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http://www.acmerblog.com/hdu-3144-gokigen-naname-4961.html | 2014
03-03
# Gokigen Naname
Gokigen Naname is a Japanese puzzle game played on a square grid in which numbers in circles appear at some of the intersections on the grid.
The objective is to draw diagonal lines in each cell of the grid, such that the number in each circle equals the number of lines extending from that circle. Additionally, it is forbidden for the diagonal lines to form an enclosed loop.
The first figure shows the start position of a puzzle. The second figure shows the solution to the same puzzle. A Gokigen Naname puzzle always has exactly one solution.
The first line of the input contains a single integer n (2 <= n <= 7), the number of cells along each of the sides in the square grid. Then follow n + 1 lines containing the contents of the intersections of the grid cells. Each such line will contain a string of n+1 characters, either a digit between 0 and 4, inclusive, or a period (‘.’) indicating that there is no number at this intersection (arbitrarily many lines may connect to it).
The first line of the input contains a single integer n (2 <= n <= 7), the number of cells along each of the sides in the square grid. Then follow n + 1 lines containing the contents of the intersections of the grid cells. Each such line will contain a string of n+1 characters, either a digit between 0 and 4, inclusive, or a period (‘.’) indicating that there is no number at this intersection (arbitrarily many lines may connect to it).
3
1.1.
...0
.3..
..2.
5
.21...
..33.0
......
..33..
0..33.
....11
\//
\\\
/\/
/\\//
//\\\
\\\//
\/\\/
///\\
#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <stack>
#include <bitset>
#include <algorithm>
#include <sstream>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <string.h>
using namespace std;
char ans[10][10];
char mat[10][10];
char input[10][10];
bool vis[10][10];
int tR, tC,N;
bool isin(int r,int c){
return r>=0&&r<=N&&c>=0&&c<=N;
}
bool dfs(int r, int c) {
if (vis[r][c]) return false;
if (r==tR && c==tC) return true;
vis[r][c]=true;
if (isin(r+1,c+1)&& ans[r][c]=='\\') if (dfs(r+1,c+1)) return true;
if (isin(r-1,c-1)&& ans[r-1][c-1]=='\\') if (dfs(r-1,c-1)) return true;
if (isin(r-1,c+1)&& ans[r-1][c]=='/') if (dfs(r-1,c+1)) return true;
if (isin(r+1,c-1)&& ans[r][c-1]=='/') if (dfs(r+1,c-1)) return true;
return false;
}
bool have(int r, int c, int tr, int tc) {
int i,j;
for (i=0;i<=N;i++)
for (j=0;j<=N;j++) vis[i][j] = false;
tR=tr;
tC=tc;
return dfs(r,c);
}
bool canput(int r, int c, char s) {
if (s=='/' &&have(r,c+1,r+1,c)) return false;
if (s=='\\' &&have(r,c,r+1,c+1)) return false;
return true;
}
bool legal(int r,int c){
if (input[r][c]!='.'&&mat[r][c]!=0) return false;
if (r==N-1){
if (input[r+1][c]!='.'&&mat[r+1][c]!=0) return false;
if (c==N-1){
if (input[r+1][c+1]!='.'&&mat[r+1][c+1]!=0) return false;
}
}
if (c==N-1) {
if (input[r][c+1]!='.'&&mat[r][c+1]!=0) return false;
}
if (mat[r][c+1]<0||mat[r+1][c]<0||mat[r+1][c+1]<0) return false;
return true;
}
bool dfs(int now){
if (now==N*N){
return true;
}
int r = now/N,c = now%N;
ans[r][c] = '.';
if (canput(r,c,'/')){
ans[r][c] = '/';
mat[r][c+1]--;
mat[r+1][c]--;
if (legal(r,c)&&dfs(now+1))
return true;
mat[r][c+1]++;
mat[r+1][c]++;
ans[r][c] = '.';
}
if (canput(r,c,'\\')){
ans[r][c] = '\\';
mat[r][c]--;
mat[r+1][c+1]--;
if (legal(r,c)&&dfs(now+1))
return true;
mat[r][c]++;
mat[r+1][c+1]++;
ans[r][c]='.';
}
return false;
}
int main(){
int tt;
int i,j,st,next;
scanf("%d",&tt);
for (int tcas = 1;tcas<=tt;tcas++){
scanf("%d",&N);
for (i=0;i<=N;i++)
scanf("%s",&input[i]);
for (i=0;i<=N;i++)
for (j=0;j<=N;j++)
if (input[i][j]=='.') mat[i][j] = 4;
else mat[i][j]=input[i][j]-'0';
memset(ans,0,sizeof(ans));
dfs(0);
for (i=0;i<N;i++)
{
for (j=0;j<N;j++) printf("%c",ans[i][j]);
printf("\n");
}
}
}
1. 有限自动机在ACM中是必须掌握的算法，实际上在面试当中几乎不可能让你单独的去实现这个算法，如果有题目要用到有限自动机来降低时间复杂度，那么这种面试题应该属于很难的级别了。
2. Good task for the group. Hold it up for every yeara??s winner. This is a excellent oppotunity for a lot more enhancement. Indeed, obtaining far better and much better is constantly the crucial. Just like my pal suggests on the truth about ab muscles, he just keeps obtaining much better. | 2017-04-29 11:45:15 | {"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": 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": 0.3005174398422241, "perplexity": 9071.722567440673}, "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-2017-17/segments/1492917123491.68/warc/CC-MAIN-20170423031203-00498-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/306008/given-the-expectation-and-standard-deviation-of-a-lognormal-how-can-i-calculate | Given the expectation and standard deviation of a lognormal, how can I calculate as normal
Bear with me, I cannot format on Stack Exchange, but I will do my best to explain.
I need to calculate the proportion of claims above a certain point.
I have my expectation and variance hats.
I'm being told I should calculate this data from a lognormal distribution above a given point, but to do it via TI 84 I must convert the lognormal parameters and boundaries to normal. I do not know how to do this.
So specifically I have lognormal ML estimators as mu hat = 5.75, sigma squared hat = .16, and I need to know what the proportion that exceed 400 is.
EDIT: I found a calculator online, and I knew I would, but asked this anyway in case I had to calculate the cdf on a test.
The lognormalCDF(400, 5.75, .4) = 0.726965598
So the complement is 1 - 0.726965598 = .273034402 = P(X>400)
Conventionally the $\mu$ and $\sigma$ parameters refer to the mean and standard deviation of the log of the lognormal random variable. If $Y$ is lognormal$(\mu,\sigma^2)$ then $X=\log(Y)$ is normal$(\mu,\sigma^2)$.
• what's the $e^{99}$ in there doing? Note that if you have a general normal cdf function you don't need to standardize, you can compare log(400) with a normal$(\mu,\sigma^2)$. Note also that you're finding the probability in the upper tail (above 400), so the normal cdf itself would give you the complement of the tail probability. (Note that there are many discussions of the lognormal on our site - some may have useful insights for you) – Glen_b Oct 3 '17 at 4:43 | 2020-10-20 03:22:13 | {"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": 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": 0.7511047720909119, "perplexity": 387.26344098700844}, "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-2020-45/segments/1603107869785.9/warc/CC-MAIN-20201020021700-20201020051700-00399.warc.gz"} |
https://physics.stackexchange.com/questions/473331/about-the-rigour-of-replacing-spins-by-hardcore-bosons | # About the rigour of replacing spins by hardcore Bosons
In literature one sometimes find that spins are replaced by hardcore bosons. Formally one replaces spin operators $$\sigma^- \leftrightarrow a$$, $$\sigma^+ \leftrightarrow a^\dagger$$, $$\sigma_z \leftrightarrow a^\dagger a - 1/2$$ and appends a term $$U a^\dagger a^\dagger aa$$ to the spin Hamiltonian. The limit $$U \to \infty$$ in the end splits the energy of states with more than 2 bosons away from the $$\{|0\rangle,|1\rangle\}$$ manifold. Identifying $$|0\rangle \leftrightarrow |\downarrow\rangle$$, $$|1\rangle \leftrightarrow |\uparrow\rangle$$ then gives a mapping between the bosonic and spin system.
My question is: how rigorous is this mapping? Can there be systems (i.e. Hamiltonians) or states for which the spin treatment yields different results than the bosonic treatment?
A possible example I had in mind: Consider some Hamiltonian $$H(\sigma^-, \sigma^+, \sigma_z)$$ and do the mapping to $$H_B = H(a, a^\dagger, a^\dagger a - 1/2) + U a^\dagger a^\dagger aa$$. Maybe time evolution $$H_B$$ can be easily solved using coherent states, thus I decompose some initial state into the coherent state basis, do the time evolution there, project the results onto the $$\{|0\rangle, |1\rangle\}$$-manifold and then take the limit $$U\to \infty$$. However, the coherent states always have a non-zero overlap with the "bad manifold" $$\{|n\rangle | n\ge 2\}$$. Thus this procedure works only, if no probability flows from the $$\{|n\rangle | n\ge 2\}$$-manifold to the $$\{|0\rangle, |1\rangle\}$$-manifold.
No, it is easy to show, that all states in $$\{|n\rangle | n\ge 2\}$$ are approximative eigenstates of $$H_B$$ with eigenenergy $$U$$ and the full eigenstates with eigenenergy $$\mathcal{O}(U)$$ have an overlap $$\mathcal{O}(1/U)$$ with the states from $$\{|0\rangle, |1\rangle\}$$. Thus, in the limit $$U\to\infty$$ the states $$\{|0\rangle, |1\rangle\}$$ and $$\{|n\rangle | n\ge 2\}$$ belong to disjoint subspaces of $$H_B$$'s eigenspace. Formally, this would suffice for any time evolution for some time $$t$$.
However, for asymptotic states (i.e., $$t\to\infty$$) im not so sure. Comming from a pertubation theory perspective I can argue that for each infinitesimal time step states from the $$\{|0\rangle, |1\rangle\}$$-manifold can be mapped to the $$\{|n\rangle | n\ge 2\}$$-manifold with some rate $$\mathcal{O}(1/U)$$ and vice versa. Doing infinitely many of these infinitesimal time steps can allow that these transition add up to something $$\mathcal{O}(1)$$. This is, we don't necessarily know whether $$\lim_{t\to\infty}\lim_{U\to\infty} e^{-iHt} = \lim_{U\to\infty}\lim_{t\to\infty} e^{-iHt}$$.
• What is the difference other than using a different symbol for your creation operators? What matters are the CCR of spins and hardcore bosons, and these are---by definition---the same. Maybe your question is instead: starting from the CCR for (softcore) bosons, how does one derive the effective CCR for hardcore bosons in the limit of on-site repulsion? Apr 25, 2019 at 10:25
• I'm not sure, but I think your reasoning makes sense. Superexchange processes would go as ~$J^2/U$ (where $J$ is the boson hopping), so if you took limits so that $U,t \rightarrow \infty$ but $J^2 t /\hbar U \neq 0$ these processes might still occur. But maybe all this says is that you have to take your limits carefully. Apr 26, 2019 at 0:10
• @RubenVerresen I guess this is probably one way to phrase it Apr 26, 2019 at 7:01
• @RubenVerresen 🍻 I interpreted it kind of like what we were talking about with softening constraints... are there any phase separations at infinite $U$ that don't persist to any finite $U$? Apr 28, 2019 at 16:40
• @RyanThorngren That is not something I have looked into yet, so I must admit that I don't know. However, my model at hand is essentially non-interacting driven spins, similar to Jaynes-Cummings, so I assume there is no phase separation present Apr 29, 2019 at 7:48 | 2022-06-29 07:08:51 | {"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": 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": 33, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8687767386436462, "perplexity": 536.1427208580096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2022-27/segments/1656103624904.34/warc/CC-MAIN-20220629054527-20220629084527-00024.warc.gz"} |
https://www.nature.com/articles/s41598-019-38482-1?error=cookies_not_supported&code=8dedc043-24e7-419e-a950-1213667335c8 | # Towards calibration-invariant spectroscopy using deep learning
## Abstract
The interaction between matter and electromagnetic radiation provides a rich understanding of what the matter is composed of and how it can be quantified using spectrometers. In many cases, however, the calibration of the spectrometer changes as a function of time (such as in electron spectrometers), or the absolute calibration may be different between different instruments. Calibration differences cause difficulties in comparing the absolute position of measured emission or absorption peaks between different instruments and even different measurements taken at different times on the same instrument. Present methods of avoiding this issue involve manual feature extraction of the original signal or qualitative analysis. Here we propose automated feature extraction using deep convolutional neural networks to determine the class of compound given only the shape of the spectrum. We classify three unique electronic environments of manganese (being relevant to many battery materials applications) in electron energy loss spectroscopy using 2001 spectra we collected in addition to testing on spectra from different instruments. We test a variety of commonly used neural network architectures found in the literature and propose a new fully convolutional architecture with improved translation-invariance which is immune to calibration differences.
## Introduction
A trained human spectroscopist is able to look at an unknown spectrum, which can be thought of as energy-series data, overlay a proposed candidate reference spectrum and determine (qualitatively) if there is a match. The human brain can perform this task with ease despite translational shifts in the spectrum or varying levels of noise between the reference spectrum and acquired spectrum. The rigor of matching unknown spectra to references can be improved using generalized linear models, fitting procedures, or cross-correlation functions but great difficulty arises with these methods in situations with high noise or translational shifts. There is a need to develop new generalizable and automated methods which can remove qualitative interpretation in spectroscopy analysis. Qualitative interpretation in spectroscopy adds bias to the analysis which includes any overlaying of reference spectra, manual peak shifting, and manual feature selection such as using the full-width half-maximum of peaks, or the intensity ratio between peaks. In addition to these issues, many present methods of quantification/identification require reference standards from the same instrument or from instruments with the same calibration which severely limits the amount of data available to a human spectroscopist. In this study, we apply advances in statistical learning algorithms (also called machine learning, or narrow artificial intelligence) to better identify important characteristics of a spectrum.
Domain experts have used feature engineering in the past to develop useful predictor variables (also known as features) which can be used to differentiate spectra. Such features include metrics like the width of peaks, ratio of peaks and distance between peaks. Feature engineering is the manual process of a domain expert performing dimensionality reduction by using domain knowledge to isolate key pieces of information from the original data. This is in contrast to using statistical learning algorithms which are able to find features automatically by aggregating statistics across large datasets. These algorithms learn to find features automatically by viewing many examples of spectra and deciding which pieces of the spectra are the most useful for differentiating between signals. Statistical learning methods in spectroscopy have been slowly percolating the literature for the last couple decades. Gallagher and Deacon, in a pioneering study in 2002, used single layer dense neural networks to predict experimental X-ray spectra for the automated classification of minerals1. Timoshenko et al. used multi-layer dense neural networks to predict the coordination numbers in metallic nanoparticles given a full simulated x-ray absorption spectrum2 and Zheng et al. used an ensemble-learned matching scheme to characterize simulated x-ray absorption spectra generated from the Materials Project where one of the steps explicitly involves peak shifting3. Carey et al. used unsupervised methods such as nearest neighbor clustering approaches and studied the effect of preprocessing on the classification of Raman spectra4. Lopez-Reyes et al. developed unsupervised (principal component analysis) and supervised methods (dense neural networks) to classify minerals on the ExoMars rover with Raman spectroscopy5.
For the classification of the entire Raman spectroscopy database RRUFF, Liu et al. used a convolutional neural network feature extractor6. Their methodology was two-fold. First a convolutional neural network feature extractor, then for classification, a dense neural network were used. They compared these results to other common machine learning classifiers such as boosting, random forest, and support vector machines. To teach the network to understand small translational (chemical) shifts, they used data augmentation with small random crops/shifts of the spectra.
Convolutional neural networks have not been used presently in electron spectroscopy. Yedra et al. created a script in Digital Micrograph (a software environment for acquiring and processing electron energy loss spectroscopy (EELS) data and images), named Oxide Wizard, to differentiate the oxidation states for various metal oxides in EELS7. Using domain knowledge, they engineer features such as the ratios of peaks, the full width at half maximum (FWHM) of peaks and the distances between oxygen and the metal edges. Zhang et al. used a multiple linear least-squares technique to determine the valence of Mn in EELS8. With reference spectra, they fit a generalized linear model using known reference spectra. Tan et al. also showed a comparison between common methods in EELS to determine the oxidation state, such as least-squares fitting and feature engineering using peak-ratios. They determined that different methods are ideal for different transition metals9.
We develop here a challenging dataset to be used as a test-case to probe the effectiveness and generalizability of convolutional neural networks in spectroscopy classification. As a model system, we investigated the valence identification of Mn, this being relevant in many fields of materials research, particularly interesting in the study of valence in battery materials. We acquired 2001 EELS spectra for Mn2+, Mn3+, and Mn4+ to be used as a model training and model validation dataset and we also digitize 31 reference spectra for Mn2+, Mn3+, and Mn4+ from published articles to be used as a test-set. We test three architectures: a densely connected neural network, a convolutional neural network feature extractor connected with a densely connected neural network and finally a fully convolutional neural network without any dense connections.
## Results
Two key benchmarks have been developed which were found to predict performance on the out-of-distribution test set. The first benchmark is to manually translate the validation-set spectra. This was found to approximate the effect of calibration and chemical shift. The second was by adding and removing noise from the data by modeling the noise distribution using combinations of principal components and measuring the resulting change in accuracy.
Three types of neural network architectures are explored in this study. The dense network has 11,000 weights, the convolutional and dense network has 1100 weights and the fully convolutional architecture has 650 weights. The dense neural network is similar to the dense network used in Gallagher and Deacon’s study1. More recent advances in neural networks were added to increase performance such as: neuron dropout to prevent overfitting10, batch normalization between hidden layers for activation normalization11, rectified linear units (ReLU) as activation function for the hidden units, and a softmax activation function for the output neurons. The second architecture is a convolutional neural network feature extractor attached to a densely connected neural network which is similar to the one used by Liu et al.6. The convolutional neural network architecture was inspired by Liu et al.6 and the choice for the number of convolutional filters and layers was inspired by the feature extractors used in the ImageNet competition like the well-established AlexNet and VGG16 architectures12,13. The third architecture has the same convolutional neural network architecture as previously described, but it is attached to a custom classification architecture inspired by the MobileNet and SqueezeNet architectures14,15.
A full description of the neural architectures can be found in the Methods section.
The graph of the fully convolutional neural network architecture can be found in Fig. 1. The feature extraction architecture is also the same used as the convolutional and dense containing network. The output of each operation can be found in Fig. 1b–d for the fully trained network visualized on the training set.
In Fig. 1a, the input spectrum is passed through the 5 successive feature extraction blocks where each block contains a convolution, batch-normalization and down-sampling (1D average pooling). The output of the 5th block (orange) is considered to be a representation of the original input data that is optimized for discrimination (i.e. discriminative features). These features are then used to classify the valence of the inputted spectrum using the single classification block which contains: dropout, convolution (kernel size, 1), global average pooling, and softmax. A description of every operation and relevant parameters can be found in the Methods section.
### Data Collection
A total of 2001 electron energy-loss spectra of Mn2+, Mn3+, and Mn4+ were acquired using a FEI Titan transmission electron microscope in a variety of conditions. This includes 448 Mn2+ spectra, 765 Mn3+ spectra, and 788 Mn4+ spectra cropped between 635 and 665 eV (300 bins at a dispersion of 0.1 eV/bin). The microscope parameters and spectrum image preprocessing steps can be found in the Methods section.
To obtain spectra from a wide variety of instruments and resolutions to prove generalizability, reference spectra were digitized from three studies by Garvie et al., Zhang et al. and Tan et al.8,9,16. These spectra were not used for model training and are instead used as a withheld test-set representing signals that are well outside of the distribution of the acquired data. They are of significantly higher resolution and the differences in instrument calibration is clear, with the onsets of the peaks being different as shown in Fig. 2. The energy range of the digitized spectra is between 635 and 658 eV due to many of them being cropped for their respective publications.
A qualitative inspection of Fig. 2 highlights that Mn2+ is narrower than both Mn3+ and Mn4+. In addition, Mn4+ can be differentiated from Mn3+ by a small shoulder located on the low energy side of the larger peak.
A flow-chart describing the pipe-line of going from acquired data to a functioning model can be found in the Supporting Information (Figure S1).
### Model Training
Stratified 10-fold cross-validation was used for model validation and to estimate the error of the validation set. The acquired Mn dataset was divided into 10 roughly equal folds where each class is stratified. 9 folds (i.e. 90% of the data) was used for finding model parameters while the last fold was used for calculating withheld-set validation accuracy. This was repeated for every fold in the model to produce 10 models trained on different partitions of the data.
In batches of 128 randomly selected augmented training spectra at a time, the spectra are non-linearly transformed over all edges and nodes in the directed graph to produce predicted class probabilities $$p({y}_{k}^{(i)}|z,\theta )$$ for all possible classes. The cost function being minimized C(θ) is the categorical cross-entropy loss between the true one-hot encoded label $${y}_{k}^{(i)}$$ and the predicted class probability $$p({y}_{k}^{(i)}|x,\theta )$$ calculated over all N training examples in the batch and for all possible K classes.
$$C(\theta )=-\sum _{i=1}^{N}\sum _{k=1}^{K}\,{y}_{k}^{(i)}\,\mathrm{log}\,p({y}_{k}^{(i)}|x,\theta )$$
The error from the cost function can be back-propagated backwards using the chain-rule through each weight and bias in the network to assign blame as to which parameters were unhelpful in the classification. With this scheme, the weights and biases can be nudged in a direction that minimizes error using stochastic gradient descent (SGD). The specific SGD algorithm used was the adaptive moment estimation optimizer, the Adam optimizer commonly used for the training of neural networks (default parameters)17.
### Model Validation – Effect of translation
Depending on the calibration of the spectrometer, it is possible for the peaks in the measured spectrum to be translated. In electron spectroscopy, the electronic environment of the atom being measured can also cause translational shifts (i.e. different compounds with the same oxidation state can be shifted). Because of this, it is crucial that an electron spectroscopy classifier must understand the shape of the spectra and not simply memorize the absolute onset of the peak.
Translation-invariance was measured by cropping each validation example and moving the Mn ionization edge into different positions into the 300-length input vector. A 5 eV shift in this context is equivalent to moving the peak 50 bins out of the total 300 bins. To test translation-invariance, the validation-set spectra, for each cross-validation fold, were shifted left and right up to 5 eV and the resulting validation-accuracy was measured. To further probe if translation-invariance can be learned, data augmentation was used to randomly translate training examples between −5 and 5 eV. For each architecture, the training examples were randomly shifted (data augmentation) and this was performed for various scalar multiples of the training data ranging from 1–10x the original amount of training data. The results of the effect of translation-invariance with respect to neural network architecture and scalar multiples of training data can be found in Fig. 3.
As the amount of training data augmentation increases for the fully convolutional network, the performance increases (albeit subtly) (Fig. 3). This can be observed by looking at the gradient of color going from purple to red. However, this is not the case for the other two networks with no smooth increase in validation accuracy as the amount of augmentation increases.
Without applying translation data augmentation to the training set (dataset identified as “None”, dark purple colour in Fig. 3) and without shifting the validation data (i.e. zero-shift), all three networks are able to exceed 99% 10-fold cross-validation validation-set accuracy. When shifted, however, even without training data augmentation, the fully convolutional network is shown to produce a small decrease in validation accuracy as the spectra are translated, whereas the other two architectures have a sharp (<60%) decrease in accuracy near that of a random guess.
When applying translation data augmentation to the training set, the dense containing networks have a sharp increase in validation accuracy as the validation spectra are translated, suggesting that translation-invariance can be brute-force learned. When using the randomly shifted training data (“1× ” size, with random shifts) to train the dense containing networks, the fully convolutional network without any sort of data augmentation (dataset labeled as “None”) is still superior (this was measured by integrating the validation accuracy with respect to translation). It is only when large scalar multiples of training data are applied to the Conv. +Dense network that it is able to outperform the non-augmented fully convolutional architecture. However, when comparing the 10x trained fully convolutional network to 10x trained Conv. +Dense network, the fully convolutional network is still superior.
It is equally worth noting that the fully convolutional network is superior under these conditions despite containing only 60% as many weights as the convolutional and dense network, and only 6% as many weights as the fully dense network. The fully convolutional neural network is also much more constrained by using global average pooling as opposed to dense connections. This may aid in translation invariance. However, it was shown by Azulay and Weiss18 that popular pre-trained ImageNet classifiers (e.g. Inception19, VGGNet13) are not completely translation-invariant even when using global average pooling.
### Model Validation – Effect of noise
To test the robustness of convolutional neural networks to the noise frequently found in electron energy loss spectra acquired using transmission electron microscopes, a noise test was implemented using principal component analysis (PCA) for the fully convolutional neural network. PCA has grown increasingly popular at denoising EELS spectra recently20,21,22,23,24. Removing low variance principal components is effective at eliminating some types of noise frequently detected in EELS with minimal loss of signal (typically <10−2% of the signal is removed). Using these low variance principal components as a noise distribution, they were added to each input spectrum in scalar multiples ranging between zero and five times the baseline noise level. In this context, a scalar multiple of zero would be PCA cleaned data (removing low variance components), and a scalar multiple of 1 is the original signal.
PCA was performed on each validation-set during cross-validation and low variance principal components were calculated on the validation-set and added back to the validation data in different scalar multiples. This measures how well the classifier can predict the oxidation state in the presence of extreme noise that a trained human spectroscopist would have great difficulty in classifying. As an example of what the signal-to-noise ratio looks like qualitatively, the effect of different scalar multiples of low variance principal components on spectra is shown Fig. 4b.
An additional test was also performed to see if the model could be made robust to high levels of noise by deliberating adding noise to the training data. This is called training data augmentation. PCA was performed on every training fold during cross-validation and low variance principal components were added to each training example. The neural network was then evaluated using the data augmented folds of the validation-set to measure how the validation accuracy changes with increasing validation-set noise. The comparison between the fully convolutional classifier trained with or without training data augmentation is shown in Fig. 4a.
This test shows that, in the presence of noise that is a 5x scalar multiple of low variance principal components, the data augmented classifier is able to exceed 93% validation-set accuracy. Without data augmentation the accuracy decreases moderately as noise is added with a validation-set accuracy of 78% at 5x the base-line noise level. It should be noted that a 1x scalar multiple of the low variance principal components is indeed just the original signal. A 0x scalar multiple in this context is a PCA-cleaned spectrum with low variance components being removed.
### Model Testing - Digitized Reference Spectra
The three neural networks were tested against the 31 digitized spectra taken from the publications by Zhang et al., Tan et al., and Garvie et al.8,9,16 to probe generalizability in the presence of different instruments, calibration, and resolution. This dataset contains 12 Mn2+ spectra (tetrahedral, octahedral, and dodecahedral coordination), 10 Mn3+ spectra (octahedral), and 9 Mn4+ spectra (octahedral). These spectra are also shifted significantly (~3 eV) compared to the acquired spectra, likely the result of different instrument calibration, and have different levels of noise (Fig. 2).
The three architectures tested in this study tested against the 31 digitized reference spectra. The test accuracies on the digitized reference spectra dataset are shown in Table 1.
In analyzing Table 1, the fully convolutional neural network is proven to be extremely successful on the digitized reference spectra dataset even without data augmentation. In contrast, dense layer containing architectures fail to generalize to data outside of their training distributions and incorrectly classify Mn2+ and Mn4+ when data augmentation is not used. These results agree with the translation-invariance test in Fig. 3 that the dense layer containing networks have difficulty performing classification on datasets too dissimilar from their training data since the digitized reference spectra are shifted significantly (Fig. 2). It is only with data augmentation that the dense containing networks are accurate.
Neural networks are apparently capable of classifying Mn compounds that do not have the same coordination (octahedral) as the acquired training data such as tetrahedral and dodecahedral compounds. This is due to the fact that, as demonstrated by Garvie et al. and Tan et al., there are strong similarities in the shape of the fine-structure of the core-loss edges between various Mn, Fe and V compounds of the same valence9,16 for different coordinations, at least at the energy resolution used for the experiments.
The activations for each layer of the fully convolutional neural network on the digitized reference spectra dataset can be found in the Supporting Information.
### Visualizing the feature space using t-SNE
To help demonstrate the success of the fully convolutional network on the digitized reference spectra test-set in the absence of training data augmentation, the 300-length preprocessed spectra (both acquired and also the reference spectra) were projected onto a 2D plane using a t-distributed stochastic neighbor embedding (t-SNE) technique. t-SNE developed by van der Maaten and Hinton25 is a popular non-linear dimensionality reduction technique which attempts to preserve the distribution of clusters in the original high-dimensional space when projecting the data onto a 2D plane for visualization purposes. Note that the analysis presented here is entirely qualitative to aid in exploratory visualization. The visualizations shown in Fig. 5 were initialized with PCA, have a perplexity of 100, and were run for 10 000 iterations. A variety of visualizations with perplexities between 20 and 100 can be found in the Supporting Information.
The features produced by the final feature extraction layer prior to global average pooling of the fully convolutional network, consisting of 12 vectors of length 6, was flattened into a 72-length vector and also visualized using t-SNE.
The classes can be easily differentiated on the feature-space shown in Fig. 5b. In the feature-space, it is evident that the reference spectra (the triangles in Fig. 5) are contained within the clusters of acquired spectra. This is in contrast to the t-SNE visualization of the input space (Fig. 5a) where all Mn4+ are incorrectly classified as Mn3+.
The digitized reference spectra dataset is very different from the acquired dataset (cleaner signal and shifted 3 eV). It is evident from the t-SNE visualization of the full input spectra space that these two datasets lie in different distributions. We can gauge this qualitatively by observing that reference Mn4+ are more closely similar to acquired Mn3+ than they are to acquired Mn4+.
## Discussion
In the field of electron energy loss spectroscopy (EELS), spectrometer calibration is often very difficult and leads to non-reproducible data between research groups. In addition to this, the noise profile is very different between instruments so, typically, dimensionality reduction (unsupervised learning methods or manual feature extraction) of the data needs to be performed to compare results between instruments. In this study, we create a new method of spectroscopy analysis using deep convolutional neural networks. These networks are proven to have significant advantages over all other methods in EELS analysis. In addition, our proposed fully convolutional neural network also has significant advantages over other neural networks used in the chemometrics literature with respect to translation-invariance. Our convolutional neural network is immune to calibration differences and have high noise tolerances which exceeds the ability of the currently used methods.
Convolutional neural networks are automated feature extractors. They require no domain knowledge, and require no manual feature collection, such as taking the FWHM of peaks or determining peak on-set. It is clear from the t-SNE visualization in Fig. 5 that the feature “fingerprints” produced by neural networks successfully embed discriminating differences between the three oxidation state clusters from both reference data digitized from publications and spectra that we acquired. The digitized reference spectra dataset is very different from the acquired dataset (cleaner signal and shifted 3 eV) and it is evident from the t-SNE visualization of the full input spectra space that these two datasets lie in different distributions. Despite this, however, our fully convolutional neural network is proven to successfully perform classification.
## Conclusions
We have demonstrated that traditional neural architectures, such as densely connected neural networks, or convolutional neural network feature extractors attached to a densely connected neural network, have limited translation-invariance for energy-series classification tasks. In contrast, our developed fully convolutional neural network uses global average pooling on the final feature layer and retains translation-invariance. Translation-invariance is particularly important in spectroscopy because a practical classifier would be one trained from one data set acquired from one spectrometer but potentially tested on another instrument with different calibration or different sessions where stability of the instrument is not perfectly controlled. Therefore, fully convolutional classifiers have the potential to remove some qualitative methodologies from spectroscopy interpretation and can be extended to additional classes (valences, elements, bonding states etc.) provided there is training data. These classifiers may be able to be applied to many other spectroscopic methods where discrimination between spectra is determined by peak shape, instances with high noise, or cases where calibration is difficult and unreliable over time.
## Methods
### Architecture – Fully Dense Network
The densely connected neural network receives the (300,1) length input spectra and passes it through two layers of 32 hidden units each. Each hidden unit uses a rectified linear unit (ReLU) activation function. Every layer uses batch-normalization after activation and has a 50% neuronal dropout in between layers to limit over-fitting.
To re-normalize the data between hidden layers, a per-batch normalization is performed to mean center the data and to standardize the data to unit variance. The data is originally preprocessed to be normalized before model training. However, after subsequent affine transformations and activations, this will no longer be the case. This phenomenon is called the internal covariate shift and this batch normalization procedure developed by Ioffe and Szegedy is a commonly used method for normalization between operations in many popular neural network architectures11.
Overfitting occurs when the network has learned to memorize the training distribution and no longer can generalize to the validation or test distributions. One commonly used technique to limit large amounts of overfitting is to use the stochastic method of neuron dropout developed by Srivastava et al.10. Dropout is used to prevent overfitting by randomly selecting a percentage of weights per batch between the weights connecting the feature extractor neurons and the classifier neurons and setting them equal to zero. Dropout is used so that, during model training, the optimum parameter configuration will be one that does not heavily rely on a small subset of parameters.
The last hidden layer is connected to three neurons (one for each class of Mn) and activated via a softmax function (i.e. multinomial logistic regression) to draw a decision boundary to determine which class is likely present.
The predicted class probability $$p({y}_{k}|z,\theta )$$ of the correct label yk given a set of parameters θ with an input spectrum feature vector z is shown below. This softmax function is the final layer in all of the neural architectures studied here and is a commonly used classification operation used in neural networks.
$$p({y}_{k}|z,\theta )=\frac{{e}^{{z}_{j}}}{{\sum }_{k=1}^{K}{e}^{{z}_{k}}}$$
### Architecture – Convolutional and Dense Network
The second network can be considered to have two parts. The first part is used to perform automated feature extraction by using a convolutional neural network. The convolutional neural network is used to project the spectra onto a vector space that more clearly encodes discriminating differences between the classes. The second part is a densely connected neural network which uses these features to perform classification and to draw a decision boundary.
The feature extraction architecture is composed of 5 layers. Each feature extraction layer has first a 1D convolution, activation using a rectified linear unit (ReLU), followed by batch-normalization, then finally an average pooling layer (i.e. 2x averaged down-sampling).
Formally, the convolution in a convolutional graph network can be written as follows,
$${z}_{j}^{(L+1)}=(\sum _{k}{w}_{j}\,\ast \,{a}_{k}^{(L)})+{b}_{j}$$
where $${z}_{j}^{(L+1)}$$ is the output in the hidden layer (L + 1) at neuron index j of that layer, receiving inputs from previous layer neurons indexed by k. The convolution kernel (also known as a filter) wj is of fixed length and slides across the previous layer activations $${a}_{k}^{(L)}$$ at a stride of 1 to perform a sliding dot product. This is summed across all layer (L) neurons indexed by k before a bias term bj is added.
The input into each feature-extraction layer is convolved with a sliding (stride of 1) kernel of length 9, 7, 7, 5, or 3 for the 1st through 5th blocks respectively. The number of filters per block gradually expands from 2, 2, 4, 8 and 12 filters for the 1st through 5th blocks respectively. The premise behind architecture construction was to start with a large number of filters similar to VGG16 (which contains millions of free parameters), and shrink the network until validation and test set accuracies started significantly falling.
The output shape after the 5th feature extraction layer is 12 activations of size (6,1). These features are then flattened to (72,1) before being put into a dense neural network to perform classification.
### Architecture – Fully Convolutional Neural Network
The most successful architecture studied here was the fully convolutional architecture which avoids using dense layers. It is comprised of the same feature extraction architecture as the network above to produce 12 vectors of size (6,1) as features. To perform classification on these features, a classification architecture was created similar to the dense-layer free architecture which have been used in recent architectures such as MobileNet, and SqueezeNet14,15. This style of classification architecture consists of: dropout, convolution (without non-linear activation), global average pooling and activation via softmax. A dropout rate of 0.8 (determined via 10-fold cross validation) was used to enforce a strict information bottleneck.
Three convolution filters (one per class) with a stride of 1 and kernel size of 1 are used to create a shape of 3 vectors of size (6,1). Each of the three vectors is then averaged into 1 number (i.e. global average pooling) to produce 1 number per filter. This value is then passed into the logistic regressor (softmax) function to convert it into predicted class probabilities to perform classification.
### Computational Hardware and Software
Model training occurred on a GPU (GTX 1060, 6 GB video card RAM) and it took 3 seconds per epoch for 100 000 data augmented training spectra to be passed forward and backwards through the graph in batches of 2048. All neural network training was performed using Tensorflow and Keras in Python. A full description of dependencies and the training and testing scripts can be found in this repository on GitHub (https://github.com/MichaelChatzidakis/Mn_Classifier_CNNs). Data is available upon request.
### Preprocessing
Every spectrum image was unzipped and appended to the same array per valence state using the Python library HyperSpy to open the Digital Micrograph .dm3 files. The EELS spectra were then spectrally cropped to a length of 300 histogram bins (dispersion of 0.1 eV/channel) near the Mn L2,3 core-loss edges so that only Mn L2,3 edge data was included. There were a total of 1273, 1342 and 1687 spectra for Mn2+, Mn3+ and Mn4+ respectively.
Dark-Field images corresponding to the Spectrum Images can be found in the Supporting Information. Each image was taken over a Mn oxide particle, so part of most images are solely the supporting substrate and not just the Mn oxide in question. To remove pixels containing only substrate signals from the coalesced dataset, k-Means clustering was performed on each class of the data (Mn2+, Mn3+ and Mn4+). By qualitatively inspecting the cluster center, it is apparent which cluster center belongs to the Mn L2,3 core-loss edges and which cluster center belongs to the substrate (where no Mn L2,3 edge is present). The cluster centers are shown in the Supporting Information.
In electron energy loss spectroscopy, background removal is required to isolate only one core-loss ionization edge. It is a common practice to fit a power-law spectrum to the points before the edge in question. To remove the background prior to the onset of the core-loss edge, a generalized power-law was least-squares fitted to the indices (50 bins) prior to the edge onset and subtracted from each spectrum. Next, each spectrum was individually normalized between 0.0 and 1.0 and mean-subtracted. The choice of these preprocessing steps was empirical and determined via cross-validation.
There is a sizeable portion of low-quality data present in the dataset as a result of trying to get a wide range of varying thickness and crystallographic orientations in the samples during acquisition. If samples are too thick then they are not electron transparent and no ionization edge is visible. Typically, only samples under 100 nm are thin enough for the beam to transmit through. Many of the Mn oxide crystals which we acquired data from are “wedge” shaped, so only a portion of the collected data is actually thin and electron transparent. To remove these low-quality spectra, an iterative k-Means clustering approach was used. Each class of Mn was clustered with two or three cluster centers and, through qualitative inspection of the cluster center mean, thick and thin samples could be discriminated by simply seeing if the peaks were broadened compared to the literature.
### Sample Preparation and Microscope Parameters
All samples were acquired using a monochromated FEI Titan 80–300 transmission electron microscope (TEM) operating at an accelerating voltage of 80 keV. The FWHM of the zero-loss peak was ~0.2 eV. Reference Mn oxide samples above 99% purity (obtained from Sigma-Aldrich) were crushed between SiO2 glass slides to produce a fine nanoparticulate powder. A holey carbon TEM grid was tapped onto the glass slide to adsorb fine particulates. In STEM mode, spectrum-images were collected over a 2D area on the edges of the nanoparticulate Mn oxides to capture the thinnest areas. 10, 10 and 15 spectrum images of each of the MnO, Mn2O3 and MnO2 powder samples were acquired for a total of 1604, 1512 and 1863 individual spectra respectively. Spectrum images were acquired on a variety of thicknesses and crystallographic orientations. See the Supporting Information for annular dark-field images of a selection of spectrum images.
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## Acknowledgements
This work is supported by NSERC under the Discovery Grants program and the Canada Foundation for Innovation. The experimental data was acquired at the Canadian Centre for Electron Microscopy, a National Facility supported by NSERC, the Canada Foundation for Innovation under the Major Science Initiative program and McMaster University.
## Author information
Authors
### Contributions
M.C. and G.A.B. discussed the concepts. M.C. developed the idea and built the model, collected the data, and carried out the detailed processing. M.C. and G.A.B. wrote the manuscript.
### Corresponding author
Correspondence to G. A. Botton.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
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Chatzidakis, M., Botton, G.A. Towards calibration-invariant spectroscopy using deep learning. Sci Rep 9, 2126 (2019). https://doi.org/10.1038/s41598-019-38482-1
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International Journal of Food Science & Technology (2020) | 2021-02-27 01:37:35 | {"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": 0, "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": 0.6784366965293884, "perplexity": 1980.0683515462692}, "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-2021-10/segments/1614178358033.38/warc/CC-MAIN-20210226234926-20210227024926-00298.warc.gz"} |
https://www.sparrho.com/item/decoherence-of-spin-echoes/94ed8a/ | # Decoherence of spin echoes
Research paper by Tomaz Prosen, Thomas H. Seligman
Indexed on: 21 Jan '02Published on: 21 Jan '02Published in: Nonlinear Sciences - Chaotic Dynamics
#### Abstract
We define a quantity, the so-called purity fidelity, which measures the rate of dynamical irreversibility due to decoherence, observed e.g in echo experiments, in the presence of an arbitrary small perturbation of the total (system + environment) Hamiltonian. We derive a linear response formula for the purity fidelity in terms of integrated time correlation functions of the perturbation. Our relation predicts, similarly to the case of fidelity decay, faster decay of purity fidelity the slower decay of time correlations is. In particular, we find exponential decay in quantum mixing regime and faster, initially quadratic and later typically gaussian decay in the regime of non-ergodic, e.g. integrable quantum dynamics. We illustrate our approach by an analytical calculation and numerical experiments in the Ising spin 1/2 chain kicked with tilted homogeneous magnetic field where part of the chain is interpreted as a system under observation and part as an environment. | 2021-10-19 16:01:17 | {"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": 0.8242588043212891, "perplexity": 1571.1088651624727}, "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/1634323585270.40/warc/CC-MAIN-20211019140046-20211019170046-00419.warc.gz"} |
https://mathoverflow.net/questions/159525/probability-that-convex-hull-of-multivariate-gaussian-sample-contains-a-given-po | Probability that convex hull of multivariate Gaussian sample contains a given point
I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls within the convex hull of the sample (N > d).
I can't find any result concerning this problem, apart from this answer which covers only a specific point in $R^d$ (the mean). Is anybody aware of any work on this topic?
My final aim is finding the point $y$ at which the $P(y \in \text{ConHull}(X_1, \dots, X_N))$ is maximal. Thanks for any suggestion.
$y=\mu$ will maximize the probability of $y$ being in the convex hull of the sample, since the level sets of the normal distribution are ellipsoids centered at $\mu$.
• thanks for your answer. I agree that the probability of being in the convex hull is maximal at the $\mu$, but I have no idea about how to prove it. Do think that your statement regarding the level sets can be the base for a formal argument? Mar 9 '14 at 16:44
• A good lemma could be that if a sample leads to $\mu$ not being in the convex hull, translate the sample (maybe add $\mu-c$ where $c$ is the centroid of the sample) and argue that the joint pdf $f_{X_1,\ldots,X_n}$ thereby increases. Mar 9 '14 at 17:04
This is a partial answer, too long for a comment.
Asymptotically, the convex hull converges (after rescaling) to an ellipsoid and thus the inclusion probability tends to $1$ for any point in $R^d$ (as long as $\Sigma$ is non degenerate). So I assume you do not ask about asymptotics as $N\to \infty$.
Also, by performing a linear transformation you can always put yourself in the situation where $\Sigma=I$, so I will assume in what follows that this is the case.
A general answer for d=2 is given by Jewell and Romano (J. Appl. Prob 19 (1982) pp. 546-561); They show that the probability in question is equal to the coverage problem of the unit circle by random arcs of length $\pi$ whose midpoints are taken from a distribution $G$ that can be computed from your initial data: the midpoint is distributed according to the marginal of $\tan^{-1}(y-y_0)/(x-x_0)$ where $(x_0,y_0)$ is the point that you are trying to cover. In the case of $\Sigma=I$ and $(x_0,y_0)=0$, this gives the uniform distribution which is optimal for the arc covering problem.
I don't know about exact expressions for higher dimension, maybe you can find relevant stuff in http://arxiv.org/pdf/0912.0631.pdf.
Here are some exact answers for the one-dimensional case $(d=1)$: $$N=2\negthinspace:\ \frac{1}{2}(1-a^2)$$
$$N=3\negthinspace:\ \frac{3}{4}(1-a^2)$$
$$N=4\negthinspace:\ \frac{1}{8}(1-a^2)(7+a^2)$$
$$N=5\negthinspace:\ \frac{5}{16}(1-a^2)(3+a^2)$$
$$N=6\negthinspace:\ \frac{1}{32}(1-a^2)(31+16a^2+a^4)$$
where $$a=\text{erf}\left(\frac{x-\mu}{\sqrt{2}\sigma}\right)$$
I got these using Mathematica, with Expectation[ Boole[Min[a, b] < x < Max[a, b]], {a [Distributed] NormalDistribution[], b [Distributed] NormalDistribution[]}] // FullSimplify and obvious variants.
Perhaps someone else will see a pattern in the results or extend them to higher dimensions.
Update: Exact formulas for higher dimensions do not look promising.
Consider the toy question: what is the probability that $(1/2, 3)$ lies in the convex hull of $(0,1)$, $(1,2)$, and $(a,b)$, where $a$ and $b$ are both normally distributed and independent? The answer is
which Mathematica does not simplify further. The answer to the original question with $N=3, d=2$ would require four more integrals beyond that.
First of all, represent $$X_i=\Sigma^{1/2} Y_i + \mu$$, where $$Y_1, \ldots, Y_N$$ are i.i.d. standard Gaussian vectors in $$R^d$$. Then, assuming that $$\Sigma$$ is non-denegerate, we get $$P(y \in \text{Conv}(X_1, \ldots, X_N)) = P(\Sigma^{-1/2} (y-\mu) \in \text{Conv}(Y_1, \ldots, Y_N)) = f(\| \Sigma^{-1/2} (y-\mu)\|),$$ where $$f$$ is a positive function on $$[0,\infty)$$.
This function is computed explicitly by Kabluchko and Zaporozhets arxiv.org/abs/1704.04968. I believe its maximum is at $$0$$. | 2021-10-20 23:34:31 | {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8931694626808167, "perplexity": 153.14712394671133}, "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-2021-43/segments/1634323585353.52/warc/CC-MAIN-20211020214358-20211021004358-00436.warc.gz"} |
https://stackoverflow.com/questions/6229274/how-doyou-amend-the-last-commit-message-in-git | # How doyou Amend the last commit message in git? [duplicate]
I am using git-svn and I know that svn does not support changing commits
if I haven't yet run git svn dcommit can i still change the last commit message, i left something out of it.
If so, how do you amend the last commit message?
EDIT:
i figured out i can do git commit --amend but is there any problem doing this using svn backend?
## marked as duplicate by Bhargav Rao♦Aug 17 at 3:47
• This question has been answered in another stackoverflow post – JoshuaRogers Jun 3 '11 at 15:41
• is there any problem doing that with git-svn. My guess says no, since i haven't dcommit-ed it yet, but I'm unsure how it handles history with that. – loosecannon Jun 3 '11 at 15:44
• As long as you haven't dcommitted yet, there shouldn't be any problem. – JoshuaRogers Jun 3 '11 at 15:46
• @JoshuaRogers: I would accept that as an answer. – loosecannon Jun 3 '11 at 15:49
• No. When you git svn dcommit, the svn bridge will take all of the changes that have been made since you have last dcommitted and post them back. As long as you have not already dcommitted, svn will be unaware that you have changed anything. You can use git as normal. – JoshuaRogers Jun 3 '11 at 18:32 | 2019-10-16 05:58:05 | {"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": 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": 0.1955087035894394, "perplexity": 2271.5348825058704}, "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/1570986664662.15/warc/CC-MAIN-20191016041344-20191016064844-00134.warc.gz"} |
https://quiz.techlanda.com/2020/01/which-of-following-characters-are-most.html | # Quiz
## Which of the following characters are the MOST efficient to use for a comment line when writing DOS batch files?
### Which of the following characters are the MOST efficient to use for a comment line when writing DOS batch files?
• **
• ::
• REM
• ##
#### EXPLANATION
:: is essentially a blank label that can never be jumped to, whereas REM is an actual command that just does nothing. In neither case (at least on Windows 7) does the presence of redirection operators cause a problem.
However, :: is known to misbehave in blocks under certain circumstances, being parsed not as a label but as some sort of drive letter. I'm a little fuzzy on where exactly but that alone is enough to make me use REM exclusively. It's the documented and supported way to embed comments in batch files whereas :: is merely an artifact of a particular implementation.
Here is an example where :: produces a problem in a FOR loop.
This example will not work in a file called test.bat on your desktop:
@echo off
for /F "delims=" %%A in ('type C:\Users\%username%\Desktop\test.bat') do (
)
pause
While this example will work as a comment correctly:
@echo off
for /F "delims=" %%A in ('type C:\Users\%username%\Desktop\test.bat') do (
The problem appears to be when trying to redirect output into a file. My best guess is that it is interpreting :: as an escaped label called :echo. | 2021-04-22 20:42:09 | {"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": 0.6541769504547119, "perplexity": 2413.961314833421}, "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-2021-17/segments/1618039604430.92/warc/CC-MAIN-20210422191215-20210422221215-00636.warc.gz"} |
https://www.semanticscholar.org/paper/Exact-factorizations-and-extensions-of-fusion-Gelaki/dfb3e89b4e361b74a065538880be1593ff1d62f7 | # Exact factorizations and extensions of fusion categories
@article{Gelaki2016ExactFA,
title={Exact factorizations and extensions of fusion categories},
author={Shlomo Gelaki},
journal={arXiv: Quantum Algebra},
year={2016}
}
• S. Gelaki
• Published 4 March 2016
• Mathematics
• arXiv: Quantum Algebra
14 Citations
### A class of prime fusion categories of dimension $2^N$
• Mathematics
• 2019
We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, \zeta}$, where $N \geq 1$ is a natural number and $\zeta$ is a $2^N$th root of unity, that we call $N$-Ising fusion
### Hopf Algebras which Factorize through the Taft Algebra Tm2(q) and the Group Hopf Algebra K[Cn]
We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$: they are
### Algebraic structures in group-theoretical fusion categories
• Mathematics
• 2020
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable,
### Extensions of tensor categories by finite group fusion categories
• S. Natale
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2019
Abstract We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual
### Exact factorizations and extensions of finite tensor categories
• Mathematics
• 2022
We extend [G1] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A ,C ⊂ B, and relate exact
### Subalgebras of etale algebras and fusion subcategories
. In [7, Rem. 3.4] the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided
### The factorization problem for Jordan algebras: applications
• Mathematics
Collectanea Mathematica
• 2022
. We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products
### Slightly trivial extensions of a fusion category
We introduce and study the notion of slightly trivial extensions of a fusion category which can be viewed as the first level of complexity of extensions. We also provide two examples of slightly
## References
SHOWING 1-10 OF 19 REFERENCES
### Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories
Abstract.Given an action of a finite group G on a fusion category $${\mathcal{C}}$$ we give a criterion for the category of G-equivariant objects in $${\mathcal{C}}$$ to be group-theoretical, i.e.,
### Central exact sequences of tensor categories, equivariantization and applications
• Mathematics
• 2011
We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization
### Classifying complements for groups. Applications
• Mathematics
• 2012
Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the
### Classifying Bicrossed Products of Hopf Algebras
• Mathematics
• 2014
Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object ${\mathcal H}^{2} (A, H)$.
### Exact sequences of tensor categories
• Mathematics
• 2010
We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as
### On fusion categories
• Mathematics
• 2002
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show
### Module categories over equivariantized tensor categories
• Mathematics
• 2014
For a finite tensor category $\mathcal C$ and a Hopf monad $T:\mathcal C\to \mathcal C$ satisfying certain conditions we describe exact indecomposable left $\mathcal C^T$-module categories in terms | 2022-12-02 13:51:50 | {"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": 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": 0.8645514845848083, "perplexity": 989.6332470413905}, "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/1669446710902.80/warc/CC-MAIN-20221202114800-20221202144800-00572.warc.gz"} |
http://math.stackexchange.com/questions/335914/is-there-efficient-way-of-finding-last-number-in-following-sequence | # Is there efficient way of finding last number in following sequence
"Imagine the sequence lying on a circle. Take every second number in the sequence. Continue the process until you finish"
Is there efficient way of finding last number in following sequence :
we have numbers $1,2,...n$, we delete $2,4,...,$ and start it again so
$n=2$ gives $2$
$n=10$ gives $5$, because : $2,4,6,8,10,3,7,1,9,5$
$n=25$ gives $19 2,4,6,8,10,12,14,16,18,20,22,24,1,5,9,13,17,21,25,7,15,23,11,3,19$
is there any short way to for calculationg last number for given $n$ ? Ive some clues when I look at $n=2^k$
edition : http://mathworld.wolfram.com/JosephusProblem.html really good article
and here is solution :
http://oeis.org/A032434
-
Could you explain a bit more about what exactly the algorithm does? It is not clear to me. – Tobias Kildetoft Mar 20 '13 at 15:19
Imagine the sequence lying on a circle. Start from 2, Take every second number in the sequence and delete them at each step. Continue the process until you finish. – muzzlator Mar 20 '13 at 15:21
I feel kind of bad for saying "delete them at each step" after reading the anecdote – muzzlator Mar 20 '13 at 15:26 | 2014-09-22 14:44:26 | {"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": 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": 0.7067968845367432, "perplexity": 522.4538336895134}, "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-2014-41/segments/1410657137056.60/warc/CC-MAIN-20140914011217-00280-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://brilliant.org/problems/kinematicswork-both-fine/ | # Kinematics? Work? Both fine
Level pending
A ball of mass 2 kg is projected with a speed of 20m/s from the top of a tower of height 20mas shown below. Determine the speed of the ball when the ball is at a vertical distance of 10m below the point of projection.
× | 2017-05-27 17:49:58 | {"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": 0.8057469129562378, "perplexity": 333.4396742074097}, "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-2017-22/segments/1495463608984.81/warc/CC-MAIN-20170527171852-20170527191852-00046.warc.gz"} |
http://tex.stackexchange.com/questions/185091/tikz-arrow-looking-for-style | # TikZ, arrow, looking for ++++++> style
I'm wondering if it's possible to draw, by \tikzstyle, this style of arrow : +++++>. Thanks, Simon
-
Welcome to TeX.SX! Could you provide more information on what + symbols should represent in paths? – Claudio Fiandrino Jun 16 '14 at 10:47
Is [decorate=crosses] close enough? If so you can maybe obtain plusses using rotation... – Bordaigorl Jun 16 '14 at 10:50
@silama, that is what Bordaigorl suggests with his comment. – zeroth Jun 16 '14 at 10:55
Maybe something like decoration={markings, mark=between positions 0 and .9 step 4pt with {\draw[-] (2pt,-2pt) -- (2pt,2pt);} } ? – Bordaigorl Jun 16 '14 at 11:03
have you included the decorations library? \usetikzlibrary{decorations.shapes} – Bordaigorl Jun 16 '14 at 11:09
Here is an idea: use dashed, and define a decoration which draws the vertical line which crosses each dash to produce a plus:
\usetikzlibrary{decorations.markings}
\tikzset{
pluses/.style={
dashed,
decoration={markings,
mark=between positions 1.5pt and 1 step 6pt with {
\draw[-] (0,1.5pt) -- (0,-1.5pt);
}
},
postaction=decorate,
}
}
\begin{tikzpicture}
\draw[very thin, pluses, ->] (0,0) to[bend left=45] (2,2);
\draw[very thick, red, pluses, ->] (-1,1) -- (2,0);
\end{tikzpicture}
Result:
-
It works very well. Thanks everybody, thank you JLDiaz. – silama Jun 16 '14 at 11:18
@silama: you can even define a ++++> style for this; according to JLDiaz's answer you need: \tikzset{++++>/.style={pluses,->}}, then \draw[very thick, red, ++++>].... – Claudio Fiandrino Jun 16 '14 at 12:10 | 2016-06-25 10:29:18 | {"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": 0.9414762258529663, "perplexity": 6167.033806581982}, "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-2016-26/segments/1466783393093.59/warc/CC-MAIN-20160624154953-00198-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://starlink.eao.hawaii.edu/docs/sun139.htx/sun139se11.html | ### 11 Background processing
The easiest way to do CCDPACK processing in the background is to produce a file just containing the CCDPACK commands you want performed3. The content of such files is made much simpler if use of CCDPACK’s image list accessing facility is made. The best way to do this is to get your data files organised. This organisation can be performed in several different ways, I prefer the first method…
• Use a naming scheme which allows differentiation between the data types.
• Organise all your files into related subdirectories (as in many of the previous examples), i.e. put all your bias frames into a subdirectory, put all your flatfield and target data into colour related directories etc.
• Make up ASCII lists of all the names of the different file types (i.e. use an editor to create say a list of the names of your bias frames, a list of your R flatfields, R data etc.).
• None of the above, just supply all the relevant names on the command line, or in response to the prompts.
The command file which controls CCDPACK can be written as if responding to the C shell. Examples of such CCDPACK command files are shown in §7.2.9.
The next step after creating your command file is to run:
CCDFORK saves the current environment and writes a script which when activated restores it, ensures that CCDPACK is started and executes the commands in the command file. The point in saving the current environment is that any global or current values which you have set (by using CCDSETUP) are restored to the job, without interference with any other processes.
CCDFORK has three parameters, the first is the name of the input script, the second the name of the output script (ccdpack_fork by default) the final is the name of the directory in which to save the current environment. If you do not supply a name for the last option then a unique one will be generated as a subdirectory of the parent of the directory that holds the environment ($ADAM_USER or $HOME/adam). Using this command results in a script file which may be directly executed or forked (hopefully at nice priority) into the background.
Since the execution environment of the current process is saved when CCDFORK is run any previous CCDPACK global and current values, which are in force, will be restored to the background process. Thus one labour saving strategy would be to set the global parameters for a CCD device using CCDSETUP interactively, this command does not then need to be repeated in the background job. So the chances of making a mistake in this crucial stage are reduced. A typical preparation sequence is:
% ccdsetup etc.
% edit ccdpack_back
<make modifications to script>
% ccdfork ccdpack_back
% nice ccdpack_fork >&ccdpack_fork.log &
3This section does not apply to IRAF/CL users | 2021-10-21 14:51:51 | {"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": 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": 0.5102903842926025, "perplexity": 1178.6983558103268}, "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/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00178.warc.gz"} |
http://mathhelpforum.com/statistics/161807-stats-problem.html | 1. ## stats problem
Hey im having some problems with probability and here are the two homework questions.
1)What is the probability that you will pass a test and get at least 4 questions correct by randomly guessing on each question of a 6 question quiz multiple choice quiz with 5 choices for each question.
2) Every year at least 5 million people die of tobacco related causes, assume that the distribution is normal with a population mean of 5 million and a standard deviation of 2 million...find the probability that more than 4 million people will die of tobacco related causes and find the 75th percentile of the distribution of tobacco related deaths.
any help is appreciated!
2. for no.1 use binomial distribution.
n = 6
$p = \frac{1}{5}$
and then find
$P(X \geq 4)$
3. thanks... how about number 2?
4. Originally Posted by coheedfan1990 | 2017-10-18 06:17:20 | {"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": 2, "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": 0.6388895511627197, "perplexity": 821.5915698392479}, "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/1508187822747.22/warc/CC-MAIN-20171018051631-20171018071631-00075.warc.gz"} |
https://wiki.booleantrader.com/index.php?title=Exponential_Moving_Average | # Exponential Moving Average
## Description
The exponential moving average, or exponential smoothing function, works by calculating each bar as a portion of the current input and a portion of the previous exponential moving average.
It takes one parameter, the period n, a positive integer. Larger values for n will have a greater smoothing effect on the input data but will also create more lag.[1]
## Syntax
${\displaystyle fx=BT\_I\_EMA(Input1,Period)}$ | 2019-05-23 08:09:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.5326767563819885, "perplexity": 1168.6510124242166}, "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/1558232257156.50/warc/CC-MAIN-20190523063645-20190523085645-00402.warc.gz"} |
https://cstheory.stackexchange.com/questions/25260/rl-l-progress-since-2006 | # $RL=L$ Progress Since 2006
Reingold, Trevisan, and Vadhan's breakthrough 2006 paper (http://dl.acm.org/citation.cfm?id=1132583) reduced the problem of showing $RL=L$ to producing pseudorandom walks on regular digraphs that are not consistently labeled.
Has any further progress on this problem been made since then?
• One line of work has shown how to improve the analysis of the classic INW PRG for the special cases of fooling regular and permutation branching programs. The seed length is only $$\widetilde{O}(\log n)$$ or $$O(\log n)$$ in some cases. [LRTV09] [BV10] [De11] [KNP11] [Ste12] [BRRY14] [HPV21]
• Another line of work has developed a PRG framework based on iterated pseudorandom restrictions. This framework has been used to fool various classes of functions that can be computed by "arbitrary-order" read-once branching programs. Again, in some cases the seed length is $$\widetilde{O}(\log n)$$ or even $$O(\log n)$$. [GMRTV12] [GLS12] [RSV13] [CSV15] [SVW17] [CHRT18] [HLV18] [FK18] [MRT19] [DHH19] [Lee19] [LV20] [DHH20] [DMRTV21] | 2021-08-01 14:23:20 | {"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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4436154067516327, "perplexity": 2969.769599909726}, "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-2021-31/segments/1627046154214.36/warc/CC-MAIN-20210801123745-20210801153745-00501.warc.gz"} |
http://wiki.eeros.org/getting_started/tutorials/controlsystem5 | # Real-Time Robotics Framework
### Sidebar
getting_started:tutorials:controlsystem5
# Control System with Remote Connection
In the EEROS library you will find a directory with examples. Open a shell in the build directory of your EEROS library and run examples/socket/eerosServer/socketServerExample. Open a second terminal and run examples/socket/standaloneClient/standaloneClient. As soon as the connection is established the socket data block periodically sends its input signal of type Vector4. The transmission speed is given by the period of the time domain the socket data block is assigned to. The stand alone client prints these values ten times per second.
At the same time the standalone block periodically transmit six double values, whose values increase over time. The socket data block receives these values and bundles them into its output signal of type Matrix<6,1,double>
Control system with socket data block and external standalone client
Another example shows how two independent EEROS applications can pass data among each other by using a socket data block each. Open a shell in the build directory of your EEROS library and run examples/socket/eerosServer/socketServerExample. Open a second terminal and run examples/socket/eerosClientExample. While the socket data block of the former creates a socket server the latter creates a socket client which connects to the server.
You can alter the data types passed among them by commenting / uncommenting the corresponding lines directly in the source code. | 2019-03-20 21:26:59 | {"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": 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": 0.1875670850276947, "perplexity": 2778.170716030523}, "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-2019-13/segments/1552912202471.4/warc/CC-MAIN-20190320210433-20190320232433-00540.warc.gz"} |
https://bookdown.org/MathiasHarrer/Doing_Meta_Analysis_in_R/rob-plots.html | # 15 Risk of Bias Plots
by Luke A. McGuinness
McGuinness, L. A. (2021). Risk of Bias Plots. In Harrer, M., Cuijpers, P., Furukawa, T.A., & Ebert, D.D., Doing Meta-Analysis with R: A Hands-On Guide (online version). https://bookdown.org/MathiasHarrer/Doing_Meta_Analysis_in_R/rob-plots.html.
I n this chapter, we will describe how to create risk of bias plots in R, using the {robvis} package.
## 15.1 Introduction
As part of a systematic review and meta-analysis, you may also want to examine the internal validity (risk of bias) of included studies using the relevant domain-based risk of bias assessment tool, and present the results of this assessment in a graphical format.
The Cochrane Handbook recommends two types of figure: a summary barplot figure showing the proportion of studies with a given risk of bias judgement within each domain, and a traffic light plot which presents the domain level judgments for each study.
However, the options available to researchers when creating these figures are limited. While RevMan has the functionality to create the plots, many researchers do not use it to conduct their systematic review and so copying the relevant data into the system is an inefficient solution.
Similarly, producing the graphs by hand, using software such as MS PowerPoint, is time consuming and means the figures have to manually updated if changes are needed. Additionally, journals usually require figures to be of publication quality (above ~300-400dpi), which can be hard to achieve when exporting the risk of bias figures from RevMan or creating them by hand.
To avoid all of this, you can now easily plot the risk of bias figures yourself within R Studio, using the {robvis} package (McGuinness and Higgins 2020; McGuinness 2019) which provides functions to convert a risk of bias assessment summary table into a summary plot or a traffic-light plot.
Assuming that you have already installed the {dmetar} package (see Chapter 2.3), load the {robvis} package using:
library(robvis)
### 15.1.2 Importing Your Risk of Bias Summary Table Data
To produce our plots, we first have to import the results of our risk of bias assessment from Excel into R. Please note that {robvis} expects certain facts about the data you provide it, so be sure to follow the guidance below when setting up your table in Excel:
1. The first column is labelled “Study” and contains the study identifier (e.g. Anthony et al, 2019)
2. The second-to-last column is labelled “Overall” and contains the overall risk-of-bias judgments
3. The last column is labelled “Weight” and contains some measure of study precision e.g. the weight assigned to each study in the meta-analysis, or if no meta-analysis was performed, the sample size of each study). See Chapter 4.1.1 for more details.
4. All other columns contain the results of the risk-of bias assessment for a specific domain.
To elaborate on the above guidance, consider as an example the ROB2 tool which has 5 domains. The resulting data set that {robvis} would expect for this tool would have 8 columns:
• Column 1: Study identifier
• Column 2-6: One RoB2 domain per column
• Column 7: Overall risk-of-bias judgments
• Column 8: Weight.
In Excel, this risk of bias summary table would look like this:
Column Names
For three of the four tool templates (ROB2, ROBINS-I, QUADAS-2), what you name the columns containing the domain-level judgments is not important, as the templates within robvis will relabel each domain with the correct tool-specific heading.
Once you have saved the table you created in Excel to the working directory as a comma-separated-file (e.g. “robdata.csv”), you can either read the file into R programmatically using the command below or via the “import assistant” method as described in Chapter 2.4.
my_rob_data <- read.csv("robdata.csv", header = TRUE)
### 15.1.3 Templates
{robvis} produces the risk of bias figures by using the data you provide to populate a template figure specific to the risk of bias assessment tool you used. At present, {robvis} contains templates for the following three tools:
• ROB2, the new Cochrane risk of bias tool for randomized controlled trials;
• ROBINS-I, the Risk of Bias In Non-randomized Studies - of Interventions tool;
• QUADAS-2, the Quality and Applicability of Diagnostic Accuracy Studies, Version 2.
{robvis} also contains a special generic template, labeled as ROB1. Designed for use with the original Cochrane risk of bias tool for randomized controlled trials, it can also be used to visualize the results of assessments performed with other domain-based tools not included in the list above. See Section XXXXXXXX for more information on the additional steps required when using this template.
### 15.1.4 Example Data Sets
The {robvis} package contains an example data set for each template outlined above. These are stored in the following objects:
• data_rob2: Example data for the ROB2 tool
• data_robins: Example data for the ROBINS-I tool
• data_quadas: Example data for the QUADAS-2 tool
• data_rob1: Example data for the RoB-1 tool.
You can explore these data sets using the glimpse function (see Chapter 2.5.1). For example, once you have loaded the package using library(robvis), viewing the ROBINS-I example data set can be achieved by running the following command:
glimpse(data_robins)
## Rows: 12
## Columns: 10
## $Study <fct> Study 1, Study 2, Study 3, Study 4, Study 5, Study 6, Study 7… ##$ D1 <fct> Critical, Moderate, Moderate, Low, Serious, Critical, Critica…
## $D2 <fct> Low, Low, Low, Low, Serious, Serious, Moderate, Moderate, Low… ##$ D3 <fct> Critical, Low, Moderate, Serious, Low, Moderate, Moderate, Lo…
## $D4 <fct> Critical, Critical, Critical, Critical, Low, Critical, Seriou… ##$ D5 <fct> Low, Low, Critical, Moderate, Moderate, Critical, Critical, L…
## $D6 <fct> Low, Moderate, Low, Low, Low, Moderate, Serious, Low, Serious… ##$ D7 <fct> Serious, Low, Serious, Critical, Moderate, Serious, Serious, …
## $Overall <fct> Critical, Low, Serious, Low, Serious, Serious, Moderate, Mode… ##$ Weight <dbl> 33.3333333, 33.3333333, 0.1428571, 9.0909091, 12.5000000, 25.…
These example data sets are used to create the plots presented through the remainder of this guide.
## 15.2 Summary Plots
### 15.2.1 Basics
Once we have successfully imported the risk of bias summary table into R, creating the risk of bias figures is quite straightforward.
To get started, a simple weighted summary bar plot using the ROB2 example data set (data_rob2) is created by running the following code:
rob_summary(data = data_rob2,
tool = "ROB2")
### 15.2.2 Modifying the Plot
The rob_summary function has the following parameters:
• data. A data frame containing summary (domain) level risk-of-bias assessments, with the first column containing the study details, the second column containing the first domain of your assessments, and the final column containing a weight to assign to each study. The function assumes that the data includes a column for overall risk-of-bias. For example, a ROB2.0 dataset would have 8 columns (1 for study details, 5 for domain level judgments, 1 for overall judgments, and 1 for weights, in that order).
• tool. The risk of bias assessment tool used. RoB2.0 ("ROB2"), "ROBINS-I", and "QUADAS-2" are currently supported.
• overall. An option to include an additional bar for overall risk-of-bias in the figure. Default is FALSE.
• weighted. An option to specify whether weights should be used in the bar plot. Default is TRUE, in line with current Cochrane Collaboration guidance.
• colour. An argument to specify the colour scheme for the plot. Default is "cochrane", which used the ubiquitous Cochrane colours, while a preset option for a colour-blind friendly palette is also available (colour = "colourblind").
• quiet. A logical option to quietly produce the plot without displaying it. Default is FALSE.
Examples of the functionality of each argument are described below.
#### 15.2.2.1 Tool
An argument to define the tool template you wish to use. In the example above, the ROB2 template is used. The two other primary templates - the ROBINS-I and QUADAS-2 templates - are demonstrated below:
rob_summary(data = data_robins,
tool = "ROBINS-I")
rob_summary(data = data_quadas,
tool = "QUADAS-2")
#### 15.2.2.2 Overall
By default, an additional bar representing the overall risk of bias judgments is not included in the plot. If you would like to include this, set overall = TRUE. For example:
rob_summary(data = data_rob2,
tool = "ROB2",
overall = TRUE)
#### 15.2.2.3 Weighted or Unweighted Bar Plots
By default, the bar plot is weighted by some measure of study precision, so that the bar plot shows the proportion of information rather than the proportion of studies that is at a particular risk of bias. This approach is in line with the Cochrane Handbook.
You can turn off this option by setting weighted = FALSE to create an unweighted bar plot. For example, compare the following two plots:
rob_summary(data = data_rob2,
tool = "ROB2")
rob_summary(data = data_rob2,
tool = "ROB2",
weighted = FALSE)
#### 15.2.2.4 Colour Scheme
British English Spelling
Please note the non-US English spelling of colour!
The colour argument of both plotting functions allows users to select from two predefined colour schemes, "cochrane" (default) or "colourblind", or to define their own palette by providing a vector of hex codes. For example, to use the predefined "colourblind" palette:
rob_summary(data = data_rob2,
tool = "ROB2",
colour = "colourblind")
And to define your own colour scheme:
rob_summary(data = data_rob2,
tool = "ROB2",
colour = c("#f442c8","#bef441","#000000"))
When defining your own colour scheme, you must ensure that the number of discrete judgments (e.g. “Low”, “Moderate”, “High”, “Critical”) and the number of colours specified are the same. Additionally, colours must be specified in order of ascending risk-of-bias (e.g. “Low” to “Critical”), with the first hex corresponding to “Low” risk of bias.
## 15.3 Traffic Light Plots
Frequently, researchers will want to present the risk of bias in each domain for each study assessed. The resulting plots are commonly called traffic light plots, and can be produced with {robvis} via the rob_traffic_light function.
### 15.3.1 Basics
To get started, a traffic light plot using the ROB2 example dataset (data_rob2) is created by running the following code:
rob_traffic_light(data = data_rob2,
tool = "ROB2")
### 15.3.2 Modifying the Plot
The rob_summary function has the following parameters:
• data. A data frame containing summary (domain) level risk-of-bias assessments, with the first column containing the study details, the second column containing the first domain of your assessments, and the final column containing a weight to assign to each study. The function assumes that the data includes a column for overall risk-of-bias. For example, a ROB2.0 data set would have 8 columns (1 for study details, 5 for domain level judgments, 1 for overall judgments, and 1 for weights, in that order).
• tool. The risk of bias assessment tool used. RoB2.0 ("ROB2"), "ROBINS-I", and "QUADAS-2" are currently supported.
• colour. An argument to specify the colour scheme for the plot. Default is "cochrane" which used the ubiquitous Cochrane colours, while a preset option for a colour-blind friendly palette is also available ("colourblind").
• psize. An option to change the size of the “traffic light” points. Default is 20.
• quiet. A logical option to quietly produce the plot without displaying it. Default is FALSE.
##### 15.3.2.0.1 Tool
An argument to define the tool template you wish to use. The ROB2 template is demonstrated and the two other primary templates - the ROBINS-I and QUADAS-2 templates - are displayed below:
rob_traffic_light(data = data_robins,
tool = "ROBINS-I")
rob_traffic_light(data = data_quadas,
tool = "QUADAS-2")
##### 15.3.2.0.2 Colour Scheme
British English Spelling
Please note the non-US English spelling of colour!
The colour argument of both plotting functions allows users to select from two predefined colour schemes, "cochrane" (default) or "colourblind", or to define their own palette by providing a vector of hex codes.
For example, to use the predefined "colourblind" palette:
rob_traffic_light(data = data_rob2,
tool = "ROB2",
colour = "colourblind")
And to define your own colour scheme:
rob_traffic_light(data = data_rob2,
tool = "ROB2",
colour = c("#f442c8","#bef441","#000000"))
When defining your own colour scheme, you must ensure that the number of discrete judgments (e.g. “Low”, “Moderate”, “High”, “Critical”) and the number of colours specified are the same. Additionally, colours must be specified in order of ascending risk-of-bias (e.g. “Low” to “Critical”), with the first hex corresponding to “Low” risk of bias.
##### 15.3.2.0.3 Point Size
Occasionally, when a large number of risk of bias assessment have been performed, the resulting traffic light plot may be too long to be useful. Users can address this by modifying the psize argument of the rob_traffic_light function to a smaller number (default is 20). For example:
# Create bigger dataset (18 studies)
new_rob2_data <- rbind(data_rob2, data_rob2)
new_rob2_data$Study <- paste("Study", seq(1:length(new_rob2_data$Study)))
# Plot bigger dataset, reducing the psize argument from 20 to 8
rob_traffic_light(data = new_rob2_data,
tool = "ROB2",
psize = 8)
##### 15.3.2.0.4 The “ROB1” Generic Template
$\tag*{\blacksquare}$ | 2022-08-12 09:59:18 | {"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": 2, "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": 0.386140912771225, "perplexity": 4868.262918485152}, "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/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00673.warc.gz"} |
https://proofwiki.org/wiki/Negative_Infinity_is_Minimal | # Negative Infinity is Minimal
Let $\left({\overline \R, \le}\right)$ be the extended real numbers with the usual ordering.
Then $-\infty$ is a minimal element of $\overline \R$.
$\blacksquare$ | 2019-11-21 11:54:26 | {"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": 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": 0.983025312423706, "perplexity": 342.7459072494139}, "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-2019-47/segments/1573496670770.21/warc/CC-MAIN-20191121101711-20191121125711-00243.warc.gz"} |
https://physics.stackexchange.com/questions/460647/how-do-i-connect-a-four-way-key-in-a-potentiometer | # How do I connect a four way key in a potentiometer?
How do I connect (which wire to which terminal) a four way key to a Potentiometer to compare the EMF of two cells. Circuit diagram and a four wayway key is shown in the pictures.
When you want to measure the deflections on the galvanometer produced by $$E_1$$, remove the middle knob of the path along $$E_2$$. By removing a key knob, you are leaving it in an open circuit, and hence no current passes through. If you want to measure the deflections due to $$E_2$$, then remove the middle knob in $$E_1$$. | 2019-09-23 00:52:56 | {"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": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.449788898229599, "perplexity": 752.4981474493561}, "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/1568514575844.94/warc/CC-MAIN-20190923002147-20190923024147-00350.warc.gz"} |
https://math.stackexchange.com/questions/747789/how-to-tell-if-algebraic-set-is-a-variety | # How to tell if algebraic set is a variety?
I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite collection of polynomials $f_1, \dots, f_m$ in $n$ variables without any more restrictions. Then they define an algebraic variety as an algebraic set where $(f_1, \dots, f_m)$ is a prime ideal in $k[x_1, \dots, x_n]$.
My question has two parts:
1. I'm guessing the distinction is like any other area of math where you try to break things up into the "irreducible" case and deduce the general case from patching those together. How does that happen with varieties and algebraic sets? Is it correct to conclude that every algebraic set is somehow built from algebraic varieties since the ideal $(f_1, \dots, f_m)$ is contained in some prime (maximal) ideal?
2. How can one tell whether or not an algebraic set is a variety intuitively? I know formally you'd have to prove $(f_1, \dots, f_m)$ is prime (or perhaps there are some useful theorems out there?), but many times in texts the author simply states something is a variety without any justification. Is there a way to sort of "eye-ball" varieties in the sense that there are tell-tale signs of algebraic sets which are not varieties?
Perhaps this is all a moot discussion since modern algebraic geometry is done with schemes and this is perhaps a petty discussion in light of that, but nonetheless, I'd like to understand the foundations before pursuing that.
Thanks.
• «Modern algebraic geometry is done with schemes» that is a rather misguided assessment, really. – Mariano Suárez-Álvarez Apr 10 '14 at 4:32
• @Mariano Yes, perhaps it's a bit misguided, but would you agree that schemes are studied moreso than algebraic varieties today? Have there been further generalizations that now occupy the center of geometric study? – Supersingularity Apr 10 '14 at 4:45
• Schemes are studied in algebric geometry, mostly, as a way to study varieties. In fact, in lots of contexts varieties are defined to be schemes of a certain type, because they are equivalent objects; schemes are just varieties with certain information which is implicit in the variety made explicit as part of the structure, pretty much nothing else. – Mariano Suárez-Álvarez Apr 10 '14 at 4:46
• This is done in pretty much every sensible exposition of modern algebraic geometry! For example, Hartshorne's book, Chapter II, section 4. – Mariano Suárez-Álvarez Apr 10 '14 at 4:50
• Scheme theory is now the universal language of algebraic geometry. Although it is still a great tool for studying classical algebraic varieties, it has an infinitely wider scope and permits one to attack problems which couldn't be touched by previous methods. An impressive lot of Fields medals were awarded to those able to use these tools to solve extremely hard problems: Deligne, Faltings, Grothendieck, Lafforgue, Ngô, Voevodski,... – Georges Elencwajg Apr 10 '14 at 8:50
It is true that every algebraic set is a finite union of algebraic varieties (irreducible algebraic sets), and this union is unique up to reordering. These irreducible pieces of an algebraic set are called the irreducible components. This all follows from the fact that a polynomial ring over a field is Noetherian, so that an algebraic set with the Zariski topology is a Noetherian topological space.
As an example, I always think of the algebraic set defined by the ideal $(xz,yz),$ which is not prime. The real picture of this algebraic set is a line through a plane, and these two objects are exactly the irreducible components of the algebraic set. Here is the picture:
$\hspace{2.2in}$
The components are defined by the prime ideals $(z)$ and $(x,y)$ which are the two minimal prime ideals containing $(xz,yz)$. This may be the eyeball test you desire, as most people would look at this set and say it is made of two parts. In general, the irreducible components of an algebraic set defined by an ideal $I$ correspond exactly to the minimal prime ideals containing $I$.
Concerning your second question, it is not easy in general to determine when an ideal is prime. I asked a question here seeking different techniques to detect when ideals are prime. It is often easier to see that an ideal is not prime, as in the example I've given.
• Thank you for this! As a quick follow-up, are there important examples of algebraic sets for which it is difficult to answer questions about them (any type of typical algebraic geometry questions) and looking at its variety components does not help? (For example, when studying integers you tend to look at its prime factorization. But if you want to study Z/n for composite n, this can be very different from studying Z/p for p prime!) – Supersingularity Apr 10 '14 at 4:57
a) A useful trick for showing irreducibility of an algebraic set $X$ is to exhibit an open dense subset $X_0\subset X$ which is known to be irreducible.
In particular the closure $X\subset \mathbb P^n$ of any algebraic irreducible subset $X_0\subset \mathbb A^n$ is irreducible.
For example the intersection $C$ of the three quadrics $xw-yz=0, y^2-xz=0, z^2-yw=0$ in $\mathbb P^3$ (="twisted cubic curve") is irreducible because it is the closure of the intersection $C_0\subset \mathbb A^3$ of the three affine quadrics $x-yz=0, y^2-xz=0, z^2-y=0$ in $\mathbb A^3$ and $C_0$ is clearly irreducible as it is parametrized by $x=z^3, y=z^2,z=z$ , i.e. is the image of $\mathbb A^1$ under $z\mapsto (z^3,z^2,z)$ [See below].
Note that this is far from trivial: the intersection of any two of the three projective quadrics above is reducible!
b) Another useful trick is that the image of an irreducible algebraic set under a morphism is irreducible too.
For example the above twisted cubic curve $C$ is irreducible because it is the image of $\mathbb P^1$ under the morphism $\mathbb P^1\to \mathbb P^3: (u:v)\mapsto (u^3:u^2v:uv^2:v^3)$
Here are some personal reflections on the role of schemes in algebraic geometry, commenting on your very interesting remark: "Perhaps this is all a moot discussion since modern algebraic geometry is done with schemes".
Grothendieck introduced scheme theory in the late nineteen fifties and the high level of abstraction of that theory had a discouraging effect on even great mathematicians like Néron.
Soon however the incredible power of these new tools allowed people like Deligne Grothendieck and Faltings to solve problems untouchable by classical methods: Grothendieck and Deligne solved all of the Weil conjectures and Faltings solved conjectures of Mordell, of Shafarevich and of Tate.
Fields medals were of course awarded to Grothendieck, Deligne and Faltings and several other medals were won by mathematicians whose work involved in a fundamental way scheme theory: Lafforgue, Mumford, Ngô, Voevodski,...
For us lesser mortals scheme theory has now become quite accessible thanks to the didactic efforts of pioniers like Mumford and Hartshorne and then thanks to the excellent textbooks by Eisenbud-Harris, Liu, Görtz-Wedhorn,... and on-line notes like Vakil's splendid Foundations of Algebraic Geometry.
Algebraic varieties retain an enormous place in contemporary algebraic geometry.
However an overwhelming part of the results in that field are obtained with the help of scheme theory.
So every algebraic geometer must at some time learn scheme theory, but the best way to approach algebraic geometry is through one of the many introductory books written in the classical language by experts (who of course also master scheme theory!) like Fulton, Hulek, Perrin, Reid, ... | 2019-09-15 13:58:05 | {"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": 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": 0.7746985554695129, "perplexity": 373.65020637030375}, "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/1568514571360.41/warc/CC-MAIN-20190915114318-20190915140318-00400.warc.gz"} |
http://math.stackexchange.com/questions/447334/asymptotics-of-the-lower-approximation-of-a-pair-of-natural-numbers-by-a-coprime | # Asymptotics of the lower approximation of a pair of natural numbers by a coprime pair
When we are working, for instance, in combinatorics or graph theory, sometimes we can have the following situation. For each number $m$ from an infinite set $\mathbb M\subset\mathbb N$ we can construct an example $\mathcal E(m)$ having the required properties. Moreover, for each $n\ge m$ we can use the example $\mathcal E(m)$ for the construction of the example $\mathcal E(n)$ having properties similar to these of $\mathcal E(m)$. If we wish to obtain the respective asymptotic bound for $E(n)$ for all natural $n$, we encounter a problem to asymptotically estimate the approximation from below of the natural number $n$ by a number $m$ from the set $\mathbb M$.
For instance, in my old and small work "On graphs without 4-cycles" I investigated the problem posed by Erich Friedman here: what is the maximal number $E(n)$ of edges an $n$-vertex graph without 4-cycles? I found the asymptotics $E(n)=\frac{n^{3/2}}2\left(1+O\left(\frac 1{\ln n}\right)\right)$ as follows. We can easily prove that $E(n)\le\frac{n+n\sqrt{4n-3}}4$. We can obtain, using projective planes over finite fields, that $E(n)\ge\frac{(n-1)(\sqrt{4n-3}+1)}4-1$ provided $n=q^2+q+1$ where $q$ is a power of a prime. Then, using Rosser's bounds [Ros] $\frac n{\ln n+2}<\pi(n)<\frac n{\ln n-4}$ for $n\ge 55$, where $\pi(n)$ is the quantity of prime numbers which are not greater than $n$, I was able to show that for every natural $n\ge 2$ there exists a prime number $p\in\left[n-\frac {6n}{\ln n};n\right]$. I finally obtained the asymptotics for $E(n)$ from the above results.
At the last week I met my old coauthor, Oleg Verbitsky who proposed me the following problem. Let $n$ be a natural number. What is the minimal number $d=d(n)$ such that for each number $n'\ge n$ there exist coprime natural numbers $p,p'$ such that $n-d\le p\le n$ and $n'-d\le p'\le n'$? In his research where the question appeared, it is well enough that $d(n)=o(n)$, hence Oleg do not need any specific bound for $d(n)$. However, he thinks that the function $d(n)$ is of independent interest.
To obtain the upper bound for $d(n)$, Oleg simply took a largest prime number not greater than $n$ (my number intuition immediately said that this bound should be too weak), and using the result by Baker, Harman and Pintz [BHP], saying that there is a prime between, roughly, $n$ and something like $n-\sqrt n$ (by the way, which is asymptotically better than my above bound $n-\frac {6n}{\ln n}$), he obtained the bound $d(n)=o(n)$. But both of us are not number theorists, so my efforts to improve the bound may be an invention of a bicycle. So we decided that it is better to pose the question here. As usually, we are interested mainly in asymptotics of the function $d(n)$.
What have I tried? I expect that $d(n)$ is asymptotically very small (but not an independent on $m$ constant). I have the following evidence for this.
Let $k(l)$ be a number of different prime divisors of a number $l$. Then $k(l)\le \log_2 l$ and this bound can be (essentially) improved using the inequality $l\ge p_1 p_2\dots p_{k(l)}$ instead of $l\ge 2\times 2\times\dots 2$ ($k(l)$ times), where $p_i$ is the $i$-th prime number (that is $p_1=2$, $p_2=3$ and so forth). Moreover, slightly decreasing $n$ to $m$ we should obtain $k(m)$ even essentially smaller than $k(n)$. For finding such a number $m$ we can use (the above) results on the prime numbers density.
So, let $\{q_1,\dots, q_{k(m)}\}$ be the set of all prime divisors of the number $m$. Then among $d+1$ numbers $n', n'-1,\dots n'-d$ about $(d+1)(1-q_1)\ge (d+1)(1-p_1)$ are not divisible by $q_1$. Among these numbers about $(d+1)(1-q_1)(1-q_2)\ge (d+1)(1-p_1)(1-p_2)$ are not divisible by $q_2$ and so on. Therefore if $(d(n)+1)(1-p_1)(1-p_2)\dots (1-p_{k(m)})>1$ (which can be assured by a respectively small $d(n)$) then there should exist a coprime pair $p,p'$ such that $n-d\le p\le n$ and $n'-d\le p'\le n'$.
Thanks.
References
[BHP] R. Baker, G. Harman, J. Pintz, The difference between consecutive primes. II. Proc. Lond. Math. Soc., III. Ser. 83(3) (2001) 532-562.
[Ros] B. Rosser, Proc. London Math. Soc, 1939, v.45(2), p. 21-44.
- | 2014-04-19 13:03:21 | {"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": 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": 0.9024388790130615, "perplexity": 145.0572318168849}, "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/1397609537186.46/warc/CC-MAIN-20140416005217-00550-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://hssliveguru.com/plus-two-computer-application-model-question-paper-1/ | # Plus Two Computer Application Model Question Papers Paper 1
Kerala State Board New Syllabus Plus Two Computer Application Previous Year Question Papers and Answers.
