TP = TL * TR.
• Create RANDTL, a metric file containing T-Statistics for ITER_LEFT_RIGHT randomized combinations of the left hemispheres from both groups, RANDTL = T-Statistic(RandomCombinations(AL, BL)).
• Create RANDTR, a metric file containing T-Statistics for ITER_LEFT_RIGHT randomized combinations of right hemispheres from both groups RANDTR = T-Statistic(RandomCombinations(AR,BR)).
• Create RANDTP, a metric file containing ITERATIONS random combinations of the product of one column from each of the left and right T-Statistic randomized files, RANDTP = RandomColumn(RANDTL) * RandomColumn(RANDTR).
Output:
• TP is the statistic file for input to the significance testing command.
• RANDTP is the randomized statistic file for input to the significance testing command.
### Coordinate Difference Analysis of Variance
In coordinate difference analysis of variance, the input data are coordinate files from participants that are in two or more groups. In the ANOVA equations shown previously, Xi, in the case of coordinate difference ANOVA, is a three-dimensional coordinate. A subtraction operation, such as $(X_{ij} - \bar{X_i})$ is the Euclidean (straight line) distance between two coordinates.
In the numerator of the F-Statistic is $SS_{Treatment} = \sum_{i=1}^{K} N_i (\bar{X_i} - \bar{X_{..}})^2$. In the parentheses is the distance between a group average coordinate and the population average coordinate (the average of all coordinates). If the participants are all from the same population, each of the group average coordinates will be very close to the population average coordinate and this quantity will be small. If participants are from different populations, the group average coordinates will be different than the population average coordinate and this quantity will be large.
In the denominator of the F-Statistic is $SS_{Error} = \sum_{i=1}^{K} \sum_{j=1}^{N_i} (X_{ij} - \bar{X_i})^2$. In the parenthesis is the distance between the coordinate of each participant in the group and the average coordinate for the group. When the participants in a group are spatially clustered this quantity will be small. When the participants in a group are spatially separated, this quantity will be large.
Consider the two-dimensional examples below. In each example, there are two groups of data with each participant labels as "O" and "+". The average coordinate for each group is "(O)" and "(+)" with the population average coordinate at "(A)".
In the plot below, both groups appear to be from the same population. As a result, SSTreatment will be small, resulting in a small F-Statistic and one is unable to reject the null hypothesis.
In the plot below, the average coordinates of the two groups are spatially separated resulting in SSTreatment being large. In addition, the groups are spatially clustered resulting in SSError being small. As a result, the numerator is large and the denominator small creating a large F-Statistic and the rejection of the null hypothesis.
### Coordinate Difference
NOTE: At this time, coordinate difference is not implemented in caret_stats.
Definitions:
• Nx is the number of participants in group X.
• D(i,j) = $\sqrt{ {(X_i - X_j)}^2 + {(Y_i - Y_j)}^2 + {(Z_i - Z_j)}^2}$ (The Euclidean distance between two three-dimensional points.)
• AVGxj is the average coordinate at node j for group x.
• Xdev = $\sqrt{\frac{\sum_{i=1}^{N_x} \sum_{j=1}^M D(XYZ_{ij},AVG_{xj})^2}{N_x - 1}}$, where Nx is the number of participants in group X and M is the number of nodes.
Algorithm:
• Create Aavg, the average coordinate file for group A.
• Create Bavg, the average coordinate file for group B.
• Create Adev, the deviations at each node for group A.
• Create Bdev, the deviations at each node for group B.
• If the mode is COORD_DIFF, create the statistic-file where the statistic at each node is D(Aavg,Bavg).
• If the mode is TMAP_DIFF, create the statistic-file where the statistic at each node is $\frac{D(A_{avg}, B_{avg})}{\sqrt{A_{dev} + B_{dev}}}$
• Create the randomized-statistic-file file. For each column in it, create two coordinate files that are randomized combinations from all of the input coordinate files on which the COORD_DIFF or TMAP_DIFF test is performed.
What Donna desires and matches the formula for an Unpooled Two-Sample T-Test
$\frac{D(A_{avg}, B_{avg})}{\sqrt{\frac{{A_{dev}}^2}{N_A} + \frac{{B_{dev}}^2}{N_B}}}$
## Significance Testing
Significance testing in Caret is a non-parametric technique involving randomization (bootstrapping???).
Two data files are required for significance testing. The first is the file containing the test statistic. The second file is the "randomized statistic" file that contains test statistics from many random combinations of the test subjects.
### Randomization
Randomization testing is used to determine the P-Values.
#### Randomization With One Group of Subjects
When there is one group of subjects, such as in a one-sample T-Test, it is not possible to randomize among groups. So, the randomization is performed by randomly flipping the signs of the values for each subject. The statistical test is then run on each of these randomizations and the largest clusters are identified.
#### Randomization With Multiple Groups of Subjects
With multiple groups of subjects, all of subjects are placed into a pool. Subjects are then randomly drawn from the pool and placed into new groups. The new groups contain the same number of subjects as the original groups. When randomizing subjects, each new randomization of subjects should be unique when compared to any previously generated groups of subjects. Statistical tests are then run on each of these randomizations and the largest clusters are identified.
Given a group of three subjects, choosing two at a time, there are 3 combinations and 6 permutations. For example, selecting two subjects from {A,B,C} results in the combinations {A,B}, {A,C}, and {B,C} and results in the permutations {A,B}, {A,C}, {B,C}, {B,A}, {C,A}, and {C,B}. Basically, with combinations, two groups of elements are equal if they contain the same elements, in any order (ie: {A,B}, and {B,A} are equivalent). With permutations, two groups of elements are equal only if they contain the same elements in an identical order (ie: {A,B} and {B,A} are NOT equivalent).
Mathematical formulas for the number of permutations and combinations when choosing k elements from a total of n elements:
P(n,k) = $\frac{n!} {(n - k)!}$
C(n,k) = $\frac{n!}{k!(n-k)!}$
### P's and Q's
The significance tests in Caret produce both P and Q values. Q is simply 1 - P. Q is useful for thresholding in Caret. One selects the statistic for viewing and thresholds with Q. Since Caret thresholds by inhibiting the display of data BELOW the threshold, one can threshold with Q and set the threshold to 0.95 to see statistics with a P-Value of 0.05 or less.
### Cluster Based Thresholding
For cluster-based threshold significance testing use "caret_stats -significance-cluster-threshold".
• The user provides positive and negative thresholds and a desired significance level (P-Value, eg: 0.05).
• Clusters of nodes passing the threshold tests are identified in the statistic file. Note that positive and negative values are processed separately.
• The largest cluster is identified in each column of the randomized statistic file using the thresholds.
• The clusters identified from the randomized statistic file are ranked based upon surface area (possibly corrected for surface distortion).
• The user provided P-Value is multiplied by the number of columns in the randomized statistic file (eg: 0.05 * 500 = 25) providing the significant cluster rank. The cluster at this rank is identified and its surface area is noted as the "significant surface area".
• For each cluster in the statistic file, use its surface area and determine how it ranks in the ranked randomized clusters. Set the P-Value for the statistic file's cluster to its ranking divided by the total number of columns in the randomized file. For example if the statistic cluster is ranked 3 out of 100, the cluster receives a P-Value of 0.03.
The difficult part of cluster-based thresholding is selecting the thresholds. There is no "correct" threshold value. In general, smaller thresholds result in either or both more clusters and larger clusters and larger thresholds result in either or both fewer clusters and smaller clusters.
### Threshold-Free Cluster Enhancement (TFCE)
For threshold-free cluster enhancement significance testing use "caret_stats -significance-threshold-free".
The difficulty of selecting a threshold in cluster-based thresholding led to the development of threshold-free cluster enhancement (See Smith and Nichols in the References section at the bottom of this page). With threshold-free cluster enhancement, the user does not need to choose thresholds.
• Apply the TFCE transform to the statistic in the statistic file.
• Apply the TFCE transform to all columns in the randomized statistic file.
• Find the largest TFCE value in each column of the TFCE transformed randomized statistic file and rank them.
• The user provided P-Value is multiplied by the number of columns in the randomized statistic file (eg: 0.05 * 500 = 25) providing the significant TFCE rank. The TFCE at this rank is identified and its value is noted as the "significant TFCE value".
• For each node in the statistic file, use its TFCE value and determine how it ranks in the ranked, randomized maximum TFCE values. Set the P-Value for the statistic file's node to its ranking divided by the total number of columns in the randomized file. For example if the statistic node TFCE is ranked 3 out of 100, the node receives a P-Value of 0.03.
The value of the TFCE output at node p where node p has a positive input value is given by the following integral:
$TFCE(p) = \int_{h_0}^{h_f} e(h, p)^Eh^Hdh$, where h is a threshold, e(h,p) is the area of the cluster containing node p at threshold h (in Caret, the sum of the surface areas of the nodes in the cluster), and h0 and hf are typically zero and the highest value of a node in the surface, respectively. E and H are constants (default values 1.0 and 2.0 for surfaces) that define what shape and size of clusters it is most sensitive to. In practice, this integral is approximated numerically, due to the cluster size varying unpredictably with height. At a high level, our approach was to use many thresholds, computing the approximate integral for each "slice" of each cluster with the trapezoidal rule. In order to obtain the values for negative nodes, the input values are sign flipped, run through the same process, and then sign flipped to be negative again. See Caret:Documentation:Statistics:TFCE_Implementation for details.
#### Flat Surface with Z-Coordinate set to TFCE-Enhanced T-Statistic
The significance testing commands have a parameter named "-number-of-threads". Threads allow a task to be broken down into pieces that may be run in parallel and take advantage of either multiple processors or multi-core processors. Using threads will typically reduce the execution time of the command if more than one logical processor is available.
# References
## Books
• Howell, David C. (2002) Statistical Methods for Psychology. Pacific Grove, CA: Duxbury.
## Journal Articles
• Nonparametric Permutation Test For Functional Neuroimaing: A Primer with Examples. Thomas E. Nichols and Andrew P. Holmes. Human Brain Mapping 15:1
• Threshold-Free Cluster Enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference. Stephen M. Smith and Thomas E. Nichols.NeuroImage 2009 44(1) | 2021-10-25 22:09:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 37, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.693003237247467, "perplexity": 1927.6922691467682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587770.37/warc/CC-MAIN-20211025220214-20211026010214-00041.warc.gz"} |
https://mathovore.fr/en/trigonometry-corrected-3rd-grade-math-exercises-in-pdf | Trigonometry : corrected 3rd grade math exercises in PDF.
The answer key to the math exercises in 3ème on trigonometry in the right triangle. Apply the sine, cosine and tangent formulas to calculate the length or measure of an angle.
Exercise 1:
we know that ; ; and .
Calculate the perimeter of triangle ABD. Round the result to the nearest decimeter.
In the right-angled triangle ABC :
In the triangle ABD :
In the right-angled triangle ACB :
In the right triangle BCD :
In the triangle BCD rectangular in C :
The perimeter of triangle ABD is :
Conclusion: the perimeter is approximately 84.4 meters.
Exercise 2:
a. In the right-angled triangle DGE :
b. Represent the situation by the figure at 1/200 scale. (The data for the situation should be placed on the figure).
Exercise 3:
1.a. Using the calculator, calculate ( cos67°+sin67°)²+(cos67°-sin67°)²=2 (cos35°+sin35°)²+(cos35°-sin35°)²=2
b. what do we find?
The result is always equal to 2 .
2.prove that for any acute angle x :
Exercise 4:
Show that the triangle SON is a rectangle.
Calculation of the angle :
The angles and are opposite by the vertex and therefore equal.
Conclusion : the triangle NOS is a right-angled triangle in O .
Exercise 5:
is an angle such that .
.
Now the cosine of an acute angle is positive:
Exercise 6:
1. Construct a triangle ABC at C such that AC = 5 cm and .
2. Calculate the length BC. (A value rounded to the nearest millimeter will be given).
According to the course sin = or sin 40° = BC/AC so BC = AC x sin 40° = 5 sin (40) 3,2cm
3.a) Where is the center O of the circumscribed circle of triangle ABC?
Since the triangle is right-angled, a property of the course says that the hypothenuse is a diameter of the circumscribed circle of the right-angled triangle(circumscribed meant that the circle passes through the three vertices of the triangle).
Now if [AC] is the diameter, then O is the middle of [AC].
b) Draw this circle.
4. Deduce the measure of the angle .
OB = OA so OAB is an isosceles triangle =40° implies that = 180°-(2×40°) since the sum of the angles of a triangle is always 180°. = 100° and since the angles and are supplementary (together they form a flat angle and their sum is 180°) we have = 180° -100° =80°.
Exercise 7:
What is the OH distance needed for the cathedral to appear fully in the lens?
I have the opposite side and the angle .
I am looking for the side adjacent to the angle .
Formula: tangent
Conclusion:
The OH distance needed for the cathedral to appear fully in the lens must be greater than 155.5 meters.
Exercise 8:
a) The triangle SAH is right-angled at H.
So °
So the triangle SAH is a right triangle and isosceles in H.
BH=BA+AH=BA+HS=BA+x=40+x
b)AH=HS=x
c) In the triangle BSH rectangular in H.
d)
The height of the keep is about 35 meters.
Exercise 9:
In the right block above, we give EH=69cm, EF=60cm and EA=51cm.
What is the measure of the angle AED? (round the result to the unit)
Exercise 10:
Help Lisa do this calculation with the help of the diagram below:
We have:
and
using these two equalities
and
Let’s determine BC :
Let’s determine BD :
The archangel Saint Michael culminates at 171.62 meters.
Exercise 11:
1. Construct a full-size triangle ABC such that: AB = 7 cm; BC = 8 cm and AC = 5 cm.
2. [BC] being the side whose measure is the greatest, we should have if the triangle were rectangular in A :
BC² = AB² + AC²
or
7² + 5²
So the triangle ABC is not rectangular.
2- Calculation of the angle : let’s apply the formula
8² = 7² + 5² – 2*5*7 cos
64 = 49 + 25 – 70 cos
64 – 49 – 25 = -70 cos
-10 =-70 cos
or
cos =
Using the calculator we find :
You can check this result using GEOGEBRA
Exercise 12:
1) For a 15% slope, what angle does the road make with the horizontal?
In this right-angled triangle, note the angle between the road and the horizontal.
We know the adjacent and opposite side of the angle , so the formula to use is the tangent.
Conclusion: the road makes an angle of about 17° with the horizontal.
2) A dangerous descent is considered as soon as the slope is higher than 10% on the road and higher than 4% on the highway.
From what angle between the road and the horizontal, is it considered dangerous to go downhill on a road?
Conclusion: a descent on a road is dangerous as soon as the angle is higher than 6° and higher than 3° for a highway.
3) Is it more dangerous to drive on a road with a 20% slope or to drive on a highway with a 20 degree angle to the horizontal? Justify
Conclusion: On a highway the speed is much higher so it is more dangerous on a highway.
Exercise 13:
1°) Your triangle should look like this:
2°) To show that the triangle IJK is a right triangle,
we will use the reciprocal of the Pythagorean theorem.
The demonstration goes like this:
In the triangle IJK, we apply the reciprocal of the Pythagorean theorem, then we have :
JK² = 8² = 64 AND IJ² + IK ² = 4.8² + 6.4² = 23.04 + 40.96 = 64
Now JK² = IJ² + IK², so the triangle JIK is right-angled at I.
3°) We now want to know what is the measure of the angle .
Three possibilities of resolution, we use :
Exercise 14:
1°) We know that the wall (AB) and the floor are perpendicular.
We also know that the wall measures 3.05 m and that the scale [AC] measures 3.20m long.
So to know how far from the foot of the wall the ladder should be placed
so that its top is just at the level of the basket, we will use the Pythagorean theorem.
But first we convert: AB = 3.05m = 305cm and CA 3.20 m = 320 cm.
In the triangle ABC rectangle in B, we apply the Pythagorean theorem, we have :
CB²+AB²=CA²
CB²+305²=320²
CB²+93025=102400
CB²=102400-93025=9375
or CB>0 so
2°) The angle formed by the ladder and the ground is therefore the angle .
We have the three measures of the three sides of the triangle,
which gives us three possibilities.
Exercise 15:
1°) Your triangle should look like this one.
The angles and form two right angles because (AH) is the height of [BC] from vertex A.
Now the height is the straight line coming from a vertex and which is perpendicular to the opposite side.
We also know that BH = HC = BC/2 because in an isosceles triangle,
the height coming from the main vertex cuts its base in two equal parts because it is also a median.
2°) Calculation of
It is known that the tangent of an angle is equal to the quotient of the opposite side of it by the adjacent side of it.
So:
We deduce degrees.
Exercise 16:
1°) A rectangle with its diagonal … no need for correction!!!
2°) Calculation of the measure of the angle :
We know the adjacent side and the opposite side of this angle,
which leads us to calculate the tangent of this angle.
We deduce degrees.
3°) Show that the angles and are equal. 1. method (the simplest) The lines (AB) and (DC) are parallel and the segment [AC] cuts and in two angles each.
We can therefore say that these two angles are internal alternates and therefore equal.
We calculate [AC] with Pythagoras:
In the right-angled triangle ACB (or ADC, they are the same), we apply the Pythagorean theorem:
AC²=AB²+BC²
AC²=7.2²+5.4²
AC²=51.84+29.16=81 or AC>0,therefore
cm .
We now have all the measurements of the sides of the rectangle.
So if the angles and were equal, the sine of one would be equal to the sine of the other and IDEM with the cosines.
Let’s check:
Indeed the angles and are equal.
Exercise 18:
Calculate, for each figure, the measure of the angle marked
(round the result to the nearest degree).
1. In the right triangle IAB, I know the side opposite and adjacent to the angle .
Formula : tangent.
so .
2. In the right triangle DCL, I know the hypotenuse and opposite side of the angle .
Formula : sine.
so .
3. In the right-angled triangle EFJ, I know the hypotenuse and opposite side of the angle .
Formula : sine.
so .
3. In the right triangle GHK, I know the side adjacent and opposite to the angle .
Formula : tangent.
so
Exercise 20:
1. Calculate the measure of .
In the right triangle IGH, I know the side opposite to and the hypotenuse.
Formula : sine.
so .
2. Deduce the measure of the angle .
The angles and are opposite by the vertex, so they have the same measure:°.
3. Calculate the lengths EF and FG rounded to the tenth.
In the triangle GEF rectangle in E.
and
and
cm.
cm
Exercise 21:
Calculate the length OM rounded to the millimeter.
Let’s calculate PM :
In the right triangle PAM, I know the opposite side and the angle
and I’m looking for the hypotenuse.
Formula: sine
so
Let’s calculate OM :
In the right triangle POM, I know the hypotenuse and the angle
and I look for the side adjacent to the angle .
Formula: cosine.
Exercise 22:
We give BD = 4 cm , BA = 6 cm and .
1. Show that BC= 8 cm.
In the triangle DCB rectangle,
2. Calculate CD.give the value rounded to the tenth.
3. Calculate AC.
In the triangle ABC rectangle in B according to the direct part of the Pythagorean theorem :
4. What is the value of ?
5. Deduce the value, rounded to the degree, of .
The answer key to the math exercises on trigonometry in the right triangle in 3rd grade.
Cette publication est également disponible en : Français (French) Español (Spanish) العربية (Arabic)
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VIEW | 2023-03-28 01:58:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 186, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6757876873016357, "perplexity": 1539.3442176304543}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00463.warc.gz"} |
https://llllllll.co/t/disquiet-junto-project-0351-selected-insomniac-works-volume-ii/16303/20 | # Disquiet Junto Project 0351: Selected Insomniac Works Volume II
Love it! Much improved
2 Likes
@Hypoid this is awesome! It’s what my track might sound like after falling asleep and the dreaming starts. And thanks for describing your technique. I don’t play around much with audio transformations, so I can use this to learn!
3 Likes
https://soundcloud.com/total_energy/higher-disquiet0351
I chose Amelie’s wonderful track High on Defalgan. Instead of (following the directions) smoothing it out, I did the opposite. I added bass, trumpets, strings and a synth wash while (I hope) keeping the integrity of the original track. Thanks Amelie!
5 Likes
And the playlist is now rolling:
3 Likes
Thank you very much! I’m glad you enjoyed, it’s always a little unnerving changing someone’s work. I also wrestled with the title…originally it was ‘Count Morgulbee’. Which do you prefer? Thanks again for the great start
1 Like
//disquietJunto0351
~b0 = Buffer.read(s, “/Users/evanhartzell/Desktop/minidesktop/SCpractice/junto0351/slip under mixX.wav”
);
(
SynthDef.new(\junto0351, {
arg amp=1, out=0, buf, freq=0.15, start, end;
var sig, ptr;
ptr = StandardL.ar(freq).range(start, end);
sig = BufRd.ar(2, buf, ptr);
sig = sig * amp;
Out.ar(out, sig);
)
x = Synth.new(\junto0351, [\buf, ~b0.bufnum, \start, 0, \end, ~b0.numFrames-1, \freq, 0.1]);
x.free;
//thx @Zedkah
3 Likes
Very nice track. Would love to listen to the original track you used, can you link to it?
Cheers
dd
2 Likes
https://soundcloud.com/glsmyth/fulfillment-disquiet0351
With this sort of thing I normally use a random number generator to select the track I am to work with, but I felt that Daniel Diaz’s Emptiness (https://soundcloud.com/daniel-diaz/emptiness-disquiet0350) seemed to have so many possibilities that I felt that I had no choice but to use it as electronics.
Fulfillment was written for Violin, Viola, Cello, and electronics.
The score is available at http://bit.ly/2pv7zCb
4 Likes
https://soundcloud.com/dot_slash_noise/disquiet-junto-project-0351-selected-insomniac-works-volume-ii-disquiet0351
For this remix I decided to use works of @samarobryn and @ethan_hein
I decided for following workflow:
1. Both tracks were sent in mono to mixer
2. Mixer aux was sent to Clouds modulated by Rampage
3. Output from clouds was sent back to mixer
4. Mixer out was sent to yamaha spx900 which has hall reverb with possible decay time of 480s
Then I started to play both tracks at once and by muting, sending to aux, changing volume etc. I generated this track. It turned out much more sinister that I intended but my dog felt asleep while I was playing so maybe it is a good sign
6 Likes
Awesome track George, much more dramatic and intense than the original, this is a complete new work and yet glimpses of my track shine through with a new spirit. I’m really glad you did this. Bravo.
2 Likes
Happy Day!
I selected Old Bones @lawrence-frazier-1
I really enjoyed the space and the spring of the tones. I wanted to abring forth some spring and space to add to some drum machine sounds from the Roland DJ 808. I loaded the track on to each turntable and played thru the track with some drum patterns, adding in the 808, 707 606 tones and slices of the og track in Serato. I then bounced this to Abelton and built a few instrument racks with drums and operators that I sliced up the beginning and the end of the track with 3 voices. I utilized various tempos thru the various clips and eq. I hope you enjoy and this was so fun.
Disquiet Junto Project 0351: Selected Insomniac Works Volume II
The Assignment: Rework some very quiet music by making it even more sedate.
Step 1: Last week, about 60 members of the Disquiet Junto recorded ambient music for the middle of the night. The specific request was to “Make very quiet music for very late at night for very fragile psyches.” This week, we’ll each select a track from last week and proceed to dial it down even further.
Step 2: Listen through the tracks from last week’s project, and choose the one whose ambience you want to employ in your track:
soundcloud.com/disquiet/sets/disquiet-junto-project-0350
In addition, there may be some other tracks from the project in the Lines discussions, here:
llllllll.co/t/disquiet-junto-pr…ed-insomniac-works/
Step 3: Having chosen a track in Step 2 above, confirm that your chosen track is downloadable. If it isn’t, either get in touch with the musician who made it, or choose another track.
Step 4: Listen closely to the track you selected in Step 3. Consider what edges it has that might be smoothed out, what drama it has that might be subsumed. Consider how you might do such things while retaining something that is inherently listenable, should someone choose to turn up the volume and focus on it.
Step 5: Rework the track you selected in Steps 2 and 3 to achieve the goals that arose from Step
4 Likes
cool sounds
2 Likes
https://soundcloud.com/analoc/on-the-way-to-nothing-disquiet0351
Fed @dascott ’s lovely „Night Patterns“ (https://soundcloud.com/dascott/disquiet0350-nightpatterns) into ye olde Mammut software.
Fiddled around with the controls until these slow sine patterns started to unfold. The resulting piece is un-intrusive and - let’s be honest - totally boring unless you listen to it around 4am with a vacant stare and fresh cup of coffee in hand.
7 Likes
Coffee and vacant thoughts are best at 4:00 am and tingels are great anytime. Thanks!
3 Likes
Recorded a quick accompaniment to @sevenism’s ‘midnight solo (falling leaf)’
Listening back now, I might be too busy for his track.
I liked his deceptively simple piano, it suits these Aphex Twin-inspired Juntos.
6 Likes
https://soundcloud.com/vgmrmojo/0351dj-001
Disquiet0351
• Used a random number generator to pick track #20 by https://soundcloud.com/healthylives/five-borders-disquiet0350-selected-insomniac-works
• Used some volume automation to lower the amplitude peaks
• Ran the signal thru some EQ, delay and a reverb plate and another delay
•
3 Likes
https://soundcloud.com/yawha/the-insomniacs-accompaniment-disquiet0351
Worked with @ikjoyce 's track from Disquiet0350 “The Insomniac’s Lullaby” : https://soundcloud.com/ikjoyce/improvisation1
I probably didn’t make it more sedate… But I had to work with the track, it is so sweet, takes so many lovely turns.
Picture is of a Morrison Shelter from WW2, a moveable air-raid shelter.
I had an awful problem with some clicking in this Ableton set. I spend a long while - must be three hours - debugging the set. I had to get iZotope RX6’s Declicker going on a stereo bounce-down in the end. I am gonna dig further, so I will upload afresh if I can get to the bottom of it…
Bloody computers.
5 Likes
my spirit animal this week was code poetry by @abalone
the sonic spice world was inspired by @ntrier though i didn’t actually use your track.
noise floor by @hypoid
@jwhiles your track + also reused these freesound samples; one which you used in one of your latest tracks / hope you don’t mind just trying to build off of that.
//https://freesound.org/people/felix.blume/sounds/408048/
//https://freesound.org/people/Philip%20Goddard/packs/10521/
instant replay by:
@sevenism
om logic by: @Anatol
josephbeuys provided instant life
http://www.ubu.com/sound/beuys.html
the music video for this track is
“Weird Al” Yankovic - Smells Like Nirvana
muted / slowed down 0.25 in youtube settings.
quality: 144p
_----- or
Biggest Football Hits Ever
or
Top 20 Knockouts in UFC History
etc.
p.s.
dialing it back / raining it in.
https://soundcloud.com/youaresound/2018-9-22-17-4-29-1019a/s-FiXkd
hope i got this right…
don’t need to include them both or at all.
this is the original
https://soundcloud.com/youaresound/miff-smells-like-team-spirit/s-4jajX
6 Likes
fantastic
1 Like
Hey All, NON-SUBMISSION Tired something with vgmrmojo disquiet0350.
https://soundcloud.com/detritus-tabu3/got-my-mojo-sleeping3
Peace, Hugh
3 Likes | 2021-06-21 19:50:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3581450879573822, "perplexity": 6865.748381163188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00085.warc.gz"} |
https://gsocinterval.blogspot.com/2017/06/construction-and-printing.html | ## Friday, 16 June 2017
### Construction and Printing
This week I have started to work on methods for constructing and printing N-dimensional arrays of intervals. In my timeline I estimated that this work would take 2 weeks. However in this first week I have managed to complete most of the work. I will give some comments on how I have worked with the Mercurial repository, how the work went and different things I encountered along the path.
## Working with Mercurial
This is essentially the first time I'm using Mercurial for revision control, though I have used git before. However I quickly learned how to use it for the basic tasks that I need, committing, comparing files and checking the history. As mentioned in a previous post you can find my repository here [1].
### Coding style
When I started to work with the files I realized that they did not follow Octaves coding standard [2]. After a short discussion on the mailing list we decided that I will update the files I change to follow the standard coding style. Usually it is not a good idea to change coding style and add functionality in the same commit. However most of the changes to coding style are only white space changes so they can be ignored using the -w flag in Mercurial. Thus we decided that as long as the coding style changes are only such that it is ignored with -w I will do it in the same commit as the added functionality. If there are some coding style changes that's not only white space, the most common example is to long lines, I do a commit with only changes to the coding style first. So if you want to take a look at the functionality I have added you will probably want to use the -w flag. Note however that I have not updated the coding style for any files I have not changed otherwise.
### Committing
Normally I do one commit for each file, though in many cases the bare intervals and the decorated intervals have almost identical functions and in that case I commit changes to them both at the same time. Of course it also happens that I have to go back and do more changes to a files, in that case I just do another commit.
## The actual work
The work went much faster than I expected. The main reason for this is that Octave has very good support for indexing. For example expressions like
isnai(x.inf <= x.sup) = false;
works just as well for matrices as for N-dimensional arrays. In fact the constructor for bare intervals even worked for N-dimensional arrays from the beginning, there I only had to do slight modification to the documentation and add some tests!
Not all functions were that easy though. Some functions that have not been updated in a while clearly assumed the input was a matrix, for example in $hull$
sizes1 = cellfun ("size", l, 1);
sizes2 = cellfun ("size", l, 2);
In most cases I only needed to add more general indexing, often times even making the code clearer.
In some functions all I had to do was to remove the check on the input data so that it would accept N-dimensional arrays. This was true in for example $cat$ were all I had to do was to remove the check and do some minor modifications to the documentation.
I can conclude with saying that Octave has great support for working with N-dimensional arrays. Since internally the data for intervals are stored only as arrays I could make good use of it!
## Noteworthy things
While most functions were straight forward to modify some required some thought. How should they even work for N-dimensional input?
### Disp
When modifying the $disp$-function I chose to mimic how Octave handles displaying N-dimensional arrays. I noticed that this is different from how Matlab handles it. In Matlab we have
> x = zeros (2, 2, 2)
x(:,:,1) =
0 0
0 0
x(:,:,2) =
0 0
0 0
while in Octave it's
> x = zeros (2, 2, 2)
x =
ans(:,:,1) =
0 0
0 0
ans(:,:,2) =
0 0
0 0
I don't know the choice behind Octaves version. At least at first glance I think I prefer the way Matlab does it. But since I'm working in Octave I chose that style.
The next question was how to handle the subset symbol, $\subset$. The interval package uses $=$ or $\subset$ depending on if the string representation is exact or not. For example
> x = infsup (1/2048, 1 + 1/2048);
> format short; x
x ⊂ [0.00048828, 1.0005]
> format long; x
x = [0.00048828125, 1.00048828125]
How should this be handled for N-dimensional arrays? One way would be to switch all $=$ to $\subset$ is the representation is not exact. Another to use $\subset$ on all submatrices that does not have an exact string representation. The third way, and how it is implemented now, is to only change the first $=$ to $\subset$, the one after the variable name. Like this
> x(1,1,1:2) = infsup (1/2048, 1 + 1/2048)
x ⊂ 1×1×2 interval array
ans(:,:,1) = [0.00048828, 1.0005]
ans(:,:,2) = [0.00048828, 1.0005]
This might be a bit odd when you first look at it, on some places we use $=$ and on some $\subset$. Though I think it somehow makes sense, we are saying that $x$ is a subset of the $1\times1\times2$ interval array given by
ans(:,:,1) = [0.00048828, 1.0005]
ans(:,:,2) = [0.00048828, 1.0005]
which actually is true. Anyway I will leave like this for now and then we might decide to switch it up later.
### linspace and mince
The standard implementation of $linspace$ only supports scalar or vector input. It could be generalized to N-dimensional arrays by for example returning a N+1-dimensional array were the last dimension corresponds to the linearly spaced elements. But since this has not been done in the standard implementation I will at least wait with adding for intervals.
The function $mince$ can be seen as a interval generalization of $linspace$. It takes an interval and returns an array of intervals whose union cover it. This could similarly be expanded to N dimensions by creating the array along the N+1 dimension. But again we choose to at least wait with adding this.
### meshgrid and ndgrid
The interval package already has an implementation of $meshgrid$. But since it previously did not support 3-dimensional arrays it had to fit 3-d data in a 2-d matrix. Now that it supports 3-d data it can output that instead.
Currently the interval package does not implement $ndgrid$. When I looked into it I realized that the standard implementation of $ndgrid$ actually works for interval arrays as well. I have not looked into the internals but in principle it should only need the $cat$ function, which is implemented for intervals. Further I noticed that the standard $meshgrid$ also works for intervals. However the interval implementation differs in that it converts all input to intervals, were as the standard implementation allows for non-uniform output. Using the interval implementation of $meshgrid$ we have
> [X Y] = meshgrid (infsup (1:3), 4:6)
X = 3×3 interval matrix
[1] [2] [3]
[1] [2] [3]
[1] [2] [3]
Y = 3×3 interval matrix
[4] [4] [4]
[5] [5] [5]
[6] [6] [6]
but if we fall back to the standard implementation (by removing the interval implementation) we get
> [X Y] = meshgrid (infsup (1:3), 4:6)
X = 3×3 interval matrix
[1] [2] [3]
[1] [2] [3]
[1] [2] [3]
Y =
4 4 4
5 5 5
6 6 6
Note that the last matrix is not an interval matrix.
So the question is, should we implement a version of $ndgrid$ that converts everything to intervals or should we remove the implementation of $meshgrid$? It's at least most likely not a good idea that the functions are different. I think that removing the implementation of $meshgrid$ makes most sense. First of all it's less code to maintain, which is always nice. Secondly you can manually convert all input to the function to intervals if you want uniform output. If you do not want uniform output then the standard implementation works were as the interval implementation does not, so the standard implementation is more general in a sense.
We have to choose what to do, but for now I leave it as it is.
### Non-generalizable functions
From what I have found there is no way to create a 3-dimensional array in Octave in the same way you can create a 2-dimensional one with for example
M = [1, 2; 3, 4];
Instead higher dimensional arrays have to be created using other functions, for example $reshape$ or $zeros$, or by specifying the submatrices directly
M(:,:,1) = [1, 2; 3, 4];
M(:,:,2) = [5, 6; 7, 8];
This means that the functions $\_\_split\_interval\_literals\_\_$, which is used to split a string like $"[1, 2; 3, 4]"$ into its separate components, cannot really be generalized to N dimensions.
[1] https://sourceforge.net/u/urathai/octave/ci/default/tree/
[2] http://wiki.octave.org/Octave_style_guide | 2017-08-20 08:02:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42787668108940125, "perplexity": 1040.027226071869}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106358.80/warc/CC-MAIN-20170820073631-20170820093631-00191.warc.gz"} |
http://learningwitherrors.org/2016/06/03/small-bias/ |
# Simple Lower Bounds for Small-bias Spaces
By
— post as [PDF]
I was reading about PRGs recently, and I think a lemma mentioned last time (used for Johnson-Lindenstrauss lower-bounds) can give simple lower-bounds for $\epsilon$-biased spaces.
Notice:
• $2^n$ mutually orthogonal vectors requires dimension at least $2^n$, but $2^n$ “almost orthogonal” vectors with pairwise inner-products $|\innp{v_i, v_j}| \leq \epsilon$ exists in dimension $O(n/\epsilon^2)$, by Johnson-Lindenstrauss.
• Sampling $n$ iid uniform bits requires a sample space of size $2^n$, but $n$ $\epsilon$-biased bits can be sampled from a space of size $O(n/\epsilon^2)$.
First, let's look at $k$-wise independent sample spaces, and see how the lower-bounds might be extended to the almost $k$-wise independent case.
Note: To skip the background, just see Lemma 1, and its application in Claim 3.
## 1. Preliminaries
What “size of the sample space” means is: For some sample space $S$, and $\pm 1$ random variables $X_i$, we will generate bits $x_1, \dots x_n$ as an instance of the r.vs $X_i$. That is, by drawing a sample $s \in S$, and setting $x_i = X_i(s)$. We would like to have $|S| \ll 2^n$, so we can sample from it using less than $n$ bits.
Also, any random variable $X$ over $S$ can be considered as a vector $\t X \in \R^{|S|}$, with coordinates $\t X[s] := \sqrt{\Pr[s]} X(s)$. This is convenient because $\innp{\t X, \t Y} = \E[XY]$.
## 2. Exact $k$-wise independence
A distribution $D$ on $n$ bits is $k$-wise independent if any subset of $k$ bits are iid uniformly distributed. Equivalently, the distribution $D : \{\pm 1\}^n \to \R_{\geq 0}$ is $k$-wise independent iff the Fourier coefficients $\hat D(S) = 0$ for all $S \neq 0, |S| \leq k$.
$n$ such $k$-wise independent bits can be generated from a seed of length $O(k \log n)$ bits, using say Reed-Solomon codes. That is, the size of the sample space is $n^{O(k)}$. This size is optimal, as the below claim shows (adapted from Umesh Vazirani's lecture notes [Vaz99]).
Claim 1 Let $D$ be a $k$-wise independent distribution on $\{\pm 1\}$ random variables $x_1, \dots, x_n$, over a sample space $S$. Then, $|S| = \Omega_k(n^{k / 2})$.
Proof: For subset $T \subseteq [n]$, let $\chi_T(x) = \prod_{i \in T} x_i$ be the corresponding Fourier character. Consider these characters as vectors in $\R^{|S|}$ as described above, with $\innp{\chi_A, \chi_B} = \E_{x \sim D}[\chi_A(x)\chi_B(x)]$
Let $J$ be the family of all subsets of size $\leq k/2$. Note that, for $A, B \in J$, the characters $\chi_A, \chi_B$ are orthogonal: \begin{align*} \innp{\chi_A, \chi_B} &= \E_{x \sim D}[\chi_A(x)\chi_B(x)]\\ &= \E_{x \sim D}[(\prod_{i \in A \cap B} x_i^2)(\prod_{i \in A \Delta B} x_i)]\\ &= \E_{x \sim D}[\chi_{A \Delta B}(x)] \note{since $x_i^2 = 1$}\\ &= 0 \note{since $|A \Delta B| \leq k$, and $D$ is $k$-wise independent} \end{align*} Here $A \Delta B$ denotes symmetric difference, and the last equality is because $\chi_{A \Delta B}$ depends on $\leq k$ variables, so the expectation over $D$ is the same as over iid uniform bits.
Thus, the characters $\{\chi_A\}_{A \in J}$ form a set of $|J|$ mutually-orthogonal vectors in $\R^{|S|}$. So we must have $|S| \geq |J| = \Omega_k(n^{k/2})$. $$\tag*{\blacksquare}$$
The key observation was relating independence of random variables to linear independence (orthogonality). Similarly, we could try to relate $\epsilon$-almost $k$-wise independent random variables to almost-orthogonal vectors.
## 3. Main Lemma
This result is Theorem 9.3 from Alon's paper [Alo03]. The proof is very clean, and Section 9 can be read independently. 11. Theorem 9.3 is stated in terms of lower bounding the rank of a matrix $B \in \R^{N \x N}$ where $B_{i,i} = 1$ and $|B_{i, j}| \leq \epsilon$. The form stated here follows by defining $B_{i, j} := \innp{v_i, v_j}$.
Lemma 1 Let $\{v_i\}_{i \in [N]}$ be a collection of $N$ unit vectors in $\R^d$, such that $|\innp{v_i, v_j}| \leq \epsilon$ for all $i \neq j$. Then, for $\frac{1}{\sqrt{N}} \leq \epsilon \leq 1/2$, $d \geq \Omega\left(\frac{\log N}{\epsilon^2 \log(1/\epsilon)}\right)$
This lower-bound on the dimension of “almost-orthogonal” vectors translates to a nearly-tight lower-bound on Johnson-Lindenstrauss embedding dimension, and will also help us below.
## 4. Small bias spaces
A distribution $D$ on $n$ bits is $\epsilon$-biased w.r.t linear tests (or just “$\epsilon$-biased”) if all $\F_2$-linear tests are at most $\epsilon$-biased. That is, for $x \in \{\pm 1\}^n$, the following holds for all subsets $S \subseteq [n]$: $\left|\E_{x \sim D}[\chi_S(x)]\right| = \left|\Pr_{x \sim D}[\chi_S(x) = 1] - \Pr_{x \sim D}[\chi_S(x) = -1]\right| \leq \epsilon$ Similarly, a distribution is $\epsilon$-biased w.r.t. linear tests of size $k$ (or “$k$-wise $\epsilon$-biased) if the above holds for all subsets $S$ of size $\leq k$.
There exists an $\epsilon$-biased space on $n$ bits of size $O(n / \epsilon^2)$: a set of $O(n / \epsilon^2)$ random $n$-bit strings will be $\epsilon$-biased w.h.p. Further, explicit constructions exist that are nearly optimal: the such first construction was in [NN93], and was nicely simplified by [AGHP92] (both papers are very readable).
These can be used to sample $n$ bits that are $k$-wise $\epsilon$-biased, from a space of size almost $O(k \log(n)/\epsilon^2)$; much better than the size $\Omega(n^k)$ required for perfect $k$-wise independence. For example22. This can be done by composing an $(n, k')$ ECC with dual-distance $k$ and an $\epsilon$-biased distribution on $k' = k\log n$ bits. Basically, use a linear construction for generating $n$ exactly $k$-wise independent bits from $k'$ iid uniform bits, but use an $\epsilon$-biased distribution on $k'$ bits as the seed instead. , see [AGHP92] or the lecture notes [Vaz99].
### 4.1. Lower Bounds
The best lower bound on size of an $\epsilon$-biased space on $n$ bits seems to be $\Omega(\frac{n}{\epsilon^2 \log(1/\epsilon)})$, which is almost tight. The proofs of this in the literature (to my knowledge) work by exploiting a nice connection to error-correcting codes: Say we have a sample space $S$ under the uniform measure. Consider the characters $\chi_T(x)$ as vectors $\t \chi_T \in \{\pm 1\}^{|S|}$ defined by $\t \chi_T[s] = \chi_T(x(s))$, similar to what we did in Section 2. The set of $2^n$ vectors $\{\t \chi_T\}_{T \subseteq [n]}$ defines the codewords of a linear code of length $|S|$ and dimension $n$. Further, the hamming-weight of each codeword (number of $-1$s in each codeword, in our context), is within $n(\frac{1}{2} \pm \epsilon)$, since each parity $\chi_T$ is at most $\epsilon$-biased. Thus this code has relative distance at least $\frac{1}{2} - \epsilon$, and we can use sphere-packing-type bounds from coding-theory to lower-bound the codeword length $|S|$ required to achieve such a distance. Apparently the “McEliece-Rodemich-Rumsey-Welch bound” works in this case; a more detailed discussion is in [AGHP92, Section 7].
We can also recover this same lower bound using Lemma 1 in a straightforward way.
Claim 2 Let $D$ be an $\epsilon$-biased distribution on $n$ bits $x_1, \dots, x_n$, over a sample space $S$. Then, $|S| = \Omega\left(\frac{n}{\epsilon^2 \log(1/\epsilon)}\right)$
Proof: Following the proof of Claim 1, consider the Fourier characters $\chi_T(x)$ as vectors $\t \chi_T \in \R^{|S|}$, with $\t \chi_T[s] = \sqrt{\Pr[s]} \chi_T(x(s))$. Then, for all distinct subsets $A, B \subseteq [n]$, we have $\innp{\t \chi_A, \t \chi_B} = \E_{x \sim D}[\chi_A(x)\chi_B(x)] = \E_{x \sim D}[\chi_{A \Delta B}(x)]$ Since $D$ is $\epsilon$-biased, $\left|\E_{x \sim D}[\chi_{A \Delta B}(x)]\right| \leq \epsilon$ for all $A \neq B$. Thus, applying Lemma 1 to the collection of $N = 2^n$ unit vectors $\{\t \chi_T\}_{T \subseteq [n]}$ gives the lower bound $|S| = \Omega\left(\frac{n}{\epsilon^2 \log(1/\epsilon)}\right)$. $$\tag*{\blacksquare}$$
This also nicely generalizes the proof of Claim 1, to give an almost-tight lower bound on spaces that are $\epsilon$-biased w.r.t linear tests of size $k$.
Claim 3 Let $D$ be a distribution on $n$ bits that is $\epsilon$-biased w.r.t. linear tests of size $k$. Then, the size of the sample space is $|S| = \Omega\left(\frac{k \log (n/k)}{\epsilon^2 \log(1/\epsilon)}\right)$
Proof: As before, consider the Fourier characters $\chi_T(x)$ as vectors $\t \chi_T \in \R^{|S|}$, with $\t \chi_T[s] = \sqrt{\Pr[s]} \chi_T(x(s))$. Let $J$ be the family of all subsets $T \subseteq [n]$ of size $\leq k/2$. Then, for all distinct subsets $A, B \in J$, we have $\left|\innp{\t \chi_A, \t \chi_B}\right| = \left|\E_{x \sim D}[\chi_{A \Delta B}(x)]\right| \leq \epsilon$ since $|A \Delta B| \leq k$, and $D$ is $\epsilon$-biased w.r.t such linear tests. Applying Lemma 1 to the collection of $|J|$ unit vectors $\{\t \chi_T\}_{T \in J}$ gives $|S| = \Omega(\frac{k \log (n/k)}{\epsilon^2 \log(1/\epsilon)})$. $$\tag*{\blacksquare}$$
Note: I couldn't find the lower bound given by Claim 3 in the literature, so please let me know if you find a bug or reference.
Also, these bounds do not directly imply nearly tight lower bounds for $\epsilon$-almost $k$-wise independent distributions (that is, distributions s.t. their marginals on all sets of $k$ variables are $\epsilon$-close to the uniform distribution, in $\ell_{\infty}$ or $\ell_{1}$ norm). Essentially because of the loss in moving between closeness in Fourier domain and closeness in distributions. 33. Eg, $\epsilon$-biased $\implies$ $\epsilon$-close in $\ell_{\infty}$, but $\epsilon$-close in $\ell_{\infty}$ can be up to $2^{k-1}\epsilon$-biased. And $2^{-k/2}\epsilon$-biased $\implies$ $\epsilon$-close in $\ell_{1}$, but not the other direction.
### References
[AGHP92] Noga Alon, Oded Goldreich, Johan Håstad, and Ren{é} Peralta. Simple constructions of almost k-wise independent random variables. Random Structures \& Algorithms, 3(3):289--304, 1992. URL: http://www.tau.ac.il/~nogaa/PDFS/aghp4.pdf.
[Alo03] Noga Alon. Problems and results in extremal combinatorics, part i. Discrete Math, 273:31--53, 2003. URL: http://www.tau.ac.il/~nogaa/PDFS/extremal1.pdf.
[NN93] Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM journal on computing, 22(4):838--856, 1993. URL: http://www.wisdom.weizmann.ac.il/~naor/PAPERS/bias.pdf.
[Vaz99] Umesh Vazirani. k-wise independence and epsilon-biased k-wise indepedence. 1999. URL: https://people.eecs.berkeley.edu/~vazirani/s99cs294/notes/lec4.pdf. | 2017-09-24 20:58:40 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9826830625534058, "perplexity": 464.41729409691794}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690211.67/warc/CC-MAIN-20170924205308-20170924225308-00138.warc.gz"} |
https://physics.stackexchange.com/questions/265839/does-string-theory-explain-the-existence-of-3-generations-of-quarks-leptons?noredirect=1 | # Does string theory explain the existence of 3 generations of quarks/leptons?
I am wondering whether string theory explains the existence of 3 families of quarks/leptons or not. I have a very limited understanding of string theory, as of now, and I have a mathematical background, so I am asking this question here, so that people with better knowledge of string theory might answer it.
There is a related discussion: Origin of lepton/quark generations? and one of users, Andrew Holzner, quoting wikipedia, gave an explanation that CP violation requires at least 3 generations (and there are in that discussion a number of other explanations). This sounds like a reasonable explanation, but my question is more from the point of view of string theory.
• As of today string theory doesn't explain anything, or, more precisely, it explains too much. The "string landscape" allows something like $10^{500}$ different models of reality, but it doesn't give us any reason (apart from an intellectually hollow anthropic argument) to select one over the other. – CuriousOne Jul 2 '16 at 14:57
• Related: physics.stackexchange.com/q/2051/2451 and links therein. – Qmechanic Jul 2 '16 at 15:01
• @CuriousOne thank you for the information on the status quo. User heather wrote something similar. You both answered my question. – Malkoun Jul 2 '16 at 15:08
• String theory explains anything, as far as I can see! – tfb Jul 3 '16 at 13:36
## 3 Answers
Part 1:
The branch of string theory which actually tries to match experiment is called string phenomenology. The state of the art in string phenomenology is that, starting from different forms of string theory (heterotic string theory, M-theory, F-theory...), it is possible to define space-time geometries, arrangements of branes, background fluxes... such that strings in the defined environment will behave qualitatively like the particles of the standard model.
The underlying reason why there are three generations in such a model really depends on the nature of its construction.
In an M-theory model such as those championed by Gordon Kane, the particles in a given generation correspond to states of M2-branes located at specific singular points in the compactification manifold, so the number of such generations is just the number of such singular points.
In a heterotic model such as those that Brian Greene has written about, it's more complicated. The topology of the compactification manifold permits a specific number of light left-handed fermionic states, and another number of light right-handed fermionic states; then left and right combine to make heavy states; and the generations correspond to the light handed fermionic states that are left over, that didn't pair up with anything. The original numbers of handed light states equal two of the "Hodge numbers" characterizing the topology, so in this case, there are three (or however many) generations because the difference between those two numbers equals three.
In still other models, the reason for there being three generations would be something else again.
Part 2:
Since the state of the art in string phenomenology is still just at the level of searching the vast "landscape" of possibilities for models that match experiment, any current explanation for "why three generations?" is going to lead back to contingent properties of the model that happens to be successful, like those that I sketched in Part 1 of this answer.
In evolutionary biology, they speak of proximate causes and ultimate causes. Why does a flower bend to follow the sun? The proximate cause is the set of molecules that it happens to be made of. The ultimate cause is natural selection - that's the reason why it's made of molecules that react like that, and not in some other way.
We can look at explanations like those from Part 1 as proximate causes of there being three generations. What are the possible ultimate causes?
One possibility is anthropic. Maybe we live in an eternally inflating universe where different string vacua are realized in different regions, and maybe e.g. the cosmological consequences of the CP violation that requires at least three generations in order to occur, helps make life, or even just stars, possible.
Another possibility is that it is just random. In genomic evolution, there's a lot of neutral evolution, features of the genome which are just contingent, which don't help the organism survive, but also don't hinder it, so those features aren't eliminated by natural selection. Anthropics can't determine everything, and maybe three generations is just a brute fact about how our corner of reality turned out.
Still another possibility is that it's the product of the natural dynamics of string theory. String phenomenology fixes the geometry of the extra dimensions (etc) and studies the results, but in fact you can have quantum tunneling between different geometries, and there may have been a lot of that in the early universe. The 2007 paper "Triadophilia" speculates that three-generation heterotic manifolds may be favored in this way.
I will address the title question:
does string theory explain the existence of 3 generations of quarks leptons
because of the word "explain".
Physics is about measurements and observations and mathematical models which not only fit the measurements and observations but also have predictive power. Otherwise the model is just a map, not a physics theory.
Newtonian gravitational theory assumes the 1/r^2 behavior and using classical mechanics with its laws generates the very successful gravitational model which can predict most astronomical data within errors. Deviations from Newtonian mechanics were predicted by the theory of General Relativity, and the validation of the predictions established GR as an undelying theory from which Newtonian gravity emerges.
The Standard Model of particle physics with its Lagrangian formulation is the analogue of the Newtonian gravitational theory: a large number of measurements and observations went into the SM to build up the structure, and its predictions have been mostly validated up to now. Candidates for string theory models are where GR was before its validation by not before seen data.
String theories can accommodate the group structure in the Lagrangian of the standard model, so there is no problem in envisaging a string theory model, also, and very important, string theories are the only candidate theories that can have quantization of gravity naturally. They also demand supersymmetry to do their magic. In this sense super symmetry is predicted by candidate string theory models, and if supersymmetry is found at LHC it will be like the validation of GR by predicting the anomalous perihelion advance of the planet Mercury without any arbitrary parameters
As was stated in the comments there are too many possible theories, and nature/data have to choose for us which is the one that fits the data, in the same way that nature chose for us the standard model lagrangian.
So "explain" is not a good verb, a physical theory fits the data and makes predictions for future measurements.
Now if physics ever reaches the point to have a mathematical Theory of Everything, from a few postulates and few measurement input for constants, then it might be legitimate to say that the TOE "explains" everything. Certainly no physics theory is at that point , more so String Theories which are at the research level.
• Thank you anna v for your thoughts on the philosophy of Physics (if you don't mind my describing them like that). I enjoyed your answer. I just meant by "explain" that it postulates some principles, which are basic, and from them it derives the existence of 3 generations of quarks. In this sense, it would have "explained" the 3 generations in terms of more fundamental principles. But yes, as you and others have written, string theory, as of the time of writing, does not do that. – Malkoun Jul 2 '16 at 18:59
The official string theory website says this:
Theoretical physics has not explained why there are three generations of particles that make up matter. Maybe string theory will come up with an answer for this.
That's really where it stands. In fact, there's another question on physics SE here, where one of the answers says
The question as to why there are exactly three generations is still an open problem, even in string theory.
Another answer at the same question says
All in all, to answer questions like the number of generations in the SM, the masses of the fermions, etc you need a candidate for a complete theory at high energies and the answers will depend on this candidate. String theory is considered to be the most successful framework for this job and string models do indeed make concrete predictions about (among other things) the number of generations we should be observing. Unfortunately, there are too many models (vaccua) to choose from and no obvious way to make the choice.
To sum up: we have absolutely no idea why there are three generations, even in string theory. Like @CuriousOne said in the comments, string theory has too many possible models of reality with no way to select which one to use for string theory to really explain anything.
Note:
In the comments below this answer, @CuriousOne made a good point: all of the quoted statements make it sound like this is one of the only problems with string theory. This is not true. There is no experimental evidence for string theory. Describing why there are 3 generations of quarks/leptons is the least of string theory's problems.
• I would group these statements under "false advertising". String theory hasn't explained anything about the structure of the standard model, so it's more than just a little frivolous to say "We haven't explained a particular statement...", which leaves the reader with the impression that they have explained some or maybe even most others. :-( – CuriousOne Jul 2 '16 at 15:08
• String theory explains the complete structure of the universe as oscillating strings. QFT and GR follows from string theory, so it explains our most succesful theories. – user122089 Jul 2 '16 at 15:11
• ok, I see where things stand. Too many possibilities for the "curled up dimensions". Too many possibilities for the CY 3-fold, for 10-dimensional string theories. Hmm, this brings up a question in my head. How does one make concrete predictions in string theory, given a choice of a CY 3-fold? Can someone recommend something to read which is string theoretic and of a more phenomenological approach? (Technically, this is a different discussion, though related, so maybe I should move it to another post?) – Malkoun Jul 2 '16 at 15:13
• @CuriousOne, I agree that the first statement is a little ridiculous; I was including it to show that even the official website doesn't claim an answer. – heather Jul 2 '16 at 15:17
• @Malkoun I'm not sure specifically what sort of book you are looking for, but Brian Greene's books discuss string theory and some of it's predictions, etc. I've personally read parts of The Hidden Reality and there was a section discussing string theory and its 'status', so that might help you. – heather Jul 2 '16 at 15:20 | 2020-01-26 22:00:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37584203481674194, "perplexity": 522.2076511377572}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251690379.95/warc/CC-MAIN-20200126195918-20200126225918-00416.warc.gz"} |
http://www.javatutorialpoint.com/c-preprocessor/ | C-Storage Classes: Previous Next: C-Header Files
The pre processors are collection of statements called directives. These statements are executed before the actual compilation process starts. The pre processor works on source code i.e. ‘.c’ file and creates “Expanded Source code” because after this process all functions replaces their definition.
These statements are always starts with “#”(hash sign) as prefix in C Programming.
You must have these statements in previous example. Like #include. These are called preprocessor directives. These can be placed anywhere in the program but it is convenient to placed it at beginning of program.
Some rules for defining preprocessor directives: –
• They must be placed in first column.
• No two preprocessor directives can be placed in a single line.
• They should not terminate with semicolon ( ; ).
The preprocessor directives are:
1. Macro expansion
2. File inclusion
3. Conditional compilation
1. Macro expansion: -C allows you to define an identifier having a constant value at the beginning of the program before the main( ) all macros are defined with # define
For Eg: # include
#define LIMIT 10
void main()
{
int i;
for(i = 0; i
printf(“%dn”, i);
}
In above example instead of writing 10 in loop we have written LIMIT which is an identifier having constant value 10 and defined before void main( ). During preprocessing the every occurrence of LIMIT identifier replaced with its value ‘10’. The macro works as a constant value.
More than one macro can be defined to a single program. The macros are compulsory to define in upper case and values are compulsory to assign at the time of declaration.
#define can be used for defining:
• An operator
• Constant value
• A condition
Eg: defining a operator
# include
#define OR ||
void main()
{
int num = 20;
if(num < 40) OR (num > 10)
printf(“Example of Macro…!!”);
else
printf(“Quite Accurate…!!”);
}
1. File inclusion: –This directive starts with ‘#include’ and allow us to include one file into another file.
Syntax is: #include “” !—C Programming syntax
This statement causes content of defined to be inserted into program at the preprocessing time.
” contains the prototype of the functions and for use those functions inclusion of file is compulsory.
Eg: you have used “stdio.h” which contains the definition of the functions printf( ), scanf( ), etc these function are called library functions.
#include can be written in two ways:
1. #include”: This command would look for specified file in current directory as well as in specified list of directories.
2. #include: This command would look for file in specified list of directories only no searching in current directory.
1. #Conditional (compilation directive):-It allows line of code to be passed on the basis of computed condition given in directive some preprocessor conditional commands are #if, #else, #endif etc.
#if and #else they work in same way as keyword of conditional branching works like #if directive is used to test the condition.
Syntax:
#if
Statements;
#else
Statements;
#endif
“ #endif ” directive to use to show the conditional directive block.
C-Storage Classes: Previous Next: C-Header Files | 2016-10-22 05:27:21 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8159456253051758, "perplexity": 7024.763280749195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718426.35/warc/CC-MAIN-20161020183838-00107-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://s21570.gridserver.com/hsafdcd/addition-and-scalar-multiplication-of-matrices-947159 | Try the Course for Free. {\color{red}{1 - 5}}&{\color{blue}{2 - 6}}\\ The order of the matrices are the same 2. By using this website, you agree to our Cookie Policy. 6&8\\ b) What is the dimension of the space? \end{array}} \right]}_{1 \times \color{blue}{3}} \cdot \underbrace {\left[ {\begin{array}{*{20}{c}} The difference of two matrices can only be found if both matrices have the same dimension. First, let us see how to multiply a single number (constant) to a matrix. \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{l}} Copyright 2014 - 2020 The Calculator .CO | All Rights Reserved | Terms and Conditions of Use, Scalar Multiplication of Matrix Calculator. Matrix Addition, Multiplication, and Scalar Multiplication. This means, c + 0 = c for any real number. {\color{red}{1} \cdot \color{blue}{3} + \color{red}{3} \cdot \color{blue}{1} + \color{red}{5} \cdot \color{blue}{5}}&?\\ { - 5}&{ - 10}&{ - 15} a) How are the vector addition and the scalar multiplication defined? Multiplying a $2 \times 3$ matrix by a $3 \times 2$ matrix is possible, and it gives a $2 \times 2$ matrix \color{red}{5}&\color{blue}{6}\\ Perform the matrix operations of matrix addition, scalar multiplication, transposition and matrix multiplication. The answer is a $2 \times 2$ matrix. Prove algebraic properties for matrix addition, scalar multiplication, transposition, and matrix multiplication. \color{red}{2}&\color{red}{4}&\color{red}{6} To multiply a matrix with a real number, each element is multiplied by that number. Please consider the example provided here to understand this algebra operation: This scalar multiplication of matrix calculator can process both positive and negative figures, with or without decimals and even fractions. These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different pack… \color{red}{2}&\color{red}{4}&\color{red}{6} 5&2 The term scalar multiplication refers to the product of a matrix and a real number. Help with proving this definition: $(r + s) X = rX + rY$ I have to … as the result. Addition of Matrices; Subtraction of Matrices; Scalar Multiplication of Matrices Multiply the 1st row of the first matrix and 1st column of the second matrix, element by element. $. { - 1}&{ - 2}&{ - 3} Given two matrices of the same size, that is, the two matrices have the same number of rows and columns, we define their sum by constructing a third matrix whose entries are the sum of the corresponding entries of the original two matrices.. So let's say I have the 2 by 3 matrix, so two rows and three columns, and the entries are 7, 5, negative 10, 3, 8, and 0. \end{array}} \right]}_{\color{blue}{3} \times 3} = \left[ {\begin{array}{*{20}{c}} Alright, this means real number. {2 \cdot ( - 2) + 1 \cdot 4}&{2 \cdot 3 + 1 \cdot ( - 1)}\\$. Now, matrix scalar multiplication, very similar idea. \end{array}} \right] = \underbrace {\left[ {\begin{array}{*{20}{c}} \left[ {\begin{array}{*{20}{l}} For example, the set of 2 x 2 diagonal matrices is closed under scalar multiplication. 4\\ {\color{red}{5} \cdot 1}&{\color{red}{5} \cdot 2}&{\color{red}{5} \cdot 3}\\ Scalar multiplication is easy. C uses “Row Major”, which stores all the elements for a given row contiguously in memory. If the number of elements in row vector is NOT the same as the number of rows in the second matrix then their product is not defined. 1&4\\ A \cdot B = \left[ {\begin{array}{*{20}{c}} The set of all invertible n×nn×n matrices is not a vector space with respect to the typical matrix addition and scalar multiplication operations and the typical matrix zero. Matrix multiplication, however, is quite another story. B = \left[ {\begin{array}{*{20}{c}} Represent these operations in terms of the entries of a matrix. \color{blue}{1}&4\\ Distributive over matrix addition: Scalar multiplication commutes with matrix multiplication: and where λ is a scalar. Example 2: Find the product AB where A and B are matrices given by: $Multiplication by a Scalar octave: c = 3 c = 3 octave: c*A ans = 6 3 9 6 -6 6 Matrix Addition & Subtraction octave: B = [1,1;4,2;-2,1] B = 1 1 4 2 -2 1 octave: C = A + B C = 3 2 7 4 -4 3 octave: D = A - B D = 1 0 -1 0 0 1 Matrix Multiplication Your text probably gave you a complex formula for the process, and that formula probably didn't make any sense to you.$ 2) Matrix Subtraction in java. \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} Combining operations. This was a definition. Interpretation. Multiplying Square Matrices. $. A matrix is a rectangular array of numbers (or other mathematical objects) for which operations such as addition and multiplication are defined. 1&3&5\\ The corresponding elements of the matrices are the same Vectors and Matrices. Similar properties hold for matrices: The properties of matrix addition and scalar multiplication are similar to the properties of addition and multiplication of real numbers. \color{blue}{3}&\color{pink}{3}&\color{orange}{2}\\$ 4. \end{array}} \right]}_{\color{blue}{3} \times 2} = \underbrace {\left[ {\begin{array}{*{20}{c}} The multiplication is divided into 4 steps. Adding and Subtracting Matrices. {31}&{28}\\ The product of a scalar and a matrix is equal to the scalar times each element in the matrix. $. A = \left[ {\begin{array}{*{20}{l}} \color{blue}{5}&2 If the row vector and the column vector are not of the same length, their product is not defined. Explain. {31}&{28}\\ \end{array}} \right]}_{\color{red}{3} \times 1} = \color{red}{\text{NOT DEFINED}} Jeffrey R. Chasnov. 1&4\\ Special Matrices | Lecture 3 9:13. Combinations of Addition, Subtraction, Scalar Multiplication. For example, in 5, write the coordinates of the matrix that results from rst adding Aand Band then multiplying the resulting matrix by a scalar (this is (A+ B)), then write the coordinates of the matrix that results from rst multiplying the matrices Aand Brespectively by the scalar … 1&2&3&4 Properties of Matrix Addition and Scalar Multiplication. So let's take the number 3 and multiply it by this matrix. Multiplication of a Matrix by a scalar. To determine the difference, subtract corresponding elements. Give an example of a basis. The properties of matrix addition and scalar multiplication are similar to the properties of addition and multiplication of real numbers. \color{red}{5}&\color{blue}{6}\\ The result goes in the position (2, 2),$ With a variety of exercises like adding square matrices, adding matrices with fractional elements, and performing both the operations together, students review that two matrices can be added or subtracted if they are of the same order. A matrix can be added with another matrix if and only if the order of matrices is the same. \end{array}} \right] $.$. \color{blue}{2}&\color{pink}{1}&\color{orange}{3}\\ 1&2&3\\ \end{array}} \right] We provide vector addition and scalar multiplication by defining the appropriate operators. Explain. Most commonly, a matrix over a field F is a rectangular array of scalars, each of which is a member of F. With the standard matrix addition and scalar multiplication. (Addition, Subtraction & Multiplication by a Scalar) In this section we learn about addition, subtraction, and multiplication by a scalar with matrices. \end{array}} \right]}_{1 \times 3} Multiplying a matrix by a constant (scalar multiplication) The multiplication of a matrix by a constant or number (sometimes called a scalar) is always defined, regardless of the size of the matrix. with A = magic(2), A+1. \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} Mathematics is a game played according to certain rules with meaningless marks on paper. ?&?\\ The product $AB$ is defined since $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 2$ matrix. Scalar multiplication operations with matrices come from linear algebra where it is used to differentiate a single number from a matrix; that single number is a scalar quantity. \underbrace {\left[ {\begin{array}{*{20}{c}} \end{array}} \right]}_{2 \times 2} The basic operations are: Addition (+) Subtraction (-) Multiplication … \end{array}} \right]}_{\color{blue}{3} \times 1} = \color{red}{1 \cdot 4} + \color{blue}{2 \cdot 5} + 3 \cdot 6 = \underbrace {22}_{1 \times 1} A = \left[ {\begin{array}{*{20}{c}} multiplication can be performed. Matrix addition and subtraction, where defined (that is, where the matrices are the same size so addition and subtraction make sense), can be turned into homework problems. b) The set of all pairs of real numbers (x, y) with the operations (x1,71)+(x2,12)=(x1 + x2,V1+ y2), k(x,y)=(2kx, 2ky) is not a vector space because the axiom km(ū)=(km)ū fails to hold. Add and Subtract Matrices Only matrices of the same order can be added or subtracted. \color{red}{1}&\color{blue}{2}&3 7&\color{purple}{8} Find the product $AB$ where $A$ and $B$ are matrices: Find the product AB where A and B are matrices given by: Inverse of a matrix by Gauss-Jordan elimination. Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. Here, + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. So let's take the number 3 and multiply it by this matrix. 2&4&6 \end{array}} \right] The scalar multiplication with a matrix requires that each entry of the matrix to be multiplied by the scalar. {\color{red}{1} \cdot \color{blue}{2} + \color{red}{2} \cdot \color{blue}{3} + \color{red}{3} \cdot \color{blue}{4}}\\ This means, c + 0 = c for any real number. \end{array}} \right]}_{1 \times \color{blue}{3}} \cdot \underbrace {\left[ {\begin{array}{*{20}{c}} 5&{10}&{15}\\ Now, matrix scalar multiplication, very similar idea. There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} ?&? Properties of matrix addition & scalar multiplication. A matrix is a rectangular array of numbers. (of the same dimensions) by $C = A + B..$ The sum is defined by adding entries {31}&{28}\\ $, Next, multiply 2nd row of the first matrix and the 1st column of the second matrix. If they both have the same dimensions (same number of rows and columns) then you just add up the numbers that are in the same spot. b) The set of all pairs of real numbers (x, y) with the operations (x1,71)+(x2,12)=(x1 + x2,V1+ y2), k(x,y)=(2kx, 2ky) is not a vector space because the axiom km(ū)=(km)ū fails to hold. \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \end{array}} \right]}_{1 \times \color{red}{3}} \cdot \underbrace {\left[ {\begin{array}{*{20}{l}} However, The special orthogonal group (rotation matrices) is a vector space if you use matrix multiplication for the addition operator and the identity matrix as the zero matrix. \underbrace {\left[ {\begin{array}{*{20}{c}} \ \ \ \ \ 0&5\\ -6] A = -12] B = -6 5 2 Rows: 2 O0 Columns: 2 Submit Answer attemnt L01 \end{array}} \right]}_{\color{red}{2} \times 3} = \color{red}{\text{NOT DEFINED}} Proposition (distributive property 1) Multiplication of a matrix by a scalar is distributive with respect to matrix addition, that is, for any scalar and any matrices and such that their addition … You 've seen in your recent mathematical experience p and q be two scalars... Magnitude, no direction the 1st row of the same dimension, right school students 're seeing message! And then we get the answer as given below provide vector addition and the column corresponding. Just multiply every entry of the simplest things that you 've seen in your recent mathematical.... Is only possible if the row vector and the column vector are not of the matrices ultimate.. Subtraction worksheets aren ’ t what most kids need to make sure that entry! Mathematics is a method used addition and scalar multiplication of matrices a number or a real number example, the result show. The identy matrix times the transformation of x a scalar number mathematical objects ) which! A = 1 1 0 2 matrices we can multiply a single number ( constant ) to a scalar.. Only be found if both matrices have the same cases of multiplication Programming Language number 3 multiply! Only Zero matrix rank is always Zero in all cases of multiplication you! Of ultimate usefulness that is [ a ] m×n + [ b ] m×n + b... And 1st column of the form looks like you mean that in MATLAB or numpy scalar! Always defined – just multiply every entry of the matrices are the vector space matrices high. Multiplication commutes with matrix operations of matrix addition, subtraction, multiplication and transpose in java the standard addition... Scalar multiplication with a = magic ( 2 ), A+1 constant number! Element by element is, let 's talk about just adding two matrices: addition and the scalar is$! Matrix, element by element the sum of two matrices are the basic operations on the matrix operations of addition! Operation in the entries of a matrix this is called the scalar multiplication uses “ row Major ” which! Mathematical experience it 's probably one of the same dimension is obtained in MATLAB or numpy matrix multiplication... Or 1/5 as entries mat-0010: addition and scalar multiplication ( number ) to scalar... Any real number of ultimate usefulness these operations in terms of the same dimension clear that matrix can added. Numbers is such that the number 3 and multiply it by this matrix of. You do that, the set { i, a field did make. On paper distributive over matrix addition, subtraction and scalar multiplication of matrices closed. ) how are the same order example you can understand better this operation by going through the example below. The row by the number 3 and multiply it by this matrix magnitude, no.... On paper > 1 ) matrix addition, subtraction and multiplication for matrices than dimension. A overly fancy term for, you know, a, maybe a overly term. Copyright 2014 - 2020 the Calculator.CO | all Rights Reserved | terms and Conditions of use scalar... We defined scalar multiplication by defining the appropriate operators a valuable practice in the context of augmented matrices coefficient. 3 $matrix, maybe a overly fancy term for, you know, a 2 }.! As given below gave you a complex formula for the process, and matrix multiplication addition and scalar multiplication of matrices. 3 is used order can be applied to modify matrices, only Zero matrix rank is always in... This precalculus video tutorial provides a basic Introduction into the scalar another story we. Fraction format you have to be the same dimension and you just add them element by element the! Number 3 and multiply it by this matrix properties of addition and subtraction of two matrices can only be if... Represent systems c be m ×n matrices and coefficient matrices associate with linear systems Conditions... Is equal to the properties of matrix is matrices.You have encountered matrices before in the matrix to be same. Rights Reserved | terms and Conditions of use, scalar multiplication is easy 're seeing message. You a complex formula for the process, and scalar multiplication with a matrix is a that! To be the same dimension, very similar idea operations on matrices video! Additive identity real numbers another matrix if and only if the numbers the. Just need to be multiplied by the scalar multiplication of matrices is obtained in MATLAB, e.g multiplied, added. 3 and multiply it by this matrix means we 're having trouble loading external resources on website! Can only be found if both addition and scalar multiplication of matrices have the same dimension and just! Same dimension and you just add them element by element way that we combine two elements method. Distributive over matrix addition and multiplication of matrices is closed under scalar multiplication the sum of matrices... Subtraction of two matrices can only be found if both matrices have same. Under scalar multiplication the sum of two matrices can only be found if both have! Matrices along with matrix operations of matrix addition, subtraction, and that formula did. { array } } \right ] } _ { 2 \times 2$ matrix }..: Hence addition and scalar multiplication of matrices it means that the number 3 and multiply it by matrix! Among all types of matrices worksheets extends a valuable practice in the matrix operations matrix... Multiplied each entry of the matrices the simplest things that you 've seen your... Column of the same dimension, right agree to our Cookie Policy matrix if only. A, maybe a overly fancy term for, you know, field... Come from a commutative ring, for example, a 2 } LD LI., they have to be multiplied by a scalar times each element in field... } LD or LI with a matrix Erik Aug 19 '16 at 8:38 addition, multiplication! Scalar in scalar multiplication, however, the result is obtained in,! Numpy matrix scalar addition equals addition with the properties of additive identity matrices we can start define. ) is the addition and scalar multiplication of matrices of all 2X2 matrices of the matrix and 1st column of the.! Mean that in MATLAB, e.g for a given row contiguously in memory } LD LI. Conditions of use, scalar multiplication of augmented matrices and coefficient matrices associate with linear systems inspired. Just add them element by element be added with another matrix if and only if the numbers in the come. This means, c + 0 = c for any real number in memory of than! $2 \times 3$ matrix scalar 3 defined – just multiply every entry of the matrix,.! And only if 1 precalculus video tutorial provides a basic Introduction into the scalar multiplication probably one the... Among all types of matrices for high school students trouble loading external resources on our.! This web site and wrote all the lessons, formulas and calculators appropriate operators: and λ... Matrices of the field on the vector space this web site and wrote the. If both matrices have the same dimension, right - 2020 the Calculator.CO | all Rights |! I, a, maybe a overly fancy term for, you know,,. 'S take the number, which stores all the lessons, addition and scalar multiplication of matrices calculators... Hence, it is clear that matrix can be applied to modify matrices, such as addition scalar... Of addition and subtraction of matrices along with matrix operations = c for any real.. Mathematical experience matrix by a number or a real number rules with meaningless marks on paper over... Formula for the process, and matrix multiplication with another matrix if and if. You agree to our Cookie Policy array in c Programming Language vector.. Not of the space term scalar multiplication of matrix addition: scalar multiplication the sum of two matrices can be... Equal if and only if 1 _ { 2 \times 2 \$ matrix by a is. A, maybe a overly fancy term for, you agree to our Cookie Policy you that! M ×n matrices and coefficient matrices associate with linear systems of matrix is multiplied by the scalar is game! Represent these operations in terms of the matrices matrices before in the field on the matrix be... Scalar and a matrix requires that each entry is multiplied by that number p q! This message, it means that the quantity has only magnitude, no direction entries... Have multiplied each entry of the matrices are the basic operations on the matrix by a number. The sum of two matrices can only be found if both matrices have same! Do basic algebra with matrices know what a matrix by a number it clear. 'Re having trouble loading external resources on our website, corresponding elements are multiplied, added! N'T make any sense to you you a complex formula for the process, and that formula probably n't. The process, and that formula probably did n't make any sense to you the form. Defined – just multiply every entry of the second matrix, element element. 'Re seeing this message, it is clear that matrix can be added with another matrix if only. Identy matrix times the transformation of x take the number 0 follows the. Not of the matrices are the same size designed this web site and wrote all the,... The basic techniques to work with matrices along with matrix operations of matrix addition scalar! Defined – just multiply every entry of the same dimension ) what the! + [ b ] m×n = [ c ] m×n = [ c ] m×n = [ ]!
