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https://www.physicsforums.com/threads/dimension-of-the-metric-of-a-projection-of-a-sphere.962962/ | # B Dimension of the metric of a projection of a sphere
#### jk22
Let $(x_1,x_2,x_3)=\vec{r}(\theta,\phi)$ the parametrization of a usual sphere.
If we consider a projection in two dimension $(a,b)=\vec{f}(x_1,x_2,x_3)$
Then I don't understand how to use the metric, since it is $g_{ij}=\langle \frac{\partial\vec{f}}{\partial x_i}|\frac{\partial\vec{f}}{\partial x_j}\rangle$ which is a 3x3 matrix but we have only two coordinates $a,b$ in the projection.
Related Differential Geometry News on Phys.org
#### WWGD
Gold Member
AFAIK, you use the pullback metric from the polar coordinates. This is the way I have seen it done.
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• Solo and co-op problem solving | 2019-09-18 15:26:12 | {"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": 0.8416401147842407, "perplexity": 1963.027527649784}, "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-2019-39/segments/1568514573309.22/warc/CC-MAIN-20190918151927-20190918173927-00434.warc.gz"} |
https://math.stackexchange.com/questions/3069465/integral-convergence-estimate-with-cut-off-function | # Integral Convergence Estimate with cut-off function
Let $$u \in L^{\infty}(\Omega)\cap H_{0}^{1}(\Omega)$$ and define a cut-off function $$\eta_{R} \in C_{0}^{\infty}(\mathbb{R})$$ for $$\Omega \subset \mathbb{R}$$ an unbounded (interval) domain as follows
$$\eta_{R}(x):=\begin{cases} 1 &, \text{if }|x|\leq R\\ 0 &, \text{if } x\geq R+1\text{ or }x\leq-R-1\\ 0<\eta_{R}(x)<1 &, \text{ other }x \end{cases}$$
Now define a function $$u_R = u\eta_{R}$$. Define a functional $$I[u] = ||u||_{H_{0}^{1}(\Omega)}^{2} - ||u||_{L^{p}(\Omega)}^{p}$$ for $$2\leq p <\infty$$.
I would like to show that $$I[u_{R}] = I[u]+o(1)$$ as $$R\to\infty$$. This is my attempt so far:
\begin{align*} |I[u]-I[u_{R}]| &\leq |\, ||u||_{H_{0}^{1}(\Omega)}^{2}-||u_{R}||_{H_{0}^{1}(\Omega)}^{2}\,| + |\, ||u||_{L^{p}(\Omega)}^{p} - ||u_{R}||_{L^{p}(\Omega)}^{p}\,| \\ &\leq |\, ||u||_{L^{2}(\Omega)}^{2} - ||u||_{L^{2}(\Omega)}^{2}\,| + |\, ||\nabla u||_{L^{2}(\Omega)}^{2}- ||\nabla u_{R}||_{L^{2}(\Omega)}^{2}\,|+ |\, ||u||_{L^{p}(\Omega)}^{p} - ||u_{R}||_{L^{p}(\Omega)}^{p}\,|\\ &= A + B + C \end{align*}
So, I would like to see the estimate one by one. First, I will start from $$A$$. Observe that \begin{align*} A &\leq \int_{\Omega}|u^{2}\eta_{R}^{2}-u^{2}|dx \\ &\leq \sup\limits_{\Omega}u^{2}\int_{\Omega}|1-\eta_{R}^{2}|dx\\ &\leq 2\sup\limits_{\Omega}u^{2}\int_{\Omega}|1-\eta_{R}|dx \end{align*}
Similarly for $$C$$, I will obtain the following estimate $$C \leq p\sup\limits_{\Omega}|u|^{p}\int_{\Omega}|1-\eta_{R}|dx$$
Finally, for $$B$$, I would obtain $$B \leq \int_{\Omega}|\partial_{x}(u\eta_{R})|^{2} - (\partial_{x}u)^{2}|dx$$
So, I have two main problems here.