## Kerala Plus Two Computer Application Model Question Papers Paper 1 with Answers
Board SCERT Class Plus Two Subject Computer Application Category Plus Two Previous Year Question Papers
Time: 2 Hours
Cool off time : 15 Minutes
General Instructions to candidates
• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.
Part – A
Answer all the questions from 1 to 5 carry one score each: (Scores: 5 × 1 = 5)
Question 1.
Write a C++ statement to declare an array with size 25 to accept the name of a student.
char name[25];
Question 2.
Name the following tags:
1. To include a button in HTML
2. To partition the browser window
1. <input type=”button”>
2. <frameset>
Question 3.
Define Web Hosting
Buying or renting storage space to store website in a web server and provide service(made available 24 × 7) to all the computers connected to the Internet. This is called web hosting.
Question 4.
Define BPR.
Business Process Re-engineering: In general BPR is the series of activities such as rethinking and redesign the business process to enhance the enterprise’s performance such as reducing the cost(expenses), improve the quality, prompt, and speed(time-bound) service. BPR enhances the productivity and profit of an enterprise.
Question 5.
Name the intellectual property represented by the symbols ®, ©.
Part – B
Answer any nine questions from 6 to 16 carry 2 scores each: (Scores : 9 × 2 = 18)
Question 6.
Define built-in functions. Give two examples.
Some functions that are already available in C++ are called pre-defined or built-in functions.
• strlen() – to find the number of characters in a string(i.e. string length).
• strcpy() – It is used to copy the second string into the first string.
• strcat() – It is used to concatenate the second string into the first one.
• strcmp() – It is used to compare two strings and returns an integer.
Question 7.
Write the use of the following in HTML:
(a) <A Href=”http://www.dhsekerala.gov.in>DHSE</A>.
(b) <EMBEDsrc=song1.mp3></EMBED>
(a) To link dhse portal. This is an example for external linking.
(b) This is used to include an audio file in our web page.
Question 8.
Name the tag and attribute needed to create the following lists in HTML:
a) <OL type=”1″>
b) <UL type=”square”>
Question 9.
Write a short note on free hosting.
The name implies it is free of cost service and the expense is met by the advertisements. Some service providers allow limited facility such as limited storage space do not allow multimedia(audio and video) files.
A paid service website’s address is as follows
eg: www.bvmhsskalparamba.com
Usually, two types of free web hosting services as follows:
1) as a directory service.
2) as a Subdomain
Question 10.
Write the type of web hosting that is most suitable:
1. For hosting a school website with a database.
2. For hosting a website for a firm
3. For creating a blog to share pictures and posts
4. For creating a low-cost personal website with a unique domain name.
1. Shared Hosting
2. Dedicated Hosting
3. Free Hosting
4. VPS/Shared Hosting
Question 11.
Define the following:
(a) Field
(b) Record
(a) Fields: the smallest unit of stored data.
eg: Regno, name, batch etc
(b) Record: Collection of related fields
eg: 101, Jose, Science
Question 12.
Write the name of any two-column constraints and their usage.
Constraints are used to ensure database integrity.
1. Not Null – It ensures that a column can never have NULL values.
2. Unique – It ensures that no two rows have the same value in a column.
3. Primary key – Similar to unique but it can be used only once in a table.
4. Default – We can set a default value.
5. Autojncrement – This constraint is used to perform auto_increment the values in a column. That automatically generates serial numbers. Only one auto_increment column per table is allowed.
Question 13.
Define the following ERP related technologies:
(a) CRM
(b) SCM
(a) Customer Relationship Management (CRM): As we know the customer is the king of the market. The existence of a company mainly the customers. CRM consists of programs to enhance the customer’s relationship with the company.
(b) Supply Chain Management (SCM): This is deals with moving raw materials from suppliers to the company as well as finished goods from the company to customers. The activities include are inventory(raw materials, work in progress and finished goods) management, warehouse management, transportation management, etc.
Question 14.
Write a short note on SAP.
SAP stands for Systems, Applications, and Products for data processing. It is a German MNCin Walldorf and founded in 1972. Earlier they developed ERP packages for large MNC. But nowadays they developed for small scale industries also.
Question 15.
Define the following terms:
(a) Trademark: This is a unique, simple, and memorable sign to promote a brand and hence increase the business and goodwill of a company. It must be registered. The period of registration is for 10 years and can be renewed. The registered trademark under Controller General of Patents Design and Trademarks cannot use or copy by anybody else.
(b) Copyright: The trademark is ©, copyright is the property right that arises automatically when a person creates a new work on his own, and by Law, it prevents the others from the unauthorized or intentional copying of this without the permission of the creator for 60 years after the death of the author.
Question 16.
Name the following:
2. Service used to send messages with Multimedia content
3. Packet oriented mobile data service on GSM.
4. Smart card technology used only in GSM phone systems.
1. GPS
2. MMS
3. GPRS
4. SIM
Part – C
Answer any nine questions from 17 to 27 carry 3 scores each: (Scores: 9 × 3 = 27)
Question 17.
Define Jump Statements. Explain any two.
The execution of a program is sequential but we can change this sequential manner by using jump statements. The jump statements are
1. goto statements: By using goto we can transfer the control anywhere in the program without any condition. The syntax is the goto label;
2. break statement: It is used to skip over a part of the code i.e. we can premature exit from a loop such as while, do-while, for, or switch.
3. continue statement: It bypasses one iteration of the loop.
4. exit(0) function: It is used to terminate the program. For this, the header file cstdlib must be included.
Question 18.
Nested Loop: A loop contains another loop completely then it is called a nested loop.
Eg.
#include<iostream>
using namespace std;
int main()
{
int i, j;
for(i=1; i<5; i++)
{
for(j=1; j<=i; j++)
cout<<"*";
cout<<endl;
}
}
Question 19.
Compare call-by-value and call-by-reference methods for calling functions.
Call by Value Call by Reference Ordinary variables are used as formal parameter Reference variables are used as formal parameters A copy of the original value is passed The original value is passed Any change made by the function will not affect the original value Any change made by the function will affect the original value Separate memory location is needed for actual and formal variables Memory of actual arguments is shared by formal arguments.
Question 20.
Differentiate between local and global variables.
Local Variable Global Variable 1. Declared inside a block. 1. Declared outside of all blocks. 2. It cannot be used in any other block. 2. It can be used anywhere in the program. 3. Memory is allocated when the block is active. 3. Memory is allocated when the program begins. 4. Memory is de-allocated when the block is completed. 4. Memory is de-allocated when the program terminates.
Question 21.
Name any two attributes of the following tags:
(a) <HTML>
(b) <MARQUEE>
(c) <FONT>
(a) Attributes of <HTML> are dir(direction ltr or rtl) and Language
(b) Attributes of <MARQUEE> (Any Two)
Height – Sets the height of the Marquee text
Width – Sets the width of the Marquee text
Direction – Specifies the scrolling direction of the text such as up, down, left or right
Behavior – Specifies the type such as Scroll, Slide(Scroll and stop) and alternate(to and fro).
<marquee behavior-“scroll”
scrollamount=”100″>hello</marquee>
<marquee behavior-“slide”
scrollamount=”100″>hello</marquee>
<marquee behavior-“alternate”
scrollamount=”100″>hello</marquee>
Scrolldelay – Specifies the time delay in seconds between each jump.
scrollamount – Specifies the speed of the text
loop – This specifies the number of times the marquee scroll. Default infinite.
bgcolor – Specifies the back ground colour.
Hspace – Specifies horizontal space around the marquee
Vspace – Specifies vertical space around the marquee
(c) <Font> used to specify the font characteristics. Its attributes are size, face, and color.
Question 22.
Name the three essential tags for creating a table in HTML. Write the purpose of each tag.
• <Table> is used to create a table.
• <TR> is used to create a row.
• <TH> is used to create heading cells.
• <TD> is used to create data cells.
Question 23.
Rewrite the following C++ code in JavaScript:
void length ()
{
char str[ ]="WELCOME";
cout<<strl;
}
<html>
<Script Language="JavaScript">
function lengths()
{
var str1;
str1="WELCOME";
document.write(str1);
}
</script>
<body>
<center>
<form name="frm">
<input type="button" value="print" onClick="lengths()">
</form>
</center>
</body>
</html>
Question 24.
Explain any three operators used in Relational algebra.
Relational Algebra(Any Three)
1. Select operation(s): It is used to select tuples in a relation that satisfies a condition.
2. Project Operation (p): It is used to select certain columns while discards some other columns.
3. Cartesian Product (X): All possible combinations of tuples from two relations.
4. Union Operation (E): All tuples appearing in either or both of two relations.
5. Intersection operation (Q): All tuples appearing in both relations.
6. Set difference operation (-): All tuples appearing in the first relation and not in the second.
Question 25.
Explain any three advantages of DBMS.
1. Data Redundancy: It means duplication of data. DBMS eliminates redundancy. DBMS does not store more than one copy of the same data.
2. Inconsistency can be avoided: If redundancy occurs there is a chance to inconsistency. If redundancy is removed then inconsistency cannot occur.
3. Data can be shared: The data stored in the database can be shared by the users or programs.
4. Standards can be enforced – The data in the database follows some standards.
Eg: a field ‘Name’ should have 40 characters long. Some standards are ANSI, ISO, etc.
5. Security restrictions can be applied – The data is of great value so it must be kept secure and private. Data security means the protection of data against accidental or intentional disclosure or unauthorized destruction or modification by an unauthorized person.
6. Integrity can be maintained: It ensures that the data is to be entered in the database is correct.
7. Efficient data access: It stored a huge amount of data efficiently and can be retrieved whenever a need arise.
8. Crash recovery: Sometimes all or a portion of the data is lost when a system crashes. A good DBMS helps to recover data after the system crashed.
Question 26.
Define the following:
(a) DML
(b) DDL
(c) DCL
(a) DML – DML means Data Manipulation Language.
It is used to insert records into a table, modify the records of a table, delete the records of a table, and retrieve the records from a table.DML commands are select, insert, delete, and update.
(b) DDL – DDL means Data Definition Language.
It is used to create the structure of a table, modify the structure of a table, and delete the structure of a table. DDL commands are created, alter and drop
(c) DCL – DCL means Data Control Language.
It is used to control access to the database. Commands are Grant, Revoke, etc
Question 27.
Write the result of the following:
(a) ALTER TABLE <table name>
Drop <column name>
(b) DELETE FROM <table name>
(c) DROP TABLE <table name>
(a) This command is used to delete a column from a table.
(b) This command is used to delete all the records from a table.
(c) This command is used to delete a table from the memory.
Part – D
Answer any two questions from 28 to 30 with 5 scores each: (Scores: 2 × 5 = 10)
Question 28.
Consider the following C++ code
int main()
{
char str[20];
cout<<"Enter a String";
cin>>str;
puts(str);
return 0;
}
1. Write the value of str if the string ‘HELLO WORLD’ is input to the code. Justify.
2. Write the amount of memory allocated for storing the array str. Give reason.
3. Write an alternative we can use to input string in place of cin
1. The output is “HELLO”.Here the white space after HELLO is treated as the delimiter hence the remaining(“WORLD”) will be truncated.
2. 20 bytes of memory is allocated for storing the array str.
3. gets() or getline() can be used to input string for accepting white spaces.
Question 29.
In HTML
(a) Differentiate client-side script and server-side script
(b) Name the tag and its attribute used to include a script in a web page
(c) Name any two server-side scripting language.
(a)
Client-Side Scripting Server Side Scripting The script is copied to the client browser The script is copied to the webserver Executed by the client Executed by the server and result is get back to the browser window Used for Client level validation Connect to the database in the server It is possible to block by the user Cannot possible Client-side scripts depend on the type and version of the browser It does not depend on the type and version of the browser
(b) <SCRIPT Language=”JavaScript”>
(c) ASP, JSP, PHP, etc.
Question 30.
In JavaScript:
(a) Explain any three types of operators used.
(b) Describe any two datatypes used. | 2022-11-29 11:56:26 | {"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": 0.2068648636341095, "perplexity": 4421.1360361405905}, "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/1669446710691.77/warc/CC-MAIN-20221129100233-20221129130233-00416.warc.gz"} |
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-molecular-approach-4th-edition/chapter-16-exercises-page-771/18 | ## Chemistry: Molecular Approach (4th Edition)
When calculating $[H_3O^+]$ for weak acid solutions, we use the $K_a$ expression: $$K_a = \frac{[H_3O^+][A^-]}{[HA]}$$ Substituting for the equilibrium concentrations (we can use an ICE table): $$K_a = \frac{x^2}{[HA]_{initial} - x}$$ In cases where the ionization is not very significant ( less than 5% ), when the acid is very weak or the initial acid concentration is large, we can assume that $$[HA]_{initial} \gt \gt x$$ Meaning that $[HA]_{initial} - x \approx [HA]_{initial}$, which is true considering x is small compared to the acid concentration and the use of significant figures. If we did not use that 'x is small approximation', the expression turns into a quadratic equation; if we use the assumption, the equation is simple and can easily be solved for x.
When calculating $[H_3O^+]$ for weak acid solutions, we use the $K_a$ expression: $$K_a = \frac{[H_3O^+][A^-]}{[HA]}$$ Substituting for the equilibrium concentrations (we can use an ICE table): $$K_a = \frac{x^2}{[HA]_{initial} - x}$$ In cases where the ionization is not very significant ( less than 5% ), when the acid is very weak or the initial acid concentration is large, we can assume that $$[HA]_{initial} \gt \gt x$$ Meaning that $[HA]_{initial} - x \approx [HA]_{initial}$, which is true considering x is small compared to the acid concentration and the use of significant figures. If we did not use that 'x is small approximation', the expression turns into a quadratic equation; if we use the assumption, the equation is simple and can easily be solved for x. | 2023-01-29 15:43:06 | {"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": 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": 0.9178295135498047, "perplexity": 384.1169519357035}, "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/1674764499744.74/warc/CC-MAIN-20230129144110-20230129174110-00575.warc.gz"} |
http://weijiawu.com.cn/2019/04/19/%E6%96%87%E6%9C%AC%E6%A3%80%E6%B5%8B-PSENet-1s/ | # 文本检测-PSENet-1s
Shape Robust Text Detection with Progressive Scale Expansion Network
KeyWords Plus: CVPR2019 Curved Text Face++
## Introduction
PSENet 分好几个版本，最新的一个是19年的CVPR，这是一篇南京大学和face++合作的文章（好像还有好几个机构的人），19年出现了很多不规则文本检测算法，TextMountain、Textfield等等，不过为啥我要好好研究这个（因为这篇文章开源了代码。。。）
### 1、论文创新点
1、Propose a novel kernel-based framework, namely, Progressive Scale Expansion Network (PSENet)
（1）、Starting from the kernels with minimal scales (instances can be distinguished in this step)
（2）、Expanding their areas by involving more pixels in larger kernels gradually
（3）、Finish- ing until the complete text instances (the largest kernels) are explored.
这个文章主要做的创新点大概就是预测多个分割结果，分别是S1,S2,S3…Sn代表不同的等级面积的结果，S1最小，基本就是文本骨架，Sn最大。然后在后处理的过程中，先用最小的预测结果去区分文本，再逐步扩张成正常文本大小。。。
### 2、算法主体
We firstly get four 256 channels feature maps (i.e. P2, P3, P4, P5) from the backbone. To further combine the semantic features from low to high levels, we fuse the four feature maps to get feature map F with 1024 channels via the function C(·) as:
先backbone下采样得到四层的feature maps，再通过fpn对四层feature分别进行上采样2,4,8倍进行融合得到输出结果。
如上图所示，网络有三个分割结果，分别是S1,S2,S3.首先利用最小的kernel生成的S1来区分四个文本实例，然后再逐步扩张成S2和S3
### 3、label generation
产生不同尺寸的S1….Sn需要不同尺寸的labels
不同尺寸的labels生成如上图所示，缩放比例可以用下面公式计算得出：
这个$d_{i}$表示的是缩小后mask边缘与正常mask边缘的距离，缩放比例rate $r_{i}$可以由下面计算得出：
### 4、Loss Function
Loss 主要分为分类的text instance loss和shrunk losses，L是平衡这两个loss的参数。分类loss主要用了交叉熵和dice loss。
The dice coefficient D(Si, Gi) 被计算如下：
$L_{s}$ 被计算如下：
### 4、Datasets
SynthText
TotalText
Newly-released benchmark for text detection. Besides horizontal and multi-Oriented text instances.The dataset is split into training and testing sets with 1255 and 300 images, respectively.
CTW1500
CTW1500 dataset mainly consisting of curved text. It consists of 1000 training images and 500 test images. Text instances are annotated with polygons with 14 vertexes.
ICDAR 2015
Icdar2015 is a commonly used dataset for text detection. It contains a total of 1500 pictures, 1000 of which are used for training and the remaining are for testing. The
ICDAR 2017 MLT
ICDAR 2017 MIL is a large scale multi-lingual text dataset, which includes 7200 training im- ages, 1800 validation images and 9000 testing images.
### 5、Experiment Results
Implementation Details
All the networks are optimized by using stochastic gradient descent (SGD).The data augmentation for training data is listed as follows: 1) the images are rescaled with ratio {0.5, 1.0, 2.0, 3.0} randomly; 2) the images are horizon- tally flipped and rotated in the range [−10◦, 10◦] randomly; 3) 640 × 640 random samples are cropped from the trans- formed images.
Total-Text
CTW1500
ICDAR 2015
IC17-MLT | 2020-05-29 02:01:08 | {"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": 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": 0.475680410861969, "perplexity": 8971.93862502669}, "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-2020-24/segments/1590347401004.26/warc/CC-MAIN-20200528232803-20200529022803-00075.warc.gz"} |
https://math.stackexchange.com/questions/686453/relation-between-the-weighted-matrix-norm-and-the-weights | # Relation between the weighted matrix norm and the weights
For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the original vector norm $||\overrightarrow{x}||$, and $||A||_W$ denote the induced matrix norm by the weighted vector norm $||\overrightarrow{x}||_W$. Prove that if $A \in \mathbb{C}^{m\times{}m}$ then $||A||_W = ||WAW^{-1}||$.
• Hello, R.K. Please see this post about how to ask for help with homework. Also, excellent advice on how to ask a good question in general can be found here. – AnonSubmitter85 Feb 22 '14 at 22:14
By definition $$\|A\|_W=\sup_{x\ne 0}\frac{\|WAx\|}{\|Wx\|}=\sup_{x\ne 0}\frac{\|WAW^{-1}(Wx)\|}{\|Wx\|}.$$ But as $W$ is non-singular $$\big\{Wx:x\in\mathbb R^n\smallsetminus\{0\}\big\}=\big\{y:y\in\mathbb R^n\smallsetminus\{0\}\big\},$$ and hence $$\sup_{x\ne 0}\frac{\|WAW^{-1}(Wx)\|}{\|Wx\|}=\sup_{y\ne 0}\frac{\|WAW^{-1}y\|}{\|y\|}=\|WAW^{-1}\|.$$ | 2019-08-23 09:39:49 | {"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9967974424362183, "perplexity": 156.97437251480733}, "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-2019-35/segments/1566027318243.40/warc/CC-MAIN-20190823083811-20190823105811-00424.warc.gz"} |
https://www.zbmath.org/?q=an%3A1314.46026 | # zbMATH — the first resource for mathematics
Orthogonality in $$\ell _p$$-spaces and its bearing on ordered Banach spaces. (English) Zbl 1314.46026
Summary: We introduce a notion of $$p$$-orthogonality in a general Banach space for $$1 \leq p \leq \infty$$. We use this concept to characterize $$\ell _p$$-spaces among Banach spaces and also among complete order smooth $$p$$-normed spaces as (ordered) Banach spaces with a total $$p$$-orthonormal set (in the positive cone). We further introduce a notion of $$p$$-orthogonal decomposition in order smooth $$p$$-normed spaces. We prove that if the $$\infty$$-orthogonal decomposition holds in an order smooth $$\infty$$-normed space, then the $$1$$-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
##### MSC:
46B40 Ordered normed spaces 46L07 Operator spaces and completely bounded maps 47L25 Operator spaces (= matricially normed spaces)
Full Text:
##### References:
[1] Birkhoff, G, Orthogonality in linear metric spaces, Duke Math. J., 1, 169-172, (1935) · Zbl 0012.30604 [2] James, RC, Orthogonality in normed linear spaces, Duke Math. J., 12, 291-302, (1945) · Zbl 0060.26202 [3] Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, I. Academic Press, New York (1983) · Zbl 0518.46046 [4] Karn, AK, A $$p$$-theory of ordered normed spaces, Positivity, 14, 441-458, (2010) · Zbl 1225.46014 [5] Karn, AK; Vasudevan, R, Approximate matrix order unit spaces, Yokohama Math. J., 44, 73-91, (1997) · Zbl 0902.46030 [6] Karn, AK; Vasudevan, R, Matrix duality for matrix ordered spaces, Yokohama Math. J., 45, 1-18, (1998) · Zbl 0944.46011 [7] Karn, AK; Vasudevan, R, Characterization of matricially Riesz normed spaces, Yokohama Math. J., 47, 143-153, (2000) · Zbl 0965.46002 [8] Oikhberg, T., Peralta, A.M.: Automatic continuity of $$M$$-norms on $$C^*$$-algebras. J. Math. Anal. Appl. 381(2), 799-811 (2011) · Zbl 1225.46043 [9] Oikhberg, T; Peralta, AM, Automatic continuity of orthogonality preservers on a non-commutative $$L_p (τ )$$ space, J. Funct. Anal., 264, 1848-1872, (2013) · Zbl 1288.47034 [10] Pedersen, G.K.: $$C^{⁎ }$$-algebras and their automorphism groups. Academic Press, London (1979) · Zbl 0416.46043 [11] Raynaud, Y; Xu, Q, On subspaces of non-commutative $$L^p$$-spaces, J. Funct. Anal., 203, 149-196, (2003) · Zbl 1056.46056
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. | 2021-04-21 05:33:28 | {"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": 0.8608784079551697, "perplexity": 2324.2130746491616}, "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-2021-17/segments/1618039508673.81/warc/CC-MAIN-20210421035139-20210421065139-00038.warc.gz"} |
http://dbhart-wntr.readthedocs.io/en/latest/controls.html | # Water network controls¶
One of the key features of water network models is the ability to control pipes, pumps, and valves using simple and complex conditions. EPANET uses “controls” and “rules” to define conditions [Ross00]. A control is a single action (i.e., closing/opening a link or changing the setting) based on a single condition (i.e., time based or tank level based). A rule is more complex; rules take an IF-THEN-ELSE form and can have multiple conditions and multiple actions in each of the logical blocks. WNTR supports EPANET’s rules and controls when generating a water network model from an EPANET INP file and simulating hydraulics using either the EpanetSimulator or the WNTRSimulator. WNTR includes additional options to define controls that can be used by the WNTRSimulator.
The basic steps to define a control for a water network model are:
1. Define the control action(s)
2. Define condition(s) (i.e., define what should cause the action to occur)
3. Define the control or rule using the control action(s) and condition(s)
4. Add the control or rule to the network
These steps are defined below. Examples use the “Net3.inp” EPANET INP file to generate the water network model object, called wn.
## Control actions¶
Control actions tell the simulator what to do when a condition becomes “true.” Control actions are created using the ControlAction class. A control action is defined by a target link, the property to change, and the value to change it to. The following example creates a control action that opens pipe 330:
>>> import wntr.network.controls as controls
>>> act1 = controls.ControlAction(l1, 'status', 1)
>>> print(act1)
set Pipe('330').status to Open
## Conditions¶
Conditions define when a control action should occur. The condition classes are listed in Table 8.
Table 8 Condition Classes
Condition class Description
TimeOfDayCondition Time-of-day or “clocktime” based condition statement
SimTimeCondition Condition based on time since start of the simulation
ValueCondition Compare a network element attribute to a set value
TankLevelCondition Compare the level in a tank to a set value.
RelativeCondition Compare attributes of two different objects (e.g., levels from tanks 1 and 2)
OrCondition Combine two WNTR Conditions with an OR
AndCondition Combine two WNTR Conditions with an AND
All of the above conditions are valid EPANET conditions except RelativeCondition.
## General Controls and Rules¶
All controls and rules may be created in WNTR with the Control class, which takes an instance of any of the above conditions, an iterable of ControlAction instances that should occur when the condition is true, and an optional iterable of ControlAction instances that should occur when the condition is false. The Control class also takes optional priority and name arguments. If multiple controls with conflicting actions should occur at the same time, the control with the highest priority will override all others. The priority argument should be an element of the ControlPriority enum. The default priority is medium (3). The name argument should be a string.
The following examples illustrate the creation of controls/rules in WNTR:
>>> n1 = wn.get_node('1')
>>> cond1 = controls.ValueCondition(n1, 'level', '>', 46.0248)
>>> print(cond1)
Tank('1').level > 46.0248
>>> rule1 = controls.Control(cond1, [act1], name='control1')
>>> print(rule1)
rule control1 := if Tank('1').level > 46.0248 then set Pipe('330').status to Open with priority 3
>>> cond2 = controls.SimTimeCondition(wn, '=', '121:00:00')
>>> print(cond2)
sim_time = 435600 sec
>>> act2 = controls.ControlAction(pump2, 'status', 1)
>>> rule2 = controls.Control(cond2, [act2], name='control2')
>>> print(rule2)
rule control2 := if sim_time = 435600 sec then set HeadPump('10').status to Open with priority 3
More complex controls/rules can be written using one of the Boolean logic condition classes. The following example creates a new rule that will open pipe 330 if both conditions are true, and otherwise it will open pipe 10. This rule will behave very differently from the rules above:
>>> cond3 = controls.AndCondition(cond1, cond2)
>>> print(cond3)
( Tank('1').level > 46.0248 && sim_time = 435600 sec )
>>> rule3 = controls.Control(cond3, [ act1 ], [ act2 ], priority=3, name='complex_control')
>>> print(rule3)
rule complex_control := if ( Tank('1').level > 46.0248 && sim_time = 435600 sec ) then set Pipe('330').status to Open else set HeadPump('10').status to Open with priority 3
Actions can also be combined, as shown in the following example:
>>> cond4 = controls.OrCondition(cond1, cond2)
>>> rule4 = controls.Control(cond4, [act1, act2])
>>> print(rule4)
rule := if ( Tank('1').level > 46.0248 || sim_time = 435600 sec ) then set Pipe('330').status to Open and set HeadPump('10').status to Open with priority 3
The flexibility of the Control class combined with the different ControlCondition classes and ControlAction instances provides an extremely powerful tool for defining complex network operations.
## Simple controls¶
Simple controls (contols that emulate EPANET’s [CONTROLS] section) may be defined more simply and concisely using the class methods of Control: time_control and conditional_control.
Conditional controls: Control objects created with the conditional_control class method define tank level and junction pressure based controls. Conditional controls require a source, attribute, operation, threshold, and a control action. The source is a water network model component and the attribute is any valid attribute for that object. The operation is defined using NumPy functions such as np.greater and np.less or elements of the Comparison enum. The threshold is the value that triggers the condition to be true. The control action is defined above.
In the following example, a conditional control is defined that opens pipe 330 if the level of tank 1 goes above 46.0248 m. The source is the tank n1 and the attribute is the level. To specify that the condition should be true when the level is greater than the threshold, the operation is set to np.greater and the threshold is set to 46.0248. The control action act1 from above is used in the conditional control:
>>> n1 = wn.get_node('1')
>>> thresh1 = 46.0248
>>> ctrl1 = controls.Control.conditional_control(n1, 'level', np.greater, thresh1, act1)
>>> print(ctrl1)
pre_and_postsolve := if Tank('1').level > 46.0248 then set Pipe('330').status to Open with priority 3
Time-based controls: Control objects created with the time_control class method define time-based controls. Time-based controls require a water network model object, a time at which the action should occur, a control action, and additional flags to specify the time reference point and recurrence. The time flag is either SIM_TIME or SHIFTED_TIME; these indicate simulation or clock time, respectively. The daily flag is either True or False and indicates if the control should be repeated every 24 hours.
In the following example, a time-based control is defined that opens Pump 10 at hour 121. The time flag is set to SIM_TIME and the daily flag is set to False. A new control action is defined that opens the pump:
>>> time2 = 121 * 60 * 60
>>> timeflag2 = 'SIM_TIME'
>>> dailyflag2 = False
>>> ctrl2 = controls.Control.time_control(wn, time2, timeflag2, dailyflag2, act2)
>>> print(ctrl2)
presolve := if sim_time = 435600.0 sec then set HeadPump('10').status to Open with priority 3
Note that the EpanetSimulator is limited to use the following pairs: time_flag=’SIM_TIME’ with daily_flag=False, and time_flag=’SHIFTED_TIME’ with daily_flag=True. The WNTRSimulator can use any combination of time flag and daily flag.
## Adding controls to a network¶
Once a control is created, they can be added to the network. This is accomplished using the add_control method of the water network model object. The control should be named so that it can be retrieved and modified if desired:
>>> wn.add_control('NewTimeControl', ctrl2)
>>> wn.get_control('NewTimeControl')
<Control: '', <SimTimeCondition: model, 'Is', '5-01:00:00', False, 0>, [<ControlAction: 10, status, Open>], [], priority=3> | 2018-04-26 23:00:53 | {"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": 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": 0.5236786603927612, "perplexity": 5197.884910670174}, "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-2018-17/segments/1524125948617.86/warc/CC-MAIN-20180426222608-20180427002608-00006.warc.gz"} |
https://math.stackexchange.com/questions/3021181/what-is-the-significance-of-group-automorphism | # What is the significance of Group Automorphism?
I have understood the definition of group automorphism and have studied various examples for the same. But what is the significance of an automorphism? When we study isomorphisms, we try to investigate how similar is a group to another group. What do we get from establishing isomorphisms from a group to itself?
• We can realise other groups as automorphism groups, e.g., $Aut(Q_8)\cong S_4$, see here. – Dietrich Burde Dec 1 '18 at 12:01
• You can make rings out of group by taking its automorphism group, Ex, Integer ring is the automorphism ring of $(\mathbb{Z},+)$. – mathpadawan Dec 1 '18 at 13:01
• @mathnoob That needs further explanation to be valid. A straightforward reading would suggest you mean to say that the integers are the endomorphism ring of $(\Bbb{Z},+)$, not an "automorphism ring," which is undefined. – jgon Dec 1 '18 at 17:50
An automorphism on a structure describes a symmetry on that structure - a way in which certain elements of the structure play identical roles within the structure.
For example, a graph isomorphism is a bijection between the sets of nodes of two graphs such that $$x$$ and $$y$$ are adjacent (have an edge connecting them) if and only if $$f(x)$$ and $$f(y)$$ are adjacent. This means that the two graphs are really the same graph (they can be made to look identical if they're drawn in the right way).
Now look at this graph:
I'm sure you would understand what I meant if I said that the vertices 5, 6, 8 and 9 all "play the same role" in the graph. Vertices 0 and 2 also "play the same role". But vertices 5 and 1 play very different roles, they don't fit into the graph in identical ways (for example, 5 has one neighbor while 1 has two). This is due to the obvious symmetries which exist in this graph, corresponding to the two automorphisms of the graph given by:
1. $$f$$ such that $$f(5)=6, f(6)=5, f(1)=3, f(3)=1, f(8)=f(9)$$, and $$f(x)=x$$ for all other nodes.
2. $$g$$ such that $$g(5)=8, g(8)=5, g(4)=7, g(7)=4, g(0)=2,g(2)=0, g(6)=9, g(9)=6$$, and $$g(x)=x$$ for all other nodes.
The fact that 5 and 1 do not play the same role corresponds to the fact that there is no automorphism mapping 5 to 1 or vice-versa.
To me the best motivation for studying group automorphisms is their application to semi-direct products. For example, in the classification of groups of order $$pq$$ for distinct primes $$p$$ and $$q$$, you can use the Sylow theorems to show that $$G$$ is some semi-direct product of $$C_p$$ and $$C_q$$. Knowing the structure of $$\text{Aut}(C_p)$$ is then crucial in proving that in fact there is only one non-Abelian semi-direct product of $$C_p$$ and $$C_q$$ up to isomorphism. In general, when trying to classify groups of a given order understanding semi-direct products can be very useful. To do this, we first should understand automorphism groups. | 2019-06-17 23:31:04 | {"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": 0.8393104672431946, "perplexity": 141.99468504764164}, "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-2019-26/segments/1560627998581.65/warc/CC-MAIN-20190617223249-20190618005249-00532.warc.gz"} |
https://web2.0calc.com/questions/hulp | +0
# hulp
+1
42
3
+63
A rectangular building is $$4$$ meters by $$6$$ meters. A dog is tied to a rope that is $$10$$ meters long, and the other end is tied to the midpoint of one of the long sides of the building. Find the total area of the region that the dog can reach (not including the inside of the building), in square meters.
If you get $$26.704, 314, 111, \text{or } 10\Pi -24$$ check your work those are wrong.
The rope cannot pass through the walls of the building.
Feb 21, 2023
edited by ItsFree Feb 21, 2023
edited by ItsFree Feb 21, 2023
#1
+118448
+1
79pi m^2
Feb 22, 2023
#2
+63
0
THank you so much Melody.
ItsFree Feb 24, 2023
#3
+118448
0
I guess that means you were only interested in the answer?
Melody Feb 24, 2023 | 2023-03-21 07:50:40 | {"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": 0.923249363899231, "perplexity": 1596.481937882935}, "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/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00042.warc.gz"} |
http://haustechnik-hasselbusch.de/series-divergence-test-calculator.html | ### Series Divergence Test Calculator
Tests for convergence or divergence from chapter 11 of AP Calculus BC. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. is divergent. the TI-SmartView™ CE emulator software for the TI-84 Plus graphing family. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. The limit of the series terms isn't zero and so by the Divergence Test the series diverges. If is divergent then is divergent. If r > 1 (including infinity), then the series is divergent. (Calculator Permitted) (a) What is the sum of 1 11 n nn13 f §· ¨¸ ©¹ ¦ (b) Using your calculator, calculate S 500 to verify that the SOPS (sum. The sum of an arithmetic progression from a given starting value to the nth term can be calculated by the formula: Sum(s,n) = n x (s + (s + d x (n - 1))) / 2. Therefore, one typically applies it for series that look divergent right from the start. Besides finding the sum of a number sequence online, server finds the partial sum of a series online. f x = 1 x p x > 0. Includes the nth-Term, geometric series, p-Series, integral test, ratio test, comparison, nth-Root, and the alternating series test. In our Series blogs, we've gone over four types of series, Geometric, p, Alternating, and Telescoping, and their convergence tests. Note: Alternating Series Test can only show convergence. A convergent series is a mathematical series in which the sequence of partial sums converges to 1. Corrected a couple of typing errors. Compute the numerical divergence of the vector field. Divergence Calculator - eMathHelp. Integral Test X∞ n=0 a n with a n ≥ 0 and a n decreasing Z ∞ 1 f(x)dx and X∞ n=0 a n both converge/diverge where f(n) = a n. Contrasting that is the situation in Europe, where the larger. Hints help you try the next step on your own. Use the nth term test to determine whether the following series converges or diverges. The Jensen-Shannon divergence, or JS divergence for short, is another way to quantify the difference (or similarity) between two probability distributions. Share Bookmark. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Find more Mathematics widgets in Wolfram|Alpha. If a n = arn then the series is a geometric series. 0 ≤ y ≤ f x x Divergent Series. The test is named after 19th-century German mathematician Peter Gustav Lejeune Dirichlet. and therefore, This means. Since the integral converges, so does the series. Test the series for convergence or divergence using the Alternating Series Test. While the integral test is a nice test it does force us to do improper integrals which aren t always easy and in some cases may be impossible to determine the. Convergent and divergent thinking require two different parts of the brain. State what the test is. Hints help you try the next step on your own. Input first term ( ), common ratio ( ), number of terms () and select what to compute. We usage the integral test. Otherwise, you must use a different test for convergence. Free Series Root Test Calculator - Check convergence of series using the root test step-by-step This website uses cookies to ensure you get the best experience. The comparison tests we consider below are just the sufficient conditions of convergence or divergence of series. The mnemonic, 13231, helps you remember ten useful tests for the convergence or divergence of an infinite series. We first start with the plot of { a n } n = 0. While convergent thinking relies more on logic, divergent thinking relies more on creativity. The applet did not load, and the above is. Determine whether or not the series converge using the appropriate convergence test (there may be more than one applicable test. If then the "tail" of the series looks like a geometric series of ratio , and follows the same convergence and divergence behavior as a geometric series when. If r = 1, then the series could either be divergent or convergent. Using divergence test to determine if the series is divergent or state that the test is inconclusive. That is an alternating series is a series of the form P 1k1a k where a k 0 for all k. This test has 95% accuracy, just because you. You can use integers ( 10 ), decimal numbers ( 10. Calculus on the Web was. Convergence or divergence of a series is proved using sufficient conditions. Because it meets both convergent and divergent conditions, tes is inclusive when L=1. \square! \square!. Root Test: Suppose that the terms of the sequence in question are non-negative, and that there exists r such that. The comparison tests we consider below are just the sufficient conditions of convergence or divergence of series. Taylor and Maclaurin (Power) Series Calculator. SolveMyMath's Taylor Series Expansion Calculator. This tool calculates the overall capacitance value for multiple capacitors connected either in series or in parallel. Now, we will focus on convergence tests for any type of infinite series, as long as they meet the tests' criteria. For example, consider the series \[\sum_{n=1}^∞\dfrac{1}{n^2+1}. Can you believe that, through our time loving Tris and Four, wishing we could pass initiation, and generally living vicariously through Veronica Roth's words, that we have never had a Divergent quiz designed to find what faction we belong in?! The fact that we've been living in the shadow of this YA classic thinking we know where we fit in without really knowing—well, let's just say it. If, in the limit, this ratio is less than 1, the series converges; if it's more than 1 (this includes infinity), the series diverges; and if it. Compute the numerical divergence of the vector field. We first start with the plot of { a n } n = 0. Use a space to separate values. This means the infinite series sums up to infinity. Series and Sum Calculator with Steps. Improved robustness of the 2 Comparison Tests and fixed bug in Raabe's Test. Divergence Test for Series. Stage #1: Aptitude Test. Divergent Series. a n = a r n − 1. Since the integral converges, so does the series. Dirichlet's test is one way to determine if an infinite series converges to a finite value. It will have difficult mathematical operations and it consumes your time and energy. Step 2: Now click the button "Calculate" to get the sum. Since the ratio test is user-friendly and used by the calculator on this page, we learn how to use it here. The list may have finite or infinite number of terms. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. Power Series. limbo n n00 Since lim bn ? O and by 'n + 1 ? v b, for all n > 2, ---Select--- n>. Lecture 25/26 : Integral Test for p-series and The Comparison test In this section, we show how to use the integral test to decide whether a series of the form X1 n=a 1 np (where a 1) converges or diverges by comparing it to an improper integral. Another important test is the Ratio test. The limit of the series terms isn’t zero and so by the Divergence Test the series diverges. With the ratio test, we use a ratio of the power series and a modified n + 1 version of itself to solve for the values of x that satisfy the convergence criteria. Ratio Test: (a) If. The p series test, geometric series test, telescoping series test, root test, ratio test, integral test, alternating series test, comparison test, divergence test to name a few. (Calculator permitted) To five decimal places, approximate the sum of 1n 2 1 1 n n f ¦ using S 5. Geometric Sequences Calculator. Use the divergence test to determine whether a series converges or diverges. The geometric series test determines the convergence of a geometric series. (Calculator Permitted) a. National Science Foundation. Either this series converges or diverges. The Ratio Test states: then if, 1) L<1 the series converges absolutely. If the limit of a[n] is not zero, or …. Mar 10, 2016 · Use Gauss’ test (from the previous exercise, Section 10. What is the nth Term Test? The nth term test is probably the test that can be applied to the most series. Example 2 Use the comparison test to determine if the following series converges or diverges: X1 n=1 21=n n I First we check that a n >0 { true since 2 1=n n >0 for n 1. The ratio test works by looking only at the nature of the series you're trying to figure out (as opposed to the tests which compare the test you're investigating to a known, benchmark series). A series of events featuring lectures, service awards and volunteer opportunities will be held Jan. Theorem: If ∑ n = 1 ∞ a n and ∑ n = 1 ∞ b n are series with non-negative terms, then: If ∑ n = 1 ∞ b n converges and a n ≤ b n for all n, then ∑ n = 1 ∞ a n converges. Calculating of the sum of series online. This utility helps solve equations with respect to given variables. I used the partial sum test and found that the nth partial sum of this series is, ln. If the limit as the series (an+1/an) approaches infinity is less than 1, the series converges. In general, in order to specify an infinite series, you need to specify an infinite number of terms. Caution: This test does not detect all divergent series; for example, the harmonic series ∞ ∑ n=1 1 n diverges even though lim n→∞. Calculate limits, integrals, derivatives and series step-by-step. This script finds the convergence, sum, partial sum graph, radius and interval of convergence, of infinite series. series-calculator. Improper Integral Calculator is a free online tool that displays the integrated value for the improper integral. Check out all of our online calculators here! Enter a problem. Practice your math skills and learn step by step with our math solver. Explanation of Each Step Step (1) To apply the divergence test, we replace our sigma with a limit. developed with the support of the. Serioes of this type are called p-series. a n = a r n − 1. The Ratio Test is used extensively with power series to find the radius of convergence, but it may be used to determine convergence as well. Formally, Dirichlet's test states that the infinite series. * Remainder: | 𝑛|ᩣ 𝑛+1 5 Integral Test Series: ∑. Determine whether the series X∞ n=2 1 n(lnn)2 is convergent or divergent. Then the alternating series ∞ ∑ n=1(−1)nan and ∞ ∑ n=1(−1)n−1an both converge. By using …. For the series 1/x, 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8 +… If you divide up the series into groups of fractions with the last denominator powers of 2 1+1/2+ (1/3+1/4) +(1. Test for Divergence to see if lim n!1 a n = 0: If this limit is not zero then the series P a n diverges. Integral Test and p-Series. The limit of 1/n! as n approaches infinity is zero. Switching back and forth between the two may not seem like multitasking, but it is a form of multitasking. Solution: Let us take Cn=2 n /nx(4x-8) n. Letting we apply Gauss’ test to conclude that converges if and diverges if. Caution: This test does not detect all divergent series; for example, the harmonic series ∞ ∑ n=1 1 n diverges even though lim n→∞. BYJU’S online improper integral calculator tool makes the calculation faster, and it displays an integrated value in a fraction of seconds. Nth Term Test for Divergence This is the only test used to determine divergence of a series. Alter in traditional wedding may look inviting and explain these to anyone. You should consult a calculus text for descriptions of tests for convergence and divergence for infinite series. The series above is thus an example of an alternating series and is called the alternating harmonic series. : It's a straight forward process that focuses on figuring out the most effective answer. From the definition of the series we have the th and st terms, Therefore, Using the Taylor expansion, So we have, where is bounded. The nth term for Divergence states that if lim n → ∞ a n does not exist, or if lim n → ∞ (a n ≠ 0), then the series ∑ n …. It will have difficult mathematical operations and it consumes your time and energy. Calculates the sum of a convergent or finite series. Root Test: Suppose that the terms of the sequence in question are non-negative, and that there exists r such that. BYJU’S online improper integral calculator tool makes the calculation faster, and it displays an integrated value in a fraction of seconds. Dirichlet's test is one way to determine if an infinite series converges to a finite value. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. Unfortunately, if the limit does turn out to be zero, then the test is inconclusive. absolute value should be less than 1. Aristotelian definition is new development and service work. In the case of the geometric series, you just need to specify the first term. Divide the given equation by the highest denominator power which is n 3. Learn more about this test in this video. Hints help you try the next step on your own. This series converges by the alternating series test. Test for Divergence to see if lim n!1 a n = 0: If this limit is not zero then the series P a n diverges. Free Online Test Series & Mock Tests for SSC CGL, SBI PO, RRB NTPC, IBPS PO & Other Govt. Test the series for convergence or divergence. ∑ n=0 ∞ [5n 2 – n 3 ] / [3 + 8n 3 ] = lim n → ∞ [5n 2 – n 3 ] / [3 + 8n 3 ]. Consider a series S a n such that a n > 0 and a n > a n+1 We can plot the points (n,a n) on a graph and construct rectangles whose bases are of length 1 and whose heights are of length a n. And multitasking is not as effective as you may think. The limit comparison test ( LCT) differs from the direct comparison test. Unfortunately, if the limit does turn out to be zero, then the test is inconclusive. You can specify the order of the Taylor polynomial. Calculate limits, integrals, derivatives and series step-by-step. If is divergent, then is divergent. term test of divergence. ) lim (-1)" (7n - 1) 6n + 1 Since lim (-1) (7-1) does not exist the series is divergent n-00 6n + 1 Need Help? Read it Watch It Master It Talk to a Tutor. Convergence or divergence tests AP Calculator BC. Divergence and Curl calculator. Power series Calculator. The program will determine what test to use and if the series converges or diverges. Definition: The series (that is, the p -series where p =1) is known as the harmonic series. ∫ integral calculator online with steps. This test is the sufficient convergence test. By the convergence of the alternating series divergence test for sequence, mostly in the series test returns a series may not misuse this. Since the ratio test is user-friendly and used by the calculator on this page, we learn how to use it here. Geometric Sequence Calculator. This website uses cookies to ensure you get the best experience. Check out all of our online calculators here! Enter a problem. term test of divergence. Now the eyes diverge, accommodation is inhibited, and the. then notice that the area of these rectangles (light blue plus purple) is an upper Reimann sum. If r < 1, then the series converges. The beam divergence is in direct relation to the beam size at aperture: By increasing. New Resources. The Comparison Tests Let \$$\\sum\\limits_{n = 1}^\\infty {{a_n}} \$$ and \$$\\sum\\limits_{n = 1}^\\infty {{b_n. The Divergent Faction quiz starts with the Aptitude Test—just like the original story. Estimate the value of a series by finding bounds on its remainder term. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series …. Use of the Geometric Series calculator. Using divergence test to determine if the series is divergent or state that the test is inconclusive. This method becomes easier just by using the Convergence Calculator. Calculadora gratuita para test de divergencia - comprobar divergencia de una serie paso a paso utilizando el test de divergencia. The calculator will find the Taylor (or power) series expansion of the given function around the given point, with steps shown. diverges if lim(n→∞)≠0. is divergent. Suppose that there exists r such that. and the nth term an = a1 r n - 1. Integral Test - Basic Idea. The formula for calculating a Taylor series for a function is given as: Where n is the order, f(n) (a) is the nth order derivative of f (x) as evaluated at x = a, and a is where the series is centered. powered by. Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student. Which "Divergent" Faction Do You Actually Belong In? Choose your fate, Initiates. Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1 / 2, 1 / 3, 1 / 4, etc. Includes the …. Infinite Series. By using this website, you agree to our Cookie Policy. Series Calculator computes sum of a series over the given interval. A series convergence calculator is used to find out the sum of the sequence and for determining convergence and divergence among series. For each of the following series, apply the divergence test. net › Top Education From www. Convergence and Divergence Tests for Series Test When to Use Conclusions Divergence Test for any series X∞ n=0 a n Diverges if lim n→∞ |a n| 6= 0. {eq}\sum_{k=1}^{\infty}\frac{3k^2+2k+1}{5k^2+1} {/eq}. A series ∑ un of positive terms is convergent if from and after some fixed term un + 1 un < r < 1 , where r is a fixed number. Some convergent ones are X1 n2, X 1 n p n, and X 1 n1:001. Check out all of our online calculators here! Enter a problem. Parallel Capacitor. Null Hypothesis (H0): alpha=1. I used the partial sum test and found that the nth partial sum of this series is, ln. In addition, when the calculator fails to find series sum is the strong indication that this series is divergent (the calculator prints the message like "sum diverges"), so our calculator also indirectly helps to. If ∑ n = 1 ∞ b n diverges and a n ≥ b n for all n, then ∑ n = 1 ∞ a n diverges. It is important to realise that this test only states that if as n → ∞, a n 6→0, the series will diverge. It is defined in milli-radiant (mrad), which usually describes a part of the circumcircle. When you use the comparison test or the limit comparison test, you might be able to use the harmonic series to compare in order to establish the divergence of the series in question. 4 11 4 9 4 7 4 5 1 n 4 2n 3 1 n 1 5 1 4 1 3 1 2 4 2n 3. This program tests the convergence or divergence of a series. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. P Series Test. Either this series converges or diverges. If possible, state the value to which it converges. Convergent Series 4. Get detailed solutions to your math problems with our Power series step-by-step calculator. Calculating of the sum of series online. For F = ( x y 2, y z 2, x 2 z), use the divergence theorem to evaluate. But going back today?. If, in the limit, this ratio is less than 1, the series converges; if it's more than 1 (this includes infinity), the series diverges; and if it. This test is actually a special case of the The Integral Test for Positive Series and is as follows: Theorem 1 (The p-Series Test): The special series is convergent if and divergent if. I Therefore 2 1=n n >1 n for n 1. Statement of D'Alembert Ratio Test. On the bright side, this method is a lot more plug-and-chug: once you pick the series to compare, you just throw them into a limit problem and execute. If the terms of an infinite series don't approach zero, the series must diverge. org Education Series convergence calculator - mathforyou. To see that the series does not converge absolutely, it suffices to show that the series X∞ n=0 (−1) n √ 1 n2 +1 = X∞ n=0 1 √ n2 +1 diverges. Browse other questions tagged calculus sequences-and-series convergence-divergence factorial or ask your own question. whether a series is convergent or divergent. If then the series converges. Since the ratio test is user-friendly and used by the calculator on this page, we learn how to use it here. The formula used by taylor series calculator for calculating a series for a function is given as: F (x) = ∑^ ∞_ {n=0} f^k (a) / k! (x - a) ^k . Unfortunately, if the limit does turn out to be zero, then the test is inconclusive. Apr 07, 2021 · Infinite Series Calculator: Finding the sum of an infinite series of a function is not so simple or easy for any one. it gives more information than) Euclid's 3rd-century-BC result that there are infinitely many prime numbers. In particular, the sum is equal to the natural logarithm of 2: + + = The alternating harmonic series, while conditionally convergent, is not absolutely convergent: if the terms in the series are systematically rearranged, in general the sum becomes different and, dependent. then notice that the area of these rectangles (light blue plus purple) is an upper Reimann sum. The formula for the ratio test is:. Since the harmonic series is known to diverge, we can use it to compare with another series. BYJUS online remainder theorem calculator tool makes the calculation faster and it displays the result in a fraction of seconds. To see this, do a limit comparison with the divergent series P 1 n: lim n→∞. Divergence Calculator. EXAMPLE 10: Does the following series converge or diverge? SOLUTION: The limit does not exist, therefore, the series diverges by the nth term test for divergence. Limit Calculator. Test the series for convergence or divergence. State what the test is. First, unlike the Integral Test and the Comparison/Limit Comparison Test, this test will only tell us when a series converges and not if a series will diverge. Browse other questions tagged calculus sequences-and-series convergence-divergence factorial or ask your own question. Online Integral Calculator » Solve integrals with Wolfram|Alpha. But going back today?. The mnemonic, 13231, helps you remember ten useful tests for the convergence or divergence of an infinite series. What is the sum of ∑ ( ) b. Hints help you try the next step on your own. 3) - Integrate the series formula as an improper integral and if the integral converges the so does. Stage #1: Aptitude Test. 2) L>1 the series diverges. Convergent Vs Divergent Thinking, Differences; Convergent Thinking Divergent Thinking; The process of figuring out a concrete solution to any problem is called Convergent Thinking. Problem 1: Test for convergence Answer: Since we have a power n in the series, we will use the Root-Test. Use the comparison test to determine whether the following series are convergent or divergent. p-Series Convergence The p-series is given by 1/n p = 1/1 p + 1/2 p. The Divergence Theorem is critically important as it provides us with a test to see whether a series is divergent. Therefore, the series diverges by the Integral Test. ∫ integral calculator online with steps. Free Series Divergence Test Calculator - Check divergennce of series usinng the divergence test step-by-step This website uses cookies to ensure you get the best …. This data series is composed of three primary series - the marketing bill series, the industry group series, and the primary factor series - that shed light on different aspects of the food supply chain. Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student. (Submitted on 17 Jun 2007) Abstract: The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. Improved robustness of the 2 Comparison Tests and fixed bug in Raabe's Test. Convergence or divergence of a series is proved using sufficient conditions. com allows you to find the sum of a series online. The beam divergence describes the widening of the beam over the distance. If , then the series is divergent by the divergence theorem. Online Integral Calculator » Solve integrals with Wolfram|Alpha. Conic Sections Transformation. Geometric …. Solar radiative forcing of tropical forest to pay more. Login help. Updated the Power Series Test for R2020b. Get step-by-step solutions from expert tutors as fast as 15-30 minutes. The Basic Comparison Test. Some convergent ones are X1 n2, X 1 n p n, and X 1 n1:001. Aug 30, 2021 · Online Integral Calculator » Solve integrals with Wolfram|Alpha. Now the eyes diverge, accommodation is inhibited, and the. Featured on Meta Join me in Welcoming Valued Associates: #945 - Slate - and #948 - Vanny. The ratio test has three possibilities: converge, diverges, or cannot determine. Terms in this set (10) test for divergence. A proof of this test is at the end of the section. ) State the test used. Input the function you want to expand in Taylor serie : Variable : Around the Point a = (default a = 0) Maximum Power of the Expansion: How to Input. Test For Divergence 1. It incorporates the fact that a series converges if and only if a constant multiple of it converges. The sum of the reciprocals of all prime numbers diverges; that is: = + + + + + + + = This was proved by Leonhard Euler in 1737, and strengthens (i. Caution: This test does not detect all divergent series; for example, the harmonic series ∞ ∑ n=1 1 n diverges even though lim n→∞. If r = 1, then the series could either be divergent or convergent. Again, because of the expansion, we can conclude that div. In this case we find Therefore, because does not tend to zero as k tends to infinity, the divergence test tells us that the infinite series diverges. series-divergence-test-calculator. Infinite series can be very useful for computation and problem. Parallel Capacitor. Series Divergence Test Calculator - Symbola. If lim n!1a n 6= 0 then P 1 n=1 a n diverges. Does P bn converge? Is 0 ≤ an ≤ bn? YES P YES an Converges Is 0 ≤ bn ≤ an? NO NO P YES an Diverges LIMIT COMPARISON TEST Pick {bn}. This is one of the frequently occurring limits, and since it is not equal to zero, the series diverges by the nth term test for divergence. This utility helps solve equations with respect to given variables. What is Special about a Geometric Series. series-calculator. Parallel Capacitor. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Gerardo Mendoza and Dan Reich. If is divergent, then is divergent. Because of lim n ∞ a n. For instance, the series is telescoping. The geometric series and the ratio test Today we are going to develop another test for convergence based on the interplay between the limit comparison test we developed last time andthe geometric series. Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student. Series Calculator computes sum of a series over the given interval. Series and Sum Calculator with Steps. To improve this 'Infinite geometric series Calculator', please fill in questionnaire. If , then nothing can be said about the series , that is the series may be convergent or divergent. The comparison tests we consider below are just the sufficient conditions of convergence or divergence of series. You go through three primary stages to expose your inner virtues. Use the nth Term Divergence Test to determine whether or not the following series converge: (a) 23 3 1 13 n 4 5 2 nn nn f 1 ¦ (b) 2 1 n n f ¦ (c) 1! n 2 ! 1 n n f ¦ (d) 1 2! n 10 ! n n ¦ 4. For example, the sum from the 1-st to the 5-th term of a sequence starting. Where do you belong? Are you selfless (Abnegation), intelligent (Erudite), brave (Dauntless), honest (Candor) or peaceful (Amity)? Take this Divergent personality test now to find out! Please share with friends who also love this - thank you!. Otherwise, you must use a different test for convergence. Alter in traditional wedding may look inviting and explain these to anyone. Again, because of the expansion, we can conclude that div. Free Series Divergence Test Calculator - Check divergennce of series usinng the divergence test step-by-step This website uses cookies to ensure you get the best experience. Series Divergence Test Calculator. Otherwise is called divergent series. Some convergent ones are X1 n2, X 1 n p n, and X 1 n1:001. Divergent shading (Orange) 14. The limit of the series terms isn't zero and so by the Divergence Test the series diverges. 1 - Enter the first term A1 in the sequence, the common ratio r and n n the number of terms in the sum then press enter. Hints help you try the next step on your own. \square! \square!. If , then nothing can be said about the series , that is the series may be convergent or divergent. Since the harmonic series is known to diverge, we can use it to compare with another series. the harmonic series), it diverges. Use of the Geometric Series calculator. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. We know exactly when these series converge and when they diverge. (Calculator permitted) To five decimal places, approximate the sum of 1n 2 1 1 n n f ¦ using S 5. Calculadora gratuita para test de divergencia - comprobar divergencia de una serie paso a paso utilizando el test de divergencia. e Radius of Convergence; Example. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Convergent and divergent thinking require two different parts of the brain. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. Now the eyes diverge, accommodation is inhibited, and the. This test is actually a special case of the The Integral Test for Positive Series and is as follows: Theorem 1 (The p-Series Test): The special series is convergent if and divergent if. Gerardo Mendoza and Dan Reich. It will have difficult mathematical operations and it consumes your time and energy. She tells you to drink it, and you do. Either this series converges or diverges. The list may have finite or infinite number of terms. If r > 1 (including infinity), then the series is divergent. Because it meets both convergent and divergent conditions, tes is inclusive when L=1. Our online calculator, build on Wolfram Alpha system is able to test convergence of different series. In order to use either test the terms of the infinite series must be positive. The Divergent Faction quiz starts with the Aptitude Test—just like the original story. The Divergence Theorem is critically important as it provides us with a test to see whether a series is divergent. It's also known as the Leibniz's Theorem for alternating series. will work if a n ≤ b n for. Stage #1: Aptitude Test. There are a couple of things to note about this test. It is defined in milli-radiant (mrad), which usually describes a part of the circumcircle. To use the comparison test we must first have a good idea as to convergence or divergence and pick the sequence for comparison accordingly. The alternating series test for convergence lets us say whether an alternating series is converging or diverging. Includes the …. Unfortunately, if the limit does turn out to be zero, then the test is inconclusive. 1 Use the divergence test to determine whether a series converges or diverges. Basically if r = 1, then the ratio test fails and would require a different test to determine the convergence or divergence of the series. A divergent series is just the opposite — the sums do not meet a finite limit. In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. The limit of the series terms isn't zero and so by the Divergence Test the series diverges. By using this website, you agree to our Cookie Policy. In our Series blogs, we've gone over four types of series, Geometric, p, Alternating, and Telescoping, and their convergence tests. Divide the given equation by the highest denominator power which is n 3. The program will determine what test to use and if the series converges or diverges. In fact, 1. Switch to Parallel and Series Resistor Calculator. Apr 07, 2021 · Infinite Series Calculator: Finding the sum of an infinite series of a function is not so simple or easy for any one. Online Integral Calculator » Solve integrals with Wolfram|Alpha. Corrected a couple of typing errors. Convergence Calculator. Series Divergence Test Calculator. An alternating series can be identified because terms in the series will "alternate" between + and -, because of. Using Partial Fraction. We know that this power series will converge for x=2. Switching back and forth between the two may not seem like multitasking, but it is a form of multitasking. Integral Test (10. Solution: Let us take Cn=2 n /nx(4x-8) n. is divergent. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. In order to use this test, we first need to know what a converging series and a diverging series is. The Geometric Series Test is the obvious test to use here, since this is a geometric series. Lets look at some examples of convergent and divergence series. This tool calculates the overall capacitance value for multiple capacitors connected either in series or in parallel. Look at the partial sums: because of cancellation of adjacent terms. State the nth term test. Free Online Test Series & Mock Tests for SSC CGL, SBI PO, RRB NTPC, IBPS PO & Other Govt. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. the test of divergence is very similar to that of convergence: Go to Using a Scientific Calculator for. It will have difficult mathematical operations and it consumes your time and energy. developed with the support of the. These series converge for p > 1 and diverge for p 1. a n has a form that is similar to one of the above, see whether you can use the comparison test: ∞. We first start with the plot of { a n } n = 0. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. If you want the Maclaurin polynomial, just set the point to. Series Capacitor. The applet did not load, and the above is. CALCULUS CONVERGENCE AND DIVERGENCE TEST NAME SERIES CONVERGES DIVERGES ADDITIONAL INFO nth TERM TEST X1 n=1 an if lim n!1 an 6=0 One should perform this test first for divergence. This test is the sufficient convergence test. a n has a form that is similar to one of the above, see whether you can use the comparison test: ∞. com allows you to find the sum of a series online. Convergence or divergence of a series is proved using sufficient conditions. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. This test, according to Wikipedia, is one of the easiest tests to apply; hence it is the first "test" we check when trying to determine whether a series converges or diverges. The \\(N\$$th term test, generally speaking, does not guarantee convergence of a series. Use the nth term test to determine whether the following series converges or diverges. The Fund does not include the swap for purposes of the 80% investment test described above. 0 ≤ y ≤ f x x ≥ 1. Get full mock tests, Sectional mock tests, previous paper mock tests. Since the harmonic series is known to diverge, we can use it to compare with another series. Conic Sections Transformation. Condition of Divergence: 𝑝ᩣ1 4 Alternating Series Test 𝑛+1 𝑛 ∞ 𝑛=1 Condition of Convergence: and 0 < decreasing 𝑛+1 ᩣ 𝑛 lim 𝑛→∞ 𝑛=0 or if ∑∞ | 𝑛| 𝑛=0 is …. For example, consider the series \[\sum_{n=1}^∞\dfrac{1}{n^2+1}. Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student. Step 3: Finally, the sum of the infinite geometric sequence will be displayed in the output field. The program doesn't just provide an answer, it provides a step-by-step and detailed solution. The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. Use of the Geometric Series calculator. A telescoping series is any series where nearly every term cancels with a preceeding or following term. The steps are identical, but the outcomes are different!. If , then nothing can be said about the series , that is the series may be convergent or divergent. In order to use this test, we first need to know what a converging series and a diverging series is. If you want the Maclaurin polynomial, just set the point to. The same is true for p -series and you can prove this using the integral test. Share Bookmark. Then you'd have to use additional convergence tests to figure out series convergence or divergence. then the series a n and b n either both converge or both diverge. The Divergence Test Return to the Series, Convergence, and Series Tests starting page; Return to the List of Series Tests. Which "Divergent" Faction Do You Actually Belong In? Choose your fate, Initiates. BYJUS online remainder theorem calculator tool makes the calculation faster and it displays the result in a fraction of seconds. Series Test says that the series converges. Test used to sync properly? Floral and leaf texture. powered by. 3) L=1 the series either converges or diverges. If then the series diverges. The Divergent Faction quiz starts with the Aptitude Test—just like the original story. Convergent and divergent thinking require two different parts of the brain. a a and the constant ratio. It uses the KL divergence to calculate a normalized score that is symmetrical. Series Divergence Test Calculator - Symbolab › Search The Best Online Courses at www. In our Series blogs, we've gone over four types of series, Geometric, p, Alternating, and Telescoping, and their convergence tests. We note that is a positive decreasing series and the function is continuous. We have for. e the beam divergence given 2 beam diameter values at a given separation ; Calculate Intensity Drop at Divergence. The series alternates signs, is decreasing in absolute value, and the limit of the nth term. The sum of infinite terms that follow a rule. Proofs for both tests are also given. Integral Test - Basic Idea. If the limit of a[n] is not zero,. Try the free Mathway calculator and problem solver below to practice various math topics. Test the series for convergence or divergence using the Alternating Series Test. We note that is a positive decreasing series and the function is continuous. * Remainder: | 𝑛|ᩣ 𝑛+1 5 Integral Test Series: ∑. Let's plot the terms of two sequences : { a n } n = 0, which consists of positive terms and the sequence of partial sums { s n } n = 0 for the alternating series ∑ k = 0 ∞ ( − 1) k a k. Your first 5 questions are on us!. By using this website, you agree to our Cookie Policy. What is the nth Term Test? The nth term test is probably the test that can be applied to the most series. Using your calculator, calculate to verify that the sum of the partial sums is bounded by the sum you found in part (a). On the table in front of you, there is a hunk of cheese, and a knife. Find the sum of the convergent series X∞ n=1 1 n(n+1) Solution. and the nth term an = a1 r n - 1. Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student. Infinite Series. Theorem: Let be a p -series where. 1 Equation 2 Use 3 Notes 4 Explanation 5 Graph 6 Video Explanation First, if the equation shown above is in the form of a trinomial, find the partial sums of the summation. Use the nth Term Divergence Test to determine whether or not the following series converge: (a) 23 3 1 13 n 4 5 2 nn nn f 1 ¦ (b) 2 1 n n f ¦ (c) 1! n 2 ! 1 n n f ¦ (d) 1 2! n 10 ! n n ¦ 4. After the proof has been provided, it is expected that those methods will be used for subsequent problem solving; and once an outcome has been reached a result is generally shown as: "convergence, (or divergence), for ∑ bn is demonstrated, therefore ∑ an is also convergent, (or divergent), {by the relevant series test}. On the bright side, this method is a lot more plug-and-chug: once you pick the series to compare, you just throw them into a limit problem and execute. The limit comparison test for series convergence. To see this, do a limit comparison with the divergent series P 1 n: lim n→∞. How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. The alternating series test for convergence lets us say whether an alternating series is converging or diverging. Use the comparison test to determine whether the following series are convergent or divergent. The alternating harmonic series is conditionally convergent since we saw before that it converges by the alternating series test but its absolute value (the harmonic series) diverges. Use the nth term test to determine whether the following series converges or diverges. Mar 10, 2016 · Use Gauss’ test (from the previous exercise, Section 10. Infinite series can be very useful for computation and problem. For instance, the series is telescoping. The ratio test was …. A series of events featuring lectures, service awards and volunteer opportunities will be held Jan. ∞ ∑ n=1ln( 2n +1 n + 1) diverges. People are beginning to ask about how they can obtain some of the health information that they were previously receiving from 23andMe. Practice this lesson yourself on Kh. Free series convergence calculator - test infinite series for convergence step-by-step This website uses cookies to ensure you get the best experience. , of the string's fundamental wavelength. Convergence or divergence of a series is proved using sufficient conditions. Colorful with a mixed environment. Dirichlet's test is a generalization of the alternating series test. This series converges by the alternating series test. Related Symbolab blog posts. Take the limit of the series given and use the Divergence Test in identifying if the series is divergent or convergent. Before we can learn how to determine the convergence or divergence of a geometric series, we have to define a geometric series. Series convergence calculator - mathforyou. Geometric Series Solver. Integral Test X∞ n=0 a n with a n ≥ 0 and a n decreasing Z ∞ 1 f(x)dx and X∞ n=0 a n both converge/diverge where f(n) = a n. Example 2: Find the radius of converge, then the interval of convergence, for ∑ n = 1 ∞ ( − 1) n x n n. An infinite sequence (a n) is called convergent if limit n tends to infinity a n exists and is finite. Common ratio, r: First term, a1: Show translations. : It's a straight forward process that focuses on figuring out the most effective answer. Therefore we need to evaluate, we have, therefore:. What is Special about a Geometric Series. 5867659956 Strange humanoid creature caught on sex or an introduction. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Polyhedra packing animation; Wag the dog- Harmonic Oscillator. Practice your math skills and learn step by step with our math solver. Nth Term Test (Black) Shows divergence when black line is not at y = 0; otherwise, the test is inconclusive. Otherwise is called divergent series. Radius of convergence calculator | interval of convergence. I Therefore 2 1=n n >1 n for n 1. A convergent series is a mathematical series in which the sequence of partial sums converges to 1. Keep in mind that the test does not tell whether the series diverges. Calculate the imappropriate integral = infty. The beam divergence describes the widening of the beam over the distance. Example 2: Find the radius of converge, then the interval of convergence, for ∑ n = 1 ∞ ( − 1) n x n n. Using your calculator, calculate to verify that the sum of the partial sums is bounded by the sum you found in part (a). Applicable pay rate determined? Hardly nobody can show anyone. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. The Art of Convergence Tests. To diverge, the opposite of the near triad must occur. The root test uses a similar idea in a slightly different situation. See more: 80000 Student Loan Monthly Payment S With A Free Calculator, Loan Payment Calculator. Stage #1: Aptitude Test. The comparison tests we consider below are just the sufficient conditions of convergence or divergence of series. Get detailed solutions to your math problems with our Power series step-by-step calculator. It incorporates the fact that a series converges if and only if a constant multiple of it converges. EXAMPLE 10: Does the following series converge or diverge? SOLUTION: The limit does not exist, therefore, the series diverges by the nth term test for divergence. Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Apr 06, 2021 · Problems. 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https://physics.stackexchange.com/questions/623543/different-results-for-torque-in-inertial-and-non-inertial-frames-of-reference/623797 | # Different results for torque in inertial and non-inertial frames of reference
I have a right handed coordinate system with origin O. On the plane yz there's a triangular-shaped plate with sides lying on the axes, both of length a. The plate rotates around the z axis (vertical with respect to the ground) with angular velocity ω. I want to find the external torque with respect to O needed to keep the angular velocity constant.
I've tried to solve the problem both with respect to an inertial frame of reference and to a non-inertial one.
Inertial frame of reference
Since the chosen pole is O, all the reaction forces that the rod applies on the plate have no torque. The only other force on the plate is its weight,
$$\vec{W} = -mg\hat{z}$$
Then the total torque on the plate is
$$\vec{M_O} = \vec{M_{ext}} - \frac{mga}{3}\hat{x}$$
since the plate's center of mass is in (0, a/3, a/3).
From Euler's equation, given that the angular velocity is constant, we have
$$\vec{M_O} = \vec{\omega} \wedge I\vec{\omega}$$
Since ω has only the z component, I just calculated the last column of the inertial tensor I. I found:
$$I = \begin{bmatrix} 0 \\ -\frac{ma^2}{3} \\ \frac{ma^2}{12} \end{bmatrix}$$
Now I have the equation:
$$\frac{m\omega^2a^2}{12}\hat{x} = \vec{M_{ext}} - \frac{mga}{3}\hat{x}$$
And therefore:
$$\vec{M_{ext}} = \frac{m\omega^2a^2}{12}\hat{x} + \frac{mga}{3}\hat{x}$$
Non-inertial frame of reference
First step I did was to calculate the pseudo force on the center of mass.
$$F_{app} = \frac{m\omega^2a}{3}\hat{y}$$
In this frame of reference the plate is static, so the second cardinal equation of statics must apply:
$$\vec{M_{ext}} - \frac{mga}{3}\hat{x} - \frac{m\omega^2a^2}{9}\hat{x} = 0$$
So I find:
$$\vec{M_{ext}} = \frac{mga}{3}\hat{x} + \frac{m\omega^2a^2}{9}\hat{x}$$
As you can see the two solutions are similar but no equal. Could you please explain me why?
• I understand now, you are asking about the support torques on the pivot. – JAlex Mar 25 at 20:52
I think this is the situation
$$\vec{M}_O = \vec{M}_{\rm ext} + \vec{c} \times \vec{W} = \vec{M}_{\rm ext} + \pmatrix{ -\tfrac{a}{3} m g\\ 0 \\ 0}$$
Here $$\vec{c} = \pmatrix{0 \\ \tfrac{a}{3} \\ \tfrac{a}{3} }$$ is the center of mass relative to O, and $$\vec{W} = \pmatrix{0 \\ 0 \\ -m g}$$ the weight acting through the center of mass.
The mass moment of inertia tensor about O is
$$\mathbf{I}_O = \begin{bmatrix} \tfrac{m}{3} a^2 & & \\ & \tfrac{m}{6} a^2 & -\tfrac{m}{12} a^2 \\ & -\tfrac{m}{12} a^2 & \tfrac{m}{6} a^2 \end{bmatrix}$$
Finally the rotational velocity is
$$\vec{\omega} = \pmatrix{ 0 \\ 0 \\ \dot{\theta} }$$
So the rotational torque balance is
$$\vec{M}_O = \vec{M}_{\rm ext} + \vec{c} \times \vec{W} = \vec{\omega} \times \mathbf{I}_O \vec{\omega}$$
or $$\vec{M}_{\rm ext} = \pmatrix{ \tfrac{a}{3} m g + \tfrac{a^2}{12} m \dot{\theta}^2 \\ 0 \\ 0 }$$
Which matches your first result. Hence the error is in the second method. I suspect that torque = change in angular moment isn't valid for non inertial frames. In fact I do not see anything about changing angular momentum in the second part. Even though were at a body centric coordinate system, since $$\vec{\omega}$$ is not along a principal axis of inertia the resulting angular momentum will change direction over time.
• In the non-inertial frame of reference, the velocities of the points of the plate is null, so angular momentum is constant, since it's always null (from its definition). It's the angular momentum in the inertial frame of reference to change. Finally, the proof that change in angular momentum = external torque - velocity of the pole x momentum of the center of mass doesn't depend on the frame of reference. Anyway thanks for confirming my first result. – Ubaldo Tosi Mar 29 at 17:56
• @UbaldoTosi - wait you just confirmed an expression that I have been using elsewhere $$\frac{\rm d}{{\rm d}t} L_A = \tau_A + v_A \times p$$ relating torque to change in angular momentum on non fixed frames. Do you have a reference for this since I derived it on my own and wanted external confirmation. – JAlex Mar 29 at 23:25
• My mechanics professor derived it in class. – Ubaldo Tosi Mar 31 at 18:08
I found the solution. I leave an answer here so that anyone else who wants to know what wasn't working can read this.
Every piece of the plate is subject to an apparent force, that is NOT applied on the center of mass, but on the piece itself. If this force was the same for all pieces with the same mass, then we could consider it as applied one the center of mass when calculating its torque. However in this case, the force depends on the position of the piece.
Then the torque caused by the apparent forces (let's call it the apparent torque) is
$$\tau_{App} = \sum{r_i\times F_i}$$
where
$$F_i = m_i\omega^2y_i$$
Going from discrete to continuous, we have
$$\int_{Plate} \sigma\omega^2yzdydz$$
You can immediately see that this is the only component of the inertial tensor (in the inertial frame) that survives Euler's equation, so now we have the exact same result with both methods. | 2021-05-12 21:35:30 | {"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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8554035425186157, "perplexity": 302.13844413792464}, "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-2021-21/segments/1620243989705.28/warc/CC-MAIN-20210512193253-20210512223253-00151.warc.gz"} |
https://mstone.info/posts/willpower-20130421/ | # DRAFT: Review: “Ego Depletion”, JDW 2010
Michael Stone, April 20, 2013, , (src), (all posts)
Earlier this weekend, I read a fun paper (SAGE) by Job, Dweck, and Walton questioning the “strength” model of willpower, I think as part of a much larger research program on whether and how our “self-theories” influence, e.g., our ability to perform or to persist in performing difficult, frustrating, or tiring tasks.
• it asks a valuable research question,
• it uses an initial descriptive study to motivate a randomized experiment, and
• it considers some alternate explanations of the available data.
However, there were also some parts that bothered or confused me. As a result, in the hopes that my bothers and confusions may be of some use to others, I’ve written up several “bug reports” below that may be of interest.
Okay, here we go:
• Selection bias: “People” is a great and accessible gender-neutral plural noun but I nevertheless wish that it was more clear in the abstract and conclusion that the “people” whom the results most directly describe are mostly? (all?) undergraduate university students.
• Ethical considerations: I think it much more likely than not that the described research was approved by the Stanford or University of Zurich IRBs as minimal-risk human subjects research but I still wish that the paper made it clear one way or the other, e.g., with a link to a record of the approved protocol. (Note: the 2012 PSS submission guidelines now request this information but maybe they didn’t in 2009-2010?)
• Modeling Equations: the JDW authors report that they used logistic hierarchical linear models to comprehend their data and they provide some useful coding information but their paper doesn’t contain the modeling equations for any of the models that they fit, thereby making it needlessly difficult to understand the meaning of the regression coefficients that they report. (Note: one of my favorite books, ALDA, has lots of really nice examples of how to write about hierarchical linear models like the ones that I imagine were used here.)
• Effect size: the research agenda underlying all four reported study designs seems to posit that particular patterns of differences in treatment and control group sample Stroop-test accuracy statistics can falsify (or at least, cast doubt on) the Baumeister et al. “strength model of self-control” but JDW do not explain how their measured effect size should influence our belief in the “strength” model. (Note: the 2012 PSS submission guidelines now also require this information.)
• Graphical integrity: the JDW paper contains five plots, numbered “Figure 1”, “Figure 2”, “Figure 3-A”, “Figure 3-B”, and “Figure 3-C”. Examining just plots “1” and “3-B”:
• Plot 1: According to the legend and axis labels for this plot, each record in the underlying dataset has been labeled with one of five conditions; namely: “Nondepleting + Nonlimited-Resource Theory”, “Nondepleting + Limited-Resource Theory”, “Depleting + Nonlimited-Resource Theory”, “Depleting + Limited-Resource Theory”, or “Not Labeled”, but how was this labeling done?
According to the fine print in the figure caption,
“The limited-resource-theory group represents participants 1 standard deviation above the mean on the implicit-theories measure. The nonlimited-resource-theory group represents participants 1 standard deviation below the mean on the implicit-theories measure.”
There are a couple of problems here:
1. Ambiguity: I think the claim in the figure caption is intended to mean something like “The limited-resource-theory group represents participants whose score on the implicit-theories measure was at least one standard deviation above the mean” but I can’t tell for sure.
2. Distributional Assumptions: Labeling participants based on z-scores implicitly assumes that the underlying distribution of scores is normally distributed but no evidence is given that this is so. Why should I believe it to be true?
3. Power: Assuming that I read the caption correctly, how many observations were thrown out as a result of being unlabeled?
Zooming out, though, there are even bigger problems:
1. Bad summarization: This plot only shows one number for each condition, yet each condition ostensibly labels many records. In short: “where are the box plots”?
2. Unnecessary summarization: why group the participants at all? Why not just draw the scatterplot of all participant’s mistake-frequencies as a function of their implicit-theories-measure score, perhaps faceted or colored by their depletion treatment condition? Then you could plot the fitted models as density heat-maps in the background, thereby revealing outliers or other model-fitting problems! (Note: Hadley Wickham’s ggplot2 package makes this kind of plotting super fun and easy!)
• Plot 3b: All five plots have dependent measures with labels that begin with the prefix “Probability of a Mistake” and four of the five figures have ratio scales for these measures: that is, their scales cover intervals ranging from $[0, 0.08]$ to $[0, 0.12]$. Unlike all the other plots, Plot 3b’s scale is presented as an interval scale, covering the interval $[0.20, 0.45]$. Why? (Just to devote more ink to showing the measured inter-class differences? Or is there some deeper confusion about what scale matters for measuring effect sizes?)
• Traceability: We’ve seen some otherwise interesting research brought down recently by simple slips, e.g., in calculation, model-fitting, and data entry. In the software world, we try control for this sort of problem in a bunch of ways, most notably with open source. Anyway, as many others have requested, perhaps it’s time to start providing links to the raw data and to the intermediate analysis results as part of the published supplemental materials? (Also, if the data are already up and I just couldn’t see them because of the SAGE paywall, then maybe the issue is the need for more open access, perhaps in the style of the Episciences Project (intro) or in the style of PLOS ONE (which I see that the JDW authors are already exploring; yay!)?)
• Next, a couple of smaller issues:
• Reproducibility: What font + text was on the pages used for the “stimulus detection” task? (It would only take a few words to say…)
• Validity: how does color-blindness affect the results derived from the Stroop task performance measurements?
• Blinding: Were the randomized controlled trials also blinded?
• Finally, a review of Simmons’ et al.’s researcher degrees of freedom checklist (note 1: introduced to me by Shauna; thanks Shauna!; note 2: also, amusingly, published in PSS!):
• Simmons #1: “Authors must decide the rule for terminating data collection before data collection begins and report this rule in the article.”
• No data collection stopping rules were included in the paper.
• Simmons #2: “Authors must collect at least 20 observations per cell or else provide a compelling cost-of-data-collection justification.”
• Per-cell observation counts were not included in the paper.
• Simmons #3: “Authors must list all variables collected in a study.”
• I believe that all variables collected may be reported in the supplemental material published alongside the original paper but only a subset were reported in the original paper.
• Simmons #4: “Authors must report all experimental conditions, including failed manipulations.”
• I don’t see a claim that all experimental conditions have been reported.
• Simmons #5: *“If observations are eliminated, authors must also report what the statistical results are if those observations are included.”
• I see some effort here, e.g., when the authors observe on their longitudinal study that the 59% of participants who did not complete the study were demographically similar to those who continued.
• Simmons #6: *“If an analysis includes a covariate, authors must report the statistical results of the analysis without the covariate.”
• Some effort is also made here, particularly in Note #1. (That being said, Note #2 and the “speed/accuracy tradeoff” covariates seem just like what Simmons et al. are asking about…) | 2023-01-28 10:17:48 | {"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": 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": 0.5037623643875122, "perplexity": 2619.1912589868975}, "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/1674764499541.63/warc/CC-MAIN-20230128090359-20230128120359-00363.warc.gz"} |
http://clay6.com/qa/43163/silver-iodide-has-the-same-structure-as-zinc-sulfide-and-its-density-is-5-6 | Want to ask us a question? Click here
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# Silver iodide has the same structure as zinc sulfide, and its density is $5.67gcm^{–3}$. The edge length of the unit cell is
Can you answer this question?
## 1 Answer
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$6.50 A^{\large\circ}$
Hence (C) is the correct answer.
answered Jun 5, 2014 | 2016-10-25 06:54:21 | {"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": 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": 0.8880006074905396, "perplexity": 4061.6723346139947}, "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-2016-44/segments/1476988719960.60/warc/CC-MAIN-20161020183839-00017-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.ijfis.org/journal/view.html?uid=961&vmd=Full | Article Search
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## Original Article
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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 222-232
Published online September 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.3.222
© The Korean Institute of Intelligent Systems
## A Restoration Method of Single Image Super Resolution Using Improved Residual Learning with Squeeze and Excitation Blocks
Wang-Su Jeon1 and Sang-Yong Rhee2
1Department of IT Convergence Engineering, Kyungnam University, Changwon, Korea
2Department of Computer Engineering, Kyungnam University, Changwon, Korea
Correspondence to :
Sang-Yong Rhee (syrhee@kyungnam.ac.kr)
Received: November 5, 2020; Revised: September 13, 2021; Accepted: September 16, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Techniques for single-image super-resolution have been developed through deep learning. In this paper, we propose a method using advanced residual learning and squeeze and excitation (SE) blocks for such resolution. Improving the residual learning increases the similarity between pixels by adding one skip to the existing residual, and it is possible to improve the performance while slightly increasing the number of calculations by applying the SE block of SENet. The performance evaluation was tested as part of the super-resolution generative adversarial network (SRGAN) and using three proposed modules, and the effect of the residual and the SE blocks on the super-resolution and the change in performance was confirmed. Although the results vary slightly from image, SE and residual blocks have been found to help improve the performance and are best used with blocks.
Keywords: CNN, Super-resolution, SRGAN, Residual learning, Squeeze and excitation
As living standards improve, demand for high-quality products in all fields, including video images, is increasing. High-quality video means a high resolution and good color. Compression technology is essential for the efficient transmission of high-resolution image data, and a method of restoring information lost during the compression process is required. In addition, medical images, CCTV, and thermal imaging, among other imaging types, do not have a high resolution because of the hardware limitations, and it is difficult to provide an image quality desired by users. An ultra-high-resolution restoration is therefore required to provide a good quality image. Single image super-resolution (SISR) is a technique for restoring a down-sampled image to an original super-resolution image.
Ultra-high super-resolution has made significant progress through recent deep learning technologies [14]. However, most studies have focused only on improving the performance of the complete image [5,6], and few studies have considered the image texture [7]. In particular, when an image is simply divided into high- and low-frequency regions and compressed, it is difficult to restore the image because the loss in the high-frequency region is large. In addition, the higher the frequency, the more important the information is, and it can be stated that restoring a high-frequency region well is more important than improving the overall performance.
We use a super-resolution generative adversarial network (SRGAN) [7] as the base model, and because SENet using squeeze and excitation (SE) blocks is superior to ResNet using residual blocks, we change the residual modules of SRGAN into a module network, SB blocks, or a combination of the two to improve the restoration performance of the texture area [4]. The remainder of this paper is organized as follows. Section 2 describes existing studies on high-resolution restoration methods. We propose the three models in Section 3. Section 4 describes the experimental results and an analysis of the proposed methods, and finally, Section 5 presents some concluding remarks and areas of future research.
One way to obtain a high-resolution image is to physically increase the number of pixels in the image sensor. However, this increases the size of the image sensor, which is not economical and makes it difficult to handle. There is another way to reduce the size of a single pixel to fit many pixels in a certain area. However, if the size of the pixel is restored as much as it is reduced, the quality of the final image deteriorates [8].
Therefore, research on the restoration of ultra-high-resolution images using software has been developed, and active research in this area is still ongoing [9]. Ultra-high-resolution image restoration can be broadly divided into multi-frame-based and single-image-based research. In the case of multi-frame-based research [10, 11], it is possible to restore a super-resolution image by complementing the information acquired for each frame; however, a low-resolution image may not be able to be restored, or the quality of the restored result may degrade owing to insufficient information. In the early stage of SISR image restoration studies, although research based on linear, bicubic, and Lanczon filters with a high execution speed were conducted [12], the resulting images were slightly blurred. To overcome these shortcomings, research on preserving the edges has been introduced [9].
With the recent introduction of deep learning technology, ultra-high-resolution images have been successfully restored. The relationship between high and low resolutions has been discovered in the same image. In particular, as shown in Figure 1, a convolutional neural network CNN [9], a very deep super-resolution (VDSR) [10], and a road structure refined CNN (RSRCNN) [13], and a deeply-recursive convolutional network (DRCN) [14], which are based on CNN [1517], performed better than existing networks. The difference between a low-resolution image obtained using an existing interpolation method and a high-resolution image obtained from these models is shown in Figure 2 [9].
However, CNN-based algorithms using the mean squared error (MSE) based loss functions are still insufficient in representing the textures of ultra-high-resolution images. To address this issue, it is applied to restoring ultra-high-resolution images using SRGAN. An SRGAN is a super-resolution image reconstruction method based on a generative adversarial network [18].
### 2.1 Super Resolution GAN
Similar to other GAN models, a SRGAN consists of a generative model and a discriminative model. The SRGAN generates fake images similar to the real image from random noise in the generative model, and simultaneously learns whether the image generated in the discriminative model is real or fake through the process of discrimination. The discriminant model at the beginning of the learning easily determines the authenticity of the image generated by the generative model; however, as the learning progresses, the generative model generates an image that the discriminant model cannot distinguish.
As such, two neural networks with different purposes can be used to construct a system that generates an image similar to the real one. Existing super-simple image resolution techniques have difficulty in restoring texture, which provides detailed information about high-frequency components.
However, an SRGAN provides a solution to this issue. The structure of the SRGAN used is shown in Figure 3. To increase the accuracy, the network consists of a deep model, batch normalization, and skip connection and is trained using perceptual loss. Perceptual loss is the weighted sum of the content and adversarial losses. The VGG loss function is used as the content loss to improve the image texture expression. The core idea of the VGG loss is to find the difference between the image generated by the generator and the original image, not from the resulting images, but after passing through the feature map.
The goal of the proposed system is to learn the texture information. Because most texture regions are high-frequency signals based on edges, the high-frequency information is approximated through residual learning. In this section, a brief description of a residual network and SENet is provided. In addition, the structure of the proposed method for increasing the effective performance through residual learning is described.
### 3.1 Residual Network
ResNet [19] was developed to solve the problem in which the deeper the neural network depth is, the more difficult the learning becomes. Therefore, by learning the residuals, deep networks can be learned more easily than with GoogleNet [20, 21] and VGGNet [22]. In addition, it is easy to optimize the network, and a high accuracy is obtained through a deep model [2325].