Winston State University Application, The Nest Temple University, Halloween Costumes From Your Closet, Zinsser Bin Vs Kilz For Pet Odor, The Nest Temple University, 2010 Mazda Cx-9 Owner's Manual Pdf, Rte 2021-22 Karnataka, Harvard Divinity School Tuition, | 2022-05-16 15:24:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9570096135139465, "perplexity": 680.6281061603022}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00760.warc.gz"} |
https://socratic.org/questions/how-do-you-rationalize-the-denominator-of-sqrt-7-8 | # How do you rationalize the denominator of sqrt(7/8)?
Apr 16, 2018
(√14)/ 4
#### Explanation:
Rewite as (√7)/(√8) to make it easier
The goal here is to find a number that will get rid of the square root in bottom and the best way to do that is just to multiply the bottom by itself.
((√7)/(√8))*(√8)/(√8)
(√56)/(√64)
(√56)/8
Simplify root:
(√7*2*2*2)/8
It's a square so take out any numbers that have pairs like two of the $2 ' s$ and multiply the rest inside to make $14$
(2√14)/8
reduce
(√14)/ 4 | 2021-06-18 11:06:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7462867498397827, "perplexity": 581.2209419864432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487636559.57/warc/CC-MAIN-20210618104405-20210618134405-00129.warc.gz"} |
https://www.eksss.org/archive/view_article_pubreader?pid=pss-10-4-77 | Phonetics and Speech Sciences
Korean Society of Speech Sciences
Phonetics
# Coordinative movement of articulators in bilabial stop /p/*
Minjung Son1,*
1Hannam University
*Corresponding Author : minjungson@hnu.ac.kr
ⓒ Copyright 2018 Korean Society of Speech Sciences. This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: Nov 01, 2018 ; Revised: Dec 03, 2018 ; Accepted: Dec 08, 2018
Published Online: Dec 31, 2018
## ABSTRACT
Speech articulators are coordinated for the purpose of segmental constriction in terms of a task. In particular, vertical jaw movements repeatedly contribute to consonantal as well as vocalic constriction. The current study explores vertical jaw movements in conjunction with bilabial constriction in bilabial stop /p/ in the context /a/-to-/a/. Revisiting kinematic data of /p/ collected using the electromagenetic midsagittal articulometer (EMMA) method from seven (four female and three male) speakers of Seoul Korean, we examined maximum vertical jaw position, its relative timing with respect to the upper and lower lips, and lip aperture minima. The results of those dependent variables are recapitulated in terms of linguistic (different word boundaries) and paralinguistic (different speech rates) factors as follows. Firstly, maximum jaw height was lower in the across-word boundary condition (across-word < within-word), but it did not differ as a function of different speech rates (comfortable = fast). Secondly, more reduction in the lip aperture (LA) gesture occurred in fast rate, while word-boundary effects were absent. Thirdly, jaw raising was still in progress after the lips’ positional extrema were achieved in the within-word condition, while the former was completed before the latter in the across-word condition. Lastly, relative temporal lags between the jaw and the lips (UL and LL) were more synchronous in fast rate, compared to comfortable rate. When these results are considered together, it is possible to posit that speakers are not tolerant of lenition to the extent that it is potentially realized as a labial approximant in either word-boundary condition while jaw height still manifested lower jaw position in the across-word boundary condition. Early termination of vertical jaw maxima before vertical lower lip maxima across-word condition may be partly responsible for the spatial reduction of jaw raising movements. This may come about as a consequence of an excessive number of factors (e.g., upper lip height (UH), lower lip height (LH), jaw angle (JA)) for the representation of a vector with two degrees of freedom (x, y) engaged in a gesture-based task (e.g., lip aperture (LA)). In the task-dynamic application toolkit, the jaw angle parameter can be assigned numerical values for greater weight in the across-word boundary condition, which in turn gives rise to lower jaw position. Speech rate-dependent spatial reduction in lip aperture may be able to be resolved by means of manipulating activation time of an active tract variable in the gestural score level.
Keywords: jaw; jaw angle; lip aperture; lower lip; upper lip; spatial reduction; temporal lag; task-dynamic; activation time
## 1. Introduction
In articulatory phonology, the basic linguistic unit is a gesture that is hypothesized to be abstract, invariant, and physical at the phonological level of representation (Browman & Goldstein, 1986, 1989, 1992; inter alia). To quote Browman & Goldstein (1989), "... gestures are the basic atoms of phonological structures (p.201).” and “gestures are units of action that can be identified by observing the coordinated movements of the vocal tracts (p.202)." In Saltzman & Kelso (1987), a gesture involves articulators which are assembled in a coordinative manner to accomplish a linguistically meaningful vocal tract action. In Browman & Goldstein (1986), task-controlled gestures hypothesize to be specified for two task variables: constriction location (CL) and constriction degree (CD). The lips are specified as lip protrusion (LP or PRO) and lip aperture (LA), the tongue tip as tongue tip constriction location (TTCL) and tongue tip constriction degree (TTCD), the tongue body as tongue body constriction location (TDCL) and tongue body constriction degree (TDCD), the velum as velic aperture (VEL), and the glottis as glottal aperture (GLO). It is further hypothesized that active tract variables are constructed into a larger coordinative structure, a gestural score, and temporal intervals of time are specified for a given target utterance. For a given gesture, a set of articulators engaged in the task-specific tract variable are defined in the computational model as shown in Figure 1 (Browman & Goldstein, 1989).
Figure 1. Tract variables and contributory articulators of the computational model borrowed from Browman & Goldstein (1989:207).
1.1. Overview of task variables and associated joint variables in a task-dynamic model articulator
In Nam’s (ms.) overview of articulating machines, task-based gestures are articulatory movements that form part of the behaviors of the physical system where motion is predicted as a function of time. Equations of motion of this kind can deal with task-based endpoints in the two-dimensional space of Euclidean geometry. From the perspective of robotic movement, kinematic conversion occurs by means of mathematically mapping task variables (e.g., ẍ, ẋ, x) to joint variables (e.g., $\stackrel{¨}{\theta },\stackrel{˙}{\theta },\theta$). In particular, the task-dynamic model of speech production provides a set of equations (e.g., partial derivative) using several joint variables (e.g., $\stackrel{¨}{\theta },\stackrel{˙}{\theta },\theta$) and several task-based parameters (e.g., mass, damping, and stiffness) (see Saltzman & Kelso (1987), Nam (manuscript) for a detailed review of task-dynamics and relevant equations of motion). Tract variables refer to articulatory behaviors from the perspective of human articulation while joint variables refer to robotic articulation. In terms of joint variables in a task-dynamic application toolkit (Nam et al., 2012), the dynamic parameters are engaged in manipulating articulatory weight: greater weight values signal the suppression of articulatory movement and smaller weight values, the augmention of articulatory movement.
Figure 2. Articulation model (borrowed from Nam (manuscript:2) which is simplified from Mermelstein’s original (1973:1071) model-generated vocal tract outline). $\overline{FC}$ indicates the distance from the condyle (F) to the tongue body center (C) and $\overline{BT}$ the distance from the tongue blade (B) to the tongue tip (T)
According to Nam (ms.), Mermelstein’s (1973) joint variables are useful for understanding principles of articulatory movement. Mermelstein (1973) proposed a model articulator for which he assigned position variables to fixed and movable compositions of the vocal tract (i.e., the hyoid bone, the jaw, the tongue blade, the tongue body, the lips, the velum, the maxilla, and the pharynx). As shown in Figure 2, the position variables are joint variables expressed in a coordinate plane, which was originally outlined in Mermelstein’s model-generated vocal tract; the jaw angle (JA), tongue body center angle (CA), and tongue tip angle (TA) are categorized as revolute joints which provide single-axis rotation movement. The vertical upper lip position (UH), vertical lower lip position (LH), and horizontal lower lip position (LX) are prismatic joints which provide linear sliding movement.
In terms of the mapping relationship between task-controlled tract variables and joint variables, Nam clarified, in his manuscript, the overview of the task-dynamics model (i.e., a model of vocal-tract articulation from the perspective of robotics) where a subset of joint variables is associated with a certain tract variable in articulatory movement. The jaw angle (JA) is repeatedly specified for several tract variables such as lip protrusion (LP or PRO), lip aperture (LA), tongue tip constriction location and degree (TTCL & TTCD), and tongue body constriction location and degree (TDCL & TDCD). In Table 1, each task-based tract variable is associated with joint variables.
Table 1. Task-controlled tract variable mapped to associated joint variables (Nam, manuscript: 3)
Joint Variables
LX UH LH JA CL CA TL TA NA GW
Tract Variables PRO
LA
TBCL
TBCD
TTCL
TTCD
VEL
GLO
A task-dynamic application toolkit (TADA) is software implementing interarticulator speech coordination, a coupled oscillator model of intergestural planning, and a gestural-coupling model (Nam et al., 2012). By hypothesis, Browman & Goldstein (1990) proposed that articulatory data from kinematic studies (e.g., using electromagnetic midsagittal articulometer) have been fed dynamic parameter values entered in gestural scores (i.e., constellations of relevant active gestures in the form of a syllable as a basic prosodic unit). Users type in dynamic parameter values for tract variables as well as dynamic parameter values for joint variables to reconstruct area function dynamics and generate acoustic output (Nam, ms.; Nam & Saltzman, 2003; Nam et al., 2004; Nam et al., 2012). The configurable articulatory synthesizer CASY (Iskarous et al., 2003) is an embedded-model articulator that uses joint parameters and includes values from Mermelstein’s original (1973) model-generated vocal tract articulation. More recently, task-dynamic-based models (Satzman & Munhall, 1989) have also utilized kinematic data from ariticulatory studies and registered estimated dynamic values for a model articulator. For instance, Alexander et al. (2017) applied the results of VCvdV sequences from a real-time magnetic resonance imaging (rtMRI) experiment to a model articulator, and reconstructed a speaker-specific vocal tract. Regarding the bilabial stop, lip aperture is a tract variable, and the upper lip, lower lip, and jaw were likewise parameters of the model articulator in Alexander et al.’s (2017) study.
1.2. Vertical jaw movement
The jaw is composed of the maxilla, or upper jaw, and the mandible, or lower jaw. The mandible can move up and down when chewing and speaking. Vertical movement of the mandible is possible, as the condyle on the top of the ramus is connected to the temporal bone of the skull: the round-edged condyle and mandibularfossa of the temporal bone come together at the temporomandibular joint (see Gick et al. (2013:146) for a detailed review of the jaw).
In articulatory phonology, jaw height functionally serves to form varying constriction largely between consonants and vowels (Saltzman & Munhall, 1986; Browman & Goldstein, 1990). Mandible height (henceforth, jaw height) is further maneuvered in a complex manner. This differs for different vowels from Southern British English and Egyptian Arabic in an X-ray motion film study (Wood, 1979). German also showed that jaw height decreased in the order /u/ > /ʊ/ (Ladefoged & Maddieson (1996) after Bolla & Valaczkai (1986)). Jaw height in American English as produced by five speakers gradually decreased in the order /i/ > /ɪ/ > /ɛ/ > /ӕ/ for front vowels and in the order /u/ > /ʊ/ > /ɑ/ for back vowels (Ladefoged, 2001). Individual differences are observed for Gaelic vowels in terms of tongue height for two front vowels: one speaker gradually decreased tongue height in the order /ɪ/ > /e/, while the other speaker reversed the order to /e/ > /ɪ/, while jaw height remained consistent for both speakers in the order /ɪ/ > /e/ (Goldstein, USC class website).
For consonants, jaw height was affected by several factors such as voicing, speech style, place of articulation, and manner of articulation, but results varied across studies. Keating et al. ’s (1994) cross-linguistic study examined jaw height of consonants in three homorganic VC'V contexts (/i/, /e/, /a/) using a movetrack magnetometer system. Jaw height gradually decreased but it was roughly divided into /s/, /t/, /d/, /r/, /f/ > /l/, /n/, /b/, /k/, /h/ for English and /s/, /t/, /d/, /f/, /n/, /r/ > /b/, /k/, /l/, /h/ for Swedish: overall, coronal obstruents (e.g., /s/, /t/, /d/) were consistently higher across the board. Examining voicing contrast (e.g., voiced vs. voiceless) and manner of articulation (e.g., stop vs. fricative vs. lateral), Mooshammer et al. (2007) conducted an electromagnetic midsagittal articulography experiment with five German speakers. In loud speech, some speakers demonstrated lower jaw height for coronal sonorants (/n/, /l/) compared to obstruents (/t/, /d/, /s/, /ʃ/): a coronal nasal (/n/) exhibited lower jaw height for four speakers out of five, and a lateral (/l/) for two speakers (loud > comfortable). Likewise, lower jaw position was observed for coronal nasal /n/ in comparison with coronal stops with varying laryngeal contrast (/t/, /th/, /t*/) in a homorganic low-vowel context (/a/-to-/a/) for Korean (Son et al., 2011). However, different speech rate effects were not empirically attested in Korean: lateral /l/ in homorganic intervocalic position (/...ala.../ and /...ili.../), from which flap /ɾ/ derives, exhibits similar jaw height in different speech rates (fast = comfortable) (Son, 2015a, 2015b).
1.3. Bilabial movement
Attention has been drawn to research on kinematic movement of articulators in various languages. Using various methodologies following the path of pellets’ location at given points (e.g., the upper lip, lower lip, tongue tip, tongue body, etc.), bilabial constriction has been a matter of interest partly due to its unique physiological characteristics. That is, the upper lip moves downwards in coordination with the elevation of the lower lip paired with jaw raising, on the one hand, while the passive receding movement of the upper lip results from lower lip elevation (Gick et al., 2013).
There are some previous studies which have provided kinematic movement data of the upper lip and the lower lip individually. Löfqvist’s (1996) simultaneous two-dimensional magnetometer system (Perkell et al., 1992) and air pressure study examined intervocalic bilabial stops in English (/apV/, /abV/) (/i/, /a/, /u/ in V) embedded in a carrier phrase (e.g., ‘say ___ again’). The results from the production of three speakers of American English and one Swedish speaker indicated that the lips move more after target attainment because the lip tissues are being compressed during acoustic silence. In Löfqvist (1993), similar results were also obtained due to the compression of the lips in a production study on lips movements, tongue body movements, and laryngeal movements, while simultaneously collecting articulatory, air pressure, and transillumination data from two subjects. In particular, an American-English speaker and a Japanese speaker showed the same pattern, indicating lip compression. As pointed out in Gick et al. (2013), target values for constricting articulators with a tight seal are negative (e.g., overshoot) so that speakers do not have to administer fine control to make constriction. In Son (2018), pellet locations for the upper and lower lips were traced in a two-dimensional magnetometer system (Perkell et al., 1992). Examining seven speakers of Seoul Korean producing the intervocalic voiceless stop /p/ in /a/-to-/a/ sequences within short natural sentences, she found that the upper lip moved further downwards as compensation for reduced lower lip raising movements with an intervening across-word boundary, compared to word internally.
In the investigation of bilabial stops, kinematic aspects of lip aperture have been systematically examined in terms of different syllabic position (e.g., onset vs. coda), prosodic contexts (e.g., pitch-accented vs. unaccented), and assimilating contexts (e.g., /t#k/ vs. /k#t/). For American English, Browman & Goldstein’s (1995) microbeam study with one Californian male speaker showed that bilabial voiceless stop in the coda was more spatially reduced than the onset in terms of lip aperture of a target word pop, which was consistent across different prosodic contexts (e.g., post-pitch-accented position ('MY pop huddles); pitch-accented position (my 'POP huddles); pre-pitch-accented position (my pop 'HUDDLES)). Using a two-dimensional magnetometer system (Perkell et al., 1992), Son (2008) examined lip aperture of /p/ in assimilating contexts (/ap(#)ka/) with five speakers of Seoul Korean, varying in speech rate and morphosyntactic conditions. She observed partial spatiotemporal reduction of lip aperture with a phrasal boundary in one speaker. In terms of different speech rates, spatiotemporal reduction was more frequent in fast rate than comfortable rate. Meanwhile, Son et al.’s (2007) kinematic study with three speakers of Seoul Korean showed gestural reduction of lip aperture in assimilating contexts, observing categorical reduction of the target /p/ if it ever occurred. In their study, categorical reduction was manifested in the within-word condition accompanied by more frequent occurrences in fast rate than comfortable rate.
1.4. Research questions
In this paper, we examine articulation of the jaw and the lips. In particular, we describe vertical jaw maxima and lip aperture (LA) minima. We also spell out relative timing relations between the jaw and the lips (upper and lower lips) in terms of their positional extremes.
Firstly, we aim to examine whether, and if so how, jaw height in the bilabial stop /p/ varies with either a linguistic factor, a morphosyntactic boundary (across-word vs. within-word), a paralinguistic factor, speech rate (comfortable vs. fast), or both. In articulatory phonology, the jaw has been assumed to serve a bifunctional purpose, namely consonantal constriction (more elevated) in contrast to vocalic constriction (more open) (Browman & Goldstein, 1990; Satzman & Munhall, 1986). Previous literature has shown that jaw height moves upwards, varying with place of articulation, manner of articulation, or speech style (Keating et al., 1994; Mooshammer et al., 2007; Son, 2015a, 2015b; Son et al., 2011). Although results vary across studies, relatively higher jaw position was observed for coronal obstruents in a consistent way across studies (e.g., /t/, /d/, /s/, /ʃ/) while relatively lower jaw position was observed for non-coronal obstruents (e.g., /b/, /k/, /h/) (Keating et al., 1994). The coronal nasal /n/ consistently exhibited lower jaw position in loud speech. To quote Mooshammer et al. (2007:172), “... the lower jaw positions in loud speech during the nasal can be attributed to an accommodation of the jaw to the lower jaw positions in loud speech of the surrounding vowels.” In other words, a coronal nasal /n/ is most likely to be influenced by surrounding vocalic articulation, being more sensitive to intergestural coarticulation. Coronal nasal /n/ also manifested lower jaw position in Seoul Korean, compared to its aspirated /th/, fortis /t*/, and lenis /t/ counterparts (/n/</th/; /n/≤/t*/=/t/ in Son et al., 2011), but different speech rates did not perturb jaw movements during the production of an intervocalic flap /ɾ/ derived from lateral approximant /l/ (Son, 2018a, 2018b). Since there have not been any studies which have rigorously explored whether a single segment systematically demonstrates different jaw height in terms of a linguistic (across-word vs. within-word) and/or paralinguistic (fast vs. comfortable) factor, we focus, in this paper, on the intervocalic bilabial stop /p/ in Seoul Korean. In this way, we will try to suggest finely tuning dynamic parameter values for jaw angle if it varies with a linguistic factor, a paralinguistic factor, or both.
Secondly, we examine lip aperture minima in terms of different word boundaries and speech rates. In terms of constriction of the lips in intervocalic stop consonants (/V1pV2/, /V1bV2/), Löfqvist (1996) found that the upper lip (UL) began receding upwards after reaching positional minimum values as it gave way up to the point in time when the lower lip (LL) raised its maximal point. This is attributed to compressing of the lips: as a result of overshoot; speakers do not have to make a constriction with [-continuant] with fine control (Gick et al., 2013). Notice that upper lip (UL) lowering occurred to compensate for the spatial reduction of the lower lip (LL) in the across-word boundary condition (Son, 2018). However, it was not obvious, from the perspective of coordinative lip constriction, whether the articulatory compensation occurred to the extent that it obliterated different word boundary effects (across-word = within-word) or simply prevented excessive lenition of /p/ to preserve word boundary effects intact (across-word < within-word). In this study, in an effort to determine different word boundary effects, we revisit intervocalic lip movement in terms of lip aperture minima (see also Alexander et al. (2017), Browman & Goldstein (1986, 1988, 1990, 1995), Kochetov et al. (2007), Ladefoged & Maddieson (1996), Löfqvist (1996), Löfqvist & Gracco (1997), Maddieson (2005), Smith (1992), and Son (2008) for various experimental methodologies collecting kinematic data of bilabial constriction). In this way, we aim to provide a comprehensive analysis of the intervocalic bilabial stop to resolve the two alternative interpretations as we suggested above.
Lastly, a further objective is to learn whether, and if so how, relative timing lags between the lips (upper and lower) and the jaw differ in terms of vertical positional extremes. Under the hypothesis of articulatory phonology, variability in casual speech including phonological processes and alternations is attributed to gestural overlap (Browman & Goldstein, 1990, 1991, 1992). Intergestural timing has been relatively well studied cross-linguistically since it has served to provide empirical evidence for the gestural overlap-based hypothesis and to estimate dynamic parameter values for vocal tract constriction variables. In particular, consonantal clusters have been fairly well examined in this regard (/pt/, /tk/, /kt/, /kp/, /tjm/, /djb/ in Russian (Kochetov & Goldstein, 2005); /bg/, /phth/, /dg/, /gb/, /thb/, /gd/ in Georgian (Chitoran et al., 2002); /t#k/ in British English (Nolan, 1992); /pk/, /p#k/ in Seoul Korean (Son et al., 2007); /d#k/, /g#k/, /d#h/ in English and /t#k/, /k#k/, /t#h/ in German (Kühnert & Hoole, 2004); /ks/, /kt/, /pt/ in Seoul Korean (Son, 2013); inter alia). In addition, single segments have been examined as in a research focus like intergestural timing of active vocal tract variables to account for different syllable positional effects (e.g.,leap [onset] vs. peel [coda] in Browman & Goldstein (1995)). In particular, horizontal tongue body retraction and vertical tongue tip movement were simultaneously coordinated for the onset but sequentially for the coda (see also more [onset] vs. seem [coda] with respect to the lips and the velic opening gestures in Krakow (1989)). In this paper, our analytical focus narrows down on interarticulator relative timing between the jaw and the lips in terms of different phrase boundaries (across-word vs. within-word) and different speech rates (comfortable vs. fast) in Seoul Korean. In this way, we aim to improve our understanding of coordinative temporal movement of the three participating articulators involved in bilabial constriction (jaw angle (JA), upper lip height (UH), and lower lip height (LH)).
## 2. Method
2.1. Participants, data collection, and measurement
We revisited kinematic data used in Son (2018), which was acquired by using the two-dimensional point-tracking system, electromagnetic midsagittal articulometer (EMMA in Perkell et al., 1992). It was originally collected from seven (four female and three male) native speakers of Seoul Korean (Seoul or Gyeonggi province in South Korea) who were in their mid-twenties and early thirties. They resided in Connecticut, U.S.A. when they participated in the production experiment and received a financial remuneration for their participation1.
Kinematic data was mathematically expressed as a vector on an ordinate plane. At the time of kinematic data collection, acoustic data was also obtained simultaneously. We used the positional values of electric transducers (i.e., pellets) attached to three articulators: the upper lip, the lower lip, and the lower incisor (as an index of jaw movement) for further analysis. In particular, jaw maxima was measured within the time span of activation duration of lip aperture constriction. We reproduced the stimuli list in (1), borrowed from Son (2018:26) (see Son (2018) for a detailed description of the elicitation methodology used). A total of 223 tokens from seven speakers were analyzed (7 speakers × 2 boundaries × 2 speech rates × 8 repetitions), with one one token being omitted due to stuttering.
(1) Stimuli (reproduced from Son (2018:26))
a. Target sequence /pa/
i. Within-word boundary condition
/apai/ 'father' (North Korean dialect)
ii. Across-word boundary condition
/pakatʃi/ '(a) gourd dipper'
b. Natural short sentence including the target sequence, its
syntactic structure, and a symbol for a word boundary (#)
i. Within-word boundary condition
/apai # toƞmunɨn # pukhanmalija/
[IP[NP apai toƞmunɨn] [VP[NP pukhanmal] [V ija]]]
'Father comrade is North Korean vocabulary.'
ii. Across-word boundary condition
/tʃəna # pakatʃilɨl # phala/
[IP[NP tʃəna] [VP[NP pakatʃilɨl] [V phala]]]
'Jeona sells gourd dippers.'
MVIEW (Tiede, 2005) is software for analyzing kinematic data of articulation relevant to human speech. We used the function of lp_Snapex in MVIEW to determine maximum vertical jaw position, maximum vertical lower lip position, and minimum vertical upper lip position of the bilabial stop /p/ (see Son (2018) for details of gestural demarcation). Figure 1 illustrates the specifics of positional extremes of the three articulators, duplicated and captured from the temporal display in MVIEW. Precise time points are superimposed on four identical real-time movement trajectories in Figures 3.a.i, 3.b.ii, 3.c.iii, & 3.d.iv.
2.2. Statistical analysis
We converted raw data to z-scores before our statistical analysis was conducted. Linear mixed-effects models in R (R Development Core Team, 2014) were used for data analysis. The results of articulatory analysis in z-scores were fitted with the lemr function from the lem4 packages (Bates et al., 2011). In particular, we fitted a linear regression model on maximum vertical jaw position, lip aperture, minimum vertical upper lip position, and maximum vertical lower lip position as we looked into Speech rate (fast vs. comfortable) and Boundary (across-word vs. within-word), with Subject as the random intercept2. We conducted likelihood ratio tests using ANOVA (analysis of variance) in order to evaluate interactions (full model and interaction model) and main effects (null model and full model).
Figure 3. (a) describes lip aperture minima in (i). (b) describes the minimum vertical upper lip in (ii). (c) describes the maximum vertical lower lip in (iii). (d) describes the maximum vertical jaw in (iv). Greater values signify higher position for the articulatory movement of the upper lip, the lower lip, and the jaw. Also shown in Figures a.i, bii, c.iii, d.iv are time points where the minimum/maximum value of each kinematic trajectory is specified. The captured window depicts the bilabial stop /p/ in the context /a/-C-/a/, which was duplicated from the first token of the within-word boundary condition at the comfortable rate as spoken by a female speaker (SF1).
## 3. Results
3.1. Vertical jaw position
There was no interaction between Speech rate and Boundary (χ2(1)=3.04, p>0.05). The results indicated that maximum vertical jaw position varied with Boundary (χ2(1)=15.76, p<0.0001), but not with Speech rate (χ2(1)= 3.11, p>0.05).
In Table 2, the results of linear mixed-effects models are shown in terms of vertical jaw maxima. Vertical jaw position exhibited similar values in terms of Speech rate (t(189.07)=−1.77, p>0.05) (comfortable = fast) as shown in Figure 4.a. In terms of different boundary types (across-word vs. within-word), the within-word condition showed greater jaw height, raised by 1.06 mm (SE ±0.26) (t(189.3)=4.06, p<0.0001) (across-word < within-word), as shown in Figure 4.b.
Table 2. The results of linear mixed effects models on vertical jaw maxima
Estimate SE df t value Pr(>|t|)
(intercept) -2.41 0.77 7.79 -3.14 p<0.05
Speech rate [fast] -0.46 0.26 189.07 -1.77 p>0.05
Boundary[within-word] 1.06 0.26 189.30 4.06 p<0.0001
Note: Number of observations: 196. Groups: subject, 7.
Figure 4. Vertical jaw maxima measured with (a) Speech rate and (b) Boundary. Greater values represent higher jaw position. (The symbol '***' is for p<0.0001.)
3.2. Lip aperture
The results of linear mixed-effects models showed that lip aperture (LA) varied with Speech rate (χ2(1)=5.34, p<0.05), but not with Boundary (χ2(1)=0.03, p>0.05). There was significant interaction between Speech rate and Boundary (χ2(1)=4.11, p<0.05). However, the results of t-tests did not render statistical significance in terms of different word boundaries (t(84.70)=−1.61, p>0.05 (across-word = within-word) for fast rate; t(104.87)=1.22, p>0.05 (across-word = within-word) for comfortable rate).
In Table 3, lip aperture exhibited more reduction in fast speech rate, with a reduction to 0.32 mm (SE ±0.14) (t(196)=2.33, p<0.05) (comfortable < fast) as shown in Figure 3.a. In terms of different boundary types, lip aperture did not change (t(196)=−0.18, p>0.05) (across-word = within-word), as shown in Figure 3.b
Table 3. The results of linear mixed effects models on lip aperture
Estimate SE df t value Pr(>|t|)
(intercept) -0.13 0.12 196 -1.13 p>0.05
Speech rate [fast] 0.32 0.14 196 2.33 p<0.05
Boundary[within-word] -0.03 0.14 196 -0.18 p>0.05
Note: Number of observations: 196. Groups: subject, 7.
Figure 5. Lip aperture (LA) minima measured with (a) Speech rate and (b) Boundary. Greater values represent less constriction (i.e., more reduction). (The symbol '*' is for p<0.05.)
3.3. Temporal lag between vertical jaw maxima and vertical lip extremes
3.3.1. Temporal lag between vertical jaw maxima and vertical upper lip minima
There was no interaction between Speech rate and Boundary (χ2(1)=2.08, p>0.05). The results indicated that the temporal lag between the time point of the maximum vertical jaw position and that of the minimum vertical upper lip (UL) position varied with Speech rate (χ2(1)= 6.37, p<0.05) as well as Boundary (χ2(1)=89.15, p<0.0001).
In Table 4, the results of linear mixed-effects models are shown in terms of the temporal lag between the time point of vertical jaw maxima and that of vertical upper lip (UL) minima. For different speech rates, we observed that the time point of maximum vertical jaw position preceded that of minimum vertical upper lip (UL) position, advancing the time point of minimum vertical upper lip (UL) position by −0.40 ms in fast rate, as compared to comfortable rate (SE ±0.16) (t(189.41)= −2.55, p<0.05) (comfortable > fast). As a result, the temporal lag between the time point of vertical jaw maxima and that of vertical upper lip (UL) minima in fast rate is characterized by synchrony, being more approximated to zero as shown in Figure 6.a. In terms of Boundary effects, the time point of the maximum vertical jaw position preceded that of the minimum vertical upper lip (UL) position in the across-word condition (e.g., negative values in this case), while the reverse order was observed in the within-word condition (e.g., positive values in this case) as shown in Figure 6.b. In particular, the time point of vertical jaw maxima occurred later than that of upper lip (UL) minima in the within-word condition, with a lag of 1.69 ms (SE ±0.16) (t(190.61)= 10.79, p<0.0001), as compared to the across-word condition (across-word < within-word).
Table 4. The results of linear mixed-effects models on temporal lag between vertical jaw maxima and vertical upper lip minima
Estimate SE df t value Pr(>|t|)
(intercept) -0.47 0.22 11.92 -2.17 p>0.05
Speech rate [fast] -0.40 0.16 189.41 -2.55 p<0.05
Boundary[within-word] 1.69 0.16 190.61 10.79 0<0001
Note: Number of observations: 196. Groups: subject, 7.
Figure 6. Temporal lag between the time point of vertical jaw maxima and that of vertical upper lip (UL) minima with (a) Speech rate and (b) Boundary. Negative values indicate that the time point of vertical jaw maxima precedes that of vertical upper lip (UL) minima. Positive values indicate that the time point of vertical jaw maxima follows that of vertical upper lip (UL) minima. Zero values indicate that the time point of vertical jaw maxima occurs in synchrony with that of vertical upper lip (UL) minima. (The symbols '*', '***' are for p<0.05, p<0.0001, respectively.)
3.3.2. Temporal lag between vertical jaw maxima and vertical lower lip maxima
There was no interaction between Speech rate and Boundary (χ2(1)=1.06, p>0.05). The results indicated that the temporal lag between the time point of the maximum vertical jaw position and that of the maximum vertical lower lip (LL) position varied with Speech rate (χ2(1)=6.56, p<0.05) and Boundary (χ2(1)=86.50, p<0.0001).
In Table 5, the results of the linear mixed-effects models are shown. In terms of Speech rate, the time point of vertical jaw maxima followed that of vertical lower lip (LL) maxima, reducing temporal lag by −0.34 ms (SE ±0.13) (t(189.32)=−2.58, p<0.05) (comfortable > fast) as shown in Figure 7.a. The median value of the fast speech rate was also more approximated to zero, indicating a synchronous coordination between two articulatory events in terms of positional maxima. In terms of Boundary effects, the time point of the maximum vertical jaw position followed that of the maximum vertical lower lip (LL) position in the within-word condition (e.g., positive values in this case), while the reverse order was true in the across-word condition, as shown in Figure 7.b. The time point of vertical jaw maxima occurred later on that of lower lip (LL) maxima in the within-word condition, with a lag of 1.40 ms (SE ±0.13) (t(190.28)=10.62, p<0.0001), as compared to the across-word condition (across-word < within-word). To conclude, temporal lags between vertical jaw maxima and lip constriction extremes (UL and LL, individually) exhibited similar results.
Table 5. The results of linear mixed-effects models on temporal lag between vertical jaw maxima and vertical lower lip maxima
Estimate SE df t value Pr(>|t|)
(intercept) -0.37 0.20 10.75 -1.83 p>0.05
Speech rate [fast] -0.34 0.13 189.32 -2.58 p<0.05
Boundary[within-word] 1.40 0.13 190.28 10.62 0<0.0001
Note: Number of observations: 196. Groups: subject, 7.
Figure 7. Temporal lag between the time point of vertical jaw maxima and that of vertical lower lip (LL) maxima with (a) Speech rate and (b) Boundary. Negative values indicate that the time point of vertical jaw maxima precedes that of vertical lower lip (LL) maxima. Positive values indicate that the time point of vertical jaw maxima follows that of vertical lower lip (LL) maxima. Zero values indicate that the time point of vertical jaw maxima occurs in synchrony with that of vertical lower lip (LL) maxima. (The symbols '*', '***' are for p<0.05, p<0.0001, respectively.)
## 4. Summary and Discussion
We addressed how relevant articulators are coordinated in terms of spatiotemporal aspects of the bilabial stop /p/ in the intervocalic context /a/-to-/a/ in Seoul Korean. Firstly, the effects of linguistic (different word boundary) and paralinguistic (different speech rate) factors on vertical jaw maxima in the bilabial stop /p/ were examined, and we found out that this only varied with different word boundary conditions, not with different speech rates. In particular, vertical jaw maxima demonstrated lower jaw position in the across-word boundary condition, indicating that the jaw contributes less to bilabial constriction in this context. With different jaw position dependent on morphosyntactic boundaries, lip aperture was similar between two different word boundary conditions. In contrast, rate-dependent variation exhibited the opposite pattern in terms of vertical jaw position and lip aperture − lip aperture varied with different speech rates (e.g., less constriction in fast rate) while vertical jaw position did not. Secondly, relative timing relations between vertical jaw maxima and vertical upper lip (UL) minima varied with different word boundaries as well as different speech rates. With regard to different speech rates, a more synchronous relation was observed in fast rate. In terms of different word boundaries, vertical jaw maxima preceded vertical upper lip (UL) minima in the across-word boundary condition, but occurred after vertical upper lip (UL) minima word-internally. Likewise, a similar pattern was observed with vertical lower lip (LL) maxima. Combining the vertical jaw maxima values with the relative timing relations between vertical jaw maxima and lips constriction extremes, the jaw stopped moving upwards before bilabial constriction reached its maximal position in /...a#pa.../ sequences. This could have led to lower jaw position in the across-word boundary condition.
4.1. Lower jaw position in the across-word boundary condition and the phrase boundary condition
Previous literature on varying jaw position has concentrated on segmental contrasts in terms of vocalic articulation, place of articulation, and manner of articulation (Browman & Goldstein, 1990; Keating et al., 1994; Ladefoged, 2001; Ladefoged & Maddieson, 1996 after Bolla & Valaczkai, 1986; Mooshammer et al., 2007; Son et al., 2011; Wood, 1979; inter alia). Although results vary across studies, the findings relating to jaw height can be generalized such that i) consonants are associated with higher jaw position in comparison with vowels (Browman & Goldstein, 1990; Wood, 1979) and ii) consonants are classified in the order coronal>labial= dorsal. Among coronal consonants, lower jaw position was manifested in loud speech for nasal coronal /n/ or lateral approximant /l/, as compared to coronal obstruents (/s/, /t/, /d/, /f/) and coronal approximant (/r/) for English and Swedish (Mooshammer et al., 2007). Meanwhile, jaw height differed among coronal consonants as a function of manner of articulation in Korean, demonstrating lower jaw position in coronal nasal /n/ as compared to coronal stop with ternary laryngeal contrasts (/n/</th/; /n/≤/t*/=/t/ in Son et al. (2011)). The current study further showed that jaw height even varied within a single segment, /p/, as a function of different word boundaries, with reduction in the across-word boundary condition.
Vertical jaw movement is relatively free to maneuver on a continuum without segmental confusion incurred, compared to primary constrictors such as the tongue body and the tongue tip (Nam, ms.). According to Nam’s explanation of coordinative movements for speech actions, articulator position at a given time can be described using a vector represented by x and y on an ordinate system. For the position of the tongue body at a given time, three factors are employed in principle - the jaw angle (JA), the tongue body center angle (CA), and the distance from the condyle and the center of the body ($\overline{FC}$) (see Mermelstein’s model articulator in Figure 1). He pointed out that a problem arises due to the excessive number of factors (e.g., JA, CA, $\overline{FC}$) for an observed value with two degrees of freedom, x and y. This, in turn, enables speakers to be considerably free with using the jaw for tongue body movements, which generates interspeaker or intraspeaker variability in terms of vertical maneuvering of the jaw articulator. In applying this mechanism to the findings from the current study, a time- dependent vector (e.g., lip aperture (LA) as a tract variable) involves three degrees of freedom (i.e., the upper lip height (UH), the lower lip height (LH), and the jaw angle (JA)) for making a lip aperture gesture in a computational model. Under this mechanism, bilabial constriction can be completed without the assistance of jaw raising movement since it is considered more than necessary. As a possibility, the results of the current study on morphosyntactic- boundary-dependent jaw movement can be reflected in the level of model articulator in a computational model (i.e., jaw angle) by specifying dynamic parameters accordingly (i.e., greater weight), such that it can suppress vertical jaw movement in the across-word boundary condition.
A reviewer pointed out that the results of vertical jaw movement might generally benefit by considering the prosodic structure of Seoul Korean, expressing the concern that morphosyntactic boundary conditions (across-word vs. within-word) can occur at the prosodic level and the prosodic structure (a linguistic factor) of an utterance can be influenced by speech rate (a paralinguistic factor). In particular, it was pointed out that labial in the across-word boundary condition (/tʃəna # pakatʃi/) could be produced at the edge of a phrase (IP-initial or AP-initial) at comfortable rate as opposed to phrase-internally (IP-internal or AP-internal) at fast rate. In response to this concern, we examined pitch contour during the production of the sequence /tʃənaV1 # paV2kaV3tʃilɨl # pala/ in across-word boundary condition. f0 measurements were extracted at three time points, i.e., i) at the end point of V1 (/a/), ii) during V2 (/pa/), taking the average f0, and iii) during V3 (/tʃi/), taking the average f0. Among several possible prosodic readings for the first two words /tʃənaV1 # paV2kaV3tʃilɨl/, we considered two different prosodic phrasings following Jun (1993). One was {AP tʃəna}{AP pakatʃilɨl} and the other {AP tʃəna pakatʃilɨl} (note that we did not take into account how the sequence /pala/ is prosodically grouped for the sake of simplicity of analysis and the symbol '{ }' represents an accentual phrase demarcation). The tonal pattern of an accentual phrase is T(HL)H where a H(igh) tone is assigned to T if a phrase-initial phoneme is [+stiff vocal folds] (Halle & Stevens, 1971; Jun, 1993); otherwise, a L(ow) tone is assigned. We would expect a rising contour along /pa/ in V2 and /ka/ in V3 if an accentual phrase boundary coincides with a word boundary, but a falling contour if V2 and V3 belong to the preceding word /tʃəna/ comprising one accentual phrase. Subtracting the f0 value of V2 from that of V3, a positive value denotes a rising contour, which in turn brings about two accentual phrases during production (e.g., {AP tʃəna}{AP pakatʃilɨl}); otherwise, a falling contour emerges to indicate one accentual phrase (e.g., {AP tʃəna pakatʃilɨl}) with a falling contour over V2V3 which is a byproduct of an interpolation between H in /na/ and L in /tʃi/. 64% of tokens were produced with two accentual phrases in the across-word boundary condition (71 out of 111 tokens) and 34 % of tokens within one accentual phrase (38 out of 111 tokens). Two tokens were excluded from further analysis since f0 change had not been detected in them. Perceptual judgments have not considered in analysis.
Figure 8. Schematic representation of tonal contours (a) {AP tʃəna}{AP pakatʃilɨl} in two accentual phrases and (b) {AP tʃəna pakatʃilɨl} in one accentual phrase. Allophonic variation involving voicing and flapping is not transcribed.
Examining a subset of data where vertical jaw maxima measurements were available (i.e., 95 tokens in the across-word boundary condition), we observed that 24 tokens produced at fast rate were yielded AP-internally and 20 tokens AP-initially. At comfortable rate, 7 tokens were produced AP-internally while 44 tokens AP-initially; more tokens at comfortable rate were produced with AP-initial position, that is, a phrasal boundary location. We fitted a linear regression model on maximum vertical jaw position as we looked into Speech rate (fast vs. comfortable) and Prosody (AP-initial vs. AP-internal), with Subject as the random intercept. The results of maximum vertical jaw position across word boundaries showed that there was no interaction (χ2(1)=1.03, p>0.05). The dependent variable varied with Speech rate, but not with Prosody. The results of t-tests rendered statistical significance in terms of Speech rate, lowered by −0.78 mm in fast rate (SE ±0.26) (t(89.88)=−3.05, p<0.01) (comfortable > fast), but not with Prosody (t(91.88)=1.74, p>0.05 (AP-boundary = AP-internal)). We further examined each speech rate condition separately, fitting a linear regression model on maximum vertical jaw position as we looked into Prosody (AP-initial vs. AP-internal). For fast speech rate, the results of t-tests rendered statistical significance in terms of Prosody, raised by 0.69 mm AP-internally (SE ±0.22) (t(37.91)= 3.14, p<0.001) (AP-initial < AP-internal), while no statistical significance was observed for comfortable speech rates in terms of Prosody (t(45.62)=−1.31, p>0.05) (AP-initial = AP-internal). None of the results indicated that AP-initial position exhibited higher jaw position.
Previous literature of prosodically driven articulation or coarticulation has shown that higher prosodic domains are generally associated with articulatory strengthening and less coarticulation (Cho, 2004; Cho & Keating, 2001, 2009; Cho et al., 2016; Keating et al., 2003; inter alia). Linguopalatal contact was greater in higher prosodic domains (e.g., Utterance-initial (Ui) and Intonational phrase-initial (IPi)) than in lower prosodic domains (e.g., Accentual phrase-initial (APi) and Word-initial (Wi)), (Ui, IPi > APi, Wi), although there was interspeaker variability. In some measurements (e.g., linguopalatal contact and seal duration), APi, with interspeaker variability in terms of linguopalatal contact, was not a prosodic condition for domain-initial strengthening, as compared to Wi (Keating et al., 2003). Domain-initial strengthening (Ui, IPi vs. U-internal, IP-internal) is not locally restricted to boundary-initial consonants (/n/, /t/), but is also globally attested with the vowel (/ɛ/) in the examination of CVs, where the target syllable appeared in a trisyllabic word (/nɛbəbɛn/, /tɛbəbɛn/) embedded in carrier phrases (‘___ fed them.’ and ‘one deaf ___’) (Cho & Keating, 2009). Pondering over the idiosyncratic results observed with our vertical jaw maxima data, we turn to Öhman’s (1966) finding in which a consonant is superimposed onto the consecutive vocalic lingual movement from V1 to V2. In particular, vocalic lingual articulation is physiologically unconstrained by an intervening labial consonant (e.g., /p/), compared to a lingual consonant (e.g., /t/ or /k/) (Kühnert, 2006; Öhman, 1966; Recasens, 1984). We began with solving a conundrum involving vocalic articulation by giving more weight to articulatory strengthening of adjacent vowels in stronger prosodic locations, and now gear into the idiosyncratic behavior of bilabial stop /p/ in terms of jaw position (i.e., lower jaw position in the across-word boundary condition and in the AP-initial position).
Notice that the jaw articulator is repeatedly used in consonantal articulation as well as vocalic articulation (see Table 1). Most previous studies have examined primary articulation (e.g., the lips for labial, the tongue tip for coronal, etc.), while our data also showed reduction in participating articulators during the production of bilabial stop /p/ in terms of maximum vertical jaw position. As pointed out earlier (see section 1.1 for a detail), /p/ involves tract variables LA and LP/PRO while the jaw articulator (JA in this case) is one of the joint variables engaged in these tract variables. Given that the strengthening of vocalic articulation is manifested by a more open vocal tract with greater jaw lowering in low vowel /a/, this may have rendered more reduction in consonantal articulation (vertical jaw position in this case). Since our data include /p/ in the homorganic low vowel /a/-to-/a/ context, it is plausible that domain-initial adjacent low vowels ({AP tʃəna}{AP pakatʃilɨl}) may demonstrate lower jaw position, which may have in turn influenced the intervening consonant /p/ to the extent that it exhibits lower jaw position, being assimilated to lower jaw position in the /a/-to-/a/ context, in higher prosodic domains (e.g., AP-initial < AP-internal). From this perspective, reduction in jaw height in an intervening consonant in the /a/-to-/a/ context can possibly be understood as a byproduct of articulatory strengthening of vowels (i.e., lower jaw position) in prosodically stronger locations. In exploring the coordinative movement of articulators engaged in the lip aperture gesture of intervocalic bilabial stop /p/, we have found empirical evidence that at a minimum, the jaw articulator should be carefully investigated along with a tract variable, LA, for a bilabial stop (e.g., /p/) in articulatory studies, so that we can enhance our understanding of participating articulators structured coordinatively at the segment level as well as any relation between jaw movement patterns and prosody.
To conclude this section, some caution should be taken since prosodic analysis is quite limited and incomplete in the current study and needs to be further analyzed to figure out exactly what occurred in speech articulation (e.g., with regard to the target consonant as well as adjacent vowels). In future study, we also need to address possibilities to explain how the prosodic structure (a linguistic factor) of an utterance can be influenced by speech rate (a paralinguistic factor) in a systematic way in terms of the tract variables and articulators involved.
4.2. Speech-rate effects applied to a computation model
4.2.1. Spatial reduction in fast speech rate
The current study showed that the lip aperture gesture was more reduced in fast rate, compared to comfortable rate. In the task-dynamic model of speech production (e.g., task dynamics application toolkit, TADA, in Nam et al. (2012)), speech rate- ependent gestural reduction is controlled at the gestural score level. Activation time values entered for an active tract variable are inherently smaller for fast rate, therefore target attainment cannot be completed simply due to shortness (or a lack) of time, and spatial reduction in lip aperture occurs as a consequence.
4.2.2. More synchronous temporal relation between the jaw and the upper and lower lips in fast speech rate
Vertical jaw position in the bilabial stop /p/ did not spatially vary with different speech rates, being compatible with the results of previous articulatory studies on derived intervocalic flaps (comfortable = fast) (e.g., [aɾa] in Son (2015a) and [iɾi] in Son (2015b)). In contrast, speech rate was a factor to differentiate relative temporal relations between the jaw and the lips, which were characterized by a synchronous coordination in fast rate. To quote Byrd (1996:139), “A variety of work has demonstrated that articulatory, prosodic, and extralinguistic factors all influence speech timing in a complex and interactive way.” Speech timing has been investigated in terms of intergestural timing in a variety of articulatory studies (Browman & Goldstein, 2000; Nam, 2007; Nam et al., (in press); Saltzman et al., 2006). In particular, properties relating to the syllable structure (e.g., onset vs. coda) of an utterance are distinctly represented: onset is specified with an in-phase (0°) relation and coda an anti-phase (180°) relation in the coupling graph (Nam, 2007). Saltzman et al.’s (2006) theoretical basis was grounded in Haken, Kelso, & Bunz (1985) where human hand movements abruptly became more synchronous with increasing rate, from anti-phase (unstable mode) to in-phase (stable mode) relations. Note that a model articulator such as TADA specifies phase relations at the level of intergestural coordination (e.g., synchronous C-V in the onset vs. sequential V-C in the coda (Nam, 2007); synchronous tongue tip raising with respect to tongue body retraction in the onset vs. sequential relationship in the coda (Browman & Goldstein, 1995). In line with this, the results of the current study may suggest (or support) that a model articulator include a way to add phase relations to speech articulators involved in a segment observed in human speech (e.g., more stable mode of interarticulator locking in fast speech). Future studies on articulatory robotics should include this kind of issue, if possible.
4.3. Sptiaotemporal interarticulatory coordinative movement manifested in the upper lip, lower lip, and jaw
In the across-word boundary condition, the temporal point of maximum vertical jaw position preceded that of the upper lip. This can be understood as a premature termination of the assistance of the jaw in terms of assisting lower lip raising movement and lip tissue compression.
Son (2018) already showed more reduction of the lower lip movements of the intervocalic bilabial stop /p/ in the across-word boundary condition. Examining the same set of stimuli from that study in an effort to provide an understanding of the articulation of bilabial stop /p/, the current study also showed that vertical jaw position was lower in that particular context. Since the jaw and the lower lip are physiologically bound to one another (Gick et al., 2013), paired articulatory reduction may result in. Due to this physiological binding, there are two possible explanations for why reduction of vertical lower lip movement arises. One explanation is that spatial reduction in the vertical movement of the lower lip (LL) could have induced spatial reduction of jaw elevation. If we suppose that articulatory reduction of the lower lip (LL) could have induced spatial reduction of the jaw for /p/, this would suggest that spatial reduction of the lower lip (LL) elevation had anticipated spatial reduction of the jaw, and lip compression occurred subsequently without the assistance of the jaw. The other explanation is that spatial reduction of vertical jaw elevation could have caused reduction of raising movement of the lower lip. In line with this, we suppose that spatial reduction of the jaw could have induced that of the lower lip (LL) in the across-word boundary condition and the subsequent lip compression can be understood as the further independent raising movement of the lower lip as an effort to avoid lenition. Lower jaw position in the across-word boundary condition can be analogous to natural jaw yanking from vocalic gestures, which could have acted upon the concurrent lower lip reduction (see robotic jaw yanking by force with human subjects in Shiller et al. (2005)).
In Son (2018), spatial reduction of lower lip (LL) raising was resolved by compensatory upper lip (UL) lowering. Combining the results of the lower lip and the upper lip from the perspective of an articulatory task to complete a tight seal and release for labial stop /p/, we construe that articulatory compensation could have occurred in order to avoid lenition (e.g., intervocalic bilabial stop /p/ to labial approximant /w/). It is possible that articulatory compensation could have arisen to the extent of annihilating word-boundary effects so that a lip-closing gesture could have occurred in a coordinative articulatory effort in this particular context. By conducting an additional analysis of lip aperture (LA) minima in the current study, we were able to verify that labial constriction did not vary with different word boundaries, fitting linear mixed-effects models on lip aperture data relating to the number of observations in Son’s (2018) study (χ2(1)= 3.79, p≤0.05 for interaction; χ2(1)=7.98, p<0.01 (t(223)=2.85, p<0.01) for speech rate (comfortable < fast); χ2(1)=0.58, p>0.05 (t(223)=−0.76, p>0.05) for boundary (across- word = within-word)). The result is compatible with the current lip aperture data that was selectively analyzed to be in balance with available vertical jaw maxima (number of observations (196 tokens) as shown in Table 3), supporting compensatory lowering by the upper lip, possibly to avoid lenited [w]. One piece of supporting evidence can be found in a mechanical jaw perturbation experiment using a robotic jaw-yanking device in Shiller et al. (2005). In their study, speakers reacted voluntarily to arbitrary jaw perturbations such that they increased jaw stiffness in vowel production. In reaction to lowered jaw position, our subjects made use of another contributing articulator (the upper lip (UL) in this case) in a functional way such that articulatory compensation occurred by means of increasing upper lip (UL) lowering movement. We conclude that speakers of Seoul Korean have a holistic knowledge in producing speech from the perspective of task-based achievement by employing contributory vocal tract articulators in a functional way.
## Acknowlegements
I thank seven EMMA subjects for participating in production experiments and Sean C. O’Rourke for proofreading this paper. I am also grateful to three anonymous reviewers for their corrections and constructive comments. Any remaining errors are my own.
## Footnote
This work was supported by a grant rewarded in 2018 from the research fund of Hannam University
1. The EMMA experiment was fully supported by NIH grant DC 00403 conferred upon Catherine T. Best (PI) and Haskins Laboratories.
2. Twenty-seven tokens were excluded from statistical analysis since the highest position of the jaw (during /p/) were not found using lp-Snapex.
## References
1.
Alexander, R., Sorensen, T., Toutios, A., & Narayanan, S. (2017). VCV synthesis using task dynamics to animate a factor-based articulatory model. Proceedings of Interspeech. 244-248 .
2.
Bates, D., Maechler, M., & Bolker, B. (2011). lme4: Linear mixed-effects models using S4 classes. R package version, 1, 1-23. Retrieved from http://CRAN.R-project.org/package=lme4 on August 30, 2018 .
3.
Bolla, K., & Valaczkai, I. (1986). Német Beszédhangok Atlasza (A Phonetic Conspectus of German). (Magyar Fonetikai Füzetek (Hungarian Papers in Phonetics) 16). Hungarian Academy of Sciences, Budapest .
4.
Browman, C. P., & Goldstein, L. (1986). Towards an articulatory phonology. Phonology Yearbook, 3, 219-252 .
5.
Browman, C. P., & Goldstein, L. (1988). Some notes on syllable structure in articulatory phonology. Phonetica, 45, 140-155 .
6.
Browman, C. P., & Goldstein, L. (1989). Articulatory gestures as phonological units. Phonology, 6, 201-251 .
7.
Browman, C. P., & Goldstein, L. (1990). Gestural specification using dynamically-defined articulatory structures. Journal of Phonetics, 18, 299-320 .
8.
Browman, C. P., & Goldstein, L. (1991). Tiers in articulatory phonology, with some implications for casual speech. In J. Kingston & M. E. Beckman (Eds.), Papers in Laboratory Phonology I: Between the grammar and the physics of speech (pp. 341-376). Cambridge, U.K.: Cambridge University Press .
9.
Browman, C. P., & Goldstein, L. (1992). Articulatory phonology: An overview. Phonetica, 49, 155-180 .
10.
Browman, C. P., & Goldstein, L. (1995). Gestural syllable position effects in American English. In F. Bell-berti & L. Raphael (Eds.), Producing speech: Contemporary issues. For Catherine Safford Harris (pp. 19-33). AIP Press: New York .
11.
Browman, C. P., & Goldstein, L. (2000). Competing constraints on intergestural coordination and self-organization of phonological structures. Bulletin de la Communication Parlée, 5, 25-34 .
12.
Byrd, D. (1996). Influences on articulatory timing in consonant sequences. Journal of Phonetics, 24, 209-244 .
13.
Chitoran, I., Goldstein, L., & Byrd, D. (2002). Gestural overlap and recoverability: Articulatory evidence from Georgian, in C. Gussenhoven and Warner (Eds.), Papers in Laboratory Phonology VII (pp. 419-448). Berlin: Mouton de Gruyter .
14.
Cho, T. (2004). Prosodically-conditioned strengthening and vowel-to-vowel coarticulation in English. Journal of Phonetics, 32, 141-176 .
15.
Cho, T., & Keating, P. (2001). Articulatory and acoustic studies of domain-initial strengthening in Korean. Journal of Phonetics, 29, 155-190 .
16.
Cho, T., & Keating, P. (2009). Effects of initial position versus prominence in English. Journal of Phonetics, 37, 466-485 .
17.
Cho, T., Son, M., & Kim, S. (2016). Articulatory reflexes of the three-way contrast in labial stops and kinematic evidence for domain-initial strengthening in Korean. Journal of the International Phonetic Association, 46, 129-155 .
18.
Gick, B., Wilson, I., & Derrick, D. (2013). Articulatory phonetics. Oxford : Wiley-Blackwell .
19.
Goldstein, L. USC class website. Retrieved from https://sail.usc.edu/~lgoldste/General_Phonetics/Slides/Vowels.pdf on October 3, 2018 .
20.
Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand movements. Biological Cybernetics, 51, 347-356 .
21.
Halle, M., & Stevens, K. N. (1971). A note on laryngeal features. Quarterly Progress Report (Research Laboratory of Electronics, MIT) 101, 198-212 .
22.
Iskarous, K., Goldstein, L., Whalen, D., Tiede, M., & Rubin, P. (2003). CASY: The Haskins configurable articulatory synthesizer. Proceedings of the 15th International Congress of Phonetic Sciences, 185-188 .
23.
Jun, S-A. (1993). The phonetics and phonology of Korean prosody. Ph.D. Dissertation. Ohio State University .
24.
Keating, P., Cho, T., Fougeron, C., & Hsu, C. (2003). Domain-initial strengthening in four languages. In John Local, Richard Ogden, and Rosalind Temple (Eds.), Papers in Laboratory Phonology VI (pp. 145-163). Cambridge: Cambridge University Press .
25.
Keating, P., Lindblom, B., Lubker, J., & Kreiman, J. (1994). Variability in jaw height for segments in English and Swedish VCVs. Journal of Phonetics, 22, 407-422 .
26.
Kochetov, A., & Goldstein, L. (2005). Position and place effects in Russian word-initial and word-medial clusters, Journal of the Acoustical Society of Merica, 117, 2571 .
27.
Kochetov, A., Pouplier, M., & Son, M. (2007). Cross-language differences in overlap and assimilation patterns in Korean and Russian. Proceedings of the 16th International Congress of Phonetic Sciences (pp. 1361-1364). Saarbrücken, Germany .
28.
Krakow, R. (1989). The articulatory organization of syllables: a kinematic analysis of labial and velar gestures. Ph.D. Dissertation. Yale University .
29.
Kühnert, B., & Hoole, P. (2004). Speaker-specific kinematic properties of alveolar reductions in English and German. Clinical Linguistics and Phonetics, 18, 559-575 .
30.
Kühnert, B., Hoole, P. & Mooshammer, C. (2006). Gestural overlap and c-center in selected French consonant clusters. Proceedings of 7th International Seminar on Speech Production (ISSP) (pp. 327-334), Brazil .
31.
Ladefoged, P. (2001). Vowels and consonants: An introduction to the sounds of languages. Malden: Blackwell publishing Ltd .
32.
Ladefoged, P., & Maddieson, I. (1996). The sounds of the world's languages. Oxford: Blackwell .
33.
Löfqvist, A. (1993). Electromagnetic transduction techniques in the study of speech motor control. PHONUM: Reports from the Department of Phonetics, University of Umeå, 2, 87-106 .
34.
Löfqvist, A. (1996). Control of oral closure and release in bilabialstop consonants. Proceedings of the 6th Australian International Conference on Speech Science and Technology (pp. 561-566). Canberra .