1. How to ensure that $$\int_{\Omega}|1-\eta_{R}|dx \to 0$$ as $$R\to\infty$$ rigorously?
2. What should I do to estimate $$B$$ since I am not sure what to do with the term $$\partial_{x}\eta_{R}$$ here.
Any help will much be appreciated! | 2019-01-20 12:53:36 | {"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": 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": 0.9998331665992737, "perplexity": 653.0705476391158}, "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-2019-04/segments/1547583716358.66/warc/CC-MAIN-20190120123138-20190120145138-00027.warc.gz"} |
https://data-and-the-world.onrender.com/posts/depression-preprint-part-i/ | # Depression Preprint Analysis, Part 1
This is the first post in a series focused on trying to analyze the contents of a collection of preprint papers on a topic – in this case, depression. This post involves how I scraped the (initial) website, along with some analysis of basic information from the descriptions of the preprints.
## OSF
The OSF (Open Science Foundation) is project by the Center for Open Science, dedicated to aggregating the results of multiple arXiv-like preprint services and making them searchable as a group. As a result, it makes for a potentially interesting jumping off point to get preprints from a number of different sources.
The interface for the search is pretty standard, and the site returns abstracts and other basic information on the preprints themselves, along with links to the preprints on their respective sites.
The data in these summaries is pretty well-structured, so getting the information out of the respective elements with BeautifulSoup isn’t too bad. That said, the returned results are rendered with JavaScript, so you need to render the page fully before scraping it. As a result, we also need Selenium to fully automate the process.
## Scraping the OSF Website
First, library imports and initial setup:
>> from collections import namedtuple
>> from time import sleep
>>
>> from bs4 import BeautifulSoup
>> import pandas as pd
>> from selenium import webdriver
>> from tqdm import tqdm ## just for monitoring purposes, not necessary for functionality
>>
>> ## initial URL for search results
>> SEARCH_URL = "https://osf.io/preprints/discover?q=depression"
>>
>> ## for storing results
>> OSFPaperTuple = namedtuple("OSFPaperTuple",
>> field_names=["PaperName","URL","LastEdited","Categories","Source"])
>>
>> driver = webdriver.Firefox()
>> driver.get(SEARCH_URL)
Identifying the pages is fairly simple, all pages of the search can be accessed via a URL of the form https://osf.io/preprints/discover?page=XXX&q=depression. Getting the number of pages returned in the search is necessary, but this is fortunately included in an element at the bottom of the page.
>> page_html = BeautifulSoup(driver.page_source, "lxml")
>> pagination_element = page_html.find_all("ul", attrs={"class":"pagination"})[-1]
>> number_of_pages = int(last_page_link.text.strip())
We also need to extract the information of interest from each result: the title of the preprint, the URL to its location, the date of the last edit, the categories that the preprint is labeled with, and the name of the source which is hosting the preprint. The search results are almost totally consistent – the only exception is that some preprints are missing URL links, but those can be detected by checking for the presence of an <a> element in the title element.
>> def get_OSF_paper_info(paper_element):
>> ## Collects information on the papers and returns them in a named tuple.
>> else:
>> paper_url = ""
>>
>> last_edited = paper_element.find("em").text.strip()[13:-4]
>>
>> category_elements = paper_element.find_all("span", attrs={"class":"subject-preview"})
>> if category_elements:
>> categories = [e.text.strip() for e in category_elements]
>> else:
>> categories = []
>>
>> source_name = paper_element.find("span", attrs={"class":"search-result-providers"}).text.strip()
>> return OSFPaperTuple(paper_name, paper_url, last_edited, categories, source_name)
Putting this all together is fairly simple:
>> paper_tuples = []
>> for page in tqdm(range(number_of_pages)):
>> target_page = f"https://osf.io/preprints/discover?page={page+1}&q=depression"
>> driver.get(target_page)
>> sleep(5) ## try to avoid overwhelming the site
>> page_html = BeautifulSoup(driver.page_source, "lxml")
>> papers_on_page = page_html.find_all("div", attrs={"class": "col-sm-8"})[1]
>> paper_elements = papers_on_page.find_all("div", attrs={"class":"ember-view"}, recursive=False)
>> current_page_tuples = [get_OSF_paper_info(p) for p in paper_elements]
>> paper_tuples.extend(current_page_tuples)
At the time of running this, there were 443 search pages to scrape. I added a delay as I didn’t want to hit the OSF site too heavily with search page requests, so it took about 40 minutes to complete this.