After learning and evaluating ImageNet using ResNet, the result won the ILSVRC 2015 challenge with a very small error of 3.57%, and a good performance was demonstrated in the detection and segmentation of the CIFAR-10 and COCO datasets. Residual learning can be considered an ensemble approach in which the identity mapping between layers and weights learned from shallow models is combined. This is also similar to adjusting the reflection ratio of the old and new memories in long short-term memory. Residual learning can be seen as approximating information between pixels, which improves the restoration performance [26, 27].
### 3.2 Squeeze and Excitation Block
In this study, the SE block, which is applied in SENet [28], was used to improve the performance. The SE block can be applied to any existing model, as shown in Figure 4. If the SE block shown in Figure 3 is added to VGGNet, GoogLeNet, and ResNet, among others, the performance will increase. Although the performance is greatly improved, the number of parameters does not increase significantly, and thus the increase in the number of computations is not large.
In general, to improve the performance, the number of computations increases by that amount. In the case of SENet, use of the SE block is efficient because the accuracy can be improved without significantly increasing the number of computations.
### 3.3 Proposed Module
The structure of the system module suggested in this study is shown in Figure 5. We applied the residual and SE blocks separately to the SRGAN and verified how the network affects the super-resolution by applying it concurrently. Figure 5(a) shows the conventional ResNet, and Figure 5(b) shows ResNet with the addition of an SE block. As mentioned earlier, this aims to maximize the texture information and minimize the smoothness. Figure 5(c) shows the addition of the term of the residual block to Figure 5(a) for use in learning when considering the characteristics at different high frequencies. Figure 5(d) shows the addition of the SE block to the ResNet shown in Figure 5(c).
### 4.1 Experiment Details
The experimental environment used in this study was as follows. The hardware consists of an Intel Xeon Gold 5120 CPU, 64 GB of RAM, and GPGPU is the RTX TITAN X of 24 GB. The deep learning framework is TensorFlow 1.14.0 in Ubuntu 18.04.4, and the dataset used for training is DIV2K 2017 [29]. The DIV2K data consist of high-resolution images with a pixel resolution of 2044×1404 and low-resolution images corresponding to each image. And the training set consists of 800 images, and the test set consists of 100 images.
To form the model, we apply 200 epochs and use the Adam optimizer with a learning rate of 0.0001 and a momentum of 0.9. This model has a batch size of 16. To be used for learning, the generator restores the reduced images (96 × 96 in size) to a pixel resolution of 384 × 384, and the discriminator determines the similarity by inputting the output of the generator and the high-resolution image, executing a binary classification to classify it into low- and high-resolution images, with learning progressing as a single network connecting the generator and discriminator. In the discriminator, the original high-resolution image and the resulting super-resolution image are compared and evaluated, and learning continues. During the learning, the sigmoid function included at the end of the discriminator is removed, and the loss function is calculated. The final loss value is calculated by adding the contradictory loss value calculated in the discriminator to the content loss value generated by the independent VGG at a given ratio. The builder’s formula and the SRGAN identification formula are shown in Equations (1) and (2) [7].
θG˜=argmin1NθGlSR(GθG(InLR),InHR)minθGmaxθDEIHR~Ptrain(IHR)[logDθD(IHR)]+EILR~PG(ILR)[log(1-DθD(GθG(IHR)))].
In the above equation, G is a generator receiving a low-resolution image input and therefore generates a super-resolution image. In addition, θG is the value of the G parameter, and the generated image is the value of the loss function determined by comparing it to the high-resolution image. Moreover, D is the generated image, which is a high-resolution image or a generated image, and θG denotes the augmentation of D.
To quantitatively measure the image quality, we used the peak signal-to-noise ratio (PSNR), which evaluates the image quality loss information; the structural similarity (SSIM); the visual information fidelity (VIF), which evaluates the accuracy of the visual information; and the relative polar edge coherence (RECO), which evaluates the similarity using polar coordinates, as in Equations (3)(6) [30].
In Equation (3), MAXI is the maximum pixel value of the image. In the case of an 8-bit gray image, it was 255. The higher the MSE is, the better the video quality.
The SSIM index is obtained by multiplying the brightness comparison, contrast comparison, and structure comparison values of the actual and reconstructed images. Because the human visual system is specialized in deriving the structural information of an image, as the key hypothesis, if the structural information is distorted, the perception will be significantly affected. In Equaton (4), μi is the average brightness, and σi is the variance of image i. In addition, Ci is a stabilization constant that supports a situation in which the discriminant has a value of close to zero. Moreover, (2μxμy+C1)(μx2+μy2+C1) is an equation comparing the brightness, and (2σxσy+C2)(σx2+σy2+C2) is the contrast comparison. In addition, (σxy+C3)(σxσy+C3) compares the structures of the two images.
VIF calculates the mutual information between C, which is the original image, and E, which is an image recognized by a human. That is the entropy they share. It also calculates the amount of mutual information between D, which is a distorted image, and F, which is recognized as a distorted image. The quality of the distorted image is then predicted by calculating the ratio between the two, as shown in Equation (5).
The ECO metric is an absolute measure, and thus it might be used in principle for no-Reference quality estimation, provided that a quality unit is defined. However, ECO suffers from the fact that it depends on the image content. To compensate for this, the RECO index, the ratio between the images I(x1,x2) and a reference image Ī(x1,x1) of the same scene, is defined through Equation (6).
PSNR=20log10(MAXI)-10log10(MSE),SSIM(x,y)=(2μxμy+C1)(2σxσy+C2)(σxy+C3)(μx2+μy2+C1)(σx2+σy2+C2)(σxσy+C3),VIF=iSubbandsI(CN,j;FN,jSN,j)iSubbandsI(CN,j;EN,jSN,j),RECO(σ)=ECO(σ)+CEC˜O(σ)+C.
### 4.2 Experiment Result
Table 1 shows the performance evaluation for each model, and as a result of the experiment, it can be confirmed that the residual and SE blocks are helpful in improving the performance. In particular, when using the residual and SE blocks, the performance can still be improved. As shown in Table 2, it was possible to see an increase in the operating speed of 1.1- to 2-fold compared to the existing SRGAN; however, the performance was improved, and the performance changed owing to the addition of blocks.
Thereafter, six images were tested with the proposed model, as shown in Table 2. In Table 2, Image #0 is the result of measuring the image quality of the penguin in Figure 6. In the case of Figure 5, the VIF obtained by the proposed method is shown to be the best, and it can be seen that the beak of the penguin in Figure 6 is well restored. In addition, #45 indicates the image in Figure 7, where it can be seen that the proposed the method is similar to the real image and has the smallest grid pattern.
Image #47 is shown in Figure 8, and in the case of the improved residual, a grating pattern is found, although not when applying the proposed method; in addition, the performance evaluation is also high. In the case of #62, as shown in Figure 9, the proposed method achieves the best results for pattern restoration of the hair, and the performance is also high.
In the case of #69, as shown in Figure 10, the performance was better than the proposed method when the residual improved because the pattern of hair was constant. For #82, as shown in Figure 11, the proposed method achieves the best quality evaluation, and the texture restoration is also the highest. Therefore, when using a residual block with an SE block, it was confirmed that the removal of the grid pattern results in a higher restoring force based on the similarity of the pixels.
Super-resolution image restoration is a major issue in the modern era of the storage and exchanging of image data. In this study, an SRGAN, which has contributed to this area, and three proposed models were tested using the DIV2K dataset to achieve an ultra-high-resolution restoration using a single image. A performance evaluation was conducted for PSNR, SSIM, VIF, and RECO. For the entire dataset, it depends on the model, but on average, 0.01% of VIF and SSIM and 1% of PSNR were better. Six images showed differences in the image quality according to the numerical results. However, RECO achieved worse results than the base model. With this, we know that the SE block improves the performance. The SRGAN+Residual+SE model is said to be the best because it takes a longer time than the other approaches. Therefore, the SRGAN+SE or SRGAN+Residual models can be used. However, there was a problem in that a smoothing phenomenon occurred again when the layer was deepened.
In future research, we will consider using SRGAN+SE or a combined SRGAN+Residual Markov random field to the smoothing problem for improving the restoration performance and restoring the resolution of the depth image.
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2020R1F1A107596811).
### Conflict of Interest
Fig. 1.
Type of CNN based super resolution architecture: (a) SR-CNN [9], (b) RSNCNN [13], (c) VDSR [10], and (d) DRCN [14].
Fig. 2.
Comparison of SR and bicubic image.
Fig. 3.
SRGAN overview [7].
Fig. 4.
SE block structure. Fsq: A channel descriptor is created by aggregating the feature maps across the spatial dimension (H×W). This descriptor embeds the global distribution (information) of the channel-wise feature response, and thus information that can be obtained from the global receptive field can be used in all layers of the network. Fex: Simple self-gating is applied to generate the per-channel modulation using weight embedding as the input. C1: Custom module output. C2: SE block inputs and outputs. xin, xout, W, H, and C represent the input, output, width, height, and channel, respectively.
Fig. 5.
Proposed architecture: (a) baseline, (b) SRGAN+SE (c), SRGAN+Residual and (d)SRGAN+Residual+SE.
Fig. 6.
Image #0 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
Fig. 7.
Image #45 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
Fig. 8.
Image #47 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
Fig. 9.
Image #62 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
Fig. 10.
Image #69 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE
Fig. 11.
Image #82 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
Table. 1.
Table 1. Image quality measurement from model of DIV2K 2017 dataset (unit: dB).
SRGANSRGAN +SESRGAN +ResidualSRGAN +Residual+SE
VIF0.371720.380360.3848590.388321
SSIM0.7702720.781850.7767310.783999
PSNR25.2097626.6898326.0503426.33148
RECO0.8253080.7826260.8041530.775451
Table. 2.
Table 2. Image quality measurement from DIV2K 2017 dataset.
SRGANSRGAN +SESRGAN +ResidualSRGAN +Residual+SE
Time (s)1.85642.14822.52963.9106
Image #0
VIF0.4281650.4541840.4552080.457381
SSIM0.7476100.7798860.7851170.782060
PSNR26.39637127.51896727.35776027.436177
RECO0.8542010.7261980.7788290.707260
Image #45
VIF0.2554550.2657570.2693510.270637
SSIM0.6874840.7044390.6977040.709990
PSNR21.86147222.30850522.37628122.294032
RECO0.7735650.8149060.7861660.780359
Image #47
VIF0.3740790.3898110.3937890.401365
SSIM0.7482300.7630340.7272750.763076
PSNR25.41145725.99912925.85330126.207819
RECO0.8631570.8591680.8749120.836694
Image #62
VIF0.2466990.2375770.2496780.252927
SSIM0.7724320.7679370.7737430.781695
PSNR26.34277029.91251827.48657527.825911
RECO0.8274250.7436270.7881960.836395
Image #69
VIF0.3213730.3257260.3289150.320290
SSIM0.7399720.7451780.7467680.734274
PSNR22.13970822.58686022.49036822.105132
RECO0.6587060.6394480.6461930.639967
Image #82
VIF0.6045460.6091050.6122100.627323
SSIM0.9259060.9306250.9297790.932901
PSNR29.10680331.81300230.73776932.119837
RECO0.9747950.9124060.9506200.852030
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Wang-Su Jeon received his B.S. and M.S. degrees in computer engineering and IT convergence engineering from Kyungnam University, Masan, Korea, in 2016 and 2018, respectively, and is currently pursuing a Ph.D. in IT convergence engineering at Kyungnam University. His present interests include computer vision, pattern recognition, and machine learning
E-mail: jws2218@naver.com
Sang-Yong Rhee received his B.S. and M.S. degrees in industrial engineering from Korea University, Seoul, Korea, in 1982 and 1984, respectively, and his Ph.D. in industrial engineering at Pohang University, Pohang, Korea. He is currently a professor in computer engineering, Kyungnam University, Masan, Korea. His research interests include computer vision, augmented reality, neuro-fuzzy, and human-robot interfaces.
E-mail: syrhee@kyungnam.ac.kr
### Article
#### Original Article
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 222-232
Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.222
## A Restoration Method of Single Image Super Resolution Using Improved Residual Learning with Squeeze and Excitation Blocks
Wang-Su Jeon1 and Sang-Yong Rhee2
1Department of IT Convergence Engineering, Kyungnam University, Changwon, Korea
2Department of Computer Engineering, Kyungnam University, Changwon, Korea
Correspondence to:Sang-Yong Rhee (syrhee@kyungnam.ac.kr)
Received: November 5, 2020; Revised: September 13, 2021; Accepted: September 16, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
### Abstract
Techniques for single-image super-resolution have been developed through deep learning. In this paper, we propose a method using advanced residual learning and squeeze and excitation (SE) blocks for such resolution. Improving the residual learning increases the similarity between pixels by adding one skip to the existing residual, and it is possible to improve the performance while slightly increasing the number of calculations by applying the SE block of SENet. The performance evaluation was tested as part of the super-resolution generative adversarial network (SRGAN) and using three proposed modules, and the effect of the residual and the SE blocks on the super-resolution and the change in performance was confirmed. Although the results vary slightly from image, SE and residual blocks have been found to help improve the performance and are best used with blocks.
Keywords: CNN, Super-resolution, SRGAN, Residual learning, Squeeze and excitation
### 1. Introduction
As living standards improve, demand for high-quality products in all fields, including video images, is increasing. High-quality video means a high resolution and good color. Compression technology is essential for the efficient transmission of high-resolution image data, and a method of restoring information lost during the compression process is required. In addition, medical images, CCTV, and thermal imaging, among other imaging types, do not have a high resolution because of the hardware limitations, and it is difficult to provide an image quality desired by users. An ultra-high-resolution restoration is therefore required to provide a good quality image. Single image super-resolution (SISR) is a technique for restoring a down-sampled image to an original super-resolution image.
Ultra-high super-resolution has made significant progress through recent deep learning technologies [14]. However, most studies have focused only on improving the performance of the complete image [5,6], and few studies have considered the image texture [7]. In particular, when an image is simply divided into high- and low-frequency regions and compressed, it is difficult to restore the image because the loss in the high-frequency region is large. In addition, the higher the frequency, the more important the information is, and it can be stated that restoring a high-frequency region well is more important than improving the overall performance.
We use a super-resolution generative adversarial network (SRGAN) [7] as the base model, and because SENet using squeeze and excitation (SE) blocks is superior to ResNet using residual blocks, we change the residual modules of SRGAN into a module network, SB blocks, or a combination of the two to improve the restoration performance of the texture area [4]. The remainder of this paper is organized as follows. Section 2 describes existing studies on high-resolution restoration methods. We propose the three models in Section 3. Section 4 describes the experimental results and an analysis of the proposed methods, and finally, Section 5 presents some concluding remarks and areas of future research.
### 2. Related Work
One way to obtain a high-resolution image is to physically increase the number of pixels in the image sensor. However, this increases the size of the image sensor, which is not economical and makes it difficult to handle. There is another way to reduce the size of a single pixel to fit many pixels in a certain area. However, if the size of the pixel is restored as much as it is reduced, the quality of the final image deteriorates [8].
Therefore, research on the restoration of ultra-high-resolution images using software has been developed, and active research in this area is still ongoing [9]. Ultra-high-resolution image restoration can be broadly divided into multi-frame-based and single-image-based research. In the case of multi-frame-based research [10, 11], it is possible to restore a super-resolution image by complementing the information acquired for each frame; however, a low-resolution image may not be able to be restored, or the quality of the restored result may degrade owing to insufficient information. In the early stage of SISR image restoration studies, although research based on linear, bicubic, and Lanczon filters with a high execution speed were conducted [12], the resulting images were slightly blurred. To overcome these shortcomings, research on preserving the edges has been introduced [9].
With the recent introduction of deep learning technology, ultra-high-resolution images have been successfully restored. The relationship between high and low resolutions has been discovered in the same image. In particular, as shown in Figure 1, a convolutional neural network CNN [9], a very deep super-resolution (VDSR) [10], and a road structure refined CNN (RSRCNN) [13], and a deeply-recursive convolutional network (DRCN) [14], which are based on CNN [1517], performed better than existing networks. The difference between a low-resolution image obtained using an existing interpolation method and a high-resolution image obtained from these models is shown in Figure 2 [9].
However, CNN-based algorithms using the mean squared error (MSE) based loss functions are still insufficient in representing the textures of ultra-high-resolution images. To address this issue, it is applied to restoring ultra-high-resolution images using SRGAN. An SRGAN is a super-resolution image reconstruction method based on a generative adversarial network [18].
### 2.1 Super Resolution GAN
Similar to other GAN models, a SRGAN consists of a generative model and a discriminative model. The SRGAN generates fake images similar to the real image from random noise in the generative model, and simultaneously learns whether the image generated in the discriminative model is real or fake through the process of discrimination. The discriminant model at the beginning of the learning easily determines the authenticity of the image generated by the generative model; however, as the learning progresses, the generative model generates an image that the discriminant model cannot distinguish.
As such, two neural networks with different purposes can be used to construct a system that generates an image similar to the real one. Existing super-simple image resolution techniques have difficulty in restoring texture, which provides detailed information about high-frequency components.
However, an SRGAN provides a solution to this issue. The structure of the SRGAN used is shown in Figure 3. To increase the accuracy, the network consists of a deep model, batch normalization, and skip connection and is trained using perceptual loss. Perceptual loss is the weighted sum of the content and adversarial losses. The VGG loss function is used as the content loss to improve the image texture expression. The core idea of the VGG loss is to find the difference between the image generated by the generator and the original image, not from the resulting images, but after passing through the feature map.
### 3. Proposed System
The goal of the proposed system is to learn the texture information. Because most texture regions are high-frequency signals based on edges, the high-frequency information is approximated through residual learning. In this section, a brief description of a residual network and SENet is provided. In addition, the structure of the proposed method for increasing the effective performance through residual learning is described.
### 3.1 Residual Network
ResNet [19] was developed to solve the problem in which the deeper the neural network depth is, the more difficult the learning becomes. Therefore, by learning the residuals, deep networks can be learned more easily than with GoogleNet [20, 21] and VGGNet [22]. In addition, it is easy to optimize the network, and a high accuracy is obtained through a deep model [2325].
After learning and evaluating ImageNet using ResNet, the result won the ILSVRC 2015 challenge with a very small error of 3.57%, and a good performance was demonstrated in the detection and segmentation of the CIFAR-10 and COCO datasets. Residual learning can be considered an ensemble approach in which the identity mapping between layers and weights learned from shallow models is combined. This is also similar to adjusting the reflection ratio of the old and new memories in long short-term memory. Residual learning can be seen as approximating information between pixels, which improves the restoration performance [26, 27].
### 3.2 Squeeze and Excitation Block
In this study, the SE block, which is applied in SENet [28], was used to improve the performance. The SE block can be applied to any existing model, as shown in Figure 4. If the SE block shown in Figure 3 is added to VGGNet, GoogLeNet, and ResNet, among others, the performance will increase. Although the performance is greatly improved, the number of parameters does not increase significantly, and thus the increase in the number of computations is not large.
In general, to improve the performance, the number of computations increases by that amount. In the case of SENet, use of the SE block is efficient because the accuracy can be improved without significantly increasing the number of computations.
### 3.3 Proposed Module
The structure of the system module suggested in this study is shown in Figure 5. We applied the residual and SE blocks separately to the SRGAN and verified how the network affects the super-resolution by applying it concurrently. Figure 5(a) shows the conventional ResNet, and Figure 5(b) shows ResNet with the addition of an SE block. As mentioned earlier, this aims to maximize the texture information and minimize the smoothness. Figure 5(c) shows the addition of the term of the residual block to Figure 5(a) for use in learning when considering the characteristics at different high frequencies. Figure 5(d) shows the addition of the SE block to the ResNet shown in Figure 5(c).
### 4.1 Experiment Details
The experimental environment used in this study was as follows. The hardware consists of an Intel Xeon Gold 5120 CPU, 64 GB of RAM, and GPGPU is the RTX TITAN X of 24 GB. The deep learning framework is TensorFlow 1.14.0 in Ubuntu 18.04.4, and the dataset used for training is DIV2K 2017 [29]. The DIV2K data consist of high-resolution images with a pixel resolution of 2044×1404 and low-resolution images corresponding to each image. And the training set consists of 800 images, and the test set consists of 100 images.
To form the model, we apply 200 epochs and use the Adam optimizer with a learning rate of 0.0001 and a momentum of 0.9. This model has a batch size of 16. To be used for learning, the generator restores the reduced images (96 × 96 in size) to a pixel resolution of 384 × 384, and the discriminator determines the similarity by inputting the output of the generator and the high-resolution image, executing a binary classification to classify it into low- and high-resolution images, with learning progressing as a single network connecting the generator and discriminator. In the discriminator, the original high-resolution image and the resulting super-resolution image are compared and evaluated, and learning continues. During the learning, the sigmoid function included at the end of the discriminator is removed, and the loss function is calculated. The final loss value is calculated by adding the contradictory loss value calculated in the discriminator to the content loss value generated by the independent VGG at a given ratio. The builder’s formula and the SRGAN identification formula are shown in Equations (1) and (2) [7].
$θG˜=argmin1N∑θGlSR(GθG(InLR),InHR)$$minθG maxθD EIHR~Ptrain(IHR)[logDθD(IHR)]+EILR~PG(ILR)[log(1-DθD(GθG(IHR)))].$
In the above equation, G is a generator receiving a low-resolution image input and therefore generates a super-resolution image. In addition, θG is the value of the G parameter, and the generated image is the value of the loss function determined by comparing it to the high-resolution image. Moreover, D is the generated image, which is a high-resolution image or a generated image, and θG denotes the augmentation of D.
To quantitatively measure the image quality, we used the peak signal-to-noise ratio (PSNR), which evaluates the image quality loss information; the structural similarity (SSIM); the visual information fidelity (VIF), which evaluates the accuracy of the visual information; and the relative polar edge coherence (RECO), which evaluates the similarity using polar coordinates, as in Equations (3)(6) [30].
In Equation (3), MAXI is the maximum pixel value of the image. In the case of an 8-bit gray image, it was 255. The higher the MSE is, the better the video quality.
The SSIM index is obtained by multiplying the brightness comparison, contrast comparison, and structure comparison values of the actual and reconstructed images. Because the human visual system is specialized in deriving the structural information of an image, as the key hypothesis, if the structural information is distorted, the perception will be significantly affected. In Equaton (4), μi is the average brightness, and σi is the variance of image i. In addition, Ci is a stabilization constant that supports a situation in which the discriminant has a value of close to zero. Moreover, $(2μxμy+C1)(μx2+μy2+C1)$ is an equation comparing the brightness, and $(2σxσy+C2)(σx2+σy2+C2)$ is the contrast comparison. In addition, $(σxy+C3)(σxσy+C3)$ compares the structures of the two images.
VIF calculates the mutual information between C, which is the original image, and E, which is an image recognized by a human. That is the entropy they share. It also calculates the amount of mutual information between D, which is a distorted image, and F, which is recognized as a distorted image. The quality of the distorted image is then predicted by calculating the ratio between the two, as shown in Equation (5).
The ECO metric is an absolute measure, and thus it might be used in principle for no-Reference quality estimation, provided that a quality unit is defined. However, ECO suffers from the fact that it depends on the image content. To compensate for this, the RECO index, the ratio between the images I(x1,x2) and a reference image Ī(x1,x1) of the same scene, is defined through Equation (6).
$PSNR=20log10(MAXI)-10log10(MSE),$$SSIM(x,y)=(2μxμy+C1)(2σxσy+C2)(σxy+C3)(μx2+μy2+C1)(σx2+σy2+C2)(σxσy+C3),$$VIF=∑i∈S ubbandsI(C→N,j;F→N,j∣SN,j)∑i∈S ubbandsI(C→N,j;E→N,j∣SN,j),$$RECO(σ)=ECO(σ)+CEC˜O(σ)+C.$
### 4.2 Experiment Result
Table 1 shows the performance evaluation for each model, and as a result of the experiment, it can be confirmed that the residual and SE blocks are helpful in improving the performance. In particular, when using the residual and SE blocks, the performance can still be improved. As shown in Table 2, it was possible to see an increase in the operating speed of 1.1- to 2-fold compared to the existing SRGAN; however, the performance was improved, and the performance changed owing to the addition of blocks.
Thereafter, six images were tested with the proposed model, as shown in Table 2. In Table 2, Image #0 is the result of measuring the image quality of the penguin in Figure 6. In the case of Figure 5, the VIF obtained by the proposed method is shown to be the best, and it can be seen that the beak of the penguin in Figure 6 is well restored. In addition, #45 indicates the image in Figure 7, where it can be seen that the proposed the method is similar to the real image and has the smallest grid pattern.
Image #47 is shown in Figure 8, and in the case of the improved residual, a grating pattern is found, although not when applying the proposed method; in addition, the performance evaluation is also high. In the case of #62, as shown in Figure 9, the proposed method achieves the best results for pattern restoration of the hair, and the performance is also high.
In the case of #69, as shown in Figure 10, the performance was better than the proposed method when the residual improved because the pattern of hair was constant. For #82, as shown in Figure 11, the proposed method achieves the best quality evaluation, and the texture restoration is also the highest. Therefore, when using a residual block with an SE block, it was confirmed that the removal of the grid pattern results in a higher restoring force based on the similarity of the pixels.
### 5. Conclusion
Super-resolution image restoration is a major issue in the modern era of the storage and exchanging of image data. In this study, an SRGAN, which has contributed to this area, and three proposed models were tested using the DIV2K dataset to achieve an ultra-high-resolution restoration using a single image. A performance evaluation was conducted for PSNR, SSIM, VIF, and RECO. For the entire dataset, it depends on the model, but on average, 0.01% of VIF and SSIM and 1% of PSNR were better. Six images showed differences in the image quality according to the numerical results. However, RECO achieved worse results than the base model. With this, we know that the SE block improves the performance. The SRGAN+Residual+SE model is said to be the best because it takes a longer time than the other approaches. Therefore, the SRGAN+SE or SRGAN+Residual models can be used. However, there was a problem in that a smoothing phenomenon occurred again when the layer was deepened.
In future research, we will consider using SRGAN+SE or a combined SRGAN+Residual Markov random field to the smoothing problem for improving the restoration performance and restoring the resolution of the depth image.
### Fig 1.
Figure 1.
Type of CNN based super resolution architecture: (a) SR-CNN [9], (b) RSNCNN [13], (c) VDSR [10], and (d) DRCN [14].
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 2.
Figure 2.
Comparison of SR and bicubic image.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 3.
Figure 3.
SRGAN overview [7].
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 4.
Figure 4.
SE block structure. Fsq: A channel descriptor is created by aggregating the feature maps across the spatial dimension (H×W). This descriptor embeds the global distribution (information) of the channel-wise feature response, and thus information that can be obtained from the global receptive field can be used in all layers of the network. Fex: Simple self-gating is applied to generate the per-channel modulation using weight embedding as the input. C1: Custom module output. C2: SE block inputs and outputs. xin, xout, W, H, and C represent the input, output, width, height, and channel, respectively.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 5.
Figure 5.
Proposed architecture: (a) baseline, (b) SRGAN+SE (c), SRGAN+Residual and (d)SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 6.
Figure 6.
Image #0 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 7.
Figure 7.
Image #45 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 8.
Figure 8.
Image #47 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 9.
Figure 9.
Image #62 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 10.
Figure 10.
Image #69 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
### Fig 11.
Figure 11.
Image #82 quality measurement in DIV2K dataset: (a) high resolution, (b) SRGAN, (c) SRGAN+SE, (d) SRGAN+Residual, and (e) SRGAN+Residual+SE.
The International Journal of Fuzzy Logic and Intelligent Systems 2021; 21: 222-232https://doi.org/10.5391/IJFIS.2021.21.3.222
Image quality measurement from model of DIV2K 2017 dataset (unit: dB).
SRGANSRGAN +SESRGAN +ResidualSRGAN +Residual+SE
VIF0.371720.380360.3848590.388321
SSIM0.7702720.781850.7767310.783999
PSNR25.2097626.6898326.0503426.33148
RECO0.8253080.7826260.8041530.775451
Image quality measurement from DIV2K 2017 dataset.
SRGANSRGAN +SESRGAN +ResidualSRGAN +Residual+SE
Time (s)1.85642.14822.52963.9106
Image #0
VIF0.4281650.4541840.4552080.457381
SSIM0.7476100.7798860.7851170.782060
PSNR26.39637127.51896727.35776027.436177
RECO0.8542010.7261980.7788290.707260
Image #45
VIF0.2554550.2657570.2693510.270637
SSIM0.6874840.7044390.6977040.709990
PSNR21.86147222.30850522.37628122.294032
RECO0.7735650.8149060.7861660.780359
Image #47
VIF0.3740790.3898110.3937890.401365
SSIM0.7482300.7630340.7272750.763076
PSNR25.41145725.99912925.85330126.207819
RECO0.8631570.8591680.8749120.836694
Image #62
VIF0.2466990.2375770.2496780.252927
SSIM0.7724320.7679370.7737430.781695
PSNR26.34277029.91251827.48657527.825911
RECO0.8274250.7436270.7881960.836395
Image #69
VIF0.3213730.3257260.3289150.320290
SSIM0.7399720.7451780.7467680.734274
PSNR22.13970822.58686022.49036822.105132
RECO0.6587060.6394480.6461930.639967
Image #82
VIF0.6045460.6091050.6122100.627323
SSIM0.9259060.9306250.9297790.932901
PSNR29.10680331.81300230.73776932.119837
RECO0.9747950.9124060.9506200.852030
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30. Zhai, G, and Min, X (). Perceptual image quality assessment: a survey. Science China Information Sciences. 63, 2020. article no. 211301 | 2022-11-30 16:26:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.5585975050926208, "perplexity": 2111.1940510841177}, "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/1669446710765.76/warc/CC-MAIN-20221130160457-20221130190457-00754.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=214363 | ## Same temperature and entropy
Does entropy change when temperature remains constant? What if heat is added into a system, while the volume expands and the pressure drops at a constant temperature? Is there any change in entropy?
PhysOrg.com physics news on PhysOrg.com >> Iron-platinum alloys could be new-generation hard drives>> Lab sets a new record for creating heralded photons>> Breakthrough calls time on bootleg booze
Recognitions: Gold Member Homework Help Science Advisor Does entropy change when temperature remains constant? Yes, entropy can change even at constant temperature. For example, adding material into a system increases entropy, as does any irreversible isothermal process such as free expansion of a gas into a vacuum. What if heat is added into a system, while the volume expands and the pressure drops at a constant temperature? Is there any change in entropy? Yes, heating a system always increases its entropy. Another way to view this process is that the temperature is the same, but the space for atomic motion has increased because the volume increased. There are therefore more available microstates for the system, which is equivalent to saying the entropy has increased.
Recognitions:
$$\Delta G = \Delta H - T\Delta S$$
G is the Gibbs free energy, H the enthalpy, T temeprature, S entropy. There are lots of processes that use this relationship: phase transitions, chemical reactions, etc. The sign of $$\Delta G$$ tells you if the processes is spontaneous or not- in biochemical reactions, non-spontaneous reactions are generally powered by using the chemical energy in adenosine triphosphate (ATP) or guanine triphosphate (GTP)- but not in the way elementary texts describe. | 2013-05-21 22:47:01 | {"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": 0.4509595036506653, "perplexity": 1086.178048208982}, "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-2013-20/segments/1368700842908/warc/CC-MAIN-20130516104042-00031-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/70069/parametric-form-of-an-ellipse-given-by-ax2-by2-cxy-d | # Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$
If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$.
When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I find the angular shift and from there how do I adjust the factors multiplying the sine and cosine?
-
Principal axis theorem gives you an answer in the sense that there is a worked out example there. Hopefully somebody has the time to give you a bit more theory. I gotta go :-( – Jyrki Lahtonen Oct 5 '11 at 14:03
Formulae 16-23 here are useful. – Ｊ. Ｍ. Oct 5 '11 at 16:05
You would complete squares: $\left(a x + \frac{1}{2} c y\right)^2 + \left(a b - \frac{c^2}{4} \right) y^2 = a d$.
From there: $a x + \frac{c}{2} y = \sqrt{a d} \sin(t)$ and $\sqrt{a b - \frac{c^2}{4}} y = \sqrt{a d} \cos(t)$, assuming $c^2 < 4 a b$, and $a d > 0$.
Solving for $x$ and $y$ and denoting $\mathcal{D} = 4 a b - c^2$ $$x(t) = \sqrt{\frac{d}{a}} \left( \sin(t) - \frac{c}{\sqrt{\mathcal{D}} } \cos(t) \right) \qquad y(t) = \frac{2 \sqrt{a d}}{\sqrt{\mathcal{D}}} \cos(t)$$
-
I like that. +2 if I could. Much appreciated. – jnm2 Oct 5 '11 at 15:33
+1 You didn't use the principal axes, but found a parametrization anyway :-) – Jyrki Lahtonen Oct 5 '11 at 15:48
In the Olden Days (when I was in school) there was a college course called "Analytic Geometry". It would include such things as finding the rotation to eliminate the $xy$ term in a plane conic section. – GEdgar Oct 5 '11 at 18:55
If the parametric ellipse coordinates are $\left(x(t),y(t)\right) = (X \cos\varphi \cos(t)-Y \sin\varphi \sin(t), Y \cos\varphi \sin(t)+X \sin\varphi \cos(t))$
Then the parameters $X,Y,\varphi$ are $$X =\pm \sqrt{\frac{2 d}{a+b+\sqrt{(a-b)^2+c^2}}}$$
$$Y =\pm \sqrt{\frac{2 d}{a+b-\sqrt{(a-b)^2+c^2}}}$$
$$\varphi = \frac{1}{2}\tan^{-1}\left(\frac{c}{a-b}\right)$$
Example:
$10 x^2+20 y^2-18 x y = 100$
The coefficients are ($a=10$, $b=20$, $c=-18$, $d=100$)
The parametric coefficients are $(-1.008311 \cos(t)+3.973667 \sin(t), 1.713639 \cos(t)+2.338119 \sin(t))$
-
(This is supposed to be a comment on ja72's answer, but it got too long.)
$$\tan\,2\varphi = \frac{c}{a-b}$$
I consider it a bit wasteful of effort to evaluate an arctangent and the subsequently pass it as the argument of a trigonometric unction much later; to avoid this, we can use the double angle formula for the tangent and the "Citardauq" formula in tandem to yield the relation
$$\tan\,\varphi=\frac{c}{a-b+(\mathrm{sign}\,c)\sqrt{c^2+(a-b)^2}}$$
Denoting this expression as $t$, we can then substitute this into the rotation matrix
$$\frac1{\sqrt{1+t^2}}\begin{pmatrix}1&-t\\t&1\end{pmatrix}$$
ja72's answer already gave the formulae for the axes; remember that $(p\sin\,u,q\cos\,u)$ and $(p\cos\,u,q\sin\,u)$ are the same ellipse traversed differently, so you can subsequently rotate whichever of the two expressions you pick with the rotation matrix I gave earlier.
- | 2014-10-24 18:51:10 | {"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": 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": 0.9427999258041382, "perplexity": 395.79842435158173}, "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-2014-42/segments/1414119646425.54/warc/CC-MAIN-20141024030046-00292-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.sparrho.com/item/finiteness-properties-of-the-n-equals-4-super-yang--mills-theory-in-supersymmetric-gauge/93a641/ | # Finiteness Properties of the N=4 Super-Yang--Mills Theory in Supersymmetric Gauge
Research paper by Laurent Baulieu, Guillaume Bossard, Silvio Paolo Sorella
Indexed on: 04 Feb '08Published on: 04 Feb '08Published in: High Energy Physics - Theory
#### Abstract
With the introduction of shadow fields, we demonstrate the renormalizability of the N=4 super-Yang--Mills theory in component formalism, independently of the choice of UV regularization. Remarkably, by using twisted representations, one finds that the structure of the theory and its renormalization is determined by a subalgebra of supersymmetry that closes off-shell. Starting from this subalgebra of symmetry, we prove some features of the superconformal invariance of the theory. We give a new algebraic proof of the cancellation of the $\beta$ function and we show the ultraviolet finiteness of the 1/2 BPS operators at all orders in perturbation theory. In fact, using the shadow field as a Maurer--Cartan form, the invariant polynomials in the scalar fields in traceless symmetric representations of the internal R-symmetry group are simply related to characteristic classes. Their UV finiteness is a consequence of the Chern--Simons formula. | 2020-11-30 05:21:42 | {"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": 0.6987013220787048, "perplexity": 982.3019979706064}, "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-2020-50/segments/1606141205147.57/warc/CC-MAIN-20201130035203-20201130065203-00114.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/jimo.2013.9.689 | # American Institute of Mathematical Sciences
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July 2013, 9(3): 689-701. doi: 10.3934/jimo.2013.9.689
## Convex hull of the orthogonal similarity set with applications in quadratic assignment problems
1 State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, 100191, China
Received August 2012 Revised November 2012 Published April 2013
In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while the second highly outperforms several well-known lower bounds in literature.
Citation: Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial and Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689
##### References:
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##### References:
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2021 Impact Factor: 1.411 | 2022-07-05 05:55:40 | {"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": 0.5720889568328857, "perplexity": 5090.8599218852105}, "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/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00722.warc.gz"} |
https://zbmath.org/?q=an:0755.06006 | zbMATH — the first resource for mathematics
Generalized orthomodular posets. (English) Zbl 0755.06006
The autor generalizes the notion of generalized orthomodular lattices, introduced by Janowitz, to the notion of generalized orthomodular posets GOMP. Every orthomodular poset is a GOMP. Every (weak) GOMP $$A$$ can be embedded as an order ideal in an orthomodular poset $$P$$ such that $$x\in A$$ or $$x^ \bot\in A$$ for every $$x\in P$$. For a GOMP existing suprema $$x\lor y\in A$$ are preserved. Every Rickart *-ring “is” a GOMP.
Reviewer: G.Kalmbach (Ulm)
MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets | 2021-09-21 20:12:36 | {"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": 0.8709409236907959, "perplexity": 2655.127278200397}, "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-2021-39/segments/1631780057227.73/warc/CC-MAIN-20210921191451-20210921221451-00612.warc.gz"} |
https://mathoverflow.net/questions/408015/semipositive-curvature-on-holomorphic-line-bundle | # Semipositive curvature on holomorphic line bundle
Let $$(X,\omega)$$ be a (possibly non-Kähler) compact hermitian manifold and let $$L\rightarrow X$$ be a holomorphic line bundle. Is there an algebraic characterization of (Griffiths) semi-positivity of $$L$$? i.e. such that there is a non-singular smooth hermitian metric $$h$$ on $$L$$ such that $$iF_{\nabla^C}\geq0$$?
This condition should be stronger than nef (which implies that for any $$\epsilon>0$$ there is $$h_\epsilon$$ hermitian on $$L$$ s.t. $$iF_{\nabla^C(h_\epsilon)}\geq -\epsilon \omega$$) and a regular version of pseudoeffective (for which one has a class of singular hermitian metrics with 'minimal singularities', see "Analytic methods in algebraic geometry" $$\S$$6, J. P. Demailly).
• I don't think that there is a characterization, no. Of course, you always have the implication $L$ semi-ample (i.e. $L^{\otimes m}$ is basepoint free for some $m>0$) implies $L$ admits an hermitian metric with semipositive curvature. Nov 8, 2021 at 13:25 | 2023-02-01 16:13:38 | {"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": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9618614912033081, "perplexity": 285.9976844893491}, "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/1674764499946.80/warc/CC-MAIN-20230201144459-20230201174459-00797.warc.gz"} |
https://www.infoq.com/articles/excel-lambda-turing-complete/?itm_source=articles_about_low-code&itm_medium=link&itm_campaign=low-code | Facilitating the Spread of Knowledge and Innovation in Professional Software Development
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InfoQ Homepage Articles The Excel Formula Language Is Now Turing-Complete
# The Excel Formula Language Is Now Turing-Complete
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### Key Takeaways
• The Excel team announced LAMBDA, a new Excel feature that lets users define and name custom formula functions that behave like standard Excel functions
• Custom LAMBDA functions admit parameters, can call other LAMBDA functions and recursively call themselves
• With LAMBDA, the Excel formula language is Turing-complete. User-defined functions can thus compute anything without resorting to imperative languages (e.g., VBA, JavaScript)
• The Microsoft team is also experimenting with type and dimensional inference enabled by machine learning.
• The new feature is poised to considerably simplify formulas for both power and casual Excel users. It may however take time for the community to realize the full potential of LAMBDA.
Brian Jones, head of product for Excel, recently announced LAMBDA, a new capability added to the Excel formula language. LAMBDA lets users define custom functions using Excel’s formula language, rather than JavaScript or VBA.
With the addition of custom functions that can call each other and recursively call themselves, Excel’s formula language becomes Turing-complete, effectively meaning that Excel users can compute anything without resorting to another programming language. The Excel community has already started to put the feature to use.
Simon Peyton Jones, a major contributor to the Haskell functional programming language and researcher at Microsoft Research, noted that the Excel formula language may be the most popular functional programming language in use in the world.
In a presentation from the Calc Intelligence team, Jones explained the rationale behind the new evolution of Excel’s formula language as follows:
When I first joined Microsoft 22 years ago, my first question was how can a functional programming researcher make an impact at Microsoft? So I soon zeroed in on Excel because Excel’s formula language is precisely a purely functional programming language. Moreover, it’s more widely used than any other programming language on the planet. [However,] Excel’s formula language, considered as a programming language, is terribly limited.
[...]
You can’t define new functions. You can only write formulas that call the existing built-in 600 functions that are part of Excel. I don’t really count writing new functions in JavaScript here because end users just can’t do that.
Jones mentioned that the ideas that eventually led to the implementation of LAMBDA date from the early 2000s. LAMBDA adds to Excel spreadsheets the most fundamental mechanism that programmers use to control complexity: the ability to define reusable abstractions.
A user-defined lambda function can be an argument to another lambda or its result; lambdas can return lambdas; lambdas can be named and recursively call themselves.
With the addition of LAMBDA, the Excel formula language thus becomes Turing-complete, which means that Excel users can perform any computation with Excel lambda functions.
Microsoft Research provides the following example of lambda functions that cooperate to reverse a string:
(Source: Microsoft Research blog)
The previous example shows three lambdas (HEAD, TAIL, and REVERSE) with REVERSE calling itself recursively, and using HEAD and TAIL for its computation. For comparison purposes, a function that reverses a string could be written in the purely functional language Haskell as follows:
reverse_str s = case s of "" -> s
c:cs -> reverse_str cs ++ [c]
c:cs effectively pattern-matches c to the head (the first character) of the string, with cs being the tail (the rest of characters).
Excel Lambda functions can be defined with the LAMBDA function as in the following formula, implementing the function x -> x + y:
(Source: Microsoft support article)
Lambdas can be named with Excel’s built-in name manager, making them easier to reuse in other parts of the spreadsheet. The previous illustration assumes that the lambda was named myLambda prior to its usage.
Microsoft provides plenty of examples of lambda in LAMBDA’s announcement post and documentation. The following lambda function for instance replaces forbidden characters in a string:
=LAMBDA(textString, illegalChars,
IF(illegalChars=””, textstring,
REPLACECHARS(
SUBSTITUTE(textString, LEFT(illegalChars, 1), “”),
RIGHT(illegalChars, LEN(illegalChars)-1)
)
)
)
Referring to the previous example, one Excel expert said:
Previously, you would have to use a function to remove the illegal characters one at a time. This could get very complicated because you would not always have the same number of illegal characters to be removed in each string, and it would need careful use of functions to prevent errors from occurring.
Following LAMBDA’s release in December 2020, the Excel community has also been at work producing examples of lambda functions.
LAMBDA thus makes it easier for Excel users - the vast majority of which are not programmers, to abstract commonly used functions behind a named formula. The alternatives include copy-pasting entire formulas in cells. The practice commonly generates very large, hard-to-read, hard-to-maintain formulas and has been known to generate errors that may go undetected for a long time.
As a programming language, the Excel formula language benefits from the Excel application (used as an IDE), its immediate feedback loop (achieved with interactive playgrounds, REPLs, or hot module replacement in other languages), and its dataflow programming model.
However, Excel, as a low-code application, may have underinvested in supporting large user applications. While modularity and abstraction, key enablers of development at scale, may have been improved with LAMBDA, automated testing, contracts, and typing, key features that positively impact robustness, are mostly missing, leaving plenty of room for human errors. Ray Panko, professor at the University of Hawaii, assessed that, on average, 88% of the Excel spreadsheets have 1% or more errors in their formulas. Cassotis Consulting commented on the study results as follows:
The frequency of errors generated while developing spreadsheets is similar to that of the development of programming codes. However, the latter go through many more tests and validation processes before they are officially used, while most spreadsheets are used soon after the first draft is developed.
As Doug Hudgeon, CEO of Managed Functions, explained to InfoQ in an article addressing low-code platforms:
Community developers create two types of risks. First, integration risk, which involves exposing data that shouldn’t be exposed. And second, transformation risk, which involves bugs or miscalculations in the app that lead to bad business decisions.
Typed cells could have lowered transformation risk and avoided a €750k loss that is attributed to a fund erroneously marked on the spreadsheet as a euro fund instead of as a dollar fund. The National Treasury Management Agency (NTMA), which purchased the fund, explained:
Subsequently, when the error was discovered, the dollar exchange rate had moved against the NTMA and the investment return was down €750,000.
Regarding typing, the Excel product team has already expanded the types that Excel understands. The team also experiments with logic-based type and dimensional inference, supported by machine learning.
Support for creating, versioning, publishing, importing, and debugging formulas may need to improve. Jones signaled future improvements for formula creation:
One thing that I can tell you that gets me every time is the experience of editing in the name manager ... definitely lots of room for improvement there.
The Microsoft team also works on sheet-defined functions, which let users define the parameters and outputs of functions as worksheet cells. Advait Sarkar, a senior researcher in the Calc Intelligence team, demoed how sheet-defined functions simplified the experience of creating functions. Worksheets used to contain sheet-defined functions may also be used to store tests that document the function or breakdown stages of a complex computation in miscellaneous cells. With the abstraction offered by LAMBDA, one may even dream of a library of value generators that supports checking formulas with property-based testing, like the Arbitrary class used by Haskell’s QuickCheck property-based testing tool.
The Excel formula language growing in capabilities also means that it is growing in complexity. Jones however noted:
Even if it takes greater skill and knowledge to author a lambda, it takes no extra skill to call it. LAMBDA allows skilled authors to extend Excel with application-domain-specific functions that appear seamlessly part of Excel to their colleagues, who simply call them.
The research team reported positive preliminary user feedback. One user reported:
Excel already has the opportunity to package up oft-repeated calculations as Visual Basic functions. However, what you are proposing is much to be preferred ... One [advantage] is performance; VB functions can be rather slow. [...] VB functions break the audit trail; not all of their behavior is determined by their parameters, as they can retrieve data from cells other than through the parameter list. Debugging VB functions requires programming skills; yours requires more standard spreadsheet skills.
Other users may fail to see the point. Mike James, the author of The Programmer’s Guide To Theory, objected:
So if you’re not bright enough to understand lambda calculus, you can just use it like a dumb ape ... We all know where that ends up. Spreadsheets are dangerous enough without making them super dangerous by way of academic obfuscation.
[...]
What we have here is an academic curiosity - a desk ornament or toy. It might amuse some people clever enough to recognize a lambda when they see it, but down-earth-spreadsheeting it isn’t.
In any case, Excel lambdas are just out and how they will be used by the Excel community will ultimately decide its practical usefulness. Programming experts may, for instance, implement DSLs with parsers and interpreters expressed as lambdas. Business users may then use those DSLs to achieve their business purpose with a lowered possibility of unintended mistakes, better error reporting - and the friendliness, immediate feedback of an Excel environment. Intermediate users may extend a DSL without getting into an actual programming language like VBA or JavaScript, whose integration with Excel adds security issues.
While lambda functions are likely to be abused in ways that remain to be understood, and a new class of errors is poised to see the light if no care is taken, the feature may also simplify formulas already existing and new worksheets, thereby reducing the potential for basic mistakes. Like in other programming languages, errors born from blindly copy-pasting code (formulas) will continue to occur. They may, however, be less surface for errors if the formulas are shrinking, through the reuse of properly tested user-defined lambda abstractions.
Ultimately, spreadsheets have just become more powerful for a large majority of users. As Jonathan Edwards puts it:
The simplicity and utility of spreadsheets put programming to shame. We believe the power of spreadsheets is that they offer a computational substrate: an autonomous artifact combining code and persistent data that is presented through a simple spatial metaphor.
Users interested in trying LAMBDA may join the Office Insider Program and choose the Beta Channel to get early access. Feedback and suggestions are welcome and can be posted within the program or on the Excel Tech Community forum.
Bruno Couriol holds a Msc in Telecommunications, a BsC in Mathematics and a MBA by INSEAD. Starting with Accenture, most of his career has been spent as a consultant, helping large companies addressing their critical strategical, organizational and technical issues. In the last few years, he developed a focus on the intersection of business, technology and entrepreneurship.
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Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p | 2022-07-07 16:33:56 | {"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": 0.20391961932182312, "perplexity": 3043.494841978821}, "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/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00683.warc.gz"} |
http://electricalacademia.com/basic-electrical/source-transformation/ | Home / Basic Electrical / Source Transformation Example Problems with Solutions
# Source Transformation Example Problems with Solutions
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Source Transformation
Source transformation is a circuit analysis technique in which we transform voltage source in series with resistor into a current source in parallel with the resistor and vice versa.
A highly valuable byproduct of Thevenin’s and Norton’s theorem is the technique of source transformation. Source transformation is based on the observation that if a Thevenin’s network and Norton’s network are both equivalent to a particular source network, then they must also equivalent to each other. This observation allows you to simply an analysis by converting a voltage source with series resistance to an equivalent current source with parallel resistance, or vice versa.
Source conversion may be applied to portions of a circuit to simplify intermediate calculations. Going even further, repeated source conversions may reduce a circuit to all-series or all-parallel form. Conversions may also be applied to controlled sources as well as to independent sources.
In the development of practical voltage sources and current sources, many similarities should have been noticed -such as internal resistance, voltage, and current characteristics. These similarities lead one to ask if it is possible to interchange one practical source for the other and retain the same results at the load. This forms the basis of source transformation.
If the same value of load resistance RL is connected to the circuits of the following figure, the load current flowing in circuit (a) is;
${{I}_{L}}=\frac{{{E}_{s}}}{{{R}_{s}}+{{R}_{L}}}~~~~\text{ }\cdots \text{ }~~~~~~~\left( 1 \right)$
Figure 1: Source Transformation
And the load current flowing in circuit (b) is
${{I}_{L}}=\frac{{{R}_{s}}}{{{R}_{s}}+{{R}_{L}}}*{{I}_{s}}~~~~\text{ }\cdots \text{ }~~~~~\left( 2 \right)$
To produce the same effect at the load, equations (1) and (2) must be equal.
$\frac{{{E}_{s}}}{{{R}_{s}}+{{R}_{L}}}=\frac{{{R}_{s}}}{{{R}_{s}}+{{R}_{L}}}*{{I}_{s}}$
Thus establishing the result that
${{E}_{s}}={{I}_{s}}{{R}_{s}}$
Or
${{I}_{s}}=\frac{{{E}_{s}}}{{{R}_{s}}}$
The value of internal resistance for practical voltage and the current source does not change. Although these equations have been derived for AC conditions. They are also equally applicable to DC circuits. Note that the same values of voltage, current and power may be obtained at the output terminals by either a practical voltage or a practical current source.
## Impossible Source Transformation
These are two certain situations when we cannot do source transformation.
1.When voltage source has zero resistance in this case,
${{I}_{s}}=\frac{{{E}_{s}}}{R}$
$R=0$
So;
${{I}_{s}}=\infty$
2.When current source has infinite resistance, in this case
${{V}_{s}}={{I}_{s}}R$
$R=\infty$
So,
${{V}_{s}}=\infty$
## Source Transformation Example
Find Vo using source Transformation
We convert 250V voltage source into 10A current source.
Combining both current sources, we obtain
Adding resistance in a parallel manner will give us the following circuit
Finally,
${{V}_{0}}=20V~$
## Source Transformation Example with Dependent Source
Let’s find v2 in the following circuit using source transformation.
Figure.2: Source Transformation Example with Dependent Source
Solution
The 72-V source and the 4Ω series resistance convert to a parallel structure with source current of
$~72V/4\Omega =18A$
The VCVS and the 12 Ω series resistance likewise convert to a parallel structure with source current of
$~3{{v}_{2}}/12\text{ }\Omega =0.25S*{{v}_{2}}$
So, the VCVS becomes a VCCS whose transconductance is 0.25S. The resulting diagram in figure 3 still has the control voltage v2 in place, as required.
Figure.3
But now all three resistances can be combined as 4||6||12=2 Ω. This parallel equivalent resistance carries the net current from the sources, and so
${{V}_{2}}=2\left( 18-0.25{{v}_{2}} \right)$
Solving for V2 yields
${{V}_{2}}=36/1.5=24V$
Did you find apk for android? You can find new Free Android Games and apps. | 2018-10-16 12:39:27 | {"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": 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": 0.4152519702911377, "perplexity": 1549.697097120228}, "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-2018-43/segments/1539583510749.55/warc/CC-MAIN-20181016113811-20181016135311-00222.warc.gz"} |
https://code.kx.com/q/basics/precision/ | # Precision¶
## Float precision¶
Precision of floats is a tricky issue since floats (doubles in other languages) are actually binary rational approximations to real numbers. Whenever you are concerned with precision, set \P 0 before doing anything else, so that you can see what’s really going on.
Due to the finite accuracy of the binary representation of floating-point numbers, the last decimal digit of a float is not reliable. This is not peculiar to kdb+.
q)\P 0
q)1%3
0.33333333333333331
Efficient algorithms for complex calculations such as log and sine introduce imprecision. Moreover, even basic calculations raise issues of rounding. The IEEE floating-point spec addresses many such issues, but the topic is complex.
Q takes this into account in its implementation of the equality operator =, which should actually be read as “tolerantly equal.” Roughly speaking, this means that the difference is relatively small compared to some acceptable representation error. This makes the following hold:
q)r7:1%7
q)sum 7#r7
0.99999999999999978
q)1.0=sum 7#r7
1b
Only zero is tolerantly equal to zero and you can test any two numbers for intolerant equality with 0=x-y. Thus, we find:
q)0=1.0-sum 7#r7
0b
The following example appears inconsistent with this:
q)r3:1%3
q)1=r3+r3+r3
1b
q)0=1-r3+r3+r3
1b
It is not. The quantity r3+r3+r3 is exactly 1.0. This is part of the IEEE spec, not q, and seems to be related to rounding conventions for binary floating point operations.
The = operator uses tolerant equality semantics. Not all primitives do.
q)96.100000000000009 = 96.099999999999994
1b
q)0=96.100000000000009-96.099999999999994
0b
q)deltas 96.100000000000009 96.099999999999994
96.100000000000009 -1.4210854715202004e-014
q)differ 96.100000000000009 96.099999999999994
10b
q)96.100000000000009 96.099999999999994 ? 96.099999999999994
1
q)group 96.100000000000009 96.099999999999994
96.100000000000009| 0
96.099999999999994| 1
Not transitive
Tolerant equality does not obey transitivity:
q)a:96.099999999999994
q)b:96.10000000001
q)c:96.10000000002
q)a
96.099999999999994
q)b
96.100000000009999
q)c
96.100000000020003
q)a=b
1b
q)b=c
1b
q)a=c
0b
The moral of this story is that we should think of floats as being “fuzzy” real values and never use them as keys or where precise equality is required – e.g., in group or ?.
For those interested in investigating these issues in depth, we recommend the excellent exposition by David Goldberg “What Every Computer Scientist Should Know about Floating Point Arithmetic’.
### Q SIMD sum¶
The l64 builds of kdb+ now have a faster SIMD sum implementation using SSE. With the above paragraph in mind, it is easy to see why the results of the older and newer implementation may not match.
Consider the task of calculating the sum of 1e-10*til 10000000.
The SIMD code is equivalent to the following (\P 0):
q){x+y}over{x+y}over 0N 8#1e-10*til 10000000
4999.9995000000017
While the older, “direct” code yields:
q){x+y}over 1e-10*til 10000000
4999.9994999999635
The observed difference is due to the fact that the order of addition is different, and floating-point addition is not associative.
Worth noting is that the left-to-right order is not in some way “more correct” than others, seeing as even reversing the order of the elements yields different results:
q){x+y}over reverse 1e-10*til 10000000
4999.9995000000026
If you need to sum numbers with most precision, you can look into implementing a suitable algorithm, like the ones discussed in “Accurate floating point summation” by Demmel et al.
## Comparison tolerance¶
Comparison tolerance is the precision with which two numbers are determined to be equal. It applies only where one or the other is a finite floating-point number, i.e. types real, float, and datetime (see Dates below). It allows for the fact that such numbers may be approximations to the exact values. For any other numbers, comparisons are done exactly.
Formally, there is a comparison tolerance t such that if x or y is a finite floating-point number, then x=y is 1 if the magnitude of x-y does not exceed t times the larger of the magnitudes of x and y. t is set to 2-43, and cannot be changed. In practice, the implementation is an efficient approximation to this test.
Note that a non-zero value cannot equal 0, since for any non-zero x, the magnitude of x is greater than t times the magnitude of x. Thus 0=a-b tests for strict equality between a and b.
Comparison tolerance is not transitive, and can cause problems for find and distinct. Thus, floats should not be used for database keys.
For example:
q)t:2 xexp -43 / comparison tolerance
q)a:1e12
q)a=a-1 / a is not equal to a-1
0b
q)t*a / 1 is greater than t*a
0.1136868
q)a:1e13
q)a=a-1 / a equals a-1
1b
q)t*a / 1 is less than t*a
1.136868
q)0=a-(a-1) / a is not strictly equal to a-1
0b
To see how this works, first set the print precision so that all digits of floating-point numbers are displayed.
\P 18
The result of the following computation is mathematically 1.0, but the computed value is different because the addend 0.001 cannot be represented exactly as a floating-point number.
q)x: 0 / initialize x to 0
q)do[1000;x+:.001] / increment x one thousand times by 0.001
q)x / the resulting x is not quite 1.000
1.0000000000000007
q)x=1 / does x equal 1?
1b
However, the expression x = 1 has the value 1b, and x is said to be tolerantly equal to 1:
q)x=1 / does x equal 1?
1b
Moreover, two distinct floating-point values x and y for which x = y is 1 are said to be tolerantly equal. No non-zero value is tolerantly equal to 0. Formally, there is a system constant $$E$$ called the comparison tolerance such that two non-zero values $$a$$ and $$b$$ are tolerantly equal if:
$$|a-b| ≤ E × max(|a|, |b|)$$
but in practice the implementation is an efficient approximation to this test. Note that according to this inequality, no non-zero value is tolerantly equal to 0. That is, if a=0 is 1 then a must be 0. To see this, substitute 0 for b in the above inequality and it becomes:
$$| a | ≤ E ×| a |$$
which, since $$E$$ is less than 1, can hold only if a is 0.
### Use¶
Besides Equal, comparison tolerance is used in the operators
= < <= >= > ~
differ within
And prior to V3.0
floor ceiling
It is also used by the iterators Converge, Do and While.
It is not used by other keywords that have tests for equality:
?
distinct except group in inter union xgroup
Sort keywords: asc desc iasc idesc rank xasc xdesc
### Examples¶
q)a:1f
q)b:a-10 xexp -13
In the following examples, b is treated equal to a, i.e. equal to 1:
q)a=b
1b
q)a~b
1b
q)a>b
0b
q)floor b /before V3.0, returned 1
0
In the following examples, b is treated not equal to a:
q)(a,a)?b
2
q)(a,a) except b
1 1f
q)distinct a,b
1 0.99999999999989997
q)group a,b
1 | 0
0.99999999999989997| 1
q)iasc a,b
1 0
### Dates¶
The datetime type is based on float, and hence uses comparison tolerance, for example:
q)a:2000.01.02 + sum 1000#1%86400 / add 1000 seconds to a date
q)a
2000.01.02T00:16:40.000
q)b:2000.01.02T00:16:40.000 / enter same datetime
q)a=b / values are tolerantly equal
1b
q)0=a-b / but not strictly equal
0b
Other temporal types, including the new timestamp and timespan types in V2.6, are based on int or long. These do not use comparison tolerance, and are therefore appropriate for database keys. | 2023-01-28 06:33:57 | {"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": 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": 0.6594222784042358, "perplexity": 1865.976884611026}, "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-2023-06/segments/1674764499524.28/warc/CC-MAIN-20230128054815-20230128084815-00351.warc.gz"} |
https://chem.libretexts.org/Under_Construction/Purgatory/Book%3A_Analytical_Chemistry_2.0_(Harvey)/02_Basic_Tools_of_Analytical_Chemistry/2.3%3A_Stoichiometric_Calculations | # 2.3: Stoichiometric Calculations
A balanced reaction, which gives the stoichiometric relationship between the moles of reactants and the moles of products, provides the basis for many analytical calculations. Consider, for example, an analysis for oxalic acid, $$\ce{H_2C_2O_4}$$, in which $$\ce{Fe^{3+}}$$ oxidizes oxalic acid to $$\ce{CO_2}$$.
$\ce{2Fe^{3+}}(aq) + \ce{H2C2O4}(aq) + \ce{2H2O}(l) \rightarrow \ce{2Fe^2+}(aq) + \ce{2CO2}(g) + \ce{2H3O+}(aq)$
The balanced reaction indicates that one mole of oxalic acid reacts with two moles of $$\ce{Fe^{3+}}$$. As shown in Example $$\PageIndex{1}$$, we can use this balanced reaction to determine the amount of oxalic acid in a sample of rhubarb.
Example $$\PageIndex{1}$$
The amount of oxalic acid in a sample of rhubarb was determined by reacting with $$\ce{Fe^{3+}}$$. After extracting a 10.62 g of rhubarb with a solvent, oxidation of the oxalic acid required 36.44 mL of 0.0130 M $$\ce{Fe^{3+}}$$. What is the weight percent of oxalic acid in the sample of rhubarb?
Figure $$\PageIndex{1}$$: Oxalic acid
SOLUTION
We begin by calculating the moles of $$\ce{Fe^{3+}}$$ used in the reaction
$\mathrm{\dfrac{0.0130\: mol\: Fe^{3+}}{L} \times 0.03644\: L = 4.73{\color{Red} 7} \times 10^{-4}\: mol\: Fe^{3+}}$
The moles of oxalic acid reacting with the $$\ce{Fe^{3+}}$$, therefore, is
$\mathrm{4.73{\color{Red} 7}\times10^{-4}\: mol\: Fe^{3+} \times \dfrac{1\: mol\: H_2C_2O_4}{2\: mol\: Fe^{3+}} = 2.36{\color{Red} 8} \times 10^{-4}\: mol\: H_2C_2O_4}$
Converting the moles of oxalic acid to grams of oxalic acid
$\mathrm{2.36{\color{Red} 8}\times10^{-4}\: mol\: C_2H_2O_4 \times \dfrac{90.03\: g\: H_2C_2O_4}{mol\: H_2C_2O_4} = 2.13{\color{Red} 2}\times10^{-2}\: g\: H_2C_2O_4}$
and calculating the weight percent gives the concentration of oxalic acid in the sample of rhubarb as
$\mathrm{\dfrac{2.13{\color{Red} 2}\times10^{-2}\: g\: H_2C_2O_4}{10.62\: g\: rhubarb}\times100 = 0.201\%\: \textrm{w/w}\: H_2C_2O_4}$
Note: Oxalic acid, in sufficient amounts, is toxic. At lower physiological concentrations it leads to the formation of kidney stones. The leaves of the rhubarb plant contain relatively high concentrations of oxalic acid. The stalk, which many individuals enjoy eating, contains much smaller concentrations of oxalic acid.
Note
Note that we retain an extra significant figure throughout the calculation, rounding to the correct number of significant figures at the end. We will follow this convention in any problem involving more than one step. If we forget that we are retaining an extra significant figure, we might report the final answer with one too many significant figures. In this chapter we will mark the extra digit in red for emphasis. Be sure that you pick a system for keeping track of significant figures.
The analyte in Example $$\PageIndex{1}$$, oxalic acid, is in a chemically useful form because there is a reagent, $$\ce{Fe^{3+}}$$, that reacts with it quantitatively. In many analytical methods, we must convert the analyte into a more accessible form before we can complete the analysis. For example, one method for the quantitative analysis of disulfiram, $$\ce{C_{10}H_{20}N_2S_4}$$—the active ingredient in the drug Anatbuse—requires that we convert the sulfur to $$\ce{H_2SO_4}$$ by first oxidizing it to $$\ce{SO_2}$$ by combustion, and then oxidizing the $$\ce{SO_2}$$ to $$\ce{H_2SO_4}$$ by bubbling it through a solution of $$\ce{H_2O_2}$$. When the conversion is complete, the amount of $$\ce{H_2SO_4}$$ is determined by titrating with $$\ce{NaOH}$$.
Figure $$\PageIndex{2}$$: Disulfiram
To convert the moles of $$\ce{NaOH}$$ used in the titration to the moles of disulfiram in the sample, we need to know the stoichiometry of the reactions. Writing a balanced reaction for $$\ce{H_2SO_4}$$ and $$\ce{NaOH}$$ is straightforward
$\ce{H2SO4}(aq) + \ce{2NaOH}(aq) \rightarrow \ce{2H2O}(l) + \ce{Na2SO4}(aq)$
but the balanced reactions for the oxidations of $$\ce{C_{10}H_{20}N_2S_4}$$ to $$\ce{SO_2}$$, and of $$\ce{SO_2}$$ to $$\ce{H_2SO_4}$$ are not as immediately obvious. Although we can balance these redox reactions, it is often easier to deduce the overall stoichiometry by using a little chemical logic.
Exercise $$\PageIndex{1}$$
You can dissolve a precipitate of $$\ce{AgBr}$$ by reacting it with $$\ce{Na_2S_2O_3}$$, as shown here.
$\ce{AgBr}(s) + \ce{2Na2S2O3}(aq) \rightarrow \ce{Ag(S2O3)2^3-}(aq) + \ce{Br-}(aq) + \ce{4Na+}(aq)$
How many mL of 0.0138 M $$\ce{Na_2S_2O_3}$$ do you need to dissolve 0.250 g of $$\ce{AgBr}$$?
Example $$\PageIndex{2}$$
An analysis for disulfiram, $$\ce{C_{10}H_{20}N_2S_4}$$, in Antabuse is carried out by oxidizing the sulfur to $$\ce{H_2SO_4}$$ and titrating the $$\ce{H_2SO_4}$$ with $$\ce{NaOH}$$. If a 0.4613-g sample of Antabuse is taken through this procdure, requiring 34.85 mL of 0.02500 M $$\ce{NaOH}$$ to titrate the $$\ce{H_2SO_4}$$, what is the $$\%\: \textrm{w/w}$$ disulfiram in the sample?
SOLUTION
Calculating the moles of $$\ce{H_2SO_4}$$ is easy—first, we calculate the moles of $$\ce{NaOH}$$ used in the titration
$\mathrm{(0.02500\: M)×(0.03485\: L) = 8.712{\color{Red} 5}\times10^{-4}\: mol\: NaOH}$
and then we use the balanced reaction to calcualte the corresponding moles of $$\ce{H_2SO_4}$$.
$\mathrm{8.712{\color{Red} 5}\times10^{-4}\: mol\: NaOH \times \dfrac{1\:mol\: H_2SO_4}{2\: mol\: NaOH} = 4.356{\color{Red} 2}\times10^{-4}\: mol\: H_2SO_4}$
We do not need balanced reactions to convert the moles of $$\ce{H_2SO_}$$4 to the corresponding moles of $$\ce{C_{10}H_{20}N_2S_4}$$. Instead, we take advantage of a conservation of mass—all the sulfur in $$\ce{C_{10}H_{20}N_2S_4}$$ must end up in the H2SO4(Here is where we use a little chemical logic! A conservation of mass is the essence of stoichiometry.); thus
$\mathrm{4.356{\color{Red} 2}×10^{-4}\: mol\: \ce{H2SO4} \times \dfrac{1\:mol\: S}{mol\: \ce{H2SO4}} \times \dfrac{1\:mol\: \ce{C10H20N2S4}}{4\: mol\: S} = 1.089{\color{Red} 0}\times10^{-4}\: mol\: \ce{C10H20N2S4}}$
or
$\mathrm{1.089{\color{Red} 0}×10^{-4}\: mol\: \ce{C10H20N2S4} × \dfrac{296.54\: g\: \ce{C10H20N2S4}}{mol\: \ce{C10H20N2S4}} = 0.03229{\color{Red} 3}\: g\: \ce{C10H20N2S4}}$
$\mathrm{\dfrac{0.03229{\color{Red} 3}\: g\: \ce{C10H20N2S4}}{0.4613\: g\: sample} × 100 = 7.000\%\: \textrm{w/w}\: \ce{C10H20N2S4}}$ | 2020-06-01 09:51:35 | {"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": 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": 0.7093082666397095, "perplexity": 1414.968690471556}, "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-2020-24/segments/1590347415315.43/warc/CC-MAIN-20200601071242-20200601101242-00114.warc.gz"} |
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# Widespread white matter microstructural abnormalities in bipolar disorder: evidence from mega- and meta-analyses across 3033 individuals
## Abstract
Fronto-limbic white matter (WM) abnormalities are assumed to lie at the heart of the pathophysiology of bipolar disorder (BD); however, diffusion tensor imaging (DTI) studies have reported heterogeneous results and it is not clear how the clinical heterogeneity is related to the observed differences. This study aimed to identify WM abnormalities that differentiate patients with BD from healthy controls (HC) in the largest DTI dataset of patients with BD to date, collected via the ENIGMA network. We gathered individual tensor-derived regional metrics from 26 cohorts leading to a sample size of N = 3033 (1482 BD and 1551 HC). Mean fractional anisotropy (FA) from 43 regions of interest (ROI) and average whole-brain FA were entered into univariate mega- and meta-analyses to differentiate patients with BD from HC. Mega-analysis revealed significantly lower FA in patients with BD compared with HC in 29 regions, with the highest effect sizes observed within the corpus callosum (R2 = 0.041, Pcorr < 0.001) and cingulum (right: R2 = 0.041, left: R2 = 0.040, Pcorr < 0.001). Lithium medication, later onset and short disease duration were related to higher FA along multiple ROIs. Results of the meta-analysis showed similar effects. We demonstrated widespread WM abnormalities in BD and highlighted that altered WM connectivity within the corpus callosum and the cingulum are strongly associated with BD. These brain abnormalities could represent a biomarker for use in the diagnosis of BD. Interactive three-dimensional visualization of the results is available at www.enigma-viewer.org.
## Introduction
Bipolar disorder (BD) is a severe chronic mental illness that affects ~1% of the general population [1]. There is often a long period with inadequate treatment before the diagnosis is established [2]. Consequently, there is a great need to identify biomarkers of BD. A better understanding of the neurobiology of BD could ultimately help to refine the diagnosis and guide innovative interventions. Recent advances in magnetic resonance imaging (MRI) could help to achieve this goal.
Neural models of BD suggest a role of fronto-limbic dysconnectivity in the emergence of mood symptoms of BD [3, 4]. This model is mainly supported by results from functional MRI (fMRI) studies demonstrating that emotional instability in this disorder might be underpinned by abnormal connectivity between frontal and limbic regions [5, 6]. However, results from diffusion tensor imaging (DTI) studies, a technique that allows the exploration of structural connectivity in vivo, have highlighted far more extensive brain abnormalities in BD. Indeed, the first DTI studies identified alterations in limbic tracts [7,8,9], followed by numerous studies that reported WM alterations within non-limbic regions, such as the corpus callosum [10,11,12,13,14,15] and corona radiata [16]. Meta-analyses based on whole-brain data have revealed lower fractional anisotropy (FA), a metric derived from DTI known to be positively correlated with the directionality and coherence of white matter bundles [17], in patients with BD near the parahippocampal gyrus, subgenual cingulate cortex [18], temporo-parietal junction and cingulum [19].
Inconsistencies in the location of WM microstructure alterations may be related to limited sample sizes and diversity in methods to collect data from different populations and for DTI data analysis. Indeed, differences in sample characteristics such as age of onset, disease duration, psychotic features, and lithium treatment, all of which have been associated with WM features [12, 20,21,22], may have contributed to the inconsistency in previous findings. Consequently, large harmonized multi-center studies are required to improve the reliability of case-control findings.
The ENIGMA consortium presents a framework to identify generalizable biomarkers, by analyzing large samples with a harmonized processing pipeline—a strategy that has already identified widespread cortical alterations and specific subcortical volumetric abnormalities in patients with BD [23, 24]. Thus, we analyzed DTI data from the ENIGMA-BD working group with the objectives of (i) identifying reliable generalizable WM abnormalities in BD using mega- and meta- analytics; (ii) testing if clinical characteristics modulate WM microstructure using mega- analytics. Specifically, we expected more pronounced alterations (i.e., larger FA differences with respect to healthy controls) in WM microstructure in patients with a more severe course of illness, and a significant association with psychotropic medication.
## Methods
### Samples
The ENIGMA-BD DTI working group, comprised of 26 cohorts spanning 12 countries, yielded a total of 3033 individuals (1551 healthy controls (HC) and 1482 patients with BD) included in this study. Demographic and clinical information from the whole sample is shown in Table 1; details of the contributing sites may be found in Table S1 and available clinical data for each site is provided in Table S2. Each cohort comprised a minimum of 12 subjects per group and a minimal ratio of patients to controls of 1:3, to allow for robust comparisons and meta-analysis. When needed, we randomly removed some subjects from a given group (mainly control subjects that were too numerous at 4 sites, except for one site that comprised too many patients in comparison to controls; for details, see Table S3). The current analysis includes data acquired until February 2018.
All participating sites obtained approval from their local ethics committees and all participants gave written informed consent. Participants younger than 18 or older than 65 as well as individuals with diffusion images with low quality after visual inspection (e.g., movement artifacts) were excluded from the analyses.
### Image processing
Acquisition parameters for each of the 26 sites are provided in Table S4. The pre-processing (i.e., eddy current and echo-planar corrections and tensor fitting) was performed at each site using harmonized analysis and quality control protocols from the ENIGMA consortium that have previously been applied in large-scale studies of schizophrenia [25]; recommended pipelines and procedures for the image analyses and quality control are provided online at the ENIGMA-DTI website (http://enigma.ini.usc.edu/protocols/dti-protocols/). After estimation of tensors, each site performed the image analysis and extracted the FA of each region of interest (ROI) (see description in Table S5) according to the ENIGMA-DTI protocol. The multi-subject JHU white matter parcellation atlas [26] was used to parcellate regions of interest from the ENIGMA template in MNI space. Mean FA from 43 regions of interest (ROI) as well as average whole-brain FA were then extracted for each participant across all cohorts.
### Mega-analysis
Our first aim was to identify WM microstructure differences between patients with BD and HC. We merged individual FA values of the 43 ROIs and Average FA (from each cohort) into one mega-analysis and entered them separately in a linear mixed model (using R software version 3.2.1. (R Core Team, 2015) and lme4 package [27]) including fixed effects for the diagnosis (patients vs. controls), age, sex, and random intercepts for each site:
$$\begin{array}{l}{\mathrm{FA}}\,{\mathrm{ROI}}_i = {\mathrm{Intercept}} + \beta 1 \ast {\mathrm{Diagnosis}} + \beta 2 \ast {\mathrm{Age}} \\ \,\,\,\,\,\,\,\,+ \, \beta 3 \ast {\mathrm{Sex}} + {\mathrm{random}}\,{\mathrm{effect}}\left( {{\mathrm{site}}} \right)\end{array}$$
We used Bonferroni correction to control for multiple comparisons (p< 0.05/44 = 0.0011). We also assessed the influence of average FA (per subject) across the entire TBSS FA tract skeleton (including core and periphery FA [25]) on local FA differences observed in the first analysis by running the same models including average FA as a covariate.
We performed additional analyses to assess how age, sex, illness duration, age of onset, medication at the time of scan (lithium, antipsychotics, anticonvulsants, and antidepressants), illness severity, history of psychotic symptoms and type of BD (type I vs. type II) might have modulated the main effect of diagnosis. We tested the effect of age and sex by including age-by-diagnosis and sex-by-diagnosis interaction terms. We included medication and history of psychosis as dichotomous measures in the analyses (yes/no variables) and used the density of episodes as an index of illness severity (number of mood episodes/illness duration). Importantly, each analysis controlled for age and sex, so that associations with illness duration and the age of onset would not be confounded by global age differences.
Age, sex, and diagnosis were available for all participants, whereas the remaining variables were available for some sites only (see Table S2 for details of available data for each site).
### Meta-analysis
Given previous demonstrations of the usefulness of meta-analysis for multisite neuroimaging [28], we performed a meta-analysis to allow comparisons with previous ENIGMA studies and comparison across sites. Similarly to previous ENIGMA meta-analyses, we conducted a random-effects inverse-variance weighted meta-analysis (R, metaphore package), to combine Cohen’s d effect size of each of the 26 cohorts of the study, both for right and left tracts separately and for bilateral tracts (to allow comparison with other ENIGMA DTI working groups). We calculated the I2 statistic to estimate the heterogeneity of the diagnostic effects across sites. This analysis was run following publicly available scripts on the ENIGMA-GitHub (https://github.com/ENIGMA-git).
## Results
We included 1482 patients with BD and 1551 HC. The patients were significantly older than the controls (mean age BD = 39.6 years; mean age HC = 35.1 years; t = 10.11; p < 0.001) and comprised a higher proportion of females (60.7 vs. 51.1%; χ2 = 25.77; p < 0.001). We included both age and sex as covariates in the mega- and meta-analyses, and tested for the age-by-diagnosis and sex-by-diagnosis interactions for further exploration of these effects.
### Mega-analysis
Linear mixed models revealed significantly lower FA in BD vs. HC along 29 out of 43 WM tracts and whole skeleton FA (see Table 2, Fig. 1). The largest effect sizes were found in the whole corpus callosum (CC) (R2 = 0.0441; P < 1.0 × 10−20), followed by the body (R2 = 0.0368; P < 1.0 × 10−20) and genu (R2 = 0.0331; P < 1.0 × 10−20) of the CC and the bilateral cinguli (right: R2 = 0.0281; P < 1.0 × 10−20; left: R2 = 0.0269; P < 1.0 × 10−20). Notably, we found lower FA in bilateral tracts, with the exception of the inferior fronto-occipital fasciculus, where significant difference was observed only in the right hemisphere. In a second analysis, with similar LMM but also covarying for average FA, we still observed lower FA in BD vs. HC across 19 tracts, meaning that the whole-brain average FA moderately influenced the results and that the effects were not exclusively driven by a global decrease in FA in patients (Table S6).
#### Age and sex effects
To examine differential effects of age and sex on group differences in FA values, we tested for age-by-diagnosis and sex-by-diagnosis interactions for each ROI. Results showed significant age-by-diagnosis interactions in bilateral superior corona radiata, the posterior limb of the internal capsule and left cingulum, such that there was steeper apparent age-related decline in the HC than BD group in all but the cingulate gyrus portion of the cingulum, where the opposite was found (Table S7; Figure S1). We did not find any significant sex-by-diagnosis interaction (Table S8).
#### Effects of clinical variables
Within the BD group, we found a significant positive relationship of age at onset to FA in the right inferior fronto-occipital fasciculus (Table S9) and a negative association between illness duration and FA within the left cingulum (Table S10) (Fig. S2). In addition, we observed significantly lower FA in patients receiving vs. not receiving antipsychotics within the genu of the CC and in patients receiving vs. not receiving anticonvulsants within multiple ROIs (Figs. S3 and S4; Tables S11 and S12). In contrast, we found higher FA values in several regions among patients receiving vs. not receiving lithium (Fig. S5, Table S13).
We did not observe any significant relationships between FA and antidepressant medication, illness severity, history of psychotic symptoms, or BD subtype (I or II) (see Tables S14S17).
### Meta-analysis
Results from the meta-analysis revealed lower FA among 23 out of the 44 ROIs (43 tracts and the whole-brain skeleton) analyzed (Table 3, Fig. 2). Similarly to the mega-analysis, the results showed largest effect sizes for the whole CC (d = −0.46; P = 7.86 × 10−12), body of the CC (d = −0.43; P = 5.41 × 10−11), and left cingulum (d = −0.39; P = 2.38 × 10−8). Overall, the meta-analysis showed similar effects to the mega-analysis but was slightly less sensitive. The I2 test indicates small to high heterogeneity across sites for all effect sizes (I2 = 0.002–69.24). To allow comparison with other DTI studies of the ENIGMA consortium, we also conducted a meta-analysis based on bilateral tracts (i.e., 25 ROIs). We found significant decrease FA in patients with BD compared to HC along 15 fasciculi. Similarly, the higher effect sizes were observed for the CC (d = −0.46; P = 7.86 × 10−12) and cingulum (d = −0.39; P = 4.58 × 10−8) (Figure S6, Table S18).
## Discussion
In the largest multi-center DTI study of BD to date, we found alterations of WM microstructure in patients with BD along multiple bundles, with strongest effects within the CC and the cingulum. FA was lower in patients in most ROIs, although effect sizes were small. Age, age of onset, illness duration as well as anticonvulsants and antipsychotic medications were associated with lower FA.
We collected individual data from 1482 patients and 1551 controls across 26 international cohorts, allowing a sample size considerably exceeding all prior DTI studies of BD. Unlike most studies that found localized WM alterations in BD, we identified widespread abnormalities (lower FA along 29 out of the 44 regions analyzed in the mega-analysis and 32 out of 44 ROIs in the meta-analysis). Similarly to results in the ENIGMA DTI schizophrenia project, this suggests a global profile of microstructural abnormalities in BD, which are however not specific to that disorder [25].
For both analyses (i.e., mega and meta), the largest effect sizes were observed within the CC and cingulum. This is consistent with a recent meta-analysis showing decreased FA within the CC, cingulum and the anterior superior longitudinal fasciculus in BD in comparison to controls [29]. The cingulum is a major pathway in the limbic system. Impairment of cingulum and uncinate structural integrity is in good agreement with previous models of altered fronto-limbic connectivity in BD [3, 30].
In contrast, the role of the CC in pathophysiological models of BD is less straightforward. Disconnection in patients with BD with psychotic history has been suggested [12] but there is no clear evidence for the implication of the CC in emotion processing or mood switching [31]. Reduced FA within the CC was also reported in a meta-analysis of DTI studies in schizophrenia [25] and major depressive disorder [29], suggesting an overlapping involvement in both psychosis and affective disorders. Further studies are warranted to evaluate to what extent the CC is differentially affected in these disorders. Preliminary data suggest that disruption of interhemispheric connectivity is a disease marker rather than a vulnerability marker to BD [32]. Nonetheless, we identified extensive WM abnormalities suggesting that current pathophysiological models of BD are incomplete. Future models should not be limited to fronto-limbic networks, and should perhaps consider interhemispheric disconnectivity as a key feature of BD.
Importantly, the patient group was significantly older than the control group. Although we controlled for age in all analyses, it is possible that the linear models used are not fully accounting for the age-related variance [33]. However, the assessment of the effects of age revealed a significant interaction between age and diagnosis for only 4 ROIs out of the 43 analyzed. We found a significant increase in the effect of age in patients with BD for the left CGC only, while we found the reverse association for the bilateral SCR and the left PLIC, these effects were not anticipated and should be verified when replication samples become available.
We found that lithium intake was associated with higher FA in several tracts, as well as with global FA. Prior studies have suggested neuroprotective effects of lithium, on gray matter [23, 34,35,36] and white matter [37]. Higher FA associated with lithium use could reflect a direct influence of lithium on water diffusion or a beneficial effect on myelination [38], as suggested by the observation that lithium promotes myelin gene expression, morphological maturation, and remyelination in cultured oligodendrocytes via the Wnt/β-catenin and the Akt/CREB pathways [39]. In patients with BD, lithium may increase axial diffusivity in WM tracts also influenced by genetic variation in this pathway [22]. We also found lower FA in patients who received anticonvulsants in several tracts and average global FA. Further, patients who were on antipsychotic treatment showed lower FA within the genu of the CC. This is consistent with prior results suggesting a negative relationship between anticonvulsants, antipsychotics and cortical thickness or FA [23, 37]. However, it could be possible that the choice of the medication was driven by some patients’ particularities or unknown neurobiological characteristics, which are hard to assess with a cross-sectional design, leading to confounding by indication. Longitudinal clinical trials are needed to clarify this point.
We did not find significant differences between BD type I and type II. The power of prior meta-analyses of DTI studies has also been too low to perform this comparison. However, sensitivity analyses for these meta-analyses indicated that the sub-group of patients with BD I was driving the FA difference observed between patients with BD and HC [19, 29, 40]. Although we had enough power, the comparison of BD I vs. BD II did not replicate this result. Consistent with our results, however, ENIGMA analyses of T1-weighted anatomical MRI data of patients with BD did not yield any detectable differences between BD types [23, 24].
In sum, the multisite nature of the study is a strength that allowed us to detect small but significant differences. Our results seem to challenge the hypothesis of a precise localization for the WM alterations in BD. Indeed, we have highlighted extensive abnormalities, which do not seem to be specific to this psychiatric disorder. Lower FA across multiple bundles has already been consistently observed in studies of schizophrenia, with apparently higher effect sizes (e.g., [25]). Consequently, to build more precise neurobiological models of BD future studies should benefit from new advanced neuroimaging methods such as Neurite Orientation Dispersion and Density Imaging (NODDI) [41]. This recent processing model allows fine-grained measurement of the WM microstructure, with physiological interpretation of the derived indices, and has already shown promising results in BD [42]. However, the large-scale application of such methods will only be possible with raw data sharing within international consortia. This will allow the application of advanced DTI models and whole-brain analyses, which are needed to better understand WM abnormalities observed in BD. Finally, longitudinal studies conducted in conjunction with advanced DTI protocols are essential to clarify the impact of pharmaceutical treatments on brain microstructure.
Some limitations are important to emphasize. We did not include other diffusion parameters in our analysis. Lower FA may represent abnormal fiber coherence but does not yield information on fiber density or myelination. The mean, radial and axial diffusivity measure would have added complementary information regarding the nature of WM alteration. However, we have focused on the most commonly used measure, which offers better comparability with prior studies. Also, most studies have highlighted a correlation between FA and these other measures, while their inclusion would have tripled the number of analyses. In addition, although we found “widespread” WM abnormalities in patients with BD, the robust ENIGMA DTI pipeline used to partition the ROIs involved only long and isolinear bundles. With this methodological approach (i.e., FSL TBSS), we cannot evaluate localized changes within the superficial WM, as have been previously observed in BD and schizophrenia [43]. Also, this methodological approach poorly reconstructs fiber crossings, which may have led to incomplete localization of group differences. Further studies are warranted to identify more fine-grained WM abnormalities in BD.
Importantly, retrospective multisite analyses have some limitations. Differences in the acquisition parameters, magnet strength, head coil and manufacturer provided software could have impacted the results. However, we believe that our approach, using a harmonized data processing pipeline, with a reliable procedure, allows for the first time coordinated mega- and meta-analyses with robust results.
Moreover, the effects of the covariates found here are only derived from post hoc analyses in cross-sectional studies with a somewhat limited representation of individuals with BD over age 50 (only 18% of the sample). Longitudinal studies would be more suitable to identify and predict the effect of age, illness duration/severity and medication on WM microstructure in patients with BD. In addition, despite their importance, we were not able to test the relation between FA and other covariates, such as body mass index and frequent BD comorbidities (e.g., anxiety or substance use disorder). Too few sites had collected these measures to allow robust analyses. However, we believe that our sample is ecologically valid and captures the heterogeneity of BD.
With this unprecedented sample size, we found evidence for widespread WM abnormalities in patients with BD and showed differences in BD WM microstructure that were unobserved until now. These results may inform future DTI studies with regard to expected effect sizes, and the effects of several covariates and clinical variables. We also highlighted that the CC and the cingulum had the strongest decrease in FA in patients with BD. Despite growing evidence for altered structure of the CC in BD, its specific role in the pathophysiology of BD needs to be further integrated into neural models of BD.