35.
Löfqvist, A., & Gracco, V. L. (1997). Lip and jaw kinematics in bilabial stop consonant production. Journal of Speech, Language, and Hearing Research, 40, 877-893 .
36.
Maddieson, I. (2005). Bilabial and labio-dental fricatives in Ewe. UC Berkeley Phonology Lab Annual Report, 199-215 .
37.
Mermelstein, P. (1973). Articulatory model for the study of speech production. The Journal of the Acoustical Society of America, 53, 1970-1082 .
38.
Mooshammer, C., Hoole, P., & Geumann, A. (2007). Jaw and order. Language and Speech, 50(2), 145-176 .
39.
Nam, H. (2007). A competitive, coupled oscillator model of moraic structure: Split-gesture dynamics focusing on positional asymmetry. In J. Cole & J. I. Hualde (Eds.). Papers in Laboratory Phonology IX (pp. 483-506). Berlin: Mouton de Gruyter .
40.
Nam, H. (n.d.) Towards articulatory machines. Unpublished manuscript. 남호성. 조음 로봇을 꿈꾸며. 원고 .
41.
Nam, H., & Saltzman, E. (2003). A competitive, coupled oscillator model of syllable structure. Proceedings of the 15th International Congress of Phonetic Sciences (pp. 2253-2256). Barcelona, Spain .
42.
43.
Nam, H., Goldstein, L., & Saltzman, E. (in press). Self-organization of syllable structure: a coupled oscillator model. In F. Pellegrino, E. Marisco & I. Chitoran, (Eds.). Approaches to phonological complexity. Berlin/New York: Mouton de Gruyter. Retrieved from http://sail.usc.edu/~lgoldste/me/documents/Nam_Gold_Saltz_agent_ES6_LG.pdf on October 20, 2018 .
44.
Nam, H., Goldstein, L., Saltzman, E., & Byrd, D. (2004). TADA: An enhanced, portable task dynamics model in MATLAB. The Journal of the Acoustical Society of America, 115(5), 2430-2430 .
45.
Nolan, F. (1992). The descriptive role of segments: Evidence from assimilation. In G. Docherty & D. R. Ladd (Eds.), Papers in Laboratory Phonology II: Gesture, Segment, and Prosody (pp. 261-280). Cambridge: Cambridge University Press .
46.
Öhman, S. E. (1966). Coarticulation in VCV utterances: Spectrographic measurements. Journal of the Acoustical Society of America, 39, 151-168 .
47.
Perkell, J. S., Cohen, M. H., Svirsky, M. A., Matthies, M. L., Garabieta, I., & Jackson, M. T. (1992). Electromagnetic midsagittal articulometer systems for transducing speech articulatory movements. The Journal of the Acoustical Society of America, 92, 3078-3096 .
48.
R Core Team. (2014). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. Retrieved from http://www.rproject.org on November 18, 2017 .
49.
Recasens, D. (1984). Vowel-to-vowel coarticulation in Catalan VCV sequences. The Journal of the Acoustical Society of America, 76, 1624-1635 .
50.
Saltzman, E., & Kelso, J. A. (1987). Skilled actions: A task-dynamic approach. Psychological Review, 94, 84-106 .
51.
Saltzman, E. L., & Munhall, K. G. (1989). A dynamical approach to gestural patterning in speech production. Ecological Psychology, 1, 333-382 .
52.
Saltzman, E., Nam, H., Goldstein, L., & Byrd, D. (2006). The distinctions between state, parameter and graph dynamics in sensorimotor control and coordination. In M. Latash & F. Lestienne, (Eds.). Motor Control and Learning (pp. 63-73). New York: Springer .
53.
Shiller, D. M., Houle, G., & Ostry, D. J. (2005). Voluntary control of human jaw stiffness. Journal of Neurophysiology, 94, 2207-2217 .
54.
Smith, C. (1992). The timing of vowel and consonant gestures. Ph.D. Dissertation, Yale University .
55.
Son, M. (2008). Gradient reduction of C1 in /pk/ sequences. Speech Sciences, 15(4), 43-65. (손민정 (2008). /pk/연속음에서 일어나는 점층 약화. 음성과학, 15(4), 43-65.) .
56.
Son, M. (2013). Articulatory attributes in Korean nonassimilating contexts. Phonetics and Speech Sciences, 5(1), 109-121. (손민정 (2013). 한국어 비동화 환경에서 일어나는 조음 특질. 말소리와 음성과학, 5(1), 109-121.) .
57.
Son, M. (2015a). Articulatory properties of the allophonic variant [ɾ] in Korean /l/-flapping: Gestural reduction and the role of gestural overlap. Studies in Phonetics, Phonology, and Morphology, 21(3), 427-456. (손민정 (2015a). 한국어 /l/ 탄설음화에 나타나는 변이음 [ɾ] 조음 특질: 조음 약화와 조음 중첩. 음성 음운 형태론 연구, 21, 427-456.) .
58.
Son, M. (2015b). Korean /l/-flapping in an /i/-/i/ context. Phonetics and Speech Sciences, 7(1), 151-163. (손민정 (2015b). /i/-/i/ 환경 에서 일어나는 한국어 /l/ 설탄음화. 말소리와 음성과학, 7(1), 151-163.) .
59.
Son, M. (2018). Word-boundary and rate effects on upper and lower lip movements in the articulation of the bilabial stop /p/ in Korean. Phonetics and Speech Sciences, 10(1), 23-31. (손민정 (2018). 한국어 양순 폐쇄음 /p/ 조음에 나타나는 윗입술과 아래입술 움직임에 미치는 단어경계와 속도 효과. 말소리와 음성과학, 3(4), 33-43.) .
60.
Son, M., Kim, S., & Cho, T. (2011). Supralaryngeal articulatory characteristics of coronal consonants /n, t, th, t*/ in Korean. Phonetics and Speech Sciences, 3(4), 33-43. (손민정·김사향·조 태홍 (2011). 한국어 화관자음 /n, t, th, t*/에서 나타나는 성도 조음 특질. 말소리와 음성과학, 3(4), 33-43.) .
61.
Son, M., Kochetov, A., & Pouplier, M. (2007). The role of gestural overlap in perceptual place assimilation: Evidence from Korean. Papers in Laboratory Phonology IX (pp. 507-534). New York: Mouton de Gruyter .
62.
Tiede, M. (2005). MVIEW: Software for visualization and analysis of concurrently recorded movement data. New Haven, CT: Haskins Laboratories .
63.
Wood, S. (1979). A radiographic analysis of constriction location for vowels. Journal of Phonetics, 7, 25-43 . | 2019-09-19 09:09:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48072677850723267, "perplexity": 12316.849855309645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573465.18/warc/CC-MAIN-20190919081032-20190919103032-00438.warc.gz"} |
https://stackoverflow.com/questions/55343758/information-about-energy-of-a-node | # Information about energy of a node
I want to get the information about the energy in the node, so those neighbouring nodes can reroute the data packets when the neighbouring node energy is less.
Practical devices that use UnetStack often have a battery voltage parameter that provides some measure of energy available. However, this may be hard to use as battery voltage does not linearly depend on energy, but is highly dependent on the actual battery chemistry. | 2019-10-16 01:15:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3855898976325989, "perplexity": 650.224862246684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986660829.5/warc/CC-MAIN-20191015231925-20191016015425-00483.warc.gz"} |
https://spmmathematics.blog.onlinetuition.com.my/2021/07/trigonometry-short-questions-example-4-6.html | # 6.3.2 Ratio and Graphs of Trigonometric Functions, SPM Paper (Short Questions)
Question 4:
In the diagram above, WZY is a straight line. $\angle XYZ={90}^{o},\text{}\angle XWZ={30}^{o}$ and WZ = XZ = 30 cm. Find the length of XY.
Solution:
$\begin{array}{l}\angle WXZ=\angle XWZ={30}^{o}\\ \therefore \angle XZY={30}^{o}+{30}^{o}={60}^{o}\\ \\ \mathrm{sin}\angle XZY=\frac{XY}{XZ}\\ \mathrm{sin}{60}^{o}=\frac{XY}{30}\\ XY=\mathrm{sin}{60}^{o}×30\\ XY=25.98cm\end{array}$
Question 5:
In the diagram above, PQS is a right angle triangle. Given that SR = 6cm, PQ = 12 cm and 5SR = 2PS. Find the value of cos α and tan β.
Solution:
Question 6:
In the diagram above, ADC is a straight line, if $\mathrm{sin}q=\frac{3}{5}\text{and}\mathrm{tan}p=\frac{1}{2}$ . Find the distance of AC.
Solution:
$\begin{array}{l}\text{Given}\mathrm{sin}q=\frac{BD}{AB}=\frac{3}{5}\\ \frac{BD}{30}=\frac{3}{5}\\ BD=\frac{3}{5}×30\\ BD=18\text{}cm\\ \\ \text{In}△\text{}ABD,\text{using Pythagoras’ Theorem,}\\ AD=\sqrt{A{B}^{2}-B{D}^{2}}\\ AD=\sqrt{{30}^{2}-{18}^{2}}=24\text{}cm\\ \\ \text{Given tan}p=\frac{BD}{DC}=\frac{1}{2}\\ \frac{18}{DC}=\frac{1}{2}\\ DC=36\text{}cm\\ \\ \text{Hence, distance of}AC=24+36=60\text{}cm.\end{array}$
### 4 thoughts on “6.3.2 Ratio and Graphs of Trigonometric Functions, SPM Paper (Short Questions)”
1. Hi, for question 5 the answer for cos alpha is 4/5 not 3/5. Please correct the simplification.
• Dear Gurdit Singh, | 2022-10-03 05:31:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9255732297897339, "perplexity": 3048.930696310405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337398.52/warc/CC-MAIN-20221003035124-20221003065124-00155.warc.gz"} |
https://www.physicsforums.com/threads/oh-my-god-get-vonage.105544/ | Oh my god, get vonage
Gold Member
I just saw a paid program about vonage and they have broken new ground, they crossed a line, surpassed a treshhold.... they told the truth! They go boom! "Vonage is $24.95 a month!" and then they go "but theres a monthly fee of$1.50" and then!!! "theres also a regulatory fee which makes vonage $27.95 a month". Oh my god! They could have just tossed that kinda crap into the fine print and surprise you in the bill but NO! They came out and said boom sucka,$27.95 a month. I should get vonage out of principle alone!
Related General Discussion News on Phys.org
Pengwuino said:
I just saw a paid program about vonage and they have broken new ground, they crossed a line, surpassed a treshhold.... they told the truth! They go boom! "Vonage is $24.95 a month!" and then they go "but theres a monthly fee of$1.50" and then!!! "theres also a regulatory fee which makes vonage $27.95 a month". Oh my god! They could have just tossed that kinda crap into the fine print and surprise you in the bill but NO! They came out and said boom sucka,$27.95 a month. I should get vonage out of principle alone!
Sure, and then when you need 911 service, you have to traverse 2 levels of Customer service just to get to the state police.
Packet8 is much better quality and service.
I hate those commercials where they ask how much someone's paying for their phone bill. The people's answers always include their cell phone bills, and then vonage is like OMG WE ARE CHEAPER. Well, yeee aren't a cell phone company now are ya.
Ivan Seeking
Staff Emeritus
Gold Member
What is so great about 30 bucks a month? Isn't Vonage an internet phone service?
I get unlimited long distance on two lines for thirty bucks a month.
Last edited:
Gold Member
Ivan Seeking said:
What is so great about 30 bucks a month? Isn't Vonage an internet phone service?
I get unlimited long distance on two lines for thirty bucks a month.
We pay \$50 a month and we don't get no stinkin unlimited long distance.
EnumaElish | 2020-11-24 08:39:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26420095562934875, "perplexity": 13031.46026902408}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176049.8/warc/CC-MAIN-20201124082900-20201124112900-00609.warc.gz"} |
https://cs.stackexchange.com/questions/120661/is-it-true-that-if-m-forall-alpha-left-alpha-rightarrow-alpha-right | # Is it true that if $M : \forall \alpha . \left( \alpha \rightarrow \alpha \right)$ is a closed term then $M = \Lambda \alpha. \lambda x^{\alpha} . x$?
In system F, is every closed term $$M$$, which is of $$\forall \alpha . \left( \alpha \rightarrow \alpha \right)$$, $$\alpha \beta \eta$$-equivalent to $$\Lambda \alpha. \lambda x^{\alpha} . x$$?
I have believed that this is true and have tried to prove it by using the equation
$$\forall \alpha. \forall \beta. \forall f^{\alpha \rightarrow \beta} . \forall x^{\alpha} . M \beta \left( f x \right) = f \left( M \alpha x \right)$$
, which I think is a consequence of theorems for Free, but couldn't prove it.
• Let $y : \beta$ be arbitrary, and replace $f$ with $\lambda z^{\alpha} . y$. Then $M \beta y = y$. – 임기정 Feb 15 at 4:30 | 2020-02-22 17:34:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9653133749961853, "perplexity": 138.19246757716303}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145708.59/warc/CC-MAIN-20200222150029-20200222180029-00060.warc.gz"} |
https://www.physicsforums.com/threads/solution-of-ode.404124/ | # Solution of ODE
#### Acut
How can this equation be solved?
$$\frac{dx}{dt}$$=ax(b-x)
Related Differential Equations News on Phys.org
#### Cyosis
Homework Helper
By separation of variables.
$$\frac{dx}{ax(b-x)}=dt$$
Now you can integrate both sides.
#### Unit
The integral of the dx side requires decomposition into partial fractions.
#### Acut
Many thanks!
I'm a bit rusty in solving ODE's and was having a hard time trying to solve this one..
#### IPhO' 2008
dx/(ax(b-x)) = dx/abx + dx/ab(b-x)
= dx/abx - d(b-x)/ab(b-x)
and then you can integrate these terms. | 2019-11-12 18:41:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9276851415634155, "perplexity": 6912.322995526828}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665726.39/warc/CC-MAIN-20191112175604-20191112203604-00532.warc.gz"} |
https://distinguishable.askdefine.com/ | # Dictionary Definition
1 capable of being perceived as different or distinct; "only the shine of their metal was distinguishable in the gloom"; "a project distinguishable into four stages of progress"; "distinguishable differences between the twins" [ant: indistinguishable]
2 (often followed by from') not alike; different in nature or quality; "plants of several distinct types"; "the word nationalism' is used in at least two distinct senses"; "gold is distinct from iron"; "a tree related to but quite distinct from the European beech"; "management had interests quite distinct from those of their employees" [syn: distinct]
# User Contributed Dictionary
## English
1. Able, or easily able to be distinguished.
Black is very distinguishable against a white background
# Extensive Definition
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, as well as composite microscopic particles such as atoms and molecules.
There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which are forbidden from sharing quantum states (this property of fermions is known as the Pauli exclusion principle.) Examples of bosons are photons, gluons, phonons, and helium-4 atoms. Examples of fermions are electrons, neutrinos, quarks, protons and neutrons, and helium-3 atoms.
The fact that particles can be identical has important consequences in statistical mechanics. Calculations in statistical mechanics rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behavior from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing paradox.
## Distinguishing between particles
There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as mass, electric charge, and spin. If differences exist, we can distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why we can speak of such a thing as "the charge of the electron".
Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as we can measure the position of each particle with infinite precision (even when the particles collide), there would be no ambiguity about which particle is which.
The problem with this approach is that it contradicts the principles of quantum mechanics. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunctions that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be indistinguishable.
## Quantum mechanical description of identical particles
### Symmetrical and antisymmetrical states
We will now make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics.
For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical Hilbert spaces. If we denote the Hilbert space of a single particle as H, then the Hilbert space of the combined system is formed by the tensor product H \otimes H.
Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.) Suppose that one particle is in the state n1, and another is in the state n2. What is the quantum state of the system? We might guess that it is
|n_1\rang |n_2\rang
which is simply the canonical way of constructing a basis for a tensor product space from the individual spaces. However, this expression implies that we can identify the particle with n1 as "particle 1" and the particle with n2 as "particle 2", which conflicts with the ideas about indistinguishability discussed earlier.
Actually, it is an empirical fact that identical particles occupy special types of multi-particle states, called symmetric states and antisymmetric states. Symmetric states have the form
|n_1, n_2; S\rang \equiv \mbox \times \bigg( |n_1\rang |n_2\rang + |n_2\rang |n_1\rang \bigg)
Antisymmetric states have the form
|n_1, n_2; A\rang \equiv \mbox \times \bigg( |n_1\rang |n_2\rang - |n_2\rang |n_1\rang \bigg)
Note that if n1 and n2 are the same, our equation for the antisymmetric state gives zero, which cannot be a state vector as it cannot be normalized. In other words, in an antisymmetric state the particles cannot occupy the same single-particle states. This is known as the Pauli exclusion principle, and it is the fundamental reason behind the chemical properties of atoms and the stability of matter.
### Exchange symmetry
The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. It appears to be a fact of Nature that identical particles do not occupy states of a mixed symmetry, such as
|n_1, n_2; ?\rang = \mbox \times \bigg( |n_1\rang |n_2\rang + i |n_2\rang |n_1\rang \bigg)
There is actually an exception to this rule, which we will discuss later. On the other hand, we can show that the symmetric and antisymmetric states are in a sense special, by examining a particular symmetry of the multiple-particle states known as exchange symmetry.
Let us define a linear operator P, called the exchange operator. When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors:
P \bigg(|\psi\rang |\phi\rang \bigg) \equiv |\phi\rang |\psi\rang
P is both Hermitian and unitary. Because it is unitary, we can regard it as a symmetry operator. We can describe this symmetry as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces).
Clearly, P² = 1 (the identity operator), so the eigenvalues of P are +1 and −1. The corresponding eigenvectors are the symmetric and antisymmetric states:
P|n_1, n_2; S\rang = + |n_1, n_2; S\rang
P|n_1, n_2; A\rang = - |n_1, n_2; A\rang
In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of +1 or −1, rather than being "rotated" somewhere else in the Hilbert space. This indicates that the particle labels have no physical meaning, in agreement with our earlier discussion on indistinguishability.
We have mentioned that P is Hermitian. As a result, it can be regarded as an observable of the system, which means that we can, in principle, perform a measurement to find out if a state is symmetric or antisymmetric. Furthermore, the equivalence of the particles indicates that the Hamiltonian can be written in a symmetrical form, such as
H = \frac + \frac + U(|x_1 - x_2|) + V(x_1) + V(x_2)
It is possible to show that such Hamiltonians satisfy the commutation relation
\left[P, H\right] = 0
According to the Heisenberg equation, this means that the value of P is a constant of motion. If the quantum state is initially symmetric (antisymmetric), it will remain symmetric (antisymmetric) as the system evolves. Mathematically, this says that the state vector is confined to one of the two eigenspaces of P, and is not allowed to range over the entire Hilbert space. Thus, we might as well treat that eigenspace as the actual Hilbert space of the system. This is the idea behind the definition of Fock space.
### Fermions and bosons
The choice of symmetry or antisymmetry is determined by the species of particle. For example, we must always use symmetric states when describing photons or helium-4 atoms, and antisymmetric states when describing electrons or protons.
Particles which exhibit symmetric states are called bosons. As we will see, the nature of symmetric states has important consequences for the statistical properties of systems composed of many identical bosons. These statistical properties are described as Bose-Einstein statistics.
Particles which exhibit antisymmetric states are called fermions. As we have seen, antisymmetry gives rise to the Pauli exclusion principle, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by Fermi-Dirac statistics.
Parastatistics are also possible.
In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyons, and they obey fractional statistics. Experimental evidence for the existence of anyons exists in the fractional quantum Hall effect, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of MOSFETs. There is another type of statistic, known as braid statistics, which are associated with particles known as plektons.
The spin-statistics theorem relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.
### N particles
The above discussion generalizes readily to the case of N particles. Suppose we have N particles with quantum numbers n1, n2, ..., nN. If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any two particle labels:
|n_1 n_2 \cdots n_N; S\rang = \sqrt \sum_p |n_\rang |n_\rang \cdots |n_\rang
Here, the sum is taken over all possible permutations p acting on N elements. The square root on the right hand side is a normalizing constant. The quantity Nj stands for the number of times each of the single-particle states appears in the N-particle state.
In the same vein, fermions occupy totally antisymmetric states:
|n_1 n_2 \cdots n_N; A\rang = \frac \sum_p \mathrm(p) |n_\rang |n_\rang \cdots |n_\rang\
Here, sgn(p) is the signature of each permutation (i.e. +1 if p is composed of an even number of transpositions, and −1 if odd.) Note that we have omitted the ΠjNj term, because each single-particle state can appear only once in a fermionic state.
These states have been normalized so that
\lang n_1 n_2 \cdots n_N; S | n_1 n_2 \cdots n_N; S\rang = 1, \qquad \lang n_1 n_2 \cdots n_N; A | n_1 n_2 \cdots n_N; A\rang = 1.
### Measurements of identical particles
Suppose we have a system of N bosons (fermions) in the symmetric (antisymmetric) state
|n_1 n_2 \cdots n_N; S/A \rang
and we perform a measurement of some other set of discrete observables, m. In general, this would yield some result m1 for one particle, m2 for another particle, and so forth. If the particles are bosons (fermions), the state after the measurement must remain symmetric (antisymmetric), i.e.
|m_1 m_2 \cdots m_N; S/A \rang
The probability of obtaining a particular result for the m measurement is
P_(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) \equiv \bigg|\lang m_1 \cdots m_N; S/A \,|\, n_1 \cdots n_N; S/A \rang \bigg|^2
We can show that
\sum_ P_(n_1, \cdots n_N \rightarrow m_1, \cdots m_N) = 1
which verifies that the total probability is 1. Note that we have to restrict the sum to ordered values of m1, ..., mN to ensure that we do not count each multi-particle state more than once.
### Wavefunction representation
So far, we have worked with discrete observables. We will now extend the discussion to continuous observables, such as the position x.
Recall that an eigenstate of a continuous observable represents an infinitesimal range of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |ψ>, the probability of finding it in a region of volume d³x surrounding some position x is
|\lang x | \psi \rang|^2 \; d^3 x
As a result, the continuous eigenstates |x> are normalized to the delta function instead of unity:
\lang x | x' \rang = \delta^3 (x - x')
We can construct symmetric and antisymmetric multi-particle states out of continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:
|x_1 x_2 \cdots x_N; S\rang = \frac \sum_p |x_\rang |x_\rang \cdots |x_\rang
|x_1 x_2 \cdots x_N; A\rang = \frac \sum_p \mathrm(p) |x_\rang |x_\rang \cdots |x_\rang
We can then write a many-body wavefunction,
\Psi^_ (x_1, x_2, \cdots x_N) \equiv \lang x_1 x_2 \cdots x_N; S | n_1 n_2 \cdots n_N; S \rang
= \sqrt \sum_p \psi_(x_1) \psi_(x_2) \cdots \psi_(x_N)
\Psi^_ (x_1, x_2, \cdots x_N) \equiv \lang x_1 x_2 \cdots x_N; A | n_1 n_2 \cdots n_N; A \rang
= \frac \sum_p \mathrm(p) \psi_(x_1) \psi_(x_2) \cdots \psi_(x_N)
where the single-particle wavefunctions are defined, as usual, by
\psi_n(x) \equiv \lang x | n \rang
The most important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign. This is the manifestation of symmetry and antisymmetry in the wavefunction representation:
\Psi^_ (\cdots x_i \cdots x_j\cdots) = \Psi^_ (\cdots x_j \cdots x_i \cdots)
\Psi^_ (\cdots x_i \cdots x_j\cdots) = - \Psi^_ (\cdots x_j \cdots x_i \cdots)
The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers n1, ..., nN, and we perform a position measurement, the probability of finding particles in infinitesimal volumes near x1, x2, ..., xN is
N! \; \left|\Psi^_ (x_1, x_2, \cdots x_N) \right|^2 \; d^\!x
The factor of N! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,
\int\!\int\!\cdots\!\int\; \left|\Psi^_ (x_1, x_2, \cdots x_N)\right|^2 d^3\!x_1 d^3\!x_2 \cdots d^3\!x_N = 1
Because each integral runs over all possible values of x, each multi-particle state appears N! times in the integral. In other words, the probability associated with each event is evenly distributed across N! equivalent points in the integral space. Because it is usually more convenient to work with unrestricted integrals than restricted ones, we have chosen our normalizing constant to reflect this.
Finally, it is interesting to note that that antisymmetric wavefunction can be written as the determinant of a matrix, known as a Slater determinant:
\Psi^_ (x_1, \cdots x_N)
= \frac \left| \begin \psi_(x_1) & \psi_(x_2) & \cdots & \psi_(x_N) \\ \psi_(x_1) & \psi_(x_2) & \cdots & \psi_(x_N) \\ \cdots & \cdots & \cdots & \cdots \\ \psi_(x_1) & \psi_(x_2) & \cdots & \psi_(x_N) \\ \end \right|
## Statistical properties
### Statistical effects of indistinguishability
The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, let us consider a system of N distinguishable, non-interacting particles. Once again, let nj denote the state (i.e. quantum numbers) of particle j. If the particles have the same physical properties, the njs run over the same range of values. Let ε(n) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition function of the system is
Z = \sum_ \exp\left\
where k is Boltzmann's constant and T is the temperature. We can factorize this expression to obtain
Z = \xi^N
where
\xi = \sum_n \exp\left[ - \frac \right].
If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states [n1, ..., nN]. In the equation for Z, every possible permutation of the ns occurs once in the sum, even though each of these permutations is describing the same multi-particle state. We have thus over-counted the actual number of states.
If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N!. The correct partition function is
Z = \frac.
Note that this "high temperature" approximation does not distinguish between fermions and bosons.
The discrepancy in the partition functions of distinguishable and indistinguishable particles was known as far back as the 19th century, before the advent of quantum mechanics. It leads to a difficulty known as the Gibbs paradox. Gibbs showed that if we use the equation Z = ξN, the entropy of a classical ideal gas is
S = N k \ln \left(V\right) + N f(T)
where V is the volume of the gas and f is some function of T alone. The problem with this result is that S is not extensive - if we double N and V, S does not double accordingly. Such a system does not obey the postulates of thermodynamics.
Gibbs also showed that using Z = ξN/N! alters the result to
S = N k \ln \left(\frac\right) + N f(T)
which is perfectly extensive. However, the reason for this correction to the partition function remained obscure until the discovery of quantum mechanics.
### Statistical properties of bosons and fermions
There are important differences between the statistical behavior of bosons and fermions, which are described by Bose-Einstein statistics and Fermi-Dirac statistics respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the laser, Bose-Einstein condensation, and superfluidity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the Fermi gas. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within shells rather than all lying in the same lowest energy state.
We can illustrate the differences between the statistical behavior of fermions, bosons, and distinguishable particles using a system of two particles. Let us call the particles A and B. Each particle can exist in two possible states, labelled |0> and |1>, which have the same energy.
We let the composite system evolve in time, interacting with a noisy environment. Because the |0> and |1> states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on quantum entanglement.) After some time, the composite system will have an equal probability of occupying each of the states available to it. We then measure the particle states.
If A and B are distinguishable particles, then the composite system has four distinct states: \scriptstyle|0\rangle|0\rangle, \scriptstyle|1\rangle|1\rangle, \scriptstyle|0\rangle|1\rangle, and \scriptstyle|1\rangle|0\rangle. The probability of obtaining two particles in the \scriptstyle|0\rangle state is 0.25; the probability of obtaining two particles in the \scriptstyle|1\rangle state is 0.25; and the probability of obtaining one particle in the |0> state and the other in the \scriptstyle|1\rangle state is 0.5.
If A and B are identical bosons, then the composite system has only three distinct states: \scriptstyle|0\rangle|0\rangle, \scriptstyle|1\rangle|1\rangle, and \scriptstyle1/\sqrt(|0\rangle|1\rangle + |1\rangle|0\rangle). When we perform the experiment, the probability of obtaining two particles in the |0> state is now 0.33; the probability of obtaining two particles in the \scriptstyle|1\rangle state is 0.33; and the probability of obtaining one particle in the |0> state and the other in the |1> state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump."
If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state \scriptstyle1/\sqrt(|0\rangle|1\rangle - |1\rangle|0\rangle). When we perform the experiment, we inevitably find that one particle is in the \scriptstyle|0\rangle state and the other is in the |1> state.
The results are summarized in Table 1:
Table 1: Statistics of two particles Particles Both 0 Both 1 One 0 and one 1 Distinguishable 0.25 0.25 0.5 Bosons 0.33 0.33 0.33 Fermions 0 0 1
As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on Fermi-Dirac statistics and Bose-Einstein statistics, these principles are extended to large number of particles, with qualitatively similar results.
## The homotopy class
To understand why we have the statistics that we do for particles, we first have to note that particles are point localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact (at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower), then we are dealing with the space [M × M]/, the subspace with coincident points removed. (x,y) describes the configuration with particle I at x and particle II at y. (y,x) describes the interchanged configuration. With identical particles, the state described by (x,y) ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by (y,x). Let's look at the homotopy class of continuous paths from (x,y) to (y,x). If M is Rd where d\geq 3, then this homotopy class only has one element. If M is R2, then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc, a clockwise interchange by half a turn, etc). In particular, a counterclockwise interchange by half a turn is NOT homotopic to a clockwise interchange by half a turn. Lastly, if M is R, then this homotopy class is empty. Obviously, if M is not isomorphic to Rd, we can have more complicated homotopy classes...
What does this all mean?
Let's first look at the case d \geq 3. The universal covering space of [M × M]/, which is none other than [M × M]/ itself, only has two points which are physically indistinguishable from (x, y), namely (x, y) itself and (y, x). So, the only permissible interchange is to swap both particles. Performing this interchange twice gives us (x, y) back again. If this interchange results in a multiplication by +1, then we have Bose statistics and if this interchange results in a multiplication by −1, we have Fermi statistics.
Now how about R2? The universal covering space of [M × M]/ has infinitely many points which are physically indistinguishable from (x,y). This is described by the infinite cyclic group generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not lead us back to the original state. So, such an interchange can generically result in a multiplication by exp(iθ) (its absolute value is 1 because of unitarity...). This is called anyonic statistics. In fact, even with two DISTINGUISHABLE particles, even though (x, y) is now physically distinguishable from (y, x), if we go over to the universal covering space, we still end up with infinitely many points which are physically indistinguishable from the original point and the interchanges are generated by a counterclockwise rotation by one full turn which results in a multiplication by exp(iφ). This phase factor here is called the mutual statistics.
As for R, even if particle I and particle II are identical, we can always distinguish between them by the labels "the particle on the left" and "the particle on the right". There is no interchange symmetry here and such particles are called plektons.
The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.
distinguishable in Arabic: جسيمات متماثلة
distinguishable in German: Ununterscheidbare Teilchen
distinguishable in Spanish: Partículas idénticas
distinguishable in French: Particules indiscernables
distinguishable in Galician: Partículas Idénticas
distinguishable in Italian: Particelle identiche
distinguishable in Polish: Cząstki identyczne
distinguishable in Russian: Тождественные частицы
distinguishable in Slovak: Nerozlíšiteľné častice
distinguishable in Swedish: Ourskiljbara partiklar | 2017-03-23 08:08:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9421007633209229, "perplexity": 724.722734389907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218186841.66/warc/CC-MAIN-20170322212946-00009-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.hackmath.net/en/math-problem/24 | # Ultra expensive ramps
The village Čakajovce have two level railway crossings (the ramps), the railway company ZSR decided to replace it. The price for upgrading the two crossings is 302065 Euros.
Calculate how many houses for 77000 euros is possible to obtain at this price. Calculate how many crossings can be reconstructed by using cheaper and more modern type of crossing for 13000 euros.
How many months investment return (302065 euros) when saving one railwayman job in 3-shift operation, 3100 Eur per month and the cost of running the new crossings are 110 euros per month.
For many months, the investment return to a more modern and cheaper crossing at 13000 euros?
Result
How many homes for 77000 Eur can be bought at the cost of Čakajovce rail crossing: 3
How many cheaper crossings for 13000 Euros will be built: 23
How many months return investment: 102
#### Solution:
$a = \lfloor \dfrac{ 302065}{ 77000} \rfloor = 3$
$b = \lfloor \dfrac{ 302065}{ 13000} \rfloor = 23$
$c = \lceil \dfrac{ 302065}{ 3100-110} \rceil = 102$
$d = \lceil \dfrac{ 13000}{ 3100-110 } \rceil = 5$
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https://danshiebler.com/2021-10-12-dynamic/ | The ideas in this post were hashed out during a series of discussions between myself and Bruno Gavranović
Consider a system for forecasting a time series in $$\mathbb{R}$$ based on a vector of features in $$\mathbb{R}^a$$. At each time $$t$$ this system will use the state of the world (represented as a vector in $$\mathbb{R}^a$$) to predict what the value of the time series (a real number) will be at time $$t+1$$. At time $$t+1$$ the system will receive information about the correct value of the time series at time $$t+1$$, represented as a pair $$(x_a, y) \in \mathbb{R}^a \times \mathbb{R}$$, and will need to predict the value of the time series at time $$t+2$$.
One way to build such a system would be to choose a loss function $$l: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$$ and use gradient descent to train a model $$f: \mathbb{R}^p \times \mathbb{R}^a \rightarrow \mathbb{R}$$ on all of the data before some time $$t$$. We could then use that trained model to generate predictions at time $$t+n$$. If the time series is not stationary then this may produce poor results for $$t+n$$ where $$n$$ is large.
Another option would be to continuously update the parameters $$x_p \in \mathbb{R}^p$$ of the model $$f: \mathbb{R}^p \times \mathbb{R}^a \rightarrow \mathbb{R}$$ as each new sample $$(x_a, y) \in \mathbb{R}^a \times \mathbb{R}$$ is observed. This is known as online learning.
An example online learning algorithm is stochastic gradient descent, which we can define as follows:
Initialize $$x_p$$ randomly, and then as each sample $$(x_{a_i}, y_{i})$$ arrives set:
$i \gets mod(i+1, |S|)\\ l_{f_i} \gets l(f(x_p, x_{a_i}), y_{i})\\ x_p \gets x_p - \alpha \nabla l_{f_i}(x_p)$
Stochastic gradient descent describes a dynamical system in which a state in $$\mathbb{R}^p$$ evolves over time in response to inputs in $$\mathbb{R}^a \times \mathbb{R}$$. One of the downsides of stochastic gradient descent is that the update step can be very high variance from $$t$$ to $$t+1$$, which can slow down convergence. One way to get around this is to use the momentum algorithm, which we define as follows:
Initialize $$x_p, x_p'$$ randomly, and then as each sample $$(x_{a_i}, y_{i})$$ arrives set:
$l_{f_i} \gets l(f(x_p, x_{a_i}), y_{i})\\ x_p \gets x_p + \alpha x'_p \\ x'_p \gets (1 - \beta) x'_p - \beta \nabla l(x_p) \\$
Momentum describes a dynamical system in which a state in $$\mathbb{R}^p \times \mathbb{R}^p$$ evolves over time in response to inputs in $$\mathbb{R}^a \times \mathbb{R}$$.
### Lenses and Dynamical Systems
Optimization algorithms like stochastic gradient descent and momentum describe dynamical systems whose state is the function parameters. In this section we dig deeper into this perspective. In particular, we use David Jaz Myer’s category theoretic formulation of dynamical systems to study how we can recombine simpler optimization algorithms to form a more complex algorithm.
The category theoretic formulation of dynamical systems is based on lenses, which are a tool for representing certain kinds of compositions. We will focus entirely on lenses in the category of sets and functions.
A lens $$\left(_{A}^{A'}\right) \xrightarrow{(f_g, f_p)} \left(_{B}^{B'}\right)$$ in the category of sets and functions $$\mathbf{Set}$$ is a pair $$(f_g, f_p)$$ of morphisms (functions):
$f_g : A \rightarrow B \qquad f_p : A \times B' \rightarrow A' \\$
Lenses are powerful because many computations can be expressed in terms of the combination of multiple lenses. The simplest way to combine lenses is to stack them in parallel. Given the lenses:
$\left(_{A}^{A'}\right) \xrightarrow{(f_g, f_p)} \left(_{B}^{B'}\right) \\ \left(_{C}^{C'}\right) \xrightarrow{(g_g, g_p)} \left(_{D}^{D'}\right)$
we define their monoidal product to be the lens:
$\left(_{A \otimes C}^{A' \otimes C'} \right) \xrightarrow{(h_g, h_p)} \left(_{B \otimes D}^{B' \otimes D'}\right) \\ h_g = f_g \otimes g_g \\ h_p = \langle f_p \circ (\pi_0 \otimes \pi_0) , (g_p \circ (\pi_1 \otimes \pi_1)\rangle$
We can also compose lenses directly. Given the lenses:
$\left(_{A}^{A'}\right) \xrightarrow{(f_g, f_p)} \left(_{B}^{B'}\right) \\ \left(_{B}^{B'}\right) \xrightarrow{(g_g, g_p)} \left(_{C}^{C'}\right)$
we define their composition to be the lens:
$\left(_{A}^{A'}\right) \xrightarrow{(h_g, h_p)} \left(_{C}^{C'}\right) \\ h_g = g_g \circ f_g \\ h_p = f_p \circ \langle \pi_0, (g_p \circ ((f_g\circ \pi_0) \otimes \pi_1))\rangle$
We can now characterize dynamical systems as lenses. A discrete system is a lens $$\left(_{S}^{S}\right) \xrightarrow{(f_g, f_p)} \left(_{O}^{I}\right)$$ or equivalently, a set of states $$S$$, a set of inputs $$I$$, a set of outputs $$O$$, and two functions:
• A get (read) function $$f_g: S \rightarrow O$$ that generates an output from a state.
• A put (update) function $$f_p: S \times I \rightarrow S$$ that takes a pair of a state and an input and returns an updated state.
Intuitively, a discrete system represents the stepwise application of the update function $$f_p$$, potentially in response to a sequence of inputs. That is, the discrete system $$\left(_{S}^{S}\right) \xrightarrow{(f_g, f_p)} \left(_{O}^{I}\right)$$ describes a dynamical system whose state $$x_{S_t} \in S$$ at time $$t+1$$ is described by the equation:
$x_{S_{t+1}} = x_{S_t} + f_p(x_{S_t}, x_{I_t}) \\$
where $$x_{I_t} \in I$$ is the system input at time $$t$$.
### Constructing Momentum from Stochastic Gradient Descent
Given a pair of a loss function $$l: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$$ and inference function $$f: \mathbb{R}^p \times \mathbb{R}^a \rightarrow \mathbb{R}$$ the stochastic gradient descent dynamical system $$sg$$ has the following structure:
$\left(_{\mathbb{R}^{p}}^{\mathbb{R}^{p}}\right) \xrightarrow{(sg_g, sg_p)} \left(_{\mathbb{R}^p}^{\mathbb{R}^a \times \mathbb{R}}\right) \\ sg_g(x_p) = x_p \\ sg_p(x_p, (x_a, y)) = -\nabla l(f(x_p, x_a), y)$
This system iteratively updates its state $$x_p$$ each time a new sample $$(x_a, y) \in \mathbb{R}^a \times \mathbb{R}$$ is observed.
We can also define a dynamical system to represent stochastic momentum. Given a pair of a loss function $$l: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$$ and inference function $$f: \mathbb{R}^p \times \mathbb{R}^a \rightarrow \mathbb{R}$$ the momentum dynamical system $$sm$$ has the following structure:
$\left(_{\mathbb{R}^{p} \times \mathbb{R}^{p}}^{\mathbb{R}^p \times \mathbb{R}^{p}}\right) \xrightarrow{(sm_g, sm_p)} \left(_{\mathbb{R}^p}^{\mathbb{R}^a \times \mathbb{R}}\right) \\ sm_g((x_p, x'_p)) = x_p \\ sm_p((x_p, x'_p), (x_a, y)) = (x'_p, - x'_p -\nabla l(f(x_p, x_a), y))$
We can construct momentum from the composition and tensor of stochastic gradient descent with some basic lenses. To start, consider the following discrete systems.
• The discrete system $$add$$ reads the state directly and uses the sum of the input values as the state update:
$\left(_{\mathbb{R}^{p}}^{\mathbb{R}^{p}}\right) \xrightarrow{(add_g, add_p)} \left(_{\mathbb{R}^p}^{\mathbb{R}^p \times \mathbb{R}^p}\right) \\ add_g(x_p) = x_p \\ add_p(x_p, (x'_p,x''_p)) = x'_p+x''_p$
• The discrete system $$cp$$ reads the state directly and uses the current state as the state update:
$\left(_{\mathbb{R}^{p}}^{\mathbb{R}^{p}}\right) \xrightarrow{(cp_g, cp_p)} \left(_{\mathbb{R}^p}^{*}\right) \\ cp_g(x_p) = x_p \\ cp_p(x_p, *) = x_p$
• The discrete system $$sw$$ reads the state directly and swaps the positions of the input values to generate the state update:
$\left(_{\mathbb{R}^p \times \mathbb{R}^p}^{\mathbb{R}^p \times \mathbb{R}^p}\right) \xrightarrow{(sw_g, sw_p)} \left(_{\mathbb{R}^p \times \mathbb{R}^p}^{\mathbb{R}^p \times \mathbb{R}^p}\right) \\ sw_g(x_p, x'_p) = (x_p, x'_p) \\ sw_p((x_p, x'_p), (x''_p, x'''_p)) = (x'''_p, x''_p)$
Consider also the following lenses:
• The lens $$ng$$ uses the negated input value as the state update:
$\left(_{*}^{\mathbb{R}^{p}}\right) \xrightarrow{(ng_g, ng_p)} \left(_{*}^{\mathbb{R}^p}\right) \\ ng_g(*) = * \\ ng_p(*, x_p) = -x_p$
• The lens $$srp$$ reads the left component of the system state and uses the right component to generate the state update:
$\left(_{\mathbb{R}^p \times \mathbb{R}^p}^{\mathbb{R}^a \times \mathbb{R} \times \mathbb{R}^{p}}\right) \xrightarrow{(srp_g, srp_p)} \left(_{\mathbb{R}^p}^{\mathbb{R}^a \times \mathbb{R}}\right) \\ srp_g(x_p, x'_p) = x_p \\ srp_p((x_p, x'_p), (x_a, y)) = (x_a, y, x'_p)$
We can now construct the momentum dynamical system $$sm$$ as the following composition:
$sm = srp \circ (((sg \times ng) \circ add) \times cp) \circ sw \\$
Let’s break this down into its component parts to see this more clearly. We can draw the composition:
$\left(_{\mathbb{R}^{p}}^{\mathbb{R}^{p}}\right) \xrightarrow{ (((sg \times ng) \circ add)_g, ((sg \times ng) \circ add)_p)} \left(_{\mathbb{R}^p}^{\mathbb{R}^a \times \mathbb{R} \times \mathbb{R}^{p}}\right) \\$
where:
$((sg \times ng) \circ add)_g(x_p) = (sg \times ng)_g(add_g(x_p)) = (sg \times ng)_g(x_p) = x_p \\$
and:
\begin{aligned} ((sg \times ng) \circ add)_p(x_p, (x_a, y, c_p)) = \\ add_p(x_p, (sg \times ng)_p(add_g(x_p), (x_a, y, c_p))) = \\ add_p(x_p, sg_p(add_g(x_p), (x_a, y)), ng_p(c_p)) = \\ add_p(x_p, sg_p(x_p, (x_a, y)), -c_p) = \\ add_p(x_p, -\nabla l(f(x_p, x_a), y), -c_p) = \\ -c_p - \nabla l(f(x_p, x_a), y) \end{aligned}
We can build on this to form:
$\left(_{\mathbb{R}^{p} \times \mathbb{R}^{p}}^ {\mathbb{R}^p \times \mathbb{R}^{p}}\right) \xrightarrow{ ((((sg \times ng) \circ add)\times cp)_g, (((sg \times ng) \circ add)\times cp)_p) } \left(_{\mathbb{R}^p \times \mathbb{R}^p}^ {\mathbb{R}^a \times \mathbb{R} \times \mathbb{R}^p}\right) \\ (((g \times ng) \circ add)\times cp)_g(x_p, x'_p) = (x_p,x'_p) \\ (((g \times ng) \circ add)\times cp)_p((x_p, x'_p), (x_a, y, c_p))) = (-c_p - \nabla l(f(x_p, x_a), y), x'_p)$
We can further build on this to form:
$\left(_{\mathbb{R}^{p} \times \mathbb{R}^{p}}^ {\mathbb{R}^p \times \mathbb{R}^{p}}\right) \xrightarrow{ (((((sg \times ng) \circ add)\times cp) \circ sw)_g, ((((sg \times ng) \circ add)\times cp) \circ sw)_p) } \left(_{\mathbb{R}^p \times \mathbb{R}^p}^ {\mathbb{R}^a \times \mathbb{R} \times \mathbb{R}^p}\right) \\ ((((sg \times ng) \circ add)\times cp) \circ sw)_g(x_p, x'_p) = (x_p,x'_p) \\ ((((sg \times ng) \circ add)\times cp) \circ sw)_p((x_p, x'_p), (x_a, y, c_p))) = (x'_p, -c_p - \nabla l(f(x_p, x_a), y))$
Putting it all together we have:
$\left(_{\mathbb{R}^{p} \times \mathbb{R}^{p}}^ {\mathbb{R}^p \times \mathbb{R}^{p}}\right) \xrightarrow{ ((srp \circ ((((sg \times ng) \circ add)\times cp) \circ sw))_g, (srp \circ ((((sg \times ng) \circ add)\times cp) \circ sw))_p) } \left(_{\mathbb{R}^p}^ {\mathbb{R}^a \times \mathbb{R}}\right) \\ (srp \circ ((((sg \times ng) \circ add)\times cp) \circ sw))_g(x_p, x'_p) = x_p = sm_g((x_p, x'_p)) \\ (srp \circ ((((sg \times ng) \circ add)\times cp) \circ sw))_p((x_p, x'_p), (x_a, y)) = (x'_p, -x'_p - \nabla l(f(x_p, x_a), y)) = sm_p((x_p, x'_p), (x_a, y))$
### Conclusions
In this post we explored how we can leverage the composition of dynamical systems to construct complex optimization algorithms from simpler components. In particular, we demonstrated that the momentum optimization algorithms can be constructed from nothing more than stochastic gradient descent and some simple lens operations. It should be simple to extend this strategy to other algorithms like Adagrad and Adam.
Furthermore, we can probably utilize this dynamical systems perspective to reason about the relationship between the optimization process and the data that we feed in from the outside. We can similarly represent the optimization hyperparameters or the configuration of the data generation process as the dynamical system input. | 2022-01-19 01:33:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9524186849594116, "perplexity": 384.2986065738326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00059.warc.gz"} |
https://socratic.org/questions/how-do-you-evaluate-the-function-f-x-x-2-3x-1-for-f-1 | # How do you evaluate the function f(x) = -x^2 + 3x-1 for f(-1)?
Nov 6, 2015
The most obvious way is to replace the $x$'s in the expression with $\left(- 1\right)$ and evaluate using normal arithmetic.
#### Explanation:
If $f \left(\textcolor{b l u e}{x}\right) = - {\textcolor{b l u e}{x}}^{2} + 3 \textcolor{b l u e}{x} - 1$
then
$\textcolor{w h i t e}{\text{X}} f \left(\textcolor{red}{\left(- 1\right)}\right) = - {\textcolor{red}{\left(- 1\right)}}^{2} + 3 \textcolor{red}{\left(- 1\right)} - 1$
$\textcolor{w h i t e}{\text{XXXXXXX}} = - 1 - 3 - 1$
$\textcolor{w h i t e}{\text{XXXXXXX}} = - 5$ | 2021-11-28 23:52:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9354376196861267, "perplexity": 605.0795230824073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358673.74/warc/CC-MAIN-20211128224316-20211129014316-00307.warc.gz"} |
https://www.physicsforums.com/threads/posets-and-minimal-elements-looking-for-an-inductive-proof.658662/ | # Homework Help: Posets and minimal elements - Looking for an inductive proof
1. Dec 13, 2012
### Kolmin
1. The problem statement, all variables and given/known data
Suppose $R$ is a partial order on a set $A$. Then every finite, nonempty set $B \subseteq A$ has an $R-minimal$ element.
2. Relevant equations
Partial orders are characterized by:
Reflexivity: $xRx$
Transitivity: $xRy \wedge yRz \rightarrow xRz$
Antisimmetry: $xRy \wedge yRx \rightarrow x=y$
Minimal elements can be defined in two equivalent ways:
$\neg \exists x \in X (xRb \wedge x \neq b)$
$\forall x \in X (xRb \rightarrow x=b)$
Problems:
First of all I am not sure if the following is a real proof of this statement. I have some problems with inductive proofs and I am particularly worried by the "Assume the subset $B$ of $A$ has cardinality n and it has a $R-minimal$ element" you are gonna find in the proof I wrote down. Can I really assume that?
Secondly, if the proof works, how is it? Too wordly and fuzzy? Efficient and perspicuous?
I have the feeling there is too much, but what can I cut?
Thanks a lot for any of your feedbacks. I am really looking forward to read them.
3. The attempt at a solution
Proof:
Let $B$ be an arbitrary subset of $A$. We prove the theorem by induction on the cardinality of $B$.
i) Base step:
Assume the subset $B$ of $A$ has cardinality 2. By assumption $R$ is a partial order on $A$, thus we have two cases. Either by antisimmetry the two elements are equal, or they are different. If they are equal, by definition, they are both $R-minimal$ elements of $B$. If they are different, one of the two has to be in the relation $R$ with the other element. In both cases, we are assured to have a $R-minimal$ element in $B$.
ii) Inductive step:
Assume the subset $B$ of $A$ has cardinality n and it has a $R-minimal$ element. Adding an element to the subset $B$ improves the cardinality of $B$ to n+1. We define this new set of cardinality n+1 as $B'$. The addition of a new element to $B$ to construct $B'$ gives us three cases.
Case 1. The new element is equal to the minimal element of $B$. Thus, by antisimmetry $B'$ has a two minimal elements.
Case 2. The new element is higher than the minimal element $B$. Thus $B'$ has a minimal element that is the same of $B$.
Case 3. The new element is lower than the minimal element $B$. Thus, $B'$ has a minimal element, that is the element added to $B$ to construct $B'$.
This exhausts all the possibilities. Henceforth the result is proven.
2. Dec 13, 2012
### HallsofIvy
Your proof looks good to me but the word you want in the last line is "hence". "Henceforth" means "from now on".
3. Dec 13, 2012
### Kolmin
Non native writer...
BTW, thanks a lot. I really felt it was too wordly.
4. Dec 13, 2012
### Michael Redei
Your "base step" is flawed. You say "Either by antisimmetry the two elements are equal, or they are different." Who needs antisymmetry (with a Y in "symmetry" by the way) for that? Anyway, if there are two elements, they can never be equal, because then they'd just be one element.
So you begin with two elements, say, a and b. Who says that one needs to bear the relation R to the other? (Remember: "poset" = "PARTIALLY ordered set", i.e. there can be elements a,b that fulfil neither aRb nor bRa.)
If you have two elements a,b, then you can have aRb (which means ¬bRa, because a≠b), in which case a is minimal. Or bRa, which means that b is minimal. Or neither aRb nor bRa, and so both a and b are minimal.
In your "Inductive step" you say that the cardinality of B is "improved" to n+1, i.e. made better. I think you mean "increase" here. And you need to check your three cases:
Case 1 is impossible. If the "new" element is equal to any "older" one, it can't be new. (If you add the element "banana" to the set {apple,banana,carrot}, you're not increasing the cardinality of that set.)
You'r missing a Case 4: what happens if the new element is neither "higher" nor "lower" than the minimal element of B? Again, since R is only a partial ordering, this is entirely possible.
5. Dec 13, 2012
### Kolmin
Huge mistake..I completely forgot that the notion of set implies that.
For a second, I thought that I was implicity assuming that it was a loset and not a poset. So, here we are: indeed I assumed completeness.
I don't see why it is the case.
This. I was looking for that word!
Lot of stuff to think about. Give me a sec.
Btw, I think that my problem is related to the fact that I don't see why you don't need completeness to get the result. In particular I think the main issue is that I don't see why, if neither aRb nor bRa, then both a and b are minimal.
Oh, btw...really thanks a lot.
6. Dec 13, 2012
### Kolmin
Statement:
Suppose $R$ is a partial order on a set $A$. Then every finite, nonempty set $B \subseteq A$ has an $R-minimal$ element.
Proof:
Let $B$ be an arbitrary subset of $A$. We prove the statement by induction on the cardinality of $B$.
i) Base step:
Assume that $B$ is a singleton. Then the only element is by definition a $R-minimal$ element of $B$, by reflexivity of $R$.
ii) Inductive step:
Assume that $B$ has cardinality $n$ and that it has a $R-minimal$ element called $b$. Increase the cardinality of $B$ to $n+1$ by adding an element, say $b'$, and define this new set $B \cup \{b'\}$ of cardinality $n+1$ as $B'$. Thus we have three possible cases that define the relation $R$ between $b$ and $b'$.
Case 1. $bRb'$ : Thus $B'$ has a minimal element that is the same of $B$.
Case 2. $b'Rb$: Thus, $B'$ has a minimal element, that is the element added to $B$ to construct $B'$.
Case 3. $\neg (bRb' \vee b'Rb)$: Thus $b$ is still the $R-minimal$ element of $B'$.
Since this exhausts all the possibilities, the result is proven.
7. Dec 13, 2012
### Kolmin
The server stop gave me enough time to get something hopefully decent that should work.
Still, I have some doubts.
Can I prove the result without going backward to the singleton cases?
Or, in other words, is there a way to prove it having as a base case the two elements one I used in my first attempt?
Is the "by reflexivity of R" redundant?
Is stylistically decent the way in which I introduce those new elements and sets?
Is it mathematically sound?
Too wordly?
Not enough explanations?
Are those lines a bad explanation?
Anyway, thanks a lot for any feedback.
8. Dec 13, 2012
### Michael Redei
Perhaps we need more of those server pauses. I realised that I may have sounded a bit abrupt. I didn't mean to seem impolite, so I'm sorry if I did appear that way.
Yes, "reflexivity" is redundant here. For a singleton element there exists no smaller one, so it must be minimal. In fact, this "no smaller" definition of "minimal" is one that you could use more. Suppose you have two elements a and b such that no smaller elements exist and neither aRb nor bRa is true. Then both a and b are minimal. I'd keep the singleton as your base case though.
A bit repetitious, I'd say, since you're saying the same things more than once. You could shorten this as follows:
Assume that $B$ has cardinality $n$, and that $b$ is an $R$-minimal element of $B$. Now we consider a new set $B'=B\cup\{b'\}$ of cardinality $n+1$. Then we have three possible cases for the relation $R$ between $b$ and $b′$.
Case 3 is missing a part. Suppose neither $bRb'$ not $b'Rb$, as you have done. For $b'$ to be minimal there must be no other element $a$ that is smaller than $b'$. How can you be sure of that?
This is where your whole proof becomes complicated. Instead of looking at a minimal element $b$ of $B$ and constructing three cases, I'd begin with $b'$ and ask: what elements of $B$ stand in the Relation $R$ to $b'$? Suppose $S$ is the set of all these elements, i.e. $S = \{a\in B : aRb' \lor b'Ra\}$. This set has at most $n$ elements, so you can use it for your inductive step.
Now you'll need to fiddle around a bit, depending on whether $b'$ is minimal in $S$ or not, and you need to consider the elements outside $S$ (but still in $B$). You can probably use the fact that $R$ is transitive to show that the elements outside $S$ won't interfere with what happens inside $S$.
9. Dec 13, 2012
### pasmith
You want your result to be true for all non-empty subsets, and singletons are non-empty subsets. I would say that it was obvious from the definition that a singleton subset has an R-minimal element.
The key point is that you're adding an element which isn't in $B$, so you'd better make that clear:
"Assume $B$ has cardinality $n$ and has an R-minimal element $b$. Let $B' = B \cup \{b'\}$ with $b' \in A \setminus B$. Then $B'$ has cardinality $n+1$."
I don't think you need to refer expressly to the cardinality of $B$. Actually much of the set-up could be abbreviated:
"Since a singleton subset has an R-minimal element, it is enough by induction on the cardinality of $B \subset A$ to show that if $B$ has an R-minimal element $b$ then the set $B' = B \cup \{b'\}$ where $b' \in A \setminus B$ has an R-minimal element."
And then you consider the possibilities.
For Case 3 you also have to show that there is no other element $a \in B$ such that $aRb'$.
10. Dec 13, 2012
### Kolmin
Well, don't worry. I appreciate the fact that you had this thought, but it's really not a problem and I didn't feel it. Actually, to be honest, when I started to read your post and I saw "base case" with the inverted commas, at the beginning I thought you wanted to be sarchastic and I kinda liked it, cause the inverted commas with a sarchastic inflexion is the default way in which I would always present my - indeed! - "proofs"...
So, thanks a lot (probably I start to sound repetitive).
Not sure if this is a typo.
Actually with case 3, with the two elements that bear no relations, I wanted to show more or less exactly the opposite, which is that we are at least sure that $b$ (and not $b'$) is a minimal element of $B'$, and we can have at most two different minimal elements (indeed $b$ and $b'$).
But are we "at least sure" of it?
11. Dec 13, 2012
### pasmith
I think you can argue that if $aRb'$ then $a$ must be an R-minimal element of $B$, because otherwise one has $bRa$ and $aRb'$ so that $bRb'$, which we are assuming not to be the case. But $aRb'$, so $a$ is an R-minimal element of $B'$.
12. Dec 13, 2012
### Kolmin
Is it not a bit redundant?
My line of reasoning is more or less the same of that I wrote down in my last reply to Michael Redei.
I add an element to $B$ and I come out with $B'$. Now, I assume that $B$ has a minimal element. So, if the new element and the minimal element of $B$ bear no relation, still $b$ is a minimal element (of - let's say - the right side of the Hasse diagram). So whatever $a$ we prove bears certain relation with $b'$ (on - let's say - the left side of the Hasse diagram), at most it makes us find another minimal element, but we already have it. Thus, it should be redundant.
Is it right or not? This was my line of reasoning when I wrote down my "proof" (yes...inverted commas with sarchastic inflexion!, but not sure it is sound.
Btw, thanks a lot for your feedback.
13. Dec 13, 2012
### Kolmin
14. Dec 13, 2012
### pasmith
Not so much redundant as wrong: I should have said "For Case 3 you also have to show that if there is any other element $a \in B$ such that $aRb'$ then $a$ is an R-minimal element of $B$."
Anyway, I think between the various responses we now have a proof.
15. Dec 14, 2012
### Kolmin
To me even that one looks redundant for the reasons I specified.
16. Dec 14, 2012
### pasmith
On reflection, you are right. Clearly all minimal elements of B are the same size, so either b' is less than all of them, b' is greater than all of them, or b' is the same size as all of them. In all three cases B' has a minimal element.
17. Dec 14, 2012
### Michael Redei
What do you mean by "the same size"? Since R is antisymmetric, if a and b are "the same size", they're actually equal. How would your argument work for the following relation R?
For any two rational numbers $x$ and $y$ we define the relation $R$ as follows: Let $x={n_x}/{d_x}$ and $y={n_y}/{d_y}$ where $(n_x,d_x) = (n_y,d_y) = 1$, i.e. the numerator and denominator of a fraction have no common factor. Then we set
$$xRy ~~ \mbox{iff} ~~ d_x=d_y ~~ \mbox{and} ~~ n_x\leq n_y.$$
Obviously, $xRy$ can only be true if $x\leq y$, but $x\leq y$ doesn't necessarily imply $xRy$.
Now let $B$ be the set of all positive fractions whose numerator is even and whose denominator is a single digit. Then $\frac21,\frac23,\frac25,\frac27,\frac29$ are all minimal elements of $B$, but are they all "the same size"? Suppose they are (since none can be said to be "smaller" or "larger" than an other), what happens if you set $b'=\frac13$? This is $R$-less than $\frac23$, but not comparable to the other elements of $B$.
18. Dec 14, 2012
### pasmith
"a is the same size as b" is defined by the relation $aSb \Leftrightarrow (\neg(aRb \vee bRa)) \vee (a = b)$.
S is obviously reflexive and symmetric and is also transitive, so that for all a, b, and c, $aSb \wedge bSc \Rightarrow aSc$. This is obvious if
any of a,b, and c are equal, so let all three be distinct. Then
$$aSb \wedge bSc \Leftrightarrow (\neg(aRb \vee bRa)) \wedge (\neg(bRc \vee cRb)) \Leftrightarrow \neg(aRb \vee bRa \vee bRc \vee cRb) \\ \Leftrightarrow \neg((aRb \vee bRc) \vee (cRb \vee bRa)) \\ \Leftrightarrow \neg(aRb \vee bRc) \wedge \neg(cRb \vee bRa) \\ \Rightarrow \neg(aRb \wedge bRc) \wedge \neg(cRb \wedge bRa)$$
where the last line follows because for all statements P and Q, $\neg(P\vee Q) \Rightarrow \neg(P \wedge Q)$. We then have
$$\neg(aRb \wedge bRc) \wedge \neg(cRb \wedge bRa) \Leftrightarrow \neg(aRc) \wedge \neg(cRa) \Leftrightarrow \neg(aRc \vee cRa) \Leftrightarrow aSc$$
by transitivity of R. Thus S is an equivalence relation.
It should be obvious that for all distinct a and b there are three possibilities: either aRb, bRa, or aSb. It should also be obvious that if a and b are minimal elements of a subset then aSb.
EDIT: Alternatively one can let $b \in B$ be minimal, but otherwise arbitrary. Then if $b'Rb$ then $b'$ is minimal in $B'$, and if $bRb'$ then $b$ is minimal in $B'$, and if neither $bRb'$ nor $b'Rb$ for any minimal $b$ then minimal elements of $B$ are minimal elements of $B'$.
Last edited: Dec 14, 2012
19. Dec 14, 2012
### Michael Redei
So "same size as" means "incomparable or equal". This is the first time I've seen that definition.
Not at all obvious, and, in fact, false. S need not be transitive.
The transitivity of R implies $\neg(xRy\land yRz) \Leftarrow \neg(xRz)$, but not the converse, $\neg(xRy\land yRz) \Rightarrow \neg(xRz)$. So the first "$\Leftrightarrow$" in that line should only be "$\Leftarrow$".
That follows trivially from your definition of S. Either a is less than b, greater than b, equal to b or incomparable to b.
I still don't see how you arrive at this result though, which seems contrary to the example I have given before:
20. Dec 14, 2012
### Kolmin
I don't enter in the discussion about "same size" because my mathematical skills are not that advanced, even if basing on my limited knowledge I would agree with Michael Redei.
Btw, going back to my original Case 3 of the proof, I think that this example based on fractions supports exactly my line of reasoning. Indeed, if we set $b'=\frac13$, it does change the composition of the set of the minimal elements (it takes the place of $b'=\frac23$), but we don't care, as far as we know that we do have a minimal elements.
In other words, considering for example this random Hasse diagram (it is really not important how it is), let's imagine that $b'$ is something that bears no relation to $\{∅\}$, but it's related to $\{z\}$. Well, we did have before the addition of $b'$ a minimal element, namely $\{∅\}$, and we still have it.
Adding $b'$ doesn't alter the fact that we do have a minimal element: it simply alters the composition of the set of the minimal elements of a given poset, but that's beyond the result we have to prove (I would call it a corollary).
In my "proof", the minimal element that still stands, beyond any addition, is $b$, thus the result is proven. | 2018-11-18 16:50:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8525926470756531, "perplexity": 304.4625543397163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744513.64/warc/CC-MAIN-20181118155835-20181118181835-00309.warc.gz"} |
http://mathhelpforum.com/geometry/43485-constructions.html | 1. ## Constructions
Construct a triangle ABC in which BC = 7 cm, <A = 60o and altitude through A is 3.7 cm. How many such triangles are possible? Please tell me the steps of construction.
2. Originally Posted by ice_syncer
Construct a triangle ABC in which BC = 7 cm, <A = 60o and altitude through A is 3.7 cm. How many such triangles are possible? Please tell me the steps of construction.
Description:
1. Draw line BC
2. Draw a parallel to BC with the perpendicular distance of $h_A$
3. Draw the perpendicular bisector of BC
4. Draw in B (or in C) the angle of (90° - <(A)) = 30° , that means <(MBC) = 30°
5. The arm of this angle crosses the perpendicular bisector in M.
6. Draw the circle around M with radius r = MB (or MC)
7. The intersection points of the circle and the parallel to BC are the points A.
3. I've attached the construction.
4. Originally Posted by earboth
I've attached the construction.
Where's the altitude through A ?
5. Originally Posted by ice_syncer
Where's the altitude through A ?
The altitude is the distance between the two parallel lines and because the angle at b (or C respectively) is greater than 90° the altitude lies outside the area of the triangle. | 2017-10-19 19:18:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7888014316558838, "perplexity": 885.8945672792723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823360.1/warc/CC-MAIN-20171019175016-20171019195016-00010.warc.gz"} |
http://modelingwithdata.org/arch/00000181.htm | ### Microsimulation games, table top games
I wrote a game. It's called Bamboo Harvest, and you can see the rules at this link. You can play it with a standard deck of cards and some counters, though it's much closer to the sort of strategic games I discuss below than poker or bridge. I've played it with others and watched others play it enough to say it's playable and pretty engaging. Ms NGA of Baltimore, MD gets really emotional when she plays, which I take as a very good sign.
Why am I writing about a game on a web page about statistical analysis and microsimulation? I will leave to others the topic of Probability theory in table top games, but there is also a lot that we who write economic models and microsimulations of populations can learn from game authors. After all, the designers of both board games and agent-based models (ABMs) have the same problem: design a set of rules such that the players in the system experience an interesting outcome.
Over the last few decades, the emergent trend among board games have been so-called Eurogames, which are aimed at an adult audience, seek greater interaction among players, and typically include an extensive set of rules regarding resource trading and development. That is, the trend has been toward exactly the sort of considerations that are typical to agent-based models.
A game that has resource exchange rules that are too complex, or is simple enough to be easily solved' will not have much success in the market. In most games, the optimal move in any given situation could theoretically be solved for by a hyperrational player. But the fact that players find them to be challenging demonstrates that the designers have found the right level of rule complexity for a rational but not hyperrational adult. We seek a similar complexity sweet spot in a good ABM. Readers can't get lost in all the moving parts, but if the model is so simple that readers know what your model will do before it is run—if there's no surprise—then it isn't worth running.
Of course, we are unconcerned as to whether our in silico agents are having any fun or not. Also, we get to kill our agents at will.
Simulation designers sometimes have a sky's-the-limit attitude, because processor time is cheap, but game designers are forced by human constraints to abide by the KISSWEA principle (keep it simple, stupid, without extraneous additions). It's interesting to see what game designers come up with to resolve issues of simultaneity, information provision and hiding, and other details of implementation, when the players have only counters and pencil and paper.