From there, we can convert the list of named tuples to a dataframe, save those results, and then shut down the Selenium browser.
>> paper_df = pd.DataFrame(paper_tuples)
>> paper_df.to_csv("OSF_paper_info.csv", index=False)
>>
>> driver.close()
## Analysis
The returned data consists of 4,414 preprints and their accompanying information. Unfortunately, almost a thousand are duplicates, though there are still plenty to look at:
>> # note: I read this in later, so the Categories column is loaded
>> # as a string instead of a list; that's fixed later
>> len(paper_df)
4414
# there are duplicates, unfortunately
>> paper_df = paper_df[~paper_df.duplicated()]
>> len(paper_df)
3475
With this initial data, I’m trying to answer four different questions:
1. What are the counts of each source?
2. What are the most common word stems in the titles?
3. How often do various categories appear?
4. How are the “last updated” years distributed?
### Question 1: Source Counts
This is easily done:
>> paper_df["Source"].value_counts()
RePEc 1625
PsyArXiv 696
arXiv 433
bioRxiv 297
Preprints.org 159
OSF Preprints 143
SocArXiv 38
PeerJ 28
MindRxiv 15
INA-Rxiv 9
Thesis Commons 9
SportRxiv 7
EarthArXiv 6
MetaArXiv 4
Cogprints 2
AfricArXiv 2
AgriXiv 1
NutriXiv 1
Name: Source, dtype: int64
I’ll set aside the sources with fewer than 100 preprints just due to size. Among the remainder:
• RePEc is short for Research Papers in Economics, so it’s focused more on economic depressions that the psychological condition.
• PsyArXiv is focused on “the psychological sciences” as its front page puts it, so it’s likely to be the primary source here.
• arXiv, judging from some of the titles, is using the word “depression” in several contexts, none of which are likely aligned with this analysis:
>> import random
>> random.sample(paper_df[paper_df["Source"]=="arXiv"]["PaperName"].tolist(), 6)
['Attractor Dynamics with Synaptic Depression',
'A geometrical height scale for sunspot penumbrae',
'Understanding Forbush decrease drivers based on shock-only and CME-only models using global signature of February 14, 1978 event',
'Spectral shape of the UV ionizing background and HeII absorption at redshifts 1.8 < z < 2.9',
'The nature of the light variability of the silicon star HR 7224',
'Drifting Asteroid Fragments Around WD 1145+017']
• bioRxiv is focused on genetics and biological science, and it might have some relevant preprints:
>> random.sample(paper_df[paper_df["Source"]=="bioRxiv"]["PaperName"].tolist(), 6)
['Inhibition of protein translation by the DISC1-Boymaw fusion gene from a Scottish family with major psychiatric disorders',
'Insight into the genetic architecture of back pain and its risk factors from a study of 509,000 individuals',
'Genetic correlations between pain phenotypes and depression and neuroticism',
'A spike timing-dependent plasticity rule for single, clustered and distributed dendritic spines',
'Glucocorticoid receptor-mediated amygdalar metaplasticity underlies adaptive modulation of fear memory by stress',
'Pharmacogenetics of antidepressant response: a polygenic approach']
• Preprints.org and OSF preprints have somewhat generic names compared to the other sources, and while some of the preprint titles look relevant, they seem like a more esoteric mix:
>> random.sample(paper_df[paper_df["Source"]=="Preprints.org"]["PaperName"].tolist(), 6)
['Development of a Novel Staging Model for Affective Disorders Using Partial Least Squares Bootstrapping: Effects of Lipid-Associated Antioxidant Defenses and Neuro-Oxidative Stress',
'Understanding the Demographic Predictors and Association of Comorbidities in Hospitalized Children with Conduct Disorder',
'Effect of a Comprehensive Health Care Program on Blood Pressure, Blood Glucose, Body Composition, and Depression in Older Adults Living Alone: A Quasi-experimental Pre-posttest Study',
'Impact of Service User Video Presentations on Explicit and Implicit Stigma toward Mental Illness among Medical Students in Nepal: A Randomized Controlled Trial',
'Interpolation of Small Datasets in the Sandstone Hydrocarbon Reservoirs, Case Study from the Sava Depression, Croatia',