## Change history
• ### 16 September 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
• ### 07 October 2019
This Article was originally published under NPG’s License to Publish, but has now been made available under a [CC BY 4.0] license. The PDF and HTML versions of the Article have been modified accordingly.
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Favre, P., Pauling, M., Stout, J. et al. Widespread white matter microstructural abnormalities in bipolar disorder: evidence from mega- and meta-analyses across 3033 individuals. Neuropsychopharmacol. 44, 2285–2293 (2019). https://doi.org/10.1038/s41386-019-0485-6
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Neuropsychopharmacology (2021) | 2022-06-28 16:48:25 | {"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": 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": 0.5952567458152771, "perplexity": 10783.418499527337}, "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/1656103556871.29/warc/CC-MAIN-20220628142305-20220628172305-00775.warc.gz"} |
https://delong.typepad.com/sdj/2010/07/cant-anybody-here-play-this-game-fiscal-policy-edition.html | ## Can't Anybody Here Play This Game? Fiscal Policy Edition
Niall Ferguson writes:
Today’s Keynesians have learnt nothing: When Franklin Roosevelt became president in 1933, the deficit was already running at 4.7 per cent of GDP. It rose to a peak of 5.6 per cent in 1934. The federal debt burden [in the United States] rose only slightly – from 40 to 45 per cent of GDP – prior to the outbreak of the second world war. It was the war that saw the US (and all the other combatants) embark on fiscal expansions of the sort we have seen since 2007. So what we are witnessing today has less to do with the 1930s than with the 1940s: it is world war finance without the war...
Could we please have some acknowledgement of the fact that the reason the debt-to-GDP ratio did not rise across the 1930s was because GDP rose, not because debt didn't rise? Debt more than doubled from $22.5 billion to$49.0 billion between June 30, 1933 and June 30, 1941. But nominal GDP rose from $56 billion in 1933 to$127 billion in 1941.
And could we please have some acknowledgement that our 9.4% of GDP deficit in fiscal 2010 pales in comparison to the 30.8% of GDP deficit of 1943, or the 23.3% and 22.0% deficits of 1944 and 1945?
Niall Ferguson should not do this. The Financial Times should not enable Niall Ferguson to do this. | 2021-11-28 21:11:37 | {"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": 0.23781873285770416, "perplexity": 3569.2721567960243}, "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/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00103.warc.gz"} |
https://www.physicsforums.com/threads/confusion-with-product-to-sum-trig-identities.700747/ | Confusion with product-to-sum trig identities
1. Jul 9, 2013
Jyan
I'm having some confusion with a couple trig identities. On wikipedia (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities), the following two identities are listed:
sinθcosβ = (1/2)[sin(θ+β) + sin(θ-β)]
and
sinβcosθ = (1/2)[sin(θ+β) - sin(θ-β)]
I can see the difference between them if they are used with the same variables θ and β. But, how do you know which one is valid in any given situation? I find this a hard question to phrase, but I hope you can see my confusion. If you have sin x cos y, which identity can you apply? Does it matter? So long as you apply the other one to sin y cos x?
2. Jul 9, 2013
Ackbach
They're really the same identity - one of them being superfluous. If you swap the roles of the arguments, and use the fact that $\sin(-x)=-\sin(x)$, then you will see that they are the same.
Last edited: Jul 9, 2013
3. Jul 9, 2013
Jyan
I see, thank you.
4. Jul 9, 2013
Ackbach
You're welcome! | 2018-02-24 16:59:22 | {"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": 0.9183683395385742, "perplexity": 891.1367797226783}, "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-2018-09/segments/1518891815843.84/warc/CC-MAIN-20180224152306-20180224172306-00197.warc.gz"} |
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## DanielaJohana one year ago simplify (1/x+1/y)/xy
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1. iPwnBunnies
Use distributive property.
2. DanielaJohana
$\frac{ \frac{ 1 }{ x }+\frac{ 1 }{ y } }{ xy }$
3. iPwnBunnies
OHH, now I see
4. iPwnBunnies
Combine (Add) the numerator to get one fraction, and do a simple division
5. iPwnBunnies
You add the two fractions in the numerator by giving them the same denominator, which makes this problem simple and nice
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Privacy Policy | 2017-01-23 09:18:06 | {"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": 0.8082034587860107, "perplexity": 7940.5738286566475}, "config": {"markdown_headings": true, "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-2017-04/segments/1484560282202.61/warc/CC-MAIN-20170116095122-00390-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://jobbmatc.web.app/20069/4998.html | # Druhý derivát dy dx
What is the difference between y^2 and y? Why to use chain rule in first case and not in the second one like 1(y(x))*dy/dx?
Následné deriváty sú derivátmi funkcie po druhom deriváte. Proces výpočtu po sebe nasledujúcich derivátov má nasledujúcu funkciu f, ktorú môžeme odvodiť a teda získať derivačnú funkciu f '. K tomuto derivátu f môžeme odvodiť znova, získať (f ')'. Here we look at doing the same thing but using the "dy/dx" notation (also called Leibniz's notation) instead of limits. slope delta x 2.
For derivative with power always use logarithm: y=xcos(3x) ln(y)= ln(xcos(3x)) ln(y Inverzní funkce a diferenciace - Inverse functions and differentiation z Wikipedie, otevřené encyklopedie dy dx = 1 cos2 x This can be written as sec2 x because the function secx is defined to be 1 cosx. Example Suppose we want to differentiate y = secx. The function secx is defined to be 1 cosx, that is, a quotient. Taking u = 1 v = cosx du dx = 0 dv dx = −sinx Quoting the formula: dy dx = vdu dx −udv v2 So dy dx = cosx·0− 1·(−sinx dx dy u v p streamline u v p + dp u + du v + dv V Along the streamline, we have dy dx = v u or u dy = v dx (3) We multiply the x-momentum equation (1) by dx, use relation (3) to replace vdx by udy, and combine the u-derivative terms into a du differential.
## Potravinářské aktuality 2002 Z OB SAHU Vý iva Jakost potravin Rizika z potravin Analýzy a pøístroje Zaøízení a technologie Podniky a trhy Nové výrobky Legislativa Akce Nové knihy 1 12 15 18 20 23 26 30 38 39 Vý ivové doplòky r ybího oleje napomáhají pøi chronické únavì Syndrom chronické únavy (CFS, té známý jako ME) mù e být zpùsoben chemickou nerovnováhou v mozku.
It makes it possible to measure changes in the rates of change. For example, the second derivative of the displacement is the variation of the speed (rate of variation of the displacement), namely the acceleration.
### Free implicit derivative calculator - implicit differentiation solver step-by-step
To Druhý datadisk velice pěkné strategie vytořili autoři z vývojářksé společnosti Blue Fang Games a jeho název zní Zoo Tycoon Marine Mania.
In order to differentiate dy∕dx directly we need to write t in terms of x. Jul 9, 2020 This calculus video tutorial discusses the basic idea behind derivative notations such as dy/dx, d/dx, dy/dt, dx/dt, and d/dy.My Website: Dec 2, 2012 This video explains the difference between dy/dx and d/dxLearn Math Tutorials Bookstore http://amzn.to/1HdY8vmDonate Dec 2, 2012.
století Gottfrieda Wilhelma Leibnize, používá symboly dx a dy k reprezentaci nekonečně malých přírůstků x a y (také známé jako diferenciály, typ nekonečně malých čísel). , stejně jako Δ x a Δ y představují konečné přírůstky x a y. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math problems instantly. Feb 12, 2007 · d/dx e^(x+1) = e^(x+1) * d/dx (x+1) = e^(x+1). You will find that things which make a simple movement of the graph up/down and/or left/right do not affect the derivative. Only magnifications and other things which affect the shape (and thus the rate of change, in certain areas which are affected by the change of shape) do.
The question asks for the Derivative of Sin to the power of 3, x. ( ( Sin ^3 ) x ) I ended up with something along the lines of: ( 3 ( cosx ) ^ 2 ) * 3( x ^ 2). It was wrong, and I'm pretty much up the creek without a paddle. Does anyone know the answer to this, and if so please explain how you came Oct 17, 2009 · So $dy/dx= (dy/du)(du/dx)= n u^{n-1}(-sin(x))= -n sin(x)cos^{n-1}(x)$. With a little practice, you should be able to do that with actually writing down the "u" substitution: to differentiate $cos^n(x)$ think "The "outer function" is a power so the derivative is $n cos^{n-1}(x)$ and then I multiply by its Apr 13, 2020 · Use the chain rule. The chain rule provides a method for taking the derivative of a function in which one operation happens within another.
Tap for more steps To apply the Chain Rule, set as . dx: arccot 2x = −2 4x 2 + 1 * The remaining derivatives come up rarely in calculus. Nevertheless, here are the proofs. The derivative of y = arcsec x. Again, Question: Dy Find The Derivative Of The Given Functions Dx D Sino D Coso (For C Use E- Sino To Derive D0 Do Formula For The Derivative) = Coso And A) Y = 3x Sinx + COSX B) Y = Sec?r- Tan²x C) Find The Equation Of The Tangent Line To The Graph Of Y = (sinx)(cosx) At X = 4 The chain rule for derivatives is $\\frac{dy}{dx} = \\frac{dy}{du}\\cdot \\frac{du}{dx}$ This basically means the derivative of a composite function is the derivative of the outer function with the original argument multiplied by the derivative of the inner function. Free implicit derivative calculator - implicit differentiation solver step-by-step (dy)/(dx)=-e^(-x) Here , y=e^-x Let, y=e^u and u=-x :.(dy)/(du)=e^u and (du)/(dx)=-1 Using Chain Rule: color(blue)((dy)/(dx)=(dy)/(du)*(du)/(dx) :.(dy)/(dx)=e^u xx Implicit differentiation helps us find dy/dx even for relationships like that. This is done using the chain rule, and viewing y as an implicit function of x.
d(v) (differentiation w.r.t.
20 z 28000
gladiátor nie si zabavený meme generátor
meno majiteľa účtu na karte
porovnaj prihlásenie na trh v kine
hodiny podpory zákazníkov spoločnosti moneylion
### Derivát a diferenciál; Čo je univerzálnejšie: prírastok argumentu alebo jeho rozdiel; Nahradenie prírastkov rozdielmi; Funkčný rozdiel: príklady; Diferenciálna aproximácia; …
Nevertheless, here are the proofs.
## What is the difference between y^2 and y? Why to use chain rule in first case and not in the second one like 1(y(x))*dy/dx?
If a derivative is taken n times, then the notation dnf / dxn or fn(x) is used.
Nejdříve je třeba vypočítat derivaci bitu uvnitř kosinu, a to 2x. Poté, co jsme našli derivaci kosinu (záporný sinus), můžeme ho násobit 2x. = … Derivát funkce reálné proměnné měrná citlivost na změnu hodnoty funkce (počáteční hodnota) s ohledem na změnu svého argumentu (vstupní hodnota). Deriváty jsou základním nástrojem kalkulu . | 2023-02-05 23:56:57 | {"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": 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": 0.7845698595046997, "perplexity": 7459.512399718044}, "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-2023-06/segments/1674764500294.64/warc/CC-MAIN-20230205224620-20230206014620-00429.warc.gz"} |
https://www.allanswered.com/post/avewp/heat-flux-for-a-moving-energy-source/ | ### Heat flux for a moving energy source
317
views
1
4 months ago by
Hi all
I'm a math person and currently working on some optimization algorithms for electron beam melting. The reason I'm saying that because my backgrounds in thermo mechanics is little.
Anyway, with some help I was able to construct my simulation model on FEniCS and it runs just fine. Now I'm trying to get some results out of my model and feed them to my optimization algorithms.
Currently I need to calculate the heat flux and use them in finding the thermal gradients. I did my research and now I understand that the heat flux is the energy per unit area. Many references state that the heat flux is my governing equation (flux in = flux out). The first question is that true? and if that was the case then how can I make FEniCS report those values (excerpt of my code below)
IF not true, then how can I go about calculating that?
F = (rho*c/Dt*(T-T0)*del_T \
+ kappa*T.dx(i)*del_T.dx(i) \
- rho*f*del_T)*dv \
+ h*(T-Ta)*del_T*da
F += sum(integrals_N)
T_M = Function(Space)
F += emis*st_bol*(T_M**4-Ta**4)*del_T*ds(5)
left=lhs(F)
right=rhs(F)
A = assemble(left)
b = None
T = Function(Space)
for t in numpy.arange(0, t_end, Dt):
f.t = t
line_n = int(abs(t / line_time))
if (line_n % 2) == 0 and (line_n %2 < line_time):
f.xx = (0.001 + vel*t - (length*line_n - mis))
f.yy = 0.001 + line_n * hatch
else:
f.xx = (0.009 - vel*t + (length*line_n - mis))
f.yy = 0.001 + line_n * hatch
b = assemble(right, tensor=b)
[bc.apply(A, b) for bc in bcs]
solve(A, T.vector(), b, 'cg')
timestep += 1
T0.assign(T)
In case the flux was one side of my equation, I went and tried to print that value but I get the form of the function and the value.
Community: FEniCS Project
0
4 months ago by
Your form is based on the balance of internal energy without mechanical power:
$\rho\frac{\mathrm{d}u}{\mathrm{d}t}=-\nabla\cdot\mathbf{q}+\rho r$ρdudt =·q+ρr
The non-convective flux term $\mathbf{q}$q is called the heat flux for which you used Fourier's law
$\mathbf{q}=-\kappa\nabla T$q=κT
Now the solution field of your problem is the Temperature. So after calling solve you can use the calculated field $T$T to calculate the heat flux:
# Create vector valued FunctionSpace for the temperature gradient with smaller degree
VE = VectorElement('DG',Space.mesh().ufl_cell(), Space.ufl_element().degree()-1)
Vspace = FunctionSpace(Space.mesh(), VE)
# Project gradient onto function space
q = project(-kappa*grad(T), Vspace)
You can then plot that heat flux q , print it to file, print its values etc.
As for your question about the governing equation. What is that eequation supposed to govern?
Thank you Klunkean for the response
I use Gaussian function for interaction (the term 'f' in my equation). The general form (what I call governing equation) is
$\text{ρ Cp ∂T/∂t=∇(k ∇T)+S+q}$ρ Cp ∂T/∂t=∇(k ∇T)+S+q
where S is Gaussian function, q is radiation heat loss, and there is no heat convection.
Now does that change the way you calculate the heat flux in your answer?
Thank you again for your generous response
written 4 months ago by Ashraf El Gaaly
1
It doesn't. My comment on the interaction was incorrect. I only thought of the mechanical interaction and generalized, thinking of some other stresses that might act on the strains.
Of course you can model interaction through the source term (in my eq. $\rho r$ρr ) which does not change anything regarding the heat flux.
written 4 months ago by klunkean
I see! thank you so much
One last question and I hope that is not too much.
Back to my original question, if the heat flux is the energy per unit area, then is q (in your answer) the energy per element area for all elements?
I tried to use type(q) to find how exactly the variable look like but i didn't get any output for that command. The reason I'm asking because I want to know how to take the data and represent it.
I really appreciate your help so much. This whole community has helped me big time, specially for someone like me who lacks the necessary background.
written 4 months ago by Ashraf El Gaaly
It's actually power per unit area, or better put: energy per unit area per second.
In your discrete setting - assuming you use linear shape functions - you have one temperature value per node and inbetween nodes, i.e. inside the cell, you have a linear interpolation function. The gradient of temperature, and thus also the heat flux, will be a constant vector associated with a cell.
You can interpret its absolute value as the net non-convective influx (outflux if negative) of internal energy through the facets into your cell per time step.
As for visualization you could use a quiver plot with an arrow assigned to the center of each cell, or interpolate to a nodal function space.
written 4 months ago by klunkean | 2018-08-15 20:09:29 | {"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": 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": 0.6025834679603577, "perplexity": 1488.6629897861942}, "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-2018-34/segments/1534221210304.2/warc/CC-MAIN-20180815200546-20180815220546-00103.warc.gz"} |
https://en.wikipedia.org/wiki/Talk:Small-angle_scattering | # Talk:Small-angle scattering
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## small angle light scattering
What about small angle light scattering? The theory is exactly the same. As the title of Kerker's boook (The scattering of light and other electromagnetic radiation) makes clear. It's just a question of the accessible length scales, depending on q. A table comparing the ranges of q accessible by the three techniques would be useful. AlanParkerFrance (talk) 17:19, 26 January 2013 (UTC)
## Theory - Continuum description
In the first and second sentence in "Theory - Continuum description" section there is the equation "${\displaystyle q=4\pi \sin(\theta )/\lambda }$" and it continues as "here ${\displaystyle 2\theta }$ ...". Is a ${\displaystyle 2\theta }$ missing from the first equation? If not then what does ${\displaystyle \theta }$ represent? I am not sure if it is wrong or not well explained. — Preceding unsigned comment added by Elefand (talkcontribs) 19:07, 1 March 2015 (UTC) | 2018-01-24 00:16:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.3762701749801636, "perplexity": 1547.4751044668394}, "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-2018-05/segments/1516084892802.73/warc/CC-MAIN-20180123231023-20180124011023-00764.warc.gz"} |
http://www.sciencemadness.org/talk/viewthread.php?tid=4465&page=3 | Sciencemadness Discussion Board » Fundamentals » Organic Chemistry » Piperic acid oxidation to Piperonal Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues » Detritus » Test Forum
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Author: Subject: Piperic acid oxidation to Piperonal
Hexagon
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Quote: Originally posted by cheeseandbaloney I would think the reason the oxidation of piperic acid usually fails because the strong oxidizers could be tearing up the methylenedioxy bridge. Anything strong enough to rip apart a C=C bond seems like it would do somptin fierce to a R-O-CH2-O-R bond. It would be a surprise to me if the yields on a reaction like this were consistently high...
Nope, the oxidation usually fails because the aldehyde is oxidized all the way to the acid.
Bander
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I made a thread on this a long time ago where I received very useful help from many here, but most notably Tacho's advice. Since then I have done the below to isolate piperine.
Quote: 113g Black Pepper (4oz) ~600ml Isopropyl Alcohol, anhydrous! ~120ml Acetone, anhydrous! Reflux 113g black pepper in 600ml anhydrous IPA in a 1 L round bottom flask for 2-3 hours. Filter the ~80C olive translucent IPA through coffee filters into temporary jars. Remove the partially extracted ground pepper to save for further extractions and wash the remainder of solids out of the flask with water. *DRY* the water out of the flask completely with acetone wash and hot water bath. Pour the Piperine/IPA back into the ground bottom flask. Add broken glass chips or boiling stones. Distill off and recover the IPA until the viscous remainder starts to stick to and gum up the glassware above the bubble line. This usually happens with around ~30ml (+-10) left. Pour the remainder off and let it sit for a day undisturbed and covered. A cake/mash of orange crystals under a brown oil appears. Crush the fanned out crystal cake under ~20ml acetone (dry! no water). Pour the acetone wash off the powder into a collection jar through a coffee filter for later. It will still contain substantial disolved piperine. Notice the lighter yellow color each time as you repeat this wash and crush three times. The remaining bright yellow <1mm needle crystal powdered piperine at the bottom of the jar should be set out to dry on filters. 2-4g usually results. The waste should be evaporated over 2-3 days to retrieve another crystal cake and the above repeated.
[Edited on 27-8-2010 by Bander]
cheeseandbaloney
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Quote: Originally posted by Hexagon
Quote: Originally posted by cheeseandbaloney I would think the reason the oxidation of piperic acid usually fails because the strong oxidizers could be tearing up the methylenedioxy bridge. Anything strong enough to rip apart a C=C bond seems like it would do somptin fierce to a R-O-CH2-O-R bond. It would be a surprise to me if the yields on a reaction like this were consistently high...
Nope, the oxidation usually fails because the aldehyde is oxidized all the way to the acid.
aha! I was under the impression that the reaction failed completely and brought a mess of unwanted products. Piperonylic acid still seems useful. I wouldn't consider that a failed reaction! (unless one specifically wanted piperonal obviously).
Nicodem
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Quote: Originally posted by Gualterio_Malatesta Hence this method is more promising and is favourable to kitchen chemists not because they lack knowledge. This method was tried by several skillfull chemists and failed! I must admit my info is couple of years old, maybe there were some improvements to this oxidation step which actually yielded the desired product - Hexagon's post above (he didn't reply me though).
I'll skip the part about this being a promising method to kitchen chemists and lack of knowledge, because it sounds oxymoronic, but I'm seriously interested in those reports you mentioned of "tried by several skillfull chemists and failed". I don't remember any such report from the Hive and I do more or less regularly follow the developments at the Hyperlab, but I don't remember ever reading about what you talk. If skilled chemists tried and failed, this can mean the oxidations described in the literature are fraudulent, that even ozonolysis on piperine does not work, or that it is not possible to isolate piperonal from the mixture (or a combination of more of such factors). Please provide a few citations from these skilled chemists, so that we can see where the problem resides.
…there is a human touch of the cultist “believer” in every theorist that he must struggle against as being unworthy of the scientist. Some of the greatest men of science have publicly repudiated a theory which earlier they hotly defended. In this lies their scientific temper, not in the scientific defense of the theory. - Weston La Barre (Ghost Dance, 1972)
Gualterio_Malatesta
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2 Nicodem
I will try to find links to those attempts.
Anyways, was there any success that you are aware of involving this method?
Polverone
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Here is the original Annalen paper describing the cleavage of piperine. I don't recall seeing this paper posted before in conjunction with piperine -> piperonal discussions. I know little to no German so it will require much tedious machine translation and dictionary time before I can grasp much of it. It may be useful to anyone in this thread who understands German, though.
Attachment: annalen_1869_25-58.pdf (735kB)
PGP Key and corresponding e-mail address
Bander
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Would it be possible to remove piperonal from solution as it is formed by using a bilsulfite adduct to precipitate it before it is oxidated to the acid? I have used this method after the entire oxidation, but ended up with only ~0.1g of very pale yellow precipitate.
S.C. Wack
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How is it that everyone assumes that piperine isolation and piperic acid oxidation to piperonal have never come up here before?
It's almost as mysterious to me as is staff wasting valuable time in response to obviously clueless or trolling new nyms who refuse to search google or anything, and refuse to post in relevant threads.
Naturally google books has it by now.
The problem with the .txt procedure is that it was...fucked up...and any of these alleged failures by these top notch chemists is caused by their faith in internet teks, not chemistry. No doubt the original directions would be an improvement.
Forming a solid bisulfite adduct is unnecessary and surely lowers yields. I have heard rumors that one trick to the KMnO4 procedure is in the workup.
However any tricks, and possibilities known in the art (KMnO4-related or otherwise), seem ill placed in any thread titled "Seperation of Piperine from resin", much less in response to the posts in this thread.
[Edited on 27-8-2010 by S.C. Wack]
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Hexagon
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AFAIK the adduct you are trying to make is fucked up at anything but neutral conditions, so in the clasic synth. being basic the anvioronment, it's a safe assumption that trying to isolate the aldehyde as it's corresponding bisulfite adduct is not possible. Dunno what you ended with, but I can assure that piperonal is bisulfite aduccut is a powder (when dry) white as snow.
If you have enough piperate you can try the CuSO4 and tartrate/citrate in alkaline media oxidation, dont use an excess of sodium hydroxide since it'll turn the aldehyde in to the alcohol and the acid due to cannizaro, just use enough hydroxide to precipitate de Cu(OH)2 and neutralize the tartaric/citric acid, then add an excess of sodium (bi)carbonate or what ever, it'll foam a little bit at the beggining, after refluxing for 5 hours, I think that the piperonal should be steam distilled and from the resulting aqueous mess, add (meta)bisulfite to isolate the adduct. I tried all that with unproperly hydrolized piperate (turned to be fatty acid salts with a little bit sodium piperate) and obtained a vanillaish smell
Will not try it again with piperate because the hydrolisis of the amide plainly sucks. I'd rather extract pure piperine and then chuck that on a manganese-ammonium alum cell.
Another idea would be to drip very slowly a permanganate solution to boiling aqueous piperate and carbonate, from witch the piperonal is steam distilled as it forms, hopefully.
Polverone
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27-8-2010 at 16:01
euxy
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1. Extraction of 462 g ground black pepper with boiling acetone (1.5 l + 1 l, circa 1 hr, more isn't needed) gave 39.8 g resin.
2. Boiled for 1 hr with 483 mls 95% EtOH and 117 g urea (as in US Patent 6054585) and cooled overnight in the fridge, decanted from xtals of urea/fatty acids adduct, evaporated most EtOH and added a little water. Crude semi-solid piperine precipitate weighs 31.1 g.
3. Dissolved in 50 mls EtOAc, added 50 mls pet ether, chilled overnight in the freezer. Yield - 11,3 g of nice xtalline piperine .
4. Refluxed for 19 hrs (TLC shows completion at this time) with an equal weight of 85% KOH in 5x volume of 95% EtOH. Cooled, filtered, washed with IPA on the filter. Yield - 101% K piperate.
5. Recrystallized from 5x EtOH + 1x water. After cooling in the freezer 75% yield counting on piperine was obtained. Had an impression that mother liquor still held some piperate.
6. 2 g K-piperate dissolved in 50 mls water, added 50 mls DCM. Placed in ice bath, dripped in saturated aq. solution of 6.2 g KMnO4 with stirring, temp. kept at 5-10 C. Killed remaining oxidant with ascorbic acid. Separated organic layer, extracted with DCM, washed w/5% NaOH, then brine, stripped off solvent. Yield - 500 mg (42,5% on piperate) piperonal, which was solid at RT but melted in hands (lit. mp. 36 C)
The identity of the product was further confirmed by Henry reaction, which proceeded with good yield. Reduction of the intermediate also gave the product with all of the properties that were to bee expected
[Edited on 10-5-2011 by Antoncho]
Antoncho
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Just to not bee confusing - a pound of black pepper gives about 3,5 g of piperonal via the route described in the previous post.
The quantity that the author had actually gotten was about 1.5 g, but only beecause a lot of experimentaion has been made along the way (e.g., oxidation of piperine with CuSO4 was tried and found to bee not working at all).
Thus, a kilo of black pepper could give one about 7 g of piperonal, perhaps even more.
Hexagon
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What can I say? BTW great work antoncho!
Just one thing, did you tried the cuso4 variation with a citrate or tartrate added? looks it needs a cu(ii) complexing agent added.
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Would a loaded salt of Mn+3 reduced to Mn+2 towards piperic acid which gets oxidized with cleavage to piperonal? Is this selective oxidation strong enough to cleave the double bonds of piperic acid?
Strong oxidizer is mentioned in this oxidative cleavage which KMnO4 is, but I'm sure that this also leads to oxidation all the way to the acid.
Mush
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Quote: Originally posted by euxy The rapodshare link by CherrieBaby is dead, has anyone the articles?
Code: http://www.4shared.com/zip/xDvE_KC_/200613113482_Piperolal-oxidati.html
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The following are mixed together under agitation at ambient temperature:
294.23 g (1.5 mole) of 3,4-methylenedioxy mandelic acid;
562 g of water;
258.75 g of 37% hydrochloric acid being 2.625 moles;
2.1 g of 69% nitric acid being 23 mmoles.
The suspension obtained is heated under agitation at 43°±2° C., then 103.5 mg (1.5 mmole) of sodium nitrite dissolved in 4 g of water is introduced rapidly at this temperature, then 107.5 g of 69% nitric acid, being 1.177 mole, is introduced slowly, over about three hours, in such a way so that the temperature of the reaction medium is maintained at between 40° and 50° C. without using external heating or cooling. At the end of the introduction, the reaction medium is left for one our under agitation at 43°±2° C., then it is cooled down to ambient temperature and finally extraction takes place three times with 600 g of trichloro-1,1,1-ethane. The re-united organic phases are then washed successively, once with water, three times with a saturated aqueous solution of sodium hydrogen carbonate and finally once with water before being concentrated under reduced pressure.
Thus 220 g (1.46 mole) of crude piperonal is isolated which is purified by distillation under reduced pressure.
Thus 178 g (1.186 mole) of pure piperonal is isolated distilling at 106° C. under a vacuum of 2.4 mbars and having a melting point of 37°+1° C. The yield is established at 79% of the theoretically calculated value relative to the 3,4-methylenedioxy mandelic acid used.
http://www.freepatentsonline.com/5095128.html
I expect i'll use DCM rather than trichloro-1,1,1-ethane.
maximumpat
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...thank you. long time lurker here, I shall try and make my first post of consequence a proof of concept on this.
bfesser
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20-2-2014 at 05:11
chemplayer...
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We've tried in the past to perform the permanganate oxidation of piperinic (piperic) acid, and it's possible to do a DCM extract of the reaction mixture and get a tiny tiny amount of something rather nice smelling, but until now nothing isolated. So we tried something different and random:
Preparations:
Solution P (Piperinate):
100ml of water
3.7g of piperinic acid, derived originally from black pepper (17 mmol)
0.68g of sodium hydroxide (17 mmol)
Solution O (Oxidisers):
500ml of water at ~20C
3.9g potassium periodate (17 mmol, 1 x mole equiv)
5.4g potassium permanganate (34 mmol, 2 x mole equiv)
Solution Q (Quench):
75ml of water
5.3g sodium bisulfite (51 mmol)
2.1g sodium hydroxide (51 mmol)
Procedure:
1. Solutions P, O and Q prepared using magnetic stirring in various beakers (P in the large 800ml beaker).
2. 3 ice cubes (~50g) were added to solution P with vigorous stirring.
3. Solution O was agitated with a glass rod (to ensure no residual insoluble periodate had settled).
4. Solution O was added to rapidly stirred solution P over about 15 seconds.
5. Stirring continued and reaction allowed to proceed for 2 minutes.
6. Solution Q added over about 15 seconds, and then the mixture allowed to settle.
7. Mixture filtered (heavy brown-black precipitate), and the precipitate washed with 30ml water.
8. Precipitate washed with 2 x 30ml of dichloromethane.
9. Filtrate (2 layers) placed into separating funnel, shaken, and the bottom DCM layer separated.
10. Aqueous layer extracted with a further 15ml of DCM, and DCM layers combined.
11. DCM dried using anhydrous MgSO4, then decanted and evaporated down (outside, warm day).
12. An oil remained which solidified at room temp into a crystalline pale yellow solid.
13. 0.5g obtained whose melting point range was found to be 34-37C.
Ponderings:
- This was a total guess and we figured we'd see what would happen if periodate and permanganate were combined. We're still not totally sure if the periodate is really playing a responsible role here, but this is the best yield by far we've ever got, an isolated product, and what seems to be a relatively high purity.
- Crude yield is 20% (3.3 mmol) of piperonal from starting piperinic acid.
- Aroma is an incredible vanilla + cherries + toasty buttery ('jelly belly toasted marshmallow flavour') scents. Even slightly 'soapy' and balsamic.
Complete video of the process is here: https://www.youtube.com/watch?v=6r9elLR2WjI
Watch some vintage ChemPlayer: https://www.bitchute.com/channel/chemplayer/
clearly_not_atara
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If you have any more piperic acid, there's an interesting variation where FeSO4 is used to halt the oxidation at the aldehyde stage:
Quote: The oxidation of ethyl alcohol by the action of potassium per- manganate was first reported by Morawski and Sting1 (72) who found that the alcohol was oxidized to acetic acid. The acid in turn combined with excess of the permanganate to produce po- tassium acetate and a brown unidentified precipitate was formed. Doroshevskii and Bardt (31), working under these conditions, found that with the inclusion of ferrous salts as catalysts in the oxidation of ethyl alcohol by potassium permanganate, the results varied with the catalyst selected. If ferrous sulfate was em- ployed the oxidation stopped with the formation of acetaldehyde, but if ferrous oxalate was used as the catalyst, the product was a mixture of aldehyde and acetic acid.
(^the above procedure is for ethanol, but we were speculating it could extend to alkenes)
There is also a procedure which uses triethylbenzylammonium permanganate as a homogenous oxidant in DCM solution, quenched with aqueous AcOH to yield aldehydes selectively (Et3BnNCl + KMnO4 (DCM) >> Et3BnNMnO4 sol'n):
Quote: The procedure is as follows: To a stirred solution of endo-dicyclopentadiene (2.27mmol) in dichloromethane (20ml) was added drop- wise the oxidant solution freshly prepared with KMnO4(3.41mmol),7 triethylbenzylammonium chloride (3.41mmol) and dichloromethane (40ml) at such a rate that the temperature was maintained at 0-3°C under cooling with an ice-bath (40-50min). After addition was complete, stirring was continued until permanganate ion was completely consumed (30-40min). The homogeneous dark brown solution was then treated with aqueous solutions of variant pH. When the reaction mixture was treated with 3% NaOH solution (30ml) under nitrogen atmosphere at room temperature for 18 hours, a crystalline product (2), mp 47-52° (lit. 48-51°), was obtained in 83% yield from the organic layer upon usual work-up. The product was identified as the exo,cis-diol (2) by its ir and nmr data, which was previously obtained in 28% yield by the oxidation of endo-dicyclopentadiene in EtoH with aqueous potassium permanganate.8 No other products were detected on tlc and glc. On the other hand, when the reaction mixture was treated with an acetate solution (30ml) at pH 3, the dialdehyde (3), mp 42-44° (lit. 36-42°), was obtained in 81% yield as the single product(tlc,glc). The compound (3) is known as a key intermediate for prostaglandin synthesis and has been synthesized indirectly through periodate oxidation of the diol (2).
The last procedure which I think is really cool is to use horseradish peroxidase, which cleaves anethole (and piperic acid is pretty similar!):
Quote: Mutti et al. have recently shown that some peroxidases (i.e., horseradish peroxidase, lignin peroxidase, and Coprinus cinereus peroxidase) catalyse the cleavage of a C=C double bond adjacent to an aromatic moiety for selected substrates at the expense of molecular oxygen and at an acidic pH (Scheme 5) [32]. Among the three active peroxidases, HRP turned out to be the most active when an equal concentration of enzyme was employed. A thorough study of the reaction showed that the highest activity was obtained at ambient temperature, at pH 2, and at 2 bars of pure dioxygen pressure. Addition of DMSO as cosolvent up to 15% v v?1 increased the conversion, probably due to the improved solubility of the substrates in the aqueous reaction medium, while a further addition of DMSO led to a progressive decline of the enzymatic activity. Using trans-anethole as substrate (9) (6 g L?1 ) and HRP at low catalyst loading (3 mg, equal to 0.2–0.3 mol%), quantitative conversion was achieved within 24h. The main product was para-anisaldehyde (11) (i.e., 92% chemoselectivity), whereas the side product accounted completely for the vicinal diol (12). The substrate spectrum was quite narrow, since only other two substrates, that is, isoeugenol (10) and indene (13), could be cleaved by HRP with 12% and 72% conversion, respectively.
I'm not entirely sure how to extract horseradish peroxidase from horseradish, but it offers a manganese-free -- hence nontoxic - cleavage of this substrate.
[Edited on 4-7-2016 by clearly_not_atara]
chempropharm
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Nice work.. just a question..
If oxidation of piperine will do the job.. why going to piperic acid intermediate step?
Mush
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Quote: Originally posted by chempropharm Nice work.. just a question.. If oxidation of piperine will do the job.. why going to piperic acid intermediate step?
Yes, it will do the job .
Forensic Sci Int. 2012 Nov 30;223(1-3):306-13. doi: 10.1016/j.forsciint.2012.10.006
2.4. Piperonal from pepper
2.4.1. Oxidative cleavage of piperine by ozonolysis
A stream of ozone gas. generated by a HAILEA HLO 100 Ozone Steriliser
(— 100 mg of ozone per hour), was passed through a solution of piperine (1.00 g. 331 mmol) in a 5% solution of water in acetone. After 8 h. the ozone generator was switched off and the solution extracted with two 10 mL portions of diethyl ether. The combined ether extracts were dried over anhydrous sodium sulphate and the solvent evaporated in vacuo yielding bright yellow oil. Yield: 508 mg (97%). 'H NMR: see Fig. SI3. GC-MS: see Fig. 2.
2.42. Oxidative cleavege of piperine with aqueous KMn04 in THF
An aqueous solution of KMn04 (2.00 g. 12.6 mmol in 40 ml. of water) was added dropwise over a period of 4 h to a solution of piperine (1.00 g. 3.51 mmol in 40 mL of THF] at 60 C. After the solution had been added, the mixture was stirred for 4 h before the MnO2 precipitate that had formed was filtered off leaving a pale yellow solution. The sample was extracted with diethyl ether and the combined extracts were dried over anhydrous sodium sulphate. The ether was evaporated in vacuo to
yield a dark yellow orange oil that solidified on cooling. Yield: 340 mg (65%). 'H NMR: see Fig. SI4. GC-MS: see Fig. 2.
I believe the authors used this paper as reference:
1, https://erowid.org/archive/rhodium/chemistry/alkene2aldehyde...
2,
Code: https://pubs.acs.org/doi/abs/10.1021/jo800323x
It worked as it was predicted by Hive members.
Note:
Ozonolysis yields a mixtures of piperonal (1, GC) and 3,4 methylenedioxycinnamaldehyde (6).
Aqueous KMn04 in THF results less byproducts.
[Edited on 3-8-2019 by Mush]
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Odorographia a natural history of raw materials and drugs used in the
perfume industry intended to serve growers manufacturers and consumers.
John Charles Sawer, 1892, London
Artificial Heliotrope
Piperonal, C8H6O3, , commercially known as " Heliotropine,"
has a very agreeable odour very much like that of heliotrope. The
starting-point in its manufacture is Piperine, C17H19O3. Ground
pepper, preferably the white Singapore pepper, as it contains the
largest amount of alkaloid (9.15 per cent.), is mixed, with twice
its weight of slaked lime and a sufficient quantity of water; the
solution is then evaporated to dryness on a water-bath and the
powder exhausted with commercial ether, from which the piperine
can be obtained nearly pure on evaporation, in large crystals of a
faint straw-yellow colour. To obtain it perfectly pure, it must be
dissolved in alcohol and re-crystallized. Another process of
preparing piperine is to exhaust the pepper with alcohol of sp. gr.
0*833 and distil the tincture to the consistence of an extract.
This extract is to be mixed with potash-lye, which dissolves the
resin and leaves a green powder; by washing this in water, dis-
solving in alcohol, crystallizing and re-crystallizing, it is obtained
colourless *. *Poutet, Journ. de Chim.Med. i.p. 531.
Piperine is converted into potassium piperate by boiling it for
24- hours with its own weight of caustic potash and from 5 to 6
parts of alcohol in a large retort, using an inverted Liebig's
condenser. On cooling, the potassium piperate crystallizes out in
shining yellow laminae. It is washed with cold alcohol and
re-crystallized from hot water. If coloured, it is bleached by animal
charcoal. As thus obtained, it is in nearly colourless crystals,
which become yellow under the influence of light.
One part of potassium piperate is dissolved in from 40 to 50 parts
of hot water, and a solution of 2 parts of potassium permanganate is
gradually poured into the hot liquid with constant stirring. Each
drop of the latter is almost instantly dissolved, and the solution
acquires a very pleasant odour. A pasty mass of brown manganic
hydrate separates, which is placed on a filter and washed with hot
water until the washings cease to smell of heliotropine. These
washings are added together and the whole distilled over an open
fire. The first portions of the distillate contain the largest
proportion of piperonal, the greater part of which crystallizes out
on cooling. The remainder may be obtained by agitation with
ether *. * Chemiker Zeitung, Feb. 1884, and Ann. Ch. Pharm, clii. p. 35.
Piperonal crystallizes from water in colourless, transparent,
highly lustrous prisms, an inch long. It is sparingly soluble in
cold water, easily in cold alcohol, and in all proportions in boiling
alcohol or ether. It melts exactly at 37° and boils without
decomposition at 263°, forming a vapour which has a sp. gr. of 5.18+.
Tiemann and Haarmann state" that the odour of piperonal is
possessed by 'vanillon' a kind of vanilla, which forms thick, fleshy
capsules and is obtained from the West Indies. This sort of vanilla
is employed in perfumery for the preparation of essence of
heliotrope; it contains no piperoral, but vanillin and an oil which
is not yet identified. The perfumers, in preparing essence of
heliotrope, add a little of this oil to the extract of vanillon. If a
little be added to a solution of pure vanilllin, both substances can
be recognized by their smell for some time, but after standing for
months the mixture acquires the smell of heliotrope."
The perfume of " Heliotropine " is completely destroyed by the
action of direct sunlight; it is also injured by heat; it should
therefore be stored in a cool place in the dark, such as a cool cellar,
and be kept in yellow glass bottles, the yellow glass intercepting
the chemical rays.
Attachment: heliotropine, piperonal John Charles Sawer, 1892, London, pages 187-189.rar (637kB)
Mush
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A practical treatise on the manufacture of perfumery
Carl Deite, William T. Brannt, 1892, Philadelphia
Heliotropin or piperonal is of great importance in
the manufacture of perfumes. It forms small, colorless
prismatic crystals, which have an agreeable odor of
heliotrope. Upon the tongue heliotropin produces the same
sensation as oil of peppermint under the same conditions,
the sensation being, however, more lasting. It melts at
about 104 F., and volatilizes at a higher temperature
without leaving a residue. It is soluble in alcohol and
ether, and insoluble in cold water ; in hot water it melts
to an oily liquid which floats upon the water.
Exposed to the action of heat and air, heliotropin
acquires an uncomely appearance, balls together and, under
very unfavorable circumstances, turns brown. It is
then entirely decomposed and useless, and, hence, should
be kept in summer in as cool a place as possible. A
temperature of 95 F. has already an injurious effect
upon the perfume, and it is best not to buy it at all in
the hot summer months. To preserve the perfume in
its entire freshness, it is advisable for consumers in hot
climates to at once dissolve the heliotropin in alcohol
and to keep the solution in a cool place.
Pepper serves as the initial point for heliotropin or
piperonal, the white variety being the best for the purpose.
To obtain piperine, contained in varying quanties
(7 to 9 per cent.) in pepper, the latter is repeatedly
extracted with boiling alcohol. The extract is then
evaporated to one-third its volume, or the greater portion
of the alcohol is distilled off, and the resinous mass,
obtained after the addition of water, is repeatedly washed
in water with the addition of a small quantity, of potash
or soda lye, dissolved in alcohol and purified by repeated
recrystallization. To convert the white-yellow piperine
thus obtained into potassium piperate it is, together with
equal parts of potassium hydroxide and 5 to 6 parts of
alcohol, kept gently boiling for 24 hours in a well-closed
flask provided with an ascending Liebig cooler. A
capacious flask should be used, as the mass pounds quite
vigorously. After cooling, the precipitate, which is
obtained in yellowish, lustrous lamina, is separated through
a filter from the dark-brown mother-lye, washed with
cold alcohol and several times recrystallized from hot
water. A further discoloration may be effected by the
The potassium piperate thus obtained forms nearly
colorless prisms in verucose groups, which, however,
turn yellow when exposed to light. By boiling the
alcoholic mother-lye with 1/3 of the previously used potash-
lye, further small quantities of potassium piperate may
be obtained.
To obtain piperonal from the potassium piperate,
dissolve 1 part of the latter in 40 to 50 parts of hot water,
and then slowly introduce, with constant stirring, a
solution of 2 parts potassium permanganate in 50 parts of
water. This precaution is absolutely necessary, as
otherwise the piperonal formed would be partially further
oxidized and lost. The paste-like mass formed is passed,
while still hot, through a straining cloth, and the residue
repeatedly washed with boiling water until it shows
nothing more of the characteristic odor of heliotrope.
The wash-waters are combined with the first filtrate,
and subjected to distillation over a free fire.
The first distillates are richest in piperonal, it
generally separating already in the cooler. The fractionally
caught distillate is allowed to stand one or two days in
as cool a place as possible, whereby the greater portion
of the piperonal separates in a crystalline form or in fine
lamina. To obtain the piperonal still remaining
dissolved in the water, the mother-lye, after the separation
of the crystals through a filter, may be repeatedly
agitated with ether, whereby the piperonal dissolves in the
ether. The latter is carefully distilled off at as low a
temperature as possible (104 to 122 F.) in the water-bath
or allowed naturally to evaporate.
A practical treatise on the manufacture of perfumery 1892 Piperonal See page 193.djvu
Attachment: A practical treatise on the manufacture of perfumery 1892 Piperonal See page 193.part1.rar (3MB)
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https://help.febio.org/FEBioTheory/FEBio_tm_3-4-Subsection-3.1.2.html | Theory Manual Version 3.4
$\newcommand{\lyxlock}{}$
Subsection 3.1.1: Linearization Up Section 3.1: Weak formulation for Solid Materials Section 3.2: Weak formulation for biphasic materials
### 3.1.2 Discretization
The basis of the finite element method is that the domain of the problem (that is, the volume of the object under consideration) is divided into smaller subunits, called finite elements. In the case of isoparametric elements it is further assumed that each element has a local coordinate system, named the natural coordinates, and the coordinates and shape of the element are discretized using the same functions. The discretization process is established by interpolating the geometry in terms of the coordinates of the nodes that define the geometry of a finite element, and the shape functions: where is the number of nodes and are the natural coordinates. Similarly, the motion is described in terms of the current position of the same particles: Quantities such as displacement, velocity and virtual velocity can be discretized in a similar way.
In deriving the discretized equilibrium equations, the integrations performed over the entire volume can be written as a sum of integrations constrained to the volume of an element. For this reason, the discretized equations are defined in terms of integrations over a particular element . The discretized equilibrium equations for this particular element per node is given by where The linearization of the internal virtual work can be split into a material and an initial stress component [23]: The constitutive component can be discretized as follows: The term in parentheses defines the constitutive component of the tangent matrix relating node to node in element : Here, the linear strain-displacement matrix relates the displacements to the small-strain tensor in Voigt Notation: Or, written out completely, The spatial constitutive matrix is constructed from the components of the fourth-order tensor using the following table; where
I/J i/k j/l 1 1 1 2 2 2 3 3 3 4 1 2 5 2 3 6 1 3
The initial stress component can be written as follows: For the pressure component of the external virtual work, we find where,
Subsection 3.1.1: Linearization Up Section 3.1: Weak formulation for Solid Materials Section 3.2: Weak formulation for biphasic materials | 2021-09-24 09:50:03 | {"extraction_info": {"found_math": true, "script_math_tex": 40, "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": 12, "x-ck12": 0, "texerror": 0, "math_score": 0.8827639818191528, "perplexity": 327.1919121335753}, "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-2021-39/segments/1631780057508.83/warc/CC-MAIN-20210924080328-20210924110328-00034.warc.gz"} |
https://lammps.sandia.gov/doc/Build_make.html | # 3.2. Build LAMMPS with make
Building LAMMPS with traditional makefiles requires that you have a Makefile.<machine> file appropriate for your system in either the src/MAKE, src/MAKE/MACHINES, src/MAKE/OPTIONS, or src/MAKE/MINE directory (see below). It can include various options for customizing your LAMMPS build with a number of global compilation options and features.
## 3.2.1. Requirements
Those makefiles are written for and tested with GNU make and may not be compatible with other make programs. In most cases, if the “make” program is not GNU make, then there will be a GNU make program available under the name “gmake”. If GNU make or a compatible make is not available, you may have to first install it or switch to building with CMake. The makefiles of the traditional make based build process and the scripts they are calling expect a few additional tools to be available and functioning.
• a working C/C++ compiler toolchain supporting the C++11 standard; on Linux these are often the GNU compilers. Some older compilers require adding flags like -std=c++11 to enable the C++11 mode.
• a Bourne shell compatible “Unix” shell program (often this is bash)
• a few shell utilities: ls, mv, ln, rm, grep, sed, tr, cat, touch, diff, dirname
• python (optional, required for make lib-<pkg> in the src folder). python scripts are currently tested with python 2.7 and 3.6. The procedure for building the documentation requires python 3.5 or later.
## 3.2.2. Getting started
To include LAMMPS packages (i.e. optional commands and styles) you must enable (or “install”) them first, as discussed on the Build package page. If a packages requires (provided or external) libraries, you must configure and build those libraries before building LAMMPS itself and especially before enabling such a package with make yes-<package>. Building LAMMPS with CMake can automate much of this for many types of machines, especially workstations, desktops, and laptops, so we suggest you try it first when building LAMMPS in those cases.
The commands below perform a default LAMMPS build, producing the LAMMPS executable lmp_serial and lmp_mpi in lammps/src:
cd lammps/src # change to main LAMMPS source folder
make serial # build a serial LAMMPS executable using GNU g++
make mpi # build a parallel LAMMPS executable with MPI
make # see a variety of make options
Compilation can take a long time, since LAMMPS is a large project with many features. If your machine has multiple CPU cores (most do these days), you can speed this up by compiling sources in parallel with make -j N (with N being the maximum number of concurrently executed tasks). Also installation of the ccache (= Compiler Cache) software may speed up repeated compilation even more, e.g. during code development.
After the initial build, whenever you edit LAMMPS source files, or add or remove new files to the source directory (e.g. by installing or uninstalling packages), you must re-compile and relink the LAMMPS executable with the same make <machine> command. The makefile’s dependency tracking should insure that only the necessary subset of files are re-compiled. If you change settings in the makefile, you have to recompile everything. To delete all objects you can use make clean-<machine>.
Note
Before the actual compilation starts, LAMMPS will perform several steps to collect information from the configuration and setup that is then embedded into the executable. When you build LAMMPS for the first time, it will also compile a tool to quickly assemble a list of dependencies, that are required for the make program to correctly detect which parts need to be recompiled after changes were made to the sources.
## 3.2.3. Customized builds and alternate makefiles
The src/MAKE directory tree contains the Makefile.<machine> files included in the LAMMPS distribution. Typing make example uses Makefile.example from one of those folders, if available. Thus the make serial and make mpi lines above use src/MAKE/Makefile.serial and src/MAKE/Makefile.mpi, respectively. Other makefiles are in these directories:
OPTIONS # Makefiles which enable specific options
MACHINES # Makefiles for specific machines
MINE # customized Makefiles you create (you may need to create this folder)
Simply typing make lists all the available Makefile.<machine> files with a single line description toward the end of the output. A file with the same name can appear in multiple folders (not a good idea). The order the directories are searched is as follows: src/MAKE/MINE, src/MAKE, src/MAKE/OPTIONS, src/MAKE/MACHINES. This gives preference to a customized file you put in src/MAKE/MINE. If you create your own custom makefile under a new name, please edit the first line with the description and machine name, so you will not confuse yourself, when looking at the machine summary.
Makefiles you may wish to try include these (some require a package first be installed). Many of these include specific compiler flags for optimized performance. Please note, however, that some of these customized machine Makefile are contributed by users. Since both compilers, OS configurations, and LAMMPS itself keep changing, their settings may become outdated:
make mac # build serial LAMMPS on a Mac
make mac_mpi # build parallel LAMMPS on a Mac
make intel_cpu # build with the USER-INTEL package optimized for CPUs
make knl # build with the USER-INTEL package optimized for KNLs
make opt # build with the OPT package optimized for CPUs
make omp # build with the USER-OMP package optimized for OpenMP
make kokkos_omp # build with the KOKKOS package for OpenMP
make kokkos_cuda_mpi # build with the KOKKOS package for GPUs
make kokkos_phi # build with the KOKKOS package for KNLs | 2020-12-05 00:09:01 | {"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": 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": 0.39722609519958496, "perplexity": 6837.600479090703}, "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-2020-50/segments/1606141745780.85/warc/CC-MAIN-20201204223450-20201205013450-00422.warc.gz"} |
https://gosiaborzecka.net/2020/07/12/machine-learning-courseras-notes-week-1/ | My notes from the Machine Learning Course provided by Andrew Ng on Coursera
## What is Machine Learning?
"The field of study that gives computers the ability to learn without being explicitly programmed"
~Arthur Samuel (1959)
“A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E"
~Tom Mitchell (1998)
## Supervised learning
• We know how the correct output should look like
• We know the relationship between the input and output
• Two types of supervised learning:
• Regression
• Classification
### Regression
• Predict values based on labelled data
• Continuous valued output
Example: House size -> Sell prize
### Classification
• Predict discrete valued label
Example: Email -> Spam or not
Binary or Binary Classification – group data in two types of label
Multi-class or Multinomial Classification – group data in more than two kinds of label
## Unsupervised learning
• Have only dataset given (without labelled data)
• Algorithms finding structure in data
• No feedback based on the prediction results
• Two types of unsupervised learning:
• Clustering
• Non-clustering
### Clustering
Example: Collection on 1 000 000 different genes and find in automatically group these genes into groups that are somehow similar or related by different variables, such as lifespan, location, roles, etc.
### Non-clustering
Example: Cocktail Party Algorithm – allows to find structure in a chaotic environment
## Supervised Learning vs Unsupervised Learning
• Supervised Learning will ALWAYS have an input-output pair
• Unsupervised Learning will have ONLY data with no label and will try to find some structure
## Model Representation
• x^{(i)}– input variable (input feature)
• y^{(i)} – output variable (target variable that we are trying to predict)
• x^{(i)},y^{(i)} – training example
• x^{(i)},y^{(i)});ⅈ=1,…,m – training set
• Function h:X→Y – hypothesis, the output of the learning algorithm, where h(x) is a good predictor for y
## Regression Problem (Linear Regression)
Linear Regression with one variable x and two parameters θ
h_θ (x)=θ_0+θ_1 x – linear function
Regression problem predict a continuous-valued label
## Cost Function
The occurrence of Hypothesis Function can be measured by using Cost Function
• Hypothesis: h_θ (x)=θ_0+θ_1 x
• Parameters: θ_0,θ_1
• Cost function: J(θ_0,θ_1 )=\frac{1}{2m} \displaystyle\sum_{i=1}^m (h_θ (x_i )-y_i )^2
• Goal: \frac{minimize}{θ_0,θ_1} J(θ_0,θ_1)
Other names for this J(θ_0,θ_1) cost function:
• Squared error function
• Mean squared error
Finds local minimum for cost function J(θ_0,θ_1)
repeat until convergence {
θ_j ≔ θ_j-α \frac{∂}{∂θ_0} J(θ_0,θ_1)
}
Parameter θ should be simultaneously updated at each iteration
Example:
temp0 ≔ θ_0-α \frac{∂}{∂θ_0} J(θ_0,θ_1)
temp1 ≔ θ_1-α \frac{∂}{∂θ_0} J(θ_0,θ_1)
θ_0 := temp0
θ_1 := temp1
• Finds local minimum
• Coverages to a local minimum with fixed α because derivative becomes smaller (baby steps)
## Learning Rate α
• If α is too small, Gradient Descent is slow
• If α is too large, Gradient Descent can overstep minimum and not coverage
Choosing a proper learning rate α in the beginning and stick to it at each iteration since Gradient Descent will automatically take smaller steps when approaching local minimum.
## Gradient Descent for Linear Regression
Linear Regression with one variable
repeat until convergence {
θ_j ≔ θ_j-α \frac{∂}{∂θ_0} J(θ_0,θ_1)
(for j=1 and j=0)
}
### Linear Regression Model
h_θ (x)=θ_0+θ_1 x
J(θ_0,θ_1 )=\frac{1}{2m} \displaystyle\sum_{i=1}^m (h_θ (x_i )-y_i )^2
• Calculate partial deviation $\frac{∂}{∂θ_0} J(θ_0,θ_1)$
\frac{∂}{∂θ_0}J(θ_0,θ_1 )=\frac{1}{2m} \displaystyle\sum_{i=1}^m (h_θ (x_i )-y_i )
\frac{∂}{∂θ_1}J(θ_0,θ_1 )=\frac{1}{2m} \displaystyle\sum_{i=1}^m ((h_θ (x_i )-y_i)x_i)
repeat until convergence {
θ_0 := θ_0 - \alpha \frac{1}{m} \displaystyle\sum_{i=1}^m (h_θ (x^{(i)} )-y^{(i)})
θ_1 := θ_1 - \alpha \frac{1}{m} \displaystyle\sum_{i=1}^m (h_θ (x^{(i)} )-y^{(i)})x^{(i)}
}
• m => size of the training set
• θ_0, θ_1 => constants that are changing simultaneously
• x_i, y_i => values of the given training set
Gradient descent is called batch gradient if it needs to look at every example in the entire training set at each step
### Example of gradient descent to minimalize cost function
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Step 8
Step 9 – a good time to predict (global minimum was found)
## Linear Algebra
• Matrices are two-dimensional arrays
\begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}
• Dimension: rows x columns
• 3×2 matrix => \R^{3x2}
• Entry: A_{ij} -> i^{th} row, j^{th} column
Ex. A_{12} = b
• Vector – matrix with one column and many rows
\begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}
• Dimension: n-dimensional vector
• 3-dimensional vector -> \R^3
## Convention
• Uppercase letters – matrices (A, B, C, …)
• Lowercase letters – vectors (a, b, c, …)
## Scalar
Single value – not a matrix or vector
\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} w & x \\ y & z \end{bmatrix} = \begin{bmatrix} a + w & b + x \\ c + y & d + z \end{bmatrix}
Example:
\begin{bmatrix} 1 & 2 & 4 \\ 5 & 3 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 3 & 4 \\ 1 & 1 & 3 \end{bmatrix} = \begin{bmatrix} 1 + 1 & 2 + 3 & 4 + 4 \\ 5 + 1 & 3 + 1 & 2 + 3 \end{bmatrix} = \begin{bmatrix} 2 & 5 & 8 \\ 6 & 4 & 1 \end{bmatrix}
Undefined for matrices with different dimension
## Matrix subtracting
Subtract each corresponding element
\begin{bmatrix} a & b \\ c & d \end{bmatrix} - \begin{bmatrix} w & x \\ y & z \end{bmatrix} = \begin{bmatrix} a - w & b - x \\ c - y & d - z \end{bmatrix}
Example:
\begin{bmatrix} 1 & 2 & 4 \\ 5 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 3 & 4 \\ 1 & 1 & 3 \end{bmatrix} = \begin{bmatrix} 1 - 1 & 2 - 3 & 4 - 4 \\ 5 - 1 & 3 - 1 & 2 - 3 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 0 \\ 4 & 2 & -1 \end{bmatrix}
Undefined for matrices with different dimension
## Scalar multiplication
Multiply each element by the scalar value
\begin{bmatrix} a & b \\ c & d \end{bmatrix} * x = \begin{bmatrix} a * x & b * x \\ c * x & d * x \end{bmatrix}
Example:
\begin{bmatrix} 1 & 2 & 4 \\ 5 & 3 & 2 \end{bmatrix} * 2 = \begin{bmatrix} 1 * 2 & 2 * 2 & 4 * 2 \\ 5 * 2 & 3 * 2 & 2 * 2 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 8 \\ 10 & 6 & 4 \end{bmatrix}
## Scalar division
Divide each element by the scalar value
\begin{bmatrix} a & b \\ c & d \end{bmatrix} / x = \begin{bmatrix} a / x & b / x \\ c / x & d / x \end{bmatrix}
Example:
\begin{bmatrix} 1 & 2 & 4 \\ 5 & 3 & 2 \end{bmatrix} / 2 = \begin{bmatrix} 1 / 2 & 2 / 2 & 4 / 2 \\ 5 / 2 & 3 / 2 & 2 / 2 \end{bmatrix} = \begin{bmatrix} 0.5 & 1 & 2 \\ 2.5 & 1.5 & 1 \end{bmatrix}
## Matrix-Vector Multiplication
A x X = Y
\begin{bmatrix} & & & \\ & & & \end{bmatrix} x \begin{bmatrix} & \\ & \\ & \end{bmatrix} = \begin{bmatrix} & \\ & \end{bmatrix}
mxn nx1 m-dimensional \ vector
(m rows, n columns) (n-dimensional vector)
To get y_i, multiply A’s i^{th} row with an element of vector x, and add them up:
\begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix} * \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a * x & b * y \\ c * x & d * y \\ e * x & f * y \end{bmatrix}
Example:
\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} * \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 * 3 & 2 * 2 & 3 * 1 \\ 4 * 3 & 5 * 2 & 6 * 1 \\ 7 * 3 & 8 * 2 & 9 * 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 28 \\ 46 \end{bmatrix}
## Matrix-Matrix Multiplication
A x X = Y
\begin{bmatrix} & & & \\ & & & \end{bmatrix} x \begin{bmatrix} & & & \\ & & & \end{bmatrix} = \begin{bmatrix} & & &\\ & & & \end{bmatrix}
mxn matrix nxo matrix mxo matrix
(m rows, n columns) (n rows, o columns) (m rows, o columns)
The i^{th} column of the matrix C is obtained by multiplying A with the ith column of B (for i = 1, 2,… o)
\begin{bmatrix} a & b \\ c & d \\ e & f\end{bmatrix} * \begin{bmatrix} w & x \\ y & z \end{bmatrix} = \begin{bmatrix} a * w + b * y & a * x + b * z \\ c * w + d * y & c * x + d * z \\ e * w + f * y & e * x + f * z \end{bmatrix}
\textcolor{blue}{3x2} \textcolor{blue}{2x2} \textcolor{blue}{3x2}
Example:
\begin{bmatrix} 1 & 2 & 1 \\ 3 & 4 & 3 \\ 5 & 6 & 5 \end{bmatrix} * \begin{bmatrix} 1 & 2 \\ 3 & 2 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} 1 * 1 + 2 * 3 + 1 * 1 & 1 * 2 + 2 * 2 + 1 * 3 \\ 3 * 1 + 4 * 3 + 3 * 1 & 3 * 2 + 4 * 2 + 3 * 3 \\ 5 * 1 + 6 * 3 + 5 * 1 & 5 * 2 + 6 * 2 + 5 * 3 \end{bmatrix} = \begin{bmatrix} 8 & 9 \\ 18 & 23 \\ 28 & 37 \end{bmatrix}
## Matrix Multiplication Properties
• Matrices are not commutative: A * B != B * C
• Matrices are associative: (A * B) * C = A * (B * C)
## Identity Matrix
Denoted I (or I_{nxn}) = simply – has 1’s in the diagonal (upper left to lower right diagonal) and 0’s elsewhere
Example:
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 & ... \\ 0 & 1 & 0 & 0 & ... \\ 0 & 0 & 1 & 0 & ... \\ 0 & 0 & 0 & 1 & ... \\ ... & ... & ... & ... & 1 \\ \end{bmatrix}
For any matrix A
A * I = I * A = A
## Matrix Inverse
If is an mxm matrix, and it has an inverse
A(A^{-1}) = A^{-1} * A = I
Matrices that don’t have an inverse are:
• Singular
• Degenerate
## Matrix Transpose
Let A be on mxn matrix and let B = A^T
Then B is nxm matrix and
B_{ij} = A_{ji}
Example:
A = \begin{bmatrix} 1 & 2 & 0 \\ 3 & 5 & 9 \end{bmatrix} B = A^T \begin{bmatrix} 1 & 3 \\ 2 & 5 \\ 0 & 9 \end{bmatrix}
The first row of A becomes first columns of A^T | 2021-08-01 13:56:50 | {"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": 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": 0.9999938011169434, "perplexity": 631.9403369706499}, "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-2021-31/segments/1627046154214.36/warc/CC-MAIN-20210801123745-20210801153745-00611.warc.gz"} |
https://math.stackexchange.com/questions/4237625/ivt-and-proving-continuity | # IVT and proving continuity
I am looking to prove that $$f(x) = 2xcos(2x) - (x - 2)^{2}$$, on the interval $$[2,3]$$ and $$[3,4]$$ is continuous.
I know the intermediate value theorem applies, but my instructor is also asking that I prove that it is continuous. Does the existence of this intermediate value c where f(c)=0 automatically imply that the function is continuous? I dont think so.
I think I need to show continuity before applying the IVT. How do I do this?
I would appreciate any help, thanks.
• Continuity is required to apply IVT. Using the IVT to establish continuity (a) is circular and (b) doesn't make any sense. Aug 31 at 19:01
• I agree. I'm trying to figure how to establish continuity before applying the IVT. Aug 31 at 19:02
• The IVT only goes in one direction, and there are counterexamples for the other direction. For instance, all derivative functions have the intermediate value property (by Darboux's theorem), but not all derivative functions are continuous. Aug 31 at 19:14
You can simply argue that the function $$f_1(x)=x\cdot\cos 2x$$ is continuous and the function $$f_2(x)=-(x-2)^2$$ is continuous, which makes their sum, $$f(x)=f_1(x)+f_2(x)=x\cdot\cos 2x -(x-2)^2$$ continuous as well (since sum of two continuous functions is continuous).
Note that $$f_2$$ is continuous because it is a polynomial in $$x$$ and all polynomials are continuous since $$f^*(x)=x$$ is continuous which implies $$f_n^*(x)=x^n$$ is continuous since the product of two continuous functions is continuous.
And, $$f_2$$ is continuous again because $$f^*$$ is continuous and $$f_c=\cos x$$ is continuous and the product of two continuous functions is continuous.
• I think you should explain that you are also making repeated use of the fact that sums and products of continuous functions are continuous to show that your $f_1$ and $f_2$ are continuous. Aug 31 at 20:50 | 2021-11-28 00:07:18 | {"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": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8240625858306885, "perplexity": 133.71359689909517}, "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/1637964358323.91/warc/CC-MAIN-20211127223710-20211128013710-00513.warc.gz"} |
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-1?printView=true | Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
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• # Artykuł - szczegóły
## Acta Arithmetica
2013 | 161 | 3 | 201-218
## Characterization of the torsion of the Jacobians of two families of hyperelliptic curves
EN
### Abstrakty
EN
Consider the families of curves $C^{n,A} : y² = xⁿ + Ax$ and $C_{n,A} : y² = xⁿ + A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}(ℚ)$ and $C_{3,A}(ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}(ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}(ℚ)[2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We also almost completely determine the ℚ-rational torsion of $J_{p,A}$ for all odd primes p, and all A ∈ ℚ∖{0}. We discuss the excluded case (i.e. $A ∈ (-1)^{(p-1)/2}pℕ²$).
201-218
wydano
2013
### Twórcy
autor
• Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland | 2022-08-09 21:06:12 | {"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": 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": 0.6683634519577026, "perplexity": 796.8779390581943}, "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/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00574.warc.gz"} |
https://maxov.org/notes/math202a/lec-3.html | # Review
1. Let $$(X, d_X)$$ be a metric space. We say that it is complete if every Cauchy sequence converges.
2. By a completion of $$(X, d_x)$$ we mean a complete metric space $$\overline{X}$$ together with a specific isometry $$d_x \colon X \to \overline{X}$$ whose range is dense in $$\overline{X}$$.
Example. We already looked at $$\mathbb{Q} \hookrightarrow \mathbb{R}$$.
Proposition. If $$(Y_1, j_{y_1})$$ and $$(Y_2, j_{y_2})$$ are both completions of $$(X, d_X)$$ then there is an isometry $$\varphi \colon Y_1 \to Y_2$$ onto, such that $$j_{y_2} = \varphi \circ j_{y_1}$$.
The implication of this proposition is that any two completions of a metric space are isomorphic, implying that completion is unique.
$$S$$ dense in $$X$$ means that for every $$x \in X$$, $$B(X, \epsilon) \cap S \neq \emptyset$$ for all $$\epsilon > 0$$.
# Completions
To build completions, we will essentially construct a new space out of Cauchy sequences. Note that if we have two sequences $$\{ s_n \}, \{ f_n \} \subset X$$ converging to the same point $$x \in X$$, then $$d_X(s_n, t_n) \to 0$$ by the triangle inequality.
In non-complete metric spaces we cannot compare Cauchy sequences based on where they converge, but we can still use this fact to define a notion of equivalence on them:
For any metric space $$(Z, d_z)$$ and Cauchy sequences $$\{ s_n \}, \{ t_n \} \subset Z$$, we say they are equivalent if $$d_Z(s_n, t_n) \to 0$$ as $$n \to \infty$$.
You can check that this equivalence satisfies all the properties of an equivalence relation. Reflexivity and symmetry follow easy, and the triangle inequality gives transitivity.
Proposition. Let $$(X, d_x)$$ and $$(Y, d_y$$) be metric spaces, with $$(Y, d_y)$$ complete. Let $$S$$ be a dense subset of $$X$$ with the metric from $$X$$. Let $$f \colon S \to Y$$ be uniformly continuous. Then there exists a continuous extension, $$\bar{f} \colon X \to Y$$ of $$f$$, where $$\bar{f}|_S = f$$. $$\bar{f}$$ will also be unique and uniformly continuous.
Proof. The full proof is a problem set exercise, so do yourself. We prove existence and continuity.
For $$x \in X$$, choose a sequence $$\{ s_n \} \subset S$$ converging to $$x$$. Because $$f$$ is uniformly continuous, $$\{ f(s_n) \}$$ is a Cauchy sequence in $$Y$$. Since $$Y$$ is complete, it to a point $$p \in Y$$, define $$\bar{f}(x) = p$$.
We must show $$\overline{f}$$ is well-defined, that its definition does not depend on our choice of sequence in the previous part. Formally, we must show that if $$\{ t_n \} \subset S$$ is another Cauchy sequence converging to $$x$$, then $$\{ f_n \}$$ also converges to $$p$$. Since $$\{ s_n \}, \{ t_n \}$$ both converge to $$p$$, they are equivalent Cauchy sequences, i.e. $$d_S(s_n, t_n) \to 0$$. Since $$f$$ is uniformly continuous, $$d(f(s_n), f(t_n)) \to 0$$. This implies $$\{ f(t_n) \} \to p$$.
Every metric space $$(X, d_x)$$ has a completion $$(\overline{X}, d_{\overline{x}})$$.
We will complete the proof in the next lecture. | 2019-03-26 23:10:56 | {"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": 0.9843670725822449, "perplexity": 83.05366722080642}, "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-2019-13/segments/1552912206677.94/warc/CC-MAIN-20190326220507-20190327002507-00327.warc.gz"} |
http://planetmath.org/ifanisirrationalthenaisirrational | if $a^n$ is irrational then ${a}$ is irrational
Primary tabs
Type of Math Object:
Theorem
Major Section:
Reference
Parent: | 2018-02-22 13:00:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 16, "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": 0.47580140829086304, "perplexity": 5002.317037089663}, "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-2018-09/segments/1518891814105.6/warc/CC-MAIN-20180222120939-20180222140939-00018.warc.gz"} |
https://www.hepdata.net/record/101690 | Measurements of $D^{0}$ and $D^{*}$ Production in $p+p$ Collisions at $\sqrt{s} = 200$ GeV
Phys.Rev.D 86 (2012) 072013, 2012.
The collaboration
Abstract
We report measurements of charmed-hadron ($D^{0}$, $D^{*}$) production cross sections at mid-rapidity in $p$ + $p$ collisions at a center-of-mass energy of 200 GeV by the STAR experiment. Charmed hadrons were reconstructed via the hadronic decays $D^{0}\rightarrow K^{-}\pi^{+}$, $D^{*+}\rightarrow D^{0}\pi^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and their charge conjugates, covering the $p_T$ range of 0.6$-$2.0 GeV/$c$ and 2.0$-$6.0 GeV/$c$ for $D^{0}$ and $D^{*+}$, respectively. From this analysis, the charm-pair production cross section at mid-rapidity is $d\sigma/dy|_{y=0}^{c\bar{c}}$ = 170 $\pm$ 45 (stat.) $^{+38}_{-59}$ (sys.) $\mu$b. The extracted charm-pair cross section is compared to perturbative QCD calculations. The transverse momentum differential cross section is found to be consistent with the upper bound of a Fixed-Order Next-to-Leading Logarithm calculation. | 2021-03-03 08:21:53 | {"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": 0.9772244691848755, "perplexity": 1666.0788705718678}, "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-2021-10/segments/1614178366477.52/warc/CC-MAIN-20210303073439-20210303103439-00463.warc.gz"} |
http://math.stackexchange.com/questions/241043/some-problems-about-path-connected-and-projection-maps | # Some problems about path connected and projection maps
1. Why if $\{A_\alpha\}_{\alpha \in \Omega}$ is a collection of path connected spaces. We must show that $\cup_{\alpha \in \Omega}$ is path connected.
• I think $\cap_{\alpha \in \Omega}A_\alpha$ must be nonempty, but I can't prove $\cup_{\alpha \in \Omega}$ is path connected.
2. Prove that $P_\beta : \prod_\alpha X_\alpha \to X_\beta$ is continuous, open and onto for all $\beta.$
• I can prove $P_\beta$ is continuous and open. But I can't proof $P_\beta$ is onto [I think it's easy.] Please hint me to get $P_\beta$ is onto
-
1. Let $z \in \bigcap_{\alpha\in\Omega} A_\alpha$ and let $x,y \in \bigcup_{\alpha\in\Omega} A_\alpha$. Why is there a path $\gamma_1$ from $x$ to $z$? Why is there a path $\gamma_2$ from $y$ to $z$? Can you make a new path $\gamma$ from $x$ to $y$ that combines $\gamma_1$ and $\gamma_2$ in some way?
2. What is the definition of $P_\beta$? If I have some $x_\beta \in X_\beta$ and I choose arbitrary points $x_\alpha \in X_\alpha$ for $\alpha \ne \beta$, what is $P_\beta((x_\alpha)_\alpha)$?
Please check the proof 1. Let $z \in \cap_{\alpha \in \Omega}A_\alpha$ and let $x,y \in \cup_{\alpha \in \Omega}A_\alpha.$ There exist $\alpha,\beta \in \Omega$ such that $$x \in A_\alpha, y \in A_\beta.$$ Since $z \in \cap_{\alpha \in \Omega}A_\alpha$, we have $$z \in A_\alpha, z \in A_\beta.$$ Since $A_\alpha$ is path connected for all $\alpha,$ we have there are path $f_1 : [a,b] \to A_\alpha$ from $x$ to $z$ and path $f_2 : [c,d] \to A_\alpha$ from $y$ to $z$ such that $$f_1(a) = x, f_1(b) = z , f_2(c) = y, f_2(d) = z.$$ How I define new path from $x$ to $y$. ?? – Tee Mth Nov 20 '12 at 4:52
@TeeMth: Try $f:[a,b+d-c] \rightarrow \bigcup_{\alpha\in\Omega} A_\alpha$ given by $f(t)=f_1(t)$ for $t \in [a,b]$ and $f(t)=f_2(b+d-t)$ for $t \in [b,b+d-c]$. – wj32 Nov 20 '12 at 5:50
Thank you very much. Please hint me some problems In First Countable space: F is closed iff $\forall x_n \subset F$ and $x_n \to x$, then $x \in F$ Proof. Suppose that $\forall x_n \subset F$ and $x_n \to x$, then $x \in F$. We will show that $F$ is closed. That is show that $F = \bar{F}.$ Since $F \subset \bar{F},$ I sufficient to show that $\bar{F} \subset F.$ Let $x \in \bar{F}.$ Note that in First countable space, $x \in \bar{F}$ iff $\exists \{x_n\}$ in $F$ such that $x_n \to x.$ By assumption, we have $x \in F.$ Please you check the proof. $(\rightarrow)$I can't proof – Tee Mth Nov 20 '12 at 7:02 | 2015-07-06 20:02:44 | {"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": 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": 0.9619197249412537, "perplexity": 110.30154933109031}, "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-2015-27/segments/1435375098808.88/warc/CC-MAIN-20150627031818-00234-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.jeffirwin.xyz/posts/2021-12-11-a | # Abstract interfaces
## Swapping fractal iterator functions on the fly in Fortran
### 2021 Dec 11
Fractal images are generated by calling an iterator function many times on the 2D coordinates c for each pixel in an image. Colors in the image are assigned depending on how many iterations it takes for a point to escape beyond a certain radius, if it escapes at all.
Two common fractals are the Mandelbrot set and the burning ship fractal (figures from wikipedia):
Figure 1a: the Mandelbrot set Figure 1b: the burning ship fractal
In the black areas, the iterator function stays finite, while in the color-shaded areas it escapes to infinity at various speeds.
The only difference between these fractals is the iterator function. For the Mandelbrot set, the iterator function is (wikipedia):
| (1) | |
while for the burning ship, it is:
(2)
with the initial condition z1 = c.
Under the hood, pretty much everything about a program that creates a fractal image is the same as another program, regardless of the choice of fractal iterator function.
Thus, in accordance with the DRY principle, it makes sense to have a single program that can create fractal images for any iterator function, with as little redundancy as reasonable.
Now we are faced with a problem of how to swap out one iterator function for another at runtime. A user may choose to define the iterator function either as Equation (1) or as Equation (2). How can we set the function in a DRY and computationally efficient way?
The solution is abstract interfaces, procedure pointers, and callbacks.
## Abstract interfaces
As a first step, let’s define the iterator functions from Equations (1) and (2) in Fortran:
Recall that in Fortran, the two-word keyword double complex means a 2D (i.e. complex) number with each component having double the precision as a C float.
Notice any similarities in the functions? Aside from having similar equations of the form z = y ** 2 + c (where the y term may or may not get an abs() workout), these functions have the same exact signature. That is, fmandelbrot and fship both have the same numbers of input arguments and output arguments, and the arguments are of the same types. In this case, they have:
• 2 input arguments z and c
• 1 output argument (named after the respective function)
• all arguments are of the double complex type
Let’s formalize this signature by defining an abstract interface for the iterator functions:
After all, the compiler will need to know the interface before we can pass the function as a callback or swap out a function pointer for one function or another. The abstract interface is how we define this function signature for the compiler.
Note that the name itr_func_interface and the arguments z and c in the abstract interface are all dummies. Their names don’t matter, the arguments just have to match in number and type.
## Callbacks and procedure pointers
Next, let’s define a higher-level function that counts the number of iterations that it takes for a point c to escape, with a given iterator function f:
The function f, passed to nitrescape from outside, is a callback. Its definition is determined at runtime, not compile time. In our case, it could be either the Mandelbrot or the burning ship iterator function. All that is known at compile time is that f has the same function signature as itr_func_interface from the abstract interface.
There is a halting condition maxitr to stop infinite loops for points that don’t escape. The choice of max iterations and escape radius is somewhat arbitrary, but it’s a tradeoff between the quality of the image that you want and how patient you are.
I suppose nitrescape doesn’t really need to be a function and it could just be in the main block of the program instead, but it’s nice to encapsulate things sometimes and this gives us an excuse to demonstrate callbacks and procedure pointers.
The return value from nitrescape will be used to color the image according to some linear color map like HSV.
## It’s all coming together
Finally, let’s put it all together in a top-level subroutine, where the user can choose the iterator function by entering an integer ID ifractal:
The function fiterator is a procedure pointer. Just like f, there is no actual function named fiterator. It only points to another function like fmandelbrot or fship, which is chosen at runtime by the user. The pointer is assigned to an actual function with the => operator. Recall that procedure is the generic Fortran term for either a function (with a return value) or a subroutine (void return value).
In this application, the pointer is set to an actual function once at initialization and left as that function for the duration of the program’s run. If your application requires it, the pointer could be swapped to another function later.
Other applications of these techniques could include anything involving a function of functions. Two classical examples would be evaluating derivatives or definite integrals, probably numerically. The functions being derived or integrated are the lower-level functions, while the function that evaluates the derivative at a point or an integral over a range is the higher-level function. The lower-level functions just need to share the same abstract interface, e.g. all functions of a single real variable, or all functions of two real variables.
## Discussion
It’s worth noting some other (i.e. wrong) approaches that could be used for this particular problem.
### Condition inside loop
The if/else condition that chooses the fractal iterator could be moved inside the loop:
This doesn’t require procedure pointers, just an external function passed to nitrescape. However, adding conditional branches inside big loops can have significant performance disadvantages. And this is a very big loop, going through literally millions of executions for an image with a resolution in the megapixels. There is also a time loop outside the space loop that zooms in frame-by-frame, so that could be way too slow.
However, this is the way I did it at first before I fixed a parallelization bug.
Benchmarks are left as an exercise to the reader. I don’t always follow my own advice on side projects, so there is a lot of branching image-format logic inside the loop. It’s best to disable that and all other extraneous things before benchmarking the core fractal code.
### Copied loops
To avoid the performance disadvantage from putting the condition inside the loop, you can copy the whole loop. Obviously this is dripping WET and hard to maintain:
These nearly-identical loops become very difficult to maintain, especially with all the code omitted between brackets [], and moreso if there are more than 2 possible iterator functions. | 2022-08-12 02:18:19 | {"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": 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": 0.45538368821144104, "perplexity": 953.4608152299761}, "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/1659882571538.36/warc/CC-MAIN-20220812014923-20220812044923-00383.warc.gz"} |
https://forum.allaboutcircuits.com/threads/counting-number-of-logic-1s-in-a-8-bit-binary-vector.28921/ | # Counting number of logic 1s in a 8-bit binary vector
#### Zulfi
Joined Oct 17, 2009
2
Hi,
Right now I am in a middle of an important digital assignment. I require some help with designing a digital logic circuit. I am kind of stuck.
Part of my circuit digital design requires me to design a circuit to calculate the weight of any random byte. In other words, I have to count the number of logic 1s present in a 8-bit binary vector either using adders or counters but I cannot get my head around how to do this properly. One of the conditions is that all 8 bits cannot be zeros or ones.
3 bits would be required to hold the maximum count that is 7 e.g. ABC. I approached by trying a combination of full and half but the circuit is a bit messy and complicated and not working 100%. I wonder if there is a solution using counters. I am allowed to use any ECAD tool.
Any suggestions and ideas will be appreciated. Please illustrate if any particular counter is recommended. There are eight inputs to the counter and three outputs from the counter.
I hope I was able to explain my problem.
#### LKjell
Joined Oct 17, 2009
6
I made something just for 3 bits though. When you get 7 and 0 the output is 0. See if it like this you mean.
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#### t_n_k
Joined Mar 6, 2009
5,447
Are you required to use only adders and/or counters exclusively or can you use other logic - such as an 8-to-1 bit multiplexer?
#### Zulfi
Joined Oct 17, 2009
2
I can use multiplexors if required but I have never used them before and am not very familiar with them.
Please feel free to ask if you are unclear of anything. I am also working on the assignment.
#### LKjell
Joined Oct 17, 2009
6
Well there are many ways to do this. If you want to have minimum of gates then you can try mux method. Otherwise a suggestion is to add two input together. Say you have abcd as input then.
$$A_1 = a\oplus b$$, $$A_2 = ab$$
$$B_1 = c\oplus d$$, $$B_2 = cd$$
Now you add those together using full adder. You should then get 3 output, where $$100_2$$ is the maximum number you get.
Then you repeat with efgh and get 3 output. So you add those 3 output with the previous ones. If both most significant bits are 1 then you can conclude that your input is 1s only and need to report and error. And if you want to check that all of them is zero only then use NOR or OR gates. I prefer NOR since then I can something light up. | 2020-03-29 00:07:18 | {"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": 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": 0.5634889006614685, "perplexity": 706.6427878731658}, "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/1585370493121.36/warc/CC-MAIN-20200328225036-20200329015036-00294.warc.gz"} |
https://dsp.stackexchange.com/questions/17793/segmenting-a-frequency-series-do-we-need-band-pass-filtering | # Segmenting a frequency series: do we need band-pass filtering?
I am trying to implement what's called the "second spectrum". Basically, you do this:
1. Take a time series of length $N$.
2. Divide it into $m$ segment, each of length $N'=N/m$.
3. For each segment $m$, do a Fourier Transform. The result is the 'first spectrum',$S_i^{(1)}(f_1), i=1:m$
4. Divide each spectrum into $n$ octaves, an octave starts at $f_L=f_0 \times 2^p$ and ends at $f_H=f_0 \times 2^{p+1}$, where $p=0,1,2, etc$, and $f_0$ is the lowest frequency in $S_i^{(1)}(f_1)$. For $f_0 = 1$ Hz, The octaves will be like: 1-2 Hz, 2-4 Hz, 4-8 Hz, 8-16 Hz, etc.
5. For each octave in each spectrum, sum all spectrum values in that octave.
6. Construct a time-series for each octave. There will be $n$ time series each of length $m$.
7. Take the Fourier Transform of these time series. This is the second spectrum, $S_n^{(2)}(f_2)$
I have already implemented this algorithm (in c++), but I have a question:
Q: in step 4, can I simply divide the spectrum signal into octaves "just like that"? I mean, without any kind of band-pass filtering to these octaves?
EDIT: what I mean by the "just like that" is as if I am passing the spectrum through a rectangular window (in freq. domain). | 2019-07-24 09:58:59 | {"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": 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": 0.8885538578033447, "perplexity": 1041.1377540343055}, "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-2019-30/segments/1563195532251.99/warc/CC-MAIN-20190724082321-20190724104321-00523.warc.gz"} |
https://www.nature.com/articles/s41598-019-39888-7?error=cookies_not_supported&code=0f11c95f-b378-45d2-a85a-656a957c56d2 | # ENLIVE: An Efficient Nonlinear Method for Calibrationless and Robust Parallel Imaging
## Abstract
Robustness against data inconsistencies, imaging artifacts and acquisition speed are crucial factors limiting the possible range of applications for magnetic resonance imaging (MRI). Therefore, we report a novel calibrationless parallel imaging technique which simultaneously estimates coil profiles and image content in a relaxed forward model. Our method is robust against a wide class of data inconsistencies, minimizes imaging artifacts and is comparably fast, combining important advantages of many conceptually different state-of-the-art parallel imaging approaches. Depending on the experimental setting, data can be undersampled well below the Nyquist limit. Here, even high acceleration factors yield excellent imaging results while being robust to noise and the occurrence of phase singularities in the image domain, as we show on different data. Moreover, our method successfully reconstructs acquisitions with insufficient field-of-view. We further compare our approach to ESPIRiT and SAKE using spin-echo and gradient echo MRI data from the human head and knee. In addition, we show its applicability to non-Cartesian imaging on radial FLASH cardiac MRI data. Using theoretical considerations, we show that ENLIVE can be related to a low-rank formulation of blind multi-channel deconvolution, explaining why it inherently promotes low-rank solutions.
## Introduction
Since acquisition speed is a major issue in MRI, accelerated imaging with multiple receiver coils has been an active field of research since its beginning. Quite rapidly, two main categories of parallel imaging methods emerged: image-space methods, of which sensitivity encoding (SENSE)1 is the prototypical example and k-space methods, where it is generalized autocalibrating partially parallel acquisitions (GRAPPA)2. SENSE-like methods, when the coil sensitivity profiles are known, permit a natural description as a linear inverse problem. Incorporating the estimation of coil sensitivity profiles into the reconstruction leads to a nonlinear inverse problem, as formulated in Joint Image Reconstruction and Sensitivity Estimation in SENSE (JSENSE)3 and Regularized Nonlinear Inversion (NLINV)4.
Additionally, low-rank and subspace methods5,6,7,8 have been proposed to further increase reliability and acceleration in MRI. These methods exploit prior knowledge on the structure of the matrices arising in MRI reconstruction. Recently, ESPIRiT9 has been shown to provide robustness towards data inconsistencies similar to k-space methods such as GRAPPA2. In particular, in cases where the chosen field-of-view (FOV) is smaller than the object10 and in phase-constraint imaging11, it was shown that methods based on traditional SENSE that only use a single set of coil sensitivity profiles exhibit artifacts. In ESPIRiT, robust reconstruction is possible through a relaxed SENSE-model, which uses multiple images and sets of coil sensitivity profiles.
ESPIRiT recovers accurate coil sensitivities using an eigenvalue decomposition of an image-domain operator which projects onto the signal space of the calibration matrix. In case of inconsistencies, it produces multiple sets of maps which can be used in a relaxed SENSE reconstruction. ESPIRiT requires a fully-sampled calibration region in the center of k-space. Additionally, it cannot be applied directly to non-Cartesian data, requiring an additional gridding step to generate calibration data. A more generic subspace method is SAKE5, because it can be directly applied to data without fully-sampled calibration region or non-Cartesian data. Based on the idea that the signal is contained in a sub-space of smaller dimensionality which can be recovered, SAKE uses structured low-rank matrix completion to recover a full k-space from incomplete data. Unfortunately, it is computationally extremely demanding as each iteration has to perform a singular-value decomposition (SVD). Furthermore, because it operates completely in k-space, regularization terms may require additional Fourier transforms and must be applied to all channels. Calibration-free locally low-rank encouraging reconstruction (CLEAR)8 is a related method which uses local low-rankness in the image domain instead of the global k-space rank penalty used in SAKE. This reduces the computational complexity by reducing the size of the needed SVDs, although it does increase the number of SVDs necessary. Furthermore, as it is an image space method, regularization can be integrated more easily.