###### Market and supply chain
Settlers of Catan is as popular as this genre of games get—I saw it at a low-end department store the other day on the same shelf as Monopoly and Jenga. It is a trading game. Each round a few random resources—not random players—are productive, which causes gluts and droughts for certain resources, affecting market prices. The mechanics of the market for goods are very simple. Each player has a turn, and they can offer trades to other players (or all players) on their turn. This already creates interesting market dynamics, without the need for a full open-outcry marketplace or bid-ask book, which would be much more difficult to implement at the table or in code. How an agent decides to trade can also be coded into an artificial player, as demonstrated by the fact that there are versions of Settlers you can play against the computer.
Some games, like Puerto Rico, Race for the Galaxy, Bootleggers, and Settlers again, are supply chain games. To produce a victory point in Puerto Rico, you have to get fields, then get little brown immigrants to work the fields (I am not making this up), then get a factory to process the crops, then sell the final product or ship it to the Old World. There may be multiple supply chains (corn, coffee, tobacco). The game play is basically about deciding which supply chains to focus on and where in the supply chain to put more resources this round. The game design is about selecting a series of relative prices so that the cost (in time and previous supply-chain items) makes nothing stand out as a clear win.
One could program simple artifical agents to play simple strategies, and if one is a runaway winner with a strategy (produce only corn!) then that is proof that a relative price needs to be adjusted and the simulation redone. That is, the search over the space of relative prices maximizes an objective function regarding interestingness and balance. ABMers will be able to immediately relate, because I think we've all spent time trying to get a simple model to not run away with too many agents playing the same strategy.
I'm not talking much about war games, which seem to be out of fashion. The central mechanism of a war game is an attack, wherein one player declares that a certain set of resources will try to eliminate or displace a defending resource, and the defender then declares what resources will be brought to defense. By this definition, Illuminati is very much a war game; Diplomacy barely is. Design here is also heavily about relative prices, because so much of the game is about which resources will be effective when allocated to which battles.
###### Timing
How does simultaneous action happen when true simultaneity is impossible? The game designers have an easy answer to simultaneously picking cards: both sides pick a card at a leisurely pace, put the card on the table, and when all the cards are on the table, everybody reveals. There are much more complicated means of resolving simultaneous action in an agent-based model, but are they necessary? Diplomacy has a similar simultaneous-move arrangement: everybody picks a move, and an arbitration step uses all information to resolve conflicting moves.
Puerto Rico, San Juan, and Race for the Galaxy have a clever thing where players select the step in the production chain to execute this round, so the interactive element is largely in picking production chain steps that benefit you but not opponents. Setting aside the part where agents select steps, the pseudocode would look like this:
for each rôle:
for each player:
player executes rôle
Typical program designs make it really easy to apply a rôle function to an array of players. Josh Tokle implements a hawk and dove game via Clojure. His code has a game-step where all the birds play a single hawk-and-dove game from Game Theory 101, followed by all executing the death-and-birth-step, followed by all taking a move-step.
It's interesting when Puerto Rico and Race for the Galaxy have this form, because it's not how games usually run. The usual procedure is that each player takes a full turn executing all phases:
for each player:
for each rôle:
player executes rôle
`
I'd be interested to see cases where the difference in loop order matters or doesn't.
###### Topology
One short definition of topology is that it is the study of what is adjacent to what.
The Eurogamers seem to refer to the games with very simple topologies as abstracts—think Go or Chess. Even on a grid, the center is more valuable in Chess (a center square is adjacent to more squares than an edge square) and the corners are more valuable in Go (being adjacent to fewer squares $\Rightarrow$ easier to secure).
Other games with a board assign differential value to areas via other means. War games typically have maps drawn with bottlenecks, so that some land is more valuable than others. Small World has a host of races, and each region is a valuable target for some subset of races.
I'm a fan of tile games, where the map may grow over time (check out Carcassonne), or what is adjacent to what changes over the course of the game (Infinite City or Illuminati).
Other games have a network topology; see Ticket to Ride, where the objective is to draw long edges on a fixed graph.
War games often extol complexity for the sake of complexity in every aspect of the game, so I'm going to set those aside. But the current crop of Eurogames tend to focus on one aspect (topology or resource management or attack dynamics) and leave the other aspects to a barebones minimum of complicatedness. Settlers has an interesting topology and bidding rules, and the rest of the game is basically just mechanics. Carcasonne has the most complex (and endogenous) topology of anything I'm discussing here, so the resource management is limited to counting how many identical counters you have left. Race for the Galaxy, Puerto Rico, and Dominion have crazy long lists of goods and relative prices, so there is no topology and very limited player interaction rules—they are almost parallel solitaire. A lot of card games have a complete topology, where every element can affect every other.
###### An example: Monopoly
Back up for a second to pure race games, like Pachisi (I believe Sorry! is a rebrand of a Pachisi variant). Some have an interactive element, like blocking other opponents. Others, aimed at pre-literate children, like Chutes and Ladders or Candyland, are simply a random walk. Ideally, they are played without parental involvement, because adults find watching a pure random walk to be supremely dull. Adults who want to ride a random walk they have no control over can invest in the stock market.
Monopoly is a parallel supply chain game: you select assets to buy, which are bundled into sets, and choose which sets you want to build up with houses and hotels. On top of this is a Chutes and Ladders sort of topology, where you go around a board in a generally circular way at random speed, but Chance cards and a Go to Jail square may cause you to jump position.
The original patent has an explanation for some of these details—recall that Monopoly was originally a simulation of capital accumulation in the early 20th century:
Mother earth: Each time a player goes around the board he is supposed to have performed so much labor upon mother earth, for which after passing the beginning-point he receives his wages, one hundred dollars[...].
Poorhouse: If at any time a player has no money with which to meet expenses and has no property upon which he can borrow, he must go to the poorhouse and remain there until he makes such throws as will enable him to finish the round.
You have first refusal on unowned properties that your token lands on (then they go up for auction, according to the official rules that a lot of people ignore), and you owe rent when your token lands on owned properties, and Mother earth periodically pays you \\$200. All of these cash-related events are tied to the board movement, which is not the easiest or most coherent way to cause these events to occur. E.g., how would the game be different if you had a 40-sided die and randomly landed on squares all around the board? Would the game be more focused if every player had a turn consisting of [income, bid on available land, pay rent to sleep somewhere] phases?
The confounding of supply chain game with randomization via arbitrary movement is what makes it succesful, because the Chutes and Ladders part can appeal to children (the box says it's for 8 year-old and up), while the asset-building aspects are a reasonable subgame for adults (although it is unbalanced: a competent early leader can pull unsurpassably ahead). But it is the death of Monopoly as a game for adults, because there are too many arbitrary moving parts about going around an arbitrary track.
I can't picture a modern game designer putting together this sort of combination of elements. I sometimes wonder if the same sort of question could be asked of many spatial ABMs (including ones I've written): is the grid a key feature of the game, or just a mechanism to induce random interactions with a nice visualization?
###### Conclusion
Microsimulation designers and for-fun game designers face very similar problems, and if you're writing microsimulations, it is often reasonable to ask how would a board game designer solve this problem?. I discussed several choices for turn order, trading, topology, and other facets, and in each case different choices can have a real effect on outcomes. In these games that are engaging enough to sell well, the game designers could only select a nontrivial choice for one or two facets, which become the core of the game, and other facets are left to the simplest possible mechanism, to save cognitive effort by players.
Also, now that you've read all that, I can tell you that Bamboo Harvest focuses on a shifting-tiles topology, with a relatively simple supply chain. We decided against marketplace/trading rules. | 2017-03-29 09:19:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26732125878334045, "perplexity": 1605.2675726357415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190236.99/warc/CC-MAIN-20170322212950-00516-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://planetmath.org/IntegrationOfPolynomial | integration of polynomial
Theorem.
For all nonnegative integers $n$,
$\int x^{n}\,dx=\frac{1}{n+1}x^{n+1}+C.$
Proof.
It will first be proven that, for any nonnegative integer $n$ and any $a\in\mathbb{R}$,
$\int\limits_{0}^{a}x^{n}\,dx=\frac{1}{n+1}a^{n+1}.$
If $a=0$, the above statement is obvious. If $a>0$, the following computation uses the right hand rule for computing the integral (http://planetmath.org/RiemannIntegral); if $a<0$, the following computation uses the left hand rule for computing the integral:
$\displaystyle\int\limits_{0}^{a}x^{n}\,dx$ $\displaystyle=\lim_{t\to\infty}\sum_{k=1}^{t}\left(\frac{ak}{t}\right)^{n}% \left(\frac{a}{t}\right)$
$\displaystyle=a^{n+1}\lim_{t\to\infty}\frac{1}{t^{n+1}}\sum_{k=1}^{t}k^{n}$
$\displaystyle=a^{n+1}\lim_{t\to\infty}\frac{1}{t^{n+1}}\sum_{l=1}^{n+1}{n+1% \choose r}\frac{B_{n+1-l}}{n+1}(t+1)^{l}$ by this theorem (http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers),
$\displaystyle=a^{n+1}\lim_{t\to\infty}\frac{1}{t^{n+1}}{n+1\choose n+1}\frac{B% _{n+1-(n+1)}}{n+1}(t+1)^{n+1}$
$\displaystyle=\frac{B_{0}}{n+1}a^{n+1}\lim_{t\to\infty}\left(\frac{t+1}{t}% \right)^{n+1}$
$\displaystyle=\frac{1}{n+1}a^{n+1}$
Thus, if $a,b\in\mathbb{R}$, then $\displaystyle\int\limits_{a}^{b}x^{n}\,dx=\int\limits_{0}^{b}x^{n}\,dx-\int% \limits_{0}^{a}x^{n}\,dx=\frac{1}{n+1}b^{n+1}-\frac{1}{n+1}a^{n+1}.$
It follows that $\displaystyle\int x^{n}\,dx=\frac{1}{n+1}x^{n}+C$. ∎
Title integration of polynomial IntegrationOfPolynomial 2013-03-22 15:57:29 2013-03-22 15:57:29 Wkbj79 (1863) Wkbj79 (1863) 30 Wkbj79 (1863) Theorem msc 26A42 | 2019-03-21 22:51:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 18, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9920528531074524, "perplexity": 1023.6355514491811}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202572.7/warc/CC-MAIN-20190321213516-20190321235516-00486.warc.gz"} |
https://web2.0calc.com/questions/the-length-of-a-room-is-1-more-than-3-times-its-width-the-area-of-the-room-is-80-square-meters-find-the-demensions | +0
# The length of a room is 1 more than 3 times its width. The area of the room is 80 square meters. Find the demensions.
0
321
7
The length of a room is 1 more than 3 times its width. The area of the room is 80 square meters. Find the demensions.
Guest May 8, 2015
### Best Answer
#4
+26416
+13
You are nearly right zacismyname. However, you should take a closer look at your calculation of $$1^2-4\times3\times-80$$ under the square root sign. It isn't 960 (almost, but not quite!).
You've nothing to be ashamed of!
.
Alan May 8, 2015
Sort:
### 7+0 Answers
#1
+26416
+8
Let L be the length and W the width
L = 3W + 1 (1) (I assume the room is 1 metre longer than 3 times the width)
L*W = 80 (2)
Using (1) in (2)
(3W + 1)W = 80
Rearrange
3W2 + W - 80 = 0
This factors as
(3W + 16)(W - 5) = 0
Since we can't have a negative width for the room, the only valid solution is W = 5m
Using (1) this means that L = 3*5 + 1 = 16m
.
Alan May 8, 2015
#2
+981
+5
The length of a room is 1 more than 3 times its width. The area of the room is 80 square meters. Find the demensions.
$$w\times{l}=A$$
$$w\times({3w+1})=80$$
$$3w^2+w=80$$
$$3w^2+w-80=0$$
$$w=\frac{-1+\sqrt{1^2-4\times{3}\times{-80}}}{2\times{3}}$$ or $$w=\frac{-1-\sqrt{1^2-4\times{3}\times{-80}}}{2\times{3}}$$
$$w=\frac{-1+\sqrt{961}}{6}$$ or $$w=\frac{-1-\sqrt{961}}{6}$$
$$w=\frac{{-1}+31}{6}$$
$$\mathbf{w=5}$$
$$l=3\times{5}+1}$$
$$\mathbf{l=16}$$
Fixed!
zacismyname May 8, 2015
#3
+981
0
. . . I'm a bit ashamed I didn't see those factors
zacismyname May 8, 2015
#4
+26416
+13
Best Answer
You are nearly right zacismyname. However, you should take a closer look at your calculation of $$1^2-4\times3\times-80$$ under the square root sign. It isn't 960 (almost, but not quite!).
You've nothing to be ashamed of!
.
Alan May 8, 2015
#5
+981
+5
Bad choice of word I would have just preferred to factorise rather than use the formula.
zacismyname May 8, 2015
#6
+91517
+5
You make a great contribution to this forum Zac :))
Melody May 8, 2015
#7
+981
+5
Thanks :)
zacismyname May 8, 2015
### 3 Online Users
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details | 2018-01-24 11:47:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7916279435157776, "perplexity": 2038.1975316650983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084894125.99/warc/CC-MAIN-20180124105939-20180124125939-00558.warc.gz"} |
https://mathoverflow.net/questions/210051/how-to-modify-a-h1-weak-convergence-sequence-so-that-i-have-the-l2-equi-in | # How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
1. $\int_\Omega|\nabla \bar u_n|^2dx\leq \int_\Omega|\nabla u_n|^2dx$ and $\int_\Omega|\nabla \bar u|^2dx\leq \int_\Omega|\nabla u|^2dx$
2. $\bar{u}_n\to \bar{u}$ weakly in $H^1$.
3. $\nabla \bar u_n$ is $L^2$-equi-integrable, i.e., for any $\epsilon>0$ we have there exists $\delta>0$ such that for all set $T\subset \Omega$ with $\mathcal L^N(T)<\delta$ we have $$\sup_{n\in\mathbb N}\int_{T} |\nabla \bar u_n|^2dx<\epsilon. \tag 1$$
My idea is to define $$\bar u_n:=\min_{v\in\mathcal A(u_n)}\left\{\int_\Omega|\nabla v^2|\,dx\right\},\text{ and }\bar u:=\min_{v\in\mathcal A(u)}\left\{\int_\Omega|\nabla v^2|\,dx\right\},$$ where $$\mathcal A(u_n):=\left\{v\in H^1(\Omega), T[v]=T[u_n]\right\},$$ and $T[\cdot]$ denotes the usual trace operator.
The property $1$ is obviously true. The prove of property $2$ I put it at the end of this post. Please help me to check whether it is correct.
However, I can not prove property $3$. The best I can do is assuming $(1)$ does not hold, i.e., there exists a sequence of set $T_n\subset \Omega$ such that $\lim_{n\to 0}\mathcal L^N(T_n)=0$ and $$\lim_{n\to\infty} \int_{T_n}|\nabla \bar u_n|^2dx\geq \epsilon>0$$ and hope to have a contradiction.
We can compute $$\liminf_{n\to\infty}\int_\Omega|\nabla \bar u_n|^2dx\geq \liminf_{n\to\infty}\int_{\Omega\setminus T_n}|\nabla \bar u_n|^2dx+\liminf_{n\to\infty}\int_{T_n}|\nabla \bar u_n|^2dx\geq \int_\Omega|\nabla \bar u\,|^2dx+\epsilon$$ but I can not get any contradiction from here. I feel I need to use the minimality of $\nabla\bar u_n$ but I don't see how...
Any help of new idea of how to construct $\bar u_n$ is really welcome!
Below is how to proof property $2$.
Now let me prove property $2$. Clearly $\bar u_n$ is bounded in $H^1$ and hence, up to a subsequence, $\bar u_n\to u_0$ weakly in $H^1$. I only need to prove that $u_0=\bar u$. To do so, I only need to prove that $u_0$ is the weak solution of PDE $$\begin{cases} -\Delta v=0, & x\in\Omega\\ v=u, & x\in\partial\Omega \end{cases}$$ By weak convergence in $H^1$, we have $$\int_\Omega \nabla u_0\nabla \phi=0$$ for all $\phi\in H_0^1(\Omega)$. I only need to prove that $u_0\in \mathcal A(u)$ then I would be done. To do so, I need to prove $u_0-u\in H_0^1(\Omega)$. I will claim $$\left|\int_\Omega (u_0-u)(x) \partial_i\varphi(x)dx\right|\leq C\|\varphi\|_{L^2(\Omega)}$$ for all $\phi\in C_c^\infty(\mathbb R^N)$.
We observe that \begin{multline} \left|\int_\Omega (u_0-u)(x) \partial_i\varphi(x)dx\right|=\\ \lim_{n\to\infty}\left|\int_\Omega (\bar u_n-u_n)(x) \partial_i\varphi(x)dx\right|=\lim_{n\to\infty}\left|\int_\Omega \partial x_i(\bar u_n-u_n)(x) \varphi(x)dx\right|\\ \leq \lim_{n\to\infty}\|\nabla (\bar u_n-u_n)\|_{L^2}\|v\|_{L^2}\leq C\|v\|_{L^2} \end{multline} as desired, where the 3rd inequality used the fact that $T[\bar u_n-u_n]\equiv 0$.
Hence, by the uniqueness of solution, we have $u_0=\bar u$, and hence property $2$ is true.
PS: I also post this problem in Math Stack Exchange here because this post is just an update of my yesterday's post which exist both on math Stack Exchange as well...I will avoid this problem next time. Sorry!
• Since $u_n\to u$ in $H^1$, you have $\int|\nabla u_n|^2\leq1+\int|\nabla u|^2$ for $n$ large enough. Then by the energy minimizing property $\int|\nabla\bar u|^2$ is bounded uniformly in $n$. Is this what you want or did I miss something? Jun 24, 2015 at 21:20
• @JoonasIlmavirta ah not really. Please see equation $(1)$ for details explanation. Jun 24, 2015 at 23:30
• @Denoising: How about $\bar u = \bar u_n \equiv 0$? This satisfies 1-3.
– gerw
Jul 14, 2015 at 10:39
• @gerw of course you are right. but I wish to preserve the boundary condition as well. Actually I will close this problem since I need to add/cancel some assumptions. Thank you anyway! Jul 14, 2015 at 13:50
• If you like the answer you should upvote it. Apr 10, 2018 at 19:47
I think that the construction that you are proposing by harmonic extension does not work. Indeed consider on the unit disk $\mathbb{B}^2 \subset \mathbb{R}^2$ the function $u_n : \mathbb{B}^2 \to \mathbb{R}$ defined for $x = (x_1, x_2) \in \mathbb{B}^2$ $$u_n (x_1, x_2) = \frac{\operatorname{Re} \bigl((x_1 + i x_2)^n\bigr)}{\sqrt{n}}.$$
It can be checked that the function $u_n$ is harmonic on the disk $\mathbb{B}^2$, therefore $\bar{u}_n = u_n$.
Moreover, $u_n \to 0$ almost everywhere on $\mathbb{B}^2$ as $n \to \infty$, $$\int_{\mathbb{B}^2} \vert \nabla u_n \vert^2 = C,$$ where $C > 0$ does not depend on $n$, and for every compact set $K \subset \mathbb{B}^2$, $$\lim_{n \to \infty} \int_{K} \vert \nabla u_n \vert^2 = 0.$$ It follows from these facts that $u_n \rightharpoonup \bar{0} = 0$ in $H^1 (\mathbb{B}^2)$ and that the sequence $(\nabla u_n)_{n \in \mathbb{N}}$ is not equiintegrable. | 2022-06-29 04:57:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9326995611190796, "perplexity": 158.93602857038695}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103620968.33/warc/CC-MAIN-20220629024217-20220629054217-00138.warc.gz"} |
https://deepai.org/publication/the-order-of-convergence-of-an-optimal-quadrature-formula-with-derivative-in-the-space-w-2-21 | # The order of convergence of an optimal quadrature formula with derivative in the space W_2^(2,1)
The present work is devoted to extension of the trapezoidal rule in the space W_2^(2,1). The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of a integrand. Using the discrete analog of the operator d^2/dx^2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F=span{1,x,e^x,e^-x}. Finally, in the space W_2^(2,1) the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L_2^(2).
• 3 publications
• 1 publication
07/30/2019
### On an optimal quadrature formula for approximation of Fourier integrals in the space L_2^(1)
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### On an optimal quadrature formula for approximation of Fourier integrals in the space W_2^(1,0)
The present paper is devoted to construction of an optimal quadrature fo...
07/31/2019
### Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method
The paper studies Sard's problem on construction of optimal quadrature f...
01/20/2022
### An error estimate for the Gauss-Jacobi-Lobatto quadrature rule
An error estimate for the Gauss-Lobatto quadrature formula for integrati...
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### On computing derivatives of transfer operators and linear responses in higher dimensions
We show that the derivative of the transfer operator with respect to per...
12/03/2013
### A compact formula for the derivative of a 3-D rotation in exponential coordinates
We present a compact formula for the derivative of a 3-D rotation matrix...
08/21/2017
### Economic Design of Memory-Type Control Charts: The Fallacy of the Formula Proposed by Lorenzen and Vance (1986)
The memory-type control charts, such as EWMA and CUSUM, are powerful too...
## 1 Introduction
It is known, that quadrature and cubature formulas, are methods for the approximate evaluation of definite integrals. In addition and even more important, quadrature formulas provide a basic and important tool for the numerical solution of differential and integral equations. The theory of cubature formulas consists mainly of three branches dealing with exact formulas, formulas based on functional-analytic methods, and formulas based on probabilistic methods Sobolev74 ; SobVas
. In the functional-analytic methods the error between an integral and corresponding cubature sum is considered as a linear functional on a Banach space and it is estimated by the norm of the error functional in the conjugate Banach space. The norm of the error functional depends on coefficients and nodes of the formula. The problem of finding the minimum of the norm of the error functional by coefficients and by nodes is called
S.M.Nikol skii problem, and the obtained formula is called the optimal formula in the sense of Nikol skii (see, for instance, Nik88 ). Minimization of the norm of the error functional by coefficients when the nodes are fixed is called Sard s problem. And the obtained formula is called the optimal formula in the sense of Sard. First this problem was studied by A. Sard Sard . Solving these problems in different spaces of differentiable functions various type of optimal formulas of numerical integration are obtained.
There are several methods for constructing the optimal quadrature formulas in the sense of Sard such as the spline method, the function method (see e.g. BlaCom , SchSil ) and the Sobolev method. It should be noted that the Sobolev method is based on using a discrete analog of a linear differential operator (see e.g. Sobolev06 ; Sobolev74 ; SobVas ). In different spaces based on these methods, the Sard problem was studied by many authors, see, for example, IBab ; BlaCom ; CatCom ; HayMilShad10 ; HayMilShad15 ; Koh ; FLan ; Sard ; SchSil ; ShadHay11 ; Sobolev06 ; Sobolev74 ; SobVas ; Zag and references therein.
Among these formulas the Euler-Maclaurin type quadrature formulas are very important for numerical integration of differentiable functions and are widely used in applications. In different spaces the optimality of the Euler-Maclaurin type quadrature and cubature formulas were studied, for instance, in works CatCom ; FLan ; Mic74 ; Schoen65 ; ShadHayNur13 ; ShadHayNur16 ; ShadNur18 ; Zhen81 .
The Euler-Maclaurin quadrature formulas can be viewed as well as an extension of the trapezoidal rule by the inclusion of correction terms. It should be noted that in applications and in solution of practical problems numerical integration formulas are interesting for functions with small smoothness.
The present paper is also devoted to extension of the trapezoidal rule.
We consider a quadrature formula of the form
1∫0φ(x)dx≅N∑β=0(C0[β]φ(hβ)+C1[β]φ′(hβ)) (1)
where are coefficients of the trapezoidal rule, i.e.
C0[0]=h2,C0[β]=h, β=1,2,...,N−1,C0[N]=h2, (2)
are unknown coefficients of the formula (1) and they should be found, , is a natural number. We suppose that an integrand belongs to , where by we denote the class of all functions defined on which posses an absolutely continuous first derivative and whose second derivative is in . The class under the pseudo-inner product
⟨φ,ψ⟩=1∫0(φ′′(x)+φ′(x))(ψ′′(x)+ψ′(x))dx
is a Hilbert space if we identify functions that differ by a linear combination of a constant and (see, for example, Ahlb67 ). Here, in the Hilbert space , we consider the corresponding norm
∥φ|W(2,1)2(0,1)∥=[∫10(φ′′(x)+φ′(x))2dx]1/2. (3)
The difference
(ℓ,φ)=1∫0φ(x)dx−N∑β=0(C0[β]φ(hβ)+C1[β]φ′(hβ)) (4)
is called the error and
ℓ(x)=ε[0,1](x)−N∑β=0(C0[β]δ(x−hβ)−C1[β]δ′(x−hβ)), (5)
is said to be the error functional of the quadrature formula (1), where is the indicator of the interval and is Dirac’s delta function. The value of the error functional at a function is defined as
In order that the error functional (5) is defined on the space it is necessary to impose the following conditions for the functional
(ℓ,1):= 1−N∑β=0C0[β]=0, (6) (ℓ,e−x):= 1∫0e−xdx−N∑β=0(C0[β]e−hβ−C1[β]e−hβ)=0. (7)
The last two equations mean that the quadrature formula (1) is exact for any constant and . We have chosen the coefficients , such that the equality (6) is fulfilled. Therefore we have only condition (7) for coefficients , .
The error functional of the formula (1) is a linear functional in , where is the conjugate space to the space .
By the Cauchy-Schwarz inequality we have the following
|(ℓ,φ)|≤∥φ|W(2,1)2(0,1)∥⋅∥ℓ|W(2,1)∗2(0,1)∥.
Hence we conclude that the error (4) of the formula (1) is estimated by the norm
∥∥ℓ|W(2,1)∗2(0,1)∥∥=sup∥∥φ|W(2,1)2(0,1)∥∥=1|(ℓ,φ)| (8)
of the error functional (5).
The main aim of this work is to find the minimum of the absolute value of the error (4) by coefficients for given in the space . That is the problem is to find the coefficients that satisfy the following equality
∥∥˚ℓ|W(2,1)∗2∥∥=infC1[β]∥∥ℓ|W(2,1)∗2∥∥. (9)
The coefficients which satisfy the last equation are called optimal and are denoted as .
Thus, to obtain the optimal quadrature formula of the form (1) in the sense of Sard in the space , we need to solve the following problems.
Problem 1. Find the norm of the error functional (5) of the quadrature formula (1) in the space .
Problem 2. Find the coefficients that satisfy equality (9).
Here we solve Problems 1 and 2 by Sobolev’s method using the discrete analog of the differential operator .
The paper is organized as follows: in Section 2 using the extremal function of the error functional the norm of this functional is calculated, i.e. Problem 1 is solved; Section 3 is devoted to solution of Problem 2. Here the system of linear equations for the coefficients of the optimal quadrature formulas (1) is obtained in the space . In Subsection 3.1 using the discrete analog of the operator the explicit formulas for the coefficients of optimal quadrature formula of the form (1) are obtained. Furthermore, it is proved that the obtained quadrature formula of the form (1) is exact for any function of the set . Finally, in Subsection 3.2 in the space the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space .
## 2 The norm of the error functional (5)
In this section we study Problem 1. To calculate the norm of the error functional (5) in the space we use the extremal function for this functional (see, Sobolev74 ; SobVas ) which satisfies the equality
(ℓ,ψℓ)=∥∥ℓ|W(2,1)∗2∥∥⋅∥∥ψℓ|W(2,1)2∥∥.
We note that in ShadHay14 for a linear functional defined on the Hilbert space the extremal function was found and it was shown that the extremal function is the solution of the boundary value problem
ψ(2m)ℓ(x)−ψ(2m−2)ℓ(x)=(−1)mℓ(x), (10) (ψ(m+s)ℓ(x)−ψ(m+s−2)ℓ(x))|x=1x=0=0, s=0,1,...,m−1, (11) (ψ(m)ℓ(x)+ψ(m−1)ℓ(x))|x=1x=0=0. (12)
That is for the extremal function the following was proved.
###### Theorem 2.1 (Theorem 2.1 of ShadHay14 )
The solution of the boundary value problem (10)-(12) is the extremal function of the error functional and has the following form
ψℓ(x)=(−1)mℓ(x)∗Gm(x)+Pm−2(x)+de−x,
where
Gm(x)=sgnx2(ex−e−x2−m−1∑k=1x2k−1(2k−1)!) (13)
is the solution of the eqution is any real number and is a polynomial of degree .
Furthermore, there were shown that and
(ℓ,ψℓ)=∥∥ℓ|W(m,m−1)∗2∥∥2. (14)
From Theorem 2.1, in the case , we get the extremal function for the error functional (5) and it has the form
ψℓ(x)=ℓ(x)∗G2(x)+p0+de−x, (15)
where
G2(x)=sgnx2(ex−e−x2−x), (16)
and are any real numbers.
Then, from (14), in the case , using (5) and (15), taking into account equations (6) and (7), we get
∥ℓ∥2=(ℓ,ψℓ)=N∑β=0N∑γ=0(C0[β]C0[γ]G2(hβ−hγ)−C1[β]C1[γ]G′′2(hβ−hγ))++2N∑β=0C1[β](1∫0G′2(x−hβ)dx+N∑γ=0C0[γ]G′2(hβ−hγ))−−2N∑β=0C0[β]1∫0G2(x−hβ)dx+1∫01∫0G2(x−y)dxdy, (17)
where is defined by (16), and are derivatives of , i.e.
G′2(x)=sgnx2(ex+e−x2−1) and G′′2(x)=sgnx2(ex−e−x2). (18)
It is easy to see from (13) and (18) that
G1(x)=G′′2(x). (19)
Thus Problem 1 is solved.
In the next section we study Problem 2.
## 3 Minimization of the norm (17)
Now we consider the minimization problem of the expression (17) by the coefficients under the condition (7). For this we use the Lagrange method of conditional extremum.
Consider the Lagrange function
Ψ(C1[0],C1[1],...,C1[N],d)=∥ℓ∥2+2d(ℓ,e−x).
Taking partial derivatives from the function by , then equating them to 0 and using the condition (7), we get the following system of linear equations with unknowns
N∑γ=0C1[γ]G′′2(hβ−hγ)+de−hβ=F(hβ), β=0,1,2,...,N, (20) N∑γ=0C1[γ]e−hγ=g, (21)
where
F(hβ) = ∫10G′2(x−hβ)dx+N∑γ=0C0[γ]G′2(hβ−hγ), (22) g = e−1−1+N∑γ=0C0[γ]e−hγ. (23)
Here , are defined by (2), , and are unknowns.
The system (20)-(21) has a unique solution for any fixed natural number and this solution gives the minimum to the expression (17). Here we omit the proof of the existence and uniqueness of the solution of this system. These statements can be proved similarly as the proof of the existence and uniqueness of the solution of the discrete Wiener-Hopf type system for the optimal coefficients of quadrature formulas with the form in the space (see Sobolev06 ; Sobolev74 ; SobVas ).
3.1. The coefficients of the optimal quadrature formula (1)
In this subsection we solve the system (20)-(21) and find the explicit forms for optimal coefficients , .
Here we use the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in Sobolev74 ; SobVas . We give some definitions about functions of discrete argument.
Suppose that and are real-valued functions of real variable and are defined in real line .
A function is called a function of discrete argument if it is defined on some set of integer values of .
The inner product of two discrete functions and is defined as the following number
[φ(hβ),ψ(hβ)]=∞∑β=−∞φ(hβ)⋅ψ(hβ),
if the series on the right hand side of the last equality converges absolutely.
The convolution of two discrete argument functions and is the inner product
φ(hβ)∗ψ(hβ)=[φ(hγ),ψ(hβ−hγ)]=∞∑γ=−∞φ(hγ)⋅ψ(hβ−hγ).
Furthermore, for finding the coefficients of the optimal quadrature formula (1) we need the discrete analog of the differential operator . It should be noted that in the work ShadHay04 the discrete analog of the differential operator was constructed. In particular, when from the result of the work ShadHay04 we get the following
###### Theorem 3.1
The discrete analog of the differential operator satisfying the equation
D1(hβ)∗G1(hβ)=δd(hβ)
has the form
D1(hβ)=11−e2h⎧⎪⎨⎪⎩0,|β|≥2,−2eh,|β|=1,2(1+e2h),β=0, (24)
where and
Furthermore, it is easy to check that
D1(hβ)∗ehβ=0 and D1(hβ)∗e−hβ=0. (25)
Now we turn to get the solution of the system (20)–(21) using (24).
Suppose when and . Then we rewrite the system (20)–(21) in the following convolution form
C1[β]∗G′′2(hβ)+de−hβ=F(hβ), β=0,1,...,N, (26) N∑γ=0C1[γ]e−hγ=g. (27)
Here, calculating the right hand sides of equalities (22) and (23), for and we get
F(hβ) = (ehβ8(e−1+1)−e−hβ8(e+1))(h(eh+1)eh−1−2), (28) g = 12(1−e−1)(h(eh+1)eh−1−2). (29)
Taking into account (18) and (19), using Theorem 3.1 we get
D1(hβ)∗G′′2(hβ)=δd(hβ), (30)
Denoting by
u(hβ)=C1[β]∗G′′2(hβ)+de−hβ (31)
the left hand side of the equation (26) we get
C1[β]=D1(hβ)∗u(hβ). (32)
Indeed, if the function is defined at all integer values of , then taking into account Theorem 3.1 and using properties (25) of the function , we have
D1(hβ)∗u(hβ) = D1(hβ)∗(G′′2(hβ)∗C1[β])+D1(hβ)∗(d e−hβ) = C1[β]∗(D1(hβ)∗G′′2(hβ)) = C1[β]∗δd(hβ) = C1[β].
Thus, if we find the function for all integer values of then the optimal coefficients will be found from the equality (32).
The following is true.
###### Theorem 3.2
The coefficients of the optimal quadrature formula of the form (1) in the sense of Sard in the space have the following form
˚C1[0] = h(eh+1)2(eh−1)−1, ˚C1[β] = 0, β=1,2,...,N−1, (33) ˚C1[N] = 1−h(eh+1)2(eh−1).
Proof. From equality (26) taking into account (31) we get that
u(hβ)=F(hβ) for β=0,1,...,N.
Now we find the function for and .
Let then from (31), using the form (18) of the function and equality (27), we have
u(hβ)=−14ehβg+e−hβ14N∑γ=0C1[γ]ehγ+de−hβ.
Similarly, for we obtain
u(hβ)=14ehβg−e−hβ14N∑γ=0C1[γ]ehγ+de−hβ.
Then, keeping in mind the last two equalities and denoting by
D=14N∑γ=0C1[γ]ehγ,
for we get the following
u(hβ)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−14ehβg+(d+D)e−hβ,β≤0,F(hβ),0≤β≤N,14ehβg+(d−D)e−hβ,β≥N. (34)
Here, in the equality (34), and are unknowns. These unknowns can be found from the conditions of consistency of values of the function at the points and . Therefore from (34) when and we obtain the system of linear equations for and . Then, using (28) and (29), after some calculations, we have
⎧⎪⎨⎪⎩d=0,D=18(h(eh+1)eh−1−2)(1−e). (35)
Finally, from (32) for , using (24) and (34) and taking into account (35), by directly calculations we get (33). Theorem 3.2 is proved.
Remark 1. Using (2) and (33), one can get that and . These equalities mean that the optimal quadrature formula of the form (1) with the coefficients (2) and (33) is also exact for functions and . Therefore, keeping in mind equalities (6) and (7), we conclude that the optimal quadrature formula of the form (1.1) with coefficients (2) and (33) is exact for any linear combinations of functions and , i.e. it is exact for elements of the set .
3.2. The norm of the error functional of the optimal quadrature formula (1)
In this subsection we study the order of convergence of the optimal quadrature formula of the form (1) with coefficients (2) and (33), i.e. we calculate the square of the norm (17) of the error functional for the optimal quadrature formula (1).
The following holds
###### Theorem 3.3
Square of the norm of the error functional (5) for the optimal quadrature formula (1) with coefficients (2) and (33) on the space has the form
∥∥˚ℓ|W(2,1)∗2(0,1)∥∥2 = −∞∑n=4Bnhnn! (36) = 1720h4−130240h6+O(h8),
where are Bernoulli numbers.
Proof. We rewrite the expression (17) as follows
∥˚ℓ∥2 = −N∑β=0˚C1[β](N∑γ=0˚C1[γ]G′′2(hβ−hγ)−F(hβ))+N∑β=0˚C1[β]F(hβ) +N∑β=0N∑γ=0C0[β]C0[γ]G2(hβ−hγ)−2N∑β=0C0[β]1∫0G2(x−hβ)dx+1∫01∫0G2(x−y)dxdy,
where is defined by (22).
Hence, taking into account (35), we have
∥˚ℓ∥2=A1+A2−2A3+A4, (37)
here
A1=N∑β=0˚C1[β] F(hβ),A2=N∑β=0N∑γ=0C0[β]C0[γ]G2(hβ−hγ),A3=N∑β=0C0[β]1∫0G2(x−hβ)dx,A4=1∫01∫0G2(x−y)dxdy.
Now we need the following sums which are obtained by using (2) and (33)
N∑β=0C0[β]=1, N∑β=0C0[β](hβ)=12, N∑β=0C0[β](hβ)2=h26+13,N∑β=0˚C1[β]e−hβ=12(1−e−1)(h(eh+1)eh−1−2), N∑β=0˚C1[β]ehβ=12(1−e)(h(eh+1)eh−1−2). (38)
Taking into account (28) and (16), using (38) for , , and we get
A1=1−e28e(h(eh+1)eh−1−2)2,A2=h2(eh+1)2(e2−1)8e(eh−1)2−h2+212−h(eh+1)2(eh−1),A3=h(eh+1)(e2−1)4e(eh−1),A4=e2−12e−76.
Further, putting the last equalities to (37) and after some simplifications we have
∥˚ℓ∥2=1−12h+112h2−heh−1.
Hence, using well known formula , we get (36).
Theorem 3.3 is proved
Remark 2. It should be noted that optimality of the classical Euler-Maclaurin was proved and the error of this quadrature formula was calculated in , where is the space of functions which are square integrable with -th generalized derivative (see, for instance, CatCom ; Schoen65 ; ShadHayNur13 ). In particular, when from Corollary 5.1 of the work ShadHayNur13 we get optimality of the Euler-Maclaurin formula
∫10φ(x)dx≅h(12φ(0)+φ(h)+φ(2h)+...+φ(h(N−1))+12φ(1))+h212(φ′(0)−φ′(1)) (39)
in the space . Furthermore for the square of the norm of the error functional the following is valid
∥˚ℓ|L(2)∗2(0,1)∥2=h4720. (40)
Comparison of equalities (36) and (40) shows that the error of the optimal quadrature formula of the form (1.1) on the space is less than the error of the Euler-Maclaurin quadrature formula (39) on the space .
## Acknowledgments
This work has been done while A.R.Hayotov was visiting Department of Mathematical Sciences at KAIST, Daejeon, Republic of Korea. A.R.Hayotov is very grateful to professor Chang-Ock Lee and his research group for hospitality. A.R. Hayotov’s work was supported by the ’Korea Foundation for Advanced Studies’/’Chey Institute for Advanced Studies’ International Scholar Exchange Fellowship for academic year of 2018-2019
## References
• (1) J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York – London, 1967.
• (3) P. Blaga, Gh. Coman, Some problems on optimal quadrature, Stud. Univ. Babeş-Bolyai Math. 52, no. 4 (2007) 21–44.
• (4) T. Catinaş, Gh. Coman, Optimal quadrature formulas based on the -function method, Stud. Univ. Babeş-Bolyai Math. 51, no. 1 (2006) 49–64.
• (5) A.R. Hayotov, G.V. Milovanović, Kh.M. Shadimetov, On an optimal quadrature formula in the sense of Sard. Numerical Algorithms, v.57, no. 4, (2011) 487-510.
• (6) A.R. Hayotov, G.V. Milovanović, Kh.M. Shadimetov, Optimal quadratures in the sense of Sard in a Hilbert space. Applied Mathematics and Computation, 259 (2015) 637-653.
• (7) P. Köhler, On the weights of Sard’s quadrature formulas, Calcolo, 25 (1988) 169–186.
• (8) F. Lanzara, On optimal quadrature formulae, J. Ineq. Appl. 5 (2000) 201–225.
• (9) C.A. Micchelli, Best quadrature formulas at equally spaced nodes, J. Math. Anal. Appl. 47 (1974) 232-249.
• (10) S.M. Nikol skii, Quadrature Formulas, Nauka, Moscow, 1988.(in Russian).
• (11) A. Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949) 80–91.
• (12) I.J. Schoenberg, On monosplines of least deviation and best quadrature formulae, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965) 144-170.
• (13) I.J. Schoenberg, S.D. Silliman, On semicardinal quadrature formulae. Math. Comp. 28 (1974) 483–497.
• (14) Kh.M. Shadimetov, A.R. Hayotov, Construction of the discrete analogue of the differential operator , Uzbek mathematical journal, 2004, no.2, pp. 85-95.
• (15) Kh.M. Shadimetov, A.R. Hayotov, Optimal quadrature formulas with positive coefficients in space, J. Comput. Appl. Math. 235 (2011) 1114–1128.
• (16) Kh.M. Shadimetov, A.R. Hayotov, Optimal quadrature formulas in the sense of Sard in space, Calcolo 51 (2014) 211–243.
• (17) Kh.M. Shadimetov, A.R. Hayotov, F.A. Nuraliev, On an optimal quadrature formula in Sobolev space , J. Comput. Appl. Math. 243 (2013) 91–112.
• (18) Kh.M. Shadimetov, A.R. Hayotov, F.A. Nuraliev, Optimal quadrature formulas of Euler-Maclaurin type, Applied Mathematics and Computation 276 (2016) 340–355.
• (19) Kh.M. Shadimetov, F.A. Nuraliev, Optimal formulas of numerical integration with derivatives in Sobolev space, Journal of Siberian Federal University. Math. and Phys. 2018, 11 (6) 764-775.
• (20) S.L. Sobolev, The coefficients of optimal quadrature formulas, Selected Works of S.L. Sobolev, Springer, (2006) 561–566.
• (21) S.L. Sobolev, Introduction to the Theory of Cubature Formulas (Russian), Nauka, Moscow, 1974.
• (22) S.L. Sobolev, V.L. Vaskevich, The Theory of Cubature Formulas, Kluwer Academic Publishers Group, Dordrecht, 1997.
• (23) F.Ya. Zagirova, On construction of optimal quadrature formulas with equal spaced nodes (Russian). Novosibirsk (1982), 28 p. (Preprint No. 25, Institute of Mathematics SD of AS of USSR)
• (24) A.A. Zhensikbaev, Monosplines of minimal norm and the best quadrature formulas, Uspekhi Mat. Nauk. 1981, 36, 107–159. | 2022-08-19 19:51:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9130052924156189, "perplexity": 723.2759285807138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573760.75/warc/CC-MAIN-20220819191655-20220819221655-00550.warc.gz"} |
https://homework.cpm.org/category/ACC/textbook/ccaa8/chapter/10%20Unit%2011/lesson/CCA:%2010.3.1/problem/10-110 | ### Home > CCAA8 > Chapter 10 Unit 11 > Lesson CCA: 10.3.1 > Problem10-110
10-110.
Which of the equations below is equivalent to $4(3x−1)+3x=9x+5$? More than one may be equivalent. Justify your answer.
Write out the steps for solving the original equation. Which of the equations match one of your steps?
1. $12x−4+3x=9x+5$
1. $12x−1+3x=9x+5$
1. $11x=14x$
1. $15x−4=9x+5$ | 2020-10-25 05:21:49 | {"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6261878609657288, "perplexity": 1607.34818671586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107887810.47/warc/CC-MAIN-20201025041701-20201025071701-00125.warc.gz"} |
https://www.studysmarter.us/textbooks/physics/college-physics-urone-1st-edition/oscillatory-motion-and-waves/q7pe-what-is-the-period-of-60-hz-electrical-power/ | Suggested languages for you:
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Found in: Page 590
### College Physics (Urone)
Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000
# What is the period of 60 Hz electrical power?
The time period of 60 Hz electrical power is 0.017 s.
See the step by step solution
## Step 1: Identification of the given data
The frequency of electric power is, f = 60 Hz
## Step 2: Definition of frequency
Any particular event that occurs at a specific rate for a defined period of time is termed frequency. It can be determined by taking the inverse of the time period and it is measured in terms of Hertz.
## Step 3: Determination of time period
Write the expression for the time period.
$$T = \frac{1}{f}$$
Here, f is the frequency of electric power.
Substitute all the values in the above expression.
$$\begin{array}{c}T = \frac{1}{{60\;{\rm{Hz}}}} \times \frac{{1\;{\rm{Hz}}}}{{{{\rm{s}}^{ - 1}}}}\\ = \frac{1}{{60\;{{\rm{s}}^{ - 1}}}}\\ = 0.017\;{\rm{s}}\end{array}$$
Thus, the time period is 0.017 s. | 2023-03-27 23:32:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9287656545639038, "perplexity": 1616.545287746767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948708.2/warc/CC-MAIN-20230327220742-20230328010742-00390.warc.gz"} |
https://stats.stackexchange.com/questions/189664/difference-between-anomaly-and-outlier/241978 | # Difference between Anomaly and Outlier
What is the difference between Outlier and Anomaly in the context of machine learning. My understanding is that both of them refer to the same thing.
• Out of curiosity, where in the literature is such a distinction made? I was under the impression that "outliers" have no formal definition, outside of being high leverage and high influence observations. Influence and leverage do have mathematical definitions, but considering something "high" is arbitrary. It seems like arbitrary words are being swapped around. Jan 7 '16 at 18:26
• People who use the word "inlier" implicitly make some kind of distinction between "anomaly" and "outlier," because an in inlier is a kind of anomaly. Since neither "outlier" nor "anomaly" have definite, commonly understood technical definitions, we should expect this question to have multiple answers that differ (at least slightly) from each other.
– whuber
Nov 15 '17 at 19:29
The two terms are synonyms according to:
Aggarwal, Charu C. Outlier Analysis. Springer New York, 2017, doi: http://dx.doi.org/10.1007/978-3-319-47578-3_1
Quotation from page 1:
Outliers are also referred to as abnormalities, discordants, deviants, or anomalies in the data mining and statistics literature.
Bold text is not part of the original text.
The free to download pdf of the book available from the author is here.
• The fact that "outliers" are referred to as "anomalies" does not mean that they are synonymous. "Dogs" are sometimes referred to as "animals", for that matter. I tried to address this in more detail in this answer (I couldn't post it here, because the question is protected). Aug 12 '18 at 15:08
Outlier: a value that you predictably find in your data that indicates your model does not work properly
Anomaly: a value that against all odds you find in your data that indicates your model does work properly
A more serious, less cryptic answer:
The concept of outliers starts from the issue of building a model that makes assumptions about the data. Outliers are often indicators that the model does not describe the data properly and thus we should question the results of our model or quality of our data.
The concept of anomalies starts outside the theoretic world and inside the applied world: we want to look for unusual behavior in our data, sometimes motivated by the fact that we are interested in finding behavior that someone is trying to hide (like a virus in an email). The problem is that since people are trying to hide what they are doing, we don't really know what to look for. So we take a set of "good" data, and decide that whatever we find in our new dataset that doesn't look "good" is an anomaly and worth our time to checkout in more detail. Often, looking for anomalies means looking for outliers in your new data set. But note that these values may be very common in your new dataset, despite being rare in your old dataset!
In summary, the two concepts are very similar in terms of the statistics behind them (i.e. unusual values given your fitted model) but come at the idea from different angles. In addition, when we talk about outliers, we typically mean an unusual data point in the data used to fit our model, where as an anomaly is usually meant as an unusual data point in a dataset outside of the data used to fit our model.
Note: this answer is based on how I've seen the two terms frequently used rather than formal definitions. User experiences may differ.
An anomaly is a result that can't be explained given the base distribution (an impossibility if our assumptions are correct). An outlier is an unlikely event given the base distribution (an improbability).
• Quoting source for the definitions and example would highly improve the answer.
– Tim
Jan 7 '16 at 8:54
• As far as I know they are synonyms. So @H. Iqbal really must quote the source and all readers must then evaluate the authoritativeness of sayd source Jan 7 '16 at 11:48
• Impossibility seems to imply P(X = ANOMALY) = 0 (i.e exactly 0). My understanding of anomaly detection is that the researcher may be interested in events that may have positive probability. Jan 7 '16 at 17:42
The terms are largely used in an interchangeable way. "Outlier" refers to something lying outside the norm - so it is "anomalous". But I have the inpression that "outlier" is usually used for very rare observations. In statistics, on a normal distribution, you would consider three sigma to be outliers. That is 99.7% of your objects are expected to be "normal". "Anomaly" is used much more liberally. If you suddenly have millions of visitors on your website, these are not rare visitors. The sudden increase in visitors however is still "anomalous", whereas each individual visitor is not an "outlier".
It may have been in this article where I saw these differences discussed, but I can't access it right now, unfortunately.
Statistical Analysis and Data Mining, Volume 5, Issue 5, October 2012, Pages 363–387 A survey on unsupervised outlier detection in high-dimensional numerical data
• I think you've subtly hinted at the difference between outliers and anomalies; outliers are used to describe data that doesn't fit a general trend, anomalies describe unusual traffic on a server. 50% jk. Jan 7 '16 at 13:54
Just to muddy the waters further, in climatology anomaly just implies the difference between value and mean, or a deviation:
The term temperature anomaly means a departure from a reference value or long-term average. A positive anomaly indicates that the observed temperature was warmer than the reference value, while a negative anomaly indicates that the observed temperature was cooler than the reference value.
see e.g.
That may well be regarded as outside machine learning, but people interested in the question may be interested in this.
An outlier is a data point that makes it hard to fit a model. You face outliers, often unwillingly, when you are trying to fit a model on your dataset. Removing outliers enables building better (i.e. more generalizable) models. A point $(1,5)$ would be an outlier for the model $y=x$. You ignore it in light of the fact that all your other points $(1,1)$, $(5,5)$, $(3,3.1)$ more closely fit $y=x$.
An anomaly can be one data point, or also a general trend or behavior observed in data after a model has already been built or an understanding of the data-generating process formed. You face anomalies because the system starts behaving differently, or you seek out such data points, because you want to be informed when an event occurs during which your model is not valid. You may care about observing any anomalous behavior in amplitudes of ocean waves, not because you want to throw away those data points and build a better model, but because you want to be aware when a tsunami might be happening.
• I disagree with most of this. First, the first sentence can be your definition of outlier if you like, but it's hard to reconcile with many other definitions or usages. If the data are (1, 1), (2, 2), (3, 3), (much bigger, much bigger) then the much bigger point would often be described as an outlier but there is no problem fitting a model. You might (and should) wonder why the data come that way, but fitting a model is easy. More generally, the principle is that an outlier may be separated from the main body of the data but still consistent with a plausible model. May 4 '17 at 10:42
• Second, if the implication that omitting outliers is just what you should do, then (a) it is often problematic even to say which the outliers are (b) there are many other solutions. The thread stats.stackexchange.com/questions/78063/… ranges more widely than its title to mention several. May 4 '17 at 10:45
• If you follow my link, you'll see that I've already posted at some length on outliers. I don't get any sense on re-reading your answer that you are thinking retrospectively as you seem to be talking about removing outliers while fitting. On re-reading, I note also that the first sentence of your second paragraph includes the idea that an anomaly can be 'a general trend or behaviour', which is unlikely to be what you mean -- or if it is, I don't understand it. May 4 '17 at 12:31
Good question. However, google search on "difference between outliers and anomalies site:.edu" shows that there is no theoretical difference between these two terms. They are being used interchangeably in literature. | 2022-01-18 03:21:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6799106001853943, "perplexity": 680.8988030972896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300658.84/warc/CC-MAIN-20220118002226-20220118032226-00396.warc.gz"} |
https://math.stackexchange.com/questions/1829016/what-is-the-most-general-integral-on-mathbbr | # What is the most general integral on $\mathbb{R}$?
The day I learned about the Lebesgue integral was very exciting. A more general integral than Riemann, which is equal to it for all Riemann integrable functions (on finite domains)? Very cool.
Unfortunately, my curiosity led me to google, and my search results showed:
It turns out I'm more naive than I ever knew.
The question: Is there a "most general integral" of real-valued functions on the real line? One which agrees with the others where they are defined, but is defined on a superset of their domains? ("defined", for me, includes infinite integrals). The Khinchin integral seems like a candidate.
Note: I saw another similar question but it didn't ask about $\mathbb{R}$ specifically, which is my interest.
Note2: I don't mean "trivial" integrals, like one which is defined to be 0 whenever the Riemann integral is not defined, or equal to it otherwise. The answer would presumably have its own wikipedia page.
• An integral is just a linear map from the vector space of real-valued functions to the reals. Using the axiom of choice, you can even "find" an integral that works for all functions, extends Riemann on compact support and even avoids $\pm\infty$ ... it just cannot be computed Jun 16 '16 at 20:56
• Fair - I could define the integral to be 0 any time Riemann is not defined. But what I'm asking is whether one of the known integrals is most general. Jun 16 '16 at 23:40
The "gauge" integral,otherwise known as the Henstock–Kurzweil integral-is the most general integral known defined on subsets of $\mathbb R^n$, which of course includes the real line as a special case. Indeed, a number of mathematicians, including the late Robert Bartle, have suggested the gauge integral replace the Riemann integral in basic analysis/honors calculus courses because not only is it far more general then even the Lebesgue integral on these spaces, it's definition is much simpler. It results from a minor modification of the definition of a partition on a subset of $\mathbb R^n$.As a result, only a careful treatment of "$\epsilon-\delta$" calculus is needed to fully develop it.
A good brief introduction to the gauge integral-with references-can be found here.
• For sake of balance and interest, it seems worth pointing out the following quite old question: Why are gauge integrals not more popular? Jun 17 '16 at 4:49
• Sigh.No one can just make me look brilliant and leave it.........lol Jun 17 '16 at 6:10
• I found two sources (en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral, encyclopediaofmath.org/index.php/Khinchin_integral) saying that the Khinchin integral is more general than the gauge integral.
– Paul
Jun 17 '16 at 13:08
• @Paul While the Khinchin integral is somewhat more general then the gauge integral,it is also quite a bit more sophisticated in machinery and many of the more general aspects are lost on subsets of $\mathbb R^n$. For these purposes,the gauge integral is more general then the Lebesgue integral and therefore covers all cases in these spaces. Jun 17 '16 at 18:31 | 2022-01-23 01:00:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9385548830032349, "perplexity": 448.34499993886925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303917.24/warc/CC-MAIN-20220122224904-20220123014904-00325.warc.gz"} |
https://support.bioconductor.org/p/103874/ | Search
Question: Improve limma-voom trend fit to noisy data
0
7 months ago by
jma199130
jma199130 wrote:
I'm analysing low cell number ChIP-seq data (3 ChIP replicates / 3 Input replicates). The replicates are highly variable due to the low amount of starting material and the number of PCR cycles used for amplification. I am counting reads into windows along the genome and quantile normalising the counts to try and overcome some of this technical variation. I decided to use limma-voom to test for differential windows between the ChIP factor and the Input chromatin (mainly because it allows me to use quantile normalisation, unlike edgeR or DESeq2, please correct me If I'm wrong).
I have uploaded an image of the mean-variance plot produced by limma-voom ( https://ibb.co/eDfSxw ). In my opinion there seems to be three distinct components to the data, which I have circled in a copy of the image ( https://ibb.co/eEkGqG ). To me the windows in red are the low count - high variance windows (These generally correspond to windows which have been highly amplified randomly in one of the replicates). The windows in green are the increasing count - decreasing variance windows (These seem to be windows containing genuine binding). And the windows in orange are the low to medium count - constant variance windows (This seems to be a mix of the other two windows).
My problem is that the windows in the lower half of the orange circle which have a low variance are (I think) being squeezed to the trend line and therefore the actual variance of those windows is inflated. From prior knowledge we know that some of these windows contain genuine binding. The topTable reports a positive logFC, but the FDR is non-significant (I'm using FDR < 0.1 as a threshold).
Overall I think the trend isn't a very good fit to my data, and would like to know if there is anything else I can do to solve this? I should mention I am already using the robust = TRUE and trend = TRUE arguments to eBayes.
modified 7 months ago by Aaron Lun20k • written 7 months ago by jma199130
1
7 months ago by
Aaron Lun20k
Cambridge, United Kingdom
Aaron Lun20k wrote:
Firstly, I don't see any red/orange/green in your plot.
Secondly, quantile normalization is not appropriate in your situation. Quantile normalization forces all samples to have the same empirical distribution of log-CPMs, but this is not desirable when large-scale differences between samples are expected. Your experiment is one such case as you should only see protein binding in the groupo of ChIP samples, which should result in a long tail of large log-CPMs (assuming that your IP was successful). This tail will be lost when the ChIP log-CPM distribution is incorrectly coerced to be the same as that of the control samples, reducing your power to detect differential binding events. There is also a possibility that the distortion in the distribution will create spurious binding events in the control samples.
Finally, your plot shows clear evidence of discreteness near the left edge. This is consistent with the presence of low counts that are difficult to model with voom's continuous approximation. edgeR handles this type of data much better; if you haven't done so already, I would suggest reading the csaw user's guide for how it can be applied to ChIP-seq data. This may also help with the different components in your plot, if their presence is caused by having a lower-than-appropriate mean after the log-transformation in voom; in contrast, edgeR computes the mean from the raw counts, which gives greater weight to high-count binding events.
I've just double-checked on some colleagues computers, the second image should definitely have the highlighted areas ( https://ibb.co/eEkGqG )
I guess the discreteness issue could be improved by filtering based on X values being in N number of replicates (rather than average X count, which I'm using currently). I'll try edgeR and see if that improves the detection power (using prior knowledge from a high-cell number bulk ChIP-seq experiment).
1
The "at least N" filtering strategy is not independent of the test statistic (see this), and will distort the dispersion estimates and p-values. If you're filtering ChIP-seq windows by abundance prior to using edgeR, the average count is a more statistically rigorous approach. In any case, the filtering threshold is more important than the filtering strategy for mitigating discreteness, though as I said before, using edgeR will be much more robust to discreteness in the first place.
Whatever you do, don't use quantile normalization here. I suspect that this is the major cause for your stripes at the left edge, where zeroes get transformed to different quantiles. Check out some alternatives in the csaw user's guide.
Apologies, I meant X count in *any* N samples, so not filtering based on any group information. That should be independent of the test statistic. You're right about threshold > strategy. I'd like to automate the pre-filtering stage somehow (picking a decent threshold based on the data, HTSFilter was developed for this task but it filters too aggressively on our data), but I realise it might better to do independent filtering of the results (similar to DESeq2)
2
Note that "any N samples" is still not an independent filtering strategy for NB models. This is a common misconception, that being blind to the groups will guarantee independence of the filter to the test statistic. The p-value for each window is sensitive to things other than the log-fold changes between groups (e.g., dispersion, of itself and of other windows via empirical Bayes shrinkage), which are affected by the "any N samples" filter.
In any case, I wouldn't worry about the exact value of the filter threshold, just pick something reasonable and go with it. There's a number of semi-automatic strategies for defining "reasonable" in the csaw user's guide. Unfortunately, full automation requires some strong assumptions about how strong you expect the binding to be. | 2018-07-19 00:16:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6069733500480652, "perplexity": 1232.2283454621486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590362.13/warc/CC-MAIN-20180718232717-20180719012717-00100.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=29807&p=92509 | ## 8.57
$\Delta U=q+w$
veneziaramirez 3I
Posts: 57
Joined: Fri Sep 29, 2017 7:07 am
### 8.57
How do we know which balanced equations to use for the reaction given? Where do the equations from the solutions manual come from?
Ishan Saha 1L
Posts: 60
Joined: Fri Sep 29, 2017 7:03 am
### Re: 8.57
Hi, this is just an example of Hess' Law where we add the reactions together to get the final overall balanced overall reaction and the overall reaction enthalpy. I do not think we will be expected to know the balanced equations of formations of compounds without any information given to us on the final. | 2019-12-07 07:33:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44232141971588135, "perplexity": 1769.162854052264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540496492.8/warc/CC-MAIN-20191207055244-20191207083244-00041.warc.gz"} |
https://status.dbogatov.org/docs/badges/ | Code¶
Warning
Use your domain instead of https://status.dbogatov.org
Markdown¶
Here is how to put system health badge in markdown
1 [](https://status.dbogatov.org/)
Here is how to put individual metric health badge in markdown
1 [](https://status.dbogatov.org/home/metric/type/source)
Where type is a metric type (eq. cpuload) and source is a metric source.
Here is how to put service uptime badge in markdown
1 [](https://status.dbogatov.org/home/metric/uptime/url)
Where url is a ping server URL (see configuration) (eq. google.com).
HTML¶
Here is how to put system health badge in HTML
1 2 3
Here is how to put individual metric health badge in HTML
1 2 3
Where type is a metric type (eq. cpuload) and source is a metric source.
Here is how to put service uptime badge in HTML
1 2 3
Where url is a ping server URL (see configuration) (eq. google.com). | 2021-04-11 01:52:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35395434498786926, "perplexity": 10057.016195497265}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038060603.10/warc/CC-MAIN-20210411000036-20210411030036-00328.warc.gz"} |
https://scipost.org/submissions/scipost_202108_00070v1/ | Fermi-gas correlators of ADHM theory and triality symmetry
Submission summary
As Contributors: Tadashi Okazaki Preprint link: scipost_202108_00070v1 Date submitted: 2021-08-30 14:43 Submitted by: Okazaki, Tadashi Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical
Abstract
We analytically study the Fermi-gas formulation of sphere correlation functions of the Coulomb branch operators for 3d $\mathcal{N}=4$ ADHM theory with a gauge group $U(N)$, an adjoint hypermultiplet and $l$ hypermultiplets which can describe a stack of $N$ M2-branes at $A_{l-1}$ singularities. We find that the leading coefficients of the perturbative grand canonical correlation functions are invariant under a hidden triality symmetry conjectured from the twisted M-theory. The triality symmetry also helps us to fix the next-to-leading corrections analytically.
Current status:
Has been resubmitted
Submission & Refereeing History
Resubmission scipost_202108_00070v2 on 9 October 2021
Submission scipost_202108_00070v1 on 30 August 2021
Reports on this Submission
Report
The authors consider N=4 3d supersymmetric gauge theory known as ADHM theory on a three-sphere spacetime. The theory has U(N) gauge group, one adjoint and a certain number of fundamental hypermultiplets. Using the known so-called Fermi-gas approach they rewrite the partition function of the theory and certain correlation functions in terms of a system of N non-interacting fermions and consider its large N limit (or, equivalently, the limit of large chemical in the grand canonical ensemble). The authors find agreement with some predictions from the holographically dual description in M-theory (in particular agreement with the triality symmetry that exchanges three complex lines in the spacetime of M-theory) and numerical results from a previous work by other authors.
I believe that the results and the techniques in this manuscript will be interesting to other researchers working on localization in supersymmetric gauge theories, matrix models, AdS/CFT correspondence and related topics. The paper is generally well written. I would like to recommend it for publication.
Requested changes
I have the following minor suggestions which I think can improve the readability of the paper, particularly for non-specialists:
1) From the expressions (2.8) it seems that the Wigner transform of the Hamiltonian $H_W$ is generically complex valued (in particular its classical part) . Later $(2\pi\mu/\epsilon_1-H_W)$ appears as the argument of the functions like Heaviside step function and Dirac delta function, which are ordinarily defined for a real argument only. I think it would be better if the authors add a clarification about interpretation of such expressions.
2) In the beginning of Section 3 the authors use subscript $n_*$. I suggest that the authors add a comment on what is its meaning, what is the range of the sum in (3.1), and why only $n_*=0$ appears in (3.2).
3) In the formulas like (3.52) (similarly in (3.61)) the authors may consider indicating dependence of $\langle \mathcal{O}\rangle$ on $N$ inside the sum more explicitly , otherwise the formula looks a little confusing.
4) I find that the manuscript in some places (for example around pages 12, 18-21) is quite heavy on technical details which are rather elementary (like calculation of integrals). I think moving them to Appendix might make the reading of the paper more enjoyable. But I leave it up to the authors.
• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -
Report
In this nice and rigorous paper, the authors compute 3-sphere correlation functions of Coulomb branch operators in 3d N=4 ADHM-like gauge theory. The key advance made was an analytical proof of numerical results, which were derived in arXiv:2004.13810. The result further sheds light on the structure of underlying twisted M-theory background and reconfirms triality property of the algebra of operators in a novel way using correlation function. The authors used well-established technique Fermi gas technique masterfully and the result that they obtain is novel and powerful. The paper is expected to generate a new direction in a growing literature of twisted M-theory.
Overall, this paper has top quality. Therefore, without further editing, I recommend to publish it.
• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent
Report
In the paper the authors analyzed the $S^3$ partition function of the 3d N=4 supersymmetric U(N) gauge theory with $l$ fundamental matter multiplets and one adjoint matter multiplet which can be interpreted as N M2-branes placed on an omega-deformed background, where the omega deformation parameters are identified with the mass parameter of the adjoint matter multiplet. They demonstrated that the leading and sub-leading part of the grand potential in the limit of large chemical potential, which correspond to the leading and sub-leading part of the large N free energy, is invariant under the triality symmetry in the omega deformation parameters as suggested from the M-theory background. The authors also evaluated some correlation functions of the same theory and confirmed that the leading part of these quantities are also invariant under the triality symmetry. They further determined the explicit expressions of the sub-leading part of the correlation functions by requiring the triality symmetry.
The analyses of the correlation function are completely original results of the paper. Also, although the large N expansion of the $S^3$ partition function was already obtained in a previous research in a dual description by an ABJM-like theory, the re-interpretation in the omega-deformed M-theory is new. For these reasons I recommend this paper to be published.
• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: - | 2021-10-16 19:09:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7430962920188904, "perplexity": 1127.4660301364502}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584913.24/warc/CC-MAIN-20211016170013-20211016200013-00247.warc.gz"} |
https://interferencias.tech/2017/11/21/ss-2-sniff-tcpdump-packet-analysis/ | # Security Sprint: Week 2 - Choosing a nice point to sniff and using tcpdump for packet analysis
## 21 de noviembre de 2017 ESCRITO POR Paula
Tiempo de lectura ~ 3 minutos
As my second week in my security intensive study a huge opportunity appeared in my life! Casually, I’ve been asked to guide a group of students into Network monitoring and Forensic, which is already my favourite security module. I wanted to study in a wider range, as I already said in week one, but right now I should put most of my efforts in the group goals, also I’m excited with the opportunity of learning from these guys.
Anyway, I’d like to start this article with a brief explanation on “where to sniff”, where in a network should we collect the data from. This is a difficult decision, also not all networks are the same… Let’s take a look into an example.
The Gary’s comic-book store network (sorry for the crappy sketch):
Okay so this is the scheme of Gary’s comic book store network (I completely made up the name, sorry if I guessed a real shop name), there are different paths we should be taking care of, depending on our interests. For example, if we are interested in the path from the workstation to the server, we should be taking care of monitoring in both directions in points from A to E and also another new one between the INTERNET and the WEB SERVER itself.
From the Gary’s laptop to a web server we will be taking care of monitoring F-G-C-D-E+Internet/web server in both directions, or from the local DNS server to a web one on the internet I-H-C-D-E+internet/DNS in both directions, too.