'Transpersonal Gratitude, Emotional Intelligence, Life Contentment, and Mental Health Risk Among Adolescents and Young Adults']
>> random.sample(paper_df[paper_df["Source"]=="OSF Preprints"]["PaperName"].tolist(), 6)
['Music and mood regulation during the early-stages of the COVID-19 pandemic',
'High frequency stimulation-induced plasticity in the prelimbic cortex of rats emerges during adolescent development and is associated with an increase in dopamine receptor function',
'Preprint BuckfieldSinclairGlautier (2019) Slow associative learning in alcohol dependence and the Alcohol Cue Exposure Treatment Paradox',
'Subclinical anxiety and depression are associated with deficits in attentional target facilitation, not distractor inhibition',
'Placebo response and psychosis: a putative shared mechanism',
'Combat stress in a small-scale society suggest divergent evolutionary roots for posttraumatic stress disorder symptoms']
As a side note, something largely missing from these titles is the matter antidepressant drugs and other pharmacological topics. I would guess that it’s the result of some level of sampling bias, rather than an accurate cross-section of depression research as a whole.
### Question 2: Most Common Word Stems
Investigating common elements of the titles could be illustrative as well. Examining the stems of the words is probably more useful than checking the words themselves – for example, separate counts of “depression” versus “depressive” versus “depressed” wouldn’t add much useful information than just the “depress” root.
>> from collections import Counter
>> import re
>>
>> import nltk
>> from nltk.stem.snowball import SnowballStemmer
>>
>> def title_list_to_stems(title_list, stemmer, stopwords):
>> # note: gotta use .lower(), otherwise stopwords aren't properly caught
>> word_list = [word for title in title_list for word in nltk.word_tokenize(title.lower())]
>> word_list = [word for word in word_list if re.search("[A-Za-z]", word)]
>> word_list = [word for word in word_list if word not in stopwords]
>> stems = [stemmer.stem(word) for word in word_list]
>> return stems
The above function is just used to turn a list of preprint titles into a list of word stems. The code below does this for each of the six preprint sources mentioned above, and determines the 15 most common stems:
>> stemmer = SnowballStemmer("english")
>> english_stopwords = set(nltk.corpus.stopwords.words("english"))
>>
>> num_most_common = 15
>> sources_with_enough_papers = ["RePEc", "PsyArXiv", "arXiv", "bioRxiv", "Preprints.org", "OSF Preprints"]
>> paper_stems_dict = {}
>> for s in sources_with_enough_papers:
>> titles = paper_df[paper_df["Source"] == s]["PaperName"].tolist()
>> stems = title_list_to_stems(titles, stemmer, english_stopwords)
>> most_common_stems = Counter(stems).most_common(num_most_common)
>> paper_stems_dict[s] = [m[0] for m in most_common_stems]
>>
>> pd.DataFrame(paper_stems_dict)
Unsurprisingly, “depress” is at the top for all six sources. As you could probably guess from the sampling of the preprint titles earlier, RePEc’s stems are more economics-focused, arXiv is kind of a mixed bag (and isn’t really anything we’re looking for), and the other four are focused on the psychological condition, with bioRxiv focused more on the biology and genetics perspective. Interestingly, COVID-19 actually ends up as the second most common stem in PsyArXiv’s articles and well inside the top 15 for Preprints.org and OSF Preprints. Checking how many times they appear:
>> covid19_mentions = paper_df[paper_df["PaperName"].apply(lambda x: "covid" in x.lower())]
>> covid19_mentions["Source"].value_counts()
PsyArXiv 149
Preprints.org 21
OSF Preprints 19
SocArXiv 11
AfricArXiv 2
arXiv 1
Name: Source, dtype: int64
The term “covid” appears in 149 PsyArXiv preprints, which is a little over one-fifth of their preprints in the search. For Preprints.org and OSF preprints, it’s just over one-eighth. It’s interesting to me that the second most common stem for PsyArXiv preprints only appears around 21% of the time – I’m not sure if that’s exceptional or not.