Regularized Nonlinear Inversion (NLINV)4 jointly estimates the image content and the coil sensitivity profiles using a nonlinear algorithm. Similar to SAKE, it does not require a fully-sampled Cartesian calibration region and can be applied directly to non-Cartesian data.
This work aims at combining the advantages from these different methods. Inspired by ESPIRiT, we propose an extension to NLINV that extends it beyond the original SENSE-like model. This method, termed ENLIVE (Extended NonLinear InVersion inspired by ESPIRiT), can be related to a convex relaxation of the NLINV problem subject to a low-rank constraint. From NLINV, it inherits its flexibility and suitability for calibrationless and non-Cartesian imaging; from ESPIRiT it inherits robustness to data inconsistencies. We apply ENLIVE to several imaging settings covering limited FOV, phase constraints, phase singularities, and non-Cartesian acquisition. Additionally, we present comparisons to ESPIRiT and to SAKE.
Initial results have been presented at the 25th Annual Meeting of the International Society for Magnetic Resonance in Medicine12.
## Theory
### Formulation
NLINV recovers the image m and the coil sensitivity profiles cj from measurements yj by solving the regularized nonlinear optimization problem:
$$\mathop{{\rm{\arg }}\,{\rm{\min }}}\limits_{{\boldsymbol{m}},{{\boldsymbol{c}}}_{j}}\,{\sum }_{j=1}^{{N}_{C}}{\Vert {{\boldsymbol{y}}}_{j}-{\mathscr{P}} {\mathcal F} \{{{\boldsymbol{c}}}_{j}\odot {\boldsymbol{m}}\}\Vert }_{2}^{2}+\alpha ({\sum }_{j=1}^{{N}_{C}}{\Vert {\boldsymbol{W}}{{\boldsymbol{c}}}_{j}\Vert }_{2}^{2}+{\Vert {\boldsymbol{m}}\Vert }_{2}^{2})$$
(1)
with NC coils, the two or three dimensional Fourier transform $${\mathcal F}$$, the projection $${\mathscr{P}}$$ onto the measured trajectory (or the acquired pattern in Cartesian imaging) and an invertible weighting matrix W penalizing high frequencies in the coil profiles. Here, both image $${\boldsymbol{m}}\in {{\mathscr{C}}}^{{n}_{x}\cdot {n}_{y}\cdot {n}_{z}}$$ and coils $${{\boldsymbol{c}}}_{j}\in {{\mathscr{C}}}^{{n}_{x}\cdot {n}_{y}\cdot {n}_{z}}$$ are regarded as vectors of size $${n}_{x}\cdot {n}_{y}\cdot {n}_{z}=\,:{N}_{I}$$ and $$\odot$$ is their element-wise product.
In this work, we propose to extend this model to:
$$\mathop{{\rm{\arg }}\,{\rm{\min }}}\limits_{{{\boldsymbol{m}}}^{i},{{\boldsymbol{c}}}_{j}^{i}}\,{\sum }_{j=1}^{{N}_{C}}{\Vert {{\boldsymbol{y}}}_{j}-{\mathscr{P}} {\mathcal F} \{{\sum }_{i=1}^{k}{{\boldsymbol{c}}}_{j}^{i}\odot {{\boldsymbol{m}}}^{i}\}\Vert }_{2}^{2}+\alpha \,{\sum }_{i=1}^{k}({\sum }_{j=1}^{{N}_{C}}{\Vert {\boldsymbol{W}}{{\boldsymbol{c}}}_{j}^{i}\Vert }_{2}^{2}+{\Vert {{\boldsymbol{m}}}^{i}\Vert }_{2}^{2})$$
(2)
where $${{\boldsymbol{c}}}_{j}^{i}$$ and mi are k sets of unknown coil sensitivity profiles and unknown images. This approach is inspired by ESPIRiT, which uses additional maps to account for model violations9.
In the following, we will show that this formulation automatically produces solutions with rank even smaller than k if one exits. To show this, we first relate Eq. (2) to a linear inverse problem for matrices with nuclear norm regularization.
From here on, we assume that the variable transformation $${\hat{{\boldsymbol{c}}}}_{j}={\boldsymbol{W}}{{\boldsymbol{c}}}_{j}$$ has been applied to move the weighting matrix from the regularization into the forward operator. We note that this problem is equivalent to a corresponding multi-channel blind deconvolution problem13 in k-space via the convolution theorem. Using the “lifting” approach used for such blind deconvolution problems14, which can also be applied in the image domain, we now lift the Eq. (1) into a linear inverse problem in terms of a rank-1 matrix X = uvT formed by the tensor product of u and v, where u corresponds to m and v is a stacked vector composed of the weighted coil sensitivity profiles $${\hat{{\boldsymbol{c}}}}_{j}$$. The problem then becomes:
$$\mathop{{\rm{\arg }}\,\,{\rm{\min }}}\limits_{{\boldsymbol{u}},{\boldsymbol{v}}}\,{\Vert y-{\mathscr{A}}\{{\boldsymbol{u}}{{\boldsymbol{v}}}^{T}\}\Vert }_{2}^{2}+\alpha ({\Vert {\boldsymbol{u}}\Vert }_{2}^{2}+{\Vert {\boldsymbol{v}}\Vert }_{2}^{2})$$
(3)
with a linear operator $${\mathscr{A}}$$ mapping uvT to $${\mathscr{P}} {\mathcal F} {{\boldsymbol{c}}}_{j}\odot {\boldsymbol{m}}$$ and a vector y containing measurement data of all coils. Such an $${\mathscr{A}}$$ exists because uvT contains all possible products of elements of u and v. Its explicit action is explained in more detail in the Appendix. In general, all bilinear functions can be expressed as linear functions on the tensor product of the two vector spaces involved.
As suggested by Ahmed et al.14 for blind multi-channel deconvolution, we now relax the rank-1 constraint and allow k sets of images and coil sensitivity profiles. This corresponds to using $${\boldsymbol{X}}={\boldsymbol{U}}{{\boldsymbol{V}}}^{T}\in {{\mathscr{C}}}^{{N}_{I}\times {N}_{C}\cdot {N}_{I}}$$ with $${\boldsymbol{U}}\in {{\boldsymbol{C}}}^{{N}_{I}\times k}$$ and $${\boldsymbol{V}}\in {{\mathscr{C}}}^{{N}_{C}\cdot {N}_{I}\times k}$$, which then leads to the optimization problem
$$\mathop{{\rm{\arg }}\,\,{\rm{\min }}}\limits_{{\boldsymbol{U}},{\boldsymbol{V}}}\,{\Vert {\boldsymbol{y}}-{\mathscr{A}}\{{\boldsymbol{U}}{{\boldsymbol{V}}}^{T}\}\Vert }_{2}^{2}+\alpha ({\Vert {\boldsymbol{U}}\Vert }_{F}^{2}+{\Vert {\boldsymbol{V}}\Vert }_{F}^{2})$$
(4)
with the Frobenius norm $${\Vert \cdot \Vert }_{F}$$. In the Appendix we show how this corresponds to ENLIVE as formulated in Eq. (2). Under conditions given below, Eq. (4) is equivalent to a convex optimization problem for the matrix
$$\mathop{{\rm{\arg }}\,\,{\rm{\min }}}\limits_{{\boldsymbol{X}}}\,{\Vert {\boldsymbol{y}}-{\mathscr{A}}\{{\boldsymbol{X}}\}\Vert }_{2}^{2}+2\alpha {\Vert {\boldsymbol{X}}\Vert }_{\ast }$$
(5)
with nuclear norm $${\Vert \cdot \Vert }_{\ast }$$ regularization15,16. The nuclear norm promotes low-rank solutions. Furthermore, if the solution to Eq. (5) has rank smaller than or equal to k both problems are equivalent in the sense that from a solution U, V of Eq. (4) one obtains a solution of Eq. (5) via X = UVT which attains the same value and from a solution X of Eq. (5) one can construct a solution of Eq. (4) that attains the same value. This is achieved by factorizing X using the SVD and by distributing the singular values in an optimal way, i.e. equally as square roots, to the two factors. Please note that we do not propose to use this convex formulation for computation as it is very expensive, instead we propose to use the nonlinear formulation given in Eq. (2). Nevertheless, this relationship to nuclear-norm regularization is important as it explains why ENLIVE produces solutions with low rank even smaller than k, if one exists.
### Implementation
Similar to NLINV4, we solve Eq. (2) using the iteratively regularized Gauss-Newton method (IRGNM). The IRGNM solves successive linearizations with the regularization parameter decreasing in each Newton step: Starting from α0, the regularizations in each step is reduced according to αn = α0qn−1, 0 < q < 1. As initial guess, we use mi ≡ 1 for the images and $${{\boldsymbol{c}}}_{j}^{i}\equiv 0$$ for the coil sensitivity profiles. Because we initialize images and sensitivity profiles for all sets in the same way, the problem is symmetric in the sets and the algorithm will produce degenerate solutions with identical sets. To break this symmetry, we require the k sets of coil profiles to be orthogonal using Gram-Schmidt orthogonalization after each Newton step. For orthogonalization, the coil profiles of each set are treated as stacked one-dimensional vectors.
The weighting matrix W enforcing smoothness in the coil profiles was chosen as in4. In k-space, this leads to a penalty increasing with distance from the center of k-space according to $${\mathrm{(1}+a{\Vert {\boldsymbol{k}}\Vert }^{2})}^{b\mathrm{/2}}$$. In this work, a = 240 and b = 40 were used. Furthermore, k-space is normalized so that it extends from −ni/2 to ni/2 for i {x, y, z}. As W applies weights in k-space, it is the product of a Fourier matrix transforming each coil profile to k-space an of this diagonal weight matrix.
Images and coil profiles are combined in a post-processing step. This is used to either create individual images for each set i by combining coil-weighted images $${{\boldsymbol{m}}}^{i}{{\boldsymbol{c}}}_{j}^{i}$$ using
$${{\boldsymbol{M}}}^{i}=\sqrt{\sum _{j=1}^{{N}_{C}}|{{\boldsymbol{m}}}^{i}\odot {{\boldsymbol{c}}}_{j}^{i}{|}^{2}}$$
(6)
or to create a single combined image by first combining each set to obtain a proper image for each coil and then doing a final coil combination with
$${\boldsymbol{M}}=\sqrt{\sum _{j=1}^{{N}_{C}}|\sum _{i=1}^{k}{{\boldsymbol{m}}}^{i}\odot {{\boldsymbol{c}}}_{j}^{i}{|}^{2}}\mathrm{.}$$
(7)
## Results
### Limited FOV
In the examples with a restricted FOV, both ENLIVE with a single set of maps, i.e. NLINV, and ESPIRiT reconstructions show a similar central artifact (Fig. 1). This artifact can be readily explained as a consequence of the undersampling pattern and the signal model violation at the edges of the image: Without a parallel imaging reconstruction, we expect aliasing artifacts from all pixels in the FOV. The parallel imaging reconstruction using a single set of maps can resolve this aliasing only for pixels outside of the regions of model violation. Since these edge regions alias to the image center, the artifact appears there. Both ENLIVE and ESPIRiT reconstructions allowing multiple sets of maps (Figs 1 and 2a) can resolve the aliasing everywhere. For ENLIVE, the coil profiles (Fig. 3) of the second map are sensitive in these regions. For ENLIVE using more than 2 sets of maps, the third and fourth map are close to zero (Fig. 2b). Since no thresholding is used, they cannot be exaclty zero. As is common in parallel imaging, tuning of the regularization is necessary for successful reconstruction: Fig. 4 shows that using too high regularization (too few Newton steps) does not eliminate the central infolding artifact, while too low regularization (too many Newton steps) leads to high-frequency artifacts. Added noise degrades image quality, especially in the case of too low regularization, but does not change the appearance of the infolding artifact. Additionally, Fig. 5 shows that the reconstruction is not sensitive to specific choices for the parameters a and b of the coil weighting matrix W.
### Phase-constrained Imaging
Next, reconstructions for phase-constrained imaging using virtual-conjugate coils with and without an additional partial-Fourier factor are shown in Fig. 6. In both cases, reconstruction using only a single set of maps exhibit aliasing artifacts. These are a consequence of the real-value constraint imposed by using virtual-conjugate coils together with high-frequency phase variations caused by off-resonance from fat: A single real-valued image cannot account for this high-frequency phase, therefore the aliasing cannot be resolved. Relaxing the reconstruction by allowing multiple sets of maps resolves this problem, since the second set of maps can now account for this high-frequency phase variation.
### Phase Singularities
Figure 7a shows a phantom example where the initial guess has been intentionally chosen to induce a phase singularity in the reconstruction. The phase singularity leads to signal loss using a single set of maps. Using ENLIVE allowing multiple sets of maps, the affected region can be resolved in the second map. By combining the images, a single image without signal loss can be recovered. This situation can also occur in practice. Figure 7b shows a slice through the throat with large phase variations, while Fig. 7c shows a short-axis view of the human heart acquired with radial FLASH. Using ENLIVE allowing multiple sets of maps, it is possible to reconstruct artifact-free images.
### Low-rank Property
Figures 8 and 9 show calibrationless variable-density Poisson-disc undersampled reconstructions with differing undersampling factors comparing ENLIVE to SAKE. In Fig. 8, both ENLIVE and SAKE provide artifact-free reconstruction for moderate undersampling up to R = 4.0. At R = 7.0, SAKE shows artifacts while ENLIVE is artifact free. For these undersampling factors, the second ENLIVE set image is close to zero, while the first set contains the image. For R = 8.5, both ENLIVE and SAKE show strong artifacts. Additionally, the second ENLIVE map shows some image features. Reconstruction time for R = 4.0 for this dataset using a single core of an Intel Core i5-4590 CPU was 22 s using ENLIVE and 6.3 h using SAKE. In Fig. 9, ENLIVE and SAKE provide artifact-free reconstruction up to R = 3.0. At R = 5.0, ENLIVE reconstruction is noisy while SAKE shows a large signal void. Reconstruction time for R = 2.0 for this dataset using a single core was 18.6 s using ENLIVE and 41.5 min using SAKE.
Figure 10 shows Cartesian ENLIVE reconstructions of data undersampled using CAIPIRINHA patterns with different undersampling factors. As a reference, the corresponding patterns are shown in the first column. For all undersampling factors, the second map image is close to zero wile the first map contains the entire image. With increasing undersampling, high noise starts to appear in the first map and the combned image. Still, no undersampling artifacts appear even at R = 16. Furthermore, even at this high undersampling, no image features appear in the second map, in contrast to the result in Fig. 8. We conjecture that the adequate calibration region in this datasets prevents that artifact.
## Discussion
This work introduces ENLIVE, a nonlinear reconstruction method for parallel imaging using a relaxed forward model. Using the IRGNM, ENLIVE simultaneously estimates multiple sets of images and coil sensitivity profiles, extending NLINV by ESPIRiT’s approach of using multiple sets of maps. The resulting bi-linear problem with $${\ell }_{2}$$-regularization can be related to a lifted linear formulation using nuclear norm regularization, which promotes low-rank solutions. From this, it becomes apparent that the method, while employing a different parametrization, is similar to SAKE and P-LORAKS6,7, which are based on structured low-rank matrix completion in k-space, and to CLEAR8, which locally promotes low-rankness in the image domain. Although the low-rankness of the matrix considered in the k-space methods is also caused by the fact that the signal lives in a sub-space spanned by the coil sensitivities5,9, it is constructed from many shifted copies of the signal in k-space. This leads to a huge linear reconstruction problem with a rank constraint. In contrast, CLEAR uses block-wise reconstruction in the image domain, which is more similar to ENLIVE, but still requires a large number of small SVDs. A similar concept has been used to implement other low-rank methods. For example, building on top of the work on object modeling introduced in17, several approaches using annihilating filters have recently been proposed for combining parallel imaging with compressed sensing18,19,20,21. The existence of annihilating filters implies in turn the existence of a weighted low-rank Hankel matrix which can be constructed from the k-space samples. These methods then recover missing samples by structured low-rank matrix completion. In ENLIVE, the convex matrix completion problem has been replaced by a much smaller bi-linear problem with simple quadratic penalties15,16. In some sense, this is similar to the idea of transforming linear problems with $${\ell }_{1}$$-regularization into quadratic problems with $${\ell }_{2}$$-regularization22.
Low-rank approaches have also been proposed for dynamic imaging. One method for blind compressed sensing23 estimates both the time series of images as well as a dictionary which sparsifies that series. Haldar and Liang24 introduce a method which uses partial separability of the signal into functions describing its k-space and its time dependence. Both of these approaches exploit the low rank of the time-dependent signal. While structurally similar, Haldar and Liang24 use an explicitly rank-constraint formulation while Lingala et al.23 use an $${\ell }_{1}$$-norm to induce sparsity. In contrast, ENLIVE’s $${\ell }_{2}$$-regularization achieves low-rankness even below its constraint on the maximum rank through the equivalence to a formulation with regularization of the nuclear norm outlined in the Theory, which forms the core of the proposed method.
ENLIVE can also be related to a previous extension of NLINV proposed for separation of chemical species25,26. This method is based on the idea that the signal is a superposition of different images shifted in the spatial domain according to the chemical shift. As also shown for ESPIRiT, the sensitivities for the shifted signals from different species also appear to be shifted. They therefore violate the simple SENSE model with a single set of of maps and, consequently, cause the appearance of a second set of maps. The previously proposed extension to NLINV can be understood as a version of ENLIVE with the additional constraint that different sets of sensitivities are shifted versions of each other.
As shown in this work, small FOV and phase-constrained reconstructions using a single set of maps show artifacts whenever there are inconsistencies which cannot be explained using the simple model, while ENLIVE allowing two sets of maps enables artifact-free reconstruction in all evaluated cases. When using correct regularization, added noise does not impede artifact removal either. In cases where reconstruction with a single set of maps is already free from artifacts, ENLIVE automatically only uses a single set. In general, though, the maximum number of ENLIVE maps must be specified manually. This is similar to ESPIRiT where, while theoretically the correct number of maps can automatically be estimated as the multiplicity of the eigenvalue 1, in practice a maximum number of maps is set in advance to enable efficient computation of the eigenvector maps by power iteration. However, an extension to ENLIVE to automatically adapt the number of maps during the iteration is also conceivable.
As the distribution of the phase between image and coil sensitivities cannot be determined from the data alone without additional prior knowledge, choosing a good phase is a common problem when calibrating sensitivities27,28. This fundamental problem affects different algorithms in different ways. In Walsh’s method29 or ESPIRiT a pixel-wise phase across channels simply remains undefined and has to be aligned to a reference. If the reference is not ideal, phase singularities may occur. Phase singularities imply a non-smooth phase which then reduces sparsity in compressed sensing, preventing an efficient and compact representation of the sensitivities in the Fourier domain30, or causing problems in post-processing. For example, as Li et al.31 have shown, phase singularities can appear as artifactual microhemorrhage in susceptibility weighted imaging. NLINV and ENLIVE guarantee smooth sensitivities, but this then traps the algorithm in a local minimum and creates a hole instead32. For ENLIVE, the use of a second set of maps may still avoid signal loss in the reconstruction.
Even though local minima are a general concern with nonlinear methods, in our experience, the only practically relevant examples are the phase singularities. There, although the ENLIVE reconstruction is not optimal, use of a second map may mitigate the resulting artifact.
Compared to ESPIRiT, ENLIVE is more flexible since it has fewer prerequisites for its use, e.g. no calibration region is necessary. However, in the case of an undersampled Cartesian acquisition with calibration region, ESPIRiT is still to be preferred in most cases because of its speed. Only when faced with a very large number of channels might ESPIRiT lead to longer reconstruction times due to the unfavorable scaling of its SVD with the number of channels.
In summary, ENLIVE combines different advantages of NLINV, ESPIRiT, and SAKE. As NLINV and SAKE, it utilizes all available data, can be directly applied to non-Cartesian data, and does not require a calibration region. As ESPIRiT and SAKE, it is not limited to the SENSE model but automatically adapts to certain inconsistencies in the data. As ESPIRiT and NLINV, it is computationally efficient and makes use of an explicit image-domain representation during reconstruction which facilitates the use of advanced regularization terms.
## Conclusion
In this work we propose ENLIVE, a nonlinear method for parallel imaging which seeks to combine the robustness of ESPIRiT with the flexibility of NLINV. ENLIVE can be related to a lifted formulation of blind multi-channel deconvolution with nuclear norm regularization, which show that it belongs to the class of calibrationless parallel imaging methods based on structured low-rank matrix completion. In imaging settings involving limited FOV, phase constraints, and phase singularities, it has been shown to provide artifact-free reconstruction with quality comparable to state-of-the-art methods.
## Methods
The proposed method was implemented in the Berkeley Advanced Reconstruction Toolbox (BART)33 and all other reconstructions were performed using BART as well. Process-level parallelization was achieved using GNU parallel34. To facilitate the reproducibility of our research, data and source code used to generate the results of this paper can be downloaded from https://github.com/mrirecon/enlive.
To test its robustness in case of inconsistencies, ENLIVE was applied in several different experimental settings: We selected examples for imaging with an FOV smaller than the extent of the object, phase-constrained imaging, and phase singularities. In all cases, reconstructions using ENLIVE were performed using one, i.e. NLINV, or two sets of maps with initial regularization set to α0 = 1. If not stated otherwise, 11 Newton steps and q = 1/2 were used for the IRGNM. These parameters, as well as the parameters for the other methods, were chosen according to best visual appearance.
All volunteer imaging for this study was performed with their prior informed written consent, in accordance with the relevant guidelines and regulations, and with the approval of the ethics committee of the University Medical Center Göttingen.
In an example without inconsistencies we tested whether ENLIVE produces results with only one set of maps. Additional examples show ENLIVE’s performance under high undersampling and in non-Cartesian imaging.
### Limited FOV
We applied ENLIVE to the same dataset used in9. This is a retrospectively 2-fold undersampled 2D spin-echo dataset (TR/TE = 550/14 ms, FA = 90°, BW = 19 kHz, matrix size: 320 × 168, slice thickness: 3 mm, 24 × 24 calibration region) with an FOV of 200 × 150 mm2, acquired at 1.5 T using an 8-channel head coil. The dataset was zero-padded in k-space to produce square image space pixels. This FOV is smaller than the head of the subject in the lateral direction which leads to artifacts in a traditional SENSE reconstruction. These data were reconstructed with ENLIVE using one or two sets of maps and compared to ESPIRiT using one or two sets of maps. To investigate the effect of additional sets of maps, the data were additionally reconstructed using 1, 2, 3, and 4 sets of maps. For ENLIVE, q = 2/3 was used. To investigate the sensitivity to noise and to regularization, an additional reconstruction using 13, 16, 19, 22 and 25 Newton steps and added Gaussian white noise with noise levels of 0%; 0.1%; 1%; 2.5%; 5% was performed. The noise level here is the standard deviation of the added noise as percent of the magnitude of the DC component. From this, 19 Newton steps was determined as the optimum and used for reconstruction. For ESPIRiT a kernel size of 6 × 6 and a threshold of 0.001 was used.
### Phase-constrained Imaging
Phase-constrained parallel imaging35 with virtual conjugate coils36 is equivalent to an explicit phase constraint in SENSE, but more robust in GRAPPA and ESPIRiT due to their ability to adapt to inconsistencies11,37. To assess ENLIVE’s performance in phase-constrained imaging settings with virtual conjugate coils, we applied it to the same dataset used in11. This is a single slice in readout direction of a retrospectively 3-fold undersampled 3D FLASH dataset (TR/TE = 11/4.9 ms) acquired at 3 T using a 32-channel head coil. 24 × 24 auto-calibration lines were used. Additionally, a partial Fourier factor of 5/8 was applied to the data and evaluated separately.
### Phase Singularities
Similar to other algorithms27,28,32 phase singularities can appear in coil sensitivity profiles with ENLIVE. As ENLIVE enforces smooth coil sensitivity profiles, this leads to an artifactual hole in the sensitivities around the singularity. To demonstrate this effect, we synthetically constructed an example using BART to generate 6-channel k-space data (matrix size: 256 × 256) of the numerical Shepp-Logan phantom. To get ENLIVE trapped in a local minimum with a phase singularity, we provided an initial guess already containing a phase singularity. In regions with rapid phase variation, such phase singularities can also appear in ENLIVE reconstructions of in-vivo data. A transversal slice through the throat containing such a phase singularity was selected from the same dataset used for phase-constrained imaging.
To further show that ENLIVE can be applied directly to non-Cartesian data, we reconstructed selected data containing a phase singularity from a real-time FLASH38 acquisition using a 30 channel thorax coil of a short-axis view through the heart of a volunteer with no known illnesses (TR/TE = 2.22/1.32 ms, FA = 10°, matrix size: 160 × 160, FOV = 256 × 256 mm2, slice thickness: 6 mm, field strength: 3.0T). Five consecutive frames during diastole, comprising 65 radial spokes, were selected, corrected for gradient delays39, regridded to a 1.5 times finer grid and subsequently reconstructed with ENLIVE using 1 and 2 maps. For this dataset, q = 2/3 and 17 iterations of the IRGNM were used.
### Low-rank Property
In order to show that ENLIVE automatically uses only the required number of sets of maps, we retrospectively undersampled the same 3D dataset used for phase-constrained imaging using variable-density Poisson-disc sampling40 with undersampling factors of R = 4.0, 7.0, 8.5 and without a calibration region, and then extracted the same slice in readout direction. As a comparison, these data were also reconstructed using SAKE with 50 iterations and a relative size of the signal subspace of 0.05.
Additionally, we applied SAKE and ENLIVE to a 3D fast spin-echo acquisition41 of a human knee (TR/TE = 1550/25 ms, FA = 90°, echo train length: 40, matrix size: 320 × 256, FOV = 160 × 153.6 mm2, field strength: 3.0T) from mridata.org42. This dataset was also undersampled using variable-density Poisson-disc sampling with undersampling factors of R = 2, 3, 5 and a single slice in readout direction was extracted. These data were then reconstructed using ENLIVE with 1 and 2 maps and with SAKE with 50 iterations and a relative size of the signal subspace of 0.125.
To evaluate ENLIVE in settings with high acceleration factors, we undersampled the 3D dataset used for phase-constrained imaging using Cartesian CAIPIRINHA43 patterns with undersampling factors of R = 4, 9, 16 with a 24 × 24 calibration region. These data were then reconstructed with ENLIVE using 2 maps with q = 1/3 and 8 iterations of the IRGNM.
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## Acknowledgements
Supported by the DZHK (German Centre for Cardiovascular Research). Part of this research was funded by the Physics-to-Medicine Initiative Göttingen (LM der Niedersächsischen Vorab) and DFG (UE 189/1-1). We acknowledge support by the Open Access Publication Funds of the Göttingen University.
## Author information
All authors contributed to the design of the study. H.C.M.H., S.R. and M.U. implemented the method. H.C.M.H. performed the numerical experiments. H.C.M.H., R.N.W. and M.U. contributed to the data analysis. F.O. and M.L. provided guidance on design and implementation. All authors contributed to the preparation of the manuscript.
Correspondence to H. Christian M. Holme.
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## Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. | 2019-08-21 02:39:35 | {"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": 0, "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": 0.6926681399345398, "perplexity": 2620.764851684246}, "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-2019-35/segments/1566027315750.62/warc/CC-MAIN-20190821022901-20190821044901-00290.warc.gz"} |
https://proofwiki.org/wiki/Intermediate_Value_Theorem/Corollary | Intermediate Value Theorem/Corollary
Theorem
Let $I$ be a real interval.
Let $f: I \to \R$ be a real function which is continuous on $\left({a \,.\,.\, b}\right)$.
Let $a, b \in I$ such that $\left({a \,.\,.\, b}\right)$ is an open interval.
Let $0 \in \R$ lie between $f \left({a}\right)$ and $f \left({b}\right)$.
That is, either:
$f \left({a}\right) < 0 < f \left({b}\right)$
or:
$f \left({b}\right) < 0 < f \left({a}\right)$
Then $f$ has a root in $\left({a \,.\,.\, b}\right)$.
Proof
Follows directly from the Intermediate Value Theorem and from the definition of root.
$\blacksquare$ | 2019-12-12 08:00:31 | {"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": 0.9313031435012817, "perplexity": 126.81279040954105}, "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-2019-51/segments/1575540542644.69/warc/CC-MAIN-20191212074623-20191212102623-00338.warc.gz"} |
http://frnsys.com/ai_notes/scratch/learning_causal_models.html | • complexity of joint distributions: for a multinomial distribution with $k$ states for each of its $n$ variables, the full distribution requires $k^n - 1$ parameters
• complexity can be reduced using a CPD graph structure such as a Bayes Net (BN)
• learning parameters of a BN: straightforward, like any CPD, you can use maximum likelihood estimates (MLE) or Bayesian estimates with a Dirichlet prior
## Learning the structure of a BN
Includes local and global components...
### Local: independence tests
#### Measures of deviance-from-independence between variables
For variables $x_i, x_j$ in dataset $\mathcal D$ of $M$ samples...
1. Pearson's Chi-squared ($\chi^2$) statistic:
$$d_{\chi^2}(\mathcal D) = \sum_{x_i, x_j} \frac{(M[x_i,x_j] - M \cdot \hat P(x_i) \cdot \hat P(x_j))^2}{M \cdot \hat P(x_i) \cdot \hat P(x_j)}$$
Independence increases as this value approaches 0
1. Mutual information (KL divergence) b/w joint and product of marginals:
$$d_I(\mathcal D) = \frac{1}{M} \sum_{x_i,x_j} M[x_i, x_j] \log \frac{M[x_i, x_j]}{M[x_i]M[x_j]}$$
Independence increases as this value approaches 0
### A decision rule for accepting/rejecting hypothesis of independence
Choose some p-value $t$, acept if $d(\mathcal D) <= t$, else reject.
### Global: scoring the structure
For a graph $\mathcal G$ with $n$ variables
1. Log-likelihood score:
$$\text{score}_L (\mathcal G: \mathcal D) = \sum_{\mathcal D} \sum_{i=1}^n \log \hat P (x_i | \text{parents}(x_i))$$
1. Bayesian score:
$$\text{score}_B (\mathcal G: \mathcal D) = \log p(\mathcal D | \mathcal G) + \log p(\mathcal G)$$
1. Bayes information criterion (with Dirichlet prior over graphs):
$$\text{score}_{BIC} (\mathcal G: \mathcal D) = l(\hat \theta) : \mathcal D) - \frac{\log M}{2} \text{Dim}(\mathcal G)$$
### Learning algorithms
• Constraint-based
• find best structure to explain determined dependencies
• sensitive to errors in testing individual dependencies
• Score-based
• search the space of networks to find high-scoring structure
• requires heuristics (e.g. greedy or branch-and-bound)
• Bayesian model averaging
• prediction over all structures
• may not have closed form | 2017-11-19 04:41:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 16, "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": 0.7491706609725952, "perplexity": 5980.494789161563}, "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-2017-47/segments/1510934805362.48/warc/CC-MAIN-20171119042717-20171119062717-00702.warc.gz"} |
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Which of the following can be inferred from the data above?
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Which of the following can be inferred from the data above? [#permalink]
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Which of the following statements can be inferred from the data above?
I. The Klein family's annual income more than doubled from 1985 to 1995.
II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than $40,000. A. I only B. II only C. I and III only D. II and III only E. I, II, and III Hi This is an GMAT Prep Question.Please let me know the correct answer and why it correct? Moreover, why We cannot infer II from the data above? Originally posted by vikrantgulia on 31 Dec 2013, 02:01. Last edited by Bunuel on 14 Jul 2014, 03:16, edited 1 time in total. Edited the question. Senior Manager Status: Final Lap Joined: 25 Oct 2012 Posts: 270 Concentration: General Management, Entrepreneurship GPA: 3.54 WE: Project Management (Retail Banking) Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 31 Dec 2013, 04:36 vikrantgulia wrote: Hi This is an GMAT Prep Question.Please let me know the correct answer and why it correct? Moreover, why We cannot infer II from the data above? Hi there, The answer IMO is C : Statements I and III must be true according to the figure above I- The klein family's annual income more than doubled from 1985 to 1995 : True, In 1985 it was$25K while in 1995 it was more than $60K II- The klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995 That is not true because the annual income increased by approximately$15K from 1985 to 1990 while it increased by approximately $20K from 1990 to 1995 Notice that these values are obtained by substracting the value of the annual income of the latter year from that of the former III- The klein family's average (arithmetic mean) annual income for the period shown was greater than$ 40K.
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .
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13 Jul 2014, 17:03
For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under $40,000. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8077 Location: Pune, India Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 14 Jul 2014, 02:32 7 This post received KUDOS Expert's post 1 This post was BOOKMARKED dolphinitida wrote: For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under$40,000.
What you are talking about is the median. Year 6 will give you the median income. The mean needn't lie at the middle value.
Say, for a set of numbers 4, 5, 5, 6, 7, 20, 22, mean will not be 6 (the middle value). Median will be 6 but mean will be much higher.
Another way to analyze statement III is this:
First 3 points are at most 15000 less than 40,000.
Next 4 points are almost 40,000 each.
The next point is at least 15,000 more than 40,000
The next 3 points are at least 20,000 more than 40,000.
So overall, the distance of the points higher than 40,000 is more than the distance of the points lower than 40,000. This is just your deviation concept applied visually.
For more on deviation, check: http://www.veritasprep.com/blog/2012/05 ... eviations/
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews VP Joined: 09 Jun 2010 Posts: 1196 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 02 May 2015, 07:45 this question require us to approximate. not easy. but if you practice often, it would be easy _________________ visit my facebook to help me. on facebook, my name is: thang thang thang Intern Joined: 13 Jul 2015 Posts: 37 Location: Singapore GMAT 1: 730 Q50 V39 WE: Operations (Investment Banking) Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 01 Oct 2015, 00:37 vikrantgulia wrote: Attachment: GMAT.png Which of the following statements can be inferred from the data above? I. The Klein family's annual income more than doubled from 1985 to 1995. II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than$40,000.
A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
Hi
This is an GMAT Prep Question.Please let me know the correct answer and why it correct?
Moreover, why We cannot infer II from the data above?
Bunuel, could you comment if this is an actual GMATPrep Quant question or an IR question?
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Re: Which of the following can be inferred from the data above? [#permalink]
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01 Oct 2015, 16:36
Hi SamuelWitwicky,
Most Test Takers see 1 question in the Quant section that involves a complex-looking chart, graph or table; the prompt is usually more complex than an 'average' question and takes the better part of 3 minutes to solve. Thus, this type of question COULD show up in your Quant section (but you won't see many of them).
As complex as the prompt looks, very little actual math is required to solve it. Some general estimation and pattern-matching will get you the correct answer.
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VeritasPrepKarishma wrote:
SamuelWitwicky wrote:
I feel statement II can be inferred to be wrong from the graph.
And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No.
There is a difference:
Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only.
On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995?
When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle)
This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it.
ok got it! thanks a lot Karishma
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Re: Which of the following can be inferred from the data above? [#permalink]
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26 Dec 2015, 00:16
Rock750 wrote:
vikrantgulia wrote:
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .
How did you take last three numbers as 60. they look more close to 65 to me. Deciding the xact values make me waste a lot of time on these questions. Any help will be much appreciated. Thanks
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Re: Which of the following can be inferred from the data above? [#permalink]
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NoHalfMeasures wrote:
Rock750 wrote:
vikrantgulia wrote:
Ok, lets see:
(25 + 25 + 25 + 35 + 35 + 40 + 40 + 55 + 60 + 60 +60) / 11 = 41,8 SO , that is true .
How did you take last three numbers as 60. they look more close to 65 to me. Deciding the xact values make me waste a lot of time on these questions. Any help will be much appreciated. Thanks
Use the deviation method I have discussed above.
If you do want to take values, take approximate values:
25, 26, 26, 35, 36, 38, 38, 56, 63, 63, 63
Mind you, the average you will get will not be very close to 40 so a few points here and there on your approximation will make no difference.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 29 Nov 2015 Posts: 13 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 29 Dec 2015, 11:31 VeritasPrepKarishma wrote: SamuelWitwicky wrote: I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. There is a difference: Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle) This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it. So just to be sure, in PS and DS, when the question asks 'can you infer...' it means can you get an exact value of this (true/false, yes/no, etc)? But in IR, when the question asks 'can you infer...' it means is there sufficient information provided in the paragraphs and tables for you to ARRIVE at an answer or DEDUCE an answer? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8077 Location: Pune, India Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 29 Dec 2015, 21:48 2 This post received KUDOS Expert's post sarathvr wrote: VeritasPrepKarishma wrote: SamuelWitwicky wrote: I feel statement II can be inferred to be wrong from the graph. And this is exactly how IR always asks the questions. Can the following be inferred? Yes or No. There is a difference: Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true only. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle) This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it. So just to be sure, in PS and DS, when the question asks 'can you infer...' it means can you get an exact value of this (true/false, yes/no, etc)? But in IR, when the question asks 'can you infer...' it means is there sufficient information provided in the paragraphs and tables for you to ARRIVE at an answer or DEDUCE an answer? No, the difference does not lie in whether the question is asked in PS/DS or IR. The difference lies in the way the question is asked. Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995. This is a statement. To infer it, it needs to be true. If this is not true, you say that you cannot infer this from the graph. On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995? When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199
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Re: Which of the following can be inferred from the data above? [#permalink]
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21 Nov 2016, 08:23
Responding to a pm:
Quote:
For this particular question (URL link above), I chose Option E as I can infer all the statements, two of which(I and III) are right and one (II) is false.
Kindly explain where am I going wrong?
You cannot infer statement II. Here is why:
Can you infer that: The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
If this is not true, you say that you cannot infer this from the graph. This is a statement and to be inferred, it needs to be true ONLY.
On the other hand: Did the Klein family's annual income increase by a greater amount from 1985 to 1990 than from 1990 to 1995?
When it is a question, you can answer with a Yes or No. Both answers are acceptable. The only answer not acceptable is may be. (your basic DS principle)
This question is asking about a statement and whether you can infer that. If the statement is correct, only then you can infer it.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 30 Aug 2016 Posts: 5 Location: Italy Schools: HEC Dec"18 (I) GMAT 1: 700 Q47 V41 GPA: 3.83 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 03 Jan 2017, 11:18 Just a quick note Once you realized that I. must be true and II. must not, then you can ignore statement III. and directly mark C: C is the only answer in which I. is true and II. is not! Intern Joined: 05 Oct 2016 Posts: 10 Location: India GMAT 1: 600 Q50 V30 GMAT 2: 600 Q50 V30 GMAT 3: 600 Q50 V30 GPA: 3.4 Re: Which of the following can be inferred from the data above? [#permalink] Show Tags 16 Feb 2017, 08:29 VeritasPrepKarishma wrote: dolphinitida wrote: For III - Since the data is for 11 consecutive years, why can't you consider year 6 (1990) as the mean? If you do, the annual income appears to be right under$40,000.
What you are talking about is the median. Year 6 will give you the median income. The mean needn't lie at the middle value.
Say, for a set of numbers 4, 5, 5, 6, 7, 20, 22, mean will not be 6 (the middle value). Median will be 6 but mean will be much higher.
Another way to analyze statement III is this:
First 3 points are at most 15000 less than 40,000.
Next 4 points are almost 40,000 each.
The next point is at least 15,000 more than 40,000
The next 3 points are at least 20,000 more than 40,000.
So overall, the distance of the points higher than 40,000 is more than the distance of the points lower than 40,000. This is just your deviation concept applied visually.
For more on deviation, check: http://www.veritasprep.com/blog/2012/05 ... eviations/
this saves a lot of time. i kept addiing all the values and calculating average, eating lot of time
thanks for the post
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Re: Which of the following can be inferred from the data above? [#permalink]
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07 Mar 2018, 06:02
I did a calculation mistake while evaluating statement 3 - marked answer A as option. Learning: I should not do calculation in mind, rather I should actually write calculations in a piece of paper.
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Re: Which of the following can be inferred from the data above? [#permalink]
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07 Mar 2018, 06:46
11 data points : Income in 1000
85 -> 25
86 -> 26
87 -> 26
88 -> 35
89 -> 40
90 -> 44
91 -> 44
92 -> 58
93 -> 64
94 -> 64
95 -> 64
I. The Klein family's annual income more than doubled from 1985 to 1995.
Income in 95 = 64 which is more than twice of 25 ( income from 85 ) - True : B, D are out
II. The Klein family's annual income increased by a greater amount from 1985 to 1990 than from 1990 to 1995.
90 - 85 = 44 - 25 = 19
95 - 90 = 64 - 44 = 20
so 19 > 20 - No So E is out
III. The Klein family's average (arithmetic mean) annual income for the period shown was greater than \$40,000.
to find average - Let's divide data points by 11 directly rather than summing and then dividing by 11
$$\frac{25}{11}$$ + $$\frac{26}{11}$$ + $$\frac{26}{11}$$ + $$\frac{35}{11}$$ + $$\frac{40}{11}$$ +
$$\frac{44}{11}$$ + $$\frac{44}{11}$$ + $$\frac{58}{11}$$ + $$\frac{64}{11}$$ + $$\frac{64}{11}$$ +
$$\frac{64}{11}$$
= 2 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5 + 5 + 5 [ just taking integer values ]
= 8 + 10 + 9 + 15
= 18 + 24
= 42 > 40 hence answer = C
A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
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Re: Which of the following can be inferred from the data above? [#permalink] 07 Mar 2018, 06:46
Display posts from previous: Sort by | 2018-05-24 19:35:13 | {"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": 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": 0.5581694841384888, "perplexity": 1443.454422680656}, "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-2018-22/segments/1526794866772.91/warc/CC-MAIN-20180524190036-20180524210036-00504.warc.gz"} |
https://mathoverflow.net/questions/336951/how-many-integers-below-n-can-be-expressed-as-a-sum-of-k-mth-powers | # How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?
For $$m,k \geq 2$$, let $$C_{m,k}(n)$$ denote the number of positive integers less than or equal to $$n$$ which can be expressed as a sum of $$k$$ $$m$$th powers.
I am interested in the asymptotic behavior of $$C_{m,k}(n)$$ for large $$n$$. What is currently known?
Here is what I could find.
• $$C_{2,2}(n) = \Theta(\frac{n}{\sqrt{\log{n} } } )$$. This is due to a result by Landau and Ramanujan. By Lagrange's four squares theorem , $$C_{2,k}(n) = \Theta(n)$$ for $$k \geq 4$$.
• Because even powers are in particular squares, by the previous bullet point, $$C_{2m, 2}(n) = O(\frac{n}{\sqrt{\log{n} } } )$$ for $$m \geq 1$$.
• This question is related to Waring's problem. Let $$G(m)$$ be the least positive integer $$k$$ such that every sufficiently large integer can be expressed as a sum of $$k$$ $$m$$th powers. Then $$C_{m,k}(n) = \Theta(n)$$ for every $$k \geq G(m)$$.
Intuitively, I am expecting $$C_{m,k}(n)$$ to be very close to $$\Theta(n)$$, even for large $$m$$ and small $$k$$. Still, it would be interesting to know if there are any results about the factors of $$\log$$ which appear. For example, this might come down to the difference between $$O(\frac{n}{\log{n}} )$$ and $$O(\frac{n}{\log \log{n} } )$$.
Related questions are
• For $m$ even you're certainly not going to get $\Theta(n)$ for small $k$. Since $m$-th powers are positive, there are at most $n^{1/m}$ possible summands, and so there are $O(n^{k/m})$ possible sums of $k$ $m$-th powers which are at most $n$. Thus your bound in the second bullet point can be improved a lot. – Christian Gaetz Jul 25 at 20:22 | 2019-10-15 02:58:11 | {"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": 25, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9818344712257385, "perplexity": 69.16658204164428}, "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/1570986655735.13/warc/CC-MAIN-20191015005905-20191015033405-00154.warc.gz"} |
https://www.wisdomandwonder.com/tag/mathematics | In recreational mathematics, a magic square is a square grid (equal number of rows and columns) filled with distinct numbers such that the numbers in each row, and in each column, as well as the numbers in the main and secondary diagonals, all add up to the same value, called the magic constant.
You gotta see this fun!
## The Man Who Knew Infinity
Via Wikipedia:
The Man Who Knew Infinity is a 2015 British biographical drama film based on the 1991 book of the same name by Robert Kanigel. The film stars Dev Patel as the real-life Srinivasa Ramanujan, a mathematician who after growing up poor in Madras, India, earns admittance to Cambridge University during World War I, where he becomes a pioneer in mathematical theories with the guidance of his professor, G. H. Hardy (played by Jeremy Irons).
## When is a number an integer?
This post poses a seemingly very simple question:
“When is a number an integer?”
and very quickly evolves to provide a delightful exposition of what it means when someone answers with “it depends”.
My vote is that the most intuitive and least-surprising approach is x%%1==0.
## The perennial fear revealed by a rules engine
When seeking to attain mastery of rules engines (RE), you will experience an odd phenomenon. Others, upon hearing some details of your course of study, will react in a what initially appears to be a manner angrily dismissive of the topic itself. This is strange given the fundamental role that computation plays in literally everything we do, from a manifest perspective, with hardware computers, organic computers, and other.
Upon further interaction, and reflection, it quickly becomes apparent what motivates this reaction is fundamentally, the re-experience of a long inaccessible, perennial fear.
Continue reading “The perennial fear revealed by a rules engine”
## Distinguishing models
It is important to distinguish between the mathematical numbers, the Scheme objects that attempt to model them, the machine representations used to implement the numbers, and notations used to write numbers.
— R6RS Chapter 3 | 2018-12-15 12:01:51 | {"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": 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": 0.4373644292354584, "perplexity": 1869.7346348593555}, "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/1544376826856.55/warc/CC-MAIN-20181215105142-20181215131142-00239.warc.gz"} |
http://mathhelpforum.com/pre-calculus/26653-exponential-log-function-help.html | # Math Help - Exponential/Log Function help
1. ## Exponential/Log Function help
I Can't seem to figure out how to show the work for this question
For a biology experiment, there are 50 cells present. After 2 hours there are 1600 bacteria. How many bacteria would there be after 6 hours?
The answer is 1 638 400 but I don't know how to get to that answer. Can someone help me out?
2. Originally Posted by Hockey_Guy14
I Can't seem to figure out how to show the work for this question
For a biology experiment, there are 50 cells present. After 2 hours there are 1600 bacteria. How many bacteria would there be after 6 hours?
The answer is 1 638 400 but I don't know how to get to that answer. Can someone help me out?
Hello,
we assume exponential growth.
The amount of bacteria with respect to time can be calculated by:
$a(t) = a(0) \cdot e^{k\cdot t}$.......... Thus you have to calculate the constants a(0) and k from the given conditions:
t = 0 and a(0) = 50
t = 2 and a(2) = 1600. Therefore:
$1600 = 50 \cdot e^{k \cdot 2}$
$32 = e^{k \cdot 2}$
$\ln(32) = k \cdot 2 ~\implies~ k = \frac12 \cdot \ln(32)$
Now you can calculate a(6):
$a(6)= 50 \cdot e^{\frac12 \cdot \ln(32) \cdot 6}=50 \cdot 32^3= 1,638,400$ | 2015-03-27 19:46:13 | {"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": 0, "img_math": 0, "codecogs_latex": 5, "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": 0.7680333852767944, "perplexity": 435.57152783117493}, "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-2015-14/segments/1427131296603.6/warc/CC-MAIN-20150323172136-00047-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://cybsecdiary.com/make-mobile-mockup-using-adobe-xd/ | Contents
# Make mobile app mock by Adobe XD
I used Adobe XD to make mockup application for internal hackathon-like project. Though it was my first time to use XD in my job, the process was very straight forward.
## 1 Install
Install from Adobe XD official site.
• It’s enough to select starter plan for free - that is for small size mock. You can share one link for your app.
## 2 Design
Draw just like power point. Here are some specific tips.
Art board
It’s a canvas for drawing. You can select many size including mobile devices like “iPhone 12 Pro Max”
Layers
Every object is assigned to each “layer”. After locating object, you can change the order of the object(back to front) by changing the order of layer in side menu. Layer is useful when selecting multiple specific objects at one time. Plugin
Plugin > Manage plugin and search/install plugin. Icon 4 Design worth using to embed svg icons to sketches.
## 3 Prototype
Make screen transition by connecting arrow between the origin of parts or entire screen to target screen.
## 4 Share
Make link and share that. Shared person can open the web link and can see, comment to the prototype. | 2022-07-07 01:30:09 | {"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": 0.20253701508045197, "perplexity": 6477.385867745593}, "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-20220707002618-20220707032618-00462.warc.gz"} |
https://freakyphysics.com/2018/02/22/misconceptions-101-there-is-no-gravity-in-space/ | Misconceptions 101: There is No Gravity in Space
You’ve seen it in movies. You’ve heard about it somewhere. You probably believe it yourself. You’ve even probably seen live video of astronauts in space floating around! It’s pretty much settled, there must be no gravity in outer space!
To start with, there is gravity in outer space. In fact, there is gravity all over the universe. Isaac Newton, and Albert Einstein later on, have clearly established this fact as truth. For this analysis, we are going to focus on Newton’s work.
Newton was the first to describe gravitation in detail, and came up with a beautifully short equation to describe how it works:
F_G=G\frac{Mm}{r^2}
where F_G is the gravitational force, M and m are the masses of the two objects being considered, and r is the distance between the centers of mass of the two objects.
This equation can mean many things, so let’s focus on the misconception. If there are no gravitational forces in space, then the left side of the equation must be zero.
F_G=0=G\frac{Mm}{r^2}
This implies that at least one of the terms in the right side of the equation must be zero. Let’s check out which one can become zero.
First of all, let’s say that mass of the Earth is M. This mass can never be zero, since mass cannot be destroyed (or created). This goes the same for the smaller mass m orbiting the Earth, namely the astronaut with mass .
The term G is a constant, with a value of 6.67 \times 10^{-11} N m^2 kg^{-2}. Since it is a constant, it cannot be zero as well.
This leaves us the term r. In most cases, this is the radius of the Earth, since we stay on its surface. An astronaut normally stays between 100 and 200 km (1.0 \text{ to } 2.0\times 10^{5} m) above the Earth’s surface. This increases r, which means that \frac {1}{r} becomes smaller. However, the mathematically inclined among us will realize that this value will also never reach zero, no matter how far you get.
Putting everything together, the gravitational forces due to the Earth do not completely disappear. Hence the Earth always has a gravitational effect everywhere in the universe, no matter how small! This applies to other masses as well.
So how strong is this gravitational pull from the Earth on the orbiting astronaut? The mass of the Earth is 5.97 \times 10^{24} kg, and its mean radius is 6.37 \times 10^{6} m. Using the gravitational force equation, and assuming that the mass of an astronaut with equipment is 150 \text{ kg}:
F_G=G\frac{Mm}{r^2}
F_G= 6.67 \times 10^{-11} N m^2 kg^{-2} \frac{(5.67\times10^{24})(150)}{(6.37\times 10^{6}+2.0\times 10^{5})^2}
F_G=1,383 N
The same astronaut with equipment will experience a gravitational force of F_G=1,471 N on the Earth’s surface. Upon comparison, the force in space may not be strong as on the surface, but it’s definitely comparable! So why does it seem like there is no gravity up there? To answer this, we’ll do a thought experiment similar to what Einstein himself used to come up with the general theory of relativity.
Imagine that you are floating around in a large truck container with no windows or openings to the outside, and you do not know where in the universe you are. Just by what you can see inside the container, can you determine whether you are floating in zero gravity or falling to the ground?
(Einstein used the opposite effect, that is, if you were in the container and are standing up in it, can you determine whether you are on the Earth’s surface or accelerating upward?)
Most people would say, “I wouldn’t be able to know”. Just like these people.
The people in the airplane experience what seems to be zero gravity, but they are actually falling due to gravity. It’s just that the airplane (the “container”) is falling with them. This removes any contact with the floor, and with it the removal of any contact forces that will counteract your weight. The contact forces are what make you feel your weight. Without any contact forces, you feel weightless.
A smaller effect happens when you are in an elevator. When it starts to go down, you feel lighter, since the floor does not support you as much as before moving. When you slow down and stop, you feel heavier because the contact forces on you are larger.
It’s a similar thing that happens with astronauts, but with one key difference: They are orbiting the earth instead of falling to the ground.
Orbiting may seem very different from falling, but it’s actually the same thing, with an additional motion perpendicular to falling (i.e. moving forward and falling downward at the same time). Imagine that the Earth suddenly disappears. The astronauts (and everything else) orbiting the Earth will stop moving in circular (or more accurately, elliptical) orbits and move in a straight line. The combination of falling towards the Earth due to gravity and the perpendicular motion makes the astronauts move in an approximately circular orbit around the Earth.
In either case, the overall effect is the same. The lack of contact with the floor makes you (any anything, really) feel like you are weightless.
Being in space means that there is gravity acting everywhere, just like on Earth. The sensation of weightlessness you would experience is actually not because of the lack of gravity, but because the sensation caused by moving with your container makes you think that are floating around. The fact is that you are falling, but can’t sense it because of the lack of contact forces with the floor. | 2020-09-29 16:36:25 | {"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": 0.8366075754165649, "perplexity": 388.0118749251422}, "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/1600400202418.22/warc/CC-MAIN-20200929154729-20200929184729-00645.warc.gz"} |
https://hal-cea.archives-ouvertes.fr/cea-02170472 | # Period spacings of gravity modes in rapidly rotating magnetic stars: I. Axisymmetric fossil field with poloidal and toroidal components
* Corresponding author
Abstract : $Context$. Stellar magnetic fields are often invoked to explain the missing transport of angular momentum observed in models of stellar interiors. However, the properties of an internal magnetic field and the consequences of its presence on stellar evolution are largely unknown.$Aims$. We study the effect of an axisymmetric internal magnetic field on the frequency of gravity modes in rapidly rotating stars to check whether gravity modes can be used to detect and probe such a field.$Methods$. Rotation is taken into account using the traditional approximation of rotation and the effect of the magnetic field is computed using a perturbative approach. As a proof of concept, we compute frequency shifts due to a mixed (i.e. with both poloidal and toroidal components) fossil magnetic field for a representative model of a known magnetic, rapidly rotating, slowly pulsating B-type star: HD 43317.$Results$. We find that frequency shifts induced by the magnetic field scale with the square of its amplitude. A magnetic field with a near-core strength of the order of 150 kG (which is consistent with the observed surface field strength of the order of 1 kG) leads to signatures that are detectable in period spacings for high-radial-order gravity modes.$Conclusions$. The predicted frequency shifts can be used to constrain internal magnetic fields and offer the potential for a significant step forward in our interpretation of the observed structure of gravity-mode period spacing patterns in rapidly rotating stars.
Keywords :
Document type :
Journal articles
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Cited literature [108 references]
https://hal-cea.archives-ouvertes.fr/cea-02170472
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Submitted on : Monday, July 1, 2019 - 9:25:06 PM
Last modification on : Tuesday, January 19, 2021 - 1:53:10 PM
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### Citation
V. Prat, S. Mathis, B. Buysschaert, J. van Beeck, D. M. Bowman, et al.. Period spacings of gravity modes in rapidly rotating magnetic stars: I. Axisymmetric fossil field with poloidal and toroidal components. Astronomy and Astrophysics - A&A, EDP Sciences, 2019, 627, pp.A64. ⟨10.1051/0004-6361/201935462⟩. ⟨cea-02170472⟩
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https://zbmath.org/serials/?q=se%3A00002510 | # zbMATH — the first resource for mathematics
## Journal of Statistics & Management Systems
Short Title: J. Stat. Manag. Syst. Publisher: Taru Publications, New Delhi, Delhi, India ISSN: 0972-0510 Online: http://www.tandfonline.com/toc/tsms20/currenthttp://www.tarupublications.com/journals/jsms/jsms.htmhttp://www.connectjournals.com/jsms Comments: No longer indexed
Documents Indexed: 430 Publications (1998–2011)
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#### Latest Issues
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#### Authors
10 Chen, Miaosheng 9 Artikis, Panagiotis T. 9 Chung, Kunjen 8 Huang, Yung-Fu 7 Artikis, Constantinos T. 6 Chou, Chaoyu 6 Huang, Chao-Kuei 6 Lin, Shyder 6 Masoom Ali, M. 6 Patel, R. B. 6 Wu, Hsin-Hung 5 Chang, Chen-Sung 5 Fountas, Chrysostomos E. 5 Ganesh, Siva 5 Hou, Kuo-Lung 5 Lan, Chunhsiung 5 Pao, Shih-Heng 5 Woo, Jungsoo 5 Wu, Cheng-Ru 5 Zalmai, G. J. 4 Bector, C. R. 4 Chang, Chang-Hsing 4 Chang, Shu-Hui 4 Chen, Reu-Ching 4 Cheng, Tzu-Liang 4 Chilas, John G. 4 Chiu, Yung-ho 4 Lin, Gu-Hong 4 Lin, Jyh-Horng 4 Ouyang, Liang-Yuh 4 Pan, Wen-Tsao 4 Wu, Kunshan 3 Canel, Cem 3 Chen, Chungho 3 Chen, Kuo-Chao 3 Chiou, Jer-Shiou 3 Chuang, Bor-Ren 3 Ganesalingam, Selvanayagam 3 Huang, Shian-Chang 3 Jain, Madhu 3 Koukouvinos, Christos 3 Lan, Tian-Syung 3 Lee, Mingchih 3 Lin, Yu-Chang 3 Liu, Fuhhwa Franklin 3 Pan, Jason Chao-Hsien 3 Shieh, Jiunn-I. 3 Tanaka, Teruo 3 Ting, Chih-Wen 3 Tsamadias, Constantinos 3 Yang, Ming-Feng 2 Agorastos, Kostas A. 2 Botsaris, C.-A. E. 2 Çabukoglu, Şerafettin 2 Cambini, Alberto 2 Chandra, Suresh 2 Chang, Hae-Ching 2 Chang, Hung-Teng 2 Chang, YuTeng 2 Chen, Cheng-Te 2 Chen, Chun-Da 2 Chen, Lihui 2 Chen, Roger C. Y. 2 Chiu, Yuanshyi Peter 2 Chou, Jian-Hsin 2 Chung, Kun-Jen 2 Chung, Yen Y. 2 Dalamagas, Basil 2 Dinh The Luc 2 Ferretti, Paola 2 Fey, Yeh-Jui 2 Ganeslingam, S. 2 Goulionis, John E. 2 Gupta, Rakesh 2 Gupta, Sat Narian 2 Hai, Hui Lin 2 Hsieh, Chin-Shan 2 Hu, Michael H. 2 Huang, Jianhui 2 Huang, Kuochung 2 Huang, Tien-Shou 2 Huang, Wentao 2 Katsaragakis, S. 2 Ke, Mei-Chu 2 Khan, Israr H. 2 Khumawala, Basheer M. 2 Kodama, Masanori 2 Ku, Meei-Yuh 2 Lai, Shun-Te 2 Lai, Wei-Tseng 2 Lee, Chien-Chung 2 Lee, Chuan 2 Liang, Shing-Ko 2 Liang, Tien-Fu 2 Lin, Che 2 Lin, Chieh-Yu 2 Lin, Li-Chiao 2 Lin, Tyrone T. 2 Lin, Yih-Chun 2 Lo, Chihyao ...and 491 more Authors
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#### Fields
267 Operations research, mathematical programming (90-XX) 102 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 99 Statistics (62-XX) 17 Numerical analysis (65-XX) 14 Probability theory and stochastic processes (60-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 6 Systems theory; control (93-XX) 5 Computer science (68-XX) 2 Combinatorics (05-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Real functions (26-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Measure and integration (28-XX) 1 Ordinary differential equations (34-XX) 1 Operator theory (47-XX) 1 Quantum theory (81-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX) 1 Mathematics education (97-XX)
#### Citations contained in zbMATH Open
62 Publications have been cited 171 times in 153 Documents Cited by Year
Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization. Zbl 1079.90609
Weber, Gerhard-Wilhelm
2002
An $$\text{M}^X/\text{G}/1$$ retrial queue with exhaustive vacations. Zbl 0974.60083
Aissani, A.
2000
Higher order optimality conditions in nonsmooth vector optimization. Zbl 1079.90596
Ginchev, Ivan
2002
The deterministic inventory models with shortage and defective items derived without derivatives. Zbl 1142.90306
Huang, Yung-Fu
2003
Optimality principles and duality models for a class of continuous-time generalized fractional programming problems with operator constraints. Zbl 0912.90269
Zalmai, G. J.
1998
First and second order optimality conditions in vector optimization. Zbl 1079.90595
Cambini, A.; Martein, L.
2002
Proper efficiency principles and duality models for a class of continuous-time multiobjective fractional programming problems with operator constraints. Zbl 0912.90270
Zalmai, G. J.
1998
Matrix variate generalization of a bivariate beta type 1 distribution. Zbl 1183.60005
Gupta, Arjun K.; Nagar, Daya K.
2009
Fractional programming: a recent survey. Zbl 1079.90606
Schaible, Siegfried
2002
Properties and applications in risk management operation of a stochastic discounting model. Zbl 1124.91344
Artikis, Panagiotis T.; Artikis, Constantinos T.
2005
A minimax distribution free procedure for mixed inventory model with backorder discounts and variable lead time. Zbl 1176.90013
Chuang, Bor-Ren; Ouyang, Liang-Yuh; Lin, Yu-Jen
2004
A makespan study of the two-machine flowshop scheduling problem with a learning effect. Zbl 1068.90066
Wu, Chin-Chia
2005
Risk management operations described by a stochastic discounting model incorporating a random sum of cash flows and a random maximum of recovery times. Zbl 1137.91423
Artikis, Constantinos T.; Artikis, Panagiotis T.; Fountas, Chrysostomos E.
2007
Mixture inventory model involving variable lead time and defective units. Zbl 0938.90004
Ouyang, Liang-Yuh; Wu, Kun-Shan
1999
A survey on sequences and distribution functions satisfying the first-digit-law. Zbl 1156.60302
Posch, Peter N.
2008
Optimality conditions and duality for programming problems involving set and $$n$$-set functions: a survey. Zbl 1079.90615
Stancu-Minasian, I. M.; Preda, Vasile
2002
Structural properties for a two-state partially observable Markov decision process with an average cost criterion. Zbl 1209.90345
Goulionis, John E.
2007
Normal cone method in solving linear multiobjective problems. Zbl 1079.90598
Kim, Nguyen Thi Bach; Luc, Dinh The
2002
Bass model revisited. Zbl 1219.90059
Jha, P. C.; Gupta, Anshu; Kapur, P. K.
2008
Optimal tolerance regions for some functions of multiple regression model with Student-$$t$$ errors. Zbl 1191.62121
Khan, Shahjahan
2006
Statistical inference for Markov chain European option: estimating the price, the bare risk and the theta by historical distributions of Markov chain. Zbl 1193.91153
D’Amico, Guglielmo
2006
Proposed tests for the nondecreasing alternative in a mixed design. Zbl 1179.62066
Magel, Rhonda; Terpstra, Jeff; Wen, Jun
2009
Impact of processing time and related costs on optimal lot sizing in manufacturing. Zbl 1258.90011
Ramaswamy, Kizhanatham V.; Smith, Marion
2004
Inventory model involving lead time and setup cost as decision variables. Zbl 1181.90008
Cheng, Tzu-Liang; Huang, Chao-Kuei; Chen, Kuo-Chao
2004
An alternative to Ryu et al. randomized response model. Zbl 1258.62010
Hussain, Zawar; Shabbir, Javid; Gupta, Sat
2007
Abstract convexity of positively homogeneous functions. Zbl 1079.90590
Rubinov, A.; Dzalilov, Z.
2002
The use of subdifferentials for studying generalized convex functions. Zbl 1079.90587
2002
Generalized $$b$$-invex vector valued functions. Zbl 1079.90586
Bector, C. R.; Cambini, Riccardo
2002
Generalized vector equilibrium problems. Zbl 1079.90594
Ansari, Qamrul Hasan; Yao, Jen-Chih
2002
Characterizations of pseudomonotone maps and economic equilibrium. Zbl 1079.91547
Brighi, Luigi; John, Reinhard
2002
Three types of models for stochastic scheduling with fuzzy information. Zbl 1140.90416
Peng, Jin; Iwamura, Kakuzo
2003
Estimation of finite population mean using multi-auxiliary variables. Zbl 0952.62011
Bahl, S.; Kumar, Manoj
2000
Quasi and strictly quasi $$E$$-convex functions. Zbl 0995.90074
Youness, Ebrahim A.
2001
Performance comparison between genetic algorithm and particle swarm optimization based on back propagation. Zbl 1217.90163
Chang, Jui-Fang; Lee, Ying-Jye
2010
Fair distribution of a common revenue. Zbl 1219.90098
2008
Imprecise data envelopment analysis (IDEA): a review and a new approach. Zbl 1171.90426
Derpanis, D.; Fountas, C.; Chondrocoykis, G.
2008
A novel hybrid quantum-inspired evolutionary algorithm for permutation flow-shop scheduling. Zbl 1194.90042
Zheng, Tianmin; Yamashiro, Mitsuo
2009
A stochastic model for the cost of maintaining a risk frequency reduction operation. Zbl 1099.90016
Jerwood, David; Artikis, Panagiotis T.; Fountas, Chrysostomos E.
2005
Integrated inspection strategy model with inspection errors and positive inspection time lengths. Zbl 1258.90033
Wang, Wen-Ying
2004
Optimal inventory policy under supplier credit policies depending on the ordering quantity. Zbl 1152.90306
Chung, Kun-Jen; Huang, Yung-Fu
2004
Exact optimal cycle time for the EPQ model with defective materials under inflation and time discounting. Zbl 1181.90014
Huang, Yung-Fu; Huang, Hung-Fu
2004
The ant colony system: optimization for the logistics of marine cargo in the Aegean. Zbl 1070.90138
Alexandris, Nikolaos; Fountas, Chrysostomos; Vlachos, Aristidis
2005
A Bayesian approach on monitoring process capability. Zbl 1137.62416
Lin, Gu-Hong; Jehng, Wern-Kueir; Hsieh, Kuang-Han; Wang, Lai-Wang; Lai, Shun-Te
2007
Formulating a novel stock selection model using DEA and grey situation decision model. Zbl 1137.91353
Hwang, Shiuh-Nan; Lin, Chin-Tsai; Chuang, Wang-Ching
2007
The impact of collaborative forecasting in the supply chain management of high technology products. Zbl 1146.90440
Chondrocoukis, Gregory; Nassopolous, Vassilis; Marcoulaki, Eftychia
2005
A note on single-machine group scheduling problem with a general learning function. Zbl 1155.90389
Liu, Han-Chu; Lee, Wen-Chiung; Wu, Chin-Chia
2008
Application of the theory of large deviations on error exponents in many hypotheses LAO testing. Zbl 1155.62011
2008
Duality for multiobjective fractional variational control problems with $$(F,\rho)$$-convexity. Zbl 0976.90093
Patel, R. B.
2000
A comparative study of the monitoring performance for weighted control charts. Zbl 1173.90454
Hsu, Bi-Min; Lai, Peng-Jen; Shu, Ming-Hung; Hung, Yen-Yeh
2009
Concepts of generalized concavity based on triangular norms. Zbl 1079.90589
Ramík, Jaroslav; Vlach, Milan
2002
Exponentially varying demand for integrated inventory model with varying production of deteriorating item. Zbl 1139.90324
Wu, M. Y.; Rau, H.; Wee, H. M.
2003
A real options model for establishing an electronic securities trading system under uncertain rate of Internet trading. Zbl 1140.90431
Lin, Chin-Tsai; Lin, Tyrone T.; Yeh, Lung-Chu
2003
A model for the marketing of a seasonal product with different goodwills for consumer and retailer. Zbl 1139.90351
Bykadorov, Igor; Ellero, Andrea; Moretti, Elena
2003
An inventory model with stock-dependent demand under conditions of permissible delay in payments. Zbl 0938.91037
Chang, Horng-Jinh; Dye, Chung-Yuan
1999
Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems. Zbl 0923.90135
Zalmai, G. J.
1998
On inventory models with partial backlogging. Zbl 0984.90003
Chu, Peter
2001
Control policy for $$M/E_k/1$$ queueing system. Zbl 0984.90007
2001
An investigation into the effect of a more accurate measure of distance on the detailed facility layout problem. Zbl 1024.90011
O’Muirgheasa, Conor; Kadipasaoglu, Sukran N.; Khumawala, Basheer M.
2001
A Bayesian estimator of process capability index. Zbl 1121.62114
2006
Mixture inventory model in fuzzy demand with controllable lead time. Zbl 1122.90008
Pan, Jason Chao-Hsien; Yang, Ming-Feng
2006
Discrete renewal and selfdecomposable distributions in modelling information risk management operations. Zbl 1105.90031
Artikis, Panagiotis; Artikis, Constantinos T.; Fountas, Chrysostomos E.; Hatzopoulos, Peter
2006
Analysis of required and matching loan qualities in financial institutions. Zbl 1161.91399
Lin, Tyrone T.; Lo, I-Hsuan
2006
Performance comparison between genetic algorithm and particle swarm optimization based on back propagation. Zbl 1217.90163
Chang, Jui-Fang; Lee, Ying-Jye
2010
Matrix variate generalization of a bivariate beta type 1 distribution. Zbl 1183.60005
Gupta, Arjun K.; Nagar, Daya K.
2009
Proposed tests for the nondecreasing alternative in a mixed design. Zbl 1179.62066
Magel, Rhonda; Terpstra, Jeff; Wen, Jun
2009
A novel hybrid quantum-inspired evolutionary algorithm for permutation flow-shop scheduling. Zbl 1194.90042
Zheng, Tianmin; Yamashiro, Mitsuo
2009
A comparative study of the monitoring performance for weighted control charts. Zbl 1173.90454
Hsu, Bi-Min; Lai, Peng-Jen; Shu, Ming-Hung; Hung, Yen-Yeh
2009
A survey on sequences and distribution functions satisfying the first-digit-law. Zbl 1156.60302
Posch, Peter N.
2008
Bass model revisited. Zbl 1219.90059
Jha, P. C.; Gupta, Anshu; Kapur, P. K.
2008
Fair distribution of a common revenue. Zbl 1219.90098
2008
Imprecise data envelopment analysis (IDEA): a review and a new approach. Zbl 1171.90426
Derpanis, D.; Fountas, C.; Chondrocoykis, G.
2008
A note on single-machine group scheduling problem with a general learning function. Zbl 1155.90389
Liu, Han-Chu; Lee, Wen-Chiung; Wu, Chin-Chia
2008
Application of the theory of large deviations on error exponents in many hypotheses LAO testing. Zbl 1155.62011
2008
Risk management operations described by a stochastic discounting model incorporating a random sum of cash flows and a random maximum of recovery times. Zbl 1137.91423
Artikis, Constantinos T.; Artikis, Panagiotis T.; Fountas, Chrysostomos E.
2007
Structural properties for a two-state partially observable Markov decision process with an average cost criterion. Zbl 1209.90345
Goulionis, John E.
2007
An alternative to Ryu et al. randomized response model. Zbl 1258.62010
Hussain, Zawar; Shabbir, Javid; Gupta, Sat
2007
A Bayesian approach on monitoring process capability. Zbl 1137.62416
Lin, Gu-Hong; Jehng, Wern-Kueir; Hsieh, Kuang-Han; Wang, Lai-Wang; Lai, Shun-Te
2007
Formulating a novel stock selection model using DEA and grey situation decision model. Zbl 1137.91353
Hwang, Shiuh-Nan; Lin, Chin-Tsai; Chuang, Wang-Ching
2007
Optimal tolerance regions for some functions of multiple regression model with Student-$$t$$ errors. Zbl 1191.62121
Khan, Shahjahan
2006
Statistical inference for Markov chain European option: estimating the price, the bare risk and the theta by historical distributions of Markov chain. Zbl 1193.91153
D’Amico, Guglielmo
2006
A Bayesian estimator of process capability index. Zbl 1121.62114
2006
Mixture inventory model in fuzzy demand with controllable lead time. Zbl 1122.90008
Pan, Jason Chao-Hsien; Yang, Ming-Feng
2006
Discrete renewal and selfdecomposable distributions in modelling information risk management operations. Zbl 1105.90031
Artikis, Panagiotis; Artikis, Constantinos T.; Fountas, Chrysostomos E.; Hatzopoulos, Peter
2006
Analysis of required and matching loan qualities in financial institutions. Zbl 1161.91399
Lin, Tyrone T.; Lo, I-Hsuan
2006
Properties and applications in risk management operation of a stochastic discounting model. Zbl 1124.91344
Artikis, Panagiotis T.; Artikis, Constantinos T.
2005
A makespan study of the two-machine flowshop scheduling problem with a learning effect. Zbl 1068.90066
Wu, Chin-Chia
2005
A stochastic model for the cost of maintaining a risk frequency reduction operation. Zbl 1099.90016
Jerwood, David; Artikis, Panagiotis T.; Fountas, Chrysostomos E.
2005
The ant colony system: optimization for the logistics of marine cargo in the Aegean. Zbl 1070.90138
Alexandris, Nikolaos; Fountas, Chrysostomos; Vlachos, Aristidis
2005
The impact of collaborative forecasting in the supply chain management of high technology products. Zbl 1146.90440
Chondrocoukis, Gregory; Nassopolous, Vassilis; Marcoulaki, Eftychia
2005
A minimax distribution free procedure for mixed inventory model with backorder discounts and variable lead time. Zbl 1176.90013
Chuang, Bor-Ren; Ouyang, Liang-Yuh; Lin, Yu-Jen
2004
Impact of processing time and related costs on optimal lot sizing in manufacturing. Zbl 1258.90011
Ramaswamy, Kizhanatham V.; Smith, Marion
2004
Inventory model involving lead time and setup cost as decision variables. Zbl 1181.90008
Cheng, Tzu-Liang; Huang, Chao-Kuei; Chen, Kuo-Chao
2004
Integrated inspection strategy model with inspection errors and positive inspection time lengths. Zbl 1258.90033
Wang, Wen-Ying
2004
Optimal inventory policy under supplier credit policies depending on the ordering quantity. Zbl 1152.90306
Chung, Kun-Jen; Huang, Yung-Fu
2004
Exact optimal cycle time for the EPQ model with defective materials under inflation and time discounting. Zbl 1181.90014
Huang, Yung-Fu; Huang, Hung-Fu
2004
The deterministic inventory models with shortage and defective items derived without derivatives. Zbl 1142.90306
Huang, Yung-Fu
2003
Three types of models for stochastic scheduling with fuzzy information. Zbl 1140.90416
Peng, Jin; Iwamura, Kakuzo
2003
Exponentially varying demand for integrated inventory model with varying production of deteriorating item. Zbl 1139.90324
Wu, M. Y.; Rau, H.; Wee, H. M.
2003
A real options model for establishing an electronic securities trading system under uncertain rate of Internet trading. Zbl 1140.90431
Lin, Chin-Tsai; Lin, Tyrone T.; Yeh, Lung-Chu
2003
A model for the marketing of a seasonal product with different goodwills for consumer and retailer. Zbl 1139.90351
Bykadorov, Igor; Ellero, Andrea; Moretti, Elena
2003
Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization. Zbl 1079.90609
Weber, Gerhard-Wilhelm
2002
Higher order optimality conditions in nonsmooth vector optimization. Zbl 1079.90596
Ginchev, Ivan
2002
First and second order optimality conditions in vector optimization. Zbl 1079.90595
Cambini, A.; Martein, L.
2002
Fractional programming: a recent survey. Zbl 1079.90606
Schaible, Siegfried
2002
Optimality conditions and duality for programming problems involving set and $$n$$-set functions: a survey. Zbl 1079.90615
Stancu-Minasian, I. M.; Preda, Vasile
2002
Normal cone method in solving linear multiobjective problems. Zbl 1079.90598
Kim, Nguyen Thi Bach; Luc, Dinh The
2002
Abstract convexity of positively homogeneous functions. Zbl 1079.90590
Rubinov, A.; Dzalilov, Z.
2002
The use of subdifferentials for studying generalized convex functions. Zbl 1079.90587
2002
Generalized $$b$$-invex vector valued functions. Zbl 1079.90586
Bector, C. R.; Cambini, Riccardo
2002
Generalized vector equilibrium problems. Zbl 1079.90594
Ansari, Qamrul Hasan; Yao, Jen-Chih
2002
Characterizations of pseudomonotone maps and economic equilibrium. Zbl 1079.91547
Brighi, Luigi; John, Reinhard
2002
Concepts of generalized concavity based on triangular norms. Zbl 1079.90589
Ramík, Jaroslav; Vlach, Milan
2002
Quasi and strictly quasi $$E$$-convex functions. Zbl 0995.90074
Youness, Ebrahim A.
2001
On inventory models with partial backlogging. Zbl 0984.90003
Chu, Peter
2001
Control policy for $$M/E_k/1$$ queueing system. Zbl 0984.90007
2001
An investigation into the effect of a more accurate measure of distance on the detailed facility layout problem. Zbl 1024.90011
O’Muirgheasa, Conor; Kadipasaoglu, Sukran N.; Khumawala, Basheer M.
2001
An $$\text{M}^X/\text{G}/1$$ retrial queue with exhaustive vacations. Zbl 0974.60083
Aissani, A.
2000
Estimation of finite population mean using multi-auxiliary variables. Zbl 0952.62011
Bahl, S.; Kumar, Manoj
2000
Duality for multiobjective fractional variational control problems with $$(F,\rho)$$-convexity. Zbl 0976.90093
Patel, R. B.
2000
Mixture inventory model involving variable lead time and defective units. Zbl 0938.90004
Ouyang, Liang-Yuh; Wu, Kun-Shan
1999
An inventory model with stock-dependent demand under conditions of permissible delay in payments. Zbl 0938.91037
Chang, Horng-Jinh; Dye, Chung-Yuan
1999
Optimality principles and duality models for a class of continuous-time generalized fractional programming problems with operator constraints. Zbl 0912.90269
Zalmai, G. J.
1998
Proper efficiency principles and duality models for a class of continuous-time multiobjective fractional programming problems with operator constraints. Zbl 0912.90270
Zalmai, G. J.
1998
Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems. Zbl 0923.90135
Zalmai, G. J.
1998
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https://everything.explained.today/Homology_(mathematics)/ | # Homology (mathematics) explained
In mathematics, homology[1] is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there".
There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.
## Background
### Origins
Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.
Homology itself was developed as a way to analyse and classify manifolds according to their cycles – closed loops (or more generally submanifolds) that can be drawn on a given n dimensional manifold but not continuously deformed into each other. These cycles are also sometimes thought of as cuts which can be glued back together, or as zippers which can be fastened and unfastened. Cycles are classified by dimension. For example, a line drawn on a surface represents a 1-cycle, a closed loop or
S1
(1-manifold), while a surface cut through a three-dimensional manifold is a 2-cycle.
### Surfaces
S2
, the cycle b in the diagram can be shrunk to the pole, and even the equatorial great circle a can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as c can be similarly shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and belong to the same homology class. They are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components. For example, cutting the sphere along a produces two hemispheres.
T2
has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles a, b or c can be deformed into one another. In particular, cycles a and b cannot be shrunk to a point whereas cycle c can, thus making it homologous to zero.
If the torus surface is cut along both a and b, it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along a, and the other opposite pair represents the cut along b.
The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. Up to symmetry, there are four distinct ways of gluing the sides, each creating a different surface:
K2
is the Klein bottle, which is a torus with a twist in it (The twist can be seen in the square diagram as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles a and b cannot be shrunk while c can be. But unlike the torus, following b forwards right round and back reverses left and right, because b happens to cross over the twist given to one join. If an equidistant cut on one side of b is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted Möbius strip. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable.
P2
has both joins twisted. The uncut form, generally represented as the Boy surface, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as A and A′ are identified as the same point. Again, a and b are non-shrinkable while c is. But this time, both a and b reverse left and right.
Cycles can be joined or added together, as a and b on the torus were when it was cut open and flattened down. In the Klein bottle diagram, a goes round one way and -a goes round the opposite way. If a is thought of as a cut, then −a can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so a + (−a) = 0.
But now consider two a-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the b-cycle), and it will come back as −a. This is because the Klein bottle is made from a cylinder, whose a-cycle ends are glued together with opposite orientations. Hence 2a = a + a = a + (−a) = 0. This phenomenon is called torsion. Similarly, in the projective plane, following the unshrinkable cycle b round twice remarkably creates a trivial cycle which can be shrunk to a point; that is, b + b = 0. Because b must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a b-cycle around twice in the Klein bottle gives simply b + b = 2b, since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted.
A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2n-gons) can be glued to make different manifolds. Conversely, a closed surface with n non-zero classes can be cut into a 2n-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.
The first recognisable theory of homology was published by Henri Poincaré in his seminal paper "Analysis situs", J. Ecole polytech. (2) 1. 1–121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the Betti numbers of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients.
The complete classification of 1- and 2-manifolds is given in the table.
Topological characteristics of closed, unbounded 1- and 2-manifolds
ManifoldEuler no.,
χ
OrientabilityBetti numbersTorsion coefficient
(1-dimensional)
Symbol[2] Nameb0b1b2
S1
Circle (1-manifold) 0 Orientable 1 1
S2
2 Orientable 1 0 1 None
T2
0 Orientable 1 2 1 None
P2
1 Non-orientable 1 0 0 2
K2
0 Non-orientable 1 1 0 2
2-holed torus −2 Orientable 1 4 1 None
g-holed torus (g is the genus) 2 − 2g Orientable 1 2g 1 None
Sphere with c cross-caps 2 − c Non-orientable 1 c − 1 0 2
2-Manifold with gholes and ccross-caps (c>0)2−(2g+c) Non-orientable 1 (2g+c)−1 0 2
Notes
1. For a non-orientable surface, a hole is equivalent to two cross-caps.
1. Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere
S2
, g = c = 0.
### Generalization
A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example the disk or 2-ball
B2
is bounded by a circle
S1
. It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane.
When two cycles can be continuously deformed into each other, then cutting along one produces the same shape as cutting along the other, up to some bending and stretching. In this case the two cycles are said to be or to lie in the same . Additionally, if one cycle can be continuously deformed into a combination of other cycles, then cutting along the initial cycle is the same as cutting along the combination of other cycles. For example, cutting along a figure 8 is equivalent to cutting along its two lobes. In this case, the figure 8 is said to be homologous to the sum of its lobes.
Two open manifolds with similar boundaries (up to some bending and stretching) may be glued together to form a new manifold which is their connected sum.
This geometric analysis of manifolds is not rigorous. In a search for increased rigour, Poincaré went on to develop the simplicial homology of a triangulated manifold and to create what is now called a chain complex. These chain complexes (since greatly generalized) form the basis for most modern treatments of homology.
In such treatments a cycle need not be continuous: a 0-cycle is a set of points, and cutting along this cycle corresponds to puncturing the manifold. A 1-cycle corresponds to a set of closed loops (an image of the 1-manifold
S1
). On a surface, cutting along a 1-cycle yields either disconnected pieces or a simpler shape. A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on.
Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28.[3] [4] The new combinatorial topology formally treated topological classes as abelian groups. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part.
The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".[5] Algebraic homology remains the primary method of classifying manifolds.
## Informal examples
The homology of a topological space X is a set of topological invariants of X represented by its homology groups$H_0(X), H_1(X), H_2(X), \ldots$where the
k\rm
homology group
Hk(X)
describes, informally, the number of holes in X with a k-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two components. Consequently,
H0(X)
describes the path-connected components of X.
S1
is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as$H_k\left(S^1\right) = \begin \Z & k = 0, 1 \\ \ & \text\end$where
\Z
is the group of integers and
\{0\}
is the trivial group. The group
1\right) H 1\left(S
=\Z
represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.
S2
has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are[6] $H_k\left(S^2\right) = \begin \Z & k = 0, 2 \\ \ & \text\end$
In general for an n-dimensional sphere
Sn,
the homology groups are$H_k\left(S^n\right) = \begin \Z & k = 0, n \\ \ & \text\end$
B2
is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for
2\right) H 0\left(B
=\Z
. In general, for an n-dimensional ball
Bn,
$H_k\left(B^n\right) = \begin \Z & k = 0 \\ \ & \text\end$
The torus is defined as a product of two circles
T=S1 x S1
. The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are$H_k(T) = \begin \Z & k = 0, 2 \\ \Z \times \Z & k = 1 \\ \ & \text\end$
\Z x \Z.
For the projective plane P, a simple computation shows (where
\Z2
is the cyclic group of order 2):[7] $H_k(P) = \begin \Z & k = 0 \\ \Z_2 & k = 1 \\ \ & \text\end$
H0(P)=\Z
corresponds, as in the previous examples, to the fact that there is a single connected component.
H1(P)=\Z2
is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called torsion.
## Construction of homology groups
The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first: graph homology and simplicial homology.
The general construction begins with an object such as a topological space X, on which one first defines a C(X) encoding information about X. A chain complex is a sequence of abelian groups or modules
C0,C1,C2,\ldots
. connected by homomorphisms
\partialn:Cn\toCn-1,
which are called boundary operators. That is,
...b \overset{\partialn+1
} C_n\overset C_\overset \dotsb\overset C_1\overset C_0\overset 0
where 0 denotes the trivial group and
Ci\equiv0
for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,
\partialn\circ\partialn+1=0n+1,,
i.e., the constant map sending every element of
Cn+1
to the group identity in
Cn-1.
The statement that the boundary of a boundary is trivial is equivalent to the statement that
im(\partialn+1)\subseteq\ker(\partialn)
, where
im(\partialn+1)
denotes the image of the boundary operator and
\ker(\partialn)
its kernel. Elements of
Bn(X)=im(\partialn+1)
are called boundaries and elements of
Zn(X)=\ker(\partialn)
are called cycles.
Since each chain group Cn is abelian all its subgroups are normal. Then because
\ker(\partialn)
is a subgroup of Cn,
\ker(\partialn)
is abelian, and since
im(\partialn+1)\leq\ker(\partialn)
therefore
im(\partialn+1)
is a normal subgroup of
\ker(\partialn)
. Then one can create the quotient group
Hn(X):=\ker(\partialn)/im(\partialn+1)=Zn(X)/Bn(X),
called the nth homology group of X. The elements of Hn(X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.
A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.
The reduced homology groups of a chain complex C(X) are defined as homologies of the augmented chain complex
...b \overset{\partialn+1
} C_n\overset C_\overset \dotsb\overset C_1\overset C_0\overset \Z 0
where the boundary operator
\epsilon
is
\epsilon\left(\sumini\sigmai\right)=\sumini
for a combination
\sumni\sigmai,
of points
\sigmai,
which are the fixed generators of C0. The reduced homology groups
\tilde{H}i(X)
coincide with
Hi(X)
for
i0.
The extra
\Z
in the chain complex represents the unique map
[\emptyset]\longrightarrowX
from the empty simplex to X.
Computing the cycle
Zn(X)
and boundary
Bn(X)
groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.
The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X) the free abelian group generated by the n-simplices of X. See simplicial homology for details.
The singular homology groups Hn(X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.
Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted
dn,
point in the direction of increasing n rather than decreasing n; then the groups
\ker\left(dn\right)=Zn(X)
of cocycles and
im\left(dn-1\right)=Bn(X)
of follow from the same description. The nth cohomology group of X is then the quotient group
Hn(X)=Zn(X)/Bn(X),
in analogy with the nth homology group.
## Homology vs. homotopy
Homotopy groups are similar to homology groups in that they can represent "holes" in a topological space. There is a close connection between the first homotopy group
\pi1(X)
and the first homology group
H1(X)
: the latter is the abelianization of the former. Hence, it is said that "homology is a commutative alternative to homotopy".[8] The higher homotopy groups are abelian and are related to homology groups by the Hurewicz theorem, but can be vastly more complicated. For instance, the homotopy groups of spheres are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.
As an example, let X be the figure eight. Its first homotopy group
\pi1(X)
is the group of directed loops starting and ending at a predetermined point (e.g. its center). It is equivalent to the free group of rank 2, which is not commutative: looping around the leftmost cycle and then around the rightmost cycle is different than looping around the rightmost cycle and then looping around the leftmost cycle. In contrast, its first homology group
H1(X)
is the group of cuts made in a surface. This group is commutative, since (informally) cutting the leftmost cycle and then the rightmost cycle leads to the same result as cutting the rightmost cycle and then the leftmost cycle.