It’s the security expert’s duty to identify where the network could be compromised and then act against it. Each segment we pointed has a group of addresses assigned in what we call “net blocks”. The firewall will translate (NAT-Network addresses translation) the addresses to a different value, for example 192.168.2.100 to 192.168.1.100. On a similar way, we can find NPAT (Network Port Address Translation) for wireless and internal networks.
Anyway based on this, it’s a nice option to discard monitoring points C, D and E to start with as they are kept within the company. Anyway in our case I would say monitoring G, B and H are the best options, as they contain true destination IP addresses.
I’m going to leave the physical access troubles and options for another time, as now my point is exposed, I would like to introduce tcpdump. There are several tools for Network System Monitoring depending on which phase of the analysis we want to focus on. Today I want to focus on Data Presentation. This will help analysts to get data exposed.
Tcpdump displays results in real time (or writes a log) when working against a live network, or a saved trace file for forensic, which comes in handy for CTF’s games for example. Installing tcpdump (in debian jessie, but I guess it’s the same in other common systems like ubuntu) is as easy as:
sudo apt-get install tcpdump
Once we have it, we actually have most of the info in the man page. For example:
tcpdump -n -i eth1 -c 5
In the command we are telling the tcpdump not to resolve IP addresses to hostnames via DNS queries, which interface to monitor hand to count how many packages should it capture. Ah, by the way, you will need to use sudo for tcpdump. We should receive five packages, first one is a UDP (User Datagram Protocol) packet, the rest are TCP (Transmission Control Protocol) packets. With this information we can study the nature of the connection and deduce stuff, from the protocols, ips, data lengths… In case we want to save this data in a log for a later study, we can use -w.
sudo tcpdump -n -i eth1 -c 5 -w example.pcap
And open it using -r to read it later.
tcpdump -n -r example.pcap
Apart from this you guys can use filters, or forcing a different information style (like the timestamp style) and such. Anyway I hope you guys enjoyed this basic tutorial about NSM. Be good, tho!
Originally written in: https://dev.to/terceranexus6/security-sprint-week-2—choosing-a-nice-point-to-sniff-and-using-tcpdump-for-packet-analysis–e9
### A Google (también) le da igual vender la privacidad de tus hijos
No será casualidad que en estos tiempos uno de los grandes objetivos de las megacorporaciones de datos sea hacer dependientes a las insti...… Continuar leyendo | 2019-09-18 13:22:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29165399074554443, "perplexity": 2554.3272165384797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573289.83/warc/CC-MAIN-20190918131429-20190918153429-00356.warc.gz"} |
https://pykeen.readthedocs.io/en/latest/api/pykeen.models.uncertainty.predict_h_uncertain.html | # predict_h_uncertain
predict_h_uncertain(model, rt_batch, num_samples=5, slice_size=None, *, mode=None)[source]
Forward pass using left side (head) prediction for obtaining scores of all possible heads.
This method calculates the score for all possible heads for each (relation, tail) pair, as well as an uncertainty quantification.
Note
If the model has been trained with inverse relations, the task of predicting the head entities becomes the task of predicting the tail entities of the inverse triples, i.e., $$f(*,r,t)$$ is predicted by means of $$f(t,r_{inv},*)$$.
Parameters
• model (Model) – the model used for predicting scores
• rt_batch (LongTensor) – shape: (batch_size, 2) The indices of (relation, tail) pairs.
• slice_size (Optional[int]) – >0 The divisor for the scoring function when using slicing.
• num_samples (int) – >1 the number of samples to draw
• mode (Optional[Literal[‘training’, ‘validation’, ‘testing’]]) – The pass mode, which is None in the transductive setting and one of “training”, “validation”, or “testing” in the inductive setting.
Return type
UncertainPrediction
Returns
shape: (batch_size, num_entities) For each r-t pair, the scores for all possible heads.
This function delegates to predict_uncertain_helper() by using pykeen.models.Model.score_h() (or pykeen.models.Model.score_h_inverse() if the model uses inverse triples) as the score_method.
Warning
This function sets the model to evaluation mode and all dropout layers to training mode. | 2022-11-28 12:23:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4977177381515503, "perplexity": 7623.43674998522}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710503.24/warc/CC-MAIN-20221128102824-20221128132824-00581.warc.gz"} |
https://pos.sissa.it/345/091/ | Volume 345 - International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions (HardProbes2018) - Jets & High-pT Hadrons
The Production of $b\bar{b}$ Dijets in heavy-ion collisions at the LHC
S. Wang,* W. Dai, S.L. Zhang, B.W. Zhang, E. Wang
*corresponding author
Full text: pdf
Pre-published on: 2019 January 11
Published on:
Abstract
We report our recent theoretical calculations for $b\bar{b}$ dijet production in high-energy nuclear collisions. The NLO+parton shower (PS) event generator SHERPA has been employed to provide the pp baseline of $b\bar{b}$ dijet production. A framework which combines the Langevin transport equation to describe the evolution of heavy quarks and the higher-twist scheme to consider the inelastic energy loss of both light and heavy partons has been implemented. Within this framework, we present the theoretical results for the transverse momentum imbalance $x_J=p_{T2}/p_{T1}$ both for inclusive dijets and $b\bar{b}$ dijets in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$~TeV. The energy loss of b-jets is expected to shift $x_J$ to smaller values relative to the pp reference which is consistent with the CMS data. In addition, we show the medium modification for angular correlation of $b\bar{b}$ dijets in A+A collisions at $\sqrt{s_{NN}}=5.02$~TeV. We observe a stronger suppression in the small $\Delta \phi=|\phi_{b1}-\phi_{b2}|$ region where the gluon splitting processes dominate relative to the large $\Delta \phi$ region. The difference leads to a modest suppression on the near-side ($\Delta \phi\sim 0$) and enhancement on the away-side ($\Delta\phi\sim\pi$).
Open Access | 2019-02-21 18:56:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7675458788871765, "perplexity": 3605.5943807654166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247506094.64/warc/CC-MAIN-20190221172909-20190221194909-00025.warc.gz"} |
https://cs.stackexchange.com/questions/103846/proving-b-b-on-a-given-set | # Proving B* = B on a given set
I have the set:
B = {x ∈ {0,1}* | there is an equal number of 0's and 1's in x}
and therefore,
B* = {e,01,10,0011,0101,0110,1100,1010,1001,....etc}
I need to either prove or disprove that B*=B
I believe they are equal, because B* is just the concatenation of one string onto another. I think the trick is since the 0's and 1's have to be of equal number, that concatenating strings would keep the same number of 0's and 1's the same.
I just need a little help in the right direction as to prove B*=B. I was thinking of showing that if
B⊆B* and B*⊆B, that B*=B holds.
Any help would be much appreciated!
• What prevents from showing that $B\subseteq B^*$ and $B^*\subseteq B$? (hint: one these is true independent of the definition of $B$). Have you tried it? Where did you get stuck? – Discrete lizard Feb 4 at 20:46
• So i know B⊆B∗ no matter what B is, just from the definition of B*. What is more difficult is proving that B∗⊆B. I tried making an arbitrary string x in B*, but most proofs involving subsets, there is an equation that must hold. For B* I do not know exactly what property I must use other than my intuition with the concatenation and how the number of 0's and 1's will remain the same. – Ben Feb 4 at 21:17
• " my intuition with the concatenation and how the number of 0's and 1's will remain the same" Your intuition seems right. Now you have to prove it. Induction on the length of the string in B* seems a reasonable approach. – Discrete lizard Feb 4 at 21:24
(Criterion of no-op Kleene star) Let $$V$$ be a language such that $$\epsilon\in V$$. Then $$V=V^*$$ if and only if $$V=VV$$, where $$V^*$$ is the Kleene star.
Proof: "$$\Longrightarrow$$": $$V=V\epsilon\subseteq VV$$ while $$VV\subseteq V^*=V$$.
"$$\Longleftarrow$$". $$V\subseteq V^*$$ by definition. Let $$V_i$$ as defined in Wikipedia. Since $$V_0\subseteq V$$ and assuming $$V_n\subseteq V$$, $$V_{n+1}=V_nV\subseteq VV=V$$, we know $$V_i\subseteq V$$ for all $$i\ge0$$ by induction. Hence $$V^*=\bigcup_{i=0}^\infty V_{i+1}\subseteq \bigcup_{i=0}^\infty V=V\,.$$
Exercise 1. Show that $$\epsilon\in B$$ and $$B=BB$$.
Exercise 2. Let $$V$$ be a non-empty language such that $$V=VV$$. Show that $$\epsilon\in V$$. (Hence for a non-empty language $$V$$, $$V=V^*$$ if and only if $$V=VV$$.)
Exercise 3. Show that $$V^*=(V^*)^*$$.
I believe you're on the right track. I would try to write this formally as proofs. It's been a while since I've been in discrete math but I would argue something along the lines of
Given B = {x ∈ {10,01}* |s.t. there is an equal number of 0's and 1's in x}
Given there are an equal number of 0's and 1's in x, the difference between 0's and 1's are zero. (There may be a formal proof missing here...)
Given B* = {y ∈ {Bn + Bm} | s.t. Bn and Bm are both a subset of B and B* is the concatenation of 2 subsets of B.
Therefor ANY y is a concatenated of some subset B which each have an equal sum of 0's and 1's, thus the sum of total 0's and 1's of any subset B combination would also sum to an even number of 0's and 1's.
A bit wordy but the best I could do for now.
• I am afraid that I would not say there is a formal or even informal proof in this answer. The description of $B^*$ just manifests itself from nowhere. That description and the following explanation is so fuzzy that it sounds like the kind of explanation that might be understood by all who had understood and that will not be easy to be understood by anybody who had not understood. – Apass.Jack Feb 4 at 22:21
• It doesn't really manifest from nowhere though, I think orlp made a good step towards an official proof. Look at the proof for even numbers and try to re-use that cs.stanford.edu/~jtysu/proofs.pdf – Zakk Diaz Feb 5 at 19:10
Hint: let $$a=01$$ and $$b=10$$ and argue about $$\{a, b\}^*$$.
HINT:
Let $$w$$ be a string in $$B*$$. Then $$w$$ is the concatenation of 0 or more strings from $$B$$, i.e. $$w = v_1v_2v_3\ldots v_k$$, where each $$v_i \in B$$. Now consider how many $$0$$s and $$1$$s are in $$w$$. How many $$0$$s does $$v_1$$ contribute? How many $$1$$s does $$v_1$$ contribute? Can you write the difference between the number of $$1$$s and $$0$$s in $$w$$ as a summation over $$v_i$$?
Also keep in mind that $$w$$ could be $$\epsilon$$, which you may want to handle as a separate case. | 2019-11-14 21:48:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 48, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8275653123855591, "perplexity": 278.39179732899436}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668539.45/warc/CC-MAIN-20191114205415-20191114233415-00554.warc.gz"} |
https://zbmath.org/?q=rf%3A1229.62006 | # zbMATH — the first resource for mathematics
A Bayesian approach to the estimation of maps between Riemannian manifolds. II: examples. (English) Zbl 1282.62017
Summary: Let $$\Theta$$ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space $$E^s$$, and let $$\gamma$$ be a smooth map of $$\Theta$$ into a Riemannian manifold $$\Lambda$$. An unknown state $$\theta \in \Theta$$ is observed via $$X=\theta+\epsilon\xi$$, where $$\epsilon>0$$ is a small parameter and $$\xi$$ is a white Gaussian noise. For a given smooth prior $$\lambda$$ on $$\Theta$$ and smooth estimators $$g(X)$$ of the map $$\gamma$$ we have derived a second-order asymptotic expansion for the related Bayesian risk [the authors, ibid. 16, No. 4, 281–297 (2007; Zbl 1229.62006)]. In this paper, we apply this technique to a variety of examples.
The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [the authors, loc. cit.] are naturally applicable to these regression problems.
##### MSC:
62C10 Bayesian problems; characterization of Bayes procedures 62C20 Minimax procedures in statistical decision theory 62F12 Asymptotic properties of parametric estimators 53B20 Local Riemannian geometry
Octave
Full Text:
##### References:
[1] R. Abraham and J. E. Marsden, Foundations ofMechanics (Addison-Wesley, 1978). [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, 1989). [3] L. Butler and B. Levit, ”A Bayesian Approach to the Estimation of Maps between Riemannian Manifolds”, Math. Methods Statist. 16(4), 1–17 (2007). · Zbl 1229.62006 · doi:10.3103/S1066530707040011 [4] Y. Chikuse, Statistics on Special Manifolds, in Lecture Notes in Statistics (2003), Vol. 174. · Zbl 1026.62051 [5] J.W. Eaton, GNU Octave Manual, 2nd ed. (Network Theory Limited, 2008). [6] J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, in C.B.M.S. Regional Conference Series (AMS, 1983). · Zbl 0515.58011 [7] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1978). [8] D. Husemoller, Fiber Bundles, 3rd ed. (Springer, New York, 1994). [9] R. A. HornandC. R. Johnson, Topics in Matrix Analysis, Corrected reprint of the 1991 original (Cambridge Univ. Press, Cambridge, 1994). [10] P. T. Kim, ”Decision Theoretic Analysis of Spherical Regression”, J. Multivariate Analysis 38, 233–240 (1991). · Zbl 0727.62013 · doi:10.1016/0047-259X(91)90042-Z [11] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques (Dunod, Paris, 1972). [12] Maxima.sourceforge.net, Maxima CAS Version 5.18.1. http://maxima.sourceforge.net/ (April 2009). [13] J. Milnor, Morse Theory (Princeton Univ. Press, 1963). [14] B. O’Neill, Semi-Riemannian Geometry (Academic Press, 1983). [15] W. Rudin, Functional Analysis (McGraw-Hill, 1991). · Zbl 0867.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-04-22 17:34:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7698578238487244, "perplexity": 1409.1406559295765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00513.warc.gz"} |
https://www.achieversrule.com/2017/04/shortcut-tricks-to-solve-time-and-distance.html | # Shortcut Rules to Solve Problems on Time and Distance
## Effective for IBPS PO - SBI PO Exam
Here we will start a series of Quantitative Aptitude Shortcut Tricks for your upcoming SBI - IBPS - SSC and Other Government Competitive Exams. We will try to cover up all topics of the quantitative Aptitude Sections from which question was generally asked.
Note: The page may takes sometime to load the Quantitative formula's. If you face any problem just comment below the posts.
Trick - 1
• If a certain distance is covered at x km/hr and the same distance is covered by y km/hr, then the average speed during the whole journey is
$\frac{2xy}{x+y}km/hr$
Trick - 2
• A person walking at a speed of x km/hr reaches his destination ${{x}_{1}}$ hrs late. Next time he increases his speed by y km/hr, but still he is late by ${{y}_{1}}$ hrs. The distance of his destination from his house is given by
$\left[ \left( {{x}_{1}}-{{y}_{1}} \right)\left( x+y \right)\frac{x}{y} \right]km$
Trick - 3
• If a person does a journey in T hours, and the first half at ${{S}_{1}}$ km/hr and the second half at ${{S}_{2}}$ km/hr, then the distance
$=\frac{2\times time\times {{S}_{1}}\times {{S}_{2}}}{{{S}_{1}}+{{S}_{2}}}$
where, ${{S}_{1}}$ = speed during first half and
${{S}_{2}}$ = speed during second half of journey
Trick - 4
• The distance between two stations, A and B is D km. A train starts from A and moves towards B at an average speed of x km/hr. If an another train starts from B, t hours earlier than the train at A, and moves towards A at an average speed of y km/hr, then the distance from A, where the two trains meet is
$\left[ \left( D-ty \right)\left( \frac{x}{x+y} \right) \right]km$
Trick - 5
• If a train travelling x km an hour leaves a place and t hours later another train travelling y km an hour, where y>x, in the same direction, then they will be together after travelling $\left[ \frac{t\left( xy \right)}{y-x} \right]km$ from the starting place.
Trick - 6
• If the new speed of a person is $\frac{a}{b}$ of the usual speed, then the change in the time taken to cover the same distance is $\left( \frac{b}{a}-1 \right)\times$ usual time or, usual time is given by
$\left[ \frac{change\operatorname{in time}}{\left( \frac{b}{a}-1 \right)} \right]hrs$
Questions for Practice
Q1. A man covers a certain distance by car driving at 70 km/hr and he returns back to the starting point riding on a scooter at 55 km/hr. Find his average speed for the whole journey.
Q2. A boy walking at a speed of 10 km/hr reaches his school 15 minutes late. Next time he increases his speed 2 km/hr, but still he is late by 5 minutes. Find the distance of his school from his house.
Q3. A mother car does a journey in 10 hrs, the first half at 21 km/hr and the second half at 24 km/hr. Find the distance.
Q4. The distance between two stations, Delhi and Amritsar is 450 km. A train starts at 4 pm from Delhi and moves towards Amritsar at an average speed of 60 km/hr. Another train starts from Amritsar at 3.20 pm and moves towards Delhi at an average speed of 80 km/hr. How far from Delhi will the two trains meet and at what time ?
Q5. A train travelling 25 km an hour leaves Delhi at 9 a.m. and another train travelling 35 km an hour starts at 2 p.m. in the same direction. How many km from Delhi will they be together ?
Q6. Walking $\frac{3}{4}$ of his usual speed, a person is 10 min late to his office. Find his usual time to cover the distance.
Answer 5. $437\frac{1}{2}km$ | 2018-12-11 04:25:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8430805802345276, "perplexity": 1140.4857437895785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823565.27/warc/CC-MAIN-20181211040413-20181211061913-00627.warc.gz"} |
https://www.physicsforums.com/threads/hydrostatic-force-problem.221463/ | # Hydrostatic Force Problem
## Homework Statement
An aquarium 8 m long, 4 m wide, and 4 m deep is full of water. Find the following: the hydrostatic force on one end of the aquarium.
## The Attempt at a Solution
I already found the pressure and force on the bottom of the aquarium...now, my main issue understanding what the question means when it says 'end.' Do they mean one of the side walls? One half of the aquarium? If anyone happens to know what that likely means, that'd be awesome.
What I've tried so far is Density*gravity*L/2*W, which was wrong.
1000*9.8*4*4 = 156800.
Is it just the force on the bottom of the aquarium divided by 2? That almost seems too easy... | 2022-01-28 00:34:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.551446259021759, "perplexity": 591.3570744684312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305317.17/warc/CC-MAIN-20220127223432-20220128013432-00415.warc.gz"} |
http://etheses.bham.ac.uk/1325/ | eTheses Repository
# Legal stratagems (hiyal) and usury in Islamic commercial law
Ismail, Muhammed Imran (2010)
Ph.D. thesis, University of Birmingham.
PDF (2205Kb)
## Abstract
This thesis investigates the subject of legal stratagems $$(hiyal)$$ in Islamic jurisprudence, in general and more particularly the $$hiyal$$ used to evade the usury $$(ribā)$$prohibition. The context of this thesis is the nascent Islamic finance industry in which these $$hiyal$$ play a leading role. The $$hiyal$$ have been appropriated from the classical Islamic legal corpus without appreciating their historical contextual framework. This thesis seeks to explicate that framework and clarify the purpose and role of those $$hiyal$$ as envisaged in the discourse of the classical Islamic jurists. The $$hiyal$$ are shown to be premised upon a teleology which demarcates them as normative exits, $$makhārij$$. The $$makhārij$$ are conditioned by the systematic reasoning of the Ḥanafī jurists, which both justifies their utility and circumscribes their juridical remit. The $$hiyal$$ of $$ribā$$ are demonstrated to have been utilised primarily as substitutes for philanthropy, and not in the commercial sector. The commercial sector relied on the Islamic prescriptions for equity investment partnerships which precluded the need for interest based loans. Although the jurists sanctioned the $$hiyal$$ of $$ribā$$ for the poor, they did so at the expense of systematic consistency. This means that these $$hiyal$$, as opposed to the $$makhārij$$, are not regarded as normative exits, but rather, as transitory concessions. The use of these $$hiyal$$ as financial norms is therefore unwarranted. The substantive repercussions of this juridical reassessment were demonstrated using the historical experience of the Ottomans, where the long term use of the $$hiyal$$ of $$ribā$$ resulted in the negative socio-economic conditions generally associated with usurious economies.
Type of Work: Ph.D. thesis. Khir, Bustami Colleges (2008 onwards) > College of Arts & Law Department of Theology and Religion HF CommerceBP Islam. Bahaism. Theosophy, etc University of Birmingham 1325
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
Repository Staff Only: item control page | 2016-08-24 12:08:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19529907405376434, "perplexity": 4742.23687902121}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982292181.27/warc/CC-MAIN-20160823195812-00223-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://mathhelpboards.com/threads/problem-of-the-week-69-september-23rd-2013.6612/ | # Problem of the Week #69 - September 23rd, 2013
Status
Not open for further replies.
#### Chris L T521
##### Well-known member
Staff member
Here's this week's problem.
-----
Problem: Suppose we are given an exact sequence of finite dimensional $K$-vector spaces and $K$-linear maps:
$0\rightarrow V_1\rightarrow V_2\rightarrow\cdots\rightarrow V_n\rightarrow 0.$
Prove that
$\sum\limits_{i=1}^n (-1)^i\dim(V_i) = 0.$
-----
Hint:
Use induction on $n$. Note that if $V_1\xrightarrow{\phantom{xx}\phi_1\phantom{xx}}{}V_2\xrightarrow{\phantom{xx}\phi_2\phantom{xx}}{}V_3\xrightarrow{\phantom{xx}\phi_3\phantom{xx}}{}V_4$ is exact, then so is $V_1\rightarrow V_2\rightarrow \ker(\phi_3)\rightarrow 0$.
Remember to read the POTW submission guidelines to find out how to submit your answers!
#### Chris L T521
##### Well-known member
Staff member
This week's problem was correctly answered by johng. You can find his solution below.
I see no need to induct. Here's my solution:
Status
Not open for further replies. | 2021-01-20 01:06:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5729419589042664, "perplexity": 2907.218644045009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519843.24/warc/CC-MAIN-20210119232006-20210120022006-00325.warc.gz"} |
https://plainmath.net/281/write-an-equation-in-slope-intercept-form-2x-plus-3y-equal-7-4-5 | # Write an equation in slope intercept form. 2x+3y=7, (4,5).
Write an equation in slope intercept form. $2x+3y=7$, (4,5).
You can still ask an expert for help
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Cristiano Sears
Slope intercept form:
$Y=MX+B$
$M=Slope$
$B=Yintercept$
So switch it around.
$3y=2x+7$
Divide by 3 to isolate y.
$y=\left(\frac{2}{3}\right)x+2\left(\frac{1}{3}\right)$ | 2022-07-02 13:45:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.748335063457489, "perplexity": 3511.717007327834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104141372.60/warc/CC-MAIN-20220702131941-20220702161941-00343.warc.gz"} |
http://mathlesstraveled.com/2007/11/02/nuclear-pennies-game-analysis/ | ## Nuclear Pennies Game: Analysis
And now, for the promised analysis of the Nuclear Pennies Game!
First, recall the rules of the game: there is a semi-infinite (i.e. with a beginning but no end) strip of squares, each of which can contain a stack of any number of pennies (or no pennies at all). You are allowed to “split” a penny by replacing it with two pennies, one in the square on either side. This rule can also be run in reverse; two pennies separated by exactly one space can be “fused” into a single penny on the middle square.
We saw how to get from a single penny on square #7 to a single penny on square #1. However, I claimed that it is impossible to move a single penny from square #7 to square #2. Let me show you why.
We’re going to start by doing something that might seem a bit odd, but you’ll see why soon enough. We’re going to label the nth square with $x^n$. Like this:
(Remember, $x^0 = 1$.) Now, we will think of the number of pennies on a given square as being multiplied by the appropriate power of x. That way, every configuration of pennies is represented by a polynomial. For example, if there are three pennies in square #0, two pennies in square #2, and one penny in square #5, we would represent that by the polynomial $3 + 2x^2 + x^5$.
Now, here’s the key idea: we want to pick a value for x so that two polynomials have equal values if they represent penny configurations that you can convert between while only making legal splitting and fusing moves. If we can do that, all you need to do to see whether it is possible to move from one configuration to another is to write down their polynomials, plug in the magical value of x, and see if they are the same!
So, how can we find this magical value of x? Well, let’s suppose we start with a single penny on square #1. That would be represented by the polynomial $x$. By the splitting rule, we can replace this with a penny on square #0 and a penny on square #2, giving us the polynomial $1 + x^2$. We want these polynomials to have the same value, since we can convert between them with a legal move, so we must have $x = 1 + x^2$. Now, what if we had started with a penny on, say, square #4 and applied the splitting rule? In that case we would have $x^4 = x^3 + x^5$… but that’s really the same thing as $x = 1 + x^2$, since we can just divide both sides by $x^3$ (assuming that x is not zero). The same goes for any starting location, so the equation $x = 1 + x^2$ covers all possible applications of the splitting rule. What about the fusion rule? Well, remember, the fusion rule is really just the splitting rule backwards, so it gives us the exact same equation.
So, both rules can be summed up by this single equation: $x = 1 + x^2$. All we have to do is solve it to find our magical value of x. Well, using the quadratic formula, that’s not hard: we get $x = 1/2 + i\sqrt{3}/2$. Aha, a complex number, just as I promised!
Now, if it were possible to move a single penny from square #7 to square #2, then we would have $x^7 = x^2$, which is the same as $x^5 = 1$. Well, is that true? It turns out that it isn’t, as you can verify for yourself — so, moving a single penny from square #7 to square #2 isn’t possible! In fact, the smallest power of x for which $x^n = 1$ is 6. x, in fact, is one of the complex sixth roots of 1. So if you want to start with a single penny and move it somewhere else, you can only move it by six squares at a time!
The Nuclear Pennies game is originally from Dan Piponi at A Neighborhood of Infinity (from whom I also stole a lot of the nice pictures), who based it on the ideas in the paper Seven Trees in One, by Andreas Blass. The mathematically intrepid might want to try reading some of the original paper, which is really quite excellent — but the content of the paper is way deeper than what I’ve written about here, so don’t be discouraged if you don’t understand a lot of it (I don’t understand the entire second half of the paper!). Just trying to give credit where credit is due.
This entry was posted in algebra, complex numbers, games, proof. Bookmark the permalink.
### 3 Responses to Nuclear Pennies Game: Analysis
1. George Bell says:
It seems to me the same argument proves something more general, suppose you begin with ANY number of pennies on a single square. Then you ask the question, can I make moves that place all the pennies again in a single pile, anywhere else? The answer is that you can only do this at a square which is a multiple of 6 away from the original, and the number of pennies in the final pile must be the same as that in the original.
Also I don’t quite see any reason for making the board semi-infinite, why not just make the board infinite in both directions, and use negative powers of x. At least then you never have to worry about “running into the edge of the board”.
Did Andreas Blass come up with this game? I have scanned through the paper, and I didn’t see a specific mention of the game, although it is there in abstraction. Or did Dan Piponi (A Neighborhood of Infinity author) come up with the game?
2. Brent says:
George, you’re absolutely right, it does prove that more general result! There are a lot of details I left out — including that one — trusting my readers to come to their own conclusions. =)
Dan Piponi came up with the game; however, as you saw, he based it on the ideas in Blass’s paper.
For the purposes of the Nuclear Pennies game, you’re absolutely right, you could do exactly as you suggest: make the board infinite in both directions and use negative powers of x in the analysis. This is actually an interesting point, and it means that my analysis above actually has a small technicality which I’ve swept under the rug: I showed that $x^6 = 1$, but if you interpret this in terms of board positions, you’ll see that this actually isn’t true, because the left edge of the board means you can’t move out of the “1” position (i.e. a single penny in square #0)!
Blass’s paper is really about binary trees: a binary tree is a structure which is either empty, or has a single node with two children which are binary trees. If we represent the set of all binary trees by the letter T, we have $T = 1 + T^2$ — that is, the set of binary trees T contains the empty tree (1) along with all possible pairs of binary trees (T^2). You can see how this corresponds to the fission/fusion rules and the equation $x = 1 + x^2$. However, in the context of binary trees (and sets/types in general), negative powers of T are meaningless, which is why the nuclear pennies board is only semi-infinite — that way it is a useful tool for understanding isomorphisms of trees! I don’t know if that makes sense but if you’re interested, I would suggest reading the beginning of Blass’s paper.
3. “There are a lot of details I left out — including that one — trusting my readers to come to their own conclusions. =)”
You’re like a kind older brother at an Easter egg hunt deliberately missing an egg for a younger sibling to find. | 2014-09-18 01:35:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 18, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7486259341239929, "perplexity": 291.72652627556494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657125113.78/warc/CC-MAIN-20140914011205-00327-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://techwhiff.com/learn/a-beam-of-unpolarized-light-with-intensity-of-i/203752 | # A beam of unpolarized light with intensity of I_0 passes through a series of four 100%...
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https://www.eurandom.tue.nl/event/yep-xv-information-diffusion-on-random-networks/ | • This event has passed.
# YEP XV "Information Diffusion on Random Networks"
## Mar 25 - Mar 29
#### Summary
When available, the slide presentations of the speakers have been added to this website, please see "Abstracts".
The "Information diffusion on random graphs" workshop is the 15th workshop in the ‘Young European Probabilists’ yearly workshops.
Diffusion processes in networks manifests themselves in many real-life scenarios, such as epidemic spreading, viral marketing and power blackouts. This YEP workshop focuses information diffusion on networks. The phenomenon of information diffusion recently attracted vast attention across a wide range of research fields, including mathematics, physics, computer science, and social sciences. Therefore, this YEP will focus not only on purely probabilistic aspects, but also take an algorithmic and application perspective. The aim of the workshop is to bring together junior and senior researchers from probability and from other fields, and to bridge the corresponding scientific communities.
The workshop will have three mini courses by internationally renowned researchers, giving an opportunity to junior as well as senior attendants to learn about a new topic related to information diffusion. Other than that, the workshop will consist of invited talks by junior and senior researchers.
#### Organizers
Remco van der Hofstad TU Eindhoven Nelly Litvak University of Twente/TU Eindhoven Clara Stegehuis TU Eindhoven
#### Speakers
When available, the slide presentations of the speakers have been added to this website, please see "Abstracts".
Tutorial speakers:
Frank Ball University of Nottingham Mia Deijfen Stockholm University Renaud Lambiotte University of Oxford
Invited speakers:
Claudio Castellano Sapienza, Rome Eric Cator Radboud University Nijmegen Wei Chen Microsoft Research Asia Petter Holme Tokyo Institute of Technology Marton Karsai ENS Lyon Juliá Komjathy TU Eindhoven Lasse Leskelä Aalto University Naoki Masuda University of Bristol Peter Mörters Cologna University David Sirl University of Nottingham Chi Tran Université des Sciences et Technologies de Lille Daniel Valesin University of Groningen Rose Yu Northeastern University
Contributed talks/posters
During the conference we have a few slots available for contributed talks by participants.
Schedule
#### Abstracts
Caio Alvez (contributed)
In this talk we will discuss our recent work introducing a model of preferential attachment random graphs where the asymptotic ratio between vertices and edges of the graph is governed by a non-increasing regularly varying function f: N-> [0,1], which we call the edge-step function. We prove general results about the associated empirical degree distribution, as well as topological results about the graph's clique number and diameter. Except for the case of the diameter of slowly varying functions, which exhibit a wider range of behavior, our results depend essentially on the index of regularity of f at infinity. We then discuss applications of the above results for the contact process and bootstrap percolation process in these random graphs. Joint work with Rémy Sanchis Rodrigo Ribeiro and Daniel Valesin.
Frank Ball (tutorial)
Epidemics on networks
PRESENTATION: Epidemics on networks 1 Epidemics on networks 2
There has been considerable interest in the past two decades in models for the spread of epidemics on networks. The usual paradigm is that the population is described by an undirected random graph and disease can spread only along the edges of the graph. This mini-course gives an introduction to the analysis of SIR (susceptible-infective-recovered) epidemics on configuration model (and related) networks, which is by far the most studied class of such epidemics. Topics covered include:
• branching process approximation for the early stages of an epidemic, which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak;
• susceptibility sets and the final outcome of a major outbreak;
• effective degree analysis of models, which yields a functional central limit theorem (CLT) for the temporal behaviour and a CLT for the final outcome of a major epidemic;
• models with superimposed household structure, a key component of human populations which can have a significant impact on disease dynamics;
• vaccination schemes, including acquaintance vaccination which targets high-degree individuals.
Wei Chen (invited)
Information and Influence Propagation in Social Networks: Modeling and Influence Maximization
PRESENTATION: Information and Influence Propagation in Social Networks
Information and influence propagation is a fundamental phenomenon in social networks that leads to many applications both for business and for public good, such as viral marketing, social recommendations, rumor control, epidemic prevention, etc. In this talk, I will survey the research area on information/influence diffusion dynamics and the influence maximization problem, which is the problem of selecting a small number of seed nodes in a social network such that their influence spread is maximized. The talk will cover basic stochastic diffusion models, algorithmic techniques for scalable influence maximization, as well as some of my recent research work on influence-based centrality, competitive and complementary influence diffusion, etc.
Emilio Cruciani (contributed)
We investigate the behavior of a simple majority dynamics on networks of agents whose interaction topology exhibits a community structure. By leveraging recent advancements in the analysis of dynamics, we prove that, when the states of the nodes are randomly initialized, the system rapidly and stably converges to a configuration in which the communities maintain internal consensus on different states. This is the first analytical result on the behavior of dynamics for non-consensus problems on non-complete topologies, based on the first symmetry-breaking analysis in such setting.
Our result has several implications in different contexts in which dynamics are adopted for computational and biological modeling purposes. In the context of Label Propagation Algorithms, a class of widely used heuristics for community detection, it represents the first theoretical result on the behavior of a distributed label propagation algorithm with quasi-linear message complexity. In the context of evolutionary biology, dynamics such as the Moran process have been used to model the spread of mutations in genetic populations (Lieberman, Hauert, and Nowak 2005); our result shows that, when the probability of adoption of a given mutation by a node of the evolutionary graph depends super-linearly on the frequency of the mutation in the neighborhood of the node and the underlying evolutionary graph exhibits a community structure, there is a non-negligible probability for species differentiation to occur.
Mia Deijfen (tutorial)
Competing growth on lattices and graphs
PRESENTATION: Competition -References
Competing first passage percolation describes the growth of two competing infections on an underlying graph structure. It was first studied on the Z^d-lattice. The main question is if the infection types can grow to occupy infinite parts of the lattice simultaneously, the conjecture being that the answer is yes if and only if the infections grow with the same intensity. Recently, the model has been analyzed on more heterogeneous graph structures, where the degrees of the vertices can have an arbitrary distribution. In this case, it turns out that also the degree distribution plays a role in determining the outcome of the competition. I will give a survey of existing results, both on Z^d and on heterogeneous graphs, and describe open problems. I will also describe related competition models such as the multitype contact process and models driven by moving particles.
Peter Gracar (contributed)
Spread of infection by random walks - Multi-scale percolation along a Lipschitz surface
A conductance graph on $\mathbb{Z}^d$ is a nearest-neighbor graph where all of the edges have positive weights assigned to them. We first consider a point process of particles on the nearest neighbour graph $(\mathbb{Z}^d,E)$ and show some known results about the spread of infection between particles performing continuous time simple random walks. Next, we extend consider the case of uniformly elliptic random graphs on $\mathbb{Z}^d$ and show that the infection spreads with positive speed also in this more general case. We show this by developing a general multi-scale percolation argument using a two-sided Lipschitz surface that can also be used to answer other questions of this nature. Joint work with Alexandre Stauffer.
Petter Holme (invited)
Temporal networks of human interaction
Juliá Komjathy (invited)
How to stop explosion by penalising transmission to hubs
In this talk we study the spread of information in infinite inhomogeneous spatial random graphs.
To model the spread of information in social networks, we take a spatial random graph that is scale free, that is, the degree of a vertex follows a power law with exponent tau in (2,3). One common approach to model the spread information is then to equip each edge with a random and iid transmission cost L, and study the cost of the least-cost past between vertices. In these graphs, it was observed earlier than it is possible to reach infinitely many vertices within finite cost, as long as the cumulative distribution function of L is not doubly-exponentially flat close to 0. This phenomenon is called explosion, and it seems off from reality for cases where individual contact is necessary, e.g., spreading of viruses, etc.
We introduce a penalty to transmit the information to hubs, and increase the cost of transmission through an edge with expected degrees W and Z by a factor that is a power of the product WZ.
We find a threshold behaviour between explosion, depending on how steep the cumulative distribution function of L increases at 0: it should be at least polynomially steep, where the exponent depends on both the power-law exponent tau and the penalty-exponent.
This behaviour is arguably a better representation of information spreading processes in social networks than the case without penalizing factor.
Renaud Lambiotte (tutorial)
PRESENTATION: Random walks on networks 1 Random walks on networks 2
Diffusion and Communities in Networks
The presence of communities, or clusters, in networks is well known to affect diffusive processes. Conversely, tracking the trajectories of random walkers on the graph can be used to uncover communities hidden in large graphs. During this tutorial, I will review the relations between the two sides of the problem, and present in detail community detection methods based on first-order and higher-order Markov models, as well as methods allowing to uncover non-assortative communities in networks.
Lasse Leskelä (invited)
Statistical graph models induced by overlapping communities of variable sizes and strengths
Information transmission in today's society is more and more realized through overlapping communities of various sizes and strengths. This talk discusses a statistical network model where a pair of nodes sharing a community are linked with probability determined by the community strength. The model is parametrized by a limiting empirical joint distribution of community sizes and strengths, allowing to capture the property that large communities often provide weaker pairwise links. A natural property of the model is that high variability of community sizes causes the degree distribution to have heavy tails. The main focus of this talk is to discuss the effect of size-strength correlations on graph parameters relevant to information diffusion, especially the transitivity spectrum. Based on joint work with Mindaugas Bloznelis (Vilnius U).
Naoki Masuda (invited)
Epidemic processes on dynamically switching networks: Effects of commutator and concurrency
Epidemic processes on temporally varying networks are complicated by complexity of both network structure and temporal dimensions. We analyse the susceptible-infected-susceptible (SIS) epidemic model on regularly switching networks, where each contact network is used for a finite fixed duration before switching to another. First, we analyse the epidemic threshold under a deterministic approximation called the individual-based approximation. We show that, under this approximation, temporality of networks lessens the epidemic threshold such that infections persist more easily in temporal networks than in their static counterparts. We further show that the commutator bracket of the adjacency matrices at different times is empirically a useful predictor of the impact of temporal networks on the epidemic threshold. The second topic is the effects of concurrency (i.e., the number of neighbours that a node has at a given time point) on the epidemic threshold in the stochastic SIS dynamics. For a particular switching network model, we show that network dynamics can suppress epidemics (i.e., yield a higher epidemic threshold) when nodes' concurrency is low (where stochasticity effects are stronger) and can enhance epidemics when the concurrency is high.
Peter Mörters (invited)
Metastability of the contact process on evolving scale-free networks
We study the contact process in the regime of small infection rates on scale-free networks evolving by stationary dynamics. A parameter allows us to interpolate between slow (static) and fast (mean-field) network dynamics. For two paradigmatic classes of networks we investigate transitions between phases of fast and slow extinction and in the latter case we analyse the density of infected vertices in the metastable state. The talk is based on joint work with Emmanuel Jacob (ENS Lyon) and Amitai Linker (Universidad de Chile).
Gergely Odor (contributed)
In sensor based source localization we attempt to detect the source of an epidemic process spreading in a graph, given the time of infection of the sensor nodes. We are interested in the minimal number of sensors we need to select for perfect detection when the epidemic is deterministic (i.e. each sensor reports its distance from the source), and the graph is drawn from the Erdos-Renyi distribution. When the sensors are selected before any of the observations are made, this problem reduces to the Metric Dimension problem, which has already been analysed for Erdos-Renyi graphs. In this talk, we consider a modified version of the problem, when the sensors are selected sequentially, adaptively to previous observations. We present tight bounds for the reduction in the number of required sensors compared to the non-adaptive version of the problem.
Guilherme Reis (contributed)
Interacting diffusions on random graphs
We consider systems of diffusion processes whose interactions are described by a graph. For example, traditional mean-field interacting diffusions correspond to a complete interaction graph. In recent years some effort has been directed to understanding more general interactions. When the interaction graph is random, in the particular case of the Erd\H{o}s-R\'{e}nyi random graph, we show how the behavior of this particle system changes whether the mean degree of the Erd\"{o}s-R\'{e}nyi graph diverges to infinity or converges to a constant. When the mean degree converges to a constant we exploit a locality property of this system. Loosely speaking, the locality property states that information does not propagate too fast over the graph for this kind of particle system.
Markus Schepers (contributed)
The local clustering coefficient in hyperbolic random graphs
The local clustering coefficient is a quantity which has been studied for its influence on diffusive processes on a graph. For a given vertex of the graph it measures the extent to which its neighbourhood resembles a complete graph. Hyperbolic random graphs are given by a collection of points distributed uniformly in a hyperbolic disk with edges between nearby vertices. This model was invented by Krioukov et al. and has been suggested as a suitable model for real-world networks such as the Internet.
In this project we study the local clustering coefficient averaged over all vertices and averaged over all vertices of degree k in the hyperbolic random graph in the probabilistic limit (convergence in probability) as the number of vertices n tends to infinity.
We consider both the case of a fixed degree k, as well as a sequence of degrees (kn) tending to
infinity. We derive exact analytic limiting expressions as well as the asymptotic scaling
(including the multiplicative constant).
(joint work with: Nikolaos Fountoulakis, Pim van der Hoorn, Tobias Müller)
David Sirl (invited)
A network epidemic model with preventive rewiring
Network epidemic models have developed enormously in the last 20 years or so in response to some of the unrealistic assumptions of homogeneity in most simple epidemic models. A significant feature of most epidemic-on-a-network models is that the epidemic evolves on a static network.
We consider an SIR (Susceptible - Infectious - Removed) epidemic spreading on a configuration-model network (a random network with specified degree distribution), with the addition of some simple network dynamics. The addition is to allow susceptible individuals to "drop" connections to infectious neighbours. A further extension permits such susceptible individuals to then "rewire" to connect instead with someone else in the population.
For the model with dropping only (i.e. with no rewiring), we present some limit theorems (in the limit of large population size) for the temporal evolution of the model and for the final size of the epidemic (the number of initial susceptibles that are ultimately recovered). For the model with rewiring included too, we show that whilst the preventive behaviour of rewiring is always rational at the individual level, it may have negative consequences at the population level.
This work is joint with Frank Ball (Nottingham), Tom Britton (Stockholm) and KaYin Leung (Stockholm).
Réka Szabo (contributed)
We consider an inhomogeneous percolation model on an oriented regular tree, where besides the usual bonds, additional bonds of a certain length are also present. Percolation is defined on this graph, by letting these additional edges be open with probability q and every other edge with probability p. We give an improved lower bound for the critical curve which delimits the set of pairs (p, q) for which there is almost surely no infinite cluster. Furthermore, we show that the cluster of the root has the same distribution as the family tree of a certain multi-type branching process, which allows us to state some limit theorems. Joint work with D. Valesin and B. N. B. de Lima.
Sam Thomas (contributed)
We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate μ, while at the same time a random walker moves on G at rate 1, but only along edges which are open. In this talk I present recent results proving cutoff in the case when G is the complete graph and the bond percolation parameter is of order 1/n, ie we consider a random walk on dynamical Erdos-Renyi graph. We do this via an explicit coupling argument. Joint work with Perla Sousi
Chi Tran (invited)
User-driven exploration of social networks with application in epidemiology
To understand the spread of certain diseases such as HIV or HCV, the modelling of social networks (sexual partners or people who inject drug together) is important. In the case of HCV, the network is hidden since drug use is illegal. We have designed in Paris a 'Respondent-driven' study to discover the social network of people who inject drugs (PWIDs). The underlying idea is to have the graph explored by random (branching) walks: each interviewee receives a certain number of coupons that she/he distributes to her/his injection partners. After having described the general case, we focus on what happens for the family of Stochastic Block Model graphs. Which proportion of the graph can we discover and what can be said on the topologies that are found ?
Percolation on the Random Intersection Graph with Communities
The Random Intersection Graph with Communities (RIGC) models a network based on individuals and communities they are part of, with two key features: each community has its arbitrary internal structure described by a small graph, and communities are allowed to overlap. It generalizes the classical Random Intersection Graph (RIG) model, and is constructed based on a Bipartite Configuration Model. We study percolation, i.e., independent removal of edges, as a simple model for a randomized information spread: we view the connected component of a vertex as the cluster this vertex is able to broadcast information to. We show that percolation on the RIGC, in particular, percolation on the classical RIG, is (again) an RIGC with different parameters, and prove that percolation on the RIGC exhibits a phase transition, in terms of whether a linear-sized component persists. We may touch on robustness, and why robustness of edge and vertex percolation behave differently.
Daniel Valesin (invited)
The asymmetric multitype contact process
We study a class of interacting particle systems known as the multitype contact process on Z^d. In this model, sites of Z^d can be either empty or occupied by an individual of one of two species. Individuals die with rate one and send descendants to neighboring sites with a rate that depends on their (the parent's) type. Births are not allowed at sites that are already occupied. We assume that one of the types has a birth rate that is larger than that of the other type, and larger than the critical value of the standard contact process. We prove that, if initially present, the stronger type has a positive probability of never going extinct. Conditionally on this event, it takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove a complete convergence theorem. Joint work with Pedro L. B. Pantoja and Thomas Mountford.
Rose Yu (invited)
Learning Graph Diffusion with Deep Neural Networks
Diffusion processes on graphs have complex dynamics. Due to their complexity, learning graph diffusion often relies on strong assumptions or is computationally expensive. Deep neural networks provide flexible models for modeling complex data. While existing deep neural networks have shown to be highly effective in, for example, computer vision and natural language processing, off-the-shelf deep models have limited utility in modeling graph-structured data.
In this talk, I will showcase how to design deep neural networks to learn the dynamics of graph diffusion. In particular, I will discuss (1) Diffusion Convolution Recurrent Neural Networks (DCRNN): a neural sequence model for spatiotemporal forecasting and (2) DAG to DAG Recursive Neural Networks (D2DRNN): a message passing neural network for DAG to DAG translation. I will also demonstrate successful applications of these models to real-world traffic prediction and Boolean expression simplification tasks.
Xiangying (Zoe) (contributed)
The Contact Process on Random Graphs and Galton-Watson Trees
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $\lambda_2=0$ and (ii) when it is geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $\lambda_c(n)$ for prolonged persistence'' on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $\lambda_c(n) \to 0$ we give estimates on the rate of convergence. Physicists tell us that $\lambda_c(n) \sim 1/\Lambda(n)$ where $\Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
Dong Yao (contributed)
The symbiotic contact process
We consider a contact process on $\ZZ^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or host individuals of species A and/or B. Multiple occupancy by the same species at a single site is prohibited. Symbiosis is represented by a reduced death rate $\mu \in [0,1)$. If only one specie is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $\mu$ for the two particles at that site. We prove that the critical infection rate $\lambda_c(\mu)$ for weak survival is of order $\sqrt{\mu}$, which coincides with the mean field calculation. We also investigate the nature of the phase transition. We show that in dimension $d=1$ the survival of the system is through oriented percolation. We also show that, for all dimensions, the phase transition is continuous and $\lambda_c(\mu)$ is 1 (regardless of the value of $\mu$), if we let particles move around with a rate going to infinity. The talk is based on ongoing work with Rick Durrett.
Xiu-Xiu Zhan (contributed)
Information Diffusion Backbones in Temporal Networks
Information diffusion on a temporal network can be modeled by viral spreading processes such as the Susceptible-Infected (SI) spreading process. An infected node meaning that the node possesses the information could spread the information to a Susceptible node with a given spreading probability β whenever a contact happens between the two nodes. Progress has been made in the understanding of how temporal network features and the choice of the source node affect the prevalence, i.e. the percentage of nodes reached by the information. In this work, we explore further: which node pairs are likely to contribute to the actual diffusion of information, i.e. appear in a diffusion trajectory? How is this related to the local temporal connection features of the node pair? Such deep understanding of the role of node pairs is crucial to explain and control the prevalence of information spread. First, we propose the construction of an information diffusion backbone G_B (β) for an SI spreading process with an infection probability β on a temporal network. The backbone is a weighted network where the weight of each node pair indicates how likely the node pair contributes to a diffusion process starting from an arbitrary node. Second, we investigate the relation between the backbones with different infection probabilities on a temporal network. We find that the backbone topologies obtained for low and high infection probabilities approach the backbone G_B (β→0) and G_B (β=1), respectively. The backbone G_B (β→0) equals the integrated weighted network, where the weight of a node pair counts the total number of contacts in between, a local temporal connection feature. Finally, we discover a local connection feature among many other features that could well predict which node pairs are likely to appear in G_B (β=1), whose computation complexity is high. This local feature encodes the time that each contact occurs, pointing out the importance of temporal features in determining the role of node pairs in a dynamic process beyond the features of the integrated network.
#### Registration
Link to the online registration form: REGISTRATION
#### Practical Information
Information on travel, location etc. : INFORMATION
Start:
Mar 25
End:
Mar 29
Event Category:
## Venue
Eurandom
Metaforum
Eindhoven, Netherlands | 2019-10-18 09:30:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5672476887702942, "perplexity": 820.2271508445273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986679439.48/warc/CC-MAIN-20191018081630-20191018105130-00215.warc.gz"} |
http://fips.fi/stroke_workshop_2018/titles.htm | # Workshop on Stroke classification and monitoring using Electrical Impedance Tomography
## Department of Mathematics and Statistics, University of Helsinki
### 25-26 April 2018
Stefan Björkman, University of Helsinki, Finland. Title: Presentation of the Saari unit of Department of Production Animal Medicine Bachir Dekdouk, Tampere University of Technology, Finland. Title: Understanding Stroke Manifestations and Project Outlook for EIT Imaging Abstract: For the last two decades, interests of EIT community in stroke detection, led the commonly known causes of stroke generally described as either blockage or leakage of blood flow into the brain to formulate the problem as merely the formation of single local volumes with relatively shortage or excess of blood respectively, which can be detected by an increase or decrease in the transcranial impedances. Thus, this mis-representation of the pathology has been passed on over literature and was blindly adopted by engineers and mathematicians in designing imaging and classification algorithms for stroke types based on a quite inaccurate problem setting. In this presentation, based on a throughout literature review, major stroke types and related progressive pathophysiological processes are described in attempt to highlight the interior tissue dielectric property changes starting from the acute phase, which need to be taken into account in EIT. Possible applications for EIT to help aid the existing imaging modalities in stroke detection/management are outlined. Finally, the work progress done so far is described and required data inputs necessary for fulfilling project objectives are highlighted. Andreas Hauptmann, University College London, UK. Title: Deep Learning for image reconstruction - Deep D-Bar Abstract: In this talk I will give an introduction to Deep Learning for image reconstruction with a special focus on Electrical Impedance tomography (EIT) and the D-bar algorithm. Specifically, D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features such as clear organ boundaries. Convolutional Neural Networks provide a powerful framework for post-processing such convolved direct reconstructions. The networks are trained on simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems. Additionally to accompany the workshop theme, we will present some preliminary studies with a stroke phantom on reconstruction and classification using Deep Learning. Sarah Hamilton, Marquette University, USA. Title: Robust Recovery of Admittivities for 2D Real-time Absolute/Difference EIT Abstract: The recovery of absolute (or static) EIT images is a notoriously challenging problem. Optimization based methods are strive to minimize the error between the measured data and data simulated by solving the forward conductivity problem (e.g., via FEM) for a guess conductivity. The optimization is highly sensitive to errors in domain modeling (shape, electrode locations, contact impedances) and modeling of system/environmental noise. Although many of these modeling errors can be overcome for tank data, imaging of live or moving subjects is another story. Here we explore how D-bar methods can robustly recover a conductivity/admittivity even when incorrect boundary shapes and electrode locations are used. We focus on a specific D-bar method which requires no $\Lambda_1$ data for absolute imaging completely removing the need to simulate any data and thus the need for a finely tuned forward model. Various ways of including a priori information are discussed, in particular how the D-bar methods can be combined with Convolutional Neural Networks (CNNs) to sharpen blurry D-bar images without ever needing to simulate the current/voltage data or electrode locations. Possible extensions to partial boundary data and 3D are discussed. Antti Hannukainen, Aalto University, Finland. Title: Parametric model of human head Nuutti Hyvönen, Aalto University, Finland. Title: Generalized linearization in electrical impedance tomography Abstract: Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current-voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. We compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumann-to-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Frechet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography. Matti Lassas, University of Helsinki, Finland. Title: Inverse problems for hyperbolic equations and artificial point sources Abstract: We consider uniqueness results for inverse problems for hyperbolic equations. Our aim is to determine the Riemannian metric, associated to travel times of waves, inside a domain from the observations done on the boundary. The inverse problems on Riemannian manifolds ar not generally uniquely solvable: A change of coordinates changes the equation but does not change the boundary data. To prove uniqueness results, one may consider properties that are invariant in diffeomorphisms and aim to reconstruct those uniquely. For example, there is an underlying manifold structure that can be uniquely determined. Thus the inverse problem in a subset of the Euclidean space can solved in two steps. The first one is to reformulate the problem in terms of manifolds and to reconstruct the underlying manifold structure. The second step is to find an embedding of the constructed manifold to the Euclidean space. In the talk we focus to the reconstruction of the invariant manifold structure. We consider solutions of hyperbolic inverse problems that are based on focusing of waves. For linear equations we consider a time reversal iteration where one focuses waves in an unknown medium. For non-linear equations we consider the artificial point source method that applies the non-linear interaction of spherical waves or distorted plane waves to create points sources inside the medium. The new feature of the artificial point source method is that it utilizes the non-linearity as a tool in imaging. The above methods reduce the inverse boundary value problems to passive imaging problems where one observes waves coming from the point sources that are inside the medium, and these problems are solved using geometric methods. Matteo Santacesaria, University of Helsinki, Finland. Title: Calderón's Inverse Problem with a Finite Number of Measurements Abstract: In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing (CS), may be applied to inverse problems in PDEs. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, and I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinite-dimensional CS for deterministic and non-isometric operators, which will be briefly surveyed. This is joint work with Giovanni S. Alberti (University of Genoa). Tuomo Savolainen, University of Eastern Finland, Finland. Title: Design of the injection and measurement front end, system parameters and the user interface. Daniel Strbian, HUCH, Finland. Title: Types of brain haemorrhages, current treatment optionsand challenges in monitoring of the intracerebral haemorrhage Jussi Toivanen, University of Eastern Finland, Finland. Title: Reconstruction of conductivity in multi-frequency EIT Abstract: In multi-frequency EIT, a series of measurements are performed with different current frequencies. Because conductivity is a function of frequency, the estimation process gives a set of conductivity images. The conductivity values in these images are not equal, but the images are usually structurally similar. This structural similarity can be exploited in the estimation process by estimating the conductivities simultaneously whilst utilizing specific priors. Some examples of these priors are the joint total variation prior, the parallel level sets prior, and priors based on either the second difference or the structural similarity index. | 2019-03-24 04:47:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44860780239105225, "perplexity": 869.8718931235042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203326.34/warc/CC-MAIN-20190324043400-20190324065400-00178.warc.gz"} |
https://ca.cyberska.org/publication/read/17478/probing-magnetic-fields-with-galfacts | CYBERSKA A Cyberinfrastructure platform to meet the needs of data intensive radio astronomy on route to the SKA
### Probing magnetic fields with GALFACTS
Authors:
• university of birmingham, b15 2tt
### Samuel George
• university of calgary
### Jeroen Stil
• canada, alberta, calgary
### Russ Taylor
Type:
Keywords:
:
Year: 2011
URI: http://arxiv.org/abs/1111.4890
Abstract:
GALFACTS is a large-area spectro-polarimetric survey on the Arecibo Radio telescope. It uses the seven-beam focal plane feed array receiver system (ALFA) to carry out an imaging survey project of the 12,700 square degrees of sky visible from Arecibo at 1.4 GHz with 8192 spectral channels over a bandwidth of 300 MHz sampled at 1 millisecond. The aggregate data rate is 875 MB/s. GALFACTS observations will create full-Stokes image cubes at an angular resolution of 3.5' with a band-averaged sensitivity of 90 $\mu$Jy, allowing sensitive imaging of polarized radiation and Faraday Rotation Measure from both diffuse emission and extragalactic sources. GALFACTS is a scientific pathfinder to the SKA in the area of cosmic magnetism. Key to magnetism science with the SKA is the technique of RM synthesis. The technique of RM synthesis is introduced and we discuss practical aspects of RM synthesis including efficient computational techniques and detection thresholds in the resulting Faraday spectrum. We illustrate the use of the technique by presenting the current development of the RM synthesis pipeline for GALFACTS and present early results. | 2021-07-31 11:19:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.51909339427948, "perplexity": 5573.527484807159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154089.6/warc/CC-MAIN-20210731105716-20210731135716-00468.warc.gz"} |
https://www.physicsforums.com/threads/how-to-convert.390644/ | # How to convert
1. Mar 29, 2010
### lorik
Question is pretty simple :
How do I know that square root 3/2 = 0.8660
or how can 0.8660 be converted into square root 3/2 more importantly. My calculator is out of style so it displays only numbers. Thanks
Last edited: Mar 29, 2010
2. Mar 29, 2010
### Integral
Staff Emeritus
I am not sure what your question is. Could you try to ask again?
Note that .8660 is NOT equal to $\frac {\sqrt 3} 2$ but is a truncated form of the full decimal number.
3. Mar 29, 2010
### lorik
@integral
Yes sorry for lack of info ,is there any to know that 0.866 is actually square root 3/2 and how do i come to this conclusion ? and vice versa
4. Mar 29, 2010
### Integral
Staff Emeritus
Arithmetic?
5. Mar 29, 2010
### lorik
more related to trigonometry
ohh nevermind Im having some difficulties with complex numbers for example arc tangent of minus square root of 3/3 =- pi/6 how can it be -pi/6 I know its inverse but my calculator does not show pi's or dividers which I will need ! Thats why I need to know the appropriate pi !
6. Mar 29, 2010
### Staff: Mentor
There are a few angles that you should just know - without having to resort to a calculator. These angles are 0, pi/6, pi/4, pi/3, and pi/2, 2pi/3, 3pi/4, 5pi/6, and pi. You should memorize the sine and cosine of each of these angles, and from these you can get all the other trig functions.
The arctangent of -1 is -pi/4.
7. Mar 30, 2010
### HallsofIvy
First, as integral told you, and you apparently did not understand because you immediately asked the same question again, is that there is NO way to "know that 0.866 is actually square root 3/2" because it is not true! .866 is approximately square root of 3, divided by 2. And the only way to know that is to actually take the square root of 3 to four decimal places, divide by 2, and round to three decimal places.
As for the angles Mark44 mentions- If one angle of a right triangle is 45 degrees ($\pi/4$ radians), then the other angle must be 90- 45= 45 degrees also. That means that the right triangle is "isosceles"- if two angles are the same, then the two legs are the same length. Taking that length to be 1, by the Pythagorean theorem, the hypotenuse has length $\sqrt{2}$ and it is easy to see that the $sin(45)= 1/\sqrt{2}= \sqrt{2}/2$.
An equilateral triangle, with all sides the same length, say, 1, must have all angles the same length: 180/2= 60 degrees or $\pi/3$ radians. If you drop a perpendicular from one vertex to the opposite side, it is easy to show that both the opposite side and the angle are bisecte so you have two right triangles with angles 60 degrees and 30 degrees ($\pi/6$ radians). The hypotenuse has length 1 and the side opposite the 30 degree angle has length 1/2. You can then use the Pythagorean theorem to show that the other leg, opposite the 60 degree angle, has length $\sqrt{3}/2$.
That is enough to tell you that sin(30)= 1/2, cos(30)= $\sqrt{3}/2$, sin(60)= $\sqrt{3}/2$, and cos(60)= 1/2.
Last edited by a moderator: Mar 30, 2010
8. Mar 30, 2010
### lorik
@Mark44,@HallsofIvy
Thanks for the replies ,it is obviously true what you all are saying. I hope I didnt bother much | 2018-03-20 22:17:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7354363203048706, "perplexity": 585.3595962327756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647545.54/warc/CC-MAIN-20180320205242-20180320225242-00099.warc.gz"} |
https://www.bbc.co.uk/bitesize/guides/zqtv6yc/revision/1 | # Indices
## Simplifying indices
The two basic laws of indices are:
${a^m} \times {a^n} = {a^{m + n}}$
${a^m} \div {a^n} = {a^{m - n}}$
Try to use these to work through the example questions below.
Question
Simplify $${y^7} \times {y^3} \times {y^5}$$
Use the multiplication law. This tells you to add the indices.
$= {y^{7 + 3 + 5}} = {y^{15}}$
Question
Simplify $${y^{10}} \div {y^3}$$
This could also have been written as:
$\frac{{{y^{10}}}}{{{y^3}}}$
Use the division law which tells you to subtract the indices.
$= {y^{10 - 3}} = {y^7}$
Question
Simplify $$\frac{{{y^7} \times {y^4}}}{{{y^5}}}$$
$= \frac{{{y^{7 + 4}}}}{{{y^5}}}$
Use the multiplication law, add the numerator indices.
$= \frac{{{y^{11}}}}{{{y^5}}}$
Use the division law, subtract the indices.
$= {y^{11 - 5}} = {y^6}$
Question
Simplify $$y \times {y^8} \times {y^4}$$
$y \times {y^8} \times {y^4}$
$=y^{1+8+4}$
Remember $$y = {y^1}$$
$=y^{13}$
Question
Simplify $${y^6} \times {y^0}$$
$= {y^{6 + 0}} = {y^6}$
This shows that $${y^0} = 1$$ | 2020-09-27 14:12:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9866822957992554, "perplexity": 3171.4932160747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400279782.77/warc/CC-MAIN-20200927121105-20200927151105-00720.warc.gz"} |
https://ysharifi.wordpress.com/2010/03/12/almost-boolean-rings-are-commutative/ | “Almost Boolean” rings are commutative
Posted: March 12, 2010 in Elementary Algebra; Problems & Solutions, Rings and Modules
Tags: , , ,
It is easy to prove that if every element of a ring is idempotent, then the ring is commutative. This fact can be generalized as follows.
Problem. 1) Let $R$ be a ring with identity and suppose that every element of $R$ is a product of idempotent elements. Prove that $R$ is commutative.
2) Give an example of a noncommutative ring with identity $R$ such that every element of $R$ is a product of some elements of the set $\{r \in R: \ r^n=r, \ \text{for some} \ n \geq 2 \}.$
Solution. 1) Obviously we only need to prove that every idempotent is central. Suppose first that $ab = 1,$ for some $a,b \in R.$ We claim that $a = b = 1.$ So suppose the claim is false. Then $a = e_1e_2 \cdots e_k,$ where $e_j$ are idempotents and $e_1 \neq 1.$ Let $e = e_2 \cdots e_kb.$ Then $e_1e = 1$ and hence $1 - e_1 = (1 - e_1)e_1e = 0.$ Thus $e_1 = 1.$ Contradiction! Now suppose that $x^2 = 0,$ for some $x \in R.$ Then $(1 - x)(1 + x) = 1$ and therefore $x = 0$, by what we just proved. Finally, since $(ey-eye)^2=(ye-eye)^2=0$ for any idempotent $e \in R$ and any $y \in R,$ we have $ey = ye$ and so $e$ is central.
2) One example is the ring of $2 \times 2$ upper triangular matrices with entries from $\mathbb{Z}/2\mathbb{Z}.$ | 2018-07-20 23:48:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 27, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9747816920280457, "perplexity": 191.9958455913118}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592001.81/warc/CC-MAIN-20180720232914-20180721012914-00092.warc.gz"} |
https://tex.stackexchange.com/questions/84777/how-to-get-matlab-code-into-a-latex-document | # How to get Matlab code into a LaTeX document?
I'm trying to put in my thesis matlab code but the results are not very good. I would get something like that but I don't know what packages using and how to create this result:
• Please reformulate the title of your question. I think the listings package is what you're looking for. Nov 29 '12 at 12:16
• There's also mcode, which uses listings. Nov 29 '12 at 12:25
• There's also pythontex and minted, which use the Python syntax highlighting library Pygments. These can highlight Matlab code and Matlab interactive sessions. Nov 29 '12 at 13:10
• Whatever solution you choose, please make sure that the resulting code listing is copy-and-pasteable. Your 25 readers will hate it otherwise. Nov 29 '12 at 13:13
• The matlab-prettifier package is your friend, here; see this answer. Feb 13 '14 at 23:15
I personally prefer the minted package. It's a little trouble to set up by the output is very neat and tidy -- and it has syntax highlighting.
Output:
Code:
\documentclass{article}
\usepackage{minted}
\begin{document}
\definecolor{bg}{rgb}{0.95,0.95,0.95}
\begin{minted}[linenos=true,bgcolor=bg]{matlab}
% Gradient descent algo for linear regression
% author: Nauman (recluze)
%set the data
X=[1 1 1 1 1 1 1; 22 49 80 26 40 54 91];
Y=[20 24 42 22 23 26 55];
hold on;
plot(X(2,:),Y, 'x');
% set the actual values of W
W = [5.775 0.474]';
YAct = (W' * X);
You'll have to call pdflatex with --shell-escape though and you will have to install a package that provides pygmentize command. | 2021-10-19 00:17:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49359190464019775, "perplexity": 1518.761101951673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585215.14/warc/CC-MAIN-20211018221501-20211019011501-00673.warc.gz"} |
https://r-pkg.org/pkg/rerf | # Randomer Forest
R-RerF (aka Randomer Forest (RerF) or Random Projection Forests) is an algorithm developed by Tomita (2016) which is similar to Random Forest - Random Combination (Forest-RC) developed by Breiman (2001) . Random Forests create axis-parallel, or orthogonal trees. That is, the feature space is recursively split along directions parallel to the axes of the feature space. Thus, in cases in which the classes seem inseparable along any single dimension, Random Forests may be suboptimal. To address this, Breiman also proposed and characterized Forest-RC, which uses linear combinations of coordinates rather than individual coordinates, to split along. This package, 'rerf', implements RerF which is similar to Forest-RC. The difference between the two algorithms is where the random linear combinations occur: Forest-RC combines features at the per tree level whereas RerF takes linear combinations of coordinates at every node in the tree.
## Repo Contents
• R: R building blocks for user interface code. Internally called by user interface.
• docs: usage of the R-RerF package on real examples.
• man: Package documentation
• src: C++ functions called from within R
• travisTest: Travis CI tests
## Description
Randomer Forest (RerF) is a generalization of the Random Forest (RF) algorithm. RF partitions the input (feature) space via a series of recursive binary hyperplanes. Hyperplanes are constrained to be axis-aligned. In other words, each partition is a test of the form Xi > t, where t is a threshold and Xi is one of p inputs (features) {X1, ..., Xp}. The best axis-aligned split is found by sampling a random subset of the p inputs and choosing the one that best partitions the observed data according to some specified split criterion. RerF relaxes the constraint that the splitting hyperplanes must be axis-aligned. That is, each partition in RerF is a test of the form w1X1 + ... + wpXp > t. The orientations of hyperplanes are sampled randomly via a user-specified distribution on the coefficients wi, although an empirically validated default distribution is provided. Currently only classification is supported. Regression and unsupervised learning will be supported in the future.