### Question 3: Frequencies of Categories
To start this off, I know from looking around at the OSF’s search portal that there were a number of preprints which did not have any categories at all. It actually turns out that the majority of them lack categories:
>> # do this step only if you loaded the file from the CSV - it's read back in as
>> # a string by pd.read_csv()
>> paper_df["Categories"] = paper_df["Categories"].apply(eval)
>> (paper_df["Categories"].apply(lambda x: x == [])).mean()
0.6529496402877698
The primary offender here seeps to be RePEc:
>> pd.crosstab(paper_df["Source"], paper_df["Categories"].apply(lambda x: x == []))
On the plus side, since PsyArXiv and OSF preprints have categories for all their preprints, economics topics don’t end up in the most common categories (though it looks like one topic from arXiv sneaks into 15th place):
>> Counter([x for y in paper_df["Categories"].tolist() for x in y]).most_common(20)
[('Social and Behavioral Sciences', 760),
('Psychology', 673),
('Clinical Psychology', 433),
('Medicine and Health Sciences', 376),
('Medical Specialties', 260),
('Psychiatry', 255),
('Life Sciences', 162),
('Health Psychology', 127),
('Biology', 126),
('Neuroscience and Neurobiology', 120),
('Psychiatry and Psychology', 103),
('Child Psychology', 85),
('Personality and Social Contexts', 78),
('Cognitive Psychology', 77),
('Physics', 75),
('Quantitative Psychology', 66),
('Mental Disorders', 59),
('Social Psychology', 56),
('Developmental Psychology', 45),
('Counseling Psychology', 38)]
### Question 4: How are the “last updated” years distributed?
Finally, the question of how the preprints are distributed in time. A basic check of the year reveal that there hasn’t exactly been a steady increase:
>> paper_df["LastEdited"] = pd.to_datetime(paper_df["LastEdited"])
>> paper_df["Year"] = paper_df["LastEdited"].dt.year
>> paper_df["Year"].value_counts().sort_index()
2007 40
2008 16
2009 82
2010 11
2011 22
2012 31
2013 176
2014 458
2015 488
2016 638
2017 258
2018 309
2019 138
2020 441
2021 367
Name: Year, dtype: int64
There’s a peak in 2016, which no year before or since (or so far, in 2021’s case) has matched. I’m not really sure when some of these preprint services took off, so it’s hard to tell exactly what’s driving this. Looking at a cross-tabulation of the six sources we looked at before versus year:
>> papers_with_good_sources = paper_df[paper_df["Source"].isin(sources_with_enough_papers)]
>> pd.crosstab(papers_with_good_sources["Year"], papers_with_good_sources["Source"])
So basically all of the spike appears to come from RePEc. It’s interesting that it seems to cut out after 2018 – a bit of searching on their site turns up many more results, although a spot check suggests that at least some of those just have “depression” as a word in the text instead of as a focus. There seems to be a similar situation with bioRxiv. From 2019 onwards, PsyArXiv, OSF Preprints, and Preprints.org seem to be the only real sources.
The above table in graph form:
>> import seaborn as sns
>> sns.set(rc={"figure.figsize": (8,5.5)})
>> sns.histplot(papers_with_good_sources, x="Year", hue="Source",
>> multiple="stack", discrete=True)
## Wrapping Up
The next part is going to be focused on looking at information from the preprints themselves, and potentially diving a bit more into the preprint collections. From the look of it PsyArXiv is going to be the primary source, but bioRxiv, OSF Preprints, and Preprints.org may provide some interesting supplemental material. | 2021-09-24 21:10:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3247068226337433, "perplexity": 8090.308601084349}, "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/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00098.warc.gz"} |
https://socratic.org/questions/how-do-you-simplify-4-3n-times-8 | # How do you simplify (-4 - 3n) times -8?
Jul 15, 2016
= $32 + 24 n$
This would more commonly be written as $- 8 \left(- 4 - 3 n\right)$ but as multiplication is commutative, it means the same either way.