## Types of homology
The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.
### Simplicial homology
See main article: Simplicial homology.
The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here the chain group Cn is the free abelian group or module whose generators are the n-dimensional oriented simplexes of X. The orientation is captured by ordering the complex's vertices and expressing an oriented simplex
\sigma
as an n-tuple
(\sigma[0],\sigma[1],...,\sigma[n])
of its vertices listed in increasing order (i.e.
\sigma[0]<\sigma[1]<<\sigma[n]
in the complex's vertex ordering, where
\sigma[i]
is the
i
th vertex appearing in the tuple). The mapping
\partialn
from Cn to Cn−1 is called the and sends the simplex
\sigma=(\sigma[0],\sigma[1],...,\sigma[n])
to the formal sum
\partialn(\sigma)=
n \sum i=0
(-1)i\left(\sigma[0],...,\sigma[i-1],\sigma[i+1],...,\sigma[n]\right),
which is considered 0 if
n=0.
This behavior on the generators induces a homomorphism on all of Cn as follows. Given an element
c\inCn
, write it as the sum of generators $c = \sum_ m_i \sigma_i,$ where
Xn
is the set of n-simplexes in X and the mi are coefficients from the ring Cn is defined over (usually integers, unless otherwise specified). Then define
\partialn(c)=
\sum \sigmai\inXn
mi\partialn(\sigmai).
The dimension of the n-th homology of X turns out to be the number of "holes" in X at dimension n. It may be computed by putting matrix representations of these boundary mappings in Smith normal form.
### Singular homology
See main article: Singular homology.
Using simplicial homology example as a model, one can define a singular homology for any topological space X. A chain complex for X is defined by taking Cn to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms ∂n arise from the boundary maps of simplexes.
### Group homology
See main article: Group cohomology.
In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module
F1
and a surjective homomorphism
p1:F1\toX.
Then one finds a free module
F2
and a surjective homomorphism
p2:F2\to\ker\left(p1\right).
Continuing in this fashion, a sequence of free modules
Fn
and homomorphisms
pn
can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology
Hn
of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X.
A common use of group (co)homology
H2(G,M)
is to classify the possible extension groups E which contain a given G-module M as a normal subgroup and have a given quotient group G, so that
G=E/M.
## Homology functors
Chain complexes form a category: A morphism from the chain complex (
dn:An\toAn-1
) to the chain complex (
en:Bn\toBn-1
) is a sequence of homomorphisms
fn:An\toBn
such that
fn-1\circdn=en\circfn
for all n. The n-th homology Hn can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).
If the chain complex depends on the object X in a covariant manner (meaning that any morphism
X\toY
induces a morphism from the chain complex of X to the chain complex of Y), then the Hn are covariant functors from the category that X belongs to into the category of abelian groups (or modules).
The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by Hn) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.
## Properties
If (
dn:An\toAn-1
) is a chain complex such that all but finitely many An are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the Euler characteristic
\chi=\sum(-1)nrank(An)
(using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:
\chi=\sum(-1)nrank(Hn)
and, especially in algebraic topology, this provides two ways to compute the important invariant
\chi
for the object X which gave rise to the chain complex.
Every short exact sequence
0ABC0
of chain complexes gives rise to a long exact sequence of homology groups
\toHn(A)\toHn(B)\toHn(C)\toHn-1(A)\toHn-1(B)\toHn-1(C)\toHn-2(A)\to
All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps
Hn(C)\toHn-1(A)
The latter are called and are provided by the zig-zag lemma. This lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of relative homology and .
## Applications
### Application in pure mathematics
Notable theorems proved using homology include the following:
a\inBn
with
f(a)=a.
\Rn
and
f:U\to\Rn
is an injective continuous map, then
V=f(U)
is open and f is a homeomorphism between U and V.
• The Hairy ball theorem: any continuous vector field on the 2-sphere (or more generally, the 2k-sphere for any
k\geq1
) vanishes at some point.
U\subseteq\Rm
and
V\subseteq\Rn
are homeomorphic, then
m=n.
### Application in science and engineering
In topological data analysis, data sets are regarded as a point cloud sampling of a manifold or algebraic variety embedded in Euclidean space. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of persistent homology.[9]
In sensor networks, sensors may communicate information via an ad-hoc network that dynamically changes in time. To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the network topology to evaluate, for instance, holes in coverage.[10]
In dynamical systems theory in physics, Poincaré was one of the first to consider the interplay between the invariant manifold of a dynamical system and its topological invariants. Morse theory relates the dynamics of a gradient flow on a manifold to, for example, its homology. Floer homology extended this to infinite-dimensional manifolds. The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids that can be investigated using Floer homology.[11]
In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on topologically nontrivial domains, for example, in electromagnetic simulations. In these simulations, solution is aided by fixing the cohomology class of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.[12] [13]
## Software
Various software packages have been developed for the purposes of computing homology groups of finite cell complexes. Linbox is a C++ library for performing fast matrix operations, including Smith normal form; it interfaces with both Gap and Maple. Chomp, CAPD::Redhom and Perseus are also written in C++. All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra. Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of homotopy groups of finite simplicial complexes. Gmsh includes a homology solver for finite element meshes, which can generate Cohomology bases directly usable by finite element software.
## References
• Book: Henri Cartan . Samuel Eilenberg . Cartan . Henri Paul . Eilenberg . Samuel . Homological Algebra . Princeton University Press . 1956 . 9780674079779 . Princeton mathematical series . 19 . 529171.
• Book: Eilenberg . Samuel . Moore . J.C. . Foundations of relative homological algebra . American Mathematical Society . 1965 . 9780821812556 . Memoirs of the American Mathematical Society number . 55 . 1361982.
• .
• . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
• .
• .
• .
• .
## Notes and References
1. in part from Greek ὁμός homos "identical"
2. Book: Weeks, Jeffrey R. . The Shape of Space . 2001 . CRC Press . 978-0-203-91266-9 .
3. For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), in French, note 41, explicitly names Noether as inventing the homology group.
4. Hirzebruch, Friedrich, Emmy Noether and Topology in .
5. http://math.vassar.edu/faculty/McCleary/BourbakiAlgTop.pdf Bourbaki and Algebraic Topology by John McCleary (PDF)
6. Web site: Wildberger. Norman J.. 2012. More homology computations. . https://ghostarchive.org/varchive/youtube/20211211/l7QWg0UzBRA. 2021-12-11 . live.
7. Web site: Wildberger. Norman J.. 2012. Delta complexes, Betti numbers and torsion. . https://ghostarchive.org/varchive/youtube/20211211/NgrIPPqYKjQ . 2021-12-11 . live.
8. Web site: Wildberger. N. J.. 2012. An introduction to homology. . https://ghostarchive.org/varchive/youtube/20211211/ShWdSNJeuOg. 2021-12-11 . live.
9. Web site: CompTop overview. 16 March 2014.
10. Web site: Robert Ghrist: applied topology. 16 March 2014.
11. van den Berg. J.B.. Ghrist. R.. Vandervorst. R.C.. Wójcik. W.. 16865053. Braid Floer homology. Journal of Differential Equations. 2015. 259. 5. 1663–1721. 10.1016/j.jde.2015.03.022. 2015JDE...259.1663V. free.
12. Pellikka. M. S. Suuriniemi . L. Kettunen . C. Geuzaine . Homology and Cohomology Computation in Finite Element Modeling. SIAM J. Sci. Comput.. 2013. 35. 5. B1195–B1214. 10.1137/130906556. 10.1.1.716.3210.
13. Arnold. Douglas N. . Richard S. Falk . Ragnar Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numerica. 16 May 2006. 15. 1–155. 10.1017/S0962492906210018. 2006AcNum..15....1A . 122763537 . | 2022-11-29 05:33:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "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": 0.8579444885253906, "perplexity": 708.501889132494}, "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-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00276.warc.gz"} |
http://harvard.voxcharta.org/tag/snr/ | # Posts Tagged snr
## Recent Postings from snr
### Detailed study of SNR G306.3-0.9 using XMM-Newton and Chandra observations
We used combined data from XMM-Newton and Chandra observatories to study the X-ray morphology of SNR G306.3-0.9. A spatially-resolved spectral analysis was used to obtain physical and geometrical parameters of different regions of the remnant. Spitzer infrared observations were also used to constrain the progenitor supernova and study the environment in which the SNR evolved. The X-ray morphology of the remnant displays a non-uniform structure of semi-circular appearance, with a bright southwest region and very weak or almost negligible X-ray emission in its northern part. These results indicate that the remnant is propagating in a non-uniform environment as the shock fronts are encountering a high-density medium, where enhanced infrared emission is detected. The X-ray spectral analysis of the selected regions shows distinct emission-line features of several metal elements, confirming the thermal origin of the emission. The X-ray spectra are well represented by a combination of two absorbed thermal plasma models: one in equilibrium ionization with a mean temperature of ~0.19 keV, and another out of equilibrium ionization at a higher temperature of ~1.1 or 1.6-1.9 keV. For regions located in the northeast, central, and southwest part of the SNR, we found elevated abundances of Si, S, Ar, Ca, and Fe, typical of ejecta material. The outer regions located northwest and south show values of the abundances above solar but lower than to those found in the central regions. This suggests that the composition of the emitting outer parts of the SNR is a combination of ejecta and shocked material of the interstellar medium. The comparison between the S/Si, Ar/Si, and Ca/Si abundances ratios (1.75, 1.27, and 2.72 in the central region, respectively), favor a Type Ia progenitor for this SNR, a result that is also supported by an independent morphological analysis using X-ray and IR data.
### Gravitational wave quasinormal mode from Population III massive black hole binaries in various models of population synthesis
Focusing on the remnant black holes after merging binary black holes, we show that ringdown gravitational waves of Population III binary black holes mergers can be detected with the rate of $5.9-500~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$ for various parameters and functions. This rate is estimated for the events with SNR$>8$ for the second generation gravitational wave detectors such as KAGRA. Here, ${\rm SFR_p}$ and ${\rm f_b}$ are the peak value of the Population III star formation rate and the fraction of binaries, respectively. When we consider only the events with SNR$>35$, the event rate becomes $0.046-4.21~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$. This suggest that for remnant black hole's spin $q_f>0.95$ we have the event rate with SNR$>35$ less than $0.037~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$, while it is $3-30~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$ for the third generation detectors such as Einstein Telescope. If we detect many Population III binary black holes merger, it may be possible to constrain the Population III binary evolution paths not only by the mass distribution but also by the spin distribution.
### Gravitational wave quasinormal mode from Population III massive black hole binaries in various models of population synthesis [Cross-Listing]
Focusing on the remnant black holes after merging binary black holes, we show that ringdown gravitational waves of Population III binary black holes mergers can be detected with the rate of $5.9-500~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$ for various parameters and functions. This rate is estimated for the events with SNR$>8$ for the second generation gravitational wave detectors such as KAGRA. Here, ${\rm SFR_p}$ and ${\rm f_b}$ are the peak value of the Population III star formation rate and the fraction of binaries, respectively. When we consider only the events with SNR$>35$, the event rate becomes $0.046-4.21~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$. This suggest that for remnant black hole's spin $q_f>0.95$ we have the event rate with SNR$>35$ less than $0.037~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$, while it is $3-30~{\rm events~yr^{-1}}~({\rm SFR_p}/ (10^{-2.5}~M_\odot~{\rm yr^{-1}~Mpc^{-3}})) \cdot ({\rm [f_b/(1+f_b)]/0.33})$ for the third generation detectors such as Einstein Telescope. If we detect many Population III binary black holes merger, it may be possible to constrain the Population III binary evolution paths not only by the mass distribution but also by the spin distribution.
### Study of young stellar objects around SNR G18.8+0.3
In recent works, through observations of molecular lines, we found that the supernova remnant (SNR) G18.8+0.3 is interacting with a molecular cloud towards its southern edge. Also it has been proven the presence of several neighboring HII regions very likely located at the same distance as the remnant. The presence of dense molecular gas and the existence of shock fronts generated by the SNR and HII regions make this region an interesting scenario to study the population of young stellar objects. Thus, using the most modern colour criteria applied to the emission in the mid-infrared bands obtained from IRAC and MIPS on board Spitzer, we characterized all the point sources lying in this region. We analyzed the spectral energy distributions of sources that show signs of being young stellar objects in order to confirm their nature and derive stellar parameters. Additionally, we present a map of the $^{12}$CO J=3-2 emission obtained with the ASTE telescope towards one of the HII regions embedded in the molecular cloud. The molecular gas was studied with the aim to analyze whether the cloud is a potential site of star formation.
### Direct Ejecta Velocity Measurements of Tycho's Supernova Remnant
We present the first direct ejecta velocity measurements of Tycho's supernova remnant (SNR). Chandra's high angular resolution images reveal a patchy structure of radial velocities in the ejecta that can be separated into distinct redshifted, blueshifted, and low velocity ejecta clumps or blobs. The typical velocities of the redshifted and blueshifted blobs are <~ 7,800 km/s and <~ 5,000 km/s, respectively. The highest velocity blobs are located near the center, while the low velocity ones appear near the edge as expected for a generally spherical expansion. Systematic uncertainty on the velocity measurements from gain calibration was assessed by carrying out joint fits of individual blobs with both the ACIS-I and ACIS-S detectors. We identified an annular region (~3.3'-3.5'), where the surface brightness in the Si, S, and Fe K lines reaches a peak while the line width reaches a minimum value. These minimum line widths correspond to ion temperatures of ~1 MeV for each of the three species, in excellent agreement with one-dimensional model calculations. We determine the three-dimensional kinematics of the Si- and Fe-rich clumps in the southeastern quadrant and show that these knots form a distinct, compact, and kinematically-connected structure, possibly even a chain of knots strung along the remnant's edge. By examining the viewing geometries we conclude that the knots in the southeastern region are unlikely to be responsible for the high velocity Ca II absorption features seen in the light echo spectrum of Tycho's SNR.
### Supernova Remnants in the Local Group I: A model for the radio luminosity function and visibility times of supernova remnants
Supernova remnants (SNRs) in Local Group galaxies offer unique insights into the origin of different types of supernovae. In order to take full advantage of these insights, one must understand the intrinsic and environmental diversity of SNRs in the context of their host galaxies. We introduce a semi-analytic model that reproduces the statistical properties of a radio continuum-selected SNR population, taking into account the detection limits of radio surveys, the range of SN kinetic energies, the measured ISM and stellar mass distribution in the host galaxy from multi-wavelength images and the current understanding of electron acceleration and field amplification in SNR shocks from first-principle kinetic simulations. Applying our model to the SNR population in M33, we reproduce the SNR radio luminosity function with a median SN rate of $\sim 3.1 \times 10^{-3}$ per year and an electron acceleration efficiency, $\epsilon_{\rm{e}} \sim 4.2 \times 10^{-3}$. We predict that the radio visibility times of $\sim 70\%$ of M33 SNRs will be determined by their Sedov-Taylor lifetimes, and correlated with the measured ISM column density, $N_H$ ($t_{\rm{vis}} \propto N_H^{-a}$, with $a \sim 0.33$) while the remaining will have visibility times determined by the detection limit of the radio survey. These observational constraints on the visibility time of SNRs will allow us to use SNR catalogs as `SN surveys' to calculate SN rates and delay time distributions in the Local Group.
### Expanding molecular bubble surrounding Tycho's supernova remnant (SN 1572) observed with IRAM 30 m telescope: evidence for a single-degenerate progenitor
Whether the progenitors of Type-Ia Supernovae, single degenerate or double-degenerate white dwarf (WD) systems, is a highly debated topic. To address the origin of the Type Ia Tycho's supernova remnant (SNR), SN 1572, we have carried out a 12CO J=1-0 mapping and a 3-mm line survey towards the remnant using the IRAM 30 m telescope. We show that Tycho is surrounded by a clumpy molecular bubble at the local standard of rest velocity $\sim 61$ km s$^{-1}$ which expands at a speed $\sim 4.5$ km s$^{-1}$ and has a mass of $\sim 220$ $M_\odot$ (at the distance of 2.5 kpc). Enhanced 12CO J=2-1 line emission relative to 12CO J=1-0 emission and possible line broadenings (in velocity range -64-- -60 km s$^{-1}$) are found at the northeastern boundary of the SNR where the shell is deformed and decelerated. These features, combined with the morphological correspondence between the expanding molecular bubble and Tycho, suggest that the SNR is associated with the bubble at velocity range -66-- -57 km s$^{-1}$. The most plausible origin for the expanding bubble is the fast outflow (with velocity $> 100$ km s$^{-1}$) driven from the vicinity of a WD as it accreted matter from a non-degenerate companion star. The SNR has been expanding in the low-density wind-blown bubble and the shock wave has just reached the molecular cavity wall. This is the first unambiguous detection of the expanding bubble driven by the progenitor of the Type-Ia SNR, which constitutes evidence for a single degenerate progenitor for this Type-Ia supernova.
### A young SNR illuminating nearby Molecular Clouds with cosmic rays
The Supernova Remnant (SNR) HESS J1731-347 displays strong non-thermal TeV gamma-ray and X-ray emission, thus the object is at present time accelerating particles to very high energies. A distinctive feature of this young SNR is the nearby (~30 pc in projection) extended source HESS J1729-345, which is currently unidentified but is in spatial projection coinciding with known molecular clouds (MC). We model the SNR evolution to explore if the TeV emission from HESS J1729-345 can be explained as emission from runaway hadronic cosmic rays (CRs) that are illuminating these MCs. The observational data of HESS J1729-345 and HESS J1731-347 can be reproduced using core-collapse SN models for HESS J1731-347. Starting with different progenitor stars and their pre-supernova environment, we model potential SNR evolution histories along with the CR acceleration in the SNR and the diffusion of the CRs. A simplified 3-dimensional structure of the MCs is introduced based on 12CO data, adopting a distance of 3.2 kpc to the source. A Monte Carlo-based diffusion model for the escaping CRs is developed to deal with the inhomogeneous environment. The fast SNR forward shock speed as implied from the X-ray data can easily be explained when employing scenarios with progenitor star masses between 20 and 25 solar masses, where the SNR shock is still expanding inside the main sequence (MS)-bubble at present time. The TeV spectrum of HESS J1729-345 is satisfactorily fitted by the emission from the highest-energy CRs that have escaped the SNR, using a standard galactic CR diffusion coefficient in the inter-clump medium. The TeV image of HESS J1729-345 can be explained with a reasonable 3-dimensional structure of MCs. The TeV emission from the SNR itself is dominated by leptonic emission in this model. We also explore scenarios where the shock is starting to encounter the dense MS progenitor wind bubble shell.
### Revisiting the Contributions of Supernova and Hypernova Remnants to the Diffuse High-Energy Backgrounds: Constraints on Very-High-Redshift Injections [Replacement]
Star-forming and starburst galaxies are considered as one of the viable candidate sources of the high-energy cosmic neutrino background detected in IceCube. We revisit contributions of supernova remnants (SNRs) and hypernova remnants (HNRs) in such galaxies to the diffuse high-energy neutrino and gamma-ray backgrounds, in light of the latest Fermi data above 50GeV. We also take into account possible time dependent effects of the cosmic-ray (CR) acceleration during the SNR evolution. CRs accelerated by the SNR shocks can produce high-energy neutrinos up to $\sim100$ TeV energies, but CRs from HNRs can extend the spectrum up to PeV energies. We show that, only if HNRs are dominant over SNRs, the diffuse neutrino background above 100 TeV can be explained without contradicting the gamma-ray data. However, the neutrino data around 30 TeV remain unexplained, which might suggest a different population of gamma-ray dark CR sources. Alternatively, we consider possible contributions of Pop-III HNRs up to $z\lesssim10$, and show that they are not constrained by the gamma-ray data, and thus could contribute to the diffuse high-energy backgrounds if their explosion energy reaches ${\mathcal E}_{\rm POP-III}\sim{\rm a~few}\times10^{53}$erg. More conservatively, our results suggest that the explosion energy of POP-III HNRs is ${\mathcal E}_{\rm POP-III}\lesssim7\times{10}^{53}$erg.
### Revisiting the Contributions of Supernova and Hypernova Remnants to the Diffuse High-Energy Backgrounds: Constraints on Very-High-Redshift Injections
Star-forming and starburst galaxies are considered as one of the viable candidate sources of the high-energy cosmic neutrino background detected in IceCube. We revisit contributions of supernova remnants (SNRs) and hypernova remnants (HNRs) in such galaxies to the diffuse high-energy neutrino and gamma-ray backgrounds, in light of the latest Fermi data above 50GeV. We also take into account possible time dependent effects of the cosmic-ray (CR) acceleration during the SNR evolution. CRs accelerated by the SNR shocks can produce high-energy neutrinos up to $\sim100$ TeV energies, but CRs from HNRs can extend the spectrum up to PeV energies. We show that, only if HNRs are dominant over SNRs, the diffuse neutrino background above 100 TeV can be explained without contradicting the gamma-ray data. However, the neutrino data around 30 TeV remain unexplained, which might suggest a different population of gamma-ray dark CR sources. Alternatively, we consider possible contributions of Pop-III HNRs up to $z\lesssim10$, and show that they are not constrained by the gamma-ray data, and thus could contribute to the diffuse high-energy backgrounds if their explosion energy reaches ${\mathcal E}_{\rm POP-III}\sim{\rm a~few}\times10^{53}$erg. More conservatively, our results suggest that the explosion energy of POP-III HNRs is ${\mathcal E}_{\rm POP-III}\lesssim7\times{10}^{53}$erg.
### Revisiting the Contributions of Supernova and Hypernova Remnants to the Diffuse High-Energy Backgrounds: Constraints on Very-High-Redshift Injections [Cross-Listing]
Star-forming and starburst galaxies are considered as one of the viable candidate sources of the high-energy cosmic neutrino background detected in IceCube. We revisit contributions of supernova remnants (SNRs) and hypernova remnants (HNRs) in such galaxies to the diffuse high-energy neutrino and gamma-ray backgrounds, in light of the latest Fermi data above 50GeV. We also take into account possible time dependent effects of the cosmic-ray (CR) acceleration during the SNR evolution. CRs accelerated by the SNR shocks can produce high-energy neutrinos up to $\sim100$ TeV energies, but CRs from HNRs can extend the spectrum up to PeV energies. We show that, only if HNRs are dominant over SNRs, the diffuse neutrino background above 100 TeV can be explained without contradicting the gamma-ray data. However, the neutrino data around 30 TeV remain unexplained, which might suggest a different population of gamma-ray dark CR sources. Alternatively, we consider possible contributions of Pop-III HNRs up to $z\lesssim10$, and show that they are not constrained by the gamma-ray data, and thus could contribute to the diffuse high-energy backgrounds if their explosion energy reaches ${\mathcal E}_{\rm POP-III}\sim{\rm a~few}\times10^{53}$erg. More conservatively, our results suggest that the explosion energy of POP-III HNRs is ${\mathcal E}_{\rm POP-III}\lesssim7\times{10}^{53}$erg.
### Revisiting the Contributions of Supernova and Hypernova Remnants to the Diffuse High-Energy Backgrounds: Constraints on Very-High-Redshift Injections [Replacement]
Star-forming and starburst galaxies are considered as one of the viable candidate sources of the high-energy cosmic neutrino background detected in IceCube. We revisit contributions of supernova remnants (SNRs) and hypernova remnants (HNRs) in such galaxies to the diffuse high-energy neutrino and gamma-ray backgrounds, in light of the latest Fermi data above 50GeV. We also take into account possible time dependent effects of the cosmic-ray (CR) acceleration during the SNR evolution. CRs accelerated by the SNR shocks can produce high-energy neutrinos up to $\sim100$ TeV energies, but CRs from HNRs can extend the spectrum up to PeV energies. We show that, only if HNRs are dominant over SNRs, the diffuse neutrino background above 100 TeV can be explained without contradicting the gamma-ray data. However, the neutrino data around 30 TeV remain unexplained, which might suggest a different population of gamma-ray dark CR sources. Alternatively, we consider possible contributions of Pop-III HNRs up to $z\lesssim10$, and show that they are not constrained by the gamma-ray data, and thus could contribute to the diffuse high-energy backgrounds if their explosion energy reaches ${\mathcal E}_{\rm POP-III}\sim{\rm a~few}\times10^{53}$erg. More conservatively, our results suggest that the explosion energy of POP-III HNRs is ${\mathcal E}_{\rm POP-III}\lesssim7\times{10}^{53}$erg.
### Interacting Large-Scale Magnetic Fields and Ionised Gas in the W50/SS433 System
The W50/SS433 system is an unusual Galactic outflow-driven object of debatable origin. We have used the Australia Telescope Compact Array (ATCA) to observe a new 198 pointing mosaic, covering $3^\circ \times 2^\circ$, and present the highest-sensitivity full-Stokes data of W50 to date using wide-field, wide-band imaging over a 2 GHz bandwidth centred at 2.1 GHz. We also present a complementary H$\alpha$ mosaic created using the Isaac Newton Telescope Photometric H$\alpha$ Survey of the Northern Galactic Plane (IPHAS). The magnetic structure of W50 is found to be consistent with the prevailing hypothesis that the nebula is a reanimated shell-like supernova remnant (SNR), that has been re-energised by the jets from SS433. We observe strong depolarization effects that correlate with diffuse H$\alpha$ emission, likely due to spatially-varying Faraday rotation measure (RM) fluctuations of $\ge48$ to 61 rad m$^{-2}$ on scales $\le4.5$ to 6 pc. We also report the discovery of numerous, faint, H$\alpha$ filaments that are unambiguously associated with the central region of W50. These thin filaments are suggestive of a SNR's shock emission, and almost all have a radio counterpart. Furthermore, an RM-gradient is detected across the central region of W50, which we interpret as a loop magnetic field with a symmetry axis offset by $\approx90^{\circ}$ to the east-west jet-alignment axis, and implying that the evolutionary processes of both the jets and the SNR must be coupled. A separate RM-gradient is associated with the termination shock in the Eastern ear, which we interpret as a ring-like field located where the shock of the jet interacts with the circumstellar medium. Future optical observations will be able to use the new H$\alpha$ filaments to probe the kinematics of the shell of W50, potentially allowing for a definitive experiment on W50's formation history.
### The Impact of a Supernova Remnant on Fast Radio Bursts [Replacement]
Fast radio bursts (FRBs) are millisecond bursts of radio radiation whose progenitors so far remain mysterious. Nevertheless, the timescales and energetics of the events have lead to many theories associating FRBs with young neutron stars. Motivated by this, I explore the interaction of FRBs with young supernova remnants (SNRs), and I discuss the potential observational consequences and constraints of such a scenario. As the SN ejecta plows into the interstellar medium (ISM), a reverse shock is generated that passes back through the material and ionizes it. This leads to a dispersion measure (DM) associated with the SNR as well as a time derivative for DM. Times when DM is high are generally overshadowed by free-free absorption, which, depending on the mass of the ejecta and the density of the ISM, may be probed at frequencies of $400\,{\rm MHz}$ to $1.4\,{\rm GHz}$ on timescales of $\sim100-500\,{\rm yrs}$ after the SN. Magnetic fields generated at the reverse shock may be high enough to explain Faraday rotation that has been measured for one FRB. If FRBs are powered by the spin energy of a young NS (rather than magnetic energy), the NS must have a magnetic field $\lesssim10^{11}-10^{12}\,{\rm G}$ to ensure that it does not spin down too quickly while the SNR is still optically thick at radio frequencies. In the future, once there are distance measurements to FRBs and their energetics are better understood, the spin of the NS can also be constrained.
### Supernova Remnants In The Magellanic Clouds
We present initial results of an ongoing study of the supernova remnants (SNRs) and candidates in the Magellanic Clouds. Some 108 objects in both Clouds are considered to be either an SNR or a reliable candidate. This represents the most complete sample of known SNRs in any galaxy. therefore, this study allows us to study SNR population properties such as the size and spectral index distribution. Here, we also show 12 known Large Magellanic Cloud SNRs from type Ia SN explosions and briefly comment on their importance.
### TRIPPy: Trailed Image Photometry in Python
Photometry of moving sources typically suffers from reduced signal-to-noise (SNR) or flux measurements biased to incorrect low values through the use of circular apertures. To address this issue we present the software package, TRIPPy: TRailed Image Photometry in Python. TRIPPy introduces the pill aperture, which is the natural extension of the circular aperture appropriate for linearly trailed sources. The pill shape is a rectangle with two semicircular end-caps, and is described by three parameters, the trail length and angle, and the radius. The TRIPPy software package also includes a new technique to generate accurate model point-spread functions (PSF) and trailed point-spread functions (TSF) from stationary background sources in sidereally tracked images. The TSF is merely the convolution of the model PSF, which consists of a moffat profile, and super sampled lookup table. From the TSF, accurate pill aperture corrections can be estimated as a function of pill radius with a accuracy of 10 millimags for highly trailed sources. Analogous to the use of small circular apertures and associated aperture corrections, small radius pill apertures can be used to preserve signal-to-noise of low flux sources, with appropriate aperture correction applied to provide an accurate, unbiased flux measurement at all SNR.
### Radio SNRs in the Magellanic Clouds as probes of shock microphysics
A large number of radio supernova remnants (SNRs) have been resolved in our Galaxy and nearby ones. These remnants are thought to be produced via synchrotron emission from electrons accelerated by the shock that the supernova ejecta drives into the external medium. Here we consider the sample of radio SNRs in the Magellanic Clouds. Given the size of a radio SNR and its flux, we can constrain $\sim \epsilon_e \epsilon_B \sim 10^{-3}$, which are the fractions of dissipated energy that goes into non-thermal electrons and magnetic field, respectively. These estimates do not depend on the largely uncertain values of the external density and the age of the SNR. We use this theory to develop a Monte Carlo scheme that reproduces the observed distribution of radio fluxes and sizes of the population of radio SNRs in the Magellanic Clouds. This simple model provides a framework that could potentially be applied to other galaxies with complete radio SNRs samples.
### Radio Weak Lensing Shear Measurement in the Visibility Domain - I. Methodology
The high sensitivity of the new generation of radio telescopes such as the Square Kilometre Array (SKA) will allow cosmological weak lensing measurements at radio wavelengths that are competitive with optical surveys. We present an adaptation to radio data of "lensfit", a method for galaxy shape measurement originally developed and used for optical weak lensing surveys. This likelihood method uses an analytical galaxy model and makes a Bayesian marginalisation of the likelihood over uninteresting parameters. It has the feature of working directly in the visibility domain, which is the natural approach to adopt with radio interferometer data, avoiding systematics introduced by the imaging process. As a proof of concept, we provide results for visibility simulations of individual galaxies with flux density S >= 10muJy at the phase centre of the proposed SKA1-MID baseline configuration, adopting 12 frequency channels in the band 950-1190 MHz. Weak lensing shear measurements from a population of galaxies with realistic flux and scalelength distributions are obtained after uniform gridding of the raw visibilities. Shear measurements are expected to be affected by 'noise bias': we estimate the bias in the method as a function of signal-to-noise ratio (SNR). We obtain additive and multiplicative bias values that are comparable to SKA1 requirements for SNR > 18 and SNR > 30, respectively. The multiplicative bias for SNR > 10 is comparable to that found in ground-based optical surveys such as CFHTLenS, and we anticipate that similar shear measurement calibration strategies to those used for optical surveys may be used to good effect in the analysis of SKA radio interferometer data.
### The Likely Fermi detection of the supernova remnant SN 1006
We report the likely detection of gamma-ray emission from the northeast shell region of the historical supernova remnant (SNR) SN 1006. Having analyzed 7 years of Fermi LAT Pass 8 data for the region of SN 1006, we found a GeV gamma-ray source detected with 4 sigma significance. Both the position and spectrum of the source match those of HESS J1504-418 respectively, which is TeV emission from SN 1006. Considering the source as the GeV gamma-ray counterpart to SN~1006, the broadband spectral energy distribution is found to be approximately consistent with the leptonic scenario that has been proposed for the TeV emission from the SNR. Our result has likely confirmed the previous study of the SNRs with TeV shell-like morphology: SN 1006 is one of them sharing very similar peak luminosity and spectral shape.
### Deep morphological and spectral study of the SNR RCW 86 with Fermi-LAT
RCW 86 is a young supernova remnant (SNR) showing a shell-type structure at several wavelengths and is thought to be an efficient cosmic-ray (CR) accelerator. Earlier \textit{Fermi} Large Area Telescope results reported the detection of $\gamma$-ray emission coincident with the position of RCW 86 but its origin (leptonic or hadronic) remained unclear due to the poor statistics. Thanks to 6.5 years of data acquired by the \textit{Fermi}-LAT and the new event reconstruction Pass 8, we report the significant detection of spatially extended emission coming from RCW 86. The spectrum is described by a power-law function with a very hard photon index ($\Gamma = 1.42 \pm 0.1_{\rm stat} \pm 0.06_{\rm syst}$) in the 0.1--500 GeV range and an energy flux above 100 MeV of ($2.91$ $\pm$ $0.8_{\rm stat}$ $\pm$ $0.12_{\rm syst}$) $\times$ $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. Gathering all the available multiwavelength (MWL) data, we perform a broadband modeling of the nonthermal emission of RCW 86 to constrain parameters of the nearby medium and bring new hints about the origin of the $\gamma$-ray emission. For the whole SNR, the modeling favors a leptonic scenario in the framework of a two-zone model with an average magnetic field of 10.2 $\pm$ 0.7 $\mu$G and a limit on the maximum energy injected into protons of 2 $\times$ 10$^{49}$ erg for a density of 1 cm$^{-3}$. In addition, parameter values are derived for the North-East (NE) and South-West (SW) regions of RCW 86, providing the first indication of a higher magnetic field in the SW region.
### Temporal Variability of Interstellar Na I Absorption Toward The Monoceros Loop
We report the first evidence of temporal variability in the interstellar Na I absorption toward HD 47240, which lies behind the Monoceros Loop supernova remnant (SNR). Analysis of multi-epoch Kitt Peak coud\'{e} feed spectra from this sightline taken over an eight-year period reveals significant variation in both the observed column density and the central velocities of the high-velocity gas components in these spectra. Given the $\sim$1.3 mas yr$^{-1}$ proper motion of HD 47240 and a SNR distance of 1.6 kpc, this variation would imply $\sim$10 AU fluctuations within the SNR shell. Similar variations have been previously reported in the Vela supernova remnant, suggesting a connection between the expanding supernova remnant gas and the observed variations. We speculate on the potential nature of the observed variations toward HD 47240 in the context of the expanding remnant gas interacting with the ambient ISM.
### Temporal Variability of Interstellar Na I Absorption Toward The Monoceros Loop [Replacement]
We report the first evidence of temporal variability in the interstellar Na I absorption toward HD 47240, which lies behind the Monoceros Loop supernova remnant (SNR). Analysis of multi-epoch Kitt Peak coud\'{e} feed spectra from this sightline taken over an eight-year period reveals significant variation in both the observed column density and the central velocities of the high-velocity gas components in these spectra. Given the $\sim$1.3 mas yr$^{-1}$ proper motion of HD 47240 and a SNR distance of 1.6 kpc, this variation would imply $\sim$10 AU fluctuations within the SNR shell. Similar variations have been previously reported in the Vela supernova remnant, suggesting a connection between the expanding supernova remnant gas and the observed variations. We speculate on the potential nature of the observed variations toward HD 47240 in the context of the expanding remnant gas interacting with the ambient ISM.
### Discovery of X-ray Emission from the Galactic Supernova Remnant G32.8-0.1 with Suzaku
We present the first dedicated X-ray study of the supernova remnant (SNR) G32.8-0.1 (Kes 78) with Suzaku. X-ray emission from the whole SNR shell has been detected for the first time. The X-ray morphology is well correlated with the emission from the radio shell, while anti-correlated with the molecular cloud found in the SNR field. The X-ray spectrum shows not only conventional low-temperature (kT ~ 0.6 keV) thermal emission in a non-equilibrium ionization state, but also a very high temperature (kT ~ 3.4 keV) component with a very low ionization timescale (~ 2.7e9 cm^{-3}s), or a hard non-thermal component with a photon index Gamma~2.3. The average density of the low-temperature plasma is rather low, of the order of 10^{-3}--10^{-2} cm^{-3}, implying that this SNR is expanding into a low-density cavity. We discuss the X-ray emission of the SNR, also detected in TeV with H.E.S.S., together with multi-wavelength studies of the remnant and other gamma-ray emitting SNRs, such as W28 and RCW 86. Analysis of a time-variable source, 2XMM J185114.3-000004, found in the northern part of the SNR, is also reported for the first time. Rapid time variability and a heavily absorbed hard X-ray spectrum suggest that this source could be a new supergiant fast X-ray transient.
### Fermi LAT Discovery of Extended Gamma-Ray Emissions in the Vicinity of the HB3 Supernova Remnant
We report the discovery of extended gamma-ray emission measured by the Large Area Telescope (LAT) onboard the Fermi Gamma-ray Space Telescope in the region of the supernova remnant (SNR) HB3 (G132.7+1.3) and the W3 HII complex adjacent to the southeast of the remnant. W3 is spatially associated with bright 12CO (J=1-0) emission. The gamma-ray emission is spatially correlated with this gas and the SNR. We discuss the possibility that gamma rays originate in interactions between particles accelerated in the SNR and interstellar gas or radiation fields. The decay of neutral pions produced in nucleon-nucleon interactions between accelerated hadrons and interstellar gas provides a reasonable explanation for the gamma-ray emission. The emission from W3 is consistent with irradiation of the CO clouds by the cosmic rays accelerated in HB3.
### Optical observations of the nearby galaxy IC342 with narrow band [SII] and H$\alpha$ filters. II - Detection of 16 Optically-Identified Supernova Remnant Candidates
We present the detection of 16 optical supernova remnant (SNR) candidates in the nearby spiral galaxy IC342. The candidates were detected by applying [SII]/H$\alpha$ ratio criterion on observations made with the 2 m RCC telescope at Rozhen National Astronomical Observatory in Bulgaria. In this paper, we report the coordinates, diameters, H$\alpha$ and [SII] fluxes for 16 SNRs detected in two fields of view in the IC342 galaxy. Also, we estimate that the contamination of total H$\alpha$ flux from SNRs in the observed portion of IC342 is 1.4%. This would represent the fractional error when the star formation rate (SFR) for this galaxy is derived from the total galaxy's H$\alpha$ emission.
### A Systematic Survey for Broadened CO Emission Toward Galactic Supernova Remnants
We present molecular spectroscopy toward 50 Galactic supernova remnants (SNRs) taken at millimeter wavelengths in 12CO and 13CO J=2-1 with the Heinrich Hertz Submillimeter Telescope as part of a systematic survey for broad molecular line (BML) regions indicative of interactions with molecular clouds (MCs). These observations reveal BML regions toward nineteen SNRs, including nine newly identified BML regions associated with SNRs (G08.3-0.0, G09.9-0.8, G11.2-0.3, G12.2+0.3, G18.6-0.2, G23.6+0.3, 4C-04.71, G29.6+0.1, G32.4+0.1). The remaining ten SNRs with BML regions confirm previous evidence for MC interaction in most cases (G16.7+0.1, Kes 75, 3C 391, Kes 79, 3C 396, 3C 397, W49B, Cas A, IC 443), although we confirm that the BML region toward HB 3 is associated with the W3(OH) HII region, not the SNR. Based on the systemic velocity of each MC, molecular line diagnostics, and cloud morphology, we test whether these detections represent SNR-MC interactions. One of the targets (G54.1+0.3) had previous indications of a BML region, but we did not detect broadened emission toward it. Although broadened 12CO J=2-1 line emission should be detectable toward virtually all SNR-MC interactions we find relatively few examples; therefore, the number of interactions is low. This result favors mechanisms other than SN feedback as the basic trigger for star formation. In addition, we find no significant association between TeV gamma-ray sources and MC interactions, contrary to predictions that SNR-MC interfaces are the primary venues for cosmic ray acceleration.
### The role of the diffusive protons in the gamma-ray emission of supernova remnant RX J1713.7$-$3946 --- a two-zone model [Replacement]
RX~J1713.7$-$3946 is a prototype in the $\gamma$-ray-bright supernova remnants (SNRs) and is in continuing debates on its hadronic versus leptonic origin of the $\gamma$-ray emission. We explore the role played by the diffusive relativistic protons that escape from the SNR shock wave in the $\gamma$-ray emission, apart from the high-energy particles' emission from the inside of the SNR. In the scenario that the SNR shock propagates in a clumpy molecular cavity, we consider that the$\gamma$-ray emission from the inside of the SNR may arise either from the inverse Compton scattering or from the interaction between the trapped energetic protons and the shocked clumps. The dominant origin between them depends on the electron-to-proton number ratio. The diffusive protons that escaped from the shock wave during the expansion history can provide an outer hadronic $\gamma$-ray component by bombarding the surrounding dense matter. The broadband spectrum can be well explained by this two-zone model, in which the $\gamma$-ray emission from the inside governs the TeV band, while the outer emission component substantially contributes to the GeV $\gamma$-rays. The two-zone model can also explain the TeV $\gamma$-ray radial brightness profile that significantly stretches beyond the nonthermal X-ray-emitting region. In the calculation, we present a simplified algorithm for Li & Chen's (2010) "accumulative diffusion" model for escaping protons and apply the Markov Chain Monte Carlo method to constrain the physical parameters.
### New Identification of the Mixed-Morphology Supernova Remnant G298.6-0.0 with Possible Gamma-ray Association
We present an X-ray analysis on the Galactic supernova remnant (SNR) G298.6-0.0 with Suzaku. The X-ray image shows a center-filled structure inside the radio shell, implying this SNR is categorized as a mixed-morphology (MM) SNR. The spectrum is well reproduced by a single temperature plasma model in ionization equilibrium, with a temperature of 0.78 (0.70-0.87) keV. The total plasma mass of 30 solar mass indicates that the plasma has interstellar medium origin. The association with a GeV gamma-ray source 3FGL J1214.0-6236 on the shell of the SNR is discussed, in comparison with other MM SNRs with GeV gamma-ray associations. It is found that the flux ratio between absorption-corrected thermal X-rays and GeV gamma-rays decreases as the MM SNRs evolve to larger physical sizes. The absorption-corrected X-ray flux of G298.6-0.0 and the GeV gamma-ray flux of 3FGL J1214.0-6236 closely follow this trend, implying that 3FGL J1214.0-6236 is likely to be the GeV counterpart of G298.6-0.0.
### Radio emission from Supernova Remnants
The explosion of a supernova releases almost instantaneously about 10^51 ergs of mechanic energy, changing irreversibly the physical and chemical properties of large regions in the galaxies. The stellar ejecta, the nebula resulting from the powerful shock waves, and sometimes a compact stellar remnant, constitute a supernova remnant (SNR). They can radiate their energy across the whole electromagnetic spectrum, but the great majority are radio sources. Almost 70 years after the first detection of radio emission coming from a SNR, great progress has been achieved in the comprehension of their physical characteristics and evolution. We review the present knowledge of different aspects of radio remnants, focusing on sources of the Milky Way and the Magellanic Clouds, where the SNRs can be spatially resolved. We present a brief overview of theoretical background, analyze morphology and polarization properties, and review and critical discuss different methods applied to determine the radio spectrum and distances. The consequences of the interaction between the SNR shocks and the surrounding medium are examined, including the question of whether SNRs can trigger the formation of new stars. Cases of multispectral comparison are presented. A section is devoted to reviewing recent results of radio SNRs in the Magellanic Clouds, with particular emphasis on the radio properties of SN 1987A, an ideal laboratory to investigate dynamical evolution of an SNR in near real time. The review concludes with a summary of issues on radio SNRs that deserve further study, and analyzing the prospects for future research with the latest generation radio telescopes.
### Optical discovery and multiwavelength investigation of supernova remnant MCSNR J0512-6707 in the Large Magellanic Cloud
We present optical, radio and X-ray data that confirm a new supernova remnant (SNR) in the Large Magellanic Cloud (LMC) discovered using our deep H-alpha imagery. Optically, the new SNR has a somewhat filamentary morphology and a diameter of 56 x 64 arcsec (13.5 x 15.5 pc at the 49.9 kpc distance of the LMC). Spectroscopic follow-up of multiple regions show high [SII]/H-alpha emission-line ratios ranging from 0.66+/-0.02 to 0.93+/-0.01, all of which are typical of an SNR. We found radio counterparts for this object using our new Australia Telescope Compact Array (ATCA) 6cm pointed observations as well as a number of available radio surveys at 8 640 MHz, 4 850 MHz, 1 377 MHz and 843 MHz. With these combined data we provide a spectral index (alpha) = -0.5 between 843 and 8640 MHz. Both spectral line analysis and the magnetic field strength, ranging from 124 - 184 mG, suggest a dynamical age between 2,200 and 4,700 yrs. The SNR has a previously catalogued X-ray counterpart listed as HP 483 in the ROSAT Position Sensitive Proportional Counter (PSPC) catalogue.
### Optical discovery and multiwavelength investigation of supernova remnant MCSNR J0512-6707 in the Large Magellanic Cloud [Replacement]
We present optical, radio and X-ray data that confirm a new supernova remnant (SNR) in the Large Magellanic Cloud (LMC) discovered using our deep H-alpha imagery. Optically, the new SNR has a somewhat filamentary morphology and a diameter of 56 x 64 arcsec (13.5 x 15.5 pc at the 49.9 kpc distance of the LMC). Spectroscopic follow-up of multiple regions show high [SII]/H-alpha emission-line ratios ranging from 0.66+/-0.02 to 0.93+/-0.01, all of which are typical of an SNR. We found radio counterparts for this object using our new Australia Telescope Compact Array (ATCA) 6cm pointed observations as well as a number of available radio surveys at 8 640 MHz, 4 850 MHz, 1 377 MHz and 843 MHz. With these combined data we provide a spectral index (alpha) = -0.5 between 843 and 8640 MHz. Both spectral line analysis and the magnetic field strength, ranging from 124 - 184 mG, suggest a dynamical age between 2,200 and 4,700 yrs. The SNR has a previously catalogued X-ray counterpart listed as HP 483 in the ROSAT Position Sensitive Proportional Counter (PSPC) catalogue.
### 3D Hydrodynamic Simulations of the Galactic Supernova Remnant CTB 109
Using detailed 3D hydrodynamic simulations we study the nature of the Galactic supernova remnant (SNR) CTB 109 (G109.1-1.0), which is well-known for its semicircular shape and a bright diffuse X-ray emission feature inside the SNR. Our model has been designed to explain the observed morphology, with a special emphasis on the bright emission feature inside the SNR. Moreover, we determine the age of the remnant and compare our findings with X-ray observations. With CTB 109 we test a new method of detailed numerical simulations of diffuse young objects, using realistic initial conditions derived directly from observations. We performed numerical 3D simulations with the RAMSES code. The initial density structure has been directly taken from $^{12}$CO emission data, adding an additional dense cloud, which, when it is shocked, causes the bright emission feature. From parameter studies we obtained the position $(\ell , b)=(109.1545^\circ , -1.0078^\circ)$ for an elliptical cloud with $n_\text{cloud}=25~\text{cm}^{-3}$ based on the preshock density from Chandra data and a maximum diameter of 4.54 pc, whose encounter with the supernova (SN) shock wave generates the bright X-ray emission inside the SNR. The calculated age of the remnant is about 11,000 yr according to our simulations. In addition, we can also determine the most probable site of the SN explosion. Hydrodynamic simulations can reproduce the morphology and the observed size of the SNR CTB 109 remarkably well. Moreover, the simulations show that it is very plausible that the bright X-ray emission inside the SNR is the result of an elliptical dense cloud shocked by the SN explosion wave. We show that numerical simulations using observational data for an initial model can produce meaningful results.
### 3D Hydrodynamic Simulations of the Galactic Supernova Remnant CTB 109 [Replacement]
Using detailed 3D hydrodynamic simulations we study the nature of the Galactic supernova remnant (SNR) CTB 109 (G109.1-1.0), which is well-known for its semicircular shape and a bright diffuse X-ray emission feature inside the SNR. Our model has been designed to explain the observed morphology, with a special emphasis on the bright emission feature inside the SNR. Moreover, we determine the age of the remnant and compare our findings with X-ray observations. With CTB 109 we test a new method of detailed numerical simulations of diffuse young objects, using realistic initial conditions derived directly from observations. We performed numerical 3D simulations with the RAMSES code. The initial density structure has been directly taken from $^{12}$CO emission data, adding an additional dense cloud, which, when it is shocked, causes the bright emission feature. From parameter studies we obtained the position $(\ell , b)=(109.1545^\circ , -1.0078^\circ)$ for an elliptical cloud with $n_\text{cloud}=25~\text{cm}^{-3}$ based on the preshock density from Chandra data and a maximum diameter of 4.54 pc, whose encounter with the supernova (SN) shock wave generates the bright X-ray emission inside the SNR. The calculated age of the remnant is about 11,000 yr according to our simulations. In addition, we can also determine the most probable site of the SN explosion. Hydrodynamic simulations can reproduce the morphology and the observed size of the SNR CTB 109 remarkably well. Moreover, the simulations show that it is very plausible that the bright X-ray emission inside the SNR is the result of an elliptical dense cloud shocked by the SN explosion wave. We show that numerical simulations using observational data for an initial model can produce meaningful results.
### FERMI-LAT Observations of Supernova Remnant G5.7-0.1, Believed to be Interacting with Molecular Clouds [Replacement]
This work reports on the detection of $\gamma$-ray emission coincident with the supernova remnant (SNR) SNR G5.7-0.1 using data collected by the Large Area Telescope aboard the Fermi Gamma-ray Space Telescope. The SNR is believed to be interacting with molecular clouds, based on 1720 MHz hydroxyl (OH) maser emission observations in its direction. This interaction is expected to provide targets for the production of $\gamma$-ray emission from $\pi^0$-decay. A $\gamma$-ray source was observed in the direction of SNR G5.7-0.1, positioned nearby the bright $\gamma$-ray source SNR W28. We model the emission from radio to $\gamma$-ray energies using a one-zone model. Following consideration of both $\pi^0$-decay and leptonically dominated emission scenarios for the MeV-TeV source, we conclude that a considerable component of the $\gamma$-ray emission must originate from the $\pi^0$-decay channel. Finally, constraints were placed on the reported ambiguity of the SNR distance through X-ray column density measurements made using XMM-Newton observations. We conclude SNR G5.7-0.1 is a significant $\gamma$-ray source positioned at a distance of $\sim 3$ kpc with luminosity in the 0.1--100 GeV range of $L_{\gamma} \approx 7.4 \times 10^{34}$ erg/s.
### FERMI-LAT Observations of Supernova Remnant G5.7-0.1, Believed to be Interacting with Molecular Clouds
We report the detection of $\gamma$-ray emission coincident with the supernova remnant (SNR) G5.7-0.1 using data from the Large Area Telescope on board the {\it Fermi Gamma-ray Space Telescope}. SNR shocks are expected to be sites of cosmic ray acceleration, and clouds of dense material can provide effective targets for production of $\gamma$-rays from $\pi^0$-decay. The SNR is known to be interacting with molecular clouds, as evidenced by observations of hydroxyl (OH) maser emission at 1720 MHz in its direction. The observations reveal a $\gamma$-ray source in the direction of SNR G5.7-0.1, positioned nearby the bright $\gamma$-ray source SNR W28. We model the broadband emission (radio to $\gamma$-ray) using a one-zone model, and after considering scenarios in which the MeV-TeV sources originate from either $\pi^0$-decay or leptonic emission, conclude that a considerable component of the $\gamma$-ray emission comes from the $\pi^0$-decay channel. Finally, constraints were placed on the reported ambiguity of the SNR distance through X-ray column densities measurements made using XMM-Newton observations. We conclude G5.7-0.1 is a significant $\gamma$-ray source positioned at a distance of $\sim 3$ kpc with luminosity in the 0.2-200 GeV range of $L_{\gamma} \approx 7.4 \times 10^{34}$
### XMM-Newton Large Program on SN1006 - I: Methods and Initial Results of Spatially-Resolved Spectroscopy
Based on our newly developed methods and the XMM-Newton large program of SN1006, we extract and analyze the spectra from 3596 tessellated regions of this SNR each with 0.3-8 keV counts $>10^4$. For the first time, we map out multiple physical parameters, such as the temperature ($kT$), electron density ($n_e$), ionization parameter ($n_et$), ionization age ($t_{ion}$), metal abundances, as well as the radio-to-X-ray slope ($\alpha$) and cutoff frequency ($\nu_{cutoff}$) of the synchrotron emission. We construct probability distribution functions of $kT$ and $n_et$, and model them with several Gaussians, in order to characterize the average thermal and ionization states of such an extended source. We construct equivalent width (EW) maps based on continuum interpolation with the spectral model of each regions. We then compare the EW maps of OVII, OVIII, OVII K$\delta-\zeta$, Ne, Mg, SiXIII, SiXIV, and S lines constructed with this method to those constructed with linear interpolation. We further extract spectra from larger regions to confirm the features revealed by parameter and EW maps, which are often not directly detectable on X-ray intensity images. For example, O abundance is consistent with solar across the SNR, except for a low-abundance hole in the center. This "O Hole" has enhanced OVII K$\delta-\zeta$ and Fe emissions, indicating recently reverse shocked ejecta, but also has the highest $n_et$, indicating forward shocked ISM. Therefore, a multi-temperature model is needed to decompose these components. The asymmetric metal distributions suggest there is either an asymmetric explosion of the SN or an asymmetric distribution of the ISM.
### IKT 16: the first X-ray confirmed composite SNR in the SMC
Aims: IKT 16 is an X-ray and radio-faint supernova remnant (SNR) in the Small Magellanic Cloud (SMC). A detailed X-ray study of this SNR with XMM-Newton confirmed the presence of a hard X-ray source near its centre, indicating the detection of the first composite SNR in the SMC. With a dedicated Chandra observation we aim to resolve the point source and confirm its nature. We also acquire new ATCA observations of the source at 2.1 GHz with improved flux density estimates and resolution. Methods: We perform detailed spatial and spectral analysis of the source. With the highest resolution X-ray and radio image of the centre of the SNR available today, we resolve the source and confirm its pulsar wind nebula (PWN) nature. Further, we constrain the geometrical parameters of the PWN and perform spectral analysis for the point source and the PWN separately. We also test for the radial variations of the PWN spectrum and its possible east west asymmetry. Results: The X-ray source at the centre of IKT 16 can be resolved into a symmetrical elongated feature centering a point source, the putative pulsar. Spatial modeling indicates an extent of 5.2 arcsec of the feature with its axis inclined at 82 degree east from north, aligned with a larger radio feature consisting of two lobes almost symmetrical about the X-ray source. The picture is consistent with a PWN which has not yet collided with the reverse shock. The point source is about three times brighter than the PWN and has a hard spectrum of spectral index 1.1 compared to a value 2.2 for the PWN. This points to the presence of a pulsar dominated by non-thermal emission. The expected E_{dot} is ~ 10^37 erg s^-1 and spin period < 100 ms. However, the presence of a compact nebula unresolved by Chandra at the distance of the SMC cannot completely be ruled out.
### The Properties of the Progenitor Supernova, Pulsar Wind, and Neutron Star inside PWN G54.1+0.3
The evolution of a pulsar wind nebula (PWN) inside a supernova remnant (SNR) is sensitive to properties of the central neutron star, pulsar wind, progenitor supernova, and interstellar medium. These properties are both difficult to measure directly and critical for understanding the formation of neutron stars and their interaction with the surrounding medium. In this paper, we determine these properties for PWN G54.1+0.3 by fitting its observed properties with a model for the dynamical and radiative evolution of a PWN inside an SNR. Our modeling suggests that the progenitor of G54.1+0.3 was an isolated ~15-20 Solar Mass star which exploded inside a massive star cluster, creating a neutron star initially spinning with period ~30-80ms. We also find that >99.9% of the pulsar's rotational energy is injected into the PWN as relativistic electrons and positrons whose energy spectrum is well characterized by a broken power-law. Lastly, we propose future observations which can both test the validity of this model and better determine the properties of this source -- in particular, its distance and the initial spin period of the central pulsar.
### Multi-wavelength analysis of the Galactic supernova remnant MSH 11-61A
Due to its centrally bright X-ray morphology and limb brightened radio profile, MSH 11-61A (G290.1-0.8) is classified as a mixed morphology supernova remnant (SNR). H$\textsc{i}$ and CO observations determined that the SNR is interacting with molecular clouds found toward the north and southwest regions of the remnant. In this paper we report on the detection of $\gamma$-ray emission coincident with MSH 11-61A, using 70 months of data from the Large Area Telescope on board the \textit{Fermi Gamma-ray Space Telescope}. To investigate the origin of this emission, we perform broadband modelling of its non-thermal emission considering both leptonic and hadronic cases and concluding that the $\gamma$-ray emission is most likely hadronic in nature. Additionally we present our analysis of a 111 ks archival \textit{Suzaku} observation of this remnant. Our investigation shows that the X-ray emission from MSH 11-61A arises from shock-heated ejecta with the bulk of the X-ray emission arising from a recombining plasma, while the emission towards the east arises from an ionising plasma.
### The southern molecular environment of SNR G18.8+0.3
In a previous paper we have investigated the molecular environment towards the eastern border of the SNR G18.8+0.3. Continuing with the study of the surroundings of this SNR, in this work we focus on its southern border, which in the radio continuum emission shows a very peculiar morphology with a corrugated corner and a very flattened southern flank. We observed two regions towards the south of SNR G18.8+0.3 using the Atacama Submillimeter Telescope Experiment (ASTE) in the 12CO J=3-2. One of these regions was also surveyed in 13CO and C18O J=3-2. The angular and spectral resolution of these observations were 22", and 0.11 km/s. We compared the CO emission to 20 cm radio continuum maps obtain as part of the Multi-Array Galactic Plane Imaging Survey (MAGPIS) and 870 um dust emission extracted from the APEX Telescope Large Area Survey of the Galaxy. We discovered a molecular feature with a good morphological correspondence with the SNR's southernmost corner. In particular, there are indentations in the radio continuum map that are complemented by protrusions in the molecular CO image, strongly suggesting that the SNR shock is interacting with a molecular cloud. Towards this region we found that the 12CO peak is not correlated with the observed 13CO peaks, which are likely related to a nearby \hii~region. Regarding the most flattened border of SNR G18.8+0.3, where an interaction of the SNR with dense material was previously suggested, our 12CO J=3-2 map show no obvious indication that this is occurring.
### Supernova Feedback and the Hot Gas Filling Fraction of the Interstellar Medium
Supernovae are the most energetic among stellar feedback processes, and are crucial for regulating the interstellar medium (ISM) and launching galactic winds. We explore how supernova remnants (SNRs) create a multiphase medium by performing high resolution, 3D hydrodynamical simulations at various SN rates, $S$, and ISM average densities, $n$. We find that the evolution of a SNR in a self-consistently generated three-phase ISM is qualitatively different from that in a uniform or a two-phase warm/cold medium. By traveling faster and further in the cooling-inefficient hot phase, the spatial-temporal domain of a SNR is enlarged by $>10^{2.5}$ in a hot-dominated multiphase medium (HDMM) compared to the uniform case. We then examine the resultant ISM as we vary $n$ and $S$, finding that a steady state can only be achieved when the hot gas volume fraction \fvh $\lesssim 0.6\pm 0.1$. Above that, overlapping SNRs render connecting topology of the hot gas, and such a HDMM is subjected to thermal runaway with growing pressure and \fvh. Photoelectric heating (PEH) has a surprisingly strong impact on \fvh. For $n \gtrsim 3 cm^{-3}$, a reasonable PEH rate is able to suppress the ISM from undergoing thermal runaway. Overall, we determine that the critical SN rate for the onset of thermal runaway is roughly $S_{crit} = 200 (n/1cm^{-3})^k (E_{SN}/10^{51} erg)^{-1} kpc^{-3} Myr^{-1}$, where k=(1.2,2.7) for $n$ < 1 and >1 cm$^{-3}$, respectively. We present a fitting formula of the ISM pressure $P(n, S)$, which can be used as an effective equation of state in cosmological simulations. The observed velocities of OB stars imply that the core collapse SN are almost randomly located on scales $\lesssim$ 150 pc. Despite the 5 orders of magnitude span of $(n,S)$, the average Mach number shows very small variations: $M \approx 0.5\pm 0.2, 1.2\pm 0.3, 2.3\pm 0.9$ for the hot, warm and cold phases, respectively.
### Radio spectral characteristics of the supernova remnant Puppis A and nearby sources
This paper presents a new study of the spectral index distribution of the supernova remnant (SNR) Puppis A. The nature of field compact sources is also investigated according to the measured spectral indices. This work is based on new observations of Puppis A and its surroundings performed with the Australia Telescope Compact Array in two configurations using the Compact Array Broad-band Backend centered at 1.75 GHz. We find that the global spectral index of Puppis A is -0.563 +/- 0.013. Local variations have been detected, however this global index represents well the bulk of the SNR. At the SE, we found a pattern of parallel strips with a flat spectrum compatible with small-scale filaments, although not correlated in detail. The easternmost filament agrees with the idea that the SN shock front is interacting with an external cloud. There is no evidence of the previously suggested correlation between emissivity and spectral index. A number of compact features are proposed to be evolved clumps of ejecta based on their spectral indices, although dynamic measurements are needed to confirm this hypothesis. We estimate precise spectral indices for the five previously known field sources, two of which are found to be double (one of them, probably triple), and catalogue 40 new sources. In the light of these new determinations, the extragalactic nature previously accepted for some compact sources is now in doubt.
### Kepler's Supernova: An Overluminous Type Ia Event Interacting with a Massive Circumstellar Medium at a Very Late Phase
We have analyzed XMM-Newton, Chandra, and Suzaku observations of Kepler's supernova remnant (SNR) to investigate the properties of both the SN ejecta and the circumstellar medium (CSM). For comparison, we have also analyzed two similarly-aged, ejecta-dominated SNRs: Tycho's SNR, thought to be the remnant of a typical Type Ia SN, and SNR 0509-67.5 in the Large Magellanic Cloud, thought to be the remnant of an overluminous Type Ia SN. By simply comparing the X-ray spectra, we find that line intensity ratios of iron-group elements (IGE) to intermediate-mass elements (IME) for Kepler's SNR and SNR 0509-67.5 are much higher than those for Tycho's SNR. We therefore argue that Kepler is the product of an overluminous Type Ia SN. This inference is supported by our spectral modeling, which reveals the IGE and IME masses respectively to be ~0.95 M_sun and ~0.12 M_sun (Kepler's SNR), ~0.75 M_sun and ~0.34 M_sun (SNR 0509-67.5), and ~0.35 M_sun and ~0.70 M_sun (Tycho's SNR). We find that the CSM component in Kepler's SNR consists of tenuous diffuse gas (~0.3 M_sun) present throughout the entire remnant, plus dense knots (~0.035 M_sun). Both of these components have an elevated N abundance (N/H ~ 4 times the solar value), suggesting that they originate from CNO-processed material from the progenitor system. The mass of the diffuse CSM allows us to infer the pre-SN mass-loss rate to be ~1.5e-5 (V_w/10 km/s) M_sun/yr, in general agreement with results from recent hydrodynamical simulations. Since the dense knots have slow proper motions and relatively small ionization timescales, they were likely located a few pc away from the progenitor system. Therefore, we argue that Kepler's SN was an overluminous event that started to interact with massive CSM a few hundred years after the explosion. This supports the possible link between overluminous SNe and the so-called "Ia-CSM" SNe.
### Late-time Evolution of Composite Supernova Remnants: Deep Chandra Observations and Hydrodynamical Modeling of a Crushed Pulsar Wind Nebula in SNR G327.1-1.1
In an effort to better understand the evolution of composite supernova remnants (SNRs) and the eventual fate of relativistic particles injected by their pulsars, we present a multifaceted investigation of the interaction between a pulsar wind nebula (PWN) and its host SNR G327.1-1.1. Our 350 ks Chandra X-ray observations of SNR G327.1-1.1 reveal a highly complex morphology; a cometary structure resembling a bow shock, prong-like features extending into large arcs in the SNR interior, and thermal emission from the SNR shell. Spectral analysis of the non-thermal emission offers clues about the origin of the PWN structures, while enhanced abundances in the PWN region provide evidence for mixing of supernova ejecta with PWN material. The overall morphology and spectral properties of the SNR suggest that the PWN has undergone an asymmetric interaction with the SNR reverse shock (RS) that can occur as a result of a density gradient in the ambient medium and/or a moving pulsar that displaces the PWN from the center of the remnant. We present hydrodynamical simulations of G327.1-1.1 that show that its morphology and evolution can be described by a ~ 17,000 yr old composite SNR that expanded into a density gradient with an orientation perpendicular to the pulsar's motion. We also show that the RS/PWN interaction scenario can reproduce the broadband spectrum of the PWN from radio to gamma-ray wavelengths. The analysis and modeling presented in this work have important implications for our general understanding of the structure and evolution of composite SNRs.
### Study of TeV shell supernova remnants at gamma-ray energies
The breakthrough developments of Cherenkov telescopes in the last decade have led to angular resolution of 0.1{\deg} and an unprecedented sensitivity. This has allowed the current generation of Cherenkov telescopes to discover a population of supernova remnants (SNRs) radiating in very-high-energy (VHE, E>100 GeV) gamma-rays. A number of those VHE SNRs exhibit a shell-type morphology spatially coincident with the shock front of the SNR. The members of this VHE shell SNR club are RX J1713.7-3946, Vela Jr, RCW 86, SN 1006, and HESS J1731-347. The latter two objects have been poorly studied in high-energy (HE, 0.1<E<100 GeV) gamma-rays and need to be investigated in order to draw the global picture of this class of SNRs and constrain the characteristics of the underlying population of accelerated particles. Using 6 years of Fermi P7 reprocessed data, we studied the HE counterpart of the SNRs HESS J1731-347 and SN 1006. The two SNRs are not detected in the data and given that there is no hint of detection, we do not expect any detection in the coming years from the SNRs. However in both cases, we derived upper limits that significantly constrain the gamma-ray emission mechanism and can rule out a standard hadronic scenario with a confidence level > 5 sigma. With this Fermi analysis, we now have a complete view of the HE to VHE gamma-ray emission of TeV shell SNRs. All five sources have a hard HE photon index (<1.8) suggesting a common scenario where the bulk of the emission is produced by accelerated electrons radiating from radio to VHE gamma-rays through synchrotron and inverse Compton processes. In addition when correcting for the distance, all SNRs show a surprisingly similar gamma-ray luminosity supporting the idea of a common emission mechanism. While the gamma-ray emission is likely to be leptonic dominated, this does not rule out efficient hadron acceleration in those SNRs.
### Possible golden events for ringdown gravitational waves [Replacement]
There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting simple mean of the estimated range of the event rate, we have the expected rate of 500 ${\rm yr^{-1}}$. Then, the rates of events with SNR $>20$ and SNR $>50$ are 32 ${\rm yr^{-1}}$ and 2 ${\rm yr^{-1}}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.
### Possible golden events for ringdown gravitational waves [Replacement]
There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting simple mean of the estimated range of the event rate, we have the expected rate of 500 ${\rm yr^{-1}}$. Then, the rates of events with SNR $>20$ and SNR $>50$ are 32 ${\rm yr^{-1}}$ and 2 ${\rm yr^{-1}}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.
### Possible golden events for ringdown gravitational waves
There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting simple mean of the estimated range of the event rate, we have the expected rate of 500 ${\rm yr^{-1}}$. Then, the rates of events with SNR $>20$ and SNR $>50$ are 32 ${\rm yr^{-1}}$ and 2 ${\rm yr^{-1}}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.
### Possible golden events for ringdown gravitational waves [Cross-Listing]
There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting simple mean of the estimated range of the event rate, we have the expected rate of 500 ${\rm yr^{-1}}$. Then, the rates of events with SNR $>20$ and SNR $>50$ are 32 ${\rm yr^{-1}}$ and 2 ${\rm yr^{-1}}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.
### N49: the first robust discovery of a recombining plasma in an extra galactic supernova remnant
Recent discoveries of recombining plasmas (RPs) in supernova remnants (SNRs) have dramatically changed our understanding of SNR evolution. To date, a dozen of RP SNRs have been identified in the Galaxy. Here we present Suzaku deep observations of four SNRs in the Large Magellanic Cloud (LMC), N49, N49B, N23, and DEM L71, for accurate determination of their plasma state. Our uniform analysis reveals that only N49 is in the recombining state among them, which is the first robust discovery of a RP from an extra-galactic SNR. Given that RPs have been identified only in core-collapse SNRs, our result strongly suggests a massive star origin of this SNR. On the other hand, no clear evidence for a RP is confirmed in N23, from which detection of recombination lines and continua was previously claimed. Comparing the physical properties of the RP SNRs identified so far, we find that all of them are categorized into the "mixed-morphology" class and interacting with surrounding molecular clouds. This might be a key to solve formation mechanisms of the RPs. | 2016-07-30 18:43:22 | {"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": 0.7136867642402649, "perplexity": 2072.1080700908606}, "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-2016-30/segments/1469258937291.98/warc/CC-MAIN-20160723072857-00019-ip-10-185-27-174.ec2.internal.warc.gz"} |
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## Re: implementation plan for music streams
From: Erik Sandberg Subject: Re: implementation plan for music streams Date: Fri, 12 May 2006 08:47:20 +0200 User-agent: KMail/1.9.1
On Thursday 11 May 2006 00:54, Han-Wen Nienhuys wrote:
> 2006/5/10, Erik Sandberg <address@hidden>:
> > Citerar Han-Wen Nienhuys <address@hidden>:
> > > > Known issue: unfold-repeats will probably not work for percent
> > > I don't understand this. unfold-repeats is on the front end, we can
> > > just make it replace PercentRepeatMusic with UnfoldedRepeatMusic
> > > wholly; that should work, right?
> >
> > I implemented percent repeats in a way similar to tuplet brackets, i.e.
> > by sending a parallel event. One reason for this decision is that the
> > EventChord
> > iterator is where events are supposed to be reported.
>
> Yes, and that's what I disagree with. I agree you need to put in events for
> signaling information, but I oppose to inserting them in the parser. Can
> you change the code to make the iterators generate those events on the fly.
Hm, I guess the easiest/cleanest way would be to let the
percent-repeat-iterator create an implicit SequentialMusic around the music,
with the additional percent elements, and then to let process_music pretend
that this SequentialMusic expression is its 'element. That way, all
timekeeping can be outsourced to the Sequential_music_iterator, and the
percent-repeat-iterator can more-or-less be reduced to an override of
construct_children.
I also have two slightly related questions:
- In the best of worlds, should all events always be reported to bottom
contexts? I see no technical reasons why it would need to be that way, but
it's a nice convention and it requires little work.
- If answer is yes, then I'd like to suggest that Music_iterator::try_music
automatically should descend the iterator to a bottom context. That would
eliminate the parser's need to wrap expressions inside \context Bottom. I can
implement this when I've finished some more of the music stream refactorings.
> > > I don't understand. Why don't you send TupletSpanEvents (START, STOP)
> > >
> > > from the iterator? If you do that, you might even be able to scrap a
> > > lot of the hairy timekeeping logic in the engraver.
> >
> > The nice thing about my solution is that time-scaled-music-iterator can
> > be scrapped altogether. This could also be achieved with start/stop
> > events by expanding \times <mus> to
> > { TupletSpanStartEvent <mus> TupletSpanStopEvent }
> > but I guess there would be problems with nested tuplets (how to pair
> > START and
> > STOP events?)
>
> start and stop events are nested, just like parentheses. A stop stops the
> most recently started one.
ok, fair enough. Then would it be OK to let \times expand to a SequentialMusic
and drop the iterator, as I suggested? (I think there are essential
differences between this case and percent repeats, as the required
time-keeping in the iterator would be roughly equivalent to the current
time-keeping in Tuplet_engraver)
I'm also considering to change the engraver's tuplets_ member to a list or
stack, instead of vector.
--
Erik
reply via email to
[Prev in Thread] Current Thread [Next in Thread] | 2014-07-31 21:42:23 | {"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": 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": 0.7459720373153687, "perplexity": 6487.40718411899}, "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-2014-23/segments/1406510273676.54/warc/CC-MAIN-20140728011753-00403-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/question-assume-using-figure-values-r-c-unknown-assume-time-constant-circuit-10-ms-millise-q2158009 | For this question, assume that you are using the figure above but that the values of R and C are unknown. Assume the time constant for the circuit is 10 ms (milliseconds). If the new R is half of the original R and capacitor C is increased by 10 times, calculate the new time constant.
Using the exponential formula for a charging circuit and assuming an initial voltage of zero volts across the capacitor, calculate the voltage across the capacitor at:
a.)t = 1.5 seconds
b.)t = 2.5 seconds
## Want an answer?
### Get this answer with Chegg Study
Practice with similar questions
Q:
For this question, assume that you are using Figure 6.1 but that the value or R and C are unknown. Assume the time constant for the circuit is 10 ms (milliseconds). If the new R is half of the original R and capacitor C is increased by 10 times, calculate the new time constant. Using the exponential formula for a charging circuit and assuming aninitial voltage of zero volts across the capacitor, calculate the voltage across the capacitor at t= 1.5 seconds t= 2.5 seconds
A: See answer | 2016-07-23 09:22:01 | {"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": 0.817681610584259, "perplexity": 515.3872332691038}, "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-2016-30/segments/1469257821671.5/warc/CC-MAIN-20160723071021-00129-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://usa.cheenta.com/usamo-2016/ | Categories
# USAMO 2016
Day 1
Problem 1
Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.
Problem 2
Prove that for any positive integer $k,$ $\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}$ is an integer.
Problem 3
Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY = \angle CBY$ and $\overline{BE}\perp\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ = \angle BCZ$ and $\overline{CF}\perp\overline{AB}.$
Lines $\overleftrightarrow{I_B F}$ and $\overleftrightarrow{I_C E}$ meet at $P.$ Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.
Day 2
Problem 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$
Problem 5
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB},$ $Q\in\overline{AC},$ and $N, P\in\overline{BC}.$ Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}.$ Denote by $\ell$ the angle bisector of $\angle MSQ.$
Prove that $\overline{OI}$ is parallel to $\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$
Problem 6
Integers $n$ and $k$ are given, with $n\ge k\ge 2.$ You play the following game against an evil wizard.
The wizard has $2n$ cards; for each $i = 1, ..., n,$ there are two cards labeled $i.$ Initially, the wizard places all cards face down in a row, in unknown order.
You may repeatedly make moves of the following form: you point to any $k$ of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the $k$ chosen cards and turns them back face-down. Then, it is your turn again.
We say this game is $\textit{winnable}$ if there exist some positive integer $m$ and some strategy that is guaranteed to win in at most $m$ moves, no matter how the wizard responds.
For which values of $n$ and $k$ is the game winnable? | 2021-09-18 10:15:49 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 128, "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": 0.7130307555198669, "perplexity": 149.9010049758459}, "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/1631780056392.79/warc/CC-MAIN-20210918093220-20210918123220-00035.warc.gz"} |
https://www.physicsforums.com/threads/electric-field-and-boundary-conditions.282564/ | # Electric field and boundary conditions
1. Jan 2, 2009
### KFC
1. The problem statement, all variables and given/known data
One half of the region between the plates of a spherical capacitor of inner and outer radii a and b is filled with a linear isotropic dielectric of permittivity $$\epsilon_1$$ and the other half has permittivity $$\epsilon_2$$, as shown in the figure. If the inner plate has total charge +Q and the outer plate has total charge -Q, find the field everywhere in the sphere
2. The attempt at a solution
Well, this is an old problem and I know the solution of this problem. But it is quite confusing about the boundary condition and the form of the field.
1) Since all charges are located on outer and inner spherical surface, so there is no net charged found in the contact surface, which is illustrated with the normal direction in the figure. According to the boundary condition, we have
$$D_{2n} - D_{1n} = Q_{inc} = 0$$
and the electric field along the tangential direction always be continuous, that is
$$E_{2t} \equiv E_{1t}$$
I wonder what about the tangential direction of the displacement field? Are they continuous in general?
2) The solution manual tells that according to the boundary, it takes the form of the electric field as of radial-dependent
$$\vec{E} = \frac{A\vec{r}}{r^3}$$
I am really confuse about this: how can you tell the field is of this form by boundary condition? or how can you prove the field in this form satisfies all boundary conditions?
2. Jan 3, 2009
### Thaakisfox
Due to the boundary conditions, the electric field will be radially the same in the sphere, hence due to Gausses law we have:
$$\epsilon_0\epsilon_1 E 2r^2\pi + \epsilon_0\epsilon_2 E 2r^2\pi =Q$$
So from here we obtain:
$$E=\dfrac{Q}{2\pi\epsilon_0(\epsilon_1+\epsilon_2)}\dfrac{1}{r^2}$$
In vector form (since there is only a radial component of the electric field):
$$\vec{E}=\dfrac{Q}{2\pi\epsilon_0(\epsilon_1+\epsilon_2)}\dfrac{\vec{r}}{r^3}$$
I hope its clearer noe | 2016-10-27 17:09:02 | {"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": 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": 0.8045051693916321, "perplexity": 283.6487051203797}, "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-2016-44/segments/1476988721355.10/warc/CC-MAIN-20161020183841-00330-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41467-017-00219-x?error=cookies_not_supported&code=e2e3986d-e15b-4efa-bf9b-aaf3db9e8b83 | Article | Open
# Compositionally-distinct ultra-low velocity zones on Earth’s core-mantle boundary
• Nature Communicationsvolume 8, Article number: 177 (2017)
• doi:10.1038/s41467-017-00219-x
Accepted:
Published:
## Abstract
The Earth’s lowermost mantle large low velocity provinces are accompanied by small-scale ultralow velocity zones in localized regions on the core-mantle boundary. Large low velocity provinces are hypothesized to be caused by large-scale compositional heterogeneity (i.e., thermochemical piles). The origin of ultralow velocity zones, however, remains elusive. Here we perform three-dimensional geodynamical calculations to show that the current locations and shapes of ultralow velocity zones are related to their cause. We find that the hottest lowermost mantle regions are commonly located well within the interiors of thermochemical piles. In contrast, accumulations of ultradense compositionally distinct material occur as discontinuous patches along the margins of thermochemical piles and have asymmetrical cross-sectional shape. Furthermore, the lateral morphology of these patches provides insight into mantle flow directions and long-term stability. The global distribution and large variations of morphology of ultralow velocity zones validate a compositionally distinct origin for most ultralow velocity zones.
## Introduction
Ultralow velocity zones (ULVZs) are mapped as geographically isolated zones of seismic anomalies detected on the core-mantle boundary (CMB)1, 2 with a significant reduction of seismic velocity (up to 10% for P-wave and 30% for S-wave velocities) and sometimes increased density3, 4. Mapped as thin (5–40 km) and relatively small (e.g., hundreds of kilometer laterally, or less2, 4,5,6, but sometimes up to 1000 km long7, 8), ULVZs are more commonly found within or near the large low velocity provinces (LLVPs)9. For this reason, along with the predominance of the S-velocity (Vs) drops being up to three times that of P-velocity (Vp) reductions, the ultralow wave speeds have been attributed to partial melt due to being in the hottest lowermost mantle regions2, 4.
At odds with the solely partial melt hypothesis is that some seismic studies identify ULVZs well outside of the seismically observed LLVPs10,11,12,13 including beneath subduction regions9, where temperatures are assumed to be far lower than in the presumed upwelling regions of LLVPs. Furthermore, some ULVZs do not have Vs reduction substantially greater than their Vp reduction14. Hypotheses other than solely partial melt may thus be necessary. A number of hypotheses have been proposed, including iron-enriched (Mg,Fe)O15, 16, iron-enriched post-perovskite17 (recent geodynamic modeling results show that patches of post-perovskite can be temporarily stable within LLVPs18), subducted banded iron formations19, subducted oceanic crust20 or other slab-derived materials21, 22, and products of chemical reactions between the silicate mantle and Fe-rich core23, 24. While these possibilities (as well as partial melt) may all be viable, their relationship to deep mantle flow, especially in regards to being swept towards upwelling regions and their geometrical relationship to LLVPs remains unknown. Of particular interest is the thermochemical pile hypothesis to explain LLVPs, whereby dense basal material is swept into piles to explain the seismically observed LLVPs25,26,27. Two fundamental questions are: where are the highest temperatures inside LLVPs, and are they different from accumulation locations of any additional, ultradense material that may reside at the base of the mantle? Understanding the dynamics, destinations, and morphologies of ULVZs caused by a compositionally distinct vs. partial melt origin is necessary to provide a meaningful framework for the distribution of seismic observations. We thus carried out very high resolution, three-dimensional thermochemical numerical convection calculations to study the distribution and morphology of ULVZs.
Here we explore ULVZs attributed to ultradense, compositionally distinct material, as well as ULVZs attributed to melting in the hottest deep mantle. We find that the hottest lowermost mantle regions, where partial melting could occur to explain ULVZs, are located well within the interiors of thermochemical piles. In contrast, accumulation of ultradense compositionally distinct material occurs as discontinuous patches along the margins of thermochemical piles and have an asymmetrical cross-sectional shape. The origin of ULVZs, therefore, can be constrained from their locations and shapes. The global distribution and large variations of morphology of the seismologically observed ULVZs indicate a compositionally distinct origin for most ULVZs, and that Earth’s lowermost mantle contains small-scale compositional heterogeneities with elevated intrinsic density. ULVZs within LLVPs, however, might be explained by partial melting alone.
## Results
### Description of mantle convection models
Our reference model includes thermochemical piles, motivated by the multiple lines of evidence arguing a chemically distinct origin of LLVPs28,29,30,31,32,33,34,35. The conservation equations of mass, momentum and energy are solved using our modified version of the code CitcomCU36 in the Boussinesq approximation (Methods). We employ a Rayleigh number Ra = 9.8 × 106 for most cases (Supplementary Table 1, using mantle thickness as the length-scale for non-dimensionalization). A 50× viscosity increase is employed from the upper mantle to the lower mantle (Supplementary Fig. 1). The temperature-dependent part of the viscosity is expressed as η T = exp[A(0.6 − T)], where T is non-dimensional temperature, and we use a non-dimensional activation coefficient of A = 9.21 for most cases (Supplementary Table 1), leading to a 10000× viscosity range across the mantle due to changes in temperature. We employ a three-dimensional, partial-sphere geometry (Fig. 1a) in which the longitude and colatitude span 120°, and the dimensionless radius ranges from 0.55 to 1.0 (thus, from the CMB to the surface). We utilize 512, 512 and 128 elements in longitudinal, colatitudinal, and radial directions, respectively. The mesh is refined with depth resulting in a resolution of 5 km radially and ~14.5 km laterally near the CMB. All boundaries are free-slip, isothermal at top and bottom, and insulating along the sides. The models are heated both from below and internally with a non-dimensional heat production rate of H = 60 (using Earth’s radius as the length-scale for non-dimensionalization).
We developed a hybrid tracer scheme to track composition (Methods), that simultaneously employs both ratio and absolute tracing methods37. The background mantle and the thermochemical piles are modeled with ~710 million ratio tracers and the ultradense ULVZ material is modeled with ~ 50–110 million absolute tracers, depending on the volume of ULVZ material (Supplementary Table 1). The hybrid tracer method more efficiently computes the advection of multi-scale composition, including both large-scale thermochemical piles and much smaller-scale accumulations of ultradense materials. The intrinsic density anomaly (Δρ) of each compositional component is non-dimensionalized as compositional buoyancy number B. The effective intrinsic density of each element is calculated by averaging the densities of each component, leading to an “effective buoyancy ratio”, B eff. To construct an initial condition, we carry out a calculation with two compositional components (background mantle and thermochemical pile material) and we use the quasi-steady state temperature and composition field as initial condition for models in this study.
We perform 2 types of experiments, both of which include thermochemical piles to represent LLVPs. In the first set of experiments, we explore the positions and shapes of ULVZs caused by partial melting in the hottest mantle regions. In other words, we examine the morphology of the hottest lowermost mantle regions. In the second set of experiments, we explore the positions and shapes of ULVZs caused by the accumulation of the ultradense compositional component. We then examine the morphology of these accumulations.
### ULVZs caused purely by partial melting
We first examine the locations of ULVZs due to partial melting alone (Case 1). The amount of partial melting above the CMB is controlled by the solidus temperature, liquidus temperature, and the mantle temperature above the CMB. The solidus temperature and liquidus temperature of a synthetic sample with chondritic-type composition at CMB pressure were measured by previous mineral physics experiments to be ~4150 and ~4725 K38, respectively, and the solidus and liquidus temperatures for a natural fertile peridotite at CMB pressure were measured to be ~4180 and 5375 K, respectively39. However, the solidus temperature for a pyrolitic composition with ~400 p.p.m. H2O has been reported to be as low as ~3570 K40. Largely due to our limited knowledge about the lowermost mantle composition such as the amount of H2O, the solidus temperature and liquidus temperature near the CMB pressure are not well constrained. In addition, there is large uncertainty of the CMB temperature, which has been suggested to be in the range of from ~2500–2800 K to ~3300–4300 K41, and the temperature of the thermal boundary layer above the CMB is poorly constrained. Our geodynamic models are non-dimensionalized and therefore do not independently constrain the absolute value of dimensional temperature. To convert non-dimensional to dimension temperature requires a choice for CMB temperature, which is not well constrained by observations. Because of these uncertainties, it becomes impractical to determine the amount of partial melting above the CMB in our models by comparing the dimensional lowermost mantle temperature in our models with the solidus and liquidus temperature at the CMB pressure measured in previous mineral physics experiments. Nonetheless, if there are ULVZs above the CMB caused by partial melting alone, they most likely exist in the hottest regions in the lowermost mantle. We thus focus on examining the location of hottest regions in the lowermost mantle in Case 1.
Case 1 includes 2 compositions: background mantle and piles with a buoyancy number B p = 0.8 (or 3.6% denser than the background mantle if scaled using reference temperature and thermal expansivity as given in Supplementary Table 2). Figure 1 shows a snapshot at 218 Myr. In this study, the geological time is scaled by the transit time and we assume that one transit time (the time it takes for a slab to descend from surface to the CMB) equals to 60 Myrs42, 43. Figure 1a illustrates thermochemical piles (green) with mantle plumes (red) rising from cusps along their tops. Figure 1b shows the temperature field at 5 km height above the CMB, in which it is observed that the hottest 10% regions of the piles by area (marked by light gray contours of T = 0.999) occur within pile interiors, well inward from their edges (pile edges are outlined by cyan lines). The dimensional temperature for the hottest 10% regions at this depth is in the range of ~3600–4000 K, if dimensionalized with a reference potential temperature of ΔT = 2500 K (Supplementary Table 2), after adding an adiabatic temperature increase from the surface (with a temperature of 273 K) down to 5 km above the CMB with an adiabatic thermal gradient of 0.3–0.4 K km−1. Interestingly, these hottest regions have a temperature comparable to the solidus temperature of a pyrolitic or chondritic composition near the CMB pressure, depending on the H2O content in the lowermost mantle38,39,40. However, it needs to be emphasized that the dimensional temperature in our models depends on the choice of reference temperature for scaling. We thus focus on the location of hottest regions that are the best candidate locations for partial melting.
We plot the hottest regions in the cross-section shown in Fig. 1c, with zoom-ins shown in Fig. 1d, e. The hottest regions reside within and with some distance from the pile edges because of cooling of thermochemical pile margins by the cooler, surrounding non-pile mantle (Supplementary Note 1; Supplementary Fig. 2). We find the hottest regions occur well within the interior of thermochemical piles throughout the model run (Supplementary Movie 1). We compute the lateral distance of hottest regions from the closest edges of thermochemical piles throughout the model run. We exclude the hottest regions that are within 500 km from side boundaries of the model domain. We find that the distances between the hottest regions and the edges of thermochemical piles range from a minimum of 100’s km to over 1500 km, with a peak at around 500–1000 km (Fig. 2).