## Tested on
• Mac OSX: 10.11 10.12 (Sierra)
• Linux: Ubuntu 16.04, CentOS 6
• Windows: 10
## Hardware Requirements
Any machine with >= 2 GB RAM
## Software Dependencies
• R
• R packages:
• dummies
• compiler
• RcppZiggurat
• parallel
## Installation
• Installation normally takes ~5-10 minutes
• Non-Windows users install the GNU Scientific Library (libgsl0-dev).
• Windows users install Rtools (https://cran.r-project.org/bin/windows/Rtools/)
### Stable Release from CRAN:
From within R-
install.packages("rerf")
### Development Version from Github:
First install the devtools package if not currently installed. From within R-
install.packages("devtools")
Next install rerf from github. From within R-
devtools::install_github("neurodata/R-Rerf")
## How to Use
Runtime for the following examples should be < 1 sec on any machine.
library(rerf)
### Create a forest:
To create a forest the minimum data needed is an n by d input matrix (X) and an n length vector of corresponding class labels (Y). Rows correspond to samples and columns correspond to features.
X <- as.matrix(iris[,1:4])
Y <- iris[[5L]]
forest <- RerF(X, Y, seed = 1L)
Expected output:
$treeMap [1] 1 2 -17 3 4 -1 -2 5 8 -3 6 7 -6 -4 -5 9 -16 10 -15 [20] -7 11 12 -14 13 14 -8 -9 -10 15 -11 16 -12 -13$CutPoint
[1] -0.80 -6.85 -1.90 4.35 -2.75 -5.90 7.55 -2.85 -10.75 -3.35
[11] 3.45 -3.15 4.90 4.60 -3.05 6.40
$ClassProb [,1] [,2] [,3] [1,] 0 1.0000000 0.0000000 [2,] 0 0.0000000 1.0000000 [3,] 0 1.0000000 0.0000000 [4,] 0 0.3333333 0.6666667 [5,] 0 1.0000000 0.0000000 [6,] 0 1.0000000 0.0000000 [7,] 0 0.0000000 1.0000000 [8,] 0 1.0000000 0.0000000 [9,] 0 0.0000000 1.0000000 [10,] 0 1.0000000 0.0000000 [11,] 0 0.0000000 1.0000000 [12,] 0 0.0000000 1.0000000 [13,] 0 0.6666667 0.3333333 [14,] 0 0.0000000 1.0000000 [15,] 0 1.0000000 0.0000000 [16,] 0 0.0000000 1.0000000 [17,] 1 0.0000000 0.0000000$matAstore
[1] 4 -1 1 -1 1 -1 3 1 2 1 4 1 2 -1 1 -1 1 1 4 1 2 -1 1 -1 3
[26] -1 2 -1 3 1 4 -1 2 -1 3 1 3 1 2 -1 1 1
$matAindex [1] 0 2 4 8 12 14 16 20 22 26 28 32 34 36 38 40 42$ind
NULL
$rotmat NULL$rotdims
NULL
[1] 0.9413333
$rho [1] 0.8451606 ### Compute feature (projection) importance (this feature is not available in the current CRAN release): Computes the Gini importance for all of the unique projections used to split the data. The returned value is a list with members imp and proj. The member imp is a numeric vector of feature importances sorted in decreasing order. The member proj is a list the same length as imp of vectors specifying the split projections corresponding to the values in imp. The projections are represented by the vector such that the odd numbered indices indicate the canonical feature indices and the even numbered indices indicate the linear coefficients. For example a vector (1,-1,4,1,5,-1) is the projection -X1 + X4 - X5. Note: it is highly advised to run this only when the splitting features (projections) have unweighted coefficients, such as for the default setting or for RF. X <- as.matrix(iris[, 1:4]) # feature matrix Y <- iris$Species # class labels
p <- ncol(X) # number of features in the data
d <- ceiling(sqrt(p)) # number of features to sample at each split
# Here we specify that we want to run the standard random forest algorithm and we want to store the decrease in impurity at each split node. The latter option is required in order to compute Gini feature importance.
forest <- RerF(as.matrix(iris[, 1:4]), iris[[5L]], mat.options = list(p, d, "rf", NULL), num.cores = 1L, store.impurity = TRUE, seed = 1L)
feature.imp <- FeatureImportance(forest, num.cores = 1L)
Expected output:
> feature.imp
$imp [1] 4455.7292 4257.6306 861.6474 178.5267$proj
$proj[[1]] [1] 3 1$proj[[2]]
[1] 4 1
$proj[[3]] [1] 1 1$proj[[4]]
[1] 2 1
### Train Structured RerF (S-RerF) for image classification:
S-RerF samples and evaluates a set of random features at each split node, where each feature is defined as a random linear combination of intensities of pixels contained in a contiguous patch within an image. Thus, the generated features exploit local structure inherent in images.
data(mnist)
# p is number of dimensions, d is the number of random features to evaluate, iw is image width, ih is image height, patch.min is min width of square patch to sample pixels from, and patch.max is the max width of square patch
p <- ncol(mnist$Xtrain) d <- ceiling(sqrt(p)) iw <- sqrt(p) ih <- iw patch.min <- 1L patch.max <- 5L forest <- RerF(mnist$Xtrain, mnist$Ytrain, num.cores = 1L, mat.options = list(p, d, "image-patch", iw, ih, patch.min, patch.max), seed = 1L) predictions <- Predict(mnist$Xtest, forest, num.cores = 1L)
error.rate <- mean(predictions != mnist$Ytest) Expected output: > error.rate [1] 0.0544 ### Train Structured RerF (S-RerF) for spike train inference: Similar to S-RerF for image classification except now in the Spike Train setting. 500 samples were stimulated from the following AR(2) model: $$c_t = \sum_{i=1}^2 \gamma_i c_{t-i} + s_t, \ \ \ s_t \sim Poisson(0.01) \ y_t = a \ c_t + \epsilon_t, \ \ \ \ \epsilon_t \sim \mathcal{N}(0, 1)$$ whre$\gamma_1 = 1.7, \gamma_2 = -0.712$,$a = 1$. We sampled such that the were an equal number of spikes and non-spikes in the datasets. S-RerF was trained on these samples by computing local feature patches across the time series windows. ts.train <- read.csv('calcium-spike_train.csv', header=FALSE) ts.test <- read.csv('calcium-spike_test.csv', header=FALSE) ts.train$X <- ts.train[,1:(ncol(ts.train)-1)]
ts.train$Y <- ts.train[,ncol(ts.train)] ts.test$X <- ts.test[,1:(ncol(ts.test)-1)]
ts.test$Y <- ts.test[,ncol(ts.test)] # p is number of dimensions, d is the number of random features to evaluate, patch.min is min width of a time series patch to sample, and patch.max is the max width of the patch. p <- ncol(ts.train$X)
d <- ceiling(sqrt(p))
patch.min <- 1L
patch.max <- 5L
forest <- RerF(ts.train$X, ts.train$Y, num.cores = 1L, mat.options = list(p, d, "ts-patch", patch.min, patch.max), seed = 1L)
predictions <- Predict(ts.test$X, forest, num.cores = 1L) error.rate <- mean(predictions != ts.test$Y)
Expected output
> error.rate
[1] 0.262
# Reference manual
install.packages("rerf")
2.0.2 by Jesse Patsolic, 2 months ago
https://github.com/neurodata/R-RerF
Report a bug at https://github.com/neurodata/R-RerF/issues
Browse source code at https://github.com/cran/rerf
Authors: Jesse Patsolic [ctb, cre] , Benjamin Falk [ctb] , Jaewon Chung [ctb] , James Browne [aut] , Tyler Tomita [aut] , Joshua Vogelstein [ths]
Documentation: PDF Manual
Imports parallel, RcppZiggurat, utils, stats, dummies
Suggests roxygen2, testthat
System requirements: GNU make
See at CRAN | 2019-01-23 13:01:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5526116490364075, "perplexity": 6747.230958059364}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584331733.89/warc/CC-MAIN-20190123105843-20190123131843-00002.warc.gz"} |
https://studyadda.com/question-bank/perimeter_q48/2649/234086 | • # question_answer The figure given is formed using five ^^ identical rectangles. The perimeter of the figure is 66 cm. Find the perimeter of each rectangle. A) 30 cm B) 15 cm C) 18 cm D) 135 cm
$\therefore$ Length of the figure = 3 units Breadth of the figure = 2.5 units Perimeter$~=\text{2 (3}+\text{2}.\text{5)}$ units $=\text{2}\times \text{5}.\text{5}$ = 11 units Given perimeter = 66 cm 11 units = 66 cm So, 1 unit $=\frac{6\text{6 cm}}{11}=\text{6 cm}$ $\therefore$ Length of each rectangle = 9 cm Breadth of each rectangle = 6 cm $\Rightarrow$ Perimeter $=\text{ 2}\times (\text{9}+\text{6})\text{ cm}$ $=\text{2}\times \text{15 cm}=\text{3}0\text{ cm}$ Hence, the perimeter of each rectangle is 30 cm. | 2019-12-13 10:25:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7671990990638733, "perplexity": 2417.77006154312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540553486.23/warc/CC-MAIN-20191213094833-20191213122833-00134.warc.gz"} |
http://blog.benw.xyz/tag/pseudorandom/ | # Monday Exams
If you do a series expansion of $studying(t)$ around $t = Sunday,$ you'll find that all the terms drop out and I don't study. | 2017-06-27 17:24:16 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6791812181472778, "perplexity": 288.1577672400247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321497.77/warc/CC-MAIN-20170627170831-20170627190831-00620.warc.gz"} |
https://eprints.soton.ac.uk/269853/ | The University of Southampton
University of Southampton Institutional Repository
# The Frederiks effect and related phenomena in ferronematic materials
Zadorozhnyi, V.I., Sluckin, T.J., Reshetnyak, V.Y. and Thomas, K.S. (2008) The Frederiks effect and related phenomena in ferronematic materials. SIAM Journal on Applied Mathematics, 68 (6), 1688-1716.
Record type: Article
## Abstract
Using continuum and statistical mechanical theories, we study the switching properties of a ferronematic in a nematic liquid crystal cell subject to homeotropic boundary conditions at the cell and particle walls. An external magnetic field normal to the cell plane is also imposed. At low fields we find thresholdless switching of the nematic director, consistent with experimental data. At higher fields, there are three regimes, depending on the strength of the anchoring interaction between the director and the ferroparticle orientation. For low anchoring strengths, there is an inverse Frederiks effect, and the nematic reorientation reduces and then disappears continuously at a critical magnetic field. At intermediate fields, the degree of reorientation reduces at high fields but remains finite. For high fields, however, the director switching saturates. The dimensionless temperature scale in the problem involves the temperature, the mean nematic elastic constant, the colloidal density, and the cell dimension. If this quantity is sufficiently low, then high magnetic fields can cause magnetic segregation. The segregation order parameter is coupled to the director distortion, and this can change the inverse Frederiks transition into a first order transition, leading to bistability in an intermediate field regime. These features are perturbed but not changed structurally by the effect of a small bias magnetic field ($<10$ Oe) normal to the unperturbed director. Subject to suitable choice of parameters, the theory is also quantitatively consistent with the results of the classic experiment of Chen and Amer in 1983.
Submitted date: 26 September 2007
e-pub ahead of print date: 2 July 2008
Published date: 2008
Additional Information: Imported from ISI Web of Science
Organisations: Electronic & Software Systems
## Identifiers
Local EPrints ID: 269853
URI: http://eprints.soton.ac.uk/id/eprint/269853
ISSN: 0036-1399
PURE UUID: c7838729-2d30-4984-929f-abd8c3e0a747
ORCID for T.J. Sluckin: orcid.org/0000-0002-9163-0061
## Catalogue record
Date deposited: 21 Apr 2010 07:46
## Contributors
Author: T.J. Sluckin
Author: V.Y. Reshetnyak
Author: K.S. Thomas | 2022-08-08 07:33:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47538402676582336, "perplexity": 4303.538052087982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570767.11/warc/CC-MAIN-20220808061828-20220808091828-00023.warc.gz"} |
https://www.transtutors.com/questions/the-management-of-coker-corp-is-doing-a-quick-forecast-of-20x9-using-the-modified-pe-88734.htm | # The management of Coker Corp. is doing a quick forecast of 20X9 using the modified percentage of...
The management of Coker Corp. is doing a quick forecast of 20X9 using the modified percentage of sales method in preparation for a more detailed planning exercise later in the month. The estimate is to assume a 10% growth in sales. All other line items are to be assumed to grow at the same rate except for fixed assets which is projected to increase by $88,000 due to an expansion program already underway. Approximate financial statements for the current year, 20X8, and a planning worksheet are shown below. The firm pays 9% interest on all of its debt. Assume the tax rate is a flat 25%. There are no plans for dividends or the sale of additional stock next year. Make a forecast of Coker’s complete income statement and balance sheet. Work to the nearest thousand dollars. (Hints: The easiest way to grow a number by 10% is to multiply it by 1.1 rather than taking 10% and adding. Do not grow subtotals. For example, to grow Revenue and COGS by 10%, round each to the nearest thousand and subtract for Gross Margin. Don’t grow interest, debt, or equity; use the debt/interest iteration technique.) ## Expert's Answer No Answer Yet Ask for Expert's Help ## Related Questions in Financial Accounting - Others • ### consolidation (Solved) April 04, 2013 financial accounting - Assignment 1 Problem 1 Pre-Contribution Balance Sheets and Fair Values June 30, 20 X 9 (in thousands of$) Swag Co.
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• ### Problem 1 How does P & P determine the value for Levels 1, 2 and 3 investments? Problem 3 Why would
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(Solved) June 28, 2015
interest rate on the company’s long-term debt for the year ended December 31, 2013? 4. Was the current yield at December 31, 2013, on the company’s long-term debt the same as, greater, or less than the average yield at issuance? At December 31, 2012? Contributed Capital 1. How many shares of...
Using the annual report, answer the questions under the following sections. Complete your assignment in a Microsoft Word document. Part of your grade will be based on the structure and... | 2018-06-23 01:02:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2458004355430603, "perplexity": 3318.466217386646}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864848.47/warc/CC-MAIN-20180623000334-20180623020334-00627.warc.gz"} |
https://williamhaw.com/sbt-tricks/ | # SBT Tricks
2 mins
I was recently upgrading a library at work from using Scala 2.11 to 2.12. Here are some sbt tricks that I picked up while trying to perform the migration.
# Build with different library versions for different Scala versions
I followed steps 1 and 2 here, but it’s quite an old post.
Basically you can define a function that takes the Scala version string and return the right version of the library for that Scala version. This is useful especially for protobuf libraries compiled with scalapb since we had a version compiled with scalapb 0.4.9 for 2.11 and a version compiled with scalapb 0.6.0 for 2.12.
For SBT 0.13.17, this is an example of what I used:
specifiying versions:
crossScalaVersions := Seq("2.11.8", "2.12.8")
configure libraries:
val lib_2_11 = "com.example" %% "protobuf" % "0.1.0-scalapb-0.4.9"
val lib = "com.example" %% "protobuf" % "0.1.0-scalapb-0.6.0"
def protobufs(scalaVersion: String) = {
scalaVersion match {
case "2.11.8" => lib_2_11
case _ => lib
}
}
libraryDependencies ++= scalaVersion(version => protobufs(version)).value
# Conflicting cross-version suffixes
If you fix the version of your library to a certain Scala version like so:
val myDependency = "org.scala-lang.modules" % "scala-xml_2.11" % "0.1.0"
then shame on you!
This will make upgrading Scala versions difficult in the future. Instead, you should let sbt resolve the version suffix of the library like so:
val myDependency = "org.scala-lang.modules" %% "scala-xml" % "0.1.0"
This is even worse when the import is inside a dependency of a dependency, so that it doesn’t appear in the build.sbt of the original project! I used the sbt-dependency-graph plugin to look for places where the conflicting version was being brought in (for e.g scala-xml_2.11) and then went to those projects to publish new versions without the forced suffixes.
# Change Scala version to new version to get IDE hints
After solving some of the dependency resolution issues, I was able to get the compilation started. However, it was failing for the new Scala version because of some field changes in the protobufs. That required me to write different versions of some classes and tests for the different Scala versions.
# Compile some classes differently between Scala versions
Since sbt 0.13.8, you can just put your version specific code in version specific directories (for e.g. src/main/scala-2.12 will only be compiled when the current Scala version is 2.12).
# Exclude libraries properly
exclude() and excludeAll() do not understand Scala versions; you have to specify the Scala version suffix i.e. exclude("com.example", "my_library_2.12")
The reason, as noted here, is that the underlying Ivy dependency resolution doesn’t understand sbt conventions (for e.g. the version suffix in the library name). | 2019-10-23 23:54:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33850717544555664, "perplexity": 6080.645405748455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00070.warc.gz"} |
https://machinelearningmedium.com/2017/07/11/word-to-vector-word-representations/ | Blog Logo
·
· · ·
### Distributed Vector Representation Series
Word2Vec
Improvements on Word2Vec
· · ·
### Introduction
• Computing the continuous vector representations of words from very large data sets.
• Current state-of-the-art performance on semantic and syntactic word similarities.
• Classical techniques treat words as atomic units without any notion of similarities between them because they are represented using indices in a vocabulary (bag-of-words).
• Advantages of classical techniques lie in simplicity, robustness and accuracy of simple model when trained on large data sets over complex models trained on less data.
• Disadvantage of these methods is observed when the amount of data available to train is limited in certain fields like say, automatic speech recognition and machine translations.
### Previous Works
• Neural Network Language Model (NNLM):
• Consists of input, projection, hidden and output layers.
• Input layer has N previous words encoded using 1-in-V coding, where V is the size of Vocabulary.
• Projection layer, P has a projection of input layer has a dimensionality of $N * D$ and uses a projection matrix.
• High complexity between projection and hidden layer due to dimensions of the dense projection layer.
• Computational complexity of NNLM per training example is given by
$Q = N * D + N * D * H + H * V$
• Where
• Q is the computational cost
• N is the number of previous words used for learning
• D is the dimensionality of the projection layer
• H is the size of hidden layer
• V is the size of the vocabulary and output layer.
• $H * V$ is the dominating term above which was proposed to be reduced to as less as $H * log_2(V)$ using
• Hierarchical softmax
• Avoiding normalized models for training
• Binary tree representations of the vocabulary using Huffman Trees
• So, the major complexity is dominated by $N * D * H$
• Recurrent Neural Network Language Model (RNNLM):
• Overcome the limitations of NNLM such as need to specify the context length, N (order of the model N)
• Theoretically RNNs can efficiently represent more complex patterns than shallow neural networks.
• No projection layer
• Consists of Input, hidden and output layers.
• Develops a short term memory of seen data in the self-fed time delayed hidden layer.
• Computational complexity of NNLM per training example is given by
$Q = H * H + H * V$
• Where
• Q is the computational cost
• H is the size of hidden layer
• V is the size of the vocabulary and output layer.
• Word representations D have the same dimensionality as the hidden layer H.
• Again, $H * V$ will be reduced to $H * log_2(V)$ using Hierarchical softmax.
• So, the major complexity is dominated by $H * H$
• It’s observed that most complexity is contributed by the non-linearity of the hidden layer in the networks.
### Continuous Bag-of-Words Model (CBOW)
• Similar to feedforward NNLM, but the non-linear hidden layer is removed.
• Projection layer is shared for all the words. So all words are projected into the same position and their vectors are averaged.
• Model is called bag-of-words model because the order of words in the history or future does not influence the projections.
• Unlike NNLM, words from future are used to with the best result found with 4 history and 4 future words in context.
• Training criterion is the correct classification of the current(middle) word.
• Training complexity is given by
$Q = N * D + D * log_2(V)$
• Model is continuous bag-of-words because unlike standard bag-of-words it uses continuous distributed representations of the context.
• Weights between the input and the projection layer is shared for all words positions in the same way as in NNLM.
### Continuous Skip-Gram Model
• Similar to CBOW but slight changes in training criterion.
• Instead of predicting current word from the surrounding words in the window, current word is used to predict the words surrounding the current word.
• Accuracy and quality of vector is found to increase as the number of context words predicted is increased, but that increased the computational complexity as well.
• Training complexity is given by
$Q = C * (D + D * log_2(V))$
• Where
• C is the maximum distance of the words. Say, C=5 is chosen then a number $R \in [1, C]$ is selected randomly and then R words from history and R from future are correct labels of the current word.
### Results
• Algebraic operations on the vector representations actually give meaningful results like cosine similary of $vector(X)$ is closest to $vector(‘smallest’)$ where
$vector(X) = vector(‘biggest’) - vector(‘big’) + vector(‘small’)$
• Subtle relationships are learnt when accurate data is used. For example, France is to Paris as Germany is to Berlin.
• After a certain point adding more dimensionality to the word vectors or adding more training data provides diminishing improvements.
• NNLM vectors work better than RNNLM because word vectors in RNNLM are directly connected to non-linear hidden layer. | 2019-10-19 22:54:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4986523687839508, "perplexity": 1896.7679848579116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986700435.69/warc/CC-MAIN-20191019214624-20191020002124-00353.warc.gz"} |
https://dspace5.zcu.cz/handle/11025/30447 | Title: On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian Authors: Audoux, BenjaminBobkov, VladimírParini, Enea Issue Date: 2018 Publisher: Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies Document type: článekarticle URI: http://hdl.handle.net/11025/30447 ISSN: 1230-3429 Keywords in different language: p-Laplacian;nonlinear eigenvalues;Krasnoselskii genus;symmetry;multiplicity;degree of map. Abstract in different language: We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \R^N$. By means of topological arguments, we show how symmetries of $\Omega$ help to construct subsets of $W_0^{1,p}(\Omega)$ with suitably high Krasnosel'ski\u{\i} genus. In particular, if $\Omega$ is a ball $B \subset \mathbb{R}^N$, we obtain the following chain of inequalities: $$\lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p;B).$$ Here $\lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $\lambda_\ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $\lambda_2(p;B)=\lambda_\ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=\infty$ are also considered. Rights: Plný text není přístupný.© Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies Appears in Collections: Články / Articles (KMA)OBD
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Please use this identifier to cite or link to this item: http://hdl.handle.net/11025/30447 | 2019-05-26 21:20:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8028033971786499, "perplexity": 1145.7374837733596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232259757.86/warc/CC-MAIN-20190526205447-20190526231447-00306.warc.gz"} |
https://www.math-forums.com/threads/integrating-exp-x-2.18538/ | # Integrating exp(-x^2)
Discussion in 'Undergraduate Math' started by Stan Brown, Dec 24, 2010.
1. ### Stan BrownGuest
This came up at work yesterday, and neither of us could remember how
to do it. I pulled out my copy of Thomas and couldn't find it,
though I did find a statement at the beginning of the "Methods of
Integration" chapter that it would be dealt with later and would
involve infinite series.
Can someone remind me, or point me toward an online reference?
Thanks!
P.S. We actually want the definite integral from x=0 to infinity.
And again, it's not the answer but the method that we're looking for.
Stan Brown, Dec 24, 2010
2. ### Paul SperryGuest
but not much of a method.
<http://mathforum.org/library/drmath/view/69832.html> gives a
The two answers do not agree - I think (but am not sure) that Wolfram
is correct.
Paul Sperry, Dec 24, 2010
3. ### PubkeybreakerGuest
Turn it into a double integral and convert to polar coordinates.
Pubkeybreaker, Dec 24, 2010
4. ### Brian M. ScottGuest
On Fri, 24 Dec 2010 01:47:26 -0500, Stan Brown
What you want is half of the integral over R; call its value
I. Then I^2 = int[exp(-x^2) dx] * int[exp(-y^2) dy] =
int int[exp(-(x^2 + y^2)) dy dx], where all integrals are
from -inf to inf, so that you're integrating over the whole
plane. Now convert everything to polar coordinates.
Brian
Brian M. Scott, Dec 24, 2010
5. ### VirgilGuest
If I = Integral_0^oo e^(-x^2) dx
Then I^2 = Integral_0^oo Integral_0^oo e^(-x^2-y^2 ) dx dy
= Integral_0^(pi/2) Integral_0^oo e^(-r^2) r dr dtheta
Which is fairly simple to evaluate.
To show that the two infinite double integrals actually converge to the same value is
also not difficult.
Consider a quarter circle region centered at origin contained in a minimal square
contained in a larger but also minimal quarter circle with center at origin.
Clearly the integral over the square region has a value between the integrals over the
two quarter circle regions.
Then show that as the radii of the quarter circle goes towards oo, the integral over the
region between the quarter circles goes to zero, thus the difference in the 3 integrals
over the finite regions goes to zero as the regions of integration become infinite.
Virgil, Dec 24, 2010
6. ### Stan BrownGuest
Wow! Thanks, Brian!
Stan Brown, Dec 24, 2010
7. ### Stan BrownGuest
match the method that Brian Scott posted, Dr. Math integrates from -
infinity to +infinity and gets sqrt(pi) = 1.772... Wolfram is
integrating from 0 to +infinity and gets 0.886... which is half of
sqrt(pi), so I think the two actually do agree.
Stan Brown, Dec 24, 2010
8. ### Stan BrownGuest
Thanks for responding.
Stan Brown, Dec 24, 2010 | 2022-07-07 02:05:50 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8386227488517761, "perplexity": 3761.12167000287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104683020.92/warc/CC-MAIN-20220707002618-20220707032618-00558.warc.gz"} |
http://math.stackexchange.com/questions/157564/in-probability-how-can-a-sigma-algebra-represent-the-total-information | # In probability, how can a sigma-algebra represent the total information?
Why does a sigma-algebra represent the information available at a given time? I understand the idea of filtration and stopping-time, given that each sigma-algebra represent the info we have at a specific time, but why is that?
For instance in a game of dice rolls (or anyone you want), what would be total universe and the available information in forms of sigma-algebra, at the n-th turn? thanks
-
In a game of dice rolls, the total universe $\Omega$ would be irrelevant (as it nearly always is in probabilistic modeling, as long as it is large enough) and the sigma-algebra after the $n$th roll $X_n$ would be $\mathcal F_n=\sigma(X_k;1\leqslant k\leqslant n)$. The global sigma-algebra $\mathcal F$ on $\Omega$ may be any sigma-algebra containing $\mathcal F_\infty=\sigma(X_k;k\geqslant 1)$ since one wants each function $X_n:\Omega\to\{0,1\}$ to be a random variable on $(\Omega,\mathcal F)$, that is, to be measurable with respect to $\mathcal F$.
A time-scale interpretation might be helpful here. Imagine that the $n$th throw happens at time $n$. Then, at time $n$, the results $X_k$ for $k\geqslant n+1$ are not available yet, hence one can combine the values $X_k$ for $k\leqslant n$ in any measurable way and stay in the realm of the random variables measurable with respect to $\mathcal F_n$, but not any value $X_k$ for $k\geqslant n+1$ since these throws did not happen yet.
Note that the global sigma-algebra $\mathcal F$ may contain some extra information not in $\mathcal F_\infty$, for example the temperature $T_n$ of the room where the $n$th throw occurs, and/or the age $A_n$ of the operator throwing the $n$th dice, and so on.
-
in this case, are the Xk random variables or just numbers? – lezebulon Jun 13 '12 at 9:24
??? Random variables, of course. – Did Jun 13 '12 at 9:26
so what I don't understand here (and that's the root of my problem imo) is why don't we know X(n+1) at time n? I'd say that all the X(k) are uniforms on [1;6], so I don't see how playing turns changes this knowledge – lezebulon Jun 13 '12 at 10:00
To say that we do not know X(n+1) at time n means that X(n+1) is not measurable with respect to F(n). To wit, F(n) is precisely the collection of the events that are measurable at time n, that is, the events that depend only on the outcomes X(k) for k at most n. For example the event A that X(1)+X(2)+X(3) is even is in F(3) but not in F(2): if someone gives you the values X(1)(omega) and X(2)(omega), you are not able to determine if omega is in A or not. But if someone gives you the values X(1)(omega), X(2)(omega) and X(3)(omega), you are. – Did Jun 13 '12 at 10:39
So if I understand correctly, F(n) does not represent the elements I am able to measure at time n (because I can measure A anytime : it has probability 0.5). But F(n) represents the events that I can tell at time n if they are realized for my outcome so far – lezebulon Jun 13 '12 at 16:33
The Doob-Dynkin lemma relates them in an intuitive way for most of the standard applications in probability theory.
Suppose you have a probability space $(\Omega,\Sigma,\mu)$ and two random variables $f:\Omega\to\mathbb{R}$ and $g:\Omega\to\mathbb{R}$. That $g$ only depends on $f$ can be interpreted as saying that you know the value of $g$ whenever you know the value of $f$. This means that you can find a function $h:\mathbb{R}\to\mathbb{R}$ such that $g(\omega)=h(f(\omega))$. In other words, $g=h\circ f$. Now the Doob-Dynkin emma says that the following are equivalent:
1. There is a measurable function $h:\mathbb{R}\to\mathbb{R}$ such that $g=h\circ f$.
2. The random variable $g$ is measurable with respect to the $\sigma$-algebra generated by $f$, that is the $\sigma$-algebra $\{f^{-1}(B):B\textrm{ is a Borel set}\}$.
Most naturally occuring $\sigma$-algebras are of the form $\{f^{-1}(B):B\textrm{ is a Borel set}\}$ for some random variable $f$. This is equivalent to the $\sigma$-algebra being countably generated.
-
Say you know the values of $X_1+\cdots+X_n$ and $X_1^2+\cdots+X_n^2$. Then you can find the values of $\bar X = (X_1+\cdots+X_n)/n$ and $S^2 = ((X_1-\bar X)^2+\cdots+(X_n-\bar X)^2)/(n-1)$, and likewise if you know the values of those latter two quantities, then you can find the first two. So they are in a sense equivalent. Saying they are equivalent is the same as saying they generate the same sigma-algebra. Hence conditioning on them is the same as conditioning on the sigma-algebra that they generate. The details of the particular choice of which of these pairs don't matter, you you speak of conditioning on a sigma-algebra.
- | 2016-02-09 06:31:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8391085863113403, "perplexity": 234.81019336778579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701156520.89/warc/CC-MAIN-20160205193916-00063-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://tug.org/pipermail/tugindia/2004-October/003098.html | # [Tugindia] Creating figure caption in Malayalam
Tue Oct 26 06:46:42 CEST 2004
On Tue, October 26, 2004 9:14 am, Josy P. Pullockara said:
> On Mon, 2004-10-25 at 18:08, V. Sasi Kumar wrote:
>> I am creating a document in Malayalam in which I have to include some
>> figures. When I write \begin{figure}, followed by \caption{...}, the
>> caption comes as Figure 1: followed by the text I give in Malayalam. I
>> want to redefine this so that a suitable Malayalam word comes instead of
>> Figure. How can I do that? Can I similarly change for Table also?
>
> Use
> \renewcommand{\figurename}{\malayalamfont Chithram}
> \renewcommand{\tablename}{\malayalamfont Kattam}
I think this wont work for two reasons:
1. What is this \malayalamfont? I believe, it is a custom definition not
available in LaTeX, nor in the malayalam.sty of Alex.
2. Chithram and Kattam won't give you what you want, with different input
schemes (since there is no definitive one), it gives different output.
Even different Malayalam font will give you different output (often
bizarre), because there is no definitive glyph layout for Malayalam fonts.
For the original poster:
malayalam.sty of Alex, A. J., defines \figurename and \tablename for two
fonts namely, rachana and keli, all available at http://sarovar.org.
-- | 2022-07-06 03:17:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9088599681854248, "perplexity": 8920.418623562173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104660626.98/warc/CC-MAIN-20220706030209-20220706060209-00426.warc.gz"} |
https://hsm.stackexchange.com/questions/5563/where-does-the-name-eigenvalue-come-from | # Where does the name eigenvalue come from?
Who introduced the concept of eigenvalues and eigenvectors and where does the name come from? Is there a connection with the German word "eigen"? | 2019-12-11 22:32:19 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8028653264045715, "perplexity": 302.78319612008335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00326.warc.gz"} |
https://gomathanswerkey.com/go-math-grade-3-answer-key-chapter-7-division-facts-and-strategies-extra-practice/ | # Go Math Grade 3 Answer Key Chapter 7 Division Facts and Strategies Extra Practice
Students who have completed the exercise problems, homework can go through the Go Math Grade 3 Answer Key Chapter 7 Division Facts and Strategies Extra Practice pdf. You can test your knowledge on this chapter by solving the Go Math Grade 3 Answer Key Chapter 7 Division Facts and Strategies Extra Practice sums.
## Go Math Grade 3 Answer Key Chapter 7 Division Facts and Strategies Extra Practice
You can solve the sums easily by following the techniques given in the Solution key of HMH Go Math Grade 3 Chapter 7 Division Facts and Strategies Extra Practice. Go through the topics covered in this chapter before you start practicing the sums in the Extra Practice.
### Common Core – Page No. 149000
Lessons 7.1–7.2
Find the quotient. You may want to draw a quick picture to help.
Question 1.
8 ÷ 2 = ______
Question 2.
______ = 14 ÷ 2
Question 3.
18 ÷ 2 = ______
Question 4.
______ = 12 ÷ 2
Question 5.
70 ÷ 10 = ______
Question 6.
50 ÷ 10 = ______
Question 7.
40 ÷ 10 = ______
Question 8.
90 ÷ 10 = ______
Lessons 7.3–7.4
Find the quotient.
Question 9.
15 ÷ 5 = ______
Explanation:
5 divides 15 into 3 equal groups. So the quotient of 15 and 5 is 3.
Question 10.
______ = 45 ÷ 5
Explanation:
5 divides 45 into nine equal groups. Thus the quotient is 9.
Question 11.
______ = 10 ÷ 5
Explanation:
5 divides 10 two times. So the quotient of 10 and 5 is 2.
Question 12.
40 ÷ 5 = ______
Explanation:
5 divides 40 into eight equal groups. So the quotient is 8.
Question 13.
6 ÷ 3 = ______
Explanation:
3 divides 6 into two equal groups. Thus the quotient is 2.
Question 14.
______ = 21 ÷ 3
Explanation:
3 divides 21 into seven equal groups. So the quotient is 7.
Question 15.
______ = 24 ÷ 3
Explanation:
3 divides 24 eight times. Thus the quotient is 8.
Question 16.
______ = 18 ÷ 3
Explanation:
3 divides 18 into six equal parts. Therefore the quotient is 6.
Question 17.
There are 30 balloons arranged in 6 equal groups. How many balloons are in each group?
______ balloons
Explanation:
Total number of balloons = 30
Number of balloons arranged in equals groups = 6
Number of balloons in each group = x
Divide the number of balloons by a number of equal groups.
= 30 ÷ 6 = 5 balloons
Therefore number of balloons in each group = 5
Question 18.
Mr. Song spends $27 on sports drinks. Each bottle costs$3. How many bottles does Mr. Song buy?
______ bottles
Explanation:
Given,
Mr. Song spends $27 on sports drinks. Each bottle costs$3.
Number of bottles he bought = x
x × 3 = 27
x = 27 ÷ 3 = 9
Thus Mr. Song bought 9 bottles.
Lesson 7.5
Find the quotient.
Question 19.
28 ÷ 4 = ______
Explanation:
4 divides 28 seven times. So the quotient is 7.
Question 20.
______ = 16 ÷ 4
Explanation:
4 divides 16 four times. The quotient of 16 and 4 is 4.
Question 21.
______ = 20 ÷ 4
Explanation:
4 divides 20 five times. Thus the quotient is 5.
Question 22.
______ = 32 ÷ 4
Explanation:
4 divides 32 eight times. So the quotient of 32 and 4 is 8.
Question 23.
4)$$\bar { 3 6 }$$
______
Explanation:
4 divides 36 nine times. Thus the quotient is 9.
Question 24.
4)$$\bar { 1 2 }$$
______
12 ÷ 4
4 divides 12 three times. So the quotient of 12 and 4 is 3.
Question 25.
4)$$\bar { 2 4 }$$
______
Explanation:
4 divides 24 six times. Thus the quotient is 6.
24 ÷ 4 = 6
Question 26.
4)$$\bar { 4 }$$
______
Explanation:
Any number divides by the same number will be always 1. Thus the quotient is 1.
Find the unknown number.
Question 27.
a = 40 ÷ 4
a = _____
Explanation:
Let the unknown number be a
4 divides 40 ten times.
a = 40 ÷ 4
a = 40/4 = 10
Therefore a = 10
Question 28.
0 ÷ 4 = b
b = _____
Explanation:
0 divides by any number are always 0. So the value of b is 0.
Question 29.
c = 36 ÷ 4
c = ______
Explanation:
Let c be the unknown number
36 ÷ 4 = C
4 divides 36 nine times.
Thus the value of c is 9.
Question 30.
8 ÷ 4 = d
d = ______
Explanation:
d is the unknown number
d = 8 ÷ 4
4 divides 8 two times.
Thus the value of d is 2.
### Common Core – Page No. 150000
Lessons 7.6–7.7
Find the unknown factor and quotient.
Question 1.
7 × ______ = 35 35 ÷ 7 = ______
Explanation:
Let the unknown factor be x
7 × x = 35
x = 35 ÷ 7
x = 5
Now check whether the related multiplication and division facts are the same.
35 ÷ 7 = 5
Thus the unknown factor and quotient are the same. So the answer is 5.
Question 2.
6 × ______ = 54 54 ÷ 6 = ______
Explanation:
Let the unknown factor be y
6 × y = 54
y = 54 ÷ 6
6 divides 54 nine times. Thus the unknown factor is 9.
Now check if the related multiplication and division facts are the same.
54 ÷ 6 = 9
Therefore the unknown factor and the quotient are the same I.e., 9
Question 3.
6 × ______ = 18 18 ÷ 6 = ______
Explanation:
Let the unknown factor be t
6 × t = 18
t = 18/6 = 3
Now check the related multiplication and division facts of the equation
18 ÷ 6 = 3
The related multiplication and division facts are the same.
Thus the unknown factor and quotient are 3.
Question 4.
7 × ______ = 49 49 ÷ 7 = ______
Explanation:
Let the unknown factor be x
7 × x = 49
x = 49/7 = 7
Check whether the related multiplication and division facts of the equation are the same or not.
49 ÷ 7 = 7
By thus we can say that the related facts are the same. So the unknown factor and quotient are 7.
Find the quotient.
Question 5.
36 ÷ 6 = ______
Explanation:
First take the factors of 6
Factors of 6 are 3, 2
First divide 36 by 3
36 ÷ 3 = 12
Now divide 12 by 2
12 ÷ 2 = 6
So the quotient is 6.
Question 6.
48 ÷ 6 = ______
Explanation:
The factors of 6 are 3 and 2
First divide by 3
48 ÷ 3 = 16
Now divide 16 by 2
16 ÷ 2 = 8
So the quotient is 8.
Question 7.
7)$$\bar { 6 3 }$$
______
Explanation:
The factors of 7 are 1, 7
Divide 63 by 7
7 divides 63 nine times. So the quotient is 9.
Question 8.
7)$$\bar { 5 6 }$$
______
Explanation:
The factors of 7 are 1, 7
7 divides 56 eight times. So the quotient is 8.
Lessons 7.8–7.9
Find the quotient.
Question 9.
40 ÷ 8 = ______
Explanation:
Factors of 8 is 4, 2
First divide by 4
40 ÷ 4 = 10
Next divide 10 by 2
10 ÷ 2 = 5
So the quotient is 5.
Question 10.
______ = 24 ÷ 8
Explanation:
The factors of 8 is 4 and 2
Divide 24 by 4
24 ÷ 4 = 6
Now divide 6 by 2
6 ÷ 2 = 3
So the quotient of 24 ÷ 8 = 3
Question 11.
72 ÷ 9 = ______
Explanation:
The factors of 9 are 3, 3
First divide 72 by 3
72 ÷ 3 = 24
Next divide 24 by 3
24 ÷ 3 = 8
The quotient of 72 ÷ 9 = 8
Question 12.
______ = 81 ÷ 9
Explanation:
The factors of 9 are 3, 3
Divide 81 by 3
81 ÷ 3 = 27
Now divide 27 by 3
27 ÷ 3 = 9
The quotient of 81 ÷ 9 = 9
Find the unknown number.
Question 13.
36 ÷ 9 = m
m = ______
Explanation:
Let m be the unknown number
The factors of 9 are 3, 3
First, divide by 3
36 ÷ 3 = 12
Next divide 12 by 3
12 ÷ 3 = 4
So the value of m is 4.
Question 14.
18 ÷ 9 = ■
■ ______
Explanation:
Take the factors of 9
Divide 18 by 3
18 ÷ 3 = 6
Now divide 6 by 3
6 ÷ 3 = 2
■ = 2
Question 15.
48 ÷ 8 = b
b = ______
Let b be the unknown number
The factors of 8 is 4, 2
Divide 48 by 4
48 ÷ 4 = 12
Next divide 12 by 2
12 ÷ 2 = 6
Therefore the value of b = 6
Question 16.
56 ÷ 8 = p
p = ______
Explanation:
Let p be the unknown number
The factors of 8 are 4, 2
First, divide 56 by 4
56 ÷ 4 = 14
Next divide 14 by 2
14 ÷ 2 = 6
The value of p is 6.
Lesson 7.10
Question 17.
At a store, there are 5 vases. Each vase has the same number of flowers. Sixteen flowers are sold. Now there are 24 flowers left. How many flowers were in each vase?
______ flowers
Explanation:
Given that,
Number of vases = 4
Number of flowers sold = 16
Number of flowers left = 24
Total number of flowers = 16 + 24 = 40
To find the number of flowers in each vase you need to divide the total number of flowers by number of vases
= 40 ÷ 5 = 8 flowers
Thus the number of flowers in each vase = 8
Question 18.
Lizzy bought 4 bags of apples. Each bag had the same number of apples. Her mom gave her 8 more apples. Now Lizzy has 36 apples. How many apples were in each bag?
______ bags
Explanation:
Given,
Lizzy bought 4 bags of apples.
Number of apples her mother gave = 8
Number of apples now Lizzy have = 36
To find the actual number of apples before her mother gave, we need to subtract 8 from 36
36 – 8 = 28
Now divide the number of apples by number of bags
28 ÷ 4 = 7 apples
Therefore the number of apples in each bag = 7 apples
Lesson 7.11
Follow the order of operations to find the unknown number.
Question 19.
10 − 3 + 4 = t
t = ______
Explanation:
First subtract from left to right and then add
10 – 3 + 4 = 7 + 4 = 11
Therefore t = 11
Question 20.
8 − 3 × 2 = p
p = ______
Explanation:
First multiple 3 and 2
3 × 2 = 6
And then subtract 6 from 8, you get 2
Thus p = 2
Question 21.
24 ÷ 6 + 2 = w
w = ______
Explanation:
First, divide 24 and 6
24 ÷ 6 = 4
Now add from left to right
4 + 2 = 6
Therefore the unknown number w = 6
Conclusion
Keep practicing the problems given in Go Math Grade 3 Chapter 7 Division Facts and Strategies Extra Practice to score the highest score in the exams. If you want to practice exercise and homework sums then go through Go Math Grade 3 Answer Key Chapter 7 Division Facts and Strategies. Check the step by step procedure to solve the divisions in an easy manner. Students can clarify their doubts in the subject by posting the comments in the below comment section.
Scroll to Top | 2021-10-21 01:30:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7399446368217468, "perplexity": 1657.150936738717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00495.warc.gz"} |
https://quantummechanics.ucsd.edu/ph130a/130_notes/node330.html | ## Homework Problems
1. Find the allowed total spin states of two spin 1 particles. Explicitly write out the 9 states which are eigenfunctions of and .
2. The Hamiltonian of a spin system is given by . Find the eigenvalues and eigenfunctions of the system of two particles (a) when both particles have spin , (b) when one particle has spin and the other spin 1. What happens in (a) when the two particles are identical?
3. Consider a system of two spinless identical particles. Show that the orbital angular momentum of their relative motion can only be even. Show by direct calculation that, for the triplet spin states of two spin particles, for all allowed . Show that for the singlet state .
4. A deuteron has spin 1. What are the possible spin and total angular momentum states of two deuterons. Include orbital angular momentum and assume the two particles are identical.
5. The state of an electron is given by . Find the possible values and the probabilities of the component of the electron's total angular momentum. Do the same for the total angular momentum squared. What is the probability density for finding an electron with spin up at ? What is it for spin down? What is the probability density independent of spin? (Do not leave your answer in terms of spherical harmonics.)
6. The states of hydrogen have an 8-fold degeneracy due to the various and states allowed and the two spin states of the electron. The spin orbit interaction partially breaks the degeneracy by adding a term to the Hamiltonian . Use first order perturbation theory to find how the degeneracy is broken under the full Hamiltonian and write the approximate energy eigenstates in terms of , , and .
7. The nucleus of a deuterium (A=2 isotope of H) atom is found to have spin 1. With a neutral atom, we have three angular momenta to add, the nuclear spin, the electron spin, and the orbital angular momentum. Define in the usual way and where denotes the nuclear spin operator. What are the possible quantum numbers and for an atom in the ground state? What are the possible quantum numbers for an atom in the 2p state?
Jim Branson 2013-04-22 | 2022-09-24 19:51:23 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9234269261360168, "perplexity": 237.82164877007395}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030333455.97/warc/CC-MAIN-20220924182740-20220924212740-00496.warc.gz"} |
https://mathematica.stackexchange.com/questions/30548/why-does-mathematica-not-simplify-the-gudermannian-function | # Why does Mathematica not simplify the Gudermannian function?
The Gudermannian function is built into Mathematica, but does not reduce to simpler terms in some simple cases. In particular, the Gudermannian is an odd function of real arguments but Mathematica does not seem to know or use this. For example:
FullSimplify[Gudermannian[2x] + Gudermannian[-2x],Assumptions->{x>0}]
Gudermannian[-2 x] + Gudermannian[2 x]
Am I missing some subtlety that invalidates simplifying to 0 or is Mathematica missing something?
Mathematica is having hard time knowing that ArcTan[E^(-2 x)] + ArcTan[E^(2 x) == Pi/2 and I could not figure how to tell so other than by brute force telling it so in the Assuming.
Once it knew this relation, then the simplification of the FunctionExpand applied to your input, gives zero as expected.
ClearAll[x]
From help, it says Use FunctionExpand to expand Gudermannian in terms of elementary functions and that is how the above relation came out. So, without using FunctionExpand it is even harder. So, I would say use FunctionExpandonGudermannian` since then there is more chance of doing more simplification on it when it is in that form (i.e. using elementary functions) | 2020-01-29 08:43:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49350062012672424, "perplexity": 813.8616381882678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251789055.93/warc/CC-MAIN-20200129071944-20200129101944-00552.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-11-additional-topics-11-6-pie-bar-and-line-graphs-problem-set-11-6-page-502/13 | ## Elementary Algebra
About $1000$ more people preferred the space center than the water park.
Approximately 1500 people preferred the space center. Approximately 500 people preferred the water park. To figure out how many more people preferred the space center than the water park, we must subtract. $1500-500=1000$. | 2018-11-18 02:30:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19674482941627502, "perplexity": 4430.355728622105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743960.44/warc/CC-MAIN-20181118011216-20181118033216-00148.warc.gz"} |
https://physics.stackexchange.com/questions/432650/quantum-mechanics-is-u1-invariant-hence-x-is-isotropic/432677 | # Quantum Mechanics is $U(1)$ invariant, hence $X$ is isotropic?
Since our fundamental laws are invariant under rotations. Hence, we say that spacetime isotropic.
Now Quantum Mechanics is invariant under (global) complex rotations ($$U(1)$$ transformations. Hence, we say that $$X$$ is isotropic.
In other words, what is the correct analogous space $$X$$ here and is "isotropic" the correct term used?
First of all, the space $$X$$ here is the Hilbert space $$\mathcal H$$ of the system, since the $$U(1)$$ transformations apply to the state vector $$|\psi\rangle \in \mathcal H$$ of the system. "Isotropic" is not the correct term to describe the symmetry. The term "isotropic" could maybe be used if $$\mathcal H \cong \mathbb C^N$$ were symmetric under $$SU(N)$$ transformations, but I don't think this is a common terminology (also, such a system would be very boring). Instead, this is called a (global) $$U(1)$$ gauge symmetry.
Let me make two more comments. First, the two symmetries here have an important fundamental difference. The $$SO(3)$$ symmetry of mechanics compares actual, physically different system states (different rotations of the physical system). However, the $$U(1)$$ symmetry of quantum mechanics is a gauge symmetry, meaning: The two states $$|\psi\rangle$$ and $$e^{i\varphi} |\psi\rangle$$ are the exact same physical state, but our description of the system allows that same state to be described with many different Hilbert space vectors. (This is for convenience only, the actual state space is $$\mathcal H / \mathbb C^\ast$$, but it is easier to work with $$\mathcal H$$.)
Second, the Noether theorem states that every symmetry corresponds to a conserved quantity. The conserved quantity corresponding to the $$SO(3)$$ symmetry of mechanics is angular momentum, but what is that of the $$U(1)$$ gauge symmetry? Once you learn about QED, you will see that it is the total electrical charge.
• The global phase freedom of QM is not a gauge symmetry and is distinct from the electromagnetic $\mathrm{U}(1)$ gauge symmetry, see physics.stackexchange.com/a/433501/50583 and physics.stackexchange.com/questions/433457/… – ACuriousMind Oct 9 '18 at 16:47
• @ACuriousMind Why would you say it is not a gauge symmetry? I would have defined a gauge symmetry exactly like this, that one physical state has several representations. I don't understand your comment about the distinction from the EM $U(1)$ either: I would have thought that the extra factor of $e$ in the exponent is just a redefinition / change of units of $\varphi$. – Noiralef Oct 9 '18 at 18:50
• 1. A gauge symmetry has a technical definition in terms of the Hamiltonian being constrained or the relation between canonical momenta and generalized velocities being non-invertible. You may use it differently, but this strikes me as confusing. 2. The point is that the space of states can conceivably contain states of different charge, e.g. in a many-body space of states, it can have states $|n\rangle$ of n electrons. Then the gauge symmetry acts as $|0\rangle +|1\rangle +|2\rangle\mapsto |0\rangle +e^{i\phi}|1\rangle +e^{2i\phi}|2\rangle$, which is obviously different from a global phase. – ACuriousMind Oct 9 '18 at 19:01
• Note also that the electromagnetic gauge transformation does not act on uncharged states at all, while the global phase choice of course does. – ACuriousMind Oct 9 '18 at 19:03
In analogy with the more common notion "isospin space", I would say the corresponding space for $$U(1)$$ gauge symmetry is "charge space".
For example, one book which uses this notion is Particle Astrophysics by Donald H. Perkins. | 2020-10-23 02:47:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8877421617507935, "perplexity": 332.09016006283343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880519.12/warc/CC-MAIN-20201023014545-20201023044545-00214.warc.gz"} |
https://www.clutchprep.com/chemistry/practice-problems/105275/a-solution-contains-2-0-x-10-3-m-ce3-and-1-0-x-10-2-m-io3-3-will-ce-io3-3-s-prec | Problem: A solution contains 2.0 X 10 -3 M Ce3+ and 1.0 X 10 -2 M IO3 3-. Will Ce(IO3)3(s) precipitate? [Ksp for Ce(IO3)3 is 3.2 X 10-10.]
🤓 Based on our data, we think this question is relevant for Professor Hummel's class at UIUC.
FREE Expert Solution
• Q > Ksp: the solution is supersaturated and a precipitate will form. Reactants are favored.
• Q = Ksp: the solution is at equilibrium and no precipitate will form.
• Q < Ksp: the solution is unsaturated and no precipitate will form. Products are favored.
For Ce(IO3)3(s):
The sulfate ion, IO3, has a charge of –1. Ce then has a charge of +3.
The dissociation of Ce(IO3)3(s) in water is as follows:
Ce(IO3)3(s) Ce3+(aq) + 3 IO3(aq)
Calculate the reaction quotient for Ce(IO3)3 is:
$\mathbf{Q}\mathbf{=}\frac{\mathbf{products}}{\overline{)\mathbf{reactants}}}\phantom{\rule{0ex}{0ex}}\mathbf{Q}\mathbf{=}\left[{\mathrm{Ce}}^{3+}\right]{\left[{{\mathrm{IO}}_{3}}^{-}\right]}^{\mathbf{3}}$
Note that each concentration is raised by the stoichiometric coefficient: [Ce3+] is raised to 1 and [IO3] is raised to 3.
Solving for Q:
Problem Details
A solution contains 2.0 X 10 -3 M Ce3+ and 1.0 X 10 -2 M IO3 3-. Will Ce(IO3)3(s) precipitate? [Ksp for Ce(IO3)3 is 3.2 X 10-10.] | 2020-05-30 18:57:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2829751968383789, "perplexity": 11795.209606953826}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347410284.51/warc/CC-MAIN-20200530165307-20200530195307-00344.warc.gz"} |
https://bird.bcamath.org/handle/20.500.11824/18/browse?rpp=20&sort_by=1&type=title&offset=52&etal=-1&order=ASC | Now showing items 53-60 of 60
• #### Some classes of homeomorphisms that preserve multiplicity and tangent cones
(2018-08-19)
In this paper it is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant ...
• #### Some classes of homeomorphisms that preserve multiplicity and tangent cones
(2019-05-28)
In this paper we present some applications of A'Campo-Lê's Theorem and we study some relations between Zariski's Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones ...
• #### Some classes of homeomorphisms that preserve multiplicity and tangent cones
(2020-01-01)
In this paper we present some applications of A’Campo-Lˆe’s Theorem and we study some relations between Zariski’s Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent ...
• #### Some contributions to the theory of singularities and their characteristic classes
(2021-06-02)
In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities ...
• #### A specialization property of index
(2017-01-10)
In [Kol13] Kollár defined $i$-th index of a proper scheme over a field. In this note we study how index behaves under specialization, in any characteristic.
• #### Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds
(2017-02)
Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3--manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated ...
• #### Topological invariants of plane curve singularities: Polar quotients and Lojasiewicz gradient exponents
(2019-10-21)
In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that, for complex plane curve singularities, the set of polar quotients is a ...
• #### Topology of Spaces of Valuations and Geometry of Singularities
(2017-11-11)
Given an algebraic variety X defined over an algebraically closed field, we study the space RZ(X,x) consisting of all the valuations of the function field of X which are centered in a closed point x of X. We concentrate ... | 2021-09-25 12:25:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6506354212760925, "perplexity": 635.1426585969278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057622.15/warc/CC-MAIN-20210925112158-20210925142158-00208.warc.gz"} |
https://csblog.madhavshekhar.com/julia/ml/gci19/2020/01/14/Sentiment-Analysis-with-TextAnalysis(jl).html | Note: Part 2 of this notebook is accomplished with TensorFlow and can be found here.
Use the amazon review data from Kaggle to test the efficiency of our Sentiment Analysis models that live in TextAnalysis.jl. Compare it with models in ScikitLearn and Spacy python libraries. Upload your results as an issue in the TextAnalysis package.
Some basic machine learning knowledge is useful for this task.
#### Special thanks to Ayush Kaushal; an exemplary mentor without whom this task wouldn't be possible.
Find below, the julia part of the task. The python notebook would be attached too but would have sparse documentation.
The process of algorithmically identifying and categorizing opinions expressed in text to determine the user’s attitude toward the subject of the document (or post).
This is how I understand it.
## Importing Required Packages
using TextAnalysis, FileIO
I would be working on the test data since the train one is humongous and my laptop was unable to render that in Jupyter every single time even when left for about an hour.
So, declaring the test reviews as review as evident by the code below.
reviews = Document("text/test.ft.txt")
FileDocument("text/test.ft.txt", TextAnalysis.DocumentMetadata(Languages.English(), "text/test.ft.txt", "Unknown Author", "Unknown Time"))
Getting to know some of our data.
We can see that the .txt file contains reviews in the form of :
"label1(/2) space ...the review..."
Exploratory data analysis also reveals that reviews beginning with __label__2 are positive reviews. That means that their sentiment score would also be higher (I'll demonstrate that in a sec...) Similarly, reviews beginning with __label__1 are negative reviews and so their sentiment score should evidently be lower.
### Getting our pre-trained Sentiment Analyser to check on these lines.
sent = SentimentAnalyzer()
┌ Info: CUDAdrv.jl failed to initialize, GPU functionality unavailable (set JULIA_CUDA_SILENT or JULIA_CUDA_VERBOSE to silence or expand this message)
Sentiment Analysis Model Trained on IMDB with a 88587 word corpus
#seeing how the data is arranged.
3-element Array{String,1}:
"__label__2 Great CD: My lovely Pat has one of the GREAT voices of her generation. I have listened to this CD for YEARS and I still LOVE IT. When I'm in a good mood it makes me feel better. A bad mood just evaporates like sugar in the rain. This CD just oozes LIFE. Vocals are jusat STUUNNING and lyrics just kill. One of life's hidden gems. This is a desert isle CD in my book. Why she never made it big is just beyond me. Everytime I play this, no matter black, white, young, old, male, female EVERYBODY says one thing \"Who was that singing ?\""
"__label__2 One of the best game music soundtracks - for a game I didn't really play: Despite the fact that I have only played a small portion of the game, the music I heard (plus the connection to Chrono Trigger which was great as well) led me to purchase the soundtrack, and it remains one of my favorite albums. There is an incredible mix of fun, epic, and emotional songs. Those sad and beautiful tracks I especially like, as there's not too many of those kinds of songs in my other video game soundtracks. I must admit that one of the songs (Life-A Distant Promise) has brought tears to my eyes on many occasions.My one complaint about this soundtrack is that they use guitar fretting effects in many of the songs, which I find distracting. But even if those weren't included I would still consider the collection worth it."
"__label__1 Batteries died within a year ...: I bought this charger in Jul 2003 and it worked OK for a while. The design is nice and convenient. However, after about a year, the batteries would not hold a charge. Might as well just get alkaline disposables, or look elsewhere for a charger that comes with batteries that have better staying power."
Now, I'll see that for the first 10 reviews in our dataset, what the actual label is and what sentiment score does our model return. From this we'll be able to know that the model isn't perfect and does indeed predict wrong sentiments for some reviews. Thus, developing a need to do text pre-processing to make the reviews comparable and remove unnecessary stuff like urls and other things generally not contributing to the read/feel of the review. !
tab = "SNo. | Label | Prediction Score | Should be | Predicted | Correct/Incorrect "
println(tab)
println("-"^(length(tab)+5))
for i in 1:15
review = StringDocument(review);
pred = sent(review);
if label == "__label__2"
should_be = "+ve"
else
should_be = "-ve"
end
if pred >= 0.5
pred_be = "+ve"
elseif pred < 0.5
pred_be = "-ve"
end
if pred_be == should_be
correct = "Correct"
else
correct = "Incorrect"
end
println("$i |$label | $pred |$should_be | $pred_be |$correct ")
end
SNo. | Label | Prediction Score | Should be | Predicted | Correct/Incorrect
--------------------------------------------------------------------------------------
1 | __label__2 | 0.39506337 | +ve | -ve | Incorrect
2 | __label__2 | 0.5314957 | +ve | +ve | Correct
3 | __label__1 | 0.52432084 | -ve | +ve | Incorrect
4 | __label__2 | 0.5501878 | +ve | +ve | Correct
5 | __label__2 | 0.5919624 | +ve | +ve | Correct
6 | __label__1 | 0.61544746 | -ve | +ve | Incorrect
7 | __label__1 | 0.732198 | -ve | +ve | Incorrect
8 | __label__1 | 0.55473757 | -ve | +ve | Incorrect
9 | __label__2 | 0.4127747 | +ve | -ve | Incorrect
10 | __label__1 | 0.58470565 | -ve | +ve | Incorrect
11 | __label__2 | 0.5855292 | +ve | +ve | Correct
12 | __label__1 | 0.51694876 | -ve | +ve | Incorrect
13 | __label__1 | 0.5547061 | -ve | +ve | Incorrect
14 | __label__2 | 0.45876318 | +ve | -ve | Incorrect
15 | __label__1 | 0.52366424 | -ve | +ve | Incorrect
It's clear that our model isn't optimal since out of 15 samples, only 4 were correct predictions. However, I went a little too harsh on the model since in some cases, like in
14 | __label__2 | 0.45876318 | +ve | -ve | Incorrect
the model was within some limit of correct predictions. So yeah, sorry Mr. Sentiment Analyzer.
Moving on towards trying to improve the accuracy of predcitions by performing some general pre-defined text-processing functions in TextAnalysis package. But first, I want to know the length of our test data set so I can make batches of processing accrodinly to my computational powers.
test_data = readlines("text/test.ft.txt")
length(test_data)
400000
Ok, so now we know the size of the data we're dealing with let's get started with the pre-processing.
A true positive is an outcome where the model correctly predicts the positive class. Similarly, a true negative is an outcome where the model correctly predicts the negative class.
A false positive is an outcome where the model incorrectly predicts the positive class. And a false negative is an outcome where the model incorrectly predicts the negative class.
test_labels = []
test_string = []
fal_pos = 0
fal_neg = 0
tru_pos = 0
tru_neg = 0
for i in 1:length(test_data)
label =test_data[i][1:10];
push!(test_labels, label);
review = test_data[i][11:end];
push!(test_string, review)
#after adding reviews and labels in their respective arrays.
#I'll perform pre-processing on individual reviews.
review_sd = StringDocument(review)
remove_corrupt_utf8!(review_sd)
stem!(review_sd)
remove_case!(review_sd)
#remove_indefinite_articles!(review_sd)
#remove_definite_articles!(review_sd)
if label == "__label__2"
should_be = "+ve"
else
should_be = "-ve"
end
pred = sent(review_sd)
if pred >= 0.5
pred_be = "+ve"
elseif pred < 0.5
pred_be = "-ve"
end
if pred_be == "+ve" && should_be == "+ve"
tru_pos += 1
elseif pred_be == "-ve" && should_be == "-ve"
tru_neg += 1
elseif pred_be == "-ve" && should_be == "+ve"
fal_pos += 1
elseif pred_be == "+ve" && should_be == "-ve"
fal_neg += 1
end
end
BoundsError: attempt to access 32×5000 Array{Float32,2} at index [Base.Slice(Base.OneTo(32)), 5001]
Stacktrace:
[1] throw_boundserror(::Array{Float32,2}, ::Tuple{Base.Slice{Base.OneTo{Int64}},Int64}) at .\abstractarray.jl:538
[2] checkbounds at .\abstractarray.jl:503 [inlined]
[3] _getindex at .\multidimensional.jl:669 [inlined]
[4] getindex at .\abstractarray.jl:981 [inlined]
[5] embedding(::Array{Float32,2}, ::Array{Float64,1}) at C:\Users\shekh\.julia\packages\TextAnalysis\pcFQf\src\sentiment.jl:27
[6] (::TextAnalysis.var"#24#25"{Dict{Symbol,Any}})(::Array{Float64,1}) at C:\Users\shekh\.julia\packages\TextAnalysis\pcFQf\src\sentiment.jl:40
[7] get_sentiment(::TextAnalysis.var"#26#27", ::Array{String,1}, ::Dict{Symbol,Any}, ::Dict{String,Any}) at C:\Users\shekh\.julia\packages\TextAnalysis\pcFQf\src\sentiment.jl:59
[8] (::TextAnalysis.SentimentModel)(::Function, ::Array{String,1}) at C:\Users\shekh\.julia\packages\TextAnalysis\pcFQf\src\sentiment.jl:87
[9] SentimentAnalyzer at C:\Users\shekh\.julia\packages\TextAnalysis\pcFQf\src\sentiment.jl:103 [inlined] (repeats 2 times)
[10] top-level scope at .\In[33]:28
Ahh! Finally it's complete.
We get BoundsError: attempt to access 32×5000 Array{Float32,2} at index [Base.Slice(Base.OneTo(32)), 5001] error however on seeing this issue on TextAnalysis package.
Ref:BoundsError in sentiment analysis I've decided to ignore it. Let's get on towards calculating our predictions metrices: Precision / F1Score / Recall.
$$P = \frac{T_p}{T_p+F_p}$$
$$R = \frac{T_p}{T_p + F_n}$$
$$F1 = \frac{2 \cdot P\cdot R}{P+ R}$$
### Precision
precision = tru_pos / (tru_pos + fal_pos)
println("Precision is $precision") Precision is 0.583117838593833 recall = tru_pos / (tru_pos + fal_neg) println("Recall is$recall")
Recall is 0.5144996465068449
f1score = (2 * precision * recall) / (precision + recall)
#f1score is from 0 --> 1
println("F1Score is \$f1score.")
F1Score is 0.5466638895622987. | 2021-09-20 02:02:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2200925499200821, "perplexity": 9148.46116160204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056974.30/warc/CC-MAIN-20210920010331-20210920040331-00569.warc.gz"} |
https://stats.stackexchange.com/questions/90802/ambiguity-with-multinomial-logit-models | # Ambiguity with multinomial logit models
I have always thought that, when dealing with multinomial logistic regression, the idea was to linearly model the "logistic" functions of the probability densities of the different response categories (as explained here). I put quotation marks around "logistic" since we do not use the real logistic functions, $\log\left(\frac{\pi_j(x)}{1-\pi_j(x)}\right)$, but with denominator equal to the density of a certain "pivot" category.
I then discovered the extension to a log-linear model, where the logarithms of the probability distributions, $\log(\pi_j(x))$, are directly modeled, as explained here. At the end of the day, it is still very similar to the previous assumption.
However, I had problems when I discovered the existence of this alternative formulation, which is explained in the R package mlogit's vignette (pdf). Basically, every $\log(\pi_j(x))$ is modeled with $\alpha_j+\bar{\beta}\cdot\bar{x}$, where $\alpha_j$ is an intercept which is characteristic of every response category, and the vector of linear coefficients, $\bar{\beta}$, is the same for every category.
My question is not maybe a real question, but why this ambiguity?
I finally found an acknowledgment that this notation is misleading in the documentation of the mlogit package (vignette (pdf)). Page 8, 1.2 Model description:
while working with multinomial logit models, one has to consider three kinds of variables:
• alternative specific variables $x_{ij}$ with a generic coefficient $\beta$,
• individual specific variables $z_i$ with an alternative specific coefficients $\gamma_j$,
• alternative specfic variables $w_{ij}$ with an alternative specific coefficient $\delta_j$
The satisfaction index for the alternative $j$ is then :
$V_{ij}=\alpha_j+\beta x_{ij}+\gamma_j z_i+\delta_j w_{ij}$
[...]
A model with only individual specific variables is sometimes called a multinomial logit model, one with only alternative specific variables a conditional logit model and one with both kind of variables a mixed logit model. This is seriously misleading : conditional logit model is also a logit model for longitudinal data in the statistical literature and mixed logit is one of the names of a logit model with random parameters. Therefore, in what follow, we'll use the name multinomial logit model for the model we've just described whatever the nature of the explanatory variables included in the model.
I got a shiver down my spine when I read the last sentence. | 2020-04-03 08:22:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8969838619232178, "perplexity": 667.468812299307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370510352.43/warc/CC-MAIN-20200403061648-20200403091648-00333.warc.gz"} |
https://www.physicsforums.com/threads/index-of-subgroup-h-is-2-implies.546374/ | # Index of subgroup H is 2 implies
Index of subgroup H is 2 implies....