= $32 + 24 n \text{ be careful of the signs!}$ | 2019-08-20 01:23:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "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": 0.9297551512718201, "perplexity": 1910.597372253102}, "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/1566027315174.57/warc/CC-MAIN-20190820003509-20190820025509-00444.warc.gz"} |
https://proofwiki.org/wiki/Plane_Reflection_is_Space_Rotation | Plane Reflection is Space Rotation
Theorem
Let $M$ be a straight line in the plane passing through the origin.
Let $s_M$ be the reflection of $\R^2$ in $M$.
Then $s_M$ is the rotation of the plane in space through one half turn about $M$ as an axis. | 2022-01-20 20:57:10 | {"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": 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": 0.43721941113471985, "perplexity": 79.0929591732478}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302622.39/warc/CC-MAIN-20220120190514-20220120220514-00069.warc.gz"} |
https://equityplusbd.com/price-to-earnings-ratio-p-e-ratio/ | # Price-to-Earnings Ratio – P/E Ratio
## What Is Price-to-Earnings Ratio – P/E Ratio?
The price-to-earnings ratio (P/E ratio) is the ratio for valuing a company that measures its current share price relative to its per-share earnings (EPS). The price-to-earnings ratio is also sometimes known as the price multiple or the earnings multiple.
P/E ratios are used by investors and analysts to determine the relative value of a company's shares in an apples-to-apples comparison. It can also be used to compare a company against its own historical record or to compare aggregate markets against one another or over time.
### Key Takeaways
• The price-earnings ratio (P/E ratio) relates a company's share price to its earnings per share.
• A high P/E ratio could mean that a company's stock is over-valued, or else that investors are expecting high growth rates in the future.
• Companies that have no earnings or that are losing money do not have a P/E ratio since there is nothing to put in the denominator.
• Two kinds of P/E ratios - forward and trailing P/E - are used in practice.
## P/E Ratio Formula and Calculation
Analysts and investors review a company's P/E ratio when they determine if the share price accurately represents the projected earnings per share. The formula and calculation used for this process follow.
P/E Ratio=Market value per shareEarnings per share\text{P/E Ratio} = \frac{\text{Market value per share}}{\text{Earnings per share}}
To determine the P/E value, one simply must divide the current stock price by the earnings per share (EPS). The current stock price (P) can be gleaned by plugging a stock’s ticker symbol into any finance website, and although this concrete value reflects what investors must currently pay for a stock, the EPS is a slightly more nebulous figure.
EPS comes in two main varieties. The first is a metric listed in the fundamentals section of most finance sites; with the notation "P/E (TTM)," where “TTM” is a Wall Street acronym for “trailing 12 months.” This number signals the company's performance over the past 12 months. The second type of EPS is found in a company's earnings release, which often provides EPS guidance. This is the company's best-educated guess of what it expects to earn in the future.
Sometimes, analysts are interested in long term valuation trends and consider the P/E 10 or P/E 30 measures, which average the past 10 or past 30 years of earnings, respectively. These measures are often used when trying to gauge the overall value of a stock index, such as the S&P 500 since these longer term measures can compensate for changes in the business cycle. The P/E ratio of the S&P 500 has fluctuated from a low of around 6x (in 1949) to over 120x (in 2009). The long-term average P/E for the S&P 500 is around 15x, meaning that the stocks that make up the index collectively command a premium 15 times greater than their weighted average earnings.1
## Forward Price-To-Earnings
These two types of EPS metrics factor into the most common types of P/E ratios: the forward P/E and the trailing P/E. A third and less common variation uses the sum of the last two actual quarters and the estimates of the next two quarters.
The forward (or leading) P/E uses future earnings guidance rather than trailing figures. Sometimes called "estimated price to earnings," this forward-looking indicator is useful for comparing current earnings to future earnings and helps provide a clearer picture of what earnings will look like – without changes and other accounting adjustments.
However, there are inherent problems with the forward P/E metric – namely, companies could underestimate earnings in order to beat the estimate P/E when the next quarter's earnings are announced. Other companies may overstate the estimate and later adjust it going into their next earnings announcement. Furthermore, external analysts may also provide estimates, which may