### ULVZs caused by ultradense material
A second set of experiments considers the dynamics and evolution of ultradense material as a cause of ULVZs (Case 2). These experiments have three compositions: background mantle, thermochemical piles, and a small volume of ultradense material with a buoyancy number of B u = 2.0 (or 9% denser than background mantle if scaled using reference parameters given in Supplementary Table 2). The ultradense material is initially introduced as a ubiquitous uniform layer in the lowermost 5 km of the mantle, and it quickly advects toward the pile edges, accumulating into discontinuous patches of varying size and shape (Supplementary Movie 2). Figure 3 shows a snapshot of this case at 227 Myr. Figure 3a displays the distribution of ultradense ULVZ material (red isosurfaces) underneath the thermochemical piles (partially transparent green isosurfaces). The accumulations vary in size from ~100 to ~1000 km across and ~ 5–100 km thick and have either rounded or linear map-view morphologies (discussed later). An interesting point to note is that the accumulations form into discontinuous patches, as opposed to ubiquitous, continuous ribbons along pile edges implied from 2D studies9. Figure 3b is a zoom-in of Fig. 3a that displays the effective buoyancy ratio in regions with ultradense material 5 km above the CMB, illustrating the heterogeneity of density within the accumulations, caused by stirring with the surrounding mantle. Figure 3c demonstrates that accumulations of ultradense material are typically quite thin, except for small regions within particularly large accumulations, where local heights may reach up to 100 km above the CMB. Figure 3d–f illustrates the variability in cross-sectional shape of the accumulations. The lateral width of the accumulations greatly varies from place to place, and the cross-sectional shape of the accumulations is asymmetrical, thicker on the side in contact with the background mantle. This asymmetrical shape is due to differential viscous coupling, as noted in a previous 2D study9.
Figure 4 shows the compositional field at 5 km above the CMB for a time sequence of snapshots for Case 2, illustrating the time-dependence of the distribution of ultradense material. At 121 Myr (Fig. 4a), two large patches of ultradense material (labeled U1 and U2) are located at the edge of the pile. At 160 Myr (Fig. 4b), U2 has been advected into a linear shape, whereas U1 has maintained its rounded shape. At 227 Myr (Fig. 4c), U2 has split into three parts: a remnant of U2 (still labeled as U2) migrated toward U1, another formed into a smaller accumulation with relatively rounder shape (labeled U3), and another had been entrained into the pile, up along its side, and back down again (U4). In Fig. 4c, the ultradense material in U2 and U4 has experienced higher degree of stirring with pile material, leading to a lower effective buoyancy ratio (i.e., effective intrinsic density) in these patches than U1 and U3. In general, we observe that regions of long-term, stable, horizontally convergent mantle flow produces longer-lived, rounded accumulations of ultradense material. In contrast, linear accumulations are the result of ultradense material on the move, toward a location of more-stable convergent flow. Thus, ULVZ shape can change over time scales as short as tens of Myr.
Similar to Case 1, we compute the lateral distance of regions with accumulations of ultradense material from the closest edges of thermochemical piles for Case 2 (Fig. 2). We compute the distances throughout the model run but we exclude the first 50 Myr for Case 2 when the initial global layer of ultradense material is advecting to the edges of piles. We also exclude the regions with accumulations of ultradense material that are within 500 km from side boundaries of the model domain. In contrast to the wide range of distances between hottest regions and pile edges, the compositionally distinct ultradense material generally accumulates along the edges of thermochemical piles. At depths of 40 and 68 km above the CMB, the majority of ultradense materials occurs within ~300 km from the pile edges (Fig. 2). At depths of 5 and 20 km above the CMB, the patches of ultradense material become much larger than at shallower depths (Fig. 3d), which leads to a significant amount of ultradense materials occurring between ~ 300–800 km from the pile edges. However, even at depths of 5 and 20 km above the CMB, the largest fraction of ultradense materials still occurs within ~300 km from pile edges.
### Results of other geodynamic models
One caveat is that we do not include viscous dissipation in our models, so we do not have shear heating in the piles, which we consider negligible given their low viscosities. However, it is not inconceivable that certain combinations of material properties could lead to viscous heating, and therefore possible partial melting in other parts of the pile as well, such as near the edges where flow is changing direction.
### Comparison with seismic observations of ULVZs
We show in Fig. 5a the seismic shear-wave tomography model S40RTS near the CMB46, with the edges of LLVPs marked by orange contours. We plot observations of ULVZs together with the edges of LLVPs in Fig. 5b. Here, we only select studies of ULVZs using core reflected waves (ScS, PcP, ScP), in which the locations of ULVZs have minimum uncertainties (in comparison to the core waves or long path diffracted waves). The lateral size of the ULVZs is computed based on 1/4 wavelength Fresnel zones of the CMB reflection location for the waves used in each study. A list of references for these ULVZ observations is provided in the Supplementary Table 3 and Supplementary References.
As shown in Fig. 5b and also summarized in the Fig. 1 of ref. 9, the ULVZs exhibit a variety of shapes and sizes, similar to the accumulations of ultradense material as labeled U1-4 in Fig. 4. For example, a larger-than-average, rounded ULVZ is observed near the north edges of the Pacific LLVP (Fig. 5b) and beneath Hawaii7, not unlike the large rounded U1 (Fig. 4); a linear shape ULVZ detected in the SW Pacific8 may be similar to the long linear U2 in Fig. 4b, c; and the ULVZs with small lateral-scale detected in many regions are analogous to the small U3. Our results show variable degrees of stirring of ultradense material with pile material (U2 and U4 compared to U1 and U3 in Fig. 4c), which may be analogous to the variable density increases observed in ULVZs3, 4. Our geodynamic modeling results also suggest that the accumulations of ultradense material are not ubiquitous along pile edges but form into discontinuous patches with variable morphology, demonstrating that not all LLVP margins are expected to contain ULVZs. This is supported by a recent detection of intermittent and unevenly distribution ULVZs at the northeastern margin of the Pacific LLVP47.
Similar to that shown in Fig. 2, we compute the closest lateral distances of observed ULVZs (as shown in Fig. 5b) to the edges of LLVPs along the CMB (Methods). Figure 5c shows that 55.5% of the computed ULVZ area occurs outside of the LLVPs (denoted with negative distance) and 44.5% ULVZ area occurs within the LLVPs (denoted with positive distance). We find that most ULVZs are in close proximity of LLVP margins (Fig. 5c). Outside of LLVPs, almost all the ULVZ area occurs within 800 km from the edges of LLVPs. Inside of LLVPs, we find that the ULVZ area decreases linearly with the increase of distance to LLVP edges, and there is no ULVZ area occurring more than 1200 km from the LLVP edges.
We calculated the distance of ULVZs to the edges of LLVPs for other five tomography models (Supplementary Fig. 11). The amount of ULVZ area outside and inside of LLVPs differs somewhat between models, since the locations of the LLVP edges slightly differ among models. However, the general conclusion holds that ULVZs are both outside and inside the LLVPs, with most ULVZ area occurring within ~800 km from LLVP edges of the tomographic models tested. The proximity of the observed ULVZs near LLVP edges is similar to the ultradense materials in our geodynamic models occurring near the edges of thermochemical piles (Fig. 2), suggesting a compositionally distinct component to ULVZs.
## Discussion
The seismically derived ULVZs studied here have variable lateral dimensions, morphologies, and locations, consistent with a compositionally distinct origin of ULVZs. For the Earth, the crystallization of basal magma oceanic may initially produce a thin layer of ultradense material on the CMB29, 48. The ultradense materials may be produced by the interaction between the core and the mantle and they may be produced at any location where the core and mantle interact23, 24. In addition, the subduction of slabs may bring some intrinsically dense materials to the lowermost mantle outside of the LLVPs19,20,21,22. Our experiments are geared toward understanding thermochemical convection at equilibrium conditions; however, our model setup also allows us to explore (in a limited manner) how ultradense material gets swept from the surrounding mantle to the edges of thermochemical piles. Because our initial condition consists of a thin, uniform ultradense layer ubiquitous along the CMB, the early times of the calculation exhibit the sweeping of this material toward the piles (Supplementary Movie 2; Supplementary Fig. 12). It demonstrates that any high-density compositional heterogeneity outside of piles is being advected toward the global upwelling regions (where the piles exist). Therefore, if ULVZs are caused by ultradense subduction remnants19,20,21,22 or core-mantle boundary reaction products23, 24, we expect to observe them outside of piles as they are being advected toward them. Note that if compositional ULVZs have a lower solidus than background mantle, they may also include partial melt.
Though our results suggest that ULVZs located outside or at the edges of LLVPs are compositionally distinct from their surroundings, a small number of ULVZs located well within LLVPs9 (Fig. 5b) may be caused solely by partial melting. Interestingly, partial melting within the LLVPs would likely alter composition49, perhaps producing a source of intrinsically dense heterogeneity with lower melting temperature49,50,51 that would continually advect toward LLVP edges.
## Methods
### Numerical modeling
We perform high-resolution three-dimensional calculations to investigate the morphology, distribution and dynamics of ULVZs by solving the following non-dimensional equations for conservation of mass, momentum, and energy under the Boussinesq approximation:
$∇⋅u=0$
(1)
$-∇P+∇⋅ η ϵ ° =ξRa T - B eff r$
(2)
$∂ T ∂ t + u ⋅ ∇ T= ∇ 2 T+H$
(3)
where, u is the velocity, P is the dynamic pressure, η is the viscosity, $ϵ °$ is the strain rate tensor, T is the temperature, r is the unit vector in radial direction, B eff is the effective buoyancy ratio (defined below). t is time, and H is internal heating. ξ = (R e/D)3 with R e as the Earth’s radius and D as the mantle thickness. Physical parameters in the above equations are all non-dimensional. The Eqs. (1)–(3) are solved using the CitcomCU code, which is available at https://geodynamics.org/cig/software/citcomcu/.
The thermal Rayleigh number Ra is defined as:
$Ra= ρ 0 g α 0 Δ T D 3 η 0 κ 0$
(4)
where ρ 0, α 0, ΔT, η 0, κ 0 are dimensional reference values of background mantle reference density, thermal expansivity, temperature difference between core-mantle boundary and surface, reference viscosity at temperature T = 0.6 (non-dimensional), and thermal diffusivity, respectively. g is dimensional gravitational acceleration.
The internal heating H is non-dimensionalized as:
$H= R e 2 κ 0 c P 0 Δ T H *$
(5)
where, $c P 0$ is heat capacity, H * is the dimensional heat production rate.
The buoyancy number for a compositional component (B i ) is defined as the ratio between intrinsic density anomaly and density anomaly due to thermal expansion:
$B i = Δ ρ i ρ 0 α 0 Δ T$
(6)
where, Δρ i is intrinsic density difference between an individual compositional component and the background mantle.
Similarly, the effective buoyancy ratio is defined as:
$B eff = Δ ρ el ρ 0 α 0 Δ T$
(7)
where Δρ el is effective intrinsic density anomaly on an element computed from the intrinsic density anomaly and the fraction of each compositional component in the element using the hybrid tracer method described below.
### Hybrid tracer method
In thermochemical geodynamical modeling, two methods are typically used to model the advection of compositional field: the ratio tracer method and absolute tracer method37.
In the absolute tracer method, the composition fraction (C i ) of each compositional component (except the background mantle) is proportional to the number of tracers per volume:
$C i = N i V 0 V$
(8)
where N i is the number of tracers for the ith compositional component in an element, V is the volume of the element and V 0 is a constant which equals to average volume per tracer for the ith compositional component. For background mantle, N i equals zero (i.e., no tracer in the element) and C i becomes zero. Thus, there is no need for additional tracers to simulate the background mantle. This becomes a big advantage when the volume of chemical heterogeneities is very small (e.g., the ULVZs), which could be efficiently simulated with a small amount of tracers.
For ratio tracer method, the background mantle is also represented by tracers. Usually, the density of the background mantle is the reference density and the buoyancy number for the background mantle equals zero. The compositional fraction (C i ) for each compositional component within an element is:
$C i = N i N$
(9)
where N i is the number of tracers in the element used to simulate the ith compositional component. N is the total number of tracers in that element.
The ratio tracer method is benchmarked, and compared with absolute tracer method in ref. 37. The ratio tracer method has several advantages over the absolute tracer method, such as minimal numerical diffusion and low entrainment. Thus, ratio tracer method is often used when dealing with large-scale chemical heterogeneities (i.e., LLVPs), because in this case the absolute tracer method also needs large amount of tracers and no longer has the advantage of modeling the compositional heterogeneities using less tracers.
In this study, our model is featured by both large-scale thermochemical piles and small-scale accumulations of ultradense material. We developed a hybrid tracer method which combines the advantages of ratio and absolute tracer method. Here, the background mantle and large scale compositional heterogeneities of piles are represented by ~710 million ratio tracers and the smaller scale accumulations of compositionally distinct ultradense material are simulated by ~52–110 million absolute traces (depending on the initial volume of ultradense material).
The effective intrinsic density anomaly (Δρ el) for each element in the computation domain contains two parts. One part is from background mantle and pile material which are modeled with ratio tracers, and is given by:
$Δ ρ el r =Δ ρ p C p +Δ ρ bg C bg$
(10)
where, Δρ p is the intrinsic density anomaly of pile material. C p is compositional fraction of pile material for the element which is calculated using Eq. (9). Δρ bg and C bg are the intrinsic density anomaly and compositional fraction for the background mantle for the element, respectively. The intrinsic density anomaly of the background mantle is zero, so Eq. (10) becomes:
$Δ ρ el r =Δ ρ p C p$
(11)
The other part of the effective intrinsic density anomaly on an element (Δρ el) is from ultradense (i.e., ULVZ) material which is modeled with absolute tracers, and is given as:
$Δ ρ el a =Δ ρ u C u$
(12)
where Δρ u is the intrinsic density anomaly of ultradense material. C u is compositional fraction of ultradense material for the element which is calculated using Eq. (8). We truncated C u at 1 to avoid unphysically settling of tracers37.
In the hybrid tracer method, the effective intrinsic density anomaly on an element of the computational domain (Δρ el) is given by:
$Δ ρ el =Δ ρ el a +Δ ρ el r 1 - C u$
(13)
or,
$Δ ρ el =Δ ρ u C u +Δ ρ p C p 1 - C u$
(14)
Notice that, for C u = 0 (element has no ultradense material), Δρ el is equivalently calculated using the ratio tracer method; for C u = 1 (element is saturated with ultradense material), Δρ el is equivalently calculated using the absolute tracer method.
The effective buoyancy ratio (B eff) on an element is related to the effective intrinsic density anomaly (Δρ el) on this element by:
$B eff =Δ ρ el ∕ ρ 0 α 0 Δ T$
(15)
### Core-reflected wave ULVZ studies
In this study, we survey ULVZ studies that utilized core-reflected energy waveform analyses, e.g., PcP, ScP, and ScS. These waves have the potential to detect ULVZ structure at the CMB reflection point location. This differs from studies using core waves, e.g., SPdKS which can have an uncertainty regarding mapping ULVZ structure at the core entrance or exit location of the path (similarly, PKP and PKKP have this ambiguity). Some Pdiff and Sdiff studies have evidence for ULVZs (e.g., refs. 7, 52). While these analyses indicate specific ULVZ locations, the long paths of the diffracted wave result in some uncertainty as to where along the path the ULVZ is located. For this reason, we investigate ULVZ proximity to LLVP edges with just the core-reflected data. The studies, regions, and wave type are given in Supplementary Table 3.
Each of the ULVZ Fresnel zones of the studies in Supplementary Table 3 was decimated onto a 0.5 deg by 0.5 deg grid, with the area computed for each cell. The minimum distance to the nearest LLVP boundary is computed for each cell, and the fraction of the total ULVZ area summed up as a function of that minimum distance. This is display in Fig. 5c, as well as in Supplementary Fig. 11 for six tomographic models. The models are S40RTS46, along with HMSL-S0653, S362ANI54, SEMUCB-WM155, SP12RTS56, and GyPsum57. The LLVP boundary is chosen to be the contour that surrounds 30% of the CMB by area that has the lowest shear wave speeds in the tomography model12. The results for all the tomographic models are similar in that there is a significant area percentage of ULVZs located outside the LLVPs.
### Data availability
The authors declare that all relevant data supporting the findings of this study are available within the article and its Supplementary Information file or available upon request. The code, CitcomCU, is available from https://geodynamics.org/cig/software/citcomcu/. The authors’ specific version of the code is available upon request.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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## Acknowledgements
We thank F. Deschamps, S. King and an anonymous reviwer for their constructive comments. The project is supported by NSF grants EAR-1045788, EAR-1401270 and EAR-1648817.
## Author information
### Affiliations
1. #### Arizona State University, School of Earth and Space Exploration, PO Box 871404, Tempe, AZ, 85287-1404, USA
• Mingming Li
• , Edward J. Garnero
• & Shule Yu
2. #### Michigan State University, Department of Earth and Environmental Sciences, Natural Science Building, East Lansing, MI, 48824, USA
• Allen K. McNamara
### Contributions
M.L., A.K.M. and E.J.G. contributed to conceiving the idea. M.L. carried out the numerical calculation. A.K.M. supervised the project. S.Y. and E.J.G. reviewed previous studies on ULVZs and digitized the location and size of ULVZs. All authors contributed to writing the paper.
### Competing interests
The authors declare no competing financial interests.
### Corresponding author
Correspondence to Mingming Li. | 2018-05-22 05:56:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "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": 0.7921090126037598, "perplexity": 5056.296045578141}, "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-2018-22/segments/1526794864626.37/warc/CC-MAIN-20180522053839-20180522073839-00284.warc.gz"} |
https://mtg.sg/site/volumetric-flask-uncertainty-dfd66c | $$u_R = \sqrt{\left( \frac {u_A} {A} \right)^2 +\left( \frac {u_B} {B} \right)^2}$$, $$\frac {u_R} {R} = k \times \frac {u_A} {A}$$. The concentration and uncertainty for Cu2+ is 7.820 mg/L ± 0.047 mg/L.
From Table $$\PageIndex{1}$$ the relative uncertainty in [H+] is, $\frac {u_R} {R} = 2.303 \times u_A = 2.303 \times 0.03 = 0.069 \nonumber$, The uncertainty in the concentration, therefore, is, $(1.91 \times 10^{-4} \text{ M}) \times (0.069) = 1.3 \times 10^{-5} \text{ M} \nonumber$. Solving for the uncertainty in kA gives its value as $$1.47 \times 10^{-3}$$ or ±0.0015 ppm–1. Suppose you have a range for one measurement, such as a pipet’s tolerance, and standard deviations for the other measurements. We also can use a propagation of uncertainty to help us decide how to improve an analytical method’s uncertainty. Show your work here. Hope this helps. Privacy In other words, the volume delivered by the 5-mL pipet is 5.00 mL. Missed the LibreFest?
From the discussion above, we reasonably expect that the total uncertainty is greater than ±0.000 mL and that it is less than ±0.012 mL. For a 100cm3 flask it is ± 0.1cm3. The dilution calculations for case (a) and case (b) are, $\text{case (a): 1.0 M } \times \frac {1.000 \text { mL}} {1000.0 \text { mL}} = 0.0010 \text{ M} \nonumber$, $\text{case (b): 1.0 M } \times \frac {20.00 \text { mL}} {1000.0 \text { mL}} \times \frac {25.00 \text{ mL}} {500.0 \text{mL}} = 0.0010 \text{ M} \nonumber$, Using tolerance values from Table 4.2.1, the relative uncertainty for case (a) is, $u_R = \sqrt{\left( \frac {0.006} {1.000} \right)^2 + \left( \frac {0.3} {1000.0} \right)^2} = 0.006 \nonumber$, and for case (b) the relative uncertainty is, $u_R = \sqrt{\left( \frac {0.03} {20.00} \right)^2 + \left( \frac {0.3} {1000} \right)^2 + \left( \frac {0.03} {25.00} \right)^2 + \left( \frac {0.2} {500.0} \right)^2} = 0.002 \nonumber$. The other will pipette 9 ml of the dilution liquid into the sample solution. Of course we must balance the smaller uncertainty for case (b) against the increased opportunity for introducing a determinate error when making two dilutions instead of just one dilution, as in case (a).
Now let’s look at a general case of x = fn(p,q,r,…) and we assume p, q, r,… can fluctuate randomly and be treated as independent variables. gives the analyte’s concentration as 126 ppm. J. S. Fritz and G. H. Schenk, Quantitative Analytical Chemistry, 3rd ed., Allyn & Bacon, Boston, 1974, p. 560, Sign in|Report Abuse|Print Page|Powered By Google Sites, 02. Suppose we want to decrease the percent uncertainty to no more than 0.8%. For example, to determine the mass of a penny we measure its mass twice—once to tare the balance at 0.000 g and once to measure the penny’s mass. Adding the uncertainty for the first delivery to that of the second delivery assumes that with each use the indeterminate error is in the same direction and is as large as possible. It depends on the size of the volumetric flask. As shown in the following example, we can use the tolerance values for volumetric glassware to determine the optimum dilution strategy [Lam, R. B.; Isenhour, T. L. Anal. The overall uncertainty in the final concentration—and, therefore, the best option for the dilution—depends on the uncertainty of the volumetric pipets and volumetric flasks. And while you may assume the accuracy of the 250 mL volumetric flask and the 25 mL volumetric flask would be the same, often the value will be stamped on the flask.
We also can accomplish the same dilution in two steps using a 50-mL pipet and 100-mL volumetric flask for the first dilution, and a 10-mL pipet and a 50-mL volumetric flask for the second dilution. we clearly underestimate the total uncertainty.
If we dispense 20 mL using a 10-mL Class A pipet, what is the total volume dispensed and what is the uncertainty in this volume? Using the Vernier/Ocean Optics Spectrometer, 08. $Q = (0.15 \text{ A}) \times (120 \text{ s}) = 18 \text{ C} \nonumber$, Since charge is the product of current and time, the relative uncertainty in the charge is, $u_R = \sqrt{\left( \frac {0.01} {0.15} \right)^2 + \left( \frac {1} {120} \right)^2} = 0.0672 \nonumber$, $u_R = R \times 0.0672 = (18 \text{ C}) \times (0.0672) = 1.2 \text{ C} \nonumber$.
If the pH of a solution is 3.72 with an absolute uncertainty of ±0.03, what is the [H+] and its uncertainty? where, T is the transmittance, Po is the power of radiation as emitted from the light source and P is its power after it passes through the solution. It is easy to appreciate that combining uncertainties in this way overestimates the total uncertainty. For example, if the result is given by the equation, $\frac {u_R} {R} \sqrt{\left( \frac {u_A} {A} \right)^2 + \left( \frac {u_B} {B} \right)^2 + \left( \frac {u_C} {C} \right)^2} \label{4.2}$, The quantity of charge, Q, in coulombs that passes through an electrical circuit is. In Example $$\PageIndex{3}$$, for instance, we calculated an analyte’s concentration as 126 ppm ± 2 ppm, which is a percent uncertainty of 1.6%.
The numerator, therefore, is 23.41 ± 0.028. Use the mass of green crystals from Data Table III to find the mass percent iron in the green crystals. The concentration of iron diluted solution is the concentration of iron in the 100-mL volumetric flask. When verified, record your value on the blackboard. What is the absorbance if Po is $$3.80 \times 10^2$$ and P is $$1.50 \times 10^2$$? What is the analyte’s concentration, CA, and its uncertainty if Stotal is 24.37 ± 0.02, Smb is 0.96 ± 0.02, and kA is $$0.186 \pm 0.003 \text{ ppm}^{-1}$$?
As a first guess, we might simply add together the volume and the maximum uncertainty for each delivery; thus, (9.992 mL + 9.992 mL) ± (0.006 mL + 0.006 mL) = 19.984 ± 0.012 mL. Table $$\PageIndex{1}$$ provides equations for propagating uncertainty for some of these function where A and B are independent measurements and where k is a constant whose value has no uncertainty. Uncertainty Structure of Dilution Volumetric Flask with Pipette and Yuzuru HAYASHIt and Rieko MATSUDA National ... a 10-ml volumetric flask and add dilution liquid. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. There is a …
Since the relative uncertainty for case (b) is less than that for case (a), the two-step dilution provides the smallest overall uncertainty. The spool’s initial weight is 74.2991 g and its final weight is 73.3216 g. You place the sample of wire in a 500-mL volumetric flask, dissolve it in 10 mL of HNO3, and dilute to volume. All the iron in the 100-mL flask came from a 5.00-mL sample that was removed from the 250-mL volumetric flask. When a current of 0.15 A ± 0.01 A passes through the circuit for 120 s ± 1 s, what is the total charge and its uncertainty?
In other words, the volume contained in the 100-mL volumetric flask is 100.0 mL and the volume contained in the 250-mL volumetric flask is 250.0 mL. If the uncertainty in measuring Po and P is 15, what is the uncertainty in the absorbance? $u_R = \sqrt{(0.02)^2 + (0.02)^2} = 0.028 \nonumber$. | 2022-05-21 08:20:57 | {"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": 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": 0.7445187568664551, "perplexity": 1501.3766799017228}, "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-2022-21/segments/1652662539049.32/warc/CC-MAIN-20220521080921-20220521110921-00714.warc.gz"} |
http://condensedconcepts.blogspot.com/2015/04/is-quantum-entanglement-really-needed.html | ## Friday, April 24, 2015
### Is quantum entanglement really needed?
On tuesday at UQ Carl Caves gave a Quantum Science Seminar "Quantum metrology meets Quantum Information Science".
One side point he made was that just because quantum entanglement is glamorous and beloved by luxury journals does not mean that you actually always need it to optimise any and every task. A specific example is in this paper which states:
The Heisenberg limit is thus achieved without any entanglement between the arms of the interferometer. In fact, Jiang, Lang, and Caves [4] showed that the state ψinoptis the only nonclassical product state, i.e., not a coherent state, that produces no modal entanglement after a beam splitter. These results indicate that, as in Ref. [18], modal entanglement is not a crucial resource for quantum-enhanced interferometry. | 2018-06-25 17:06:06 | {"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": 0.8958264589309692, "perplexity": 2314.1443554671578}, "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-2018-26/segments/1529267868237.89/warc/CC-MAIN-20180625170045-20180625190045-00541.warc.gz"} |
https://leanprover-community.github.io/archive/stream/113488-general/topic/example.20(x.20.3A.20.E2.84.95).20.3A.20.C2.AC.20(2.20.2B.20x.20.3D.200).20.3A.3D.20sorry.html | ## Stream: general
### Topic: example (x : ℕ) : ¬ (2 + x = 0) := sorry
#### Kevin Buzzard (Apr 04 2018 at 21:15):
It is forever taking me 3 lines to prove stuff like this. What's a slick way of doing this quickly?
#### Mario Carneiro (Apr 04 2018 at 21:17):
rw add_comm; apply succ_ne_zero
#### Kenny Lau (Apr 04 2018 at 22:19):
example (x : ℕ) : ¬ (2 + x = 0) :=
by cases x; apply nat.no_confusion
#### Kevin Buzzard (Apr 04 2018 at 22:19):
So you (Mario) decided to do it in tactic mode. Looking at this example I realise I didn't really understand what apply actually does.
#### Kenny Lau (Apr 04 2018 at 22:20):
oh hey our solutions have exactly the same length
he opened nat!
oh right
so i win
in some sense
#### Kenny Lau (Apr 04 2018 at 22:20):
well i was sleeping
#### Kevin Buzzard (Apr 04 2018 at 22:20):
solutions aren't totally ordered
#### Kevin Buzzard (Apr 04 2018 at 22:20):
you always win if you get to choose the ordering
#### Kevin Buzzard (Apr 04 2018 at 22:21):
Let me think about Mario's solution.
#### Kenny Lau (Apr 04 2018 at 22:21):
apply [xxx], where [xxx] : A -> B -> C -> D attempts to match D with the goal, and set A, B, C as three new goals if necessary
#### Kevin Buzzard (Apr 04 2018 at 22:21):
x + 2 unfolds to x + bit0 1 which unfolds to x + (1 + 1)
#### Kevin Buzzard (Apr 04 2018 at 22:22):
and I can't match this with nat.succ anything
#### Kevin Buzzard (Apr 04 2018 at 22:22):
oh I have to unfold more
#### Kenny Lau (Apr 04 2018 at 22:22):
example (x : ℕ) : x + 2 = nat.succ (x + 1) := rfl
#### Kevin Buzzard (Apr 04 2018 at 22:22):
x + succ 1, succ (x + 1)
right
#### Kevin Buzzard (Apr 04 2018 at 22:23):
but there's still more to do
how so
#### Kevin Buzzard (Apr 04 2018 at 22:23):
because apply is not exact
#### Kevin Buzzard (Apr 04 2018 at 22:23):
and this is not exact
#### Kevin Buzzard (Apr 04 2018 at 22:24):
the goal is f(succ x) and we solve it with forall n, f(n)
#### Kenny Lau (Apr 04 2018 at 22:24):
example (x : ℕ) : ¬ (2 + x = 0) :=
by rw add_comm; exact nat.succ_ne_zero (x+1)
#### Kenny Lau (Apr 04 2018 at 22:24):
apply automatically fills out the arguments for you
#### Kevin Buzzard (Apr 04 2018 at 22:24):
I've seen it not automatically fill out the arguments for me
#### Kenny Lau (Apr 04 2018 at 22:25):
well I can't say anything if you don't have an example
#### Kevin Buzzard (Apr 04 2018 at 22:25):
yeah and I can't read the definition of apply because it's in meta land
#### Kevin Buzzard (Apr 04 2018 at 22:26):
I guess I've sometimes tried to apply to partially close a goal and it's failed
#### Kevin Buzzard (Apr 04 2018 at 22:26):
I guess I just have to read the docstring more carefully.
#### Kenny Lau (Apr 04 2018 at 22:27):
/--
The apply tactic tries to match the current goal against the conclusion of the type of term. The argument term should be a term well-formed in the local context of the main goal. If it succeeds, then the tactic returns as many subgoals as the number of premises that have not been fixed by type inference or type class resolution. Non-dependent premises are added before dependent ones.
The apply tactic uses higher-order pattern matching, type class resolution, and first-order unification with dependent types.
-/
#### Mario Carneiro (Apr 05 2018 at 01:26):
I used apply in that example simply as a shorthand for exact (or refine nonterminally), it saves me a few underscores at the end. It doesn't always work, but I think it works here.
#### Nicholas Scheel (Apr 05 2018 at 03:31):
what about rw add_comm; from dec_trivial?
does it work?
you win
#### Scott Morrison (Apr 05 2018 at 03:40):
simp; from dec_trivial
ok you win
#### Scott Morrison (Apr 05 2018 at 03:40):
Or import lean-tidy and: obviously.
Last updated: May 18 2021 at 15:14 UTC | 2021-05-18 16:21:37 | {"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": 0.29173681139945984, "perplexity": 5492.363088257539}, "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-2021-21/segments/1620243991288.0/warc/CC-MAIN-20210518160705-20210518190705-00483.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/p-if-area-square-same-area-circle-then-ratio-their-perimeters-terms-area-of-circle_63974 | # P If the Area of a Square is Same as the Area of a Circle, Then the Ratio of Their Perimeters, in Terms of π, is - Mathematics
MCQ
If the area of a square is same as the area of a circle, then the ratio of their perimeters, in terms of π, is
#### Options
• π :$\sqrt{3}$
• 2 : $\sqrt{\pi}$
• 3 :$\pi$
• $\pi : \sqrt{2}$
#### Solution
We have given that area of a circle of radius r is equal to the area of a square of side a.
∴ pir^2=a^2
∴ a=sqrtpir
We have to find the ratio of the perimeters of circle and square.
∴ "perimeter of circle"/"Perimeter oof square"=(2pi r)/(4a) ..................(1)
Now we will substitute a=sqrtpir in equation (1)
∴ "perimeter of circle"/"Perimeter oof square"=(2pi r)/(4sqrtr)
∴ "perimeter of circle"/"Perimeter oof square"=pi/(2sqrtpi)
∴ "perimeter of circle"/"Perimeter oof square"=sqrtpi/2
Therefore, ratio of their perimeters is sqrtpi:2
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 10 Maths
Chapter 13 Areas Related to Circles
Q 16 | Page 70 | 2021-04-15 12:04:58 | {"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": 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": 0.7046510577201843, "perplexity": 3942.7985765192852}, "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/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00228.warc.gz"} |
https://byjus.com/maths/perfect-numbers/ | # Perfect Numbers
It is not known when Perfect Numbers were first discovered, or when they were studied. However, it is thought that they may even have been known to the Egyptians, and may have even been known before. Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers (O’Connor and Robertson, 2004).
The Pythagoreans found the number 6 interesting (more for its mystical and numerological properties than for any mathematical significance), as it is the sum of its proper factors, i.e. 6 = 1 + 2 + 3 This is the smallest Perfect Number, the next being 28 (Burton, 1980). These two numbers also had religious significance ascribed to them, as 6 is the number of days it took the Christian God to create the world, and 28 is the number of days in a Lunar Cycle.
ST Augustine even went as far to say Six is a number perfect in itself, and not because god created all things in 6 days; rather the converse is true. God created all things in 6 days because the number is perfect. This he wrote in the City of God (cited in Ellis, 2004).
Though the Pythagoreans were interested in the occult properties of Perfect Numbers, they did little of mathematical significance with them. It was around 300 BC, when Euclid wrote his Elements that the first real result was made. Although Euclid concentrated on Geometry, many number theory results can be found in his text (Burton, 1980).
We shall consider Euclid’s result in a moment, but first, let’s define Perfect Numbers more properly. There are numerous ways to define Perfect Numbers, the early definitions being given in terms of aliquot parts. This author defines Perfect Numbers as: A Perfect Number n, is a positive integer which is equal to the sum of its factors, excluding n itself.
Definition
A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128.
Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by,
N=2p-1(2p -1) where p is a prime for which 2p -1 is a prime number.
That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2p -1 is a prime. Undoubtedly Mersenne was familiar with Euclid’s book in coming up with his primes.
The following gives a table of the first nine Mersenne Primes and Perfect Numbers
Prime, p Mersenne Prime, 2p -1 Perfect Number, 2p-1(2p -1) 2 3 6 3 7 28 5 31 496 7 127 8128 13 8191 33550336 17 131071 8589869056 19 524287 137438691328 31 2147483647 2305843008139952128 61 2305843009213693951 2658455991569831744654692615953842176
Problem: Verify that 28 is a perfect number
Problem 1.2 Verify in the case 18 = 2 · 3 2 = p k q l that the sum σ(n) of all divisors satisfies the formula
$\sigma (n)\;=(1+P+P^{2}+…P^{k})(1+q+q^{2}+…q^{l})$
#### Practise This Question
Rahul divides a circular disc of radius 7 cm in two equal parts. What is the perimeter of each semi- circular shaped disc. | 2019-06-24 10:30:56 | {"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": 0.6367713809013367, "perplexity": 620.2249940282603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-26/segments/1560627999298.86/warc/CC-MAIN-20190624084256-20190624110256-00212.warc.gz"} |
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http://todaynumerically.blogspot.com/2013/03/saturday-16-march-2013.html | ## Saturday, 16 March 2013
### SATURDAY, 16 MARCH 2013
Today is the $75^{th}$ day of the year.
$75 = 3 \times 5^2$
According to A036378 there are $75$ primes between $2^9 = 512$ and $2^{10} = 1024$
The $75$ primes are:
________ ________ ________
$1) 521$$2) 523$$3) 541$
$4) 547$$5) 557$$6) 563$
$7) 569$$8) 571$$9) 577$
$10) 587$$11) 593$$12) 599$
$13) 601$$14) 607$$15) 613$
$16) 617$$17) 619$$18) 631$
$19) 641$$20) 643$$21) 647$
$22) 653$$23) 659$$24) 661$
$25) 673$$26) 677$$27) 683$
$28) 691$$29) 701$$30) 709$
$31) 719$$32) 727$$33) 733$
$34) 739$$35) 743$$36) 751$
$37) 757$$38) 761$$39) 769$
$40) 773$$41) 787$$42) 797$
$43) 809$$44) 811$$45) 821$
$46) 823$$47) 827$$48) 829$
$49) 839$$50) 853$$51) 857$
$52) 859$$53) 863$$54) 877$
$55) 881$$56) 883$$57) 887$
$58) 907$$59) 911$$60) 919$
$61) 929$$62) 937$$63) 941$
$64) 947$$65) 953$$66) 967$
$67) 971$$68) 977$$69) 983$
$70) 991$$71) 997$$72) 1009$
$73) 1013$$74) 1019$$75) 1021$
Consider writing down a sequence using the following rules
1) Write down the first odd non-negative integer, $1$
2) Write down the next two even numbers, $2, 4$
3) Write down the next three odd numbers, $5, 7, 9$
4) Write down the next four even numbers $10, 12, 14, 16$
5) Write down the next five odd numbers $17, 19, 21, 23, 25$
6) Well, you get the idea.
The first $47$ members of this sequence looks like this:
$1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81$
As you can see $75$ is the $42^{nd}$ member, of what is known as the Connell Sequence, see A001614.
The formula for this sequence is
$a(n) = 2n - \lfloor \frac {1 + \sqrt {8n - 7} } {2} \rfloor$ where $\lfloor \rfloor$ indicates the floor function.
Thus we can calculate the $75^{th}$ member of this function:
$a(75) = (2 \times 75) - \lfloor \frac {1 + \sqrt {(8 \times 75) - 7} } {2} \rfloor$
$a(75) = 150 - \lfloor \frac {1 + \sqrt {600 - 7} } {2} \rfloor$
$a(75) = 150 - \lfloor \frac {1 + \sqrt {593} } {2} \rfloor$
$a(75) = 150 - \lfloor \frac {1 + 24.35159 } {2} \rfloor$
$a(75) = 150 - \lfloor \frac {25.35159 } {2} \rfloor$
$a(75) = 150 - \lfloor 12.67579 \rfloor$
$a(75) = 150 - 12$
$\underline {\underline {a(75) = 138}}$
As an aside note that the number at the end of each sub-sequence above is the square of the index of the sub-sequence, i.e the $5^{th}$ sub-sequence ends in $25$. | 2017-10-17 01:51:03 | {"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": 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": 0.9502701759338379, "perplexity": 3369.000859102733}, "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/1508187820556.7/warc/CC-MAIN-20171017013608-20171017033608-00287.warc.gz"} |
https://johncarlosbaez.wordpress.com/category/physics/page/2/ | ## Information Geometry (Part 18)
5 August, 2021
Last time I sketched how two related forms of geometry, symplectic and contact geometry, show up in thermodynamics. Today I want to explain how they show up in probability theory.
For some reason I haven’t seen much discussion of this! But people should have looked into this. After all, statistical mechanics explains thermodynamics in terms of probability theory, so if some mathematical structure shows up in thermodynamics it should appear in statistical mechanics… and thus ultimately in probability theory.
I just figured out how this works for symplectic and contact geometry.
Suppose a system has $n$ possible states. We’ll call these microstates, following the tradition in statistical mechanics. If you don’t know what ‘microstate’ means, don’t worry about it! But the rough idea is that if you have a macroscopic system like a rock, the precise details of what its atoms are doing are described by a microstate, and many different microstates could be indistinguishable unless you look very carefully.
We’ll call the microstates $1, 2, \dots, n.$ So, if you don’t want to think about physics, when I say microstate I’ll just mean an integer from 1 to n.
Next, a probability distribution $q$ assigns a real number $q_i$ to each microstate, and these numbers must sum to 1 and be nonnegative. So, we have $q \in \mathbb{R}^n,$ though not every vector in $\mathbb{R}^n$ is a probability distribution.
I’m sure you’re wondering why I’m using $q$ rather than $p$ to stand for an observable instead of a probability distribution. Am I just trying to confuse you?
No: I’m trying to set up an analogy to physics!
Last time I introduced symplectic geometry using classical mechanics. The most important example of a symplectic manifold is the cotangent bundle $T^\ast Q$ of a manifold $Q.$ A point of $T^\ast Q$ is a pair $(q,p)$ consisting of a point $q \in Q$ and a cotangent vector $p \in T^\ast_q Q.$ In classical mechanics the point $q$ describes the position of some physical system, while $p$ describes its momentum.
So, I’m going to set up an analogy like this:
Classical Mechanics Probability Theory $q$ position probability distribution $p$ momentum ???
But what is to momentum as probability is to position?
A big clue is the appearance of symplectic geometry in thermodynamics, which I also outlined last time. We can use this to get some intuition about the analogue of momentum in probability theory.
In thermodynamics, a system has a manifold $Q$ of states. (These are not the ‘microstates’ I mentioned before: we’ll see the relation later.) There is a function
$f \colon Q \to \mathbb{R}$
describing the entropy of the system as a function of its state. There is a law of thermodynamics saying that
$p = (df)_q$
This equation picks out a submanifold of $T^\ast Q,$ namely
$\Lambda = \{(q,p) \in T^\ast Q : \; p = (df)_q \}$
Moreover this submanifold is Lagrangian: the symplectic structure $\omega$ vanishes when restricted to it:
$\displaystyle{ \omega |_\Lambda = 0 }$
This is very beautiful, but it goes by so fast you might almost miss it! So let’s clutter it up a bit with coordinates. We often use local coordinates on $Q$ and describe a point $q \in Q$ using these coordinates, getting a point
$(q_1, \dots, q_n) \in \mathbb{R}^n$
They give rise to local coordinates $q_1, \dots, q_n, p_1, \dots, p_n$ on the cotangent bundle $T^\ast Q.$ The $q_i$ are called extensive variables, because they are typically things that you can measure only by totalling up something over the whole system, like the energy or volume of a cylinder of gas. The $p_i$ are called intensive variables, because they are typically things that you can measure locally at any point, like temperature or pressure.
In these local coordinates, the symplectic structure on $T^\ast Q$ is the 2-form given by
$\omega = dp_1 \wedge dq_1 + \cdots + dp_n \wedge dq_n$
The equation
$p = (df)_q$
serves as a law of physics that determines the intensive variables given the extensive ones when our system is in thermodynamic equilibrium. Written out using coordinates, this law says
$\displaystyle{ p_i = \frac{\partial f}{\partial q_i} }$
It looks pretty bland here, but in fact it gives formulas for the temperature and pressure of a gas, and many other useful formulas in thermodynamics.
Now we are ready to see how all this plays out in probability theory! We’ll get an analogy like this, which goes hand-in-hand with our earlier one:
Thermodynamics Probability Theory $q$ extensive variables probability distribution $p$ intensive variables ???
This analogy is clearer than the last, because statistical mechanics reveals that the extensive variables in thermodynamics are really just summaries of probability distributions on microstates. Furthermore, both thermodynamics and probability theory have a concept of entropy.
So, let’s take our manifold $Q$ to consist of probability distributions on the set of microstates I was talking about before: the set $\{1, \dots, n\}.$ Actually, let’s use nowhere vanishing probability distributions:
$\displaystyle{ Q = \{ q \in \mathbb{R}^n : \; q_i > 0, \; \sum_{i=1}^n q_i = 1 \} }$
I’m requiring $q_i > 0$ to ensure $Q$ is a manifold, and also to make sure $f$ is differentiable: it ceases to be differentiable when one of the probabilities $q_i$ hits zero.
Since $Q$ is a manifold, its cotangent bundle is a symplectic manifold $T^\ast Q.$ And here’s the good news: we have a god-given entropy function
$f \colon Q \to \mathbb{R}$
namely the Shannon entropy
$\displaystyle{ f(q) = - \sum_{i = 1}^n q_i \ln q_i }$
So, everything I just described about thermodynamics works in the setting of plain old probability theory! Starting from our manifold $Q$ and the entropy function, we get all the rest, leading up to the Lagrangian submanifold
$\Lambda = \{(q,p) \in T^\ast Q : \; p = (df)_q \}$
that describes the relation between extensive and intensive variables.
For computations it helps to pick coordinates on $Q.$ Since the probabilities $q_1, \dots, q_n$ sum to 1, they aren’t independent coordinates on $Q.$ So, we can either pick all but one of them as coordinates, or learn how to deal with non-independent coordinates, which are already completely standard in projective geometry. Let’s do the former, just to keep things simple.
These coordinates on $Q$ give rise in the usual way to coordinates $q_i$ and $p_i$ on the cotangent bundle $T^\ast Q.$ These play the role of extensive and intensive variables, respectively, and it should be very interesting to impose the equation
$\displaystyle{ p_i = \frac{\partial f}{\partial q_i} }$
where $f$ is the Shannon entropy. This picks out a Lagrangian submanifold $\Lambda \subseteq T^\ast Q.$
So, the question becomes: what does this mean? If this formula gives the analogue of momentum for probability theory, what does this analogue of momentum mean?
Here’s a preliminary answer: $p_i$ says how fast entropy increases as we increase the probability $q_i$ that our system is in the ith microstate. So if we think of nature as ‘wanting’ to maximize entropy, the quantity $p_i$ says how eager it is to increase the probability $q_i.$
Indeed, you can think of $p_i$ as a bit like pressure—one of the most famous intensive quantities in thermodynamics. A gas ‘wants’ to expand, and its pressure says precisely how eager it is to expand. Similarly, a probability distribution ‘wants’ to flatten out, to maximize entropy, and $p_i$ says how eager it is to increase the probability $q_i$ in order to do this.
But what can we do with this concept? And what does symplectic geometry do for probability theory?
I will start tackling these questions next time.
One thing I’ll show is that when we reduce thermodynamics to probability theory using the ideas of statistical mechanics, the appearance of symplectic geometry in thermodynamics follows from its appearance in probability theory.
Another thing I want to investigate is how other geometrical structures on the space of probability distributions, like the Fisher information metric, interact with the symplectic structure on its cotangent bundle. This will integrate symplectic geometry and information geometry.
I also want to bring contact geometry into the picture. It’s already easy to see from our work last time how this should go. We treat the entropy $S$ as an independent variable, and replace $T^\ast Q$ with a larger manifold $T^\ast Q \times \mathbb{R}$ having $S$ as an extra coordinate. This is a contact manifold with contact form
$\alpha = -dS + p_1 dq_i + \cdots + p_n dq_n$
This contact manifold has a submanifold $\Sigma$ where we remember that entropy is a function of the probability distribution $q,$ and define $p$ in terms of $q$ as usual:
$\Sigma = \{(q,p,S) \in T^\ast Q \times \mathbb{R} : \; S = f(q), \; p = (df)_q \}$
And as we saw last time, $\Sigma$ is a Legendrian submanifold, meaning
$\displaystyle{ \alpha|_{\Sigma} = 0 }$
But again, we want to understand what these ideas from contact geometry really do for probability theory!
For all my old posts on information geometry, go here:
## Information Geometry (Part 17)
27 July, 2021
I’m getting back into information geometry, which is the geometry of the space of probability distributions, studied using tools from information theory. I’ve written a bunch about it already, which you can see here:
Now I’m fascinated by something new: how symplectic geometry and contact geometry show up in information geometry. But before I say anything about this, let me say a bit about how they show up in thermodynamics. This is more widely discussed, and it’s a good starting point.
Symplectic geometry was born as the geometry of phase space in classical mechanics: that is, the space of possible positions and momenta of a classical system. The simplest example of a symplectic manifold is the vector space $\mathbb{R}^{2n},$ with n position coordinates $q_i$ and n momentum coordinates $p_i.$
It turns out that symplectic manifolds are always even-dimensional, because we can always cover them with coordinate charts that look like $\mathbb{R}^{2n}.$ When we change coordinates, it turns out that the splitting of coordinates into positions and momenta is somewhat arbitrary. For example, the position of a rock on a spring now may determine its momentum a while later, and vice versa. What’s not arbitrary? It’s the so-called ‘symplectic structure’:
$\omega = dp_1 \wedge dq_1 + \cdots + dp_n \wedge dq_n$
While far from obvious at first, we know by now that the symplectic structure is exactly what needs to be preserved under valid changes of coordinates in classical mechanics! In fact, we can develop the whole formalism of classical mechanics starting from a manifold with a symplectic structure.
Symplectic geometry also shows up in thermodynamics. In thermodynamics we can start with a system in equilibrium whose state is described by some variables $q_1, \dots, q_n.$ Its entropy will be a function of these variables, say
$S = f(q_1, \dots, q_n)$
We can then take the partial derivatives of entropy and call them something:
$\displaystyle{ p_i = \frac{\partial f}{\partial q_i} }$
These new variables $p_i$ are said to be ‘conjugate’ to the $q_i,$ and they turn out to be very interesting. For example, if $q_i$ is energy then $p_i$ is ‘coolness’: the reciprocal of temperature. The coolness of a system is its change in entropy per change in energy.
Often the variables $q_i$ are ‘extensive’: that is, you can measure them only by looking at your whole system and totaling up some quantity. Examples are energy and volume. Then the new variables $p_i$ are ‘intensive’: that is, you can measure them at any one location in your system. Examples are coolness and pressure.
Now for a twist: sometimes we do not know the function $f$ ahead of time. Then we cannot define the $p_i$ as above. We’re forced into a different approach where we treat them as independent quantities, at least until someone tells us what $f$ is.
In this approach, we start with a space $\mathbb{R}^{2n}$ having n coordinates called $q_i$ and n coordinates called $p_i.$ This is a symplectic manifold, with the symplectic struture $\omega$ described earlier!
But what about the entropy? We don’t yet know what it is as a function of the $q_i,$ but we may still want to talk about it. So, we build a space $\mathbb{R}^{2n+1}$ having one extra coordinate $S$ in addition to the $q_i$ and $p_i.$ This new coordinate stands for entropy. And this new space has an important 1-form on it:
$\alpha = -dS + p_1 dq_i + \cdots + p_n dq_n$
This is called the ‘contact 1-form’.
This makes $\mathbb{R}^{2n+1}$ into an example of a ‘contact manifold’. Contact geometry is the odd-dimensional partner of symplectic geometry. Just as symplectic manifolds are always even-dimensional, contact manifolds are always odd-dimensional.
What is the point of the contact 1-form? Well, suppose someone tells us the function $f$ relating entropy to the coordinates $q_i.$ Now we know that we want
$S = f$
and also
$\displaystyle{ p_i = \frac{\partial f}{\partial q_i} }$
So, we can impose these equations, which pick out a subset of $\mathbb{R}^{2n+1}.$ You can check that this subset, say $\Sigma,$ is an n-dimensional submanifold. But even better, the contact 1-form vanishes when restricted to this submanifold:
$\left.\alpha\right|_\Sigma = 0$
Let’s see why! Suppose $x \in \Sigma$ and suppose $v \in T_x \Sigma$ is a vector tangent to $\Sigma$ at this point $x$. It suffices to show
$\alpha(v) = 0$
Using the definition of $\alpha$ this equation says
$\displaystyle{ -dS(v) + \sum_i p_i dq_i(v) = 0 }$
But on the surface $\Sigma$ we have
$S = f, \qquad \displaystyle{ p_i = \frac{\partial f}{\partial q_i} }$
So, the equation we’re trying to show can be written as
$\displaystyle{ -df(v) + \sum_i \frac{\partial f}{\partial q_i} dq_i(v) = 0 }$
But this follows from
$\displaystyle{ df = \sum_i \frac{\partial f}{\partial q_i} dq_i }$
which holds because $f$ is a function only of the coordinates $q_i.$
So, any formula for entropy $S = f(q_1, \dots, q_n)$ picks out a so-called ‘Legendrian submanifold’ of $\mathbb{R}^{2n+1}:$ that is, an n-dimensional submanifold such that the contact 1-form vanishes when restricted to this submanifold. And the idea is that this submanifold tells you everything you need to know about a thermodynamic system.
Indeed, V. I. Arnol’d says this was implicitly known to the great founder of statistical mechanics, Josiah Willard Gibbs. Arnol’d calls $\mathbb{R}^5$ with coordinates energy, entropy, temperature, pressure and volume the ‘Gibbs manifold’, and he proclaims:
Gibbs’ thesis: substances are Legendrian submanifolds of the Gibbs manifold.
This is from here:
• V. I. Arnol’d, Contact geometry: the geometrical method of Gibbs’ thermodynamics, Proceedings of the Gibbs Symposium (New Haven, CT, 1989), AMS, Providence, Rhode Island, 1990.
### A bit more detail
Now I want to say everything again, with a bit of extra detail, assuming more familiarity with manifolds. Above I was using $\mathbb{R}^n$ with coordinates $q_1, \dots, q_n$ to describe the ‘extensive’ variables of a thermodynamic system. But let’s be a bit more general and use any smooth n-dimensional manifold $Q.$ Even if $Q$ is a vector space, this viewpoint is nice because it’s manifestly coordinate-independent!
So: starting from $Q$ we build the cotangent bundle $T^\ast Q.$ A point in cotangent describes both extensive variables, namely $q \in Q,$ and ‘intensive’ variables, namely a cotangent vector $p \in T^\ast_q Q.$
The manifold $T^\ast Q$ has a 1-form $\theta$ on it called the tautological 1-form. We can describe it as follows. Given a tangent vector $v \in T_{(q,p)} T^\ast Q$ we have to say what $\theta(v)$ is. Using the projection
$\pi \colon T^\ast Q \to Q$
we can project $v$ down to a tangent vector $d\pi(v)$ at the point $q$. But the 1-form $p$ eats tangent vectors at $q$ and spits out numbers! So, we set
$\theta(v) = p(d\pi(v))$
This is sort of mind-boggling at first, but it’s worth pondering until it makes sense. It helps to work out what $\theta$ looks like in local coordinates. Starting with any local coordinates $q_i$ on an open set of $Q,$ we get local coordinates $q_i, p_i$ on the cotangent bundle of this open set in the usual way. On this open set you then get
$\theta = p_1 dq_1 + \cdots + p_n dq_n$
This is a standard calculation, which is really worth doing!
It follows that we can define a symplectic structure $\omega$ by
$\omega = d \theta$
and get this formula in local coordinates:
$\omega = dp_1 \wedge dq_1 + \cdots + dp_n \wedge dq_n$
Now, suppose we choose a smooth function
$f \colon Q \to \mathbb{R}$
which describes the entropy. We get a 1-form $df$, which we can think of as a map
$df \colon Q \to T^\ast Q$
assigning to each choice $q$ of extensive variables the pair $(q,p)$ of extensive and intensive variables where
$p = df_q$
The image of the map $df$ is a ‘Lagrangian submanifold‘ of $T^\ast Q:$ that is, an n-dimensional submanifold $\Lambda$ such that
$\left.\omega\right|_{\Lambda} = 0$
Lagrangian submanifolds are to symplectic geometry as Legendrian submanifolds are to contact geometry! What we’re seeing here is that if Gibbs had preferred symplectic geometry, he could have described substances as Lagrangian submanifolds rather than Legendrian submanifolds. But this approach would only keep track of the derivatives of entropy, $df,$ not the actual value of the entropy function $f.$
If we prefer to keep track of the actual value of $f$ using contact geometry, we can do that. For this we add an extra dimension to $T^\ast Q$ and form the manifold $T^\ast Q \times \mathbb{R}.$ The extra dimension represents entropy, so we’ll use $S$ as our name for the coordinate on $\mathbb{R}.$
We can make $T^\ast Q \times \mathbb{R}$ into a contact manifold with contact 1-form
$\alpha = -d S + \theta$
In local coordinates we get
$\alpha = -dS + p_1 dq_i + \cdots + p_n dq_n$
just as we had earlier. And just as before, if we choose a smooth function $f \colon Q \to \mathbb{R}$ describing entropy, the subset
$\Sigma = \{(q,p,S) \in T^\ast Q \times \mathbb{R} : \; S = f(q), \; p = df_q \}$
is a Legendrian submanifold of $T^\ast Q \times \mathbb{R}.$
Okay, this concludes my lightning review of symplectic and contact geometry in thermodynamics! Next time I’ll talk about something a bit less well understood: how they show up in statistical mechanics.
## Thermodynamics and Economic Equilibrium
18 July, 2021
I’m having another round of studying thermodynamics, and I’m running into more interesting leads than I can keep up with. Like this paper:
• Eric Smith and Duncan K. Foley, Classical thermodynamics and economic general equilibrium theory, Journal of Economic Dynamics and Control 32 (2008) 7–65.
I’ve always been curious about the connection between economics and thermodynamics, but I know too little about economics to make this easy to explore. There are people who work on subjects called thermoeconomics and econophysics, but classical economists consider them ‘heterodox’. While I don’t trust classical economists to be right about things, I should probably learn more classical economics before I jump into the fray.
Still, the introduction of this paper is intriguing:
The relation between economic and physical (particularly thermodynamic) concepts of equilibrium has been a topic of recurrent interest throughout the development of neoclassical economic theory. As systems for defining equilibria, proving their existence, and computing their properties, neoclassical economics (Mas-Collel et al., 1995; Varian, 1992) and classical thermodynamics (Fermi, 1956) undeniably have numerous formal and methodological similarities. Both fields seek to describe system phenomena in terms of solutions to constrained optimization problems. Both rely on dual representations of interacting subsystems: the state of each subsystem is represented by pairs of variables, one variable from each pair characterizing the subsystem’s content, and the other characterizing the way it interacts with other subsystems. In physics the content variables are quantities like asubsystem’s total energy or the volume in space it occupies; in economics they area mounts of various commodities held by agents. In physics the interaction variables are quantities like temperature and pressure that can be measured on the system boundaries; in economics they are prices that can be measured by an agent’s willingness to trade one commodity for another.
In thermodynamics these pairs are called conjugate variables. The ‘content variables’ are usually called extensive and the ‘interaction variables’ are usually called intensive. A vector space with conjugate pairs of variables as coordinates is a symplectic vector space, and I’ve written about how these show up in the category-theoretic approach to open systems:
• John Baez, A compositional framework for passive linear networks, Azimuth, 28 April 2015.
Continuing on:
The significance attached to these similarities has changed considerably, however, in the time from the first mathematical formulation of utility (Walras, 1909) to the full axiomatization of general equilibrium theory (Debreu, 1987). Léon Walras appears (Mirowski, 1989) to have conceptualized economic equilibrium as a balance of the gradients of utilities, more for the sake of similarity to the concept of force balance in mechanics, than to account for any observations about the outcomes of trade. Fisher (1892) (a student of J. Willard Gibbs) attempted to update Walrasian metaphors from mechanics to thermodynamics, but retained Walras’s program of seeking an explicit parallelism between physics and economics.
This Fisher is not the geneticist and statistician Ronald Fisher who came up with Fisher’s fundamental theorem. It’s the author of this thesis:
• Irving Fisher, Mathematical Investigations in the Theory of Value and Prices, Ph.D. thesis, Yale University, 1892.
Continuing on with Smith and Foley’s paper:
As mathematical economics has become more sophisticated (Debreu, 1987) the naive parallelism of Walras and Fisher has progressively been abandoned, and with it the sense that it matters whether neoclassical economics resembles any branch of physics. The cardinalization of utility that Walras thought of as a counterpart to energy has been discarded, apparently removing the possibility of comparing utility with any empirically measurable quantity. A long history of logically inconsistent (or simply unproductive) analogy making (see Section 7.2) has further caused the topic of parallels to fall out of favor. Samuelson (1960) summarizes well the current view among many economists, at the end of one of the few methodologically sound analyses of the parallel roles of dual representation in economics and physics:
The formal mathematical analogy between classical thermodynamics and mathematic economic systems has now been explored. This does not warrant the commonly met attempt to find more exact analogies of physical magnitudes—such as entropy or energy—in the economic realm. Why should there be laws like the first or second laws of thermodynamics holding in the economic realm? Why should ‘utility’ be literally identified with entropy, energy, or anything else? Why should a failure to make such a successful identification lead anyone to overlook or deny the mathematical isomorphism that does exist between minimum systems that arise in different disciplines?
The view that neoclassical economics is now mathematically mature, and that it is mere coincidence and no longer relevant whether it overlaps with any body of physical theory, is reflected in the complete omission of the topic of parallels from contemporary graduate texts (Mas-Collel et al., 1995). We argue here that, despite its long history of discussion, there are important insights still to be gleaned from considering the relation of neoclassical economics to classical thermodynamics. The new results concerning this relation we present here have significant implications, both for the interpretation of economic theory and for econometrics. The most important point of this paper (more important than the establishment of formal parallels between thermodynamics and utility economics) is that economics, because it does not recognize an equation of state or define prices intrinsically in terms of equilibrium, lacks the close relation between measurement and theory physical thermodynamics enjoys.
Luckily, the paper seems to be serious about explaining economics to those who know thermodynamics (and maybe vice versa). So, I will now read the rest of the paper—or at least skim it.
One interesting simple point seems to be this: there’s an analogy between entropy maximization and utility maximization, but it’s limited by the following difference.
In classical thermodynamics the total entropy of a closed system made of subsystems is the sum of the entropies of the parts. While the second law forbids the system from moving to a state to a state of lower total entropy, the entropies of some parts can decrease.
By contrast, in classical economics the total utility of a collection of agents is an unimportant quantity: what matters is the utility of each individual agent. The reason is that we assume the agents will voluntarily move from one state to another only if the utility of each agent separately increases. Furthermore, if we believe we can reparametrize the utility of each agent without changing anything, it makes no sense to add utilities.
(On the other hand, some utilitarian ethicists seem to believe it makes sense to add utilities and try to maximize the total. I imagine that libertarians would consider this ‘totalitarian’ approach morally unacceptable. I’m even less eager to enter discussions of the foundations of ethics than of economics, but it’s interesting how the question of whether a quantity can or ‘should’ be totaled up and then maximized plays a role in this debate.)
## The Ideal Monatomic Gas
15 July, 2021
Today at the Topos Institute, Sophie Libkind, Owen Lynch and I spent some time talking about thermodynamics, Carnot engines and the like. As a result, I want to work out for myself some basic facts about the ideal gas. This stuff is all well-known, but I’m having trouble finding exactly what I want—and no more, thank you—collected in one place.
Just for background, the Carnot cycle looks roughly like this:
This is actually a very inaccurate picture, but it gets the point across. We have a container of gas, and we make it execute a cyclic motion, so its pressure $P$ and volume $V$ trace out a loop in the plane. As you can see, this loop consists of four curves:
• In the first, from a to b, we put a container of gas in contact with a hot medium. Then we make it undergo isothermal expansion: that is, expansion at a constant temperature.
• In the second, from b to c, we insulate the container and let the gas undergo adiabatic reversible expansion: that is, expansion while no heat enters or leaves. The temperature drops, but merely because the container expands, not because heat leaves. It reaches a lower temperature. Then we remove the insulation.
• In the third, from c to d, we put the container in contact with a cold medium that matches its temperature. Then we make it undergo isothermal contraction: that is, contraction at a constant temperature.
• In the fourth, from d to a, we insulate the container and let the gas undergo adiabatic reversible contraction: that is, contraction while no heat enters or leaves. The temperature increases until it matches that of the hot medium. Then we remove the insulation.
The Carnot cycle is historically important because it’s an example of a heat engine that’s as efficient as possible: it give you the most work possible for the given amount of heat transferred from the hot medium to the cold medium. But I don’t want to get into that. I just want to figure out formulas for everything that’s going on here—including formulas for the four curves in this picture!
To get specific formulas, I’ll consider an ideal monatomic gas, meaning a gas made of individual atoms, like helium. Some features of an ideal gas, like the formula for energy as a function of temperature, depend on whether it’s monatomic.
As a quirky added bonus, I’d like to highlight how certain properties of the ideal monatomic gas depend on the dimension of space. There’s a certain chunk of the theory that doesn’t depend on the dimension of space, as long as you interpret ‘volume’ to mean the n-dimensional analogue of volume. But the number 3 shows up in the formula for the energy of the ideal monatomic gas. And this is because space is 3-dimensional! So just for fun, I’ll do the whole analysis in n dimensions.
There are four basic formulas we need to know.
First, we have the ideal gas law:
$PV = NkT$
where
$P$ is the pressure.
$V$ is the n-dimensional volume.
$N$ is the number of molecules in a container of gas.
$k$ is a constant called Boltzmann’s constant.
$T$ is the temperature.
Second, we have a formula for the energy, or more precisely the internal energy, of a monatomic ideal gas:
$U = \frac{n}{2} NkT$
where
$U$ is the internal energy.
$n$ is the dimension of space.
The factor of n/2 shows up thanks to the equipartition theorem: classically, a harmonic oscillator at temperature $T$ has expected energy equal to $kT$ times its number of degrees of freedom. Very roughly, the point is that in n dimensions there are n different directions in which an atom can move around.
Third, we have a relation between internal energy, work and heat:
$dU = \delta W + \delta Q$
Here
$dU$ is the differential of internal energy.
$\delta W$ is the infinitesimal work done to the gas.
$\delta Q$ is the infinitesimal heat transferred to the gas.
The intuition is simple: to increase the energy of some gas you can do work to it or transfer heat to it. But the math may seem a bit murky, so let me explain.
I emphasize ‘to’ because it affects the sign: for example, the work done by the gas is minus the work done to the gas. Work done to the gas increases its internal energy, while work done by it reduces its internal energy. Similarly for heat.
But what is this ‘infinitesimal’ stuff, and these weird $\delta$ symbols?
In a minute I’m going to express everything in terms of $P$ and $V.$ So, $T, N$ and $U$ will be functions on the plane with coordinates $P$ and $V.$ $dU$ will be a 1-form on this plane: it’s the differential of the function $U.$
But $\delta W$ and $\delta Q$ are not differentials of functions $W$ and $Q.$ There are no functions on the plane called $W$ and $Q.$ You can not take a box of gas and measure its work, or heat! There are just 1-forms called $\delta W$ and $\delta Q$ describing the change in work or heat. These are not exact 1-forms: that is, they’re not differentials of functions.
Fourth and finally:
$\delta W = - P dV$
This should be intuitive. The work done by the gas on the outside world by changing its volume a little equals the pressure times the change in volume. So, the work done to the gas is minus the pressure times the change in volume.
One nice feature of the 1-form $\delta W = -P d V$ is this: as we integrate it around a simple closed curve going counterclockwise, we get the area enclosed by that curve. So, the area of this region:
is the work done by our container of gas during the Carnot cycle. (There are a lot of minus signs to worry about here, but don’t worry, I’ve got them under control. Our curve is going clockwise, so the work done to our container of gas is negative, and it’s minus the area in the region.)
Okay, now that we have our four basic equations, we can play with them and derive consequences. Let’s suppose the number $N$ of atoms in our container of gas is fixed—a constant. Then we think of everything as a function of two variables: $P$ and $V.$
First, since $PV = NkT$ we have
$\displaystyle{ T = \frac{PV}{Nk} }$
So temperature is proportional to pressure times volume.
Second, since $PV = NkT$ and $U = \frac{n}{2}NkT$ we have
$U = \frac{n}{2} P V$
So, like the temperature, the internal energy of the gas is proportional to pressure times volume—but it depends on the dimension of space!
From this we get
$dU = \frac{n}{2} d(PV) = \frac{n}{2}( V dP + P dV)$
From this and our formulas $dU = \delta W + \delta Q, \delta W = -PdV$ we get
$\begin{array}{ccl} \delta Q &=& dU - \delta W \\ \\ &=& \frac{n}{2}( V dP + P dV) + P dV \\ \\ &=& \frac{n}{2} V dP + \frac{n+2}{2} P dV \end{array}$
That’s basically it!
But now we know how to figure out everything about the Carnot cycle. I won’t do it all here, but I’ll work out formulas for the curves in this cycle:
The isothermal curves are easy, since we’ve seen temperature is proportional to pressure times volume:
$\displaystyle{ T = \frac{PV}{Nk} }$
So, an isothermal curve is any curve with
$P \propto V^{-1}$
The adiabatic reversible curves, or ‘adiabats’ for short, are a lot more interesting. A curve $C$ in the $P V$ plane is an adiabat if when the container of gas changes pressure and volume while moving along this curve, no heat gets transferred to or from the gas. That is:
$\delta Q \Big|_C = 0$
where the funny symbol means I’m restricting a 1-form to the curve and getting a 1-form on that curve (which happens to be zero).
Let’s figure out what an adiabat looks like! By our formula for $Q$ we have
$(\frac{n}{2} V dP + \frac{n+2}{2} P dV) \Big|_C = 0$
or
$\frac{n}{2} V dP \Big|_C = -\frac{n+2}{2} P dV \Big|_C$
or
$\frac{dP}{P} \Big|_C = - \frac{n+2}{n} \frac{dV}{V}\Big|_C$
Now, we can integrate both sides along a portion of the curve $C$ and get
$\ln P = - \frac{n+2}{n} \ln V + \mathrm{constant}$
or
$P \propto V^{-(n+2)/n}$
So in 3-dimensional space, as you let a gas expand adiabatically—say by putting it in an insulated cylinder so heat can’t get in or out—its pressure drops as its volume increases. But for a monatomic gas it drops in this peculiar specific way: the pressure goes like the volume to the -5/3 power.
In any dimension, the pressure of the monatomic gas drops more steeply when the container expands adiabatically than when it expands at constant temperature. Why? Because $V^{-(n+2)/n}$ drops more rapidly than $V^{-1}$ since
$\frac{n+2}{n} > 1$
But as $n \to \infty$,
$\frac{n+2}{n} \to 1$
so the adiabats become closer and and closer to the isothermal curves in high dimensions. This is not important for understanding the conceptually significant features of the Carnot cycle! But it’s curious, and I’d like to improve my understanding by thinking about it until it seems obvious. It doesn’t yet.
## Nonequilibrium Thermodynamics in Biology (Part 2)
16 June, 2021
Larry Li, Bill Cannon and I ran a session on non-equilibrium thermodynamics in biology at SMB2021, the annual meeting of the Society for Mathematical Biology. You can see talk slides here!
Here’s the basic idea:
Since Lotka, physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session addresses current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.
Click on the links to see slides for most of the talks:
Persistence, permanence, and global stability in reaction network models: some results inspired by thermodynamic principles
The standard mathematical model for the dynamics of concentrations in biochemical networks is called mass-action kinetics. We describe mass-action kinetics and discuss the connection between special classes of mass-action systems (such as detailed balanced and complex balanced systems) and the Boltzmann equation. We also discuss the connection between the ‘global attractor conjecture’ for complex balanced mass-action systems and Boltzmann’s H-theorem. We also describe some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.
The principle of maximum caliber of nonequilibria
Ken Dill, Stony Brook University
Maximum Caliber is a principle for inferring pathways and rate distributions of kinetic processes. The structure and foundations of MaxCal are much like those of Maximum Entropy for static distributions. We have explored how MaxCal may serve as a general variational principle for nonequilibrium statistical physics—giving well-known results, such as the Green-Kubo relations, Onsager’s reciprocal relations and Prigogine’s Minimum Entropy Production principle near equilibrium, but is also applicable far from equilibrium. I will also discuss some applications, such as finding reaction coordinates in molecular simulations non-linear dynamics in gene circuits, power-law-tail distributions in ‘social-physics’ networks, and others.
Nonequilibrium biomolecular information processes
Pierre Gaspard, Université libre de Bruxelles
Nearly 70 years have passed since the discovery of DNA structure and its role in coding genetic information. Yet, the kinetics and thermodynamics of genetic information processing in DNA replication, transcription, and translation remain poorly understood. These template-directed copolymerization processes are running away from equilibrium, being powered by extracellular energy sources. Recent advances show that their kinetic equations can be exactly solved in terms of so-called iterated function systems. Remarkably, iterated function systems can determine the effects of genome sequence on replication errors, up to a million times faster than kinetic Monte Carlo algorithms. With these new methods, fundamental links can be established between molecular information processing and the second law of thermodynamics, shedding a new light on genetic drift, mutations, and evolution.
Nonequilibrium dynamics of disturbed ecosystems
John Harte, University of California, Berkeley
The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories, and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time-varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic, by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth, or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time-varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over
individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.
Stochastic chemical reaction networks
Supriya Krishnamurthy, Stockholm University
The study of chemical reaction networks (CRN’s) is a very active field. Earlier well-known results (Feinberg Chem. Enc. Sci. 42 2229 (1987), Anderson et al Bull. Math. Biol. 72 1947 (2010)) identify a topological quantity called deficiency, easy to compute for CRNs of any size, which, when exactly equal to zero, leads to a unique factorized (non-equilibrium) steady-state for these networks. No general results exist however for the steady states of non-zero-deficiency networks. In recent work, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict any moment of interest, for CRN’s with non-zero deficiency and non-factorizable steady states. It is however an open question how scalable these techniques are for large networks.
Heat flows adjust local ion concentrations in favor of prebiotic chemistry
Christof Mast, Ludwig-Maximilians-Universität München
Prebiotic reactions often require certain initial concentrations of ions. For example, the activity of RNA enzymes requires a lot of divalent magnesium salt, whereas too much monovalent sodium salt leads to a reduction in enzyme function. However, it is known from leaching experiments that prebiotically relevant geomaterial such as basalt releases mainly a lot of sodium and only little magnesium. A natural solution to this problem is heat fluxes through thin rock fractures, through which magnesium is actively enriched and sodium is depleted by thermogravitational convection and thermophoresis. This process establishes suitable conditions for ribozyme function from a basaltic leach. It can take place in a spatially distributed system of rock cracks and is therefore particularly stable to natural fluctuations and disturbances.
Deficiency of chemical reaction networks and thermodynamics
Matteo Polettini, University of Luxembourg
Deficiency is a topological property of a Chemical Reaction Network linked to important dynamical features, in particular of deterministic fixed points and of stochastic stationary states. Here we link it to thermodynamics: in particular we discuss the validity of a strong vs. weak zeroth law, the existence of time-reversed mass-action kinetics, and the possibility to formulate marginal fluctuation relations. Finally we illustrate some subtleties of the Python module we created for MCMC stochastic simulation of CRNs, soon to be made public.
Large deviations theory and emergent landscapes in biological dynamics
Hong Qian, University of Washington
The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer-valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ microscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington and S. Wright.
Using the maximum entropy production principle to understand and predict microbial biogeochemistry
Joseph Vallino, Marine Biological Laboratory, Woods Hole
Natural microbial communities contain billions of individuals per liter and can exceed a trillion cells per liter in sediments, as well as harbor thousands of species in the same volume. The high species diversity contributes to extensive metabolic functional capabilities to extract chemical energy from the environment, such as methanogenesis, sulfate reduction, anaerobic photosynthesis, chemoautotrophy, and many others, most of which are only expressed by bacteria and archaea. Reductionist modeling of natural communities is problematic, as we lack knowledge on growth kinetics for most organisms and have even less understanding on the mechanisms governing predation, viral lysis, and predator avoidance in these systems. As a result, existing models that describe microbial communities contain dozens to hundreds of parameters, and state variables are extensively aggregated. Overall, the models are little more than non-linear parameter fitting exercises that have limited, to no, extrapolation potential, as there are few principles governing organization and function of complex self-assembling systems. Over the last decade, we have been developing a systems approach that models microbial communities as a distributed metabolic network that focuses on metabolic function rather than describing individuals or species. We use an optimization approach to determine which metabolic functions in the network should be up regulated versus those that should be down regulated based on the non-equilibrium thermodynamics principle of maximum entropy production (MEP). Derived from statistical mechanics, MEP proposes that steady state systems will likely organize to maximize free energy dissipation rate. We have extended this conjecture to apply to non-steady state systems and have proposed that living systems maximize entropy production integrated over time and space, while non-living systems maximize instantaneous entropy production. Our presentation will provide a brief overview of the theory and approach, as well as present several examples of applying MEP to describe the biogeochemistry of microbial systems in laboratory experiments and natural ecosystems.
Reduction and the quasi-steady state approximation
Carsten Wiuf, University of Copenhagen
Chemical reactions often occur at different time-scales. In applications of chemical reaction network theory it is often desirable to reduce a reaction network to a smaller reaction network by elimination of fast species or fast reactions. There exist various techniques for doing so, e.g. the Quasi-Steady-State Approximation or the Rapid Equilibrium Approximation. However, these methods are not always mathematically justifiable. Here, a method is presented for which (so-called) non-interacting species are eliminated by means of QSSA. It is argued that this method is mathematically sound. Various examples are given (Michaelis-Menten mechanism, two-substrate mechanism, …) and older related techniques from the 50s and 60s are briefly discussed.
## Electrostatics and the Gauss–Lucas Theorem
24 May, 2021
Say you know the roots of a polynomial P and you want to know the roots of its derivative. You can do it using physics! Namely, electrostatics in 2d space, viewed as the complex plane.
To keep things simple, let us assume P does not have repeated roots. Then the procedure works as follows.
Put equal point charges at each root of P, then see where the resulting electric field vanishes. Those are the roots of P’.
I’ll explain why this is true a bit later. But first, we use this trick to see something cool.
There’s no way the electric field can vanish outside the convex hull of your set of point charges. After all, if all the charges are positive, the electric field must point out of that region. So, the roots of P’ must lie in the convex hull of the roots of P!
This cool fact is called the Gauss–Lucas theorem. It always seemed mysterious to me. Now, thanks to this ‘physics proof’, it seems completely obvious!
Of course, it relies on my first claim: that if we put equal point
charges at the roots of P, the electric field they generate will vanish at the roots of P’. Why is this true?
By multiplying by a constant if necessary, we can assume
$\displaystyle{ P(z) = \prod_{i = 1}^n (z - a_i) }$
Thus
$\displaystyle{ \ln |P(z)| = \sum_{i = 1}^n \ln|z - a_i| }$
This function is the electric potential created by equal point charges at the points ai in the complex plane. The corresponding electric field is minus the gradient of the potential, so it vanishes at the critical points of this function. Equivalently, it vanishes at the critical points of the exponential of this function, namely |P|. Apart from one possible exception, these points are the same as the critical points of P, namely the roots of P’. So, we’re almost done!
The exception occurs when P has a critical point where P vanishes. |P| is not smooth where P vanishes, so in this case we cannot say the critical point of P is a critical point of |P|.
However, when P has a critical point where P vanishes, then this point is a repeated root of P, and I already said I’m assuming P has no repeated roots. So, we’re done—given this assumption.
Everything gets a bit more complicated when our polynomial has repeated roots. Greg Egan explored this, and also the case where its derivative has repeated roots.
However, the Gauss–Lucas theorem still applies to polynomials with repeated roots, and this proof explains why:
• Wikipedia, Gauss–Lucas theorem.
Alternatively, it should be possible to handle the case of a polynomial with repeated roots by thinking of it as a limit of polynomials without repeated roots.
By the way, in my physics proof of the Gauss–Lucas theorem I said the electric field generated by a bunch of positive point charges cannot vanish outside the convex hull of these point charges because the field ‘points out’ of this region. Let me clarify that.
It’s true even if the positive point charges aren’t all equal; they just need to have the same sign. The rough idea is that each charge creates an electric field that points radially outward, so these electric fields can’t cancel at a point that’s not ‘between’ several charges—in other words, at a point that’s not in the convex hull of the charges.
But let’s turn this idea into a rigorous argument.
Suppose z is some point outside the convex hull of the points ai. Then, by the hyperplane separation theorem, we can draw a line with z on one side and all the points ai on the other side. Let v be a vector normal to this line and pointing toward the z side. Then
$v \cdot (z - a_i) > 0$
for all i. Since the electric field created by the ith point charge is a positive multiple of z – ai at the point z, the total electric field at z has a positive dot product with v. So, it can’t be zero!
### Credits
The picture of a convex hull is due to Robert Laurini.
## Parallel Line Masses and Marden’s Theorem
22 May, 2021
Here’s an idea I got from Albert Chern on Twitter. He did all the hard work, and I think he also drew the picture I’m going to use. I’ll just express the idea in a different way.
Here’s a strange fact about Newtonian gravity.
Consider three parallel ‘line masses’ that have a constant mass per length—the same constant for each one. Choose a plane orthogonal to these lines. There will typically be two points on this plane, say a and b, where a mass can sit in equilibrium, with the gravitational pull from all three lines masses cancelling out. This will be an unstable equilibrium.
Put a mass at point a. Remove the three line masses—but keep in mind the triangle they formed where they pierced your plane!
You can now orbit a test particle in an elliptical orbit around the mass at a in such a way that:
• one focus of this ellipse is a,
• the other focus is b, and
• the ellipse fits inside the triangle, just touching the midpoint of each side of the triangle.
Even better, this ellipse has the largest possible area of any ellipse contained in the triangle!
Here is Chern’s picture:
The triangle’s corners are the three points where the line masses pierce your chosen plane. These line masses create a gravitational potential, and the contour lines are level curves of this potential.
You can see that the points a and b are at saddle points of the potential. Thus, a mass placed at either a and b will be in an unstable equilibrium.
You can see the ellipse with a and b as its foci, snugly fitting into the triangle.
You can sort of see that the ellipse touches the midpoints of the triangle’s edges.
What you can’t see is that this ellipse has the largest possible area for any ellipse fitting into the triangle!
Now let me explain the math. While the gravitational potential of a point mass in 3d space is proportional to $1/r$, the gravitational potential of a line mass in 3d space is proportional to $\log r,$ which is also the gravitational potential of a point mass in 2d space.
So, if we have three equal line masses, which are parallel and pierce an orthogonal plane at points $p_1, p_2$ and $p_3,$ then their gravitational potential, as a function on this plane, will be proportional to
$\phi(z) = \log|z - p_1| + \log|z - p_2| + \log|z - p_3|$
Here I’m using $z$ as our name for an arbitrary point on this plane, because the next trick is to think of this plane as the complex plane!
Where are the critical points (in fact saddle points) of this potential? They are just points where the gradient of $\phi$ vanishes. To find these points, we can just take the exponential of $\phi$ and see where the gradient of that vanishes. This is a nice idea because
$e^{\phi(z)} = |(z-p_1)(z-p_2)(z-p_3)|$
The gradient of this function will vanish whenever
$P'(z) = 0$
where
$P(z) = (z-p_1)(z-p_2)(z-p_3)$
Since $P$ is a cubic polynomial, $P'$ is a quadratic, hence proportional to
$(z - a)(z - b)$
for some a and b. Now we use
Marden’s theorem. Suppose the zeros $p_1, p_2, p_3$ of a cubic polynomial $P$ are non-collinear. Then there is a unique ellipse inscribed in the triangle with vertices $p_1, p_2, p_3$ and tangent to the sides at their midpoints. The foci of this ellipse are the zeroes of the derivative of $P.$
For a short proof of this theorem go here:
This ellipse is called the Steiner inellipse of the triangle:
• Wikipedia, Steiner inellipse.
The proof that it has the largest area of any ellipse inscribed in the triangle goes like this. Using a linear transformation of the plane you can map any triangle to an equilateral triangle. It’s obvious that there’s a circle inscribed in any equilateral triangle, touching each of the triangle’s midpoints. It’s at least very plausible that that this circle is the ellipse of largest area contained in the triangle. If we can prove this we’re done.
Why? Because linear transformations map circles to ellipses, and map midpoints of line segments to midpoints of line segments, and simply rescale areas by a constant fact. So applying the inverse linear transformation to the circle inscribed in the equilateral triangle, we get an ellipse inscribed in our original triangle, which will touch this triangle’s midpoints, and have the maximum possible area of any ellipse contained in this triangle!
## The Koide Formula
4 April, 2021
There are three charged leptons: the electron, the muon and the tau. Let $m_e, m_\mu$ and $m_\tau$ be their masses. Then the Koide formula says
$\displaystyle{ \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} = \frac{2}{3} }$
There’s no known reason for this formula to be true! But if you plug in the experimentally measured values of the electron, muon and tau masses, it’s accurate within the current experimental error bars:
$\displaystyle{ \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} = 0.666661 \pm 0.000007 }$
Is this significant or just a coincidence? Will it fall apart when we measure the masses more accurately? Nobody knows.
Here’s something fun, though:
Puzzle. Show that no matter what the electron, muon and tau masses might be—that is, any positive numbers whatsoever—we must have
$\displaystyle{ \frac{1}{3} \le \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} \le 1}$
For some reason this ratio turns out to be almost exactly halfway between the lower bound and upper bound!
Koide came up with his formula in 1982 before the tau’s mass was measured very accurately. At the time, using the observed electron and muon masses, his formula predicted the tau’s mass was
$m_\tau$ = 1776.97 MeV/c2
while the observed mass was
$m_\tau$ = 1784.2 ± 3.2 MeV/c2
Not very good.
In 1992 the tau’s mass was measured much more accurately and found to be
$m_\tau$ = 1776.99 ± 0.28 MeV/c2
Much better!
Koide has some more recent thoughts about his formula:
• Yoshio Koide, What physics does the charged lepton mass relation tell us?, 2018.
He points out how difficult it is to explain a formula like this, given how masses depend on an energy scale in quantum field theory.
## Vincenzo Galilei
3 April, 2021
I’ve been reading about early music. I ran into Vicenzo Galilei, an Italian lute player, composer, and music theorist who lived during the late Renaissance and helped start the Baroque era. Of course anyone interested in physics will know Galileo Galilei. And it turns out Vicenzo was Galileo’s dad!
The really interesting part is that Vincenzo did a lot of experiments—and he got Galileo interested in the experimental method!
Vicenzo started out as a lutenist, but in 1563 he met Gioseffo Zarlino, the most important music theorist of the sixteenth century, and began studying with him. Vincenzo became interested in tuning and keys, and in 1584 he anticipated Bach’s Well-Tempered Clavier by composing 24 groups of dances, one for each of the 12 major and 12 minor keys.
He also studied acoustics, especially vibrating strings and columns of air. He discovered that while the frequency of sound produced by a vibrating string varies inversely with the length of string, it’s also proportional to the square root of the tension applied. For example, weights suspended from strings of equal length need to be in a ratio of 9:4 to produce a perfect fifth, which is the frequency ratio 3:2.
Galileo later told a biographer that Vincenzo introduced him to the idea of systematic testing and measurement. The basement of their house was strung with lengths of lute string materials, each of different lengths, with different weights attached. Some say this drew Galileo’s attention away from pure mathematics to physics!
You can see books by Vicenzo Galilei here:
• Internet Archive, Vincenzo Galilei, c. 1520 – 2 July 1591.
Unfortunately for me they’re in Italian, but the title of his Dialogo della Musica Antica et Della Moderna reminds me of his son’s Dialogo sopra i Due Massimi Sistemi del Mondo (Dialog Concerning the Two Chief World Systems).
Speaking of dialogs, here’s a nice lute duet by Vincenzo Galilei, played by Evangelina Mascardi and Frédéric Zigante:
It’s from his book Fronimo Dialogo, an instruction manual for the lute which includes many compositions, including the 24 dances illustrating the 24 keys. “Fronimo” was an imaginary expert in the lute—in ancient Greek, phronimo means sage—and the book apparently consists of dialogs with between Fronimo and a student Eumazio (meaning “he who learns well”).
So, I now suspect that Galileo also got his fondness for dialogs from his dad, too! Or maybe everyone was writing them back then?
## Can We Understand the Standard Model Using Octonions?
31 March, 2021
I gave two talks in Latham Boyle and Kirill Krasnov’s Perimeter Institute workshop Octonions and the Standard Model.
The first talk was on Monday April 5th at noon Eastern Time. The second was exactly one week later, on Monday April 12th at noon Eastern Time.
Here they are:
Can we understand the Standard Model? (video, slides)
Abstract. 40 years trying to go beyond the Standard Model hasn’t yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions. We explain both these approaches, and how to reconcile them.
The second is on Monday April 12th at noon Eastern Time:
Can we understand the Standard Model using octonions? (video, slides)
Abstract. Dubois-Violette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices of octonions. After an introduction to the physics of Jordan algebras, we ponder the meaning of their construction. For example, it implies that the Standard Model gauge group consists of the symmetries of an octonionic qutrit that restrict to symmetries of an octonionic qubit and preserve all the structure arising from a choice of unit imaginary octonion. It also sheds light on why the Standard Model gauge group acts on 10d Euclidean space, or Minkowski spacetime, while preserving a 4+6 splitting.
You can see all the slides and videos and also some articles with more details here. | 2021-10-20 19:45:26 | {"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": 0, "img_math": 274, "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": 0.7546650171279907, "perplexity": 589.9221666161967}, "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/1634323585348.66/warc/CC-MAIN-20211020183354-20211020213354-00074.warc.gz"} |
https://tex.stackexchange.com/questions/304284/adding-complex-phonetic-symbols-to-latex/304297 | # Adding complex phonetic symbols to LaTeX
I am trying to add symbols from the International Phonetic Alphabet to my thesis. For some of the characters, I have no idea how to add them to TeX.
The retroflex is giving me troubles.