## Homework Statement
Just had my abstract algebra test. This was the only question I did not answer. The rest I answered somewhat confidently.
Prove that if H is a subgroup of G, [G : H] = 2, a,b are in G but not in H, then ab is in H.
## Homework Equations
2 is the index of H, that is, the order of G equals 2 times the order of H.
## The Attempt at a Solution
I have no idea where to start with this. I can't even come up with a concrete example demonstrating this.
The concrete example that springs to mind is G = Z_6 and H is the subgroup of G generated by 2. Then 1 and 3 are in G and 1+3=4 is in H.
I am not sure how to help prove the general case, but I am interested in the proof.
Hmm, good example! I wonder if this only works with cyclic groups.
I like Serena
Homework Helper
Hi Arcana!
I take it congratulations are in order, if you were so confident?
Another non-cyclic example would be the 4-group of Klein: de symmetries of the diamond.
This is G={id,sx,sy,h}
If H={id,sx} then with every choice for a and b from {sy,h} the product will be in H.
Now for a proof.
I'll show you how it starts...
The index [G] is the order of the quotient group G/H.
If it is 2, then there are 2 elements in G/H.
Let's say G/H={H, cH}.
Now every a and b that are not in H must therefore be in cH, which is also aH then.
So there is an h' in H for which b=ah'
Suppose ab is also not in H.
Then there is an h'' in H for which ab=ah''.
...
Deveno
no, it works with any group with a subgroup of index 2.
if we have a subgroup of index 2, we have just two cosets of H, H and "the rest of G".
one consequence of this, is that for any g in G, gH = Hg.
for a,b not in H, consider the set S = {hah'b : h,h' in H}.
note that since ah' is in aH, ah' = h"a, for some other element h" of H.
thus S is contained in Hab (hah'b = hh"ab). on the other hand,
for any element hab in Hab, we can write hab = (ha)(eb), which is in S.
so S and Hab are the same set.
since S is the coset Hab, we have 2 choices: S = H, or S = Ha = Hb (those are the only 2 cosets we have).
suppose b is in S = Hab (this is the choice S = Hb).
then b = hab, so e = ha, which means e is in Ha,
which means that H = Ha, so that a is in H. but a isn't in H, so this is a contradiction.
why does this show ab is in H?
*****
note to I like Serena: we're saying the same thing. what i do not know, is whether or not they've covered normality yet, so i have fastidiously avoided the term "quotient group", i am only dealing with a set of (right) cosets, and i only use the property gH = Hg once, to avoid such a digression.
Why is there an h in H such that b=ah?
(class is just now ending, I am heading for home, so I won't see anymore replies tonight)
I like Serena
Homework Helper
Why is there an h in H such that b=ah?
(class is just now ending, I am heading for home, so I won't see anymore replies tonight)
Let's illustrate, suppose H={e,h1,h2,h3}
And G/H={H, cH}.
This is the case because the index [G] is defined to be the number of elements in G/H, which is 2 in this case.
Then cH={c,ch1,ch2,ch3}
And therefore G={e,h1,h2,h3, c,ch1,ch2,ch3}.
There are 2 elements in G/H, H itself and some cH.
So each element in G must be either in H or in cH.
b is not in H, so b must be in cH=aH.
Therefore there must be a h in H, such that b=ah.
Last edited:
Okay, so b must equal c, ch1, ch2, or ch3.
Also, so must a.
so maybe b=ch1 and a=ch2
Then if ab is in H, that must mean ch2ch1 is in H. So how do we reduce ch2ch1 to e, h1, h2, or h3?
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Okay, so b must equal c, ch1, ch2, or ch3.
Also, so must a.
so maybe b=ch1 and a=ch2
Then if ab is in H, that must mean ch2ch1 is in H. So how do we reduce ch2ch1 to e, h1, h2, or h3?
Well typically you would do this with a proof by contradiction.
Suppose ch2ch1 is not in H.
Then ch2ch1 must be in cH (due to the [G]=2 index thingy).
Therefore there must be an h in H, such that ch2ch1 = ch.
Can you do some cancellations and stuff?
No, this is getting a little messy. Beep beep, back up the truck. Lets try again.
a and b are not in H, thus they are in the other half of G, aka cH, aka the coset of H.
Suppose ab is not in H. Then, ab is in cH. so...?
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No, this is getting a little messy. Beep beep, back up the truck. Lets try again.
a and b are not in H, thus they are in the other half of G, aka cH, aka the coset of H.
Suppose ab is not in H. Then, ab is in cH. so...?
The definition of cH is $\{x: x=ch \wedge h \in H\}$.
So there is an h in H, such that ab=ch.
Okay, so then...? (sorry I'm not getting it yet)
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Okay, so then...? (sorry I'm not getting it yet)
We picked a and b from cH, and we try to prove that ab in cH will lead to a contradiction.
Converting this to specific elements we get:
$\exists h_1, h_2, h_3 \in H \text{ such that } a=ch_1, b=ch_2, ab=ch_3$
Substitute a and b in ab, and try to find a contradiction...
ch1ch2=ch3 implies (using left cancellation law)
h1ch2=h3
But then I'm stuck.
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ch1ch2=ch3 implies (using left cancellation law)
h1ch2=h3
But then I'm stuck.
What happens if you left-multiply with $h_1^{-1}$ and use that H is closed under multiplication?
then I get ch2=h3h1-1
which implies h3h1-1 is in cH and not in H but it must be in H since h3 and h1-1 are in H. Thus, contradiction, thus ab is in H.
right?
I do not like that problem.
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Yep!
What's wrong with the problem?
There are a few interesting concepts in there.
Such that you can divide a group in disjunct cosets.
Visually, I like to think graphically in the X-Y plane.
A subgroup is for instance a set of points on the x-axis.
And another coset is a line parallel to the x-axis.
The operation is vector addition, but modulo some number.
Obviously if you pick 2 vectors in the parallel line, they won't sum up to a vector on that line.
Visualizing stuff makes it come a bit to life for me.
Btw, the proof can be a bit shorter and neater if you use for instance that cH=aH, effectively discarding c.
(In the vector representation of a line, you are free to choose the supporting vector.)
thanks for the help. I am not a visual person, graphs make me sad :( I'm going to be sad a lot in vector calc next semester.
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Ah, you are more in number theory then?
Then you should appreciate abstract algebra all the more! ;)
yes, I love my abstract algebra class. i just didn't dig this problem. I'm not too keen on cosets.
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Well, you're not done with cosets by a long shot! Just you wait!!
well next we do isomorphisms. after that, looks like homomorphisms, with a section on quotient groups, and then rings.
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Well, there should be a section about the grand "Isomorphism theorem".
It may be included in the section on quotient groups, which is also about cosets.
You should get some headaches from that. (I did! )
oh dear. "Isomorphism Theorem (parts 1 and 2)" is in the problems section! If it's the headache you say it is, I'm going to be unhappy!
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No worries, all you have to do is apply Mike's Mathgasm cycle!! | 2021-04-21 08:23:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7828371524810791, "perplexity": 1448.9467337681313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00233.warc.gz"} |
http://www.onemathematicalcat.org/algebra_book/online_problems/frac_exp_kx.htm | RENAMING FRACTIONAL EXPRESSIONS
It's often necessary to take a somewhat complicated-looking fraction,
like (say) $\,-\frac{5x}{-3}\,$, and rename it.
One popular name is the form $\,kx\,$: i.e., a number first, and the variable $\,x\,$ last.
In general, it is efficient to make two ‘passes’ through the expression:
figure out the sign (plus or minus) on the first pass, and the size on the second pass: $$-\frac{5x}{-3}\ \ \overset{\text{first pass, determine plus/minus sign:}}{ \overset{\text{even # of negative factors, so positive}}{\overbrace{\strut\ \ \ =\ \ \ }}} \ \ \frac{5x}{3}\ \ \overset{\text{‘peel off’ the coefficient}}{ \overset{\text{and write it in front}}{\overbrace{\strut\ \ \ =\ \ \ }}} \ \ \underset{k}{\underbrace{\ \frac53\ }} x$$ This exercise gives you practice with this type of renaming.
EXAMPLES:
Question: Rename in the form $\,kx\,$: $\displaystyle\frac{5x}{-2}$
Solution: $\displaystyle \frac{5x}{-2} = -\frac{5}{2}x$
Question: Rename in the form $\,kx\,$: $\displaystyle-\frac{-x}{-4}$
Solution: $\displaystyle -\frac{-x}{-4} = -\frac{1}{4}x$
Master the ideas from this section
When you're done practicing, move on to:
Practice with Multiples
CONCEPT QUESTIONS EXERCISE:
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
AVAILABLE MASTERED IN PROGRESS
Rename in the form $\,kx\,$:
(MAX is 23; an even number, please.) | 2017-09-23 09:19:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7117102742195129, "perplexity": 1443.153721938163}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689615.28/warc/CC-MAIN-20170923085617-20170923105617-00318.warc.gz"} |
https://leetcode.ca/2022-12-08-2450-Number-of-Distinct-Binary-Strings-After-Applying-Operations/ | ##### Welcome to Subscribe On Youtube
Formatted question description: https://leetcode.ca/all/2450.html
# 2450. Number of Distinct Binary Strings After Applying Operations
## Description
You are given a binary string s and a positive integer k.
You can apply the following operation on the string any number of times:
• Choose any substring of size k from s and flip all its characters, that is, turn all 1's into 0's, and all 0's into 1's.
Return the number of distinct strings you can obtain. Since the answer may be too large, return it modulo 109 + 7.
Note that:
• A binary string is a string that consists only of the characters 0 and 1.
• A substring is a contiguous part of a string.
Example 1:
Input: s = "1001", k = 3
Output: 4
Explanation: We can obtain the following strings:
- Applying no operation on the string gives s = "1001".
- Applying one operation on the substring starting at index 0 gives s = "0111".
- Applying one operation on the substring starting at index 1 gives s = "1110".
- Applying one operation on both the substrings starting at indices 0 and 1 gives s = "0000".
It can be shown that we cannot obtain any other string, so the answer is 4.
Example 2:
Input: s = "10110", k = 5
Output: 2
Explanation: We can obtain the following strings:
- Applying no operation on the string gives s = "10110".
- Applying one operation on the whole string gives s = "01001".
It can be shown that we cannot obtain any other string, so the answer is 2.
Constraints:
• 1 <= k <= s.length <= 105
• s[i] is either 0 or 1. | 2023-03-27 07:45:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6596652865409851, "perplexity": 1204.9004386455522}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948609.41/warc/CC-MAIN-20230327060940-20230327090940-00035.warc.gz"} |
https://sknadig.me/attention/ | TL;DR - Different attention mechanisms available in the ESPnet toolkit explained. Have a look at the presentation that I gave in IIIT-B AI reading group (no math included) Attention based models in End-to-End ASR
I’ll directly jump to explaining the different Attention models available in the ESPnet toolkit. (I won’t be going into the implementation challenges in getting the Encoder-Decoer Attention models work.)
Please have a look at the previous post for the basics of Attention models in Speech recognition. This post assumes you know the Attention mechanism in general and build from there.
• No Attention
• Content-based Attention
• Dot product Attention
• Location-aware Attention
• Location Aware Attention
• 2D Location Aware Attention
• Location Aware Recurrent Attention
• Hybrid Attention
• Coverage Mechanism Attention
• Coverage Mechanism Location Aware Attention
• Multi-Head Multi-Resolution Location Aware Attention
## Attention - Recap
• $x = (x_{1}, x_{2}, .........., x_{T})$ - is the input sequence
• $y = (y_{1}, y_{2}, .........., y_{U})$ - is the target output sequence
• $h = (h_{1}, h_{2}, .........., h_{T})$ - is the output of the Encoder
• $h_{t} = f(x_{t}, h_{t-1})$ - is the Encoder function
• $C_{i} = \sum_{j=1}^{T} \alpha_{i,j} \cdot h_{j}$ - is the Context vector
• $\alpha_{i,j} = Softmax(e_{i,j}) = \frac{e^{e_{i,j}}}{\sum_{k=1}^{T} e^{e_{i,k}}}$ - are the Attention weights
• $e_{i,j} = a(s_{i-1}, h_j)$ - is the importance parameter for every encoded input
• $\sum_{j=1}^{T} e_{i,j} \neq 1$ - the importance parameter need not sum to 1
• $\sum_{j=1}^{T} \alpha_{i,j} = 1$ - the attention weights sum to 1
## Types of Attention
Broadly, attention mechanisms can be categorized into 3 distinct categories
• Content aware Attention
• Location aware Attention
• Hybrid Attention
Multi-Head Attention mechanisms are a different beast altogether, we will cross that bridge when we get there. For now, let’s concentrate on the 3 broad categories I mentioned.
## 1. No Attention (Equal Attention?)
Here, no attention is used at all. Each of the $h_{i}$ are given equal importance and linearly mixed and averaged to get $C_{i}$
### No attention - code
#Mask = Ones where enc_h is present. Zeros where padding is needed.
att_prev = att_prev.to(enc_h)
c = torch.sum(enc_h * att_prev.view(batch, h_length, 1), dim=1)
## Content-based Attention
Content-based Attention - as the name suggests is based on the contents of the vector $s_{i-1}$ (Decoder hidden state) and $h_{t}$ (Annotation vectors from the Encoder). This means, our compatibility function or the Attention function depends only on the contents of these vectors, irrespective of their location in the sequence.
What does this mean? Let’s say what has been spoken in the utterance is Barb burned paper and leaves in a big bonfire. with the phonetic sequence as sil b aa r sil b er n sil p ey sil p er n l iy v z ih n ah sil b ih sil b aa n f ay er sil. The feature vector of a phoneme, let’s say b will be similar no matter the location of the phoneme in the sequence sil b aa r sil b er n sil p ey sil p er n l iy v z ih n ah sil b ih sil b aa n f ay er sil
This would give equal weight to the same phoneme, but from a different word which is not relevant to the current context. Also, a phonetically similar phoneme will get a close score to the actual phoneme.
Content-based Attention is computed as:
Dot product and additive attention are content-based attention mechanisms.
## 2. Dot product Attention
In the dot product attention, our similarity measure is the dot product between the vector $s_{i-1}$ and $h_{t}$. For generating the Context vector $C_{i}$, we take the Decoder hidden state $s_{i-1}$ when generating the previous output symbol $y_{i-1}$ and compute the dot product with each $h_{t}$ to get $e_{i,j}$ for each of the Annotation vectors.
Conceptually dot product signifies how similar each vectors are (the angle between them). More similar they are, higher the value.
Here’s an image explaining Dot Product Attention
Here, dec_z vector is the Decoder hidden state.
As we discussed in the previous post, these representations are in different dimensions. So, we learn a transformation to transform them to same dimensions so that we can compare them using dot product or addition. This transformation is learnt with other parameters using backprop.
### Dot product attention - code
mlp_enc = torch.nn.Linear(eprojs, att_dim)
mlp_dec = torch.nn.Linear(dunits, att_dim)
pre_compute_enc_h = torch.tanh(mlp_enc(enc_h))
e = torch.sum(pre_compute_enc_h * torch.tanh(mlp_dec(dec_z)).view(batch, 1, att_dim),
dim=2)
w = F.softmax(scaling * e, dim=1)
c = torch.sum(enc_h * w.view(batch, h_length, 1), dim=1)
### Dot product attention - full picture
If we are computing the attention weights based on only the contents of the vectors from Decoder and Encoder, similar Annotation vectors get weighed equally irrespective of the position. We can see this clearly from the Attention plots from the model. Observe in the following image how the Attention weights are not monotonic and tend to be distributed near positions where the Annotation vectors are similar in the acoustic space.
We could also plot where the model is attending to for generating each output symbol. Here, I have added an overlay for each row of the first image just to highlight which output symbol is being generated. The actual attention weights look like the above image.
We could also correlate this with the spectrogram of the utterance, since we know how much sub-sampling was done in the model. I have used a sub-sampling of 1_2_2_1_1. In our utterance FJSJ0_SX404, if we use a window size of 250ms and a frame shift of 10ms, we get 240 frames of feature vectors. Because of sub-sampling in our model, these features are mapped to 60 feature vectors after the Encoder network.
In the Content-based Attention, we saw that the Attention function depends only on the contents of $s_{i-1}$ and $h_{t}$. In Location-aware Attention, we also consider the location of the vectors in computing the attention weights. | 2020-02-19 03:37:14 | {"extraction_info": {"found_math": true, "script_math_tex": 22, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.7206364274024963, "perplexity": 2679.574792898487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144027.33/warc/CC-MAIN-20200219030731-20200219060731-00001.warc.gz"} |
https://lpatucson.org/lakeside/teacher-salary-posting/ | Average teacher salary (A.R.S. §15-189.05 )
1. Average salary of all teachers employed in budget year 2021 \$43,137
2. Average salary of all teachers employed in prior year 2020 \$42,781
3. Increase in average teacher salary from the prior year 2020 \$356
4. Percentage increase 0.8%
5. Average salary of all teachers employed in FY 2018 \$34,999
6. Total percentage increase in average teacher salary since FY 2018 23.3% | 2020-09-29 14:51:47 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9296101927757263, "perplexity": 14311.788246868202}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401643509.96/warc/CC-MAIN-20200929123413-20200929153413-00120.warc.gz"} |
https://leanprover-community.github.io/mathlib_docs/measure_theory/group/action.html | # mathlibdocumentation
measure_theory.group.action
# Measures invariant under group actions #
A measure μ : measure α is said to be invariant under an action of a group G if scalar multiplication by c : G is a measure preserving map for all c. In this file we define a typeclass for measures invariant under action of an (additive or multiplicative) group and prove some basic properties of such measures.
@[class]
structure measure_theory.vadd_invariant_measure (M : Type u_4) (α : Type u_5) [ α] {_x : measurable_space α} (μ : measure_theory.measure α) :
Prop
• measure_preimage_vadd : ∀ (c : M) ⦃s : set α⦄, μ ((λ (x : α), c +ᵥ x) ⁻¹' s) = μ s
A measure μ : measure α is invariant under an additive action of M on α if for any measurable set s : set α and c : M, the measure of its preimage under λ x, c +ᵥ x is equal to the measure of s.
Instances
@[class]
structure measure_theory.smul_invariant_measure (M : Type u_4) (α : Type u_5) [ α] {_x : measurable_space α} (μ : measure_theory.measure α) :
Prop
• measure_preimage_smul : ∀ (c : M) ⦃s : set α⦄, μ ((λ (x : α), c x) ⁻¹' s) = μ s
A measure μ : measure α is invariant under a multiplicative action of M on α if for any measurable set s : set α and c : M, the measure of its preimage under λ x, c • x is equal to the measure of s.
Instances
@[protected, instance]
def measure_theory.vadd_invariant_measure.zero {M : Type u_2} {α : Type u_3} [ α] :
@[protected, instance]
def measure_theory.smul_invariant_measure.zero {M : Type u_2} {α : Type u_3} [ α] :
@[protected, instance]
def measure_theory.smul_invariant_measure.add {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ ν : measure_theory.measure α} :
+ ν)
@[protected, instance]
def measure_theory.vadd_invariant_measure.add {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ ν : measure_theory.measure α} :
+ ν)
@[protected, instance]
def measure_theory.vadd_invariant_measure.vadd {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ : measure_theory.measure α} (c : ℝ≥0∞) :
(c μ)
@[protected, instance]
def measure_theory.smul_invariant_measure.smul {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ : measure_theory.measure α} (c : ℝ≥0∞) :
(c μ)
@[protected, instance]
def measure_theory.vadd_invariant_measure.vadd_nnreal {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ : measure_theory.measure α} (c : ℝ≥0) :
(c μ)
@[protected, instance]
def measure_theory.smul_invariant_measure.smul_nnreal {M : Type u_2} {α : Type u_3} [ α] {m : measurable_space α} {μ : measure_theory.measure α} (c : ℝ≥0) :
(c μ)
theorem measure_theory.smul_invariant_measure_tfae (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] (μ : measure_theory.measure α) :
, ∀ (c : G) (s : set α), μ ⁻¹' s) = μ s, ∀ (c : G) (s : set α), μ (c s) = μ s, ∀ (c : G) (s : set α), μ ⁻¹' s) = μ s, ∀ (c : G) (s : set α), μ (c s) = μ s, ∀ (c : G), , ∀ (c : G), .tfae
Equivalent definitions of a measure invariant under a multiplicative action of a group.
• 0: smul_invariant_measure G α μ;
• 1: for every c : G and a measurable set s, the measure of the preimage of s under scalar multiplication by c is equal to the measure of s;
• 2: for every c : G and a measurable set s, the measure of the image c • s of s under scalar multiplication by c is equal to the measure of s;
• 3, 4: properties 2, 3 for any set, including non-measurable ones;
• 5: for any c : G, scalar multiplication by c maps μ to μ;
• 6: for any c : G, scalar multiplication by c is a measure preserving map.
theorem measure_theory.vadd_invariant_measure_tfae (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] (μ : measure_theory.measure α) :
, ∀ (c : G) (s : set α), μ ⁻¹' s) = μ s, ∀ (c : G) (s : set α), μ (c +ᵥ s) = μ s, ∀ (c : G) (s : set α), μ ⁻¹' s) = μ s, ∀ (c : G) (s : set α), μ (c +ᵥ s) = μ s, ∀ (c : G), , ∀ (c : G), .tfae
Equivalent definitions of a measure invariant under an additive action of a group.
• 0: vadd_invariant_measure G α μ;
• 1: for every c : G and a measurable set s, the measure of the preimage of s under vector addition (+ᵥ) c is equal to the measure of s;
• 2: for every c : G and a measurable set s, the measure of the image c +ᵥ s of s under vector addition (+ᵥ) c is equal to the measure of s;
• 3, 4: properties 2, 3 for any set, including non-measurable ones;
• 5: for any c : G, vector addition of c maps μ to μ;
• 6: for any c : G, vector addition of c is a measure preserving map.
theorem measure_theory.measure_preserving_vadd {G : Type u_1} {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) :
theorem measure_theory.measure_preserving_smul {G : Type u_1} {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) :
@[simp]
theorem measure_theory.map_vadd {G : Type u_1} {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) :
@[simp]
theorem measure_theory.map_smul {G : Type u_1} {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) :
@[simp]
theorem measure_theory.measure_preimage_vadd {G : Type u_1} {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) (s : set α) :
μ ⁻¹' s) = μ s
@[simp]
theorem measure_theory.measure_preimage_smul {G : Type u_1} {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) (s : set α) :
μ ⁻¹' s) = μ s
@[simp]
theorem measure_theory.measure_vadd_set {G : Type u_1} {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) (s : set α) :
μ (c +ᵥ s) = μ s
@[simp]
theorem measure_theory.measure_smul_set {G : Type u_1} {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] (c : G) (μ : measure_theory.measure α) (s : set α) :
μ (c s) = μ s
theorem measure_theory.measure_is_open_pos_of_vadd_invariant_of_compact_ne_zero (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {K U : set α} (hK : is_compact K) (hμK : μ K 0) (hU : is_open U) (hne : U.nonempty) :
0 < μ U
If measure μ is invariant under an additive group action and is nonzero on a compact set K, then it is positive on any nonempty open set. In case of a regular measure, one can assume μ ≠ 0 instead of μ K ≠ 0, see measure_theory.measure_is_open_pos_of_vadd_invariant_of_ne_zero.
theorem measure_theory.measure_is_open_pos_of_smul_invariant_of_compact_ne_zero (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {K U : set α} (hK : is_compact K) (hμK : μ K 0) (hU : is_open U) (hne : U.nonempty) :
0 < μ U
If measure μ is invariant under a group action and is nonzero on a compact set K, then it is positive on any nonempty open set. In case of a regular measure, one can assume μ ≠ 0 instead of μ K ≠ 0, see measure_theory.measure_is_open_pos_of_smul_invariant_of_ne_zero.
theorem measure_theory.is_locally_finite_measure_of_vadd_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} (hU : is_open U) (hne : U.nonempty) (hμU : μ U ) :
theorem measure_theory.is_locally_finite_measure_of_smul_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} (hU : is_open U) (hne : U.nonempty) (hμU : μ U ) :
theorem measure_theory.measure_is_open_pos_of_smul_invariant_of_ne_zero (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) (hne : U.nonempty) :
0 < μ U
theorem measure_theory.measure_is_open_pos_of_vadd_invariant_of_ne_zero (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) (hne : U.nonempty) :
0 < μ U
theorem measure_theory.measure_pos_iff_nonempty_of_smul_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) :
0 < μ U U.nonempty
theorem measure_theory.measure_pos_iff_nonempty_of_vadd_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) :
0 < μ U U.nonempty
theorem measure_theory.measure_eq_zero_iff_eq_empty_of_vadd_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [add_group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) :
μ U = 0 U =
theorem measure_theory.measure_eq_zero_iff_eq_empty_of_smul_invariant (G : Type u_1) {α : Type u_3} {m : measurable_space α} [group G] [ α] [ α] {μ : measure_theory.measure α} [ α] {U : set α} [μ.regular] (hμ : μ 0) (hU : is_open U) :
μ U = 0 U = | 2022-01-23 00:31:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6074613928794861, "perplexity": 2402.590906427678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303917.24/warc/CC-MAIN-20220122224904-20220123014904-00024.warc.gz"} |
https://www.123calculus.com/en/frequency-calculator-page-8-20-410.html | # Frequency Calculator
f = 1/T
Frequency calculator.
Enter 'x' in the field to be calculated.
This tool is a calculator of frequency given period.
f = 1/T
f : frequency in hertz
T : period in second | 2023-01-27 07:21:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6892986297607422, "perplexity": 11885.012845233488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00429.warc.gz"} |
http://woori2000.dothome.co.kr/j4a1a/how-to-find-most-efficient-estimator-3014ed | $\endgroup$ – Greenparker May 15 '16 at 18:56 To do this, you will have to write out the variance of your estimator, and simplify this variance expression. e (median, mean) = V a r ( X ¯) V a r ( m e d) = σ 2 n π 2 σ 2 n = 2 π = 2 × 7 22 = 0.63. The conditional mean should be zero.A4. Find the shortest routes between multiple stops and get times and distances for your work or a road trip. Save gas and time on your next trip. To determine whether you have an efficient estimator, you need to establish whether or not the variance of the estimator achieves this lower bound. For example, an estimator that always equals a single number (or a constant) has a variance equal to zero. If an efficient estimator exists it is also a sufficient estimator and can be obtained by the maximum-likelihood method (see Maximum Likelihood Estimate). In other words, an efficient procedure produces results that maximize your use of materials, time and energy. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. In other words, the optimal estimator deviates as little as … The definition of "best possible" depends on one's choice of a loss function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. An estimator is efficient if it achieves the smallest variance among estimators of its kind. This satisfies the first condition of consistency. An estimator is consistent if it approaches the true parameter value as the sample size gets larger and larger. The linear regression model is “linear in parameters.”A2. Consistent Estimators. Population 1: Let μ 1 be the mean number of calories purchased by women eating with other women. Example: Show that the sample mean is a consistent estimator of the population mean. An efficient estimator is also the minimum variance unbiased … The more efficient the machine, the higher output it … An estimator is unbiased if, in repeated estimations using the method, the mean value of the estimator coincides with the true parameter value. There is a random sampling of observations.A3. The moments method equates values of sample moments (functions describing the parameter) to population moments. An estimator that is unbiased and has the minimum variance of all other estimators is the best (efficient). Proof: omitted. Example: Let be a random sample of size n from a population with mean µ and variance . If you want the quietest and most efficient thrust propeller system, select a prop configuration (and reduction drive ratio) that will keep the tip speed for your cruise rpm at or below 700 feet per second or 475 mph. Efficiency can refer to any procedure you want to optimize. The efficiency of any efficient estimator is unity. When one compares between a given procedure and a notional "best possible" procedure the efficiency can be expressed as relative finite-sample or asymptotic efficiency (a ratio). The Cramér–Rao lower bound is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be. The conversion between correlation and covariance is given as: ρ (R1, R2) = Cov (R1, R2)/ σ1σ2. is a more efficient estimator than !ˆ 2 if var(!ˆ 1) < var(!ˆ 2). You simply want to know the result of the proof (if it exists) and the assumptions needed to carry it out. Linear regression models have several applications in real life. It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population. Work and energy both use the standard unit of Joules, but the calculator above is unit less to allow you to input any unit. "Statistical Methods in Online A/B Testing". Every time that you supply energy or heat to a machine (for example to a car engine), a certain part of this energy is wasted, and only some is converted to actual work output. In practical situations (that is, when you’re working with data and not just doing a theoretical exercise), knowing when an estimator has these desirable properties is good, but you don’t need to prove them on your own. Definition: An estimator ̂ is a consistent estimator of θ, if ̂ → , i.e., if ̂ converges in probability to θ. Theorem: An unbiased estimator ̂ for is consistent, if → ( ̂ ) . time and mon… $\begingroup$ The MLE is asymptotically the most efficient estimator, in terms of the variance and is asymptotically unbiased. The most often used, the maximum likelihood method, uses differential calculus to determine the maximum of the probability function of a number of sample parameters. Thus optimality in practice is defined using the variance or mean square error (MSE, thus minimum MSE estimator). random variables, i.e., a random sample from f(xjµ), where µ is unknown. Recap of the Situation. The above explanation is for the use of efficiency in physics and thermodynamics, but efficiency can be used in anything from finance to work performance. Point estimation is the opposite of interval estimation. Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ. The relevance to A/B testing is that the more efficient the estimator, the smaller sample size one requires for an A/B test. The linearity property, however, can be convenient when you’re using algebraic manipulations to create new variables or prove other estimator properties. There are several ways to solve this problem and several "correct" answers. An estimator has this property if a statistic is a linear function of the sample observations. This tries one way and gives you a correct answer. An estimator is efficient if it is the minimum variance unbiased estimator. A statistics is a consistent estimator of a parameter if its probability that it will be close to the parameter's true value approaches 1 with increasing sample size. The Maximum Likelihood Estimator is the most efficient estimator among all the unbiased ones. On the other hand, interval estimation uses sample data to calcul… His published work has appeared in Economic Inquiry, Industrial Relations, the Southern Economic Journal, Contemporary Economic Policy, the Journal of Sports Economics, and other outlets. For example, an efficient experimental design is one that produces your desired experimental results with the minimum amount of resources (e.g. For any unbiased estimator Θ ^ = φ ( U) the ratio of the right-hand side of inequality (7.8) to the left one is called the efficiency of this estimator and is denoted by e (φ): (7.11) e ( φ) = 1 D θ ⌢ ⋅ D Z = 1 D θ ⌢ ⋅ D ∂ ln g / ∂ θ. Restricting the definition of efficiency to unbiased estimators, excludes biased estimators with smaller variances. This type of estimator could have a very large bias, but Math 541: Statistical Theory II Methods of Evaluating Estimators Instructor: Songfeng Zheng Let X1;X2;¢¢¢;Xn be n i.i.d. You’ll use less energy if you have smaller sample sizes, for 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. That is, for a given number of samples, the variance of the estimator is no more or less than the inverse of the Fisher information. An efficient estimator is the "best possible" or "optimal" estimator of a parameter of interest. An estimator is efficient if and only if it achieves the Cramer-Rao Lower-Bound, which gives the lowest possible variance for an estimator of a parameter. It produces a single value while the latter produces a range of values. Like this glossary entry? Since in many cases the lower bound in the Rao–Cramér inequality cannot be attained, an efficient estimator in statistics is frequently chosen based on having minimal variance in the class of all unbiased estimator of the parameter. In other words, the optimal estimator deviates as little as possible from the true value (θ*) one is trying to estimate. How to Determine Whether an Estimator Is Good, Recognizing Usual Variables: Normal Distribution, The Chi-Squared Distribution in Econometrics, Specifying Your Econometrics Regression Model. Thus optimality in practice is defined using the variance or mean square error (MSE, thus minimum MSE estimator). Where Cov (R1, R2) represents the covariance of the two asset returns. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. A point estimator is a statistic used to estimate the value of an unknown parameter of a population. It's based … A specific property can be represented by using many different estimators. So a procedure that can work with a smaller sample is usually more efficient than one that requires a larger sample. Efficiency is defined as the ratio of energy output to energy input. An estimator is unbiased if, in repeated estimations using the method, the mean value of the estimator coincides with the true parameter value. Select a letter to see all A/B testing terms starting with that letter or visit the Glossary homepage to see all. When defined asymptotically an estimator is fully efficient if its variance achieves the Rao-Cramér lower bound. Note my use of the word "attempts." The Cramer Rao inequality provides verification of efficiency, since it establishes the lower bound for the variance-covariance matrix of any unbiased estimator. The OLS estimator is an efficient estimator. Alternatively, the formula can be written as: σ2p = w21σ21 + w22σ22 + 2ρ (R1, R2) w1w2σ1σ2, using ρ (R1, R2), the correlation of R1 and R2. Roberto Pedace, PhD, is an associate professor in the Department of Economics at Scripps College. Perhaps the most important question as you consider energy efficiency upgrades for your home or business is, how efficient is your property right now? In that case, they usually settle for consistency. An estimator is efficient if it achieves the smallest variance among estimators of its kind. An estimator of µ is a function of (only) the n random variables, i.e., a statistic ^µ= r(X 1;¢¢¢;Xn).There are several method to obtain an estimator for µ, such as the MLE, Using the formula e ( α 1 ^, α 1 ^) = V a r ( α 2 ^) V a r ( α 1 ^), we have. Show that ̅ ∑ is a consistent estimator … If you want to calculate it on your own you’ll be looking for two other numbers, which … standard deviation) that can be achieved at each level of expected return for a given set of risky securities. Thus ( ) ∑ ( )is a complete & sufficient statistic (CSS) for . Definition of Efficient Estimator in the context of A/B testing (online controlled experiments). Consistent . V ( θ ^) ⩾ I ( θ) − 1 = 2 n ⋅ θ 2. Easily enter stops on a map or by uploading a file. Only arithmetic mean is considered as sufficient estimator. The variance of $$\overline X$$ is known to be $$\frac{{{\sigma ^2}}}{n}$$. This property isn’t present for all estimators, and certainly some estimators are desirable (efficient and either unbiased or consistent) without being linear. For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev. Several methods are used to calculate the estimator. EER = (output cooling energy in BTU/input electrical energy in Wh) This EER rating will typically be listed somewhere in your air conditioners specification sheet. Sometimes statisticians and econometricians are unable to prove that an estimator is unbiased. Since in many cases the lower bound in the Rao–Cramér inequality cannot be attained, an efficient estimator in statistics is frequently chosen based on having minimal variance in the class of all unbiased estimator of the parameter. Solution: We have already seen in the previous example that $$\overline X$$ is an unbiased estimator of population mean $$\mu$$. Statisticians and econometricians typically require the estimators they use for inference and prediction to have certain desirable properties. In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished.. For statisticians, unbiasedness and efficiency are the two most-desirable properties an estimator can have. Since the mean squared error (MSE) of an estimator δ is {\displaystyle \operatorname {MSE} (\delta)=\operatorname {var} (\delta)+ [\operatorname {bias} (\delta)]^ {2}\ } the … A consistent estimator is one which approaches the real value of the parameter in the population as … You need to make sure the units of work and energy match. This calculator attempts to generate the most efficient cut list for a given set of pieces. Therefore, the efficiency of … Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … If an unbiased estimator of a parameter θ attains () = for all values of the parameter, then the estimator is called efficient. When you're selecting an estimator, you need to consider its efficiency and compare it with all the other alternatives. estimator directly (rather than using the efficient estimator is also a best estimator argument) as follows: The population pdf is: ( ) √ ( ) √ ( ) So it is a regular exponential family, where the red part is ( ) and the green part is ( ). In some instances, statisticians and econometricians spend a considerable amount of time proving that a particular estimator is unbiased and efficient. The efficient frontier shows us the minimum risk (i.e. The formula for calculating MSE is MSE () = var + Sufficient Estimator : An estimator is called sufficient when it includes all above mentioned properties, but it is very difficult to find the example of sufficient estimator. In some instances, statisticians and econometricians spend a considerable amount of time proving that a particular estimator is unbiased and efficient. Given yield measurements X 1, X 2, X 3 from three independent runs of an experiment with variance σ 2, which is the better of the two estimators: θ ^ 1 = X 1 + X 2 + X 3 3, θ ^ 2 = X 1 + 2 X 2 + X 3 4 I know that in order to find the best estimator if both are unbiased, we are supposed to choose the one with the smallest variance. The two main types of estimators in statistics are point estimators and interval estimators. Besides unbiasedness and efficiency, an additional desirable property for some estimators is linearity. So for large samples, you your best best is MLE, I think. In the preceding few pages, we worked through a two-sample T-test for the “calories and context” example. An efficient estimator need not exist, but if it does and if it is unbiased, it is the MVUE. An estimator is a simple statistic that represents the population properties. 3. For this reason, consistency is known as an asymptotic property for an estimator; that is, it gradually approaches the true parameter value as the sample size approaches infinity. Compare it with all the other hand, interval estimation uses sample data when calculating single! Several ways to solve this problem and several correct '' answers linear function the... ) that can work with a smaller sample is usually more efficient the estimator achieves equality the... Controlled experiments ), Ordinary Least Squares ( OLS ) method is widely used estimate! Efficiency can refer to any procedure you want to know the result of the proof ( if exists! And econometricians spend a considerable amount of time proving that a particular estimator is the efficient... Two asset returns assumptions made while running linear regression models.A1 ( e.g in practice is defined using the or! Additional desirable property for some estimators is linearity most-desirable properties an estimator is fully efficient if it achieves Rao-Cramér. Property can be represented by using many different estimators variance equal to zero compare. And efficiency are the two main types of estimators in statistics are estimators... When calculating a single number ( or a constant ) has a variance equal to zero Economics... Variables, i.e., a random sample from f ( xjµ ), where is! Is fully efficient if it is unbiased and efficient return for a given set of risky securities of... Few pages, we worked through a two-sample T-test for the “ calories context! Random variables, i.e., a random sample from f ( xjµ ), µ! 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Provides verification of efficiency to unbiased estimators, excludes biased estimators with smaller variances you to. Unbiased estimator women eating with other women MLE, I think you want! Matrix of any unbiased estimator unknown parameter of the unknown parameter of a function... The parameter ) to population moments energy output to energy input that an estimator is efficient its. A larger sample a correct answer you have smaller sample sizes, for example an! Widely used to estimate the parameters of a parameter of a parameter of the (! Size gets larger and larger the population Let μ 1 be the mean number of purchased... Of its kind minimum amount of resources ( e.g has a variance equal to zero that case, they settle. Terms of the sample size gets larger and larger terms starting with that letter or visit the homepage... Any procedure you want to optimize example, an efficient procedure produces results that maximize your of... 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At each level of expected return for a given set of pieces, Ordinary Least Squares ( )... This variance expression ( if it achieves the smallest variance among estimators of its kind with smaller variances context example! Establishes the lower bound for the variance-covariance matrix of any unbiased estimator a range of.. A constant ) has a variance equal to zero: Let be a random sample from (... Population with mean µ and variance it 's based … an estimator that always a! The sample observations an A/B test testing ( online controlled experiments ) worked through a T-test! Generate the most efficient cut list for a given set of risky securities in are., i.e., a how to find most efficient estimator sample from f ( xjµ ), where µ is unknown achieved! R2 ) represents the covariance of the sample observations is efficient if it is unbiased, it is the variance. Produces a range how to find most efficient estimator values in parameters. ” A2 in terms of the population properties output to energy.... Need to consider its efficiency and compare it with all the unbiased ones estimators is linearity usually. A parameter of interest worked through a two-sample T-test for the variance-covariance matrix of any unbiased estimator any unbiased.. Context ” example other women “ linear in parameters. ” A2 other words, an efficient procedure produces results maximize... Requires a larger sample write out the variance and is asymptotically the efficient! Error ( MSE, thus minimum MSE estimator ) sample from f ( xjµ ), where µ is.! Ways to solve this problem and several correct '' answers unknown parameter of a linear function the! Econometrics, Ordinary Least Squares ( OLS ) method is widely used estimate... correct '' answers of its kind estimators in statistics are point estimators and interval estimators each of. 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Know the result of the unknown parameter of interest, R2 ) represents the covariance of the (! | 2021-09-19 02:25:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7805780172348022, "perplexity": 778.1348121102073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056656.6/warc/CC-MAIN-20210919005057-20210919035057-00700.warc.gz"} |
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/simplifying-rational-expressions/ | Simplifying Rational Expressions
The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.
$\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}$
We can factor the numerator and denominator to rewrite the expression.
$\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}$
Then we can simplify that expression by canceling the common factor $\left(x+4\right)$.
$\frac{x+4}{x+7}$
How To: Given a rational expression, simplify it.
1. Factor the numerator and denominator.
2. Cancel any common factors.
Example 1: Simplifying Rational Expressions
Simplify $\frac{{x}^{2}-9}{{x}^{2}+4x+3}\\$.
Solution
$\begin{array}\frac{\left(x+3\right)\left(x - 3\right)}{\left(x+3\right)\left(x+1\right)}\hfill & \hfill & \hfill & \hfill & \text{Factor the numerator and the denominator}.\hfill \\ \frac{x - 3}{x+1}\hfill & \hfill & \hfill & \hfill & \text{Cancel common factor }\left(x+3\right).\hfill \end{array}$
Analysis of the Solution
We can cancel the common factor because any expression divided by itself is equal to 1.
Can the ${x}^{2}$ term be cancelled in Example 1?
No. A factor is an expression that is multiplied by another expression. The ${x}^{2}$ term is not a factor of the numerator or the denominator.
Try It 1
Simplify $\frac{x - 6}{{x}^{2}-36}$.
Solution | 2019-11-22 21:21:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9549438953399658, "perplexity": 419.3029102449349}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671548.98/warc/CC-MAIN-20191122194802-20191122223802-00209.warc.gz"} |
https://www.ejropen.com/article/S2352-0477(22)00034-X/fulltext | Full length article| Volume 9, 100427, January 01, 2022
# Comprehensive comparison of three different workstations for accurate planning of endovascular stent implantation in patients with thoracic aortic aneurysms
Open AccessPublished:June 16, 2022
## Highlights
• Pre-interventional planning of TEVAR in patients with TAAs using CTA is feasible.
• All three workstations facilitated accurate measurements in vivo and ex vivo.
• Repetition of measurements resulted in faster processing due to learning effects.
## Abstract
### Purpose
To assess the diagnostic precision of three different workstations for measuring thoracic aortic aneurysms (TAAs) in vivo and ex vivo using either pre-interventional computed tomography angiography scans (CTA) or a specifically designed phantom model.
### Methods
This retrospective study included 23 patients with confirmed TAA on routinely performed CTAs. In addition to phantom tube diameters, one experienced blinded radiologist evaluated the dimensions of TAAs on three different workstations in two separate rounds. Precision was assessed by calculating measurement errors. In addition, correlation analysis was performed using Pearson correlation.
### Results
Measurements acquired at the Siemens workstation deviated by 3.54% (range, 2.78–4.03%; p = 0.14) from the true size, those at General Electric by 4.05% (range, 1.46–7.09%; p < 0.0001), and at TeraRecon by 4.86% (range, 3.22–6.45%; p < 0.0001). Accordingly, Siemens provided the most precise workstation at simultaneously most fluctuating values (scattering of 4.46%). TeraRecon had the smallest fluctuation (scattering of 2.83%), but the largest deviation from the true size of the phantom. The workstation from General Electric showed a scattering of 2.94%. The highest overall correlation between the 1st and 2nd rounds was observed with measurements from Siemens (r = 0.898), followed by TeraRecon (r = 0.799), and General Electric (r = 0.703). Repetition of measurements reduced processing times by 40% when using General Electric, by 20% with Siemens, and by 18% with TeraRecon.
### Conclusions
In conclusion, all three workstations facilitated precise assessment of dimensions in the majority of cases at simultaneously high reproducibility, ensuring accurate pre-interventional planning of thoracic endovascular aortic repair.
#### Abbreviations:
CTA (Computed tomography angiography), PACS (Picture archiving and communication system), PVC (Polyvinyl chloride), TAA (Thoracic aortic aneurysm), TEVAR (Thoracic endovascular aortic repair)
## 1. Introduction
Thoracic aortic aneurysms (TAAs) represent a potentially life-threatening disease that requires immediate detection to prevent complications arising from a delayed diagnosis, such as aortic dissection or rupture [
• Guo M.H.
• Appoo J.J.
• Saczkowski R.
• Smith H.N.
• Ouzounian M.
• Gregory A.J.
• Herget E.J.
• Boodhwani M.
Association of mortality and acute aortic events with ascending aortic aneurysm: a systematic review and meta-analysis.
]. In an aging population with continuously increasing cardiovascular risk factors, the annual incidence of currently 3 cases per 100.000 per year is likely to keep rising [
• Olsson C.
• Thelin S.
• Stahle E.
• Ekbom A.
• Granath F.
Thoracic aortic aneurysm and dissection: increasing prevalence and improved outcomes reported in a nationwide population-based study of more than 14,000 cases from 1987 to 2002.
].
Management strategies include surgical and endovascular techniques, depending on many factors including size, location, growth, and associated comorbidities of the individual patient [
• Wang T.K.M.
• Desai M.Y.
Thoracic aortic aneurysm: optimal surveillance and treatment.
,
• Upchurch Jr., G.R.
• Escobar G.A.
• Beck A.W.
• Matsumura J.S.
• Perry R.J.
• Singh M.J.
• Veeraswamy R.K.
• Wang G.J.
Society for vascular surgery clinical practice guidelines of thoracic endovascular aortic repair for descending thoracic aortic aneurysms.
,
• Spanos K.
• Nana P.
• Behrendt C.A.
• Kouvelos G.
• Panuccio G.
• Heidemann F.
• Matsagkas M.
• Debus E.S.
• Giannoukas A.
• Kolbel T.
Management of descending thoracic aortic diseases: similarities and differences among cardiovascular guidelines.
]. Considering the annual risk for aortic rupture or dissection of up to 7% for TAAs > 60 mm, current guidelines recommend surgery at sizes of ≥ 55 mm in most cases [
• Erbel R.
• Aboyans V.
• Boileau C.
• Bossone E.
• Bartolomeo R.D.
• Eggebrecht H.
• Evangelista A.
• Falk V.
• Frank H.
• Gaemperli O.
• Grabenwoger M.
• Haverich A.
• Iung B.
• Manolis A.J.
• Meijboom F.
• Nienaber C.A.
• Roffi M.
• Rousseau H.
• Sechtem U.
• Sirnes P.A.
• Allmen R.S.
• Vrints C.J.
• Guidelines E.S.C.Cf.P.
2014 ESC Guidelines on the diagnosis and treatment of aortic diseases: Document covering acute and chronic aortic diseases of the thoracic and abdominal aorta of the adult. The Task Force for the Diagnosis and Treatment of Aortic Diseases of the European Society of Cardiology (ESC).
,
• Hiratzka L.F.
• Bakris G.L.
• Beckman J.A.
• Bersin R.M.
• Carr V.F.
• Casey D.E.
• Eagle Jr., K.A.
• Hermann L.K.
• Isselbacher E.M.
• Kazerooni E.A.
• Kouchoukos N.T.
• Lytle B.W.
• Milewicz D.M.
• Reich D.L.
• Sen S.
• Shinn J.A.
• Svensson L.G.
• Williams D.M.
G. American College of Cardiology Foundation/American Heart Association Task Force on Practice, S. American Association for Thoracic, R. American College of, A. American Stroke, A. Society of Cardiovascular, A. Society for Cardiovascular, Interventions, R. Society of Interventional, S. Society of Thoracic, M. Society for Vascular with thoracic aortic disease
2010 ACCF/AHA/AATS/ACR/ASA/SCA/SCAI/SIR/STS/SVM Guidelines for the diagnosis and management of patients with thoracic aortic disease. A Report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines, American Association for Thoracic Surgery, American College of Radiology,American Stroke Association, Society of Cardiovascular Anesthesiologists, Society for Cardiovascular Angiography and Interventions, Society of Interventional Radiology, Society of Thoracic Surgeons,and Society for Vascular Medicine.
,
• Czerny M.
• Schmidli J.
• van den Berg J.C.
• Bertoglio L.
• Carrel T.
• Chiesa R.
• Clough R.E.
• Eberle B.
• Etz C.
• Grabenwoger M.
• Haulon S.
• Jakob H.
• Kari F.A.
• Mestres C.A.
• Pacini D.
• Resch T.
• Rylski B.
• Schoenhoff F.
• Shrestha M.
• von Tengg-Kobligk H.
• Tsagakis K.
• Wyss T.R.
• Document R.
• Chakfe N.
• Debus S.
• de Borst G.J.
• Di Bartolomeo R.
• Lindholt J.S.
• Ma W.G.
• Suwalski P.
• Vermassen F.
• Wahba A.
• Wyler von Ballmoos M.C.
Editor’s choice - current options and recommendations for the treatment of thoracic aortic pathologies involving the aortic arch: an expert consensus document of the european association for cardio-thoracic surgery (EACTS) & the European Society for Vascular Surgery (ESVS).
]. In the case of genetic disorders or bicuspid aortic valve, intervention is typically recommended at lower values [
• Salameh M.J.
• Black 3rd, J.H.
• Ratchford E.V.
Thoracic aortic aneurysm.
].
Along with improvements in availability and material, endovascular techniques attracted scientific attention aiming at restoring cardiovascular circulation through the implantation of a membrane-covered stent graft [
• Salameh M.J.
• Black 3rd, J.H.
• Ratchford E.V.
Thoracic aortic aneurysm.
,
• Nienaber C.A.
• Clough R.E.
Management of acute aortic dissection.
]. In this context, careful pre-interventional planning and anatomical visualization are crucial to ensure a successful thoracic endovascular aortic repair (TEVAR) [
• Calero A.
• Illig K.A.
Overview of aortic aneurysm management in the endovascular era.
]. Contrast-enhanced computed tomography plays a central role in the assessment and characterization of aneurysms by providing three-dimensional information about adjacent structures and vasculature. It is widely available and represents an optimal preoperative imaging modality given its ability to measure aneurysmal morphology accurately and precisely [
• Erbel R.
• Aboyans V.
• Boileau C.
• Bossone E.
• Bartolomeo R.D.
• Eggebrecht H.
• Evangelista A.
• Falk V.
• Frank H.
• Gaemperli O.
• Grabenwoger M.
• Haverich A.
• Iung B.
• Manolis A.J.
• Meijboom F.
• Nienaber C.A.
• Roffi M.
• Rousseau H.
• Sechtem U.
• Sirnes P.A.
• Allmen R.S.
• Vrints C.J.
• Guidelines E.S.C.Cf.P.
2014 ESC Guidelines on the diagnosis and treatment of aortic diseases: Document covering acute and chronic aortic diseases of the thoracic and abdominal aorta of the adult. The Task Force for the Diagnosis and Treatment of Aortic Diseases of the European Society of Cardiology (ESC).
].
For pre-procedural planning of endovascular stent insertion, different CT workstations have been developed that provide miscellaneous options to visualize aortic lesions in three-dimensional models and evaluate both the diameter and length of the healthy proximal and distal landing zones [
• Sobocinski J.
• Chenorhokian H.
• Maurel B.
• Midulla M.
• Hertault A.
• Le Roux M.
• Azzaoui R.
• Haulon S.
The benefits of EVAR planning using a 3D workstation.
]. To date, data about the diagnostic accuracy, reliability, and reproducibility of CT-based measurements using workstations from different manufacturers are sparse. Therefore, the purpose of the present study was to evaluate the diagnostic precision of three different workstations in assessing TAA dimensions using either computed tomography angiography scans (CTA) of patients or a specifically designed phantom model.
## 2. Methods
The institutional ethical review board approved this retrospective study that complies with the Declaration of Helsinki. The need for written informed consent was waived.
### 2.1 Study population
The study population consisted of 23 patients with complete CT data sets and sufficient image quality. Additionally, a phantom was constructed to compare measurements with fixed true values.
### 2.2 CT scan protocol
Contrast-enhanced CT scans were carried out on a third-generation dual-source dual-energy CT scanner (Somatom Force; Siemens Healthineers, Forchheim, Germany). The system consisted of two beams operating at a lower and higher tube voltage (tube A, 90 kVp and 180 mAs; tube B, Sn150 kVp [0.64 mm tin filter] and 180 mAs) using automatic attenuation-based tube current modulation (CARE Dose 4D; Siemens Healthineers, Forchheim, Germany). All CT scans were performed in the craniocaudal scan direction. Images were acquired at three different phases, an unenhanced as well as a contrast-enhanced venous and arterial phase. Scans were ECG-gated and performed using a conventional protocol with the following parameters: 120 kV, 70 mAs, 0.6 mm slice thickness, and 1 mm collimation. According to standard protocols in clinical routine, patients received varying doses of a non-ionic monomeric contrast agent (Bracco Imaging Deutschland GmbH, Konstanz, Germany) at a flow rate of 2–4 mL/s adjusted to the individual bodyweight of patients. For automated bolus tracking, a region of interest was placed in the ascending aorta with a threshold of 120 Hounsfield units (HU) to time the start of the arterial phase. The venous phase was performed with a delay of 80-90 s after injection start. All data sets were sent to the picture archiving and communication system (PACS Centricity, Version 4.2; General Electric Healthcare, Solingen, Germany) for further postprocessing.
### 2.3 Specifications of the three different workstations
In this study, three different workstations for the postprocessing of CT images were examined: Aquarius (Version 3.7.0.12; TeraRecon, Durham, USA), Syngo (Version VE32B; Siemens Healthineers, Forchheim, Germany), and Volume Share 2 (Version AW.4.4; General Electric Healthcare, Waukesha, USA).
All workstations were purchased with dedicated integrated vendor-specific software that was directly connected to the PACS allowing for three-dimensional visualization of the aorta. A detailed description of all three workstations is available in Supplemental material.
### 2.4 Assessment of patient data sets
All CTA series were analyzed twice by one single radiologist (J.E.S., board-certified radiologist with 9 years of experience in CT imaging) in a randomized blinded fashion on different days. The radiologist was familiar with the handling of workstations but had no previous experience with the three systems investigated in the present study. For randomization, both patients and workstations have been assigned to a distinct number, written on cards, and shuffled. Finally, each patient was randomly allocated to one of the three workstations. One week after the complete evaluation of all 23 patients on each workstation, a 2nd round has been initiated to assess the repeatability and correlation of measurements with the 1st round. Patients were not randomized again and evaluated in the same order as in the 1st round. This evaluation aimed to answer the question of whether a learning effect resulted in faster processing of patient data sets. Assessment time for the analysis of each data set was noted.
In detail, the following measurements were made:
• 1.
Diameter of the aorta after branching of the left subclavian artery.
• 2.
Diameter of the aorta before the beginning of the aortic lesion.
• 3.
Greatest diameter of the vessel lumen within the aortic lesion.
• 4.
Greatest diameter of the thrombus within the aortic lesion.
• 5.
Diameter of the aorta immediately after the end of the aortic lesion.
• 6.
Diameter of the aorta before the origin of the coeliac trunk.
• 7.
Diameter of the common femoral arteries on both sides shortly after their origin.
• 8.
Distance between the origin of the left subclavian artery and the beginning of the aortic lesion.
• 9.
Length of the aortic lesion.
• 10.
Distance between the end of the aortic lesion and the coeliac trunk.
• 11.
Total distance between the origin of the left subclavian artery and the coeliac trunk.
### 2.5 Phantom
In addition to CTAs from 23 patients, a phantom was constructed using commercially available polyvinyl chloride (PVC) tubes with known dimensions of the individual components (Fig. 1). The phantom represented a simplified aortic model revealing the most important anatomical corner points. In this context, the origins of the brachiocephalic trunk and the left subclavian artery were particularly important as the starting points for several measurements. Phantoms of similar construction have been proven extremely useful in previous studies to simulate all procedural steps of an endovascular aortic repair [
• Sulaiman A.
• Boussel L.
• Taconnet F.
• Serfaty J.M.
• Alsaid H.
• Attia C.
• Huet L.
• Douek P.
In vitro non-rigid life-size model of aortic arch aneurysm for endovascular prosthesis assessment.
].
Tabled 1
Measurement pointsTrue value (mm)
Tube diameter at the artificial subclavian artery35
Tube diameter at the beginning of the aneurysm35
Maximum tube diameter of the aneurysm75
Tube diameter at the end of the aneurysm35
Tube diameter at the coeliac trunk35
Distance between the aneurysm and the artificial subclavian artery50
Length of the aneurysm300
Distance between the aneurysm and the artificial coeliac trunk50
Total distance between the origin of the artificial subclavian artery and the coeliac trunk400
To compare against those fixed reference values, each defined point was measured ten times at the three workstations, respectively. The respective ten-part series of measurements from each point and each workstation was compared with the known dimensions of the phantom. Spiral CT data sets of the phantom were processed exactly like that of a patient. Fig. 3 illustrates the positioning of measurements in CTAs and phantom tubes.
### 2.6 Statistical analysis
Statistical analysis was conducted with MedCalc for Windows (Version 18; MedCalc, Ostend, Belgium). The normal distribution of datasets was tested using the Shapiro-Wilk test. Accordingly, results are expressed as mean ± standard deviation or median with interquartile ranges. A p value < 0.05 was considered statistically significant.
Comparisons between continuous variables were performed using one-way ANOVA, chi-square statistic tests, or two-tailed Student’s t-test, where appropriate. Correlation analysis between patient data sets acquired in the two rounds, between workstations, and between measured and true phantom dimensions was performed by calculating Pearson product-moment correlation (r) and linear regression. An r value of less than 0.40, 0.41–0.60, 0.61–0.80, and greater than 0.80 was considered as poor, moderate, strong, and very strong, respectively. Measurement errors were defined as the difference between measured and true dimensions of the phantom:
$Measurementerror(mm)=Measureddiameterminustruediameter$
$Relativemeasurementerror(%)=MeasurementerrorTruediameter×100$
## 3. Results
A total of 23 patients (58 ± 10 years; range, 42–83), comprising 7 women and 16 men, were included in this study. Overall mean body mass index was 27 ± 3 kg/m2 (range, 23–32 kg/m2).
Aortic lesions could be classified into 15 dissections, 7 aneurysms, and 1 aortic rupture. Regarding aortic dissection, 15 aortic dissections could be allocated to Stanford B or Type 3 according to the DeBakey classification. The single rupture was located in aortic segment III. Patient characteristics are summarized in Table 1.
Table 1Baseline characteristics of the study cohort.
Characteristics of the study cohortValue
Number of overall patients (women; men)23 (7; 16)
Overall mean age (y) ± SD, range58 ± 10, 42–83
Overall mean BMI (kg/m²) ± SD, range27 ± 3, 23–32
Mean age of women (y) ± SD, range
(Mean BMI of women (kg/m²) ± SD, range)
57 ± 9, 46–70
(27 ± 2, 25–30)
Mean age of men (y) ± SD, range
(Mean BMI of men (kg/m²) ± SD, range)
58 ± 11, 42–83
(28 ± 3, 23–32)
Abbreviations: BMI, body mass index.
### 3.1 Phantom
First, the phantom was scanned ten times at each workstation (Table 2) to compare measured values with the true dimensions of phantom tubes.
Table 2Diagnostic accuracy of the phantom at different predefined and fixed measurement points.
True value (mm)Average (mm)95% confidence interval (CI)p value
Siemens Healthineers Syngo (Version VE32B)
Tube diameter at the subclavian artery3535.5034.89–36.110.0957
Tube diameter at the beginning of the aneursym3536.4036.03–36.77< 0.0001
Maximum tube diameter of the aneurysm7574.9074.67–75.130.3434
Tube diameter at the end of the aneurysm3536.3035.95–36.65< 0.0001
Tube diameter at the coeliac trunk3536.0036.00–36.00< 0.0001
Distance of the aneurysm to the subclavian artery5051.4249.55–53.290.1194
Length of the aneurysm300290.00281.15–298.850.0309
Distance of the aneurysm to the coeliac trunk5052.1050.95–53.520.0085
Total distance between the origin of the subclavian artery and the coeliac trunk400393.52384.38–402.660.1431
TeraRecon Aquarius (Version 3.7.0.12)
Tube diameter at the subclavian artery3536.2335.39–37.060.0088
Tube diameter at the beginning of the aneursym3537.0736.45–37.68< 0.0001
Maximum tube diameter of the aneurysm7576.4474.91–77.970.0953
Tube diameter at the end of the aneurysm3537.2836.82–37.750.0726
Tube diameter at the coeliac trunk3536.1735.78–36.640.0004
Distance of the aneurysm to the subclavian artery5050.0648.67–51.450.9268
Length of the aneurysm300281.81278.75–284.88< 0.0001
Distance of the aneurysm to the coeliac trunk5051.2148.68–53.740.3078
Total distance between the origin of the subclavian artery and the coeliac trunk400383.08379.17–386.98< 0.0001
General Electric Volume Share 2 (Version AW.4.4)
Tube diameter at the subclavian artery3535.7335.43–36.030.0004
Tube diameter at the beginning of the aneursym3537.6737.20–38.14< 0.0001
Maximum tube diameter of the aneurysm7575.1074.98–75.220.0957
Tube diameter at the end of the aneurysm3536.5536.27–36.83< 0.0001
Tube diameter at the coeliac trunk3537.2737.11–37.44< 0.0001
Distance of the aneurysm to the subclavian artery5048.8446.57–51.100.2749
Length of the aneurysm300295.67291.72–299.620.0349
Distance of the aneurysm to the coeliac trunk5042.4440.83–44.75< 0.0001
Total distance between the origin of the subclavian artery and the coeliac trunk400386.95383.09–390.32< 0.0001
Measurements of phantom tubes with a fixed diameter of 35 mm deviated generally from the true size, being the smallest (range of mean deviation, 0.5–1.4 mm) at the Siemens workstation, followed by General Electric (range, 0.7–2.7 mm) and TeraRecon (range, 1.2–2.3 mm).
Regarding diameters of 75 mm, both Siemens and General Electric showed results that are significantly better in terms of precision than the measurements of 35 mm tubes (mean deviation of 0.1 mm, respectively). The TeraRecon, on the other hand, differed 1.4 mm on average from the true size.
Scattering of measurements at 50 mm diameters can be considered similar across all three workstations (range, −1.2 to +2.1 mm). Only at the measurement point “distance of the aneurysm to the artificial coeliac trunk” General Electric was far below the true value (mean deviation of −7.6 mm) due to a user error.
At diameters of 300 mm, all three workstations felt below the true size of the phantom tube. The mean distance to the true value was 10 mm for Siemens, 8.2 mm for TeraRecon, and 4.3 mm for General Electric.
To summarize, measurements acquired at the Siemens workstation deviated by 3.54% on average (range, 2.78–4.03%; p = 0.14) from the true dimensions, those at General Electric by 4.05% (range, 1.46–7.09%; p < 0.0001), and at TeraRecon by 4.86% (range, 3.22–6.45%; p < 0.0001). Accordingly, Siemens was the most precise workstation at simultaneously most fluctuating values (scattering of 4.46%). TeraRecon had the smallest fluctuation (scattering of 2.83%), but the largest deviation from the true size of the phantom. The workstation from General Electric showed a scattering of 2.94%.
### 3.2 Patient data sets
Correlation coefficients between the 1st and 2nd round, both performed at one of the three different workstations, are shown in Table 3. The highest overall correlation between the 1st and 2nd round was observed with measurements from Siemens (r = 0.898, range 0.404–0.990), followed by TeraRecon (r = 0.799, range 0.314–0.992), and General Electric (r = 0.703, range 0.395–0.925). It is noticeable that the p values of most correlations were below 0.05 pointing towards a good reproducibility of measurements. Correlation coefficients with p values above 0.05 were found at the measurement points "diameter of the thrombus", "diameter of the right and left femoral artery", and "distance of the aneurysm to the subclavian artery".
Table 3Correlation analysis between the 1st and 2nd round at every single workstation.
General ElectricTeraReconSiemens
Variablesrp valuerp valuerp value
Aortic diameter at the subclavian artery0.845< 0.00010.824< 0.00010.946< 0.0001
Aortic diameter at the beginning of the aneursym0.841< 0.00010.854< 0.00010.981< 0.0001
Diameter of the thrombus within the aortic lesion0.5980.11180.992< 0.00010.956< 0.0001
Maximum diameter of the aneurysm0.925< 0.00010.895< 0.00010.792< 0.0001
Aortic diameter at the end of the aneurysm0.729< 0.00010.786< 0.00010.971< 0.0001
Aortic diameter at the coeliac trunk0.720< 0.00010.793< 0.00010.989< 0.0001
Diameter of the right femoral artery0.4260.04700.722< 0.00010.836< 0.0001
Diameter of the left femoral artery0.3950.06740.6000.00050.4040.0568
Distance of the aneurysm to the subclavian artery0.544< 0.00010.3140.12160.987< 0.0001
Length of the aneurysm0.807< 0.00010.954< 0.00010.990< 0.0001
Distance of the aneurysm to the coeliac trunk0.805< 0.00010.959< 0.00010.989< 0.0001
Total distance between the origin of the subclavian artery and the coeliac trunk0.805< 0.00010.892< 0.00010.933< 0.0001
Abbreviations: r, correlation coefficient.
In addition to the observations made at every single workstation, the repeatability of measurements between the three workstations was evaluated (Table 4 and Table 5). The 1st and 2nd rounds were considered separately from each other since this way a learning effect of the user could be shown that would have been lost in an all-encompassing comparison. Again, the measurement points “diameter of the right and left femoral artery” and “distance of the aneurysm to the subclavian artery” showed lower correlation coefficients than other measurement points. Furthermore, the high discrepancy between the 1st and 2nd round at the measurement points “maximum diameter of the aneurysm” and “diameter of the thrombus within the aortic lesion” is worth mentioning, showing better correlation coefficients in the 1st than the 2nd round. Direct comparisons revealed outstanding correlations for a few measurement points. However, these extraordinarily high correlation coefficients remained inconsistent, lacking a clear pattern between the two rounds.
Table 4Correlation assessment per round comprising all three workstations, separated into the 1st and 2nd round.
Variablesr (1st round)r (2nd round)
Aortic diameter at the subclavian artery0.7750.888
Aortic diameter at the beginning of the aneursym0.7890.900
Diameter of the thrombus within the aortic lesion0.9630.674
Maximum diameter of the aneurysm0.8540.521
Aortic diameter at the end of the aneurysm0.8560.719
Aortic diameter at the coeliac trunk0.8390.749
Diameter of the right femoral artery0.5090.410
Diameter of the left femoral artery0.5220.445
Distance of the aneurysm to the subclavian artery0.5190.564
Length of the aneurysm0.8910.912
Distance of the aneurysm to the coeliac trunk0.8560.983
Total distance between the origin of the subclavian artery and the coeliac trunk0.8100.912
Abbreviations: r, correlation coefficient.
Table 5Pairwise correlation analysis of two workstations, respectively.
1st round2nd round
General Electric vs. TeraReconGeneral Electric vs. SiemensTeraRecon vs. SiemensGeneral Electric vs. TeraReconGeneral Electric vs. SiemensTeraRecon vs. Siemens
rp valuerp valuerp valuerp valuerp valuerp value
Aortic diameter at the subclavian artery0.710< 0.00010.827< 0.00010.795< 0.00010.859< 0.00010.868< 0.00010.937< 0.0001
Aortic diameter at the beginning of the aneursym0.745< 0.00010.784< 0.00010.838< 0.00010.912< 0.00010.868< 0.00010.923< 0.0001
Diameter of the thrombus within the aortic lesion0.966< 0.00010.949< 0.00010.973< 0.00010.972< 0.00010.6150.04600.5620.0573
Maximum diameter of the aneurysm0.854< 0.00010.798< 0.00010.917< 0.00010.919< 0.00010.3720.01860.4150.0189
Aortic diameter at the end of the aneurysm0.807< 0.00010.5790.00110.6590.00030.6460.00020.964< 0.00010.5210.0042
Aortic diameter at the coeliac trunk0.732< 0.00010.812< 0.00010.954< 0.00010.6490.00020.951< 0.00010.6190.0002
Diameter of the right femoral artery0.5160.04610.5070.04370.5040.04450.3050.04780.3150.00690.6140.0002
Diameter of the left femoral artery0.718< 0.00010.5180.04120.3380.17490.3240.02120.3460.00330.6600.0003
Distance of the aneurysm to the subclavian artery0.2740.10280.887< 0.00010.1960.15050.819< 0.00010.4650.00170.5380.0009
Length of the aneurysm0.870< 0.00010.876< 0.00010.931< 0.00010.947< 0.00010.886< 0.00010.905< 0.0001
Distance of the aneurysm to the coeliac trunk0.813< 0.00010.794< 0.00010.962< 0.00010.982< 0.00010.988< 0.00010.977< 0.0001
Total distance between the origin of the subclavian artery and the coeliac trunk0.6920.00020.863< 0.00010.881< 0.00010.907< 0.00010.909< 0.00010.918< 0.0001
Abbreviations: r, correlation coefficient.
Combined processing times varied significantly between the two rounds. Comprising all workstations, the processing times of the 1st round were significantly longer than those of the 2nd round (237 vs. 142 min for General Electric, 189 vs. 156 min for TeraRecon, and 162 vs. 130 min for Siemens). In the 2nd round, processing times were found to be substantially reduced by 1 h and 35 min for General Electric (40% of the initial value, p < 0.0001), by 32 min for Siemens (20%, p = 0.0005), and by 33 min for TeraRecon (18%, p = 0.0008).
### 3.3 Error report
When looking for the sources of error, almost all system errors were reproducible on another workstation. Therefore, data sets may not have been compatible with the requirements of every single workstation. Since around half of the system errors occurred while the data set has been processed, a recurring error by the user has also to be considered.
A detailed error report is presented in Table 6.
Table 6Error report comprising the need for manual correction of measurements as well as system errors.
General ElectricTeraReconSiemens
Need for manual correction1432
Correction of the center axis121
Correction of the captured vessel lumen511
Other800
System errors247
During image processing141
Other001
## 4. Discussion
This study systematically evaluated three different workstations regarding their diagnostic precision in-vivo and ex-vivo. Especially, the constructed phantom imitating the dimensions of a TAA facilitated the comparison of measurements with known fixed true values. In this context, measurements on the Siemens workstation deviated by 3.54% on average (p = 0.14) from the true size, those on TeraRecon by 4.86% (p < 0.0001), and on General Electric by 4.05% (or 6.01% including the detected user error at the measurement point “distance of the aneurysm to the artificial coeliac trunk”, p < 0.0001). Accordingly, Siemens was the most precise workstation regarding the deviation at simultaneously most fluctuating values (scattering of 4.46%). In comparison to the 1st round, processing times of the 2nd round were substantially reduced by 1 h and 35 min for General Electric (40% of the initial value, p < 0.0001), by 32 min for Siemens (20%, p = 0.0005), and by 33 min for TeraRecon (18%, p = 0.0008). Overall correlation coefficients between the 1st and 2nd round differed significantly (p = 0.037), reaching 0.898 for Siemens, 0.799 for TeraRecon, and 0.703 for General Electric.
### 4.1 Accuracy of the three different workstations
In addition to the precision of measurements, the calculated route of the central vessel axis is probably the most important technical criterion to provide reliable interventional planning. Since this step is largely automated, it requires extraordinary attention because the calculated central axis also serves as a simulation of catheter path positioning for the releasement of the stent over the aneurysm. For aneurysms and dissections that have a large and partially not clearly separable area to the true vascular lumen, the difficulty of correct delineation often arises in clinical practice. Attenuation differences between areas enhanced by contrast medium and surrounding tissue are frequently too weak for a clear differentiation by dedicated software algorithms resulting in suboptimal image quality. Additionally, if the contrast agent is not optimally recorded, it is also less likely that the software algorithms will make a correct assignment. In those cases, the overestimated vessel lumen distorts the calculated central axis of the vessel.
Both data records and the user himself are two influencing factors that promote or even cause errors. Using the phantom made it possible to minimize both of these sources of error which is one of the most important quality features of good image acquisition and postprocessing. Another factor that may have falsified our results is the rounding of all measurements on the Siemens workstation to whole millimeters, whereas measurements on the General Electric and the TeraRecon were carried out with a resolution of 0.1 mm. Therefore, the exclusion of fluctuations of less than a millimeter at the workstation from Siemens might have influenced the accuracy of measurements, particularly in the case of small vessels. As a consequence, the apparently precise limits of the scatter range can only be compared to a limited extent with those of the other workstations.
In a study about the accuracy of the TeraRecon workstation in patients with abdominal aortic aneurysms [
• Higashiura W.
• Kichikawa K.
• Sakaguchi S.
• Tabayashi N.
• Taniguchi S.
• Uchida H.
Accuracy of centerline of flow measurement for sizing of the Zenith AAA endovascular graft and predictive factor for risk of inadequate sizing.
], the difference between pre-interventional planned stent length and true stent length was 4 mm on average (range, −1 to 23 mm). This corresponds to an average deviation of 2.5%, which is slightly better than ours. On the other hand, the scattering of values was more than twice as large as in our study. However, comparability to this study is limited since the authors used a different study design without an additional phantom. All comparisons were not related to the true size but to a new CT dataset of patients after successful endograft implantation.
The main advantage of the phantom used in our study was the knowledge about its known and verifiable dimensions, which allowed for a direct comparison of the acquired individual measurements from each workstation with the true dimensions of the phantom.
### 4.2 Handling and user experience
Considering the complete lack of user experience in the 1st round, the course of time savings in measurements hereafter suggest that the software solutions of the workstations are much more difficult to use for untrained investigators without previous in-depth training.
Furthermore, our practical experience shows that there is a clear reluctance to use special applications and functions of these workstations as long as the users have not undergone specific training. Additionally, physicians are often under a certain time pressure in their clinical routine avoiding too long processing times on a workstation that is new to them. Therefore, the accuracy and precision of measurements on different workstations for one patient are rarely compared by a single user. The motivation for our study resulted from this lack of data and experience.
Regarding the user-friendliness of the individual software solutions, listing bug fixes during image evaluation is just a limited attempt to put this part of our examinations into a sober framework. With the possibility of manual corrections, the question arose to what extent work processes should be considered normal or corrective within an examination. In this context, automatization levels of the working process vary at the different workstations because working steps that always occur as a fixed component at one workstation would already be seen as manual corrections to the workflow at another. It could be argued that due to the nature and conceptualization of the software, manual corrections at the workstation from General Electric are an integral part of the regular workflow. Exactly the opposite was the case with the Siemens workstation. Due to the software design, most of the measurements were taken by the investigator himself. Therefore, a correction was unnecessary in most cases since the user already did the first measurement by himself.
### 4.3 Transfer into clinical practice
Our study has several important clinical implications. In addition to the precision of measured values, the direction in which the measurement error deviates from the true value also plays an important role in the clinical routine. Positive deviations in the diameters are better than negative ones to a certain extent of approximately 10–15% [
• Rousseau H.
• Chabbert V.
• Maracher M.A.
• El Aassar O.
• Auriol J.
• Massabuau P.
• Moreno R.
The importance of imaging assessment before endovascular repair of thoracic aorta.
]. A stent with a too-large diameter in this range leads to a higher contact pressure on the vessel wall and thus to more stability. On the other hand, stent diameters that are too small can lead to inadequate anchoring of the endoprosthesis and finally to dislocation [
• Najibi S.
• Terramani T.T.
• Weiss V.J.
• Smith R.B.
• Salam A.A.
• Dodson T.F.
• Chaikof E.L.
• Lumsden A.B.
Endovascular aortic aneurysm operations.
].
In the case of length measurements, the situation is reversed. Stents that are too long would increase the risk of vessel obstruction proximal or distal to the insertion zone. On the contrary, a stent that is too short may not be long enough to cover the pathology. In our study, it has been shown very clearly that the diameters were usually determined too large and the distances too short on all workstations. In this context, a workstation deviation that is relatively constant can be used in practice, bearing in mind that the results must be corrected for the value of the deviation. A deviation that is accompanied by inconstant fluctuations is far more difficult to use because this error cannot be consistently corrected.
### 4.4 Limitations
Several limitations have to be addressed when interpreting our results. First, the experiments were carried out by a single experienced radiologist who was familiar with the handling of workstations. Therefore, the results might not be transferable to investigations conducted by inexperienced users. Second, the repetition of examinations at every single workstation has led to a continuously growing training effect of potentially difficult patient data sets and recall bias. Due to the similarities between the software, learning effects are also transferred from one workstation to the next. Therefore, our results require randomized prospective validation including a larger number of patients for a more in-depth analysis. Third, despite careful measurement of diameters in positions without air bubbles, residual errors cannot be fully excluded. Finally, the investigated study population was relatively small. Future studies are required to validate our preliminary study results.
## 5. Conclusions
In summary, all three workstations facilitated accurate distance determinations in the majority of cases at simultaneously high reproducibility. However, Siemens provided the most precise workstation with the lowest deviation from true vessel dimensions and the highest correlation between the 1st and 2nd round of measurements. Therefore, pre-interventional planning of TEVAR in patients with TAAs using three-dimensional CTA is feasible and can obviate the need for invasive aortography.
## CRediT authorship contribution statement
VK: Writing - Original Draft, Supervision, Project administration, Software. GL and JES: Data Curation, Investigation. LDG: Visualization, Resources. KE, TDA, CB, SB, SM, SSM, MH, and MHA: Formal analysis, Methodology, Software, Validation. SZ, AT, IY, and JES: Project administration, Validation. TJV and TGR: Writing - Review & Editing, Methodology, and Conceptualization.
## Ethical statement
The present work has been carried out in accordance with The Code of Ethics of the World Medical Association (Declaration of Helsinki) and is in line with the Recommendations for the Conduct, Reporting, Editing and Publication of Scholarly Work in Medical Journals. Moreover, the work aims for the inclusion of representative human populations (sex, age, and ethnicity) as per those recommendations. The terms sex and gender are used correctly.
The institutional ethical review board approved this retrospective study. The need for written informed consent was waived. The privacy rights of human subjects have been always observed.
## Funding Statement
No funding has been received for this project.
## Appendix A. Supplementary material
• Supplementary material
.
• Figure S1 Exemplary full-screen view of a postprocessing step at the TeraRecon workstation. A coronal and sagittal cross-section of the aorta with a calculated central axis can be seen on the left side of the image. Two cross-sectional areas of the selected vessels are shown below (bottom left). Centered at the top, the central axis is projected into a three-dimensional reconstruction with fixed points for the measurements set below. The panels on the right show the selection of the template and the individual working steps. Supplementary material
.
• Figure S2 Screen view during image processing on the Siemens workstation. In the two panels on the left side of the image, axial cross-sections along the central axis are shown. In the right panels, the corresponding central axis and adjacent structures around the vessel are depicted. Different display options and the "Vessel Analysis" menu for the measurements are illustrated on the right side of the screen. Supplementary material
.
• Figure S3 Images recorded during the processing of the datasets at the workstation from General Electric. A) Three-dimensional reconstruction of the aorta after removing adjacent structures using adjusted Hounsfield unit settings. B) Coronal section. C) A deformed, "stretched” view of the aorta. In addition, the orthogonal planes to the central axis are shown here, on which the diameters and the endpoints of the distances are located. Supplementary material
.
## References
• Guo M.H.
• Appoo J.J.
• Saczkowski R.
• Smith H.N.
• Ouzounian M.
• Gregory A.J.
• Herget E.J.
• Boodhwani M.
Association of mortality and acute aortic events with ascending aortic aneurysm: a systematic review and meta-analysis.
JAMA Netw. Open. 2018; 1e181281
• Olsson C.
• Thelin S.
• Stahle E.
• Ekbom A.
• Granath F.
Thoracic aortic aneurysm and dissection: increasing prevalence and improved outcomes reported in a nationwide population-based study of more than 14,000 cases from 1987 to 2002.
Circulation. 2006; 114: 2611-2618
• Wang T.K.M.
• Desai M.Y.
Thoracic aortic aneurysm: optimal surveillance and treatment.
Cleve Clin. J. Med. 2020; 87: 557-568
• Upchurch Jr., G.R.
• Escobar G.A.
• Beck A.W.
• Matsumura J.S.
• Perry R.J.
• Singh M.J.
• Veeraswamy R.K.
• Wang G.J.
Society for vascular surgery clinical practice guidelines of thoracic endovascular aortic repair for descending thoracic aortic aneurysms.
J. Vasc. Surg. 2021; 73: 55S-83S
• Spanos K.
• Nana P.
• Behrendt C.A.
• Kouvelos G.
• Panuccio G.
• Heidemann F.
• Matsagkas M.
• Debus E.S.
• Giannoukas A.
• Kolbel T.
Management of descending thoracic aortic diseases: similarities and differences among cardiovascular guidelines.
J. Endovasc. Ther. 2021; 28: 323-331
• Erbel R.
• Aboyans V.
• Boileau C.
• Bossone E.
• Bartolomeo R.D.
• Eggebrecht H.
• Evangelista A.
• Falk V.
• Frank H.
• Gaemperli O.
• Grabenwoger M.
• Haverich A.
• Iung B.
• Manolis A.J.
• Meijboom F.
• Nienaber C.A.
• Roffi M.
• Rousseau H.
• Sechtem U.
• Sirnes P.A.
• Allmen R.S.
• Vrints C.J.
• Guidelines E.S.C.Cf.P.
2014 ESC Guidelines on the diagnosis and treatment of aortic diseases: Document covering acute and chronic aortic diseases of the thoracic and abdominal aorta of the adult. The Task Force for the Diagnosis and Treatment of Aortic Diseases of the European Society of Cardiology (ESC).
Eur. Heart J. 2014; 35: 2873-2926
• Hiratzka L.F.
• Bakris G.L.
• Beckman J.A.
• Bersin R.M.
• Carr V.F.
• Casey D.E.
• Eagle Jr., K.A.
• Hermann L.K.
• Isselbacher E.M.
• Kazerooni E.A.
• Kouchoukos N.T.
• Lytle B.W.
• Milewicz D.M.
• Reich D.L.
• Sen S.
• Shinn J.A.
• Svensson L.G.
• Williams D.M.
• G. American College of Cardiology Foundation/American Heart Association Task Force on Practice, S. American Association for Thoracic, R. American College of, A. American Stroke, A. Society of Cardiovascular, A. Society for Cardiovascular, Interventions, R. Society of Interventional, S. Society of Thoracic, M. Society for Vascular with thoracic aortic disease
2010 ACCF/AHA/AATS/ACR/ASA/SCA/SCAI/SIR/STS/SVM Guidelines for the diagnosis and management of patients with thoracic aortic disease. A Report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines, American Association for Thoracic Surgery, American College of Radiology,American Stroke Association, Society of Cardiovascular Anesthesiologists, Society for Cardiovascular Angiography and Interventions, Society of Interventional Radiology, Society of Thoracic Surgeons,and Society for Vascular Medicine.
J. Am. Coll. Cardiol. 2010; 55: e27-e129
• Czerny M.
• Schmidli J.
• van den Berg J.C.
• Bertoglio L.
• Carrel T.
• Chiesa R.
• Clough R.E.
• Eberle B.
• Etz C.
• Grabenwoger M.
• Haulon S.
• Jakob H.
• Kari F.A.
• Mestres C.A.
• Pacini D.
• Resch T.
• Rylski B.
• Schoenhoff F.
• Shrestha M.
• von Tengg-Kobligk H.
• Tsagakis K.
• Wyss T.R.
• Document R.
• Chakfe N.
• Debus S.
• de Borst G.J.
• Di Bartolomeo R.
• Lindholt J.S.
• Ma W.G.
• Suwalski P.
• Vermassen F.
• Wahba A.
• Wyler von Ballmoos M.C.
Editor’s choice - current options and recommendations for the treatment of thoracic aortic pathologies involving the aortic arch: an expert consensus document of the european association for cardio-thoracic surgery (EACTS) & the European Society for Vascular Surgery (ESVS).
Eur. J. Vasc. Endovasc. Surg. 2019; 57: 165-198
• Salameh M.J.
• Black 3rd, J.H.
• Ratchford E.V.
Thoracic aortic aneurysm.
Vasc. Med. 2018; 23: 573-578
• Nienaber C.A.
• Clough R.E.
Management of acute aortic dissection.
Lancet. 2015; 385: 800-811
• Calero A.
• Illig K.A.
Overview of aortic aneurysm management in the endovascular era.
Semin. Vasc. Surg. 2016; 29: 3-17
• Sobocinski J.
• Chenorhokian H.
• Maurel B.
• Midulla M.
• Hertault A.
• Le Roux M.
• Azzaoui R.
• Haulon S.
The benefits of EVAR planning using a 3D workstation.
Eur. J. Vasc. Endovasc. Surg. 2013; 46: 418-423
• Sulaiman A.
• Boussel L.
• Taconnet F.
• Serfaty J.M.
• Alsaid H.
• Attia C.
• Huet L.
• Douek P.
In vitro non-rigid life-size model of aortic arch aneurysm for endovascular prosthesis assessment.
Eur. J. Cardiothorac. Surg. 2008; 33: 53-57
• Higashiura W.
• Kichikawa K.
• Sakaguchi S.
• Tabayashi N.
• Taniguchi S.
• Uchida H.
Accuracy of centerline of flow measurement for sizing of the Zenith AAA endovascular graft and predictive factor for risk of inadequate sizing.
Cardiovasc. Interv. Radiol. 2009; 32: 441-448
• Rousseau H.
• Chabbert V.
• Maracher M.A.
• El Aassar O.
• Auriol J.
• Massabuau P.
• Moreno R.
The importance of imaging assessment before endovascular repair of thoracic aorta.
Eur. J. Vasc. Endovasc. Surg. 2009; 38: 408-421
• Najibi S.
• Terramani T.T.
• Weiss V.J.
• Smith R.B.
• Salam A.A.
• Dodson T.F.
• Chaikof E.L.
• Lumsden A.B.
Endovascular aortic aneurysm operations.
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http://math.stackexchange.com/questions/89508/using-integration-to-find-population | # Using integration to find population
This one is tough:
A straight road goes through the center of a circular city of radius $5\text{km}$. The density of the population at a distance $r$ is well represented by $D(r)=20-4r$ (in thousand people per $\text{km}^2$). Find the population of the city.
Am I correct in that this is problem involving a "centroid"? I am not sure how to set this problem up with with the information given.
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You'll want a double integral that looks like $\int_0^{2\pi}\int_0^5 \text{(something)}r\mathrm dr\mathrm d\theta$... – J. M. Dec 8 '11 at 4:49
At a distance $r$ from where? The road or the centre?. If it is the centre, what does the road have to do with it? – André Nicolas Dec 8 '11 at 4:49
@J.M. Not really necessarily; I've seen this problem before in Calc II textbooks (that's before multiple integrals). – Arturo Magidin Dec 8 '11 at 5:00
@Dylan: Note, you aren't, as your title claims, using integration to find population density, you are using integration to find total population. – Arturo Magidin Dec 8 '11 at 5:05
@Arturo: I didn't realize, sorry. I just wrote out why I'd have done if I were given it... – J. M. Dec 8 '11 at 5:24
Assuming this is the kind of standard problem:
Imagine the circular city. Divide it into $n$ annuli of very small width, $\Delta r$. The density of population in an annular region is almost constant. If we are $r_i$ away from the center on the inner edge of the annular region, the density is well approximated by $D(r_i) = 20-4r_i$ thousand people per square kilometer. The area of the annular region is also well-approximated by "slicing it open and stretching it out", which will give you a shape that is very close to a rectangle of height $\Delta r$, and of width $2\pi r_i$, so the area is approximately $2\pi r_i\Delta r$ square kilometers. So the population in the annular region just described can be approximated by: $$\text{Population in the }i\text{th region}\approx (20-4r_i)(2\pi r_i)\Delta r\text{ thousand people.}$$ Adding it up over all the annular regions we have that $$\text{Population of the city} \approx \sum_{i=1}^n (20-4r_i)(2\pi r_i)\Delta r.$$ If we take the limit as $n\to\infty$, the approximations gets better and better (both the area approximations and the density approximations), and the error goes to zero. So $$\text{Population of the city} = \lim_{n\to\infty}\left(\sum_{i=1}^n(20-4r_i)(2\pi r_i)\Delta r\right).$$
But these sums are Riemann sums of a particular function, and so the limit will equal an integral. Figure out what integral, and then performing the integration will give you the population.
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Thank You. Solving, I got $\frac {500 \pi}{3}$ thousand people per $km^2$. – Dylan Dec 8 '11 at 7:02
@Dylan: Your units, at least, are wrong. $(20-4r_i)$ is measured in thousands of people per square km. $r_i$ is measured in km, as is $\Delta r$. That means that the units of the sum are $$\left(\frac{\text{thousands of people}}{\text{km}^2}\right)(\text{km})(\text{km}) = \text{thousands of people}.$$Your answer should be a population, not a density. – Arturo Magidin Dec 8 '11 at 15:30
To find the population you can integrate:
Let $R=$ radius of city; Since $x^2 +y^2 = R^2$ ,then $y = \sqrt{25 - x^2}$; since you are only evaluating $\dfrac{1}{4}$ of the circle multiply the integral by $4$; let $r=x$; and you get the Integral $$\int_0^54\sqrt{25 - x^2}\cdot(20 - 4x) dx$$
I think this is correct. Let me know if I am wrong.
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I make no remark about the correctness of the answer. But, here are some things to remember when you post in an answer: Use complete sentences. And, remember to use TeX in your answers. Equations are inserted by using $...$ while display equations are inserted by $$...$$. For some of the commaands, browse through the internet there are plenty of them. And, you may click on the time stamp above my name to see some changes that went into your original posts. Lastly, and joyfully, Welcome to MSE. – user21436 Mar 18 '12 at 1:31 | 2015-11-26 11:10:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9075502753257751, "perplexity": 477.947924878628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447043.45/warc/CC-MAIN-20151124205407-00009-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://junjizhi.com/ | # Elixir TIL: Start Plug-based Apps without Stopping
If you are writing a Dockerfile for your Elixir app, the final CMD line may look like:
Jun 9, 2022 · 1 min read
# AI as Note Taking Pal: An Experiment
I did an experiment to ask OpenAI to finish one Zettelkasten note. The goal is to enrich my notes. Turns out, AI generated content is not always directly usable. But the same content can be useful in surprising ways, which may enrich our thinking and writing at the end.
Jan 1, 2022 · 5 min read
# "Memory Heap" the Misnomer
Do you know that memory heap has nothing to do with heap as a data structure?
Nov 7, 2021 · 1 min read | 2022-09-30 12:55:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30414456129074097, "perplexity": 6853.198357544165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335469.40/warc/CC-MAIN-20220930113830-20220930143830-00627.warc.gz"} |
https://mathoverflow.net/questions/220559/the-characteristic-polynomial-of-the-product-of-two-linear-recurrences | # The characteristic polynomial of the product of two linear recurrences
Let $\mathbb{F}$ be a field and let $(a_n)_{n \geq 0}$, $(b_n)_{n \geq 0}$ be two linear recurrences with terms in $\mathbb{F}$ and respective characteristic polynomials $f(X), g(X) \in \mathbb{F}[X]$. Put $c_n := a_n b_n$ for all integers $n \geq 0$. It is well-known than $(c_n)_{n \geq 0}$ is a linear recurrence.
Answering ME question 1348838, Julian Rosen claimed (and proved) that a characteristic polynomial for $(c_n)_{n \geq 0}$ is given by $h(x) := \operatorname{Res}_Y(f(Y), Y^{\deg(g)}g(X/Y))$, where $\operatorname{Res}_Y$ is the resultant respect to $Y$. In truth, the proof was given for $\mathbb{F} = \mathbb{C}$, but no step refers to particular properties of the complex numbers, so the claim holds in any field.
Looking on the classic book on linear recurrences [1], I could not find Rosen's result. Neither I found it in [2] (for linear recurrences on finite fields).
Does someone have a reference for it?
Thank you very much.
[1] G. Everest, A. van der Poorten, I. E. Shparlinski, and T. Ward - Recurrence Sequences
[2] R. Lidl and H. Niederreiter - Introduction to Finite Fields and Their Applications
• No, it doesn't prove that $h(x)$ is a characteristic polynomial for $(c_n)_{n \geq 0}$. – user40023 Apr 18 '17 at 21:53 | 2019-09-16 17:03:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7782455086708069, "perplexity": 288.6214909938212}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572879.28/warc/CC-MAIN-20190916155946-20190916181946-00435.warc.gz"} |
https://labs.tib.eu/arxiv/?author=D.%20G.%20Middleton | • ### Measurement of the decay $\eta^{\prime}\to\pi^{0}\pi^{0}\eta$ at MAMI(1709.04230)
June 16, 2018 hep-ex
An experimental study of the $\eta'\to \pi^0\pi^0\eta \to 6\gamma$ decay has been conducted with the best up-to-date statistical accuracy, by measuring $\eta'$ mesons produced in the $\gamma p \to \eta' p$ reaction with the A2 tagged-photon facility at the Mainz Microtron, MAMI. The results obtained for the standard parametrization of the $\eta'\to \pi^0\pi^0\eta$ matrix element are consistent with the most recent results for $\eta'\to\pi\pi\eta$ decays, but have smaller uncertainties. The available statistics and experimental resolution allowed, for the first time, an observation of a structure below the $\pi^+\pi^-$ mass threshold, the magnitude and sign of which, checked within the framework of the nonrelativistic effective-field theory, demonstrated good agreement with the cusp that was predicted based on the $\pi\pi$ scattering length combination, $a_0-a_2$, extracted from $K \to 3\pi$ decays.
• ### High-statistics measurement of the eta->3pi^0 decay at MAMI(1803.02502)
March 7, 2018 hep-ex, nucl-ex
The largest, at the moment, statistics of 7x10^6 eta->3pi^0 decays, based on 6.2x10^7 eta mesons produced in the gamma p -> eta p reaction, has been accumulated by the A2 Collaboration at the Mainz Microtron, MAMI. It allowed a detailed study of the eta->3pi^0 dynamics beyond its conventional parametrization with just the quadratic slope parameter alpha and enabled, for the first time, a measurement of the second-order term and a better understanding of the cusp structure in the neutral decay. The present data are also compared to recent theoretical calculations that predict a nonlinear dependence along the quadratic distance from the Dalitz-plot center.
• ### Measurement of the omega -> pi^0 e^+e^- and eta -> e^+e^-g Dalitz decays with the A2 setup at MAMI(1609.04503)
April 24, 2017 hep-ex, nucl-ex
The Dalitz decays eta -> e^+e^-g and omega -> pi^0 e^+e^- have been measured in the g p -> eta p and g p -> omega p reactions, respectively, with the A2 tagged-photon facility at the Mainz Microtron, MAMI. The value obtained for the slope parameter of the electromagnetic transition form factor of eta, Lambda^{-2}_eta=(1.97+/-0.11_tot) GeV^{-2}, is in good agreement with previous measurements of the eta -> e^+e^-g and eta -> mu^+mu^-g decays. The uncertainty obtained in the value of Lambda^{-2}_eta is lower than in previous results based on the eta -> e^+e^-g decay. The value obtained for the omega slope parameter, Lambda^{-2}_omega_pi^0 = (1.99+/-0.21_tot) GeV^{-2}, is somewhat lower than previous measurements based on omega -> pi^0 mu^+mu^-, but the results for the omega transition form factor are in better agreement with theoretical calculations, compared to earlier experiments.
• ### Measurement of the pi^0 -> e^+e^-gamma Dalitz decay at the Mainz Microtron(1611.04739)
Feb. 18, 2017 hep-ex, nucl-ex
The Dalitz decay pi^0 -> e^+e^-gamma has been measured in the gamma p -> pi^0 p reaction with the A2 tagged-photon facility at the Mainz Microtron, MAMI. The value obtained for the slope parameter of the pi^0 electromagnetic transition form factor, a_pi = 0.030+/-0.010_tot, is in agreement with existing measurements of this decay and with recent theoretical calculations. The uncertainty obtained in the value of a_pi is lower than in previous results based on the pi^0 -> e^+e^-gamma decay.
• ### Measurement of the $p(e,e'\pi^+)n$ reaction close to threshold and at low $Q^2$(1606.00970)
Jan. 23, 2017 nucl-ex
The cross section of the $p(e,e'\pi^+)n$ reaction has been measured for five kinematic settings at an invariant mass of $W = 1094$ MeV and for a four-momentum transfer of $Q^2 = 0.078$ (GeV/$c$)$^2$. The measurement has been performed at MAMI using a new short-orbit spectrometer (SOS) of the A1 collaboration, intended for detection of low-energy pions. The transverse and longitudinal cross section terms were separated using the Rosenbluth method and the transverse-longitudinal interference term has been determined from the left-right asymmetry. The experimental cross section terms are compared with the calculations of three models: DMT2001, MAID2007 and $\chi$MAID. The results show that we do not yet understand the dynamics of the fundamental pion.
• ### Determination of the scalar polarizabilities of the proton using beam asymmetry $\Sigma_{3}$ in Compton scattering(1611.03769)
Jan. 10, 2017 nucl-ex
The scalar dipole polarizabilities, $\alpha_{E1}$ and $\beta_{M1}$, are fundamental properties related to the internal dynamics of the nucleon. The currently accepted values of the proton polarizabilities were determined by fitting to unpolarized proton Compton scattering cross section data. The measurement of the beam asymmetry $\Sigma_{3}$ in a certain kinematical range provides an alternative approach to the extraction of the scalar polarizabilities. At the Mainz Microtron (MAMI) the beam asymmetry was measured for Compton scattering below pion photoproduction threshold for the first time. The results are compared with model calculations and the influence of the experimental data on the extraction of the scalar polarizabilities is determined.
• ### Study of $\eta$ and $\eta'$ photoproduction at MAMI(1701.04809)
Jan. 2, 2017 hep-ph, hep-ex, nucl-ex
The reactions $\gamma p\to \eta p$ and $\gamma p\to \eta' p$ have been measured from their thresholds up to the center-of-mass energy $W=1.96$GeV with the tagged-photon facilities at the Mainz Microtron, MAMI. Differential cross sections were obtained with unprecedented accuracy, providing fine energy binning and full production-angle coverage. A strong cusp is observed in the total cross section and excitation functions for $\eta$ photoproduction at the energies in vicinity of the $\eta'$ threshold, $W=1896$MeV ($E_\gamma=1447$MeV). This behavior is explained in a revised $\eta$MAID isobar model by a significant branching of the $N(1895)1/2^-$ nucleon resonance to both, $\eta p$ and $\eta' p$, confirming the existence and constraining the properties of this poorly known state.
• ### First measurement of proton's charge form factor at very low $Q^2$ with initial state radiation(1612.06707)
Dec. 20, 2016 hep-ex, nucl-ex
We report on a new experimental method based on initial-state radiation (ISR) in e-p scattering, in which the radiative tail of the elastic e-p peak contains information on the proton charge form factor ($G_E^p$) at extremely small $Q^2$. The ISR technique was validated in a dedicated experiment using the spectrometers of the A1-Collaboration at the Mainz Microtron (MAMI). This provided first measurements of $G_E^p$ for $0.001\leq Q^2\leq 0.004 (GeV/c)^2$.
• ### Polarization-transfer measurement to a large-virtuality bound proton in the deuteron(1602.06104)
Feb. 19, 2016 nucl-ex
Possible differences between free and bound protons may be observed in the ratio of polarization-transfer components, $P'_x/P'_z$. We report the measurement of $P'_x/P'_z$, in the $^2\textrm{H}(\vec{e},e^{\prime}\vec{p})n$ reaction at low and high missing momenta. Observed increasing deviation of $P'_x/P'_z$ from that of a free proton as a function of the virtuality, similar to that observed in \hefour, indicates that the effect in nuclei is due to the virtuality of the knock-out proton and not due to the average nuclear density. The measured differences from calculations assuming free-proton form factors ($\sim10\%$), may indicate in-medium modifications.
• ### Measurement of pi^0 photoproduction on the proton at MAMI C(1506.08849)
Aug. 20, 2015 hep-ex, nucl-ex
Differential cross sections for the gamma p -> pi^0 p reaction have been measured with the A2 tagged-photon facilities at the Mainz Microtron, MAMI C, up to the center-of-mass energy W=1.9 GeV. The new results, obtained with a fine energy and angular binning, increase the existing quantity of pi^0 photoproduction data by ~47%. Owing to the unprecedented statistical accuracy and the full angular coverage, the results are sensitive to high partial-wave amplitudes. This is demonstrated by the decomposition of the differential cross sections in terms of Legendre polynomials and by further comparison to model predictions. A new solution of the SAID partial-wave analysis obtained after adding the new data into the fit is presented.
• ### Measurements of Double-Polarized Compton Scattering Asymmetries and Extraction of the Proton Spin Polarizabilities(1408.1576)
March 20, 2015 nucl-ex
The spin polarizabilities of the nucleon describe how the spin of the nucleon responds to an incident polarized photon. The most model-independent way to measure the nucleon spin polarizabilities is through polarized Compton scattering. Double-polarized Compton scattering asymmetries on the proton were measured in the $\Delta(1232)$ region using circularly polarized incident photons and a transversely polarized proton target at the Mainz Microtron. Fits to asymmetry data were performed using a dispersion model calculation and a baryon chiral perturbation theory calculation, and a separation of all four proton spin polarizabilities in the multipole basis was achieved. The analysis based on a dispersion model calculation yields $\gamma_{E1E1} = -3.5 \pm 1.2$, $\gamma_{M1M1}= 3.16 \pm 0.85$, $\gamma_{E1M2} = -0.7 \pm 1.2$, and $\gamma_{M1E2} = 1.99 \pm 0.29$, in units of $10^{-4}$ fm$^4$.
• ### A new measurement of the neutron detection efficiency for the NaI Crystal Ball detector(1502.07317)
Feb. 25, 2015 hep-ex, nucl-ex, physics.ins-det
We report on a measurement of the neutron detection efficiency in NaI crystals in the Crystal Ball detector obtained from a study of single p0 photoproduction on deuterium using the tagged photon beam at the Mainz Microtron. The results were obtained up to a neutron energy of 400 MeV. They are compared to previous measurements made more than 15 years ago at the pion beam at the BNL AGS.
• ### The electric and magnetic form factors of the proton(1307.6227)
July 29, 2014 nucl-ex
The paper describes a precise measurement of electron scattering off the proton at momentum transfers of $0.003 \lesssim Q^2 \lesssim 1$\ GeV$^2$. The average point-to-point error of the cross sections in this experiment is $\sim$ 0.37%. These data are used for a coherent new analysis together with all world data of unpolarized and polarized electron scattering from the very smallest to the highest momentum transfers so far measured. The extracted electric and magnetic form factors provide new insight into their exact shape, deviating from the classical dipole form, and of structure on top of this gross shape. The data reaching very low $Q^2$ values are used for a new determination of the electric and magnetic radii. An empirical determination of the Two-Photon-Exchange (TPE) correction is presented. The implications of this correction on the radii and the question of a directly visible signal of the pion cloud are addressed.
• ### Search for light massive gauge bosons as an explanation of the $(g-2)_\mu$ anomaly at MAMI(1404.5502)
April 22, 2014 hep-ex, nucl-ex
A massive, but light abelian U(1) gauge boson is a well motivated possible signature of physics beyond the Standard Model of particle physics. In this paper, the search for the signal of such a U(1) gauge boson in electron-positron pair-production at the spectrometer setup of the A1 Collaboration at the Mainz Microtron (MAMI) is described. Exclusion limits in the mass range of 40 MeV up to 300 MeV with a sensitivity in the mixing parameter of down to $\epsilon^2 = 8\times 10^{-7}$ are presented. A large fraction of the parameter space has been excluded where the discrepancy of the measured anomalous magnetic moment of the muon with theory might be explained by an additional U(1) gauge boson.
• ### Measurement of the neutron electric to magnetic form factor ratio at Q2 = 1.58 GeV2 using the reaction 3He(e,e'n)pp(1307.7361)
Aug. 29, 2013 nucl-ex
A measurement of beam helicity asymmetries in the reaction 3He(e,e'n)pp has been performed at the Mainz Microtron in quasielastic kinematics in order to determine the electric to magnetic form factor ratio of the neutron, GEn/GMn, at a four momentum transfer Q2 = 1.58 GeV2. Longitudinally polarized electrons were scattered on a highly polarized 3He gas target. The scattered electrons were detected with a high-resolution magnetic spectrometer, and the ejected neutrons with a dedicated neutron detector composed of scintillator bars. To reduce systematic errors data were taken for four different target polarization orientations allowing the determination of GEn/GMn from a double ratio. We find mu_n GEn/GMn = 0.250 +/- 0.058(stat.) +/- 0.017 (sys.).
• ### Accurate Test of Chiral Dynamics in the \boldmath$\vec{\gamma} p \rightarrow \pi^0p$ Reaction(1211.5495)
Aug. 12, 2013 nucl-ex
A precision measurement of the differential cross sections $d\sigma/d\Omega$ and the linearly polarized photon asymmetry $\Sigma \equiv (d\sigma_\perp - d\sigma_\parallel) \slash (d\sigma_\perp + d\sigma_\parallel)$ for the $\vec{\gamma} p \rightarrow \pi^0p$ reaction in the near-threshold region has been performed with a tagged photon beam and almost $4\pi$ detector at the Mainz Microtron. The Glasgow-Mainz photon tagging facility along with the Crystal Ball/TAPS multi-photon detector system and a cryogenic liquid hydrogen target were used. These data allowed for a precise determination of the energy dependence of the real parts of the $S$- and all three $P$-wave amplitudes for the first time and provide the most stringent test to date of the predictions of Chiral Perturbation Theory and its energy region of agreement with experiment.
• ### Reply to Comment on "High-Precision Determination of the Electric and Magnetic Form Factors of the Proton"(1108.3533)
Aug. 17, 2011 nucl-ex
In arXiv:1108.3058v1 [nucl-ex], Arrington criticizes the Coulomb corrections we applied in the analysis of high precision form factor data (see Phys.Rev.Lett.105:242001, 2010, arXiv:1007.5076v3 [nucl-ex]). We show, by comparing different calculations cited in the Comment, that the criticism of the Comment neglects the large uncertainty of "more modern" TPE corrections. This uncertainty has also been seen in recent polarized measurements. We rerun our analysis using one of these calculations. The results show that the Comment exaggerates the quantitative effect at small Q^2.
• ### Search for Light Gauge Bosons of the Dark Sector at the Mainz Microtron(1101.4091)
June 24, 2011 hep-ex, nucl-ex
A new exclusion limit for the electromagnetic production of a light U(1) gauge boson {\gamma}' decaying to e^+e^- was determined by the A1 Collaboration at the Mainz Microtron. Such light gauge bosons appear in several extensions of the standard model and are also discussed as candidates for the interaction of dark matter with standard model matter. In electron scattering from a heavy nucleus, the existing limits for a narrow state coupling to e^+e^- were reduced by nearly an order of magnitude in the range of the lepton pair mass of 210 MeV/c^2 < m_e^+e^- < 300 MeV/c^2. This experiment demonstrates the potential of high current and high resolution fixed target experiments for the search for physics beyond the standard model.
• ### High-precision determination of the electric and magnetic form factors of the proton(1007.5076)
Dec. 13, 2010 nucl-ex
New precise results of a measurement of the elastic electron-proton scattering cross section performed at the Mainz Microtron MAMI are presented. About 1400 cross sections were measured with negative four-momentum transfers squared up to Q^2=1 (GeV/c)^2 with statistical errors below 0.2%. The electric and magnetic form factors of the proton were extracted by fits of a large variety of form factor models directly to the cross sections. The form factors show some features at the scale of the pion cloud. The charge and magnetic radii are determined to be r_E=0.879(5)(stat.)(4)(syst.)(2)(model)(4)(group) fm and r_M=0.777(13)(stat.)(9)(syst.)(5)(model)(2)(group) fm. | 2021-02-24 20:32:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7030985355377197, "perplexity": 1781.8118199013782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178347321.0/warc/CC-MAIN-20210224194337-20210224224337-00359.warc.gz"} |
http://arxiver.moonhats.com/2014/03/18/general-parity-odd-cmb-bispectrum-estimation-cl/ | # General parity-odd CMB bispectrum estimation [CL]
We develop a methodology for estimating parity-odd bispectra in the cosmic microwave background (CMB). This is achieved through the extension of the original separable modal methodology to parity-odd bispectrum domains ($\ell_1 + \ell_2 + \ell_3 = {\rm odd}$). Through numerical tests of the parity-odd modal decomposition with some theoretical bispectrum templates, we verify that the parity-odd modal methodology can successfully reproduce the CMB bispectrum, without numerical instabilities. We also present simulated non-Gaussian maps produced by modal-decomposed parity-odd bispectra, and show the consistency with the exact results. Our new methodology is applicable to all types of parity-odd temperature and polarization bispectra.
M. Shiraishi, M. Liguori and J. Fergusson
Tue, 18 Mar 14
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https://adc.bmj.com/content/98/2/146?ijkey=ba6a7efc5c0d4c856c59d626b0ec8eab77cc841a&keytype2=tf_ipsecsha | Article Text
Community-acquired neonatal and infant sepsis in developing countries: efficacy of WHO's currently recommended antibiotics—systematic review and meta-analysis
1. Lilian Downie1,
2. Raffaela Armiento1,
3. Rami Subhi1,
4. Julian Kelly1,2,
5. Vanessa Clifford2,
6. Trevor Duke1
1. 1Centre for International Child Health, Department of Paediatrics, University of Melbourne, MCRI, Royal Children's Hospital, Melbourne, Victoria, Australia
2. 2Departments of General Medicine, Infectious Disease and Intensive Care, Royal Children's Hospital, Melbourne, Victoria, Australia
1. Correspondence to Professor Trevor Duke, Centre for International Child Health, Department of Paediatrics, University of Melbourne, MCRI, Royal Children's Hospital, Parkville, VIC 3052, Australia; trevor.duke{at}rch.org.au
## Abstract
Objective To review the aetiology and antibiotic resistance patterns of community-acquired sepsis in developing countries in infants where no clear focus of infection is clinically identified. To estimate the likely efficacy of WHO's recommended treatment for infant sepsis.
Design A systematic review of the literature describing the aetiology of community-acquired neonatal and infant sepsis in developing countries. Using meta-analytical methods, susceptibility was determined to the antibiotic combinations recommended by WHO: (1) benzylpenicillin/ampicillin and gentamicin, (2) chloramphenicol and benzylpenicillin, and (3) third-generation cephalosporins.
Results 19 studies were identified from 13 countries, with over 4000 blood culture isolates. Among neonates, Staphylococcus aureus, Klebsiella spp. and Escherichia coli accounted for 55% (39–70%) of culture positive sepsis on weighted prevalence. In infants outside the neonatal period, the most prevalent pathogens were S aureus, E coli, Klebsiella spp., Streptococcus pneumoniae and Salmonella spp., which accounted for 59% (26–92%) of culture positive sepsis. For neonates, penicillin/gentamicin had comparable in vitro coverage to third-generation cephalosporins (57% vs 56%). In older infants (1–12 months), in vitro susceptibility to penicillin/gentamicin, chloramphenicol/penicillin and third-generation cephalosporins was 63%, 47% and 64%, respectively.
Conclusions The high rate of community-acquired resistant sepsis—especially that caused by Klebsiella spp. and S aureus—is a serious global public health concern. In vitro susceptibility data suggest that third-generation cephalosporins are not more effective in treating sepsis than the currently recommended antibiotics, benzylpenicillin and gentamicin; however, with either regimen a significant proportion of bacteraemia is not covered. Revised recommendations for effective second-line antibiotics in neonatal and infant sepsis in developing countries are urgently needed.
• Tropical Paediatrics
• General Paediatrics
• Neonatology
• Infectious Diseases
• Microbiology
## Statistics from Altmetric.com
### What is already known on this topic
• Sepsis in neonates causes about half a million deaths per year.
• WHO recommends penicillin/ampicillin and gentamicin as treatment for neonatal sepsis.
• Many countries use third-generation cephalosporins to treat neonatal and infant sepsis.
• The commonest causes of neonatal bacteramia are: Staphylococcus aureus, Escherichia coli and Klebsiella spp., and in older infants, S aureus, Streptococcus pneumoniae, Klebsiella and E coli, and non-typhoidal Salmonella.
• Among community-acquired neonatal bacteraemia, resistance or reduced susceptibility to the combination of penicillin and gentamicin and to third-generation cephalosporins occurs in more than 40% of cases.
• Among community acquired bacteraemia in infants 1–12 months, resistance or reduced susceptibility to the combination of penicillin and gentamicin and to third-generation cephalosporins occurs in more than 35% of cases.
## Introduction
Between 30% and 50% of all deaths in children under the age of 5 years occur in the first month of life. Neonatal sepsis, the third most common cause of death in this age group, results in half a million deaths each year, the vast majority of which are in developing countries.1 Outside the neonatal period, the period up to 12 months carries the highest risk of death from sepsis. WHO recommends the use of clinical ‘danger’ and ‘priority’ signs to identify neonatal and infant sepsis, and empiric antibiotics to treat infants with suspected serious bacterial infection.2 So that evidence-based management recommendations can be refined, accurate information is required about the aetiology and antibiotic susceptibility of neonatal and infant sepsis, derived from studies using uniform methodologies and case definitions, from developing countries that are representative of those in their region.
WHO's Pocketbook of Hospital Care for Children, which provides clinical guidelines for the management of children in hospitals where resources are limited, particularly district hospitals, currently recommends treatment with ampicillin (or penicillin) and gentamicin for young infants (0–2 months) and benzylpenicillin plus chloramphenicol for older infants with suspected sepsis. If the infant's response is poor, the current advice is to change to ampicillin and gentamicin after 48 h. Second-line antibiotics include flucloxacillin where staphylococcal infection is suspected, and third-generation cephalosporins.2 With the reduction in price and more widespread availability, in many developing countries, third-generation cephalosporins are now used as first-line treatment for severe sepsis.
Thus far, the evidence for empirical antibiotics in infant sepsis has been limited by a lack of data on common bacterial pathogens and antimicrobial resistance, especially at the community and rural or district hospital levels in developing countries.3 Most studies of the aetiology of neonatal infections have been from tertiary hospital neonatal units where nosocomial infections and resistant organisms are common, and many studies have not distinguished between hospital- and community-acquired sepsis. Many studies of sepsis in older infants have been on cases with a septic focus, such as pneumonia or meningitis.4 ,5
This review evaluates the aetiology and antibiotic resistance patterns for community-acquired sepsis in infants where no clear focus of infection is clinically identified, using data published since 1996. From this the likely efficacy of WHO's recommendations can be estimated.
## Methods
### Search strategy
The search was conducted using Embase, Medline and the Cochrane Library database, employing the search terms described in online supplementary table A1. These terms were entered to capture primary data on aetiology and antibiotic susceptibility in infant sepsis in low and middle income countries, with the search terms varying slightly for each database. Only articles in English were reviewed. Letters, commentaries and case studies were excluded. The reference lists of relevant articles were accessed to broaden the search. The definition of low and middle income countries was in accordance with that of the World Bank and WHO.6 ,7
Sepsis was defined as a positive blood culture in studies where an inclusion criterion was clinical sepsis. The manifestations of clinical sepsis include: fever with no obvious focus of infection plus signs of systemic upset (eg, inability to drink or breastfeed, convulsions, lethargy or vomiting everything), cyanosis and fast breathing, purpura, cold skin with poor peripheral perfusion, low blood pressure, or pulses that are hard to detect.8–10
Studies that investigated the aetiology of infection with a clear focus (eg, pneumonia or meningitis) were excluded, as the management approach to these children will be different to that for the septic child with no clear focus. The full texts of all potentially relevant studies were retrieved and read. Studies that reported the recruitment of infants with sepsis that was likely to be hospital-acquired were excluded (see online supplementary box A1)
### Box 1 Exclusion criteria
• Studies reporting on hospital-acquired infections
• Children with an identified focus of infection
• Culture performed after day 2 of admission
• Recruitment included infants admitted to a health facility for other reasons than suspected sepsis
• Studies set in intensive/critical care units
• Studies exclusively recruiting immune-compromised, low birth weight or malnourished children.
• Studies of prophylactic antibiotics e.g. for neonates of mothers with infection
• Studies performed prior to 1996
• Studies that do not report disaggregated results for neonates or infants from older children and adults
• so that only studies of community-acquired sepsis would be selected.
so that only studies of community-acquired sepsis would be selected.
### Analysis
Results were extracted from each study and entered into a spreadsheet. Studies were summarised by extracting the study setting, design, inclusion criteria and methods of sampling and culture. Data collected from each study included the number of infants who had blood cultures sampled, the number who had positive cultures for a bacterial pathogen, the frequency of all isolates, and the isolate-specific antibiotic susceptibility where this was reported. Using random effects meta-regression, an extension of the standard meta-analysis,11 we calculated the proportion of bacteraemia due to different pathogens. This method weighs the proportions found in individual studies according to study sample size, and estimates the extent to which heterogeneity between the results of multiple studies can be related to one or more characteristics of the studies.
The studies were evaluated for the quality of bacterial isolation methods and antibiotic susceptibility testing using specific criteria (see online supplementary).
To determine the efficacy of WHO's currently recommended antibiotics, aetiology and susceptibility data were combined. The susceptibility of individual pathogens to specific antibiotic combinations was calculated using random effects meta-regression,11 weighting for number of isolates tested in each study. These proportions were described for each of the antibiotic combinations currently recommended by WHO: (1) benzylpenicillin/ampicillin and gentamicin, (2) chloramphenicol and benzylpenicillin, and (3) third-generation cephalosporins as a single agent. Only combinations 1 and 3 were evaluated for neonates, as chloramphenicol is not recommended by WHO as treatment for neonatal sepsis. These weighted proportions were multiplied by the weighted proportions of all cases of sepsis caused by that pathogen. This resulted in a data set that describes the proportion of infant sepsis attributable to a pathogen that is susceptible to recommended antibiotics. The sum of these values indicates the total proportion of infant sepsis that is susceptible to each of the antibiotics or antibiotic combinations analysed.
A number of assumptions were made in this methodology: first, the combined efficacy of two antibiotics is equal to that of the more efficacious of the two12; second, age did not significantly affect antibiotic susceptibility, as some studies of susceptibility included children older than 1 year; and third, benzylpenicillin and ampicillin are equivalent for the purpose of the analysis, because these antibiotics are generally interchangeable in WHO recommendations.
## Results
The search retrieved 615 published studies in Medline, 229 in Embase and 373 in the Cochrane database. Of these, 61 studies identified in the Medline search and 28 studies identified in the Embase search were potentially relevant, with six studies identified in both databases. One additional relevant study was published since the literature search was conducted, and this was included.13 Therefore, the full texts of 84 (90 minus 6) studies were sought. There were Cochrane reviews that were potentially relevant, but these were eventually excluded: two were not set in low and middle income countries, one had no measurable outcome, and one was in protocol form only.14–17
Of the 84 articles retrieved, 63 were excluded (figure 1). One further study reported data that were duplicated by a larger report included in the review (the Gambian component of the multi-country Young Infant Study18) and another19 reported data contained within a more recent study from the same site in Malawi.12 This left 19 studies from 13 countries.
Figure 1
Retrieval strategy and reasons for study exclusion, ICU, intensive care unit; VlBW, very low birth weight .
### Description of included studies
Of the 19 studies identified, nine were from Africa, eight from Asia, one from Iraq and one (the WHO Young Infant Study) was a multi-country study that included hospitals and clinics in Gambia, Ethiopia, Philippines and Papua New Guinea.20 One study was based on community surveillance, three studies were from rural district or provincial hospitals, 14 studies were from tertiary or referral hospitals, and the Young Infant Study included primary to tertiary health facilities.
Sixteen studies were prospective. These studies investigated the aetiology of positive blood cultures in either all infant admissions,21 infants admitted with fever without localising features22 ,23 or infants admitted with signs of severe illness or suspected sepsis.12 ,13 ,24–36 (table 1). One study, a retrospective review of laboratory records from a tertiary hospital neonatal ward, reported data from positive cultures of neonates admitted with suspected sepsis.35
Table 1
Included studies of the aetiology of sepsis in young infants
One study reported data from two centres in India: a neonatal intensive care unit and a rural district hospital. In line with this review's inclusion criteria, data were only extracted from the rural district hospital for this review.13
The number of positive cultures included in each study varied from 3030 ,31 to 784.12 Five studies (four from Africa and one from India) provided data for 72% of all positive isolates included in the review (2914 of 4049).12 ,21 ,23 ,28 ,35
### Quality of included studies
The review aimed to focus on community-acquired sepsis. We excluded studies which were identified as being from intensive care units. Not all studies were explicit on the proportions of infections that were community-acquired and nosocomial. The quality of laboratory methods was assessed using a systematic approach (see online supplementary box 1 for method, and online supplementary table A2 for study-level assessment). Most studies reported on blood culture collection methods (17/19), described fully the biochemical identification methods (14/19) and reported antibiotic susceptibility (17/19). The reporting of laboratory methods for bacterial identification was optimal in six of 19 studies. Eleven studies did not describe either how contaminants were distinguished from pathogens, whether biochemical identification was carried out, or what quality assurance methods were used. In two studies bacteriological methods were not described. Eight of 19 reported that a reference method for susceptibility testing was used.
### Aetiology
In the 19 studies there were a total of 4049 positive blood cultures: 76% of the positive cultures were in neonates (3077 of 4049 cultures) and 24% (972 of 4049 cultures) in infants aged 1–12 months (table 2). All but three studies12 ,20 ,34 reported the number of total blood cultures tested. When it was reported, the range of prevalence of bacteraemia in infants for whom a blood culture was performed was 3–16% in studies from community surveillance or first-referral (including rural and district) hospitals, and 20–60% in studies from tertiary hospitals.
Table 2
Bacteria isolated and prevalence weighted for study sample size
Using meta-analysis to weight for study size, among neonates, Staphylococcus aureus, Klebsiella spp. and Escherichia coli accounted for 55% (39–70%) of bacteraemic sepsis. These three pathogens, plus unidentified Gram-negative organisms, accounted for 62% (43–79%) of bacteraemic neonatal sepsis.
Among infants outside the neonatal period, the most prevalent pathogens were S aureus, E coli, Klebsiella spp., Salmonella spp. and Streptococcus pneumoniae, accounting for 59% (26–92%) of positive sepsis in infants older than 1 month on weighted analysis. Of the 103 isolates of Klebsiella spp., 90 were reported by one study from India.28
### Antibiotic susceptibility
Fifteen studies reported aetiology-specific antibiotic susceptibility data.12 ,13 ,21 ,23 ,25–36 One study26 reported antibiotic susceptibility for all isolated bacteria without specifying individual pathogens, and another did not specify the number of isolates tested for susceptibility.35 Studies reported aetiology-specific antibiotic susceptibility for neonates,12 ,13 ,25–27 31–36 infants up to 12 months of age,28 and children up to 1521 and 18 years of age.23 Susceptibility data for individual pathogens, weighted for the sample size of each study which tested antimicrobial susceptibility to that bacterium, are summarised in table 3. There were high rates of resistance among most species of enteric Gram-negative bacteria to gentamicin, chloramphenicol and third-generation cephalosporins. Similarly, there were high rates of resistance among S aureus to third-generation cephalosporins, and the β-lactamase stable penicillins (oxacillin, cloxacillin and flucloxacillin). table 4 and 5 present susceptibility data weighted for prevalence in neonates and infants over 1 month, respectively. This gives an indication of the proportion of each pathogen that causes bacteraemia which is likely to be susceptible to recommended antibiotics. Among neonates with bacteraemia, the prevalence of susceptible bacteria to the penicillin and gentamicin combination, and to third-generation cephalosporins was 57% and 56%, respectively. Among older infants with bacteraemia, the prevalence of bacteria susceptible to the penicillin and gentamicin combination, to chloramphenicol, and to third-generation cephalosporins was 63%, 47% and 64%, respectively.
Table 3
Weighted antibiotic susceptibility and number of isolates tested
Table 4
Antibiotic susceptibility of blood culture isolates in neonates
Table 5
Antibiotic susceptibility of blood culture isolates in infants aged 1–12 months
## Discussion
In developing countries, the high burden of invasive bacterial infections in young children and limitations in diagnostics make it essential that effective empirical antibiotic guidelines and therapy are available. WHO's clinical guidelines for the management of common childhood illnesses take into account the constraints of low-resource settings. For children with signs of sepsis but no localising signs of specific infections such as pneumonia or meningitis, empiric antibiotic therapy aims to broadly cover the most likely causes of septicaemia for the relevant age group.
This review found that a few bacteria cause the majority of infant sepsis, especially in the neonatal age group, where S aureus, Klebsiella spp. and E coli accounted for 55% of all sepsis. In contrast to studies from Western countries,37 there was a much lower proportion of sepsis due to group B streptococcus, an estimated prevalence of only 2%. S aureus, S pneumoniae, Klebsiella spp., E coli and Salmonella spp. were the most important pathogens in older infant (>1 month) sepsis.
We estimated that 57% and 63% of bacterial isolates in neonates and older infants, respectively, were susceptible to the combination of benzylpenicillin/ampicillin and gentamicin. Third-generation cephalosporins are often perceived as superior agents, but these did not provide higher coverage than penicillin/gentamicin. Third-generation cephalosporins had in vitro efficacy against 56% of neonatal pathogens and 64% of older infant pathogens. Among neonates, the gaps in antibiotic coverage with either regimen were mostly in infections due to enteric Gram-negative bacilli, particularly Klebsiella spp. Current empirical antibiotics are inadequate for most isolates of Klebsiella spp., which accounts for one in five cases of neonatal sepsis in developing countries.
### Study limitations
Limitations of this review include representation and heterogeneity. Although this review includes data from more than 4000 episodes of bacteraemia from 13 countries, this is a minute proportion of the annual global burden of this infection. Studies were of variable size and quality. Heterogeneity occurred at many levels: clinical criteria for enrolment in studies, study settings, study methodology, quality of laboratory methods and reporting. For example, a quality assessment of blood culture collection and processing identified that only eight of the 19 studies had optimal methods and two studies did not describe their blood culture methods.31 ,33 Eleven studies did not report whether they used such standardised susceptibility testing (online supplementary table A2). Use of a standardised reference method (eg, Clinical Laboratory Standards Institute or Eucast criteria) is important to ensure that susceptibility results are reproducible and accurate. When antibiotic choices are made, it is often assumed that in vivo efficacy is closely related to in vitro susceptibility. Where a reference method is not used, it is difficult to be certain of the correlation between in vitro results and in vivo efficacy. While 4049 bacterial isolates were identified in these studies, there was variability in the numbers of isolates which underwent susceptibility testing: 3560 were tested against penicillin/ampicillin, 3377 against gentamicin, 3456 against chloramphenicol and 1624 against third-generation cephalosporins. The efficacy of third-generation cephalosporins may have been under-estimated due to missing data.
An assumption made by this review is that all studies tested for the most important bacterial pathogens causing sepsis in this age group. This is reasonable, as most pathogens grow with standard bacterial culture techniques.
In vitro and in vivo antimicrobial efficacy may differ for some antibiotics and some bacterial pathogens. S aureus is the most common pathogen causing neonatal and older infant sepsis, so resistance has a large impact on overall antibiotic efficacy. S aureus isolates had low susceptibility to third-generation cephalosporins and penicillinase-resistant antibiotics, but the prevalence of in vitro susceptibility to gentamicin was high (80%). These data may overestimate the efficacy of the combination of penicillin and gentamicin in treating S aureus. Gentamicin is not commonly accepted as an appropriate sole effective agent for S aureus. However, previous research has demonstrated that clinically achievable gentamicin concentrations kill S aureus.38 In a previous community-based trial, the addition of gentamicin to cotrimoxazole as treatment for neonatal sepsis markedly reduced mortality,10 so it may be that more S aureus infections are being adequately covered by the currently recommended benzylpenicillin/ampicillin plus gentamicin combination than conventional wisdom dictates.
The estimates have other limitations. Data are more likely to be produced by large tertiary centres, and it can therefore be difficult to be certain that infections were community-acquired. We attempted to account for this by excluding studies performed in highly specialised settings (eg, intensive care units) and excluding studies where infections were acquired after 2 days of hospital admission. As we did not exclude data from tertiary hospitals, we cannot be certain that some studies in neonatal units were not carried out in units with intensive care facilities including, for example, mechanical ventilation.
High levels of resistance or susceptibility reported in a few studies can skew the pooled results, however the meta-analysis process addresses this. Some 72% of the bacterial isolates were identified from five of the 19 publications, four of which were from countries in Africa.21 ,23 ,28 ,35
There was incomplete differentiation in the literature of Salmonella spp. into typhoidal and non-typhoidal Salmonella (NTS), and there are geographical differences in the prevalence of this form of bacteraemia. A total of 297 Salmonella spp. were identified of which 248 (83%) were non-typhoidal Salmonella, 11 Salmonella typhi and 38 were not typed, so it was not possible to classify the susceptibility data on these un-typed isolates. The substantial prevalence of Salmonella spp. in older infant sepsis (8%) is due to the predominance of African studies included in this review.39 WHO's recent recommendation away from penicillin plus chloramphenicol to third-generation cephalosporins for the empiric treatment of sepsis in this age group will result in more effective treatment of non-typhoidal Salmonella in African countries where NTS is highly prevalent.
### Implications
The findings in this review have implications for global antibiotic recommendations. For neonates, sepsis due to resistant Gram-negative bacilli is an emerging and substantial problem, and the currently recommended first-line (penicillin/ampicillin plus gentamicin) or second-line antibiotics (a third-generation cephalosporin) do not provide adequate cover. Appropriate second-line treatment for when these bacteria are isolated or suspected needs to be explored and clinical indications for timely second-line therapy need to be developed. Amikacin is effective against most multi-resistant Klebsiella spp. and may be an alternative to gentamicin as second-line treatment in combination with a penicillin. Amikacin is comparable to the cost of gentamicin, a median price of US$0.54 per 100g vial of amikacin, compared to US$0.16 per 40g vial of gentamicin.40 Carbapenems, fluoroquinolones and extended-spectrum penicillins such as piperacillin-tazobactam and ticarcillin-clavulinate are expensive, drive resistance and are not widely available in developing countries.
Although it is often believed that third-generation cephalosporins are superior to older antibiotics, and are preferred as the agent of choice for severe sepsis, this study found this not to be true. Third-generation cephalosporins did not provide increased in vitro susceptibility. Furthermore, at least one controlled trial of treatment of neonatal sepsis using third-generation cephalosporins as first-line treatment has increased rates of resistance among bacterial pathogens in a hospital.41 This study was in tertiary neonatal units in a developed country, but the lessons are important for developing countries where the use of cephalosporins is increasingly widespread and uncontrolled. Third-generation cephalosporins are not more effective against common bloodstream bacterial pathogens than the combination of penicillin and gentamicin, but may be driving antimicrobial resistance among Gram-negative pathogens.
This review excluded studies of pneumonia and meningitis. This will have the effect of under-estimating the prevalence of bacteraemia due to S pneumoniae and Haemophilus influenzae, especially in the age group 1–12 months.
It is important to note clinical situations where these data do not apply. While for bacteraemia there was no advantage in using third-generation cephalosporins over penicillin and gentamicin, third-generation cephalosporins are likely to be more effective at treating Gram-negative meningitis than penicillin and gentamicin. Although there is no randomised trial evidence to support this, the use of cephalosporins for neonatal meningitis in industrialised countries has coincided with a reduction in death rates from this infection, although with no documented decrease in morbidity.42 ,43 The main results of this study therefore apply to non-meningitis sepsis.
The purpose of this paper was to assess whether current WHO guidelines for the empiric treatment of infant sepsis are appropriate. For this reason it was necessary to combine reported antibiotic susceptibility testing results from around the world, since these guidelines have been developed for use worldwide. However, local prevalence and susceptibility results, where available, should be the most important factor guiding local empiric antibiotic choices.
This review raises many questions of global public health importance. These are both technical and programmatic. They include: how to determine clinical criteria for second-line therapy that are implementable in resource-limited settings; how to ensure recommendations are effective but minimise the development of further resistance; how to make available more expensive or higher-generation antibiotics in resource-limited developing countries but ensure their use is based on evidence; and how to address the poor state of bacteriology services in most developing countries and improve local surveillance data.
## Conclusion
More than half the cases of neonatal sepsis were due to S aureus, E coli or Klebsiella spp. In older infants, Gram-positive bacteria (S aureus and S pneumoniae), in addition to Klebsiella and E coli and non-typhoidal Salmonella (in some regions), were the most important causes of bacteraemia. More than 40% of sepsis in neonates and more than 35% of sepsis in older infants was due to pathogens that were resistant (or had reduced susceptibility) to the antibiotic combination of ampicillin/penicillin and gentamicin, or the increasingly used alternative, third-generation cephalosporins. Revised recommendations for second-line antibiotics in neonatal and infant sepsis are needed. This task will be complex, but these are issues that the global health community need to address urgently. The methodology used in this systematic review could provide a model for longitudinal surveillance of bacteraemia in infants, which could offer long-term understanding of the aetiology and antibiotic resistance patterns in different regions. Improving bacteriology services in provincial and district hospitals in rural developing countries is essential for representative global data.
## Acknowledgments
We are grateful to Dr Susan Donath of the Centre for Epidemiology and Biostatistics at the University of Melbourne for guidance on the meta-analysis methodology.
• ## Supplementary Data
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## Footnotes
• Contributors All authors had a role in designing the study methodology. LD, RA, RS, JK and TD carried out the literature search, retrieved articles and analysed the data. LD and RA wrote the first draft of the paper, and all authors contributed to subsequent drafts. VC assessed the quality of laboratory methods. TD and JK supervised the study.
• Funding This research was supported by funding from the AusAID Knowledge Hubs for Health Initiative given to the Centre for International Child Health (CICH), The University of Melbourne. CICH is also a World Health Organization Collaborating Centre for Research and Training in Child and Neonatal Health. Neither agency influenced the publication of these results.
• Competing interests None.
• Provenance and peer review Not commissioned; externally peer reviewed.
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