\documentclass{article}
\usepackage{tipa}
\begin{document}
\begin{tabular}{llllllll}
~ & Labial & Dental & Alveolar & Retroflex & Palatal & Velar & Glottal \\
Stop & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiceless & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiceless aspirated & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiced & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiced aspirated & \textipa{} & ~ & ~ & ~ & ~ & ~ & ~ \\
Fricative & \textipa{} & ~ & ~ & ~ & ~ & ~ & ~ \\
Nasal & t͡ʃʰ & ~ & ~ & ~ & ~ & ~ & ~ \\
\end{tabular}
\end{document}
Can anyone tell me how to add retroflex symbols in LaTeX?
The description of the symbols can be found here.
UPDATE: I have added the symbols t͡ʃʰ and n̪. XeLaTeX is not printing them.
• According to the tipa manual retroflexes are obtained with \:. The retroflex approximant can e.g. be obtained by \textipa{\:R}. Is this what you are looking for? Apr 14, 2016 at 14:33
• I am talking about specific symbols for example voiced retro flex d
– karu
Apr 14, 2016 at 14:37
• The simplest way is to compile with XeLaTeX and enter your Unicode IPA symbols directly into your editor like you can here: ɖ ʈ etc. See Typesetting phonetic symbols: Unicode or tipa? for more info and more reasons to do this instead of using tipa. Apr 14, 2016 at 14:42
• @JasonZentz I am using Xelatex for the compilation. These symbols I find really complex.
– karu
Apr 14, 2016 at 14:44
As explained by this answer, you can either use tipa and compile using pdfLaTeX or load a Unicode IPA font using fontspec and compile using XeLaTeX or LuaLaTeX. I strongly recommend using a Unicode IPA font rather than tipa for the reasons outlined in this answer.
Here is an example that uses the IPA versions of most of the symbols shown in the image you posted (I included both the palato-alveolar affricate and palatal stop symbols). I entered these symbols directly in my TeX editor using an IPA keyboard layout, but you could also use one of many online IPA pickers and copy and paste from there into your editor. I will leave the table formatting up to you.
\documentclass{article}
\usepackage{fontspec}
\setmainfont{Charis SIL}
\begin{document}
\noindent p t̪ t ʈ t͡ʃ c k\\
pʰ t̪ʰ ʈʰ t͡ʃʰ cʰ kʰ \\
b d̪ ɖ d͡ʒ ɟ ɡ \\
bʱ d̪ʱ ɖʱ d͡ʒʱ ɟʱ ɡʱ \\
f s ʂ ʃ h \\
m n̪ n ɳ ɲ ŋ \\
r ɽ ͏ɻ\\
l ɭ \\
w v j
\end{document}
As shown below, the font you select does make a difference in whether your symbols will come out right. Some fonts simply don't have the glyphs for most IPA symbols (although the ones shown here do have all the glyphs in the example), and others do a poor job of stacking diacritics and placing things like the tie bar used in affricates. So choose your font wisely, also paying attention to how well it does with other formatting you need such as bold and small caps.
Charis SIL:
Brill:
Linux Libertine O:
Times New Roman:
• What about retroflex approximant symbol ?
– karu
Apr 14, 2016 at 15:32
• @karu, I left that out because the chart used [ʐ] but called it a retroflex approximant (that symbol is a retroflex fricative in the IPA). I wasn't sure what you would want to use -- that or the IPA retroflex approximant [ɻ]. Both of these symbols will come out fine though. I'm just trying to give you the general idea, not provide all the symbols you might possibly want. Apr 14, 2016 at 16:03
• Thanks. I understand now. Using Unicode is much better than tipa, at least for me, because I have to write characters in Malayalam.
– karu
Apr 14, 2016 at 16:11
• @kanu, I just edited the answer to add the [ɻ] symbol, just to show that it does work in all these fonts. Apr 14, 2016 at 18:58
• Just for the banter (and because I was curious), here are Gentium Plus, Heuristica, and Noto Serif, each of which perform pretty decently. The Combining Double Inverted Breve is a bit squiffy in Brill (which is a surprise) and Linux Libertine. Apr 15, 2016 at 21:12
As was told you in the comments, you can (and should if you're using xelatex) use a font that supports all IPA glyphs.
Since you're using a table, you can automatize this so that only the cells with IPA use that font, and the headers use the regular font. For the headers I'm using a particular font here just to show the difference, but you can use anything you like.
## Code
\documentclass{article}
\usepackage[margin=2.5cm]{geometry}
\usepackage{fontspec}
\usepackage{booktabs}
\usepackage{array}
\setmainfont{Century Gothic}
\newfontfamily\ipafont{Charis SIL}
\newcommand\ipa[1]{{\ipafont #1}}
% To keep the header with normal font
\makeatletter
\newcommand*{\rowstyle}[1]{% sets the style of the next row
\gdef\@rowstyle{\leavevmode#1}%
\@rowstyle\ignorespaces}
\newcolumntype{=}{>{\gdef\@rowstyle{}}}
\newcolumntype{+}{>{\@rowstyle}}
\makeatother
% Column type with ipa font
\newcolumntype{A}{+>{\ipafont}l}
\begin{document}
\begin{tabular}{=l*{7}{A}}
\toprule
\rowstyle{\normalfont}
~ & Labial & Dental & Alveolar & Retroflex & Palatal & Velar & Glottal\\ \midrule
Stop & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiceless & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiceless aspirated & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiced & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
Voiced aspirated & & ~ & ~ & ~ & ~ & ~ & ~ \\
Fricative & & ~ & ~ & ~ & ~ & ~ & ~ \\
Nasal & t͡ʃʰ & ~ & ~ & ~ & ~ & ~ & ~ \\
\bottomrule
\end{tabular}
\end{document} | 2022-05-23 11:55:53 | {"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": 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": 0.6629262566566467, "perplexity": 1487.6491917175429}, "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-2022-21/segments/1652662558015.52/warc/CC-MAIN-20220523101705-20220523131705-00708.warc.gz"} |
https://rationalwiki.org/wiki/index.php?title=User_talk:Women_are_my_superior_and_I_respect_them_as_such&diff=277407&oldid=277402 | # Difference between revisions of "User talk:Women are my superior and I respect them as such"
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LUSER:FALLDOWN IS A MISOGYNIST AND A HATE MERCHANT. IN NO WAY DOES RATIONALWIKI CONDONE SUCH HATE. PLEASE VIEW THEIR CONTRIBUTIONS (INCLUDING THEIR LUSER PAGE) WITH THIS IN MIND. ħuman 23:19, 23 November 2008 (EST)
Your welcome to RationalWiki is lukewarm at best, Women are my superior and I respect them as such. This observation is due to the nature of your initial edits. Pull up a goat and try not to make trouble. We realize it is possible that you do not understand the nature of the site or our objectives. Please see our guide for newcomers and our community standards to clarify things for you. If you're still interested in contributing, please see what our articles are intended to be.
Hi there Fall down, welcome to the hairy nude photograph of the Internet. - User $\approx$$\pi$For best results lather, rinse and repeat 01:23, 15 October 2008 (EDT)
I am taking back my welcome. If you have a problem with the site you may discuss it in a hour when your block is over. You are also in the vandal bin until you can behave. - User $\approx$$\pi$For best results lather, rinse and repeat 01:33, 15 October 2008 (EDT)
It does stager me that you complain about being blocked when your first action is to blank the main page. The comparison to CP is completely uncalled for. You may continue to edit and if you can do it in a non-destructive manner I will release the vandal break. I have better things to do then follow you around cleaning up your little messes because you have a chip on your shoulder about something you have not discussed in anyway. - User $\approx$$\pi$For best results lather, rinse and repeat 22:31, 16 October 2008 (EDT)
Yes, we women do control the world. And it was about damn time. Men have controlled the world for millennia, so now we are getting our own millennia of rule. (Note, this comment is not serious) InaVegt 08:30, 5 November 2008 (EST)
Since when have feminists been not serious? Fall down
Fall Down: You are an asshole, a moron, and an embarrassment to all your fellow penis-equipped human beings. That is all. PFoster 16:19, 7 November 2008 (EST)
A fine example of tolerance. Fall down
Hate speech is not tolerated by people with an IQ over 80, Fall Down, Fall Down. InaVegt 08:59, 8 November 2008 (EST)
Fall of the House of Usher. Lily Ta, wack! 09:05, 8 November 2008 (EST)
## Bad site link
Guys, he's an idiot, but you can't just undo it when he deletes people's talk comments from this page. It's his talk page, he should be able to do whatever he wants with it even if he's not a member of the cool crowd.--Tom Moorefiat justitia ruat coelum 09:23, 8 November 2008 (EST)
Wrong, on RationalWiki talk pages, ALL talk pages, are community property and may not be deleted unless archived. and butter 09:29, 8 November 2008 (EST)
Wow, you're right, as I see when I check the community standards. Huh. Well, I'm not happy about that, but I guess it's the rule. Carry on :) --Tom Moorefiat justitia ruat coelum 09:36, 8 November 2008 (EST)
You've only to look at what eg Conservative does on cp to see one reason. and butter 09:40, 8 November 2008 (EST)
Surely there's a rickroll exemption? Or a vandalism exception in general? DickTurpis 09:51, 8 November 2008 (EST)
Also, the Ken comparison isn't valid. Ken reverts and deletes; Fall Down doesn't have that ability. DickTurpis 09:53, 8 November 2008 (EST)
No - but it's A reason. -> Dick, common sense should prevail. :) and butter 09:56, 8 November 2008 (EST)
Common sense dictates that if someone posts am obnoxious link on your talk page, you can remove it. As someone who clicked that link and had a hell of time trying to get it to go away, I have to say it's pretty obnoxious. So is this guy, sure, but let's not be assholes too. DickTurpis 10:03, 8 November 2008 (EST)
Guys, it really doesn't matter that much. I assume, from his desire to delete it, that Fall Down has already been RicRolled, which was the intention. Reinstating it repeatedly will just result in more people getting RicRolled out of curiosity. So if he wants to delete it again, let him. Yes it is a mildly obnoxious link, but there are much worse. weaseLOId~ 10:25, 8 November 2008 (EST)
Actually, the rick roll didn't work for me, it just crashed. Fall down
Exactly, there are worse sites, and presumably this "you can't remove anything from your talk page without archiving it" rule we seem to adhere to so stringently would apply to those as well. Far from this dickhead acting like the assholes at CP, it is us this time, applying one set of rules to those we don't like and another to the rest of us. Believe me, if someone made an similar annoying post on my talk page, I would delete it, and not allow it to be readded. DickTurpis 10:33, 8 November 2008 (EST)
Fair enough, if poster & postee agree. and butter 10:42, 8 November 2008 (EST)
I think we permit deletion of obnoxious comments or links but whole pages with a multi-editor thread need to be archived. Lily Ta, wack! 11:31, 8 November 2008 (EST)
Perhaps with regard to this particular link it should have been replaced with: "A link to x site existed here, but it was removed because y". With regard to rules being applied differently - as far as I am aware the "Don't remove talk page stuff" rule is applied to everybody. If not, then it obviously should be. Equally we should remember that our RationalWiki:Community Standards page where the rules are defined states: These are not site rules but rather a list of standards we as a community want to live up to, which gives a little bit of latitude in certain situations.--Bobbing up 12:03, 8 November 2008 (EST)
Jeez, one can always leave a comment next in line - or before the toxic link - saying what it is... ħuman 19:43, 8 November 2008 (EST)
I did that (a superscripted WARNING RICKROLL) but was overruled by other editors - so it goes. and butter 19:54, 8 November 2008 (EST)
## Sharks.
Nice try on the funny. But not quite. PFoster 16:50, 8 November 2008 (EST)
I believe shars are fish. Fall down 19:09, 8 November 2008 (EST)
A shark is a fish, a shar(-pei) is a dog! and butter 19:31, 8 November 2008 (EST)
Fish are vertebrates... ħuman 19:44, 8 November 2008 (EST)
## 'sup, my pleasant-spoken friend.
Are you going to finish this? If my essay's so unreasonable, surely it should be easy to rebut?
Being bothered is a second key factor to rebuttal. Frankly, I'm not sure I can be bothered to rebut the workings of a troll so lame his idea of a website invasion is one (1) guy writing a mildly offensive essay that doesn't inspire rage so much as mild annoyance, and who receives a grand total of none (0) for his call for a crusade to 'take down this place'. So yeah, nice try; one can only fault, well, everything else. You can expect some sort of further comment, well, in time.
Oh, and why don't you unfuck your user page; I can't, for example, get to your talk from it.
lol internets
## "I'm banned from Wikipedia because of a conspiracy of female admins."
That is the stupidest thing I have ever read. And I read Conservapedia. Please, go get a life. Better yet, get a clue first, before trying to impose your lame self on "life". I hope your mother does not know how much of an asshole you are. ħuman 01:01, 23 November 2008 (EST)
Actually I thought it was quite funny. Am I seeing Poe's Law in too many places? --Bobbing up 03:15, 23 November 2008 (EST)
Please give links. We can't assess this unless we know how and why you were banned, assuming you were banned. Proxima Centauri 04:26, 23 November 2008 (EST)
## It is okay...
...just because some woman turned you down you don't have to be a wanker forever. If you had a nicer attitude to women it would help. - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 23:09, 23 November 2008 (EST)
...also, don't worry too much about your *ahem*, "size" issues. She was laughing WITH you. Really. PFoster 23:10, 23 November 2008 (EST)
...and if a girl says you're smelly... not saying anything, but showers don't actually hurt unless the water's too hot. Wazza (Not Wazzock, Wazza)Approach the Presence 23:45, 23 November 2008 (EST)
## Hi there!
You found any friends yet? - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 22:05, 28 November 2008 (EST)
Why don't you make useful edits like that more often? - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 03:24, 29 November 2008 (EST)
Where did I change your words, all I did was break the link. - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 22:41, 29 November 2008 (EST)
## Power and evil
If all power is evil, then evil is omnipotent and we who are virtuous are powerless before it. At which point you might as well just give and go away. Researcher 10:51, 5 December 2008 (EST)
Though for the record, I don't consider you at all virtuous. Just saying. Researcher 10:52, 5 December 2008 (EST)
That's right, we might as well all kill ourselves; makes as much sense as anything. More seriously, no human agency is omnipotent. And we can never abolish evil (or power). Fall down
For a guy who thinks power is evil, why would he support paedophelia, which is exertion of power over the powerless, namely children? Seems pretty clear-cut. --Kels 10:56, 5 December 2008 (EST)
Since I don't support paedophilia - that's just PFoster's slander against me - that question is moot. Fall down
You don't address the logical fallacy in your argument. By your reasoning, no moral human agency can have any power whatsoever. Researcher 16:39, 10 December 2008 (EST)
## And this is about
I've finally managed to work my way back to what seems to be at the bottom of all this. (Unless there is more even more deeply buried) Some twelve lines of insanity. True insanity. But I can't help feeling that there may be a small degree of overreaction here.--Herbert the Hamster 14:15, 5 December 2008 (EST)
I'm not sure what you found. Care to explain? Fall down— Unsigned, by: I am the truth / talk / contribs
By all means. I found this. As I said, 12 lines of insanity (though that will depend on your screen size and font.) Was that what you were asking me?--Herbert the Hamster 03:49, 6 December 2008 (EST)
You missed this where he attempts to justify some paedophiles, this which has some lovely sentiments about gays, and this, which is just weird. --Kels 10:23, 6 December 2008 (EST)
Thanks for the pointers. An interesting collections of opinions and prejudices I must admit. I rather get the impression that this user has been fought over sufficiently so I'll say no more.--Herbert the Hamster 10:38, 6 December 2008 (EST)
## Censorship
You’re right there is unofficial censorship everywhere. Free speech doesn’t force us to give a platform to those who disagree with us. You are free to express yourself on Conservative websites and neutral websites. We are free to express ourselves on Liberal websites and neutral websites.
### Does that mean Conservapedia censorship is OK?
No it doesn’t because Conservapedia is so extreme, so irrational and because Conservapedia even censors other Conservatives that are slightly different from Aschlafly. Andy uses his momma’s money to buy media space that’s only fit to be laughed at.
Christianity Knowlege Base is less extreme in its censorship. They are not attacked by vandals the way Conservapedia is. Proxima Centauri 02:46, 7 December 2008 (EST)
I think you're missing the point a bit, PC. Fall down is a pedophilia apologist and general woman-hater. Nambla or the local police station might be better places for it to spew its hatred. ħuman 02:59, 7 December 2008 (EST)
## Renames
I must admit that I'm not up to speed on all of the socks here. Should we rename then to "Fall Down 1", "Fall Down 2" etc? Or are there only a few of them? I've been away for a while.--Bobbing up 05:14, 7 December 2008 (EST)
That is possible do you have a list? - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 22:36, 8 December 2008 (EST)
He kindly put a list of them at rationalwikiwiiuiwkiwiki. Check his contribs there. ħuman 19:52, 15 December 2008 (EST)
It would be nice to get some sort of consensus first. I'm not entirely convinced with the rename to "Women are my superior and I respect them as such." The name "Fall Down" was not offensive in itself.--Bobbing up 14:43, 16 December 2008 (EST)
I think Pi renamed him just to humiliate him, a bit childish, but fun! Toast 14:46, 16 December 2008 (EST)
## I love you
I usually tolerate a troll. Your contribution to some of the main space articles would normally only warrant me vandal breaking you. I have even defended your right to free speech to have that bigoted little piece of shit you call an essay remain although buried. But your recent behaviour in harassing a user of this site is unacceptable. You no long have recourse to be unblocked and if I see one of your pathetic socks it is being thrown out. It has been explained to you how you can express your views in a rational way and you have chosen not to go down that path. You are not as you claim rational, I find it hard to believe that someone that demonstrates a hatred you do is capable of rational thought. I would argue with you but it is not worth my time, so I will simply say FUCK YOU AND GET THE FUCK OFF THIS SITE!!! - User Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \scriptstyle-i\ln(-1) 22:01, 13 December 2008 (EST)
I'm not harassing anyone, really. I've done what I did only to users that have blocked me, and thus deserve it. I have no desire to make trouble, really, only to be recognised as having the same right to my views that you have. N, you haven't explained to me 'how I could express my views in a rational way', at least not sensibly; I could not express my views without offending people that are offended by them. I hate no one; as my user page explains, I only hate lies, dishonesty, and censorship. Fall down
Your actions have already invalidated any credibility you might have had (and given your words beforehand, you didn't have any in the first place). I agree with Pi, I consider you a vandal, and anything you say will be treated the same as any other vandalism. --Kels 22:33, 13 December 2008 (EST)
Actually, no, you have engaged in a ceaseless attack on women and homosexuals, without even the slightest pretense at rationality. And, yes, you apparently do hate women and homosexuals, considering the way you constantly talk about them. Personally, I'm starting to think you are a little disturbed and need some help. But I'm not a licensed therapist, and even if I was, I probably couldn't deal with you in that capacity. Researcher 22:37, 13 December 2008 (EST)
Given that Mr. Misogynist Homophobe Stalker doesn't want to give up on this conversation, and he's in the long term ban/vandal bin anyhow, do we have support for locking the talk page? I don't have any interest in giving him a soapbox to try to defend the indefensible on. --Kels 22:55, 13 December 2008 (EST)
As just a general user I support this...I've been watching this exchange with a growing amount of disgust and am anxious to see it end Thinker 22:58, 13 December 2008 (EST)
Well, I'd say that last little childish outburst was Fall Down voting in favor as well. I'll give it a few minutes to see if we have any comment from some of the locals, but it seems pretty much a given now. --Kels 23:06, 13 December 2008 (EST)
I'm really at the point where I'm convinced the guy is actually crazy. I'm all for nuking whatever remnants he's left around. Researcher 23:07, 13 December 2008 (EST)
Sounds good. If anyone's still got issues, we can still discuss whether or not to unlock, but there seems little point. --Kels 23:10, 13 December 2008 (EST)
Actually, I don't see too much point in locking it.--Bobbing up 02:34, 14 December 2008 (EST)
## Block time
I haven't got a good enough calculator to work out how long I blocked him for; could someone else work it out? Cubic bastard Hoover!
## Manipulation
He made 2 socks called Rose Pedals. Computers could tell them apart but humans couldn't easily. I blocked both. The first got blocked after a post that looked like vandalism but was less offensive than stuff he'd been spewing for several days. I blocked the second one at once before there were any edits because it was obviously deliberately confusing. If I hadn't spotted the second account I don't know how much confusion he'd have caused with those similar names. Then I got into trouble for blocking the second account. I should have given a block comment saying he was abusing multiple accounts. I was just fed up of reverting obscene vandalism to my userpage, my talk page and generally. Therefore I blocked quickly and assumed others would know what was happening during a prolonged vandal attack. He set out to get me so angry that I'd make a mistake. If a vandal posted obscene vandalism repeatedly onto a man's userpage over several days that man could get angry and make mistakes too. He picked on me because I'm a woman. Proxima Centauri 14:51, 16 December 2008 (EST) | 2022-05-17 06:59:11 | {"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": 6, "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": 0.5366644859313965, "perplexity": 3519.6937126173216}, "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-2022-21/segments/1652662517018.29/warc/CC-MAIN-20220517063528-20220517093528-00613.warc.gz"} |
https://blancosilva.wordpress.com/puzzles/my-oldest-plays-the-piano/ | ## My oldest plays the piano!
Two old friends, Ernie and Bernie, bump into each other in the street. It was more than twenty years since their last meeting, so they decide to spend some time together in a nearby bar, chatting about their lives. At some point, Bernie asks the inevitable: “So, you got married?”
“Well, yes!”—claims enthusiastically Ernie. “And I have three beautiful children!”
“That’s great, Ernie! And how old are your kids?”—enquired Bernie.
“I know you are a sucker for puzzles, Bernie, so I will let you figure it out through a few clues. You should not have any trouble getting it. First clue: If you add the ages of my kids, the result is thirteen
“Whoa! That doesn’t help me much, does it?”—complains Bernie. “Couldn’t you give me another clue?”
“Sure, a second clue: If you multiply their ages together, the result is the same as how much we payed for these beers”
Bernie scratches his head for a minute, and cannot figure it out yet… Before he complains again, Ernie realizes the mistake, apologizes, and offers Bernie the last clue: “My oldest plays the piano!”
Bernie had no trouble finding the ages of the children this time.
These are all the possible combinations of positive integers $x$, $y$, $z$ that add up to thirteen, together with their product:
$\begin{array}{|c|c|c||r|} \hline x&y&z& x\cdot y \cdot z \\ \hline 1&1&11&11 \\ \hline 1&2&10&20 \\ \hline 1&3&9&27 \\ \hline 1&4&8&32 \\ \hline 1&5&7&35 \\ \hline 1&6&6&\mathbf{36} \\ \hline 2&2&9&\mathbf{36} \\ \hline \end{array}$ $\begin{array}{|c|c|c||r|} \hline x&y&z& x\cdot y \cdot z \\ \hline 2&3&8&48 \\ \hline 2&4&7&56 \\ \hline 2&5&6&60 \\ \hline 3&3&7&63 \\ \hline 3&4&6&72 \\ \hline 3&5&5&75 \\ \hline 4&4&5&80 \\ \hline \end{array}$
Notice that there are only two combinations that offer the same product: $(x,y,z) = (1,6,6)$ and $(x,y,z) = (2,2,9)$. The fact that Bernie did not know the answer to the riddle after the second clue, indicates that none of the other possibilities is right. He thus needs a third clue to decide between the two choices above.
I would not think of stealing the pleasure to solve the puzzle to my reader. What are the ages of Ernie’s children?
### Miscellaneous
With the aid of WolframAlpha, one should be able to find quickly all the intermediate steps to the solution. One must be careful, nevertheless, to input the right expressions. For example, the reaction of WolframAlpha to the query “all integer solutions to a+b+c=13 && a>0 && b>0 && c>0 && a<14 && b<14 && c<14” offers only one (insufficient) possibility:
A better way to go about it is, for example, to generate the solutions in python. A possible script to obtain the solutions would look like this:
for a in range(1,14):
for b in range(a,14):
for c in range(b,14):
if (a+b+c==13):
print ‘%6s %6s %6s %6s’%(a,b,c,a*b*c)
1 1 11 11
1 2 10 20
1 3 9 27
1 4 8 32
1 5 7 35
1 6 6 36
2 2 9 36
2 3 8 48
2 4 7 56
2 5 6 60
3 3 7 63
3 4 6 72
3 5 5 75
4 4 5 80
1. November 14, 2010 at 9:35 am
Thank you so much for all your help!
• November 14, 2010 at 9:48 am
How did it go in the exam?
2. June 17, 2011 at 6:13 am
I never liked this question. Even if you have two children that are both 6, that does not preclude one from being older than the other. Twins are often identified as the “older” sibling and the “younger” one. You could even have a situation where the two children were born 11 months apart, the oldest is 6 years 11+ months and the other having just turned 6.
• June 17, 2011 at 4:40 pm
Indeed! As a matter of fact, if you decide not to follow the convention for stating one’s age, then the condition “the oldest plays the piano” will not make sense at all!
I am cooking up a generalization of that problem. Here is a sneak preview:
(1) Ages are given as strictly positive real numbers (think floating point) at a given time. Say, one of the kids was 5.874235345298452093845234908 years-old at 5.00pm EST on June 17, 2011.
(2) Children are allowed to come from different mothers. That way, it is possible to have two or more kids with exactly the same age.
Even with these constraints, the problem will make sense and, instead of being solved with traditional integer handling, one would have to set it as optimization. Cool, huh?
I will post something about it soon. Thanks a lot, David! | 2018-03-23 09:00:36 | {"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": 7, "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": 0.42952507734298706, "perplexity": 1031.3718081239683}, "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/1521257648205.76/warc/CC-MAIN-20180323083246-20180323103246-00730.warc.gz"} |
http://math.stackexchange.com/questions/76488/similar-linear-transformations | # Similar Linear Transformations
This is a nice question I came across in Linear Algebra but I cant figure out how to tackle it. I need some help.
Given two linear transformations, $E$ and $F$ such that $E^2=E$ and $F^2=F$, I am supposed to determine if it is true that $E$ and $F$ are similar if and only if $rank(E)=rank(F)$.
-
Have you encountered such transformations (projections) before? Do you know some of their properties? – Olivier Bégassat Oct 27 '11 at 22:13
Hint: What are the possible eigenvalues of $E$ and $F?$ And if you put $E$ and $F$ into Jordan normal form can $1$ occur on their upper diagonals? – jspecter Oct 27 '11 at 22:13
Is there a possibility of avoiding eigenvalues in the solution to this question? – smanoos Oct 27 '11 at 22:43
Since $E^2=E$ and $F^2=F$, their minimal polynomials must divide $x^2-x=x(x-1)$. Thus their minimal polynomials cannot have repeated factors and so they are both diagonalizable.
Next, by nature of the minimal polynomials dividing $x(x-1)$, the eigenvalues of $E$ and $F$ must be $1$'s and $0$'s. Thus your answer is "Yes." If their rank is the same, the same number of $1$'s will appear in both diagonalizations. If their rank differs, they must have a different number of $1$'s in their diagonalizations and so must not be similar.
Half of this is easy - it's an exercise to show that if $E$ and $F$ don't have the same rank then they can't be similar. The other half, maybe this is a good start: let $X$ be the range of $E$, let $Y$ be the range of $F$. If $E$ and $F$ have the same rank then $X$ and $Y$ have the same dimension, so there's an isomorphism $T$ such that $TX=Y$. Maybe there's a way to combine that with $E^2=E$ and $F^2=F$ to get a proof without eigenvalues (though I don't see it just yet). – Gerry Myerson Oct 27 '11 at 23:28 | 2015-07-05 00:34:23 | {"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": 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": 0.9629862308502197, "perplexity": 121.83148683165054}, "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-2015-27/segments/1435375097038.44/warc/CC-MAIN-20150627031817-00106-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://ora.ox.ac.uk/objects/uuid:33c9ef6e-233b-47ba-a9bb-7b4bbdeb3a16 | Journal article
### Study of bb¯ correlations in high energy proton-proton collisions
Abstract:
Kinematic correlations for pairs of beauty hadrons, produced in high energy proton-proton collisions, are studied. The data sample used was collected with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV and corresponds to an integrated luminosity of 3 fb^−1 . The measurement is performed using inclusive b → J/ψX decays in the rapidity range 2 < y J/ψ < 4.5. The observed correlations are in good agreement with theoretical predictions.
Publication status:
Published
Peer review status:
Peer reviewed
Version:
Publisher's version
### Access Document
Files:
• (pdf, 1.4MB)
Publisher copy:
10.1007/JHEP11(2017)030
### Authors
More by this author
Institution:
University of Oxford
Division:
MPLS Division
Department:
Physics; Particle Physics
Role:
Author
More by this author
Institution:
University of Oxford
Division:
MPLS Division
Department:
Physics; Particle Physics
Role:
Author
Publisher:
Springer Publisher's website
Journal:
Journal of High Energy Physics Journal website
Volume:
2017
Issue:
11
Pages:
30
Publication date:
2017-11-24
Acceptance date:
2017-10-20
DOI:
EISSN:
1029-8479
ISSN:
1029-8479
Pubs id:
pubs:724348
URN:
uri:33c9ef6e-233b-47ba-a9bb-7b4bbdeb3a16
UUID:
uuid:33c9ef6e-233b-47ba-a9bb-7b4bbdeb3a16
Local pid:
pubs:724348
Keywords: | 2021-07-28 21:32:27 | {"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": 0.8659707903862, "perplexity": 14901.089512625746}, "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-2021-31/segments/1627046153791.41/warc/CC-MAIN-20210728185528-20210728215528-00429.warc.gz"} |
https://www.electro-tech-online.com/threads/glcd-wayton-mg1206e-cs1-cs2-polarity.150917/ | # GLCD (Wayton MG1206E) CS1 CS2 polarity ...
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##### Well-Known Member
I had this 2.5” 128x64 GLCD (WAYTON MIT MG1206E SGL) in my parts box , and in a crazy moment I thought I would see if it worked ( It was from a Maplin grab bag of led / lcd junk) , found some data on the www and lashed up a test bed, having realised graphics LCD require big-ish data squirted at them I used a dsPIC33EV256GM104 it is has 16k ram available. ( no details of the 20 pins on the pcb ) several days later ….. success. Well some lines on the screen, seems NT7108 / KS0108 controllers are not all the same, this one has a high level CS1 and CS2 ,( Left and right side controllers) not seen any mention of this on the files / posts found on the web. So though I would share. My intention is to write fonts / graphics into an array in PIC memory and just display that….
( colours are just my BB wire reference ).
Last edited:
#### Pommie
##### Well-Known Member
Not sure if it's useful or not but attached is some code I wrote about 10 years ago for one of these displays.
Mike.
#### Attachments
• 13.3 KB Views: 70
#### Ian Rogers
##### User Extraordinaire
Forum Supporter
Wow Mike... That's what I call a concise library.. Mine is half that size.. I do, however, use a double buffer, so alot of my code write's are to the spare canvas...
Well done though!!
##### Well-Known Member
Thank you Mike, I'm getting to understand the GLCD controller ( possibly ! )
.
#### Pommie
##### Well-Known Member
I particularly like the circle routine because it completely baffles people how it works.
Code:
void Circle(unsigned char cx,unsigned char cy,unsigned char r){
unsigned char x,y,p;
x=0;
y=r;
p=-r/2;
while(x<=y){
Plot8(cx,cy,x,y);
x=x+1;
if(p>128){
p+=2*x+1;
}else{
y--;
p+=2*(x-y)+1;
}
}
}
Mike.
#### Pommie
##### Well-Known Member
The way the circle routine works is the same as those nail and string curved paterns. The axes are at right angles in the circle case and a quarter circle is generated by working along "the nails". Using lines at different angles and lengths should give various arcs.
Mike
#### Ian Rogers
##### User Extraordinaire
Forum Supporter
The way the circle routine works is the same as those nail and string curved paterns. The axes are at right angles in the circle case and a quarter circle is generated by working along "the nails". Using lines at different angles and lengths should give various arcs.
Mike
I know.... But a arc could cross the 90 degree, that means the 8 point routine won't do the job.. So if you come across one!! I have written one, but it's a little slow, too many iterations... if this and if that!! Ill convert it to asm to see if I can boost the speed..
#### atferrari
##### Well-Known Member
I had this 2.5” 128x64 GLCD (WAYTON MIT MG1206E SGL) in my parts box , and in a crazy moment I thought I would see if it worked ( It was from a Maplin grab bag of led / lcd junk) , found some data on the www and lashed up a test bed, having realised graphics LCD require big-ish data squirted at them I used a dsPIC33EV256GM104 it is has 16k ram available. ( no details of the 20 pins on the pcb ) several days later ….. success. Well some lines on the screen, seems NT7108 / KS0108 controllers are not all the same, this one has a high level CS1 and CS2 ,( Left and right side controllers) not seen any mention of this on the files / posts found on the web. So though I would share. My intention is to write fonts / graphics into an array in PIC memory and just display that….
( colours are just my BB wire reference ).
View attachment 106034
View attachment 106031
Long time I do not use one of these. From what I recall, not all manufacturer use CS1 ans CS2 in the same way. Besides the allocation of pins, that is what distinguish GLCDs from each other.
Last edited:
#### atferrari
##### Well-Known Member
I inherited from a failed replacement (difference in pins allocation) a brand new one.
After searching for the datasheet of a quickly vanished manufacturer in Brazil, I realized that there were kind of discernible patterns among the variation for different brands.
I finally compiled the data from several tens of datasheets and based on the pins allocated to power I could use mine successfully. The data is shown in the attached pdf. With my "electronic" PC down, it is good I got a proper back up handy. Hope you can wade through it.
One day I should post my implementation of the John Conway's game "Life". A piece of judicious programming I enjoyed much. Really.
¡Buena suerte!
#### Attachments
• 9 KB Views: 73
##### Well-Known Member
Thank you at , let wading commence...
##### Well-Known Member
Mike, Not copy and paste of your code , but got lots of help etc, understand the true type working , just have to reinvent it for my project... Cheers.
##### Well-Known Member
I now have 2 more 128x64 GLCD modules to play with , a Winstar , ( blue white) , 15GBP from Rapid and an 'unknown' from China ( cheap) , this last one looks interesting, seems to be similar to the character (HD44780) LCDs , the single controller (ST7920) can do 8 bit or 4 bit parallel , or serial ... (SPB pin) and has build in character font .
#### Pommie
##### Well-Known Member
Looks like good progress. I notice the odd spacing on the font above. I used a 5 byte wide proportional font and used 0x55 for bytes that should be ignored. If you modify your code it should look a lot better.
Mike.
##### Well-Known Member
0x55 = Got you Mike .. Lashup has condensed to 28pin dsPIC33EV , but could do with some spare I/O pins , I was planning to take GLCD RST to Vdd and W/R to Gnd. as I don't think I need to read the display ram or busy status.. ? appreciate your ( anybody's) input ....
#### atferrari
##### Well-Known Member
For 2x16 and 2x20 LCDs I use a fixed delay, so no check of busy condition. Pin is, IIRC, grounded.
##### Well-Known Member
Moved up to the larger Winstar 128x64 blue display, and thanks to Pommie Mike's file , have the basics working , with RST at Vdd and R/W to gnd . Invert options and font spacing sorted, the dsPIC33 , works well only used 2% of program space ... also have a few pins to play with . | 2021-05-10 05:00:14 | {"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": 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": 0.33334389328956604, "perplexity": 5360.354830700344}, "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-2021-21/segments/1620243989030.87/warc/CC-MAIN-20210510033850-20210510063850-00352.warc.gz"} |
https://par.nsf.gov/biblio/10338975-first-measurements-subjettiness-central-pb-pb-collisions-sqrt-s_-mathrm-nn-tev | First measurements of N-subjettiness in central Pb-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ = 2.76 TeV
A bstract The ALICE Collaboration reports the first fully-corrected measurements of the N -subjettiness observable for track-based jets in heavy-ion collisions. This study is performed using data recorded in pp and Pb-Pb collisions at centre-of-mass energies of $$\sqrt{s}$$ s = 7 TeV and $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 2 . 76 TeV, respectively. In particular the ratio of 2-subjettiness to 1-subjettiness, τ 2 /τ 1 , which is sensitive to the rate of two-pronged jet substructure, is presented. Energy loss of jets traversing the strongly interacting medium in heavy-ion collisions is expected to change the rate of two-pronged substructure relative to vacuum. The results are presented for jets with a resolution parameter of R = 0 . 4 and charged jet transverse momentum of 40 ≤ p T , jet ≤ 60 GeV/ c , which constitute a larger jet resolution and lower jet transverse momentum interval than previous measurements in heavy-ion collisions. This has been achieved by utilising a semi-inclusive hadron-jet coincidence technique to suppress the larger jet combinatorial background in this kinematic region. No significant modification of the τ 2 /τ 1 observable for track-based jets in Pb-Pb collisions is observed relative to vacuum PYTHIA6 more »
Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
Award ID(s):
Publication Date:
NSF-PAR ID:
10338975
Journal Name:
Journal of High Energy Physics
Volume:
2021
Issue:
10
ISSN:
1029-8479
1. A bstract A measurement of the inclusive b-jet production cross section is presented in pp and p-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV, using data collected with the ALICE detector at the LHC. The jets were reconstructed in the central rapidity region |η| < 0 . 5 from charged particles using the anti- k T algorithm with resolution parameter R = 0 . 4. Identification of b jets exploits the long lifetime of b hadrons, using the properties of secondary vertices and impact parameter distributions. The p T -differential inclusive production cross section of b jets, as well as the corresponding inclusive b-jet fraction, are reported for pp and p-Pb collisions in the jet transverse momentum range 10 ≤ p T , ch jet ≤ 100 GeV/ c , together with the nuclear modification factor, $${R}_{\mathrm{pPb}}^{\mathrm{b}-\mathrm{jet}}$$ R pPb b − jet . The analysis thus extends the lower p T limit of b-jet measurements at the LHC. The nuclear modification factor is found to be consistent with unity, indicating that the production of b jets in p-Pb at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV is not affected bymore »
2. A bstract Jet fragmentation transverse momentum ( j T ) distributions are measured in proton-proton (pp) and proton-lead (p-Pb) collisions at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV with the ALICE experiment at the LHC. Jets are reconstructed with the ALICE tracking detectors and electromagnetic calorimeter using the anti- k T algorithm with resolution parameter R = 0 . 4 in the pseudorapidity range |η| < 0 . 25. The j T values are calculated for charged particles inside a fixed cone with a radius R = 0 . 4 around the reconstructed jet axis. The measured j T distributions are compared with a variety of parton-shower models. Herwig and P ythia 8 based models describe the data well for the higher j T region, while they underestimate the lower j T region. The j T distributions are further characterised by fitting them with a function composed of an inverse gamma function for higher j T values (called the “wide component”), related to the perturbative component of the fragmentation process, and with a Gaussian for lower j T values (called the “narrow component”), predominantly connected to the hadronisation process. The width of the Gaussian has only amore »
3. A bstract A measurement of inclusive, prompt, and non-prompt J/ ψ production in p-Pb collisions at a nucleon-nucleon centre-of-mass energy $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV is presented. The inclusive J/ ψ mesons are reconstructed in the dielectron decay channel at midrapidity down to a transverse momentum p T = 0. The inclusive J/ ψ nuclear modification factor R pPb is calculated by comparing the new results in p-Pb collisions to a recently measured proton-proton reference at the same centre-of-mass energy. Non-prompt J/ ψ mesons, which originate from the decay of beauty hadrons, are separated from promptly produced J/ ψ on a statistical basis for p T larger than 1.0 GeV/ c . These results are based on the data sample collected by the ALICE detector during the 2016 LHC p-Pb run, corresponding to an integrated luminosity $$\mathcal{L}$$ L int = 292 ± 11 μ b − 1 , which is six times larger than the previous publications. The total uncertainty on the p T -integrated inclusive J/ ψ and non-prompt J/ ψ cross section are reduced by a factor 1.7 and 2.2, respectively. The measured cross sections and R pPb are compared withmore »
4. Abstract The measurement of the azimuthal-correlation function of prompt D mesons with charged particles in pp collisions at $$\sqrt{s} =5.02\ \hbox {TeV}$$ s = 5.02 TeV and p–Pb collisions at $$\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}$$ s NN = 5.02 TeV with the ALICE detector at the LHC is reported. The $$\mathrm{D}^{0}$$ D 0 , $$\mathrm{D}^{+}$$ D + , and $$\mathrm{D}^{*+}$$ D ∗ + mesons, together with their charge conjugates, were reconstructed at midrapidity in the transverse momentum interval $$3< p_\mathrm{T} < 24\ \hbox {GeV}/c$$ 3 < p T < 24 GeV / c and correlated with charged particles having $$p_\mathrm{T} > 0.3\ \hbox {GeV}/c$$ p T > 0.3 GeV / c and pseudorapidity $$|\eta | < 0.8$$ | η | < 0.8 . The properties of the correlation peaks appearing in the near- and away-side regions (for $$\Delta \varphi \approx 0$$ Δ φ ≈ 0 and $$\Delta \varphi \approx \pi$$ Δ φ ≈ π , respectively) were extracted via a fit to the azimuthal correlation functions. The shape of the correlation functions and the near- and away-side peak features are found to be consistent in pp and p–Pb collisions, showing no modifications due to nuclear effects withinmore »
5. A bstract Measurements of the production cross-sections of the Standard Model (SM) Higgs boson ( H ) decaying into a pair of τ -leptons are presented. The measurements use data collected with the ATLAS detector from pp collisions produced at the Large Hadron Collider at a centre-of-mass energy of $$\sqrt{s}$$ s = 13 TeV, corresponding to an integrated luminosity of 139 fb − 1 . Leptonic ( τ → ℓν ℓ ν τ ) and hadronic ( τ → hadrons ν τ ) decays of the τ -lepton are considered. All measurements account for the branching ratio of H → ττ and are performed with a requirement |y H | < 2 . 5, where y H is the true Higgs boson rapidity. The cross-section of the pp → H → ττ process is measured to be 2 . 94 ± $$0.21{\left(\mathrm{stat}\right)}_{-0.32}^{+0.37}$$ 0.21 stat − 0.32 + 0.37 (syst) pb, in agreement with the SM prediction of 3 . 17 ± 0 . 09 pb. Inclusive cross-sections are determined separately for the four dominant production modes: 2 . 65 ± $$0.41{\left(\mathrm{stat}\right)}_{-0.67}^{+0.91}$$ 0.41 stat − 0.67 + 0.91 (syst) pb for gluon-gluon fusion, 0 .more » | 2022-12-08 20:34:04 | {"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": 0.916260838508606, "perplexity": 2071.361649436721}, "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/1669446711360.27/warc/CC-MAIN-20221208183130-20221208213130-00459.warc.gz"} |
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Tommaso Bolognesi: on 6/20/14 at 7:56am UTC, wrote Best regards Tommaso
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Lorraine Ford: on 6/8/14 at 4:25am UTC, wrote Dear Tommaso, Re "I don't buy the complexity viewpoint": As you know, in...
Lorraine Ford: on 6/7/14 at 1:07am UTC, wrote Dear Tommaso, Marvellous essay! It's a literary work; it's amusing; it...
Tommaso Bolognesi: on 6/4/14 at 11:08am UTC, wrote Thanks for the comments Robert. I see your point, and I agree that we may...
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CATEGORY: How Should Humanity Steer the Future? Essay Contest (2014) [back]
TOPIC: Humanity is much more than the sum of humans by Tommaso Bolognesi [refresh]
Author Tommaso Bolognesi wrote on Mar. 27, 2014 @ 15:25 GMT
Essay Abstract
Consider two roughly spherical and coextensive complex systems: the atmosphere and the upper component of the biosphere - humanity. It is well known that, due to a malicious antipodal butterfly, the possibility to accurately forecast the weather - let alone controlling it - is severely limited. Why should it be easier to predict and steer the future of humanity? In this essay we present both pessimistic and optimistic arguments about the \emph{possibility} to effectively predict and drive our future. On the long time scale, we sketch a software-oriented view at the cosmos in all of its components, from spacetime to the biosphere and human societies, borrowing ideas from various scientific theories or conjectures; the proposal is also motivated by an attempt to provide some formal foundations to Teilhard de Chardin’s cosmological/metaphysical visions, that relate the growing complexity of the material universe, and its final fate, to the progressive emergence of consciousness. On a shorter scale, we briefly discuss the possibility of using simple formal models such as Kauffman’s boolean networks, and the growing body of data about social behaviours, for simulating humanity ’in-silico’, with the purpose to anticipate problems and testing solutions.
Author Bio
Tommaso Bolognesi (Laurea in Physics, Univ. of Pavia, 1976; M.Sc. in CS, Univ. of Illinois at U-C, 1982), is senior researcher at ISTI, CNR, Pisa. His research areas have included stochastic processes in computer music composition, models of concurrency, process algebra and formal methods for software development, discrete and algorithmic models of spacetime. He has published on various international scientific journals several papers in all three areas. His essay ‘Reality is ultimately digital, and its program is still undebugged’ has obtained a 4th prize at the 2011 FQXi Essay Contest ‘Is Reality Digital or Analog’?
Janko Kokosar wrote on Mar. 29, 2014 @ 20:47 GMT
Dear Mr. Tommaso Bolognesi
In principle, we speak very similar things, (and my old reference.) You gave a new reference for me (Teilhard Chaitin), who claims "A stone has a soul... but a very small one", similarly as I. I wrote there: "If it is assumed that consciousness exists outside the biological world, panpsychism is obtained, where consciousness is everywhere." "It is possible to go still further in a matter of the non-biological world, and there can be also some very low memory (but extremely low one) and a short duration of it."
I also write about consciousness of many people, but more about consciousness of two people.
You have concentrated on Tononi's principle of complexity, but my approach is a little different: "Memory is maintaining consciousness, and the quality of two consciousnesses is the same." It is more important for me, what is the smallest basic element of consciousness. It is not easy to say, how my and Tononi's principles are distinct. Other things are also important, for instance, qualia.
I hope that "Brain Activity Map Project" and other projects will give new information about consciousness.
I think that Linde tells a lot of things, connected with your or my essay.
The topic of the contest cannon be easily connected with physics, but you succeded.
Best Regards Janko Kokosar
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Author Tommaso Bolognesi replied on Mar. 31, 2014 @ 09:15 GMT
Hi Janko,
I see you have links to two distinct essays of yours.
I am actually curious about the 'It from bit of bit from it' essay 'Proposal for Quantum Consciousness', since there you mention Tononi and the fact that you improve on his definitions. Did you find a way to come up with an alternative, simple and precise measure of the 'degree of consciousness' that can be applied to virtually any distributed dynamical system, such as, say, a network of synchronous boolean functions?
Thanks and good luck!
Joe Fisher wrote on Mar. 31, 2014 @ 16:19 GMT
My Dear Mr. Bolognesi,
I found reading your essay quite absorbing and I do hope that it does at least as well as your prior effort in the competition.
In my essay, REALITY, ONCE, I emphasize the fact that the biggest hindrance to understanding reality could very well be the English language. For instance you wrote: “Tomasi measures the amount of consciousness as the quantity of integrated information produced by a complex system.” I am somewhat at a loss wondering what information could possibly be integrated with other than it being a lesser amount of integrated information. Well what about the unintegrated information? I believe that my consciousness is unique, once. I believe that all information is not unique; therefore, all information is unrealistic.
With my very best regards,
Joe Fisher
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Author Tommaso Bolognesi replied on Apr. 3, 2014 @ 12:03 GMT
Hi Joe,
let me directly quote Tononi: In short, integrated information captures the information generated by causal interactions in the whole, over and above the information generated by the parts.
Tononi was able to find a formal way to measure the difference between the amount of information produced by a system when it enters a state, and the amount of information that its parts produce, individually, as that state change occurs. This difference must then correspond to the information produced by the interactions among the parts. These quantities can be measured by the notion of relative entropy, which essentially captures the amount of information produced when moving from one probability distribution (of system states) to another.
I see your point, that consciousness is UNIQUE, ONCE. But this does not prevent one to measure it, if you accept the idea that consciousness is a sort of side effect of the intricate structure of the brain, to which Tononis measures can be applied.
I am not sure I understand what you mean when you write that all information is not unique; therefore all information is unrealistic. Perhaps you mean that bits are generic, or all equal, and brains are specific, or all different. And you attribute the status of reality to things that are unique, specific, and not generic. In this respect you should probably conclude that the mathematical universe (as discussed by Max Tegmark) is the most unrealistic of all possible universes . . . Anyway, I tend to sympathize with Teilhards idea that the ultimate fabric of the universe is indeed made of indistinguishable, abstract, entities (the bits fit this definition), and that the evolution is such that more and more complex entities emerge, up to the brains, which are indeed unique.
Turil Sweden Cronburg wrote on Apr. 1, 2014 @ 14:30 GMT
Excellent essay! Thank you so much for writing and sharing this. Thoroughly enjoyable and thought-provoking.
I’d like to suggest that Pascal’s triangle might be the particular cellular automata that our reality functions with... This would mean that while it is unpredictable on any detailed level, without all, or nearly all, the previous information/states available, there are huge locally repeating patterns, as seen in all fractals, and so there is quite a bit of opportunity for prediction, with quite a bit of accuracy. (For example, the bell curve of variation in probability, and the triangular/linear growth and decline that we see in so many trends in all areas of behavior.)
“A binary cell cannot decide to flip itself, or change the boolean function that defines its behavior; but humans can.”
What evidence do you have of this?!
Also, I was surprised you didn’t mention Douglas Hoffstadter’s book I am a Strange Loop, which considers very similar ideas to your self-modifying code entities (and cellular automata, in general), but from a more philosophical and psychological point of view.
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Author Tommaso Bolognesi replied on Apr. 3, 2014 @ 10:45 GMT
Dear Turil,
you ask whether I have evidence that humans can flip themselves, in the sense that they can modify their behavioural rules, while cellular automata cant.
I suppose this can be seen as an instance of the tough question whether we are completely determined in our behaviour by some fixed rule, or we enjoy free will. I believe there exists a scenario in which both things are in some sense true. This is made possible by the fact that this universe is multilevel.
At the bottom level - the ultimate spacetime scale from which everything emerges computationally, including humans - rules are algorithmic, and fixed; or, if they change, this is not under our control.
But at the much higher level of our direct experiences - say, the biosphere - we feel we are able to change our own behaviour. This is illusional, since, under this scenario, even the fact that I have decided to behave better, or worse, is determined by the rules and dynamics at the lower levels, and yet the illusion works fine for us, since we cannot directly experience the causal influences of those lower dynamics on what happens to us.
I also have in mind the argument used by Wolfram for preserving a (weak) notion of free will in a fully deterministic, computational universe. His idea is that it takes no less than 10 computational steps, in a simulation, to find out how the universe, or my life, will look like in 10 universe steps from now, since the computation that the universe is performing is irreducible - no shortcut. Thus, the rule at the bottom is fixed (no free will), but the emergent behaviour appears unpredictable, thus, in a sense, rule-free, spontaneous.
If this is a bit confusing, dont worry. It is for me too. Trying to capture the notions of spontaneity, agency, creativity, free will, in a formal framework, is always troublesome. It is probably the hardest problem, when trying to formalise Teilhards cosmological views.
Turil Sweden Cronburg replied on Apr. 3, 2014 @ 14:36 GMT
Tommaso, thanks for clarifying. I fully agree with your assessment. My own approach, of using binary combinations of awareness has really helped me see both the simplicity and complexity of how things relate to one another, whether they are bits of data or humans.
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James Lee Hoover wrote on Apr. 9, 2014 @ 16:49 GMT
Tommaso,
A learned and interesting adventure conversationally between generations. Your "humanity in silico" makes me think of ants under scrutiny, tagged and studied by higher powers. The social network metaphor is interesting and disturbing at the same time since it seems to accurately represent real life, though with different motivation for observations.
Not having any easy answers for steering billions of separate humans toward a viable future, I am impressed with the images you draw but I still don't know whether to feel optimism about our future.
Jim
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Author Tommaso Bolognesi replied on Apr. 9, 2014 @ 17:28 GMT
People and companies have actually started using data from social networks, e.g. for sentiment analysis, and you can easily imagine the good and bad purposes of this.
I merely sketched a possible positive usage of humanity in silico models, but I myself do not feel particularly attracted by this type of inquiry, while I enjoy much more the ambitious goals expressed by Tommy, Tomas and Alice in the dialogue.
In any case, I am completely aligned with you in not knowing whether to feel optimism about our future . . . which in a sense makes life more interesting.
Thanks
Tommaso
PS
You write that you are impressed by the images I draw: are you referring to metaphors or the actual drawing of Tommy on the couch? This drawing was inspired by a recent novel by Michele Serra (Gli Sdraiati) - unfortunately appeared only in italian - that I would recommend to anyone who has a 19-year old son, or has been 19 very recently.
James Lee Hoover replied on May. 20, 2014 @ 21:28 GMT
Tommaso,
Time is growing short, so I am revisiting and rating. Your response to my questions and comments: "You write that you are impressed by the images I draw: are you referring to metaphors or the actual drawing of Tommy on the couch? This drawing was inspired by a recent novel by Michele Serra (Gli Sdraiati) - unfortunately appeared only in italian - that I would recommend to anyone who has a 19-year old son, or has been 19 very recently."
I was referring to the verbal images but really liked the drawing as well. Having a humanities background, writing fiction, and columns, I tend to appreciate vivid writing and imagery of all kinds, and do take note of political events that affect us. Not to do so, condemns us to the failings of delusional leaders with agendas of self-interest.
Jim
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John C Hodge wrote on Apr. 9, 2014 @ 18:15 GMT
TB
Thank you so much for writing and sharing this thought-provoking essay. Your comments on my essay were very thought-provoking.
“…humanity is not ready to face its stormy future,…”. Life is not for the squeamish. We must face the storms or find peace in the cemetery.
I suggest for all my positivism, I think we are at “life” stage, not even “thought”. And...
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Author Tommaso Bolognesi replied on Apr. 14, 2014 @ 15:52 GMT
Hi John,
similar to your essay, your message is very articulate, and dense with remarks and observations. Ill take the liberty to react only to a few of them.
You observe that the goals of my three characters, Tommy, Tomas and Alice, seem vague and perhaps unachievable. Curiously I am more convinced (and attracted) by their long term vision, involving some form of the...
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George Gantz wrote on Apr. 15, 2014 @ 11:44 GMT
Tomasso - Wonderful and compelling dialogue! Thanks for the journey. However, I did find the overall message of your essay discouraging. My essay (Tip of the Spear) offers a possible counterpoint. So here are two questions:
How does the computational universe (and cellular automata) capture the transcendental effect of self-consciousness? I find it fascinating what happens to the precision and order of mathematics when self-reference is introduced - paradoxes everywhere! Just like quantum physics. Perhaps human consciousness bootstraps free will from quantum indeterminacy - giving us the power to change the computation.
On a different note, is there no room for hope (short of de Chardin's Omega) in the evolutionary trajectory of human civilization? After all, progress seems to have occurred through an emergent process based on human altruism and without conscious human direction?
Thanks! - George
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Author Tommaso Bolognesi replied on Apr. 16, 2014 @ 08:17 GMT
Dear George,
I had already spotted your essay, with its opening by Teilhard de Chardin (for the second time in the history of the world, man will have discovered fire), but you have been faster than me in establishing the contact. Be sure that I will soon comment on your work!
You write that you find the overall message of my essay discouraging.
In fact, you are the second...
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George Gantz replied on Apr. 16, 2014 @ 18:51 GMT
Tommaso -
I suspect (but cannot prove) that the paradox of self-reference in mathematics corresponds to a paradox of self-awareness in consciousness - and that consciousness will at some point prove impervious to empirical or theoretical explanation. Interesting also that we are struggling with paradox in quantum physics as well. What explanation can we give for this most interesting feature of the universe, if not a transcendent one?
Wittgenstein famously wrote " Whereof we cannot speak, thereof we must be silent." Being human, of course, I don't think we will ever stop talking about it.
Cheers - George
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Author Tommaso Bolognesi replied on Apr. 17, 2014 @ 08:56 GMT
Oops, I erroneously wrote my answer to you in the wrong place. Find it below, as an answer to Lawrence B Crowell. Sorry.
Lawrence B Crowell wrote on Apr. 15, 2014 @ 20:31 GMT
Bolognesi,
Your essay was interesting. I have to admit that early on in reading your essay my opinion was not entirely positive. It seemed to start out rather too mysterious. However, the overall theme of your essay later one, particularly once you got into the partition function and the conditional probabilities, became clearer. I had to read your essay twice in order to dispel my initial concerns with it. There is a qubit context to quantum foundations, and further I think that we may need to look at an open world perspective. This would be to treat quantum mechanics, or quantum gravity in particular, as an open system analogous to open systems approach to statistical mechanics.
The quantum foundations of the universe or quantum cosmology involve entanglements of qubits that underlie metric structure. Your essay then suggests that networks on larger scales are then “mirrors” of this underlying system. This does have some logic to it, for the loss of entanglement on a local basis results in entropy. This entropy is then given by a partition function of states. I think this partition function is most naturally the integer partition function, at least for qubits of a black hole. In an open thermodynamic structure this can result in locally high ordered structures or networks. This is maybe a basis for the existence of biology and consciousness.
Over all I found your essay interesting and entertaining. My essay touches on some of these issues, in particular with an argument for an open world view.
Cheers LC
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Author Tommaso Bolognesi replied on Apr. 17, 2014 @ 08:40 GMT
Dear George,
we might agree on the terminology, and call transcendent any feature of the universe that we are not able to capture by the scientific method. The substantial difference is whether one considers (pessimistically) this status of affairs as permanent, or (optimistically) slowly evolving to a better and wider scientific explanation of those features. Science has progressed every time an item has been moved from box 1 (Magic - e.g. lightning) to box 2 (understood). I see this as a one way process.
I must add that the ideas about measuring consciousness expressed by Tononi (and, more abstractly, by Teilhard de Chardin) sound quite plausible to me, and I would not be too surprised if in 10, or 50, or 100 years, the phenomenon of matter that becomes able to reflect (on) itself will be explained. Progress in robotics and artificial intelligence should help a lot in this effort.
And still, human imagination is so strong that I guess we will always keep finding interesting items in box 1, to keep our scientists busy.
Ciao
Tommaso
Vladimir F. Tamari wrote on Apr. 17, 2014 @ 01:28 GMT
Tommasso
Congratulations for a really enjoyable essay and a great illustration of Tommy tied up by his devices like Gulliver (is that your drawing ?) Already my grandson of two years old is badgering everyone "wez yo ifon?" to play with.
The Internet as an emerging Noosphere is a credible scenario, and through your nice storytelling you take us to various other fascinating related concepts. One that struck me particularly is your ruminating about Wolfram and others' idea that the Universe is an evolving cellular system that runs on a bit of code.
Taking this idea furthest, at the smallest level you get to the point where the software and the hardware are one and the same. This is the root concept of my outline 2005 Beautiful Universe theory Spinning Bloch-sphere-magnet-like dipoles interact with their neighbours to create energy, radiation, matter as well as dark energy and matter. Space itself is defined by the node-node interactions, while time is not needed as a dimension but emerges when we monitor the state of the mutable Universe or local parts of it.
Your analysis conjures the frightening conclusion that if human beings are regarded as computable bits, then a Grand Programmer programs our behavior: A self-conscious Internet can steer humanity where it (the Internet) wills!
(time out for fervent prayers and supplications that it ain't gonna be so)
Best wishes from
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Author Tommaso Bolognesi replied on Apr. 17, 2014 @ 09:24 GMT
yes, thats my drawing, but I sure cant compete with your coloured pictures (including those in your 2005 Beautiful Universe essay).
And yes, one of the implications of the dialogue is that the future of humanity is in the hands of the self-conscious super-organism that emerges from the interactions of us humans. This might be the case even if one did not emphasize the computation-oriented character of the universe dynamics.
A scary picture? I am not sure. Maybe we are not exactly replicating the ant vs. anthill scenario, because a human has much more consciousness than an ant, and as such might be in a better position to interact with the emergent entity at the upper level. This is in fact what Teilhard postulates, when he talks about our relations with the Omega pole, but that, admittedly, is the most speculative part of The Human Phenomenon.
Tommaso
Vladimir F. Tamari replied on Apr. 17, 2014 @ 13:22 GMT
Thanks for your cheerful response Tommaso. All that I can add is that I sincerely hope that your optimism and faith in the human mind (or whatever is involved here) will be bourne out in the coming years, decades and centuries. Humanity has been slowly moving on a trajectory that has now become a self-propelling super- highway to a future full of possibilities and dangers. hang on tight!
Best wishes
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Georgina Woodward wrote on Apr. 21, 2014 @ 09:31 GMT
Hi Tommaso,
You wrote "It is well known that, due to a malicious antipodal butterfly, the possibility to accurately forecast the weather - let alone controlling it - is severely limited. Why should it be easier to predict and steer the future of humanity?" My immediate though was that human behavior is not chaotic and so is a lot more predictable. However a quick Google search and I found "Chaos in human behavior: the case of work motivation." Universidad de Barcelona, Spain. j.navarro PubMed Commons, 13th May 2010 Quote: This study considers the complex dynamics of work motivation. Forty-eight employees completed a work-motivation diary several times per day over a period of four weeks. The obtained time series were analysed using different methodologies derived from chaos theory (i.e. recurrence plots, Lyapunov exponents, correlation dimension and surrogate data). Results showed chaotic dynamics in 75% of cases. The findings confirm the universality of chaotic behavior within human behavior,......" Which I find really surprising.
I still don't think people are as hard to predict and steer as the weather. With training and/or conditioning they can become highly predictable. The work of Derren Brown hypnotist, illusionist, mind reader shows how easily ideas can be planted into people's minds which they then regard as their own thoughts. I don't have overt mind control in mind but subtle political -social engineering that drip feeds the desired behaviours. Smoking- not our future Its then a matter of deciding what direction will be promoted as desirable. The lesson of low fat diet advice leading to more heart disease and obesity should be heeded as cautionary tale. It shows that what we think is best for the people may not be.
Good luck, Georgina
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Author Tommaso Bolognesi replied on Apr. 23, 2014 @ 09:11 GMT
Dear Georgina,
thank you for the comments and the mention to the essay on the chaotic dynamics of human behaviour. I understand that chaos in this case refers to individual behaviour: in certain situations, we tend to behave in chaotic, unpredictable ways. Nevertheless, I agree with you that in many circumstances our behaviour, as individuals, is more regular, and predictable.
My point, however, is that, even when the members of a population have, individually, a regular, predictable behaviour, it may happen that the resulting, overall behaviour of the population, as a whole, is chaotic, due to the emergent dynamics. The obvious example is that of cellular automata: all cells behave in the same, completely defined and predictable way, and yet the patterns that emerge can be highly irregular and unpredictable.
I see that your essay is published now. Ill read it quite soon.
Ciao!
Tommaso
Thomas Howard Ray wrote on Apr. 21, 2014 @ 13:58 GMT
Nice, Tomasso!
I am also a big fan of Chaitin's metabiology and I think you hit it dead on: "In his book Chaitin mentions Wolfram and his New Kind of Science [9]. Well, one of the messages from that book is that the emergent properties of the computations of simple programs - software - might explain the complexity and creativity of the physical universe at all levels. Spacetime, before anything else, must be creative! And discrete! And algorithmic! Spacetime as a causal set [4] - an algorithmic causal set!"
While you seem to doubt that Chaitin's program either includes or can include consciousness, though, I think it is implied as an inherent characteristic of matter. (As you say, even a stone has a small soul.) This idea is formalized in Murray Gell-Mann's IGUS (information gathering and utilizing system) model of complex adaptive systems.
My own essay should be up soon. Looking forward to dialogue!
Tom
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Torsten Asselmeyer-Maluga wrote on Apr. 22, 2014 @ 11:09 GMT
Hi Tommaso,
thanks for your interest and sorry for the delay (Easter travel with my family and no internet connection...)
No, Lem is not at the origin of this idea. In his book, Lem wrote about the theoretical limits of human development. Here he discussed also the direct change of the human body (or the brain) also in the direction of genetic engineering but also as combination of technology and biology. But the main part in his argumentation is the evolutionary development of all kinds (technology, humanity and society). For me it was the first time that someone mentioned such a unifying principle and this was the main influence of Lem for me. (Quantum gravity is also such a unifying principle but this is another story....)
In the second part of your question, you mention a two-way process (technology influences humans). Yes, you are right that there is such an influence.
As model I would propose a coupling of the two evolutionary processes (also by a special rate, so not deterministic but probabilistic). Years ago we developed this strategy (and call it diochotomic strategie). Unfortunately we never published something and it is only contained in a PhD thesis (but in german). The corresponding equation was later found to be comparable to the Dyson equation in quantu field theory (but now with imaginary time, a usual trick to change from quantum field theory to statistical physics).
But your question reminds to make some work in this direction again.
Best Torsten
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Anselm Smidt wrote on Apr. 22, 2014 @ 16:40 GMT
Dies ist, was Bundesamt für Verfassungsschutz Höhle tut. Das ist nichts Neues.
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Michael Allan wrote on Apr. 24, 2014 @ 03:52 GMT
Hello Tommaso, May I offer a short appraisal of your essay, a little on the critical side? I would ask you to return the favour. - Mike
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Jonathan J. Dickau wrote on Apr. 28, 2014 @ 01:43 GMT
I liked your essay a lot Tommaso..
I did not know about some of the musings of Teilhard de Chardin before reading your essay, but his message resonates very strongly with me, and I am glad you brought it to my attention. As it turns out; he lived not far from here for a time late in his life, and his final resting place in in the same township where I reside. Of course, everything has changed; the St. Andrews monastery is now the campus for the Culinary Institute of America, but some folks still remember TdC's Hudson Valley connection.
I find the insights you discovered through de Chardin are similar to the views expressed by Arthur M. Young in his "Reflexive Universe" book. Young details how the evolution of consciousness and the cognitive faculties follows a similar pattern as the evolution of form in Physics and Biology, and unfolds in seven stages. I have attached a document supplementary to my essay, which details my adaptation of this theme to the playful process of learning, but it also speaks to the work of de Chardin - as I cover the entire evolutionary spectrum of the learning experience, or the grand arc of all learning.
We will have to compare notes again later, but for now; good luck!
All the Best,
Jonathan
attachments: Playful_Flow_of_Information.pdf
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Author Tommaso Bolognesi replied on Apr. 28, 2014 @ 09:53 GMT
Hi Jonathan,
yes, Ive read that TdC has spend his last years in New York City, and that he is buried in Poughkeepsie. Ive recently found that in this period he used to walk in Central Park, where he once met a young girl, Jean Houston, with a dog called Champ (which is the name I borrowed for the dog in my essay). They had interesting conversations, as reported by a grown up Ms. Houston:
http://tcreek1.jimdo.com/mr-tayer/
Author Tommaso Bolognesi replied on Apr. 28, 2014 @ 10:28 GMT
. . . and thank you for the pointer to Youngs Reflexive Universe, and to your addendum Playful Flow of Information. Reading your notes made me think that it would be nice to be able to see the seven steps implemented in terms of the features of some formal model - to see whether they become the essential features of an artificial universe too. I am not familiar with Youngs book, but Ive found on the web a summary of Chapter 4 that seems to provide this implementation in terms of particles, molecules, and the physical world. No mention to the computational aspect, though. This is all very interesting, and . . . time demanding!
By the way, I also liked your essay a lot; it was one of the first ones I've read, commented, and rated.
Jonathan J. Dickau replied on Apr. 28, 2014 @ 14:45 GMT
I meant to post here..
But my reply to you ended up below. Sorry for any confusion.
Regards,
Jonathan
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Jonathan J. Dickau wrote on Apr. 28, 2014 @ 14:29 GMT
May it please you to learn..
There exists an algebraic system in which that progression is already encoded, the octonion algebra. There are seven imaginary dimensions in the octonions, and if you interpret imaginary components as depicting change or motion, it is easy to see this is related to process. In fact; one might even say that octonion algebra is sequentially evolutive.
P.C. Kainen comments "Of course, multiplication in the octaval arithmetic fails to be either commutative or associative, but that could be a blessing in disguise. If multiplication depends on the order of the elements being multiplied together and even on how they are grouped, then at one fell swoop, geometry enters the calculation in an organic way."
This has been a subject of my research for a number of years, and I would be happy to compare notes, save you time by directing you to known results, ... You have already saved me time by summarizing things in your essay that strongly support my work.
All the Best,
Jonathan
attachments: 2_octophys.pdf
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Edwin Eugene Klingman wrote on Apr. 28, 2014 @ 20:24 GMT
Dear Tommaso,
Your essay has a delightful structure, and Tommy comes off looking quite good in it!
In your comment on my essay you ask about non-linear system stability. I suspect that you are addressing, by referring to nonlinearity, the Wolfram automata/Game of Life observation that with a change in one cell or automaton, as you say, "an avalanche of modification causally spreads...
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Mohammed M. Khalil wrote on May. 4, 2014 @ 12:09 GMT
Hi Tommaso,
Great essay! I find your presentation style very interesting. I learned a lot from your essay, especially about the philosophy of Teilhard de Chardin and the work of Tononi.
Thank you, and good luck.
Mohammed
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KoGuan Leo wrote on May. 6, 2014 @ 10:24 GMT
Dear Tommaso,
Sublime! I rated it a ten (10. Having self-aware Qbit like you, we have no fear of the future. Our world is Leibnitz's world, the best of all possible worlds.
Best wishes,
Leo KoGuan
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John Brodix Merryman wrote on May. 7, 2014 @ 23:53 GMT
Tommaso,
Complexity is also prone to instability and periodic crashes. While we cannot compute the exponential increase in complex systems, other than computing every step, we can, with some perspective of distance, estimate the likelihood of them crashing. Like individual lives, they can crash early, or they can crash late, but they will crash because complexity tends to only compound until it becomes unstable. Like computer programs or old blood, it just gets too much bad code built up to be able to sustain it within the platform.
So you don't necessarily have to view the human situation as completely unpredictable, due to its complexity. There are a number of issues, from finance to physics to population to economic growth, which are all showing signs of imminent crashing. One effect of a crashing system is that it provides a large amount of released energy. So the question I would ask, is there some way which these forces can be directed in an overall positive direction, or possibly one crash used to facilitate a positive direction for others and to be truly optimistic, I think there is.
Regards,
John Merryman
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George Gantz wrote on May. 9, 2014 @ 00:54 GMT
Tommaso -
The comments have slowed down but the scores are heating up. I hope you continue to do well. Not having a direct email I thought I would let you know that we just got back from a fabulous vacation in Italy - 5 days in Tuscany, 1 in Orvieto and 2 in Rome. Fabulous! Here in New England all the villages are in the valleys and the mountaintops are for viewing. Italy is upside down! This simply shows the power of the fitness landscape in directing human behavior - if you are mortally fearful of marauding bands, attacking armies or wild animals, hilltops are a place of safety and refuge. I was also overwhelmed by the sense of common purpose and commitment it took to build all those fortifications and cathedrals - or the aqueducts and magnificent fountains in Rome. If humans can do these things, then perhaps we can also determine how to steer humanity's future.
Ciao! - George
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Author Tommaso Bolognesi replied on May. 12, 2014 @ 06:58 GMT
Hi George,
I am glad that you enjoyed your 5 days in Tuscany. You did not mention Siena, which is an excellent macro-example of a city where houses tend to climb the hill. And I agree, by looking at some of the past achievements of humanity (architecture, but also music!) one may indulge in some optimism about the future.
I see that scores are heating up and I am a bit puzzled by the current ranking. This is a somewhat unusual version of the contest, for which it is harder to establish whether an essay is relevant to the theme. I am already looking forward to the next one!
Ciao
Tommaso
Anonymous wrote on May. 15, 2014 @ 22:00 GMT
Tommaso,
If only more 19-year olds and their friends were so well read, thoughtful and expressive when immersed in the digital world. Well, I too can dream, can I not?
All the dreaming aside, is this your message: As humanity isn't a totally free self-standing entity, we cannot steer its future in isolation from (a) humanity's "inside of things" like DNA and all the vast implications of DNA as software and (b) humanity's "outside of things" like "the atmosphere along with the malicious antipodal butterfly already testing humanity's steering strengths. We'll have the necessary knowledge someday in the Noosphere as a conscious entity, Omega, and only then will humanity be in a position to steer.
Your idea of "humans are social atoms" made your essay still more intriguing. I am now dreaming of a whole new genre of science fiction.
-- Ajay
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Ajay Bhatla replied on May. 15, 2014 @ 22:02 GMT
Sorry, spent too much time dreaming and got logged off!
The above is from me.
-- Ajay
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Luca Valeri wrote on May. 20, 2014 @ 22:39 GMT
Dear Tommaso,
what a beautiful essay. It took me very long time to comment on your essay. I wanted to prove you're wrong. The ultimate language of nature is not software! But I couldn't. (How would you prove you're wrong?)
In my essay I state that the generality and unity of physics is originated from formalizing the very precondition of the possibility of scientific knowledge. Where scientific knowledge is the ability to learn from the past to predict the future. If we can predict something, we can compute it. In my essay I ask if there is a being beyond physics. I say that each human being in its uniqueness does not comply with the definition of a physical object. He cannot be predicted. I confess I find the argument myself a bit cheap.
I had many thoughts reading your essay and I might post them in a later time. For now I just want to say that in my very short essay I talk about two topics that are also part of your essay ( in a very different although not such eloquent way): The creation of information/structure and how it is compatible with the growth of entropy and a derivation of relativistic space time from the qbit.
I hope you find the time to read and comment on my essay.
Best regards
Luca
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Author Tommaso Bolognesi replied on May. 22, 2014 @ 09:52 GMT
Hi Luca,
heres a first quick reaction to your post.
Your first question is, essentially: how could one disprove the computational universe conjecture? Very important question indeed, in light of the fact that any serious physical theory should be such as to be possibly disproved.
So far, this conjecture (recently termed Bit Bang) rests upon the wide experimental evidence...
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Luca Valeri replied on May. 23, 2014 @ 11:44 GMT
Hi Tommaso,
I replied on your comment in my blog.
Regards
Luca
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Conrad Dale Johnson wrote on May. 23, 2014 @ 16:38 GMT
Tommaso,
I had a very good time with your fine essay. I liked being reminded how important it was for me back in the 70's to discover Teilhard de Chardin, with his wonderfully grand view of the stages of being, the physical and biological and human. Though a lot of important ideas have emerged since he wrote, it's still a tremendous challenge to envision a perspective that includes these...
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Aaron M. Feeney wrote on May. 26, 2014 @ 02:41 GMT
Hi Tommaso,
I'm still reading your fascinating article, I will come back and comment on it soon. However, I wanted to inform you today that I've responded to your excellent questions and observations on my page, and I would very much like to receive your feedback which I know will be of the highest quality. If you do wish to comment further there, please make sure to attach your post underneath my misplaced post (i.e., place it underneath part one of my two part reply). This will ensure that part one and part two do not get separated from one another.
Also, you might enjoy my reply to George Gantz. It offers a series of very important points that I would have put in my paper if I had had more room.
You have given me a lot to think about, and I am grateful.
Warmly,
Aaron
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Member Marc Séguin wrote on May. 27, 2014 @ 01:52 GMT
Tommaso,
Thank you for a very well written and fascinating essay. I agree with you that the idea that our universe could have emerged from a computer program is quite intriguing: it resonates with Max Tegmark's thesis that the universe is nothing more than an abstract (mathematical) structure, that when "seen" from the inside acquires the emergent property of physicality. Have you ever looked at the work of Bruno Marchal of Université Libre de Bruxelles, and of other like-minded thinkers that hang around the Google Group "Everything List"? You might find it interesting.
Good luck in the contest!
Marc
P.S. Thank you for the comments you left on my essay's forum: I have answered you there.
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Author Tommaso Bolognesi replied on May. 29, 2014 @ 16:06 GMT
Oh yes, I had an interaction with Marchal when organising a workshop here in Pisa, back in 2009, and with Jurgen Schmidhuber who also was in the 'Everything List' Google Group if I well remember. However, I tend to prefer concrete simulation activity over discussions of more philosophical nature.
Thanks for pointing out. Cheers.
Anonymous wrote on May. 27, 2014 @ 21:38 GMT
Hi Tommaso,
Excellent thought provoking essay.
I like: Pierre Teilhard de Chardin (1881-1955) The appearance of the human phenomenon marks the point at which the fabric of the universe achieves the ability to reflect itself.
I question: DNA as software. I personally think DNA crosses the threshold between quantum phenomena and the classical world. Wolfram has done good work... but more insight is needed and perhaps a breakthrough is needed on his cellular life theories.
I like very much Tommy's conclusion: The next stop for humanity, to answer your question, is Superlife.
Sounds good to me.
Wishing you the best,
Don Limuti
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Member Daniel Dewey wrote on May. 29, 2014 @ 14:41 GMT
Hi Tommaso,
You have a very original style! And I'm glad to see Chaitin's book mentioned; I've just read the shortened paper version of his metabiology, but I think they're very interesting ideas. Your connection to cellular automata is also interesting. It would be great if it were possible to make substantive predictions or decisions based on this kind of model.
Best wishes,
Daniel
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Author Tommaso Bolognesi replied on May. 29, 2014 @ 16:19 GMT
Yes, Chaitin's metabiology is quite interesting, but certainly still at a preliminary stage. Yet, it proved quite useful for providing some balance in my essay, representing the missing software oriented treatment of the second of the three stages discussed by de Chardin in his book: Prelife (Wolfram), Life (Chaitin), Thought (Tononi).
Tommaso
PS
i think we have cross rated our essays. In any case, I did.
Douglas Alexander Singleton wrote on May. 29, 2014 @ 15:54 GMT
Hi Tommaso,
I just did a very quick read through of your essay (the dead line approaches :-( ) and like it very much. Part of the reason is there are *some* points of connection with things in my essay (your discussion of complex systems which has some connection with the unknown unknowns or "black swan events" of Taleb I talked about in my essay, but as you noticed not in as much detail as was maybe "promised" in the introduction). You go into much more depth on the issue of complexity as well as connecting to computation [As a side note my main area of work is field theory so I tend to view things in terms of scattering amplitudes, Feynman diagrams, path integral etc. You as a computational expert frame things in terms of computability, or the automata of S. Wolfram. If a football player -- either US football or the football played by everyone else -- were to write an essay about steering the future it would probably involve lessons learned from playing sports. In fact my HS physics teachers was also the HS football coach (this is often the case especially in small schools in the US) and most of his examples involved football].
Oh I also liked the literary device of presenting these ideas as a discussion between you and your nephew (and Alice via Skype). And also the idea of the ant-hill (and to a greater degree human societies being more than just the simple sum of their parts -- i.e. some emergent complexity. In fact if one looks at individual humans with their hosts of bacteria, viruses, fungi, parasites, symbiotic organism(useful and malicious) any individual is more than a sum of their parts. Anyway sorry I had to rush through the reading of your excellent essay, but I hope to give it a more through read later.
Best,
Doug
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Anonymous wrote on May. 29, 2014 @ 17:41 GMT
Hi Douglas,
we had already an exchange in your thread (it is hard to keep track of everything here!). Thanks for reading my essay, and for your comments. I just react to one of them, appearing in square brackets [...]. Often is has been observed that descriptions of the universe, across history, have been influenced by the current technology - from the clockwork universe of Pascal,...
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Author Tommaso Bolognesi wrote on May. 29, 2014 @ 17:43 GMT
I am not anonymous, in the sense that I am the Anonymous above.
Tommaso
Douglas Alexander Singleton wrote on May. 29, 2014 @ 18:04 GMT
Hi Tommaso,
No I already did rate you essay, I just apologize that I did not have a chance to read in more detail since there are many different themes at play. But in any case I did read enough it understand this was a very good essay and so I rated it accordingly. My statement was just to say that I might not have completely gotten all the deep details from your essay which has several levels.
Again good luck with the contest.
Best,
Doug
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Author Tommaso Bolognesi replied on May. 30, 2014 @ 08:07 GMT
Ok, great! In fact, I myself might not have completely gotten all the details from my essay! :-) What I mean by this is that, when attempting to put under the same umbrella such diverse things as spacetime, darwinian evolution and thought/consciousness, you easily run the risk of not being a professional expert in all three areas. But the effort is in part justified by a Schroedinger quote, that goes roughly like this (I only have the italian translation):
We clearly perceive that only now we begin to collect reliable material for combining in a single complex the sum of all areas of our knowledge; but, on the other hand, it has become almost impossible for a single mind to dominate more than one small specialized area. I do not see a way out from this dilemma, other than having someone trying to formulate a synthesis of facts and theories, albeit using second-hand and incomplete knowledge of some of them, running the risk of having people laugh at him/her.
Aaron M. Feeney wrote on May. 29, 2014 @ 23:48 GMT
Hi Tommaso,
I liked the concept of the benefits that may be derived by modelling different groups and institutions as cellular automata, but I wonder how this would be accomplished. Nevertheless, it is certainly an important idea to explore, and it, along with many of the other ideas you discuss in your essay have added to the richness of this forum.
I have rated your essay with these points in mind.
Warmly,
Aaron
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Anonymous wrote on Jun. 4, 2014 @ 04:53 GMT
One of the most interesting and enjoyable essays in the contest, Tommaso. It was fascinating to consider this question through the lens of Chardin, Chaitin, Wolfram, and Tononi.
I agree that the operations of physics can probably be thought of as a form of information processing. The idea of developing into a broader collective form of consciousness is very appealing.
I do think we may be in more control of our collective destiny that you seem to suggest. Complex adaptive systems like society, as you say, do exhibit order at some levels. Just as we may be able to predict and alter the climate without being able to control the weather, we may be able to shape our social evolution without being able to determine the specific fate of each individual.
Well done, in any case. Good luck in the contest!
Best,
Robert de Neufville
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Robert de Neufville replied on Jun. 4, 2014 @ 04:54 GMT
I seem to get logged out every time I write more than a short comment...
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Author Tommaso Bolognesi replied on Jun. 4, 2014 @ 11:08 GMT
Thanks for the comments Robert. I see your point, and I agree that we may be able to control to some extent the dynamics of a complex system. For some reason, though, I am much more attracted by the spontaneous dynamics that these systems may exhibit, that seems to outperform us in terms of creativity.
T
Lorraine Ford wrote on Jun. 7, 2014 @ 01:07 GMT
Dear Tommaso,
Marvellous essay! It's a literary work; it's amusing; it expounds the complexity viewpoint very well. I think this is a top essay.
Regards,
Lorraine
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Lorraine Ford replied on Jun. 8, 2014 @ 04:25 GMT
Dear Tommaso,
Re "I don't buy the complexity viewpoint":
As you know, in my essay I claim that there are at least 3 invalid assumptions underlying the ideas of physics (and that these perverse and unenlightened ideas about the nature of reality underlie the attitudes that are destroying our planet).
Well, another invalid assumption of Wolfram, Chaitin, and physics in general, is that numbers just exist, no explanations necessary. This is a Platonic viewpoint.
But I think that there is no Platonic realm - this universe is all there is. So given that restriction, what are the numbers that are found when fundamental reality is measured; what does this mean about the nature of reality? I think that there is necessarily a physical reality behind numbers (as I try to explain in my 2013 essay): I contend that numbers are (what I call) hidden information category self-relationships. I think the information category/information relationship way of looking at things is a better pointer to the nature of reality than e.g. the cellular automata viewpoint.
I contend that information is indistinguishable from/identical to physical reality; and that information is subjective experience. So, at the foundations of reality, information is subjective experience of e.g. information categories like mass and charge. I also contend that the physical outcomes of "free will" can only be represented as the creation of new (usually temporary) "rules", where law-of-nature rules are information category relationships. I contend that the views of Wolfram and Chaitin etc. imply that the universe is a very dull place, where nothing truly new ever happens: the "truly new" being new "rules".
Best wishes,
Lorraine
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Member Hector Zenil wrote on Jun. 19, 2014 @ 10:47 GMT
Hi Tommaso,
Very nice essay. I ran your Mathematica code and it looks like a fine random number generator, could you perhaps explain what is going on step by step? It is intriguing.
- Hector
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Anonymous wrote on Jun. 20, 2014 @ 07:54 GMT
Hector,
I'm glad that you ran the code. You are the first... It looks indeed like a random number generator. More specifically, it creates a random permutation of the first n integers on-the-fly. At each step the code takes a pair (pi(n), pos), where pi(n) is an n-tuple representing the current permutation of the first n integers, and pos is an index between 1 and n. The step creates a new pair (pi(n+1), pos), where pi(n+1) is a permutation of the first (n+1) integers obtained from pi(n) by inserting integer (n+1) at position pos, and pos is the number found at position pos of tuple pi(n). The computation is started from tuple pi(2) = (1, 2) and pos = 2.
You can trace the evolution of variable pos (like you do when tracing the dynamics of a Turing Machine head on the tape), or look at the whole permutation pi(n) (seen as a function from range (1..n) to itself), and you get the foggy picture of white noise, or deterministic chaos.
I found this minimal deterministic code while experimenting with algorithms for building causal sets by using permutations - an idea originally suggested by Alex Lamb, alternative to the stochastic, sprinkling` technique used by the Causal Set Programme people (Rideout, Sorkin, ...).
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Author Tommaso Bolognesi wrote on Jun. 20, 2014 @ 07:56 GMT
Best regards
Tommaso | 2019-07-18 17:57:50 | {"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": 0.40583336353302, "perplexity": 2089.946621609507}, "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-2019-30/segments/1563195525699.51/warc/CC-MAIN-20190718170249-20190718192249-00514.warc.gz"} |
https://www.projecteuclid.org/euclid.aos/1543568600 | The Annals of Statistics
Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case
Abstract
Consider a Gaussian vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$, respectively. With $n$ independent observations of $\mathbf{z}$, we study the correlation between $\mathbf{x}$ and $\mathbf{y}$, from the perspective of the canonical correlation analysis. We investigate the high-dimensional case: both $p$ and $q$ are proportional to the sample size $n$. Denote by $\Sigma_{uv}$ the population cross-covariance matrix of random vectors $\mathbf{u}$ and $\mathbf{v}$, and denote by $S_{uv}$ the sample counterpart. The canonical correlation coefficients between $\mathbf{x}$ and $\mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $\Sigma_{xx}^{-1}\Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{yx}$. In this paper, we focus on the case that $\Sigma_{xy}$ is of finite rank $k$, that is, there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_{1}\geq\cdots\geq r_{k}>0$. We study the sample counterparts of $r_{i},i=1,\ldots,k$, that is, the largest $k$ eigenvalues of the sample canonical correlation matrix $S_{xx}^{-1}S_{xy}S_{yy}^{-1}S_{yx}$, denoted by $\lambda_{1}\geq\cdots\geq\lambda_{k}$. We show that there exists a threshold $r_{c}\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_{i}\leq r_{c}$, $\lambda_{i}$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_{i}>r_{c}$, $\lambda_{i}$ possesses an almost sure limit in $(d_{+},1]$, from which we can recover $r_{i}$’s in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of $\lambda_{i}$’s under appropriate normalization. Specifically, $\lambda_{i}$ possesses Gaussian type fluctuation if $r_{i}>r_{c}$, and follows Tracy–Widom distribution if $r_{i}<r_{c}$. Some applications of our results are also discussed.
Article information
Source
Ann. Statist., Volume 47, Number 1 (2019), 612-640.
Dates
Revised: March 2018
First available in Project Euclid: 30 November 2018
https://projecteuclid.org/euclid.aos/1543568600
Digital Object Identifier
doi:10.1214/18-AOS1704
Mathematical Reviews number (MathSciNet)
MR3909944
Zentralblatt MATH identifier
07036213
Citation
Bao, Zhigang; Hu, Jiang; Pan, Guangming; Zhou, Wang. Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case. Ann. Statist. 47 (2019), no. 1, 612--640. doi:10.1214/18-AOS1704. https://projecteuclid.org/euclid.aos/1543568600
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Supplemental materials
• Supplement to “Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case”. In this supplementary material, we present some simulation results and prove Theorem 2.1 and 2.3, Lemmas 6.1–6.3, 7.3–7.4, and also Proposition 7.1. | 2019-10-20 20:34:05 | {"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": 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": 0.7713586688041687, "perplexity": 2949.7908678614212}, "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-2019-43/segments/1570986718918.77/warc/CC-MAIN-20191020183709-20191020211209-00035.warc.gz"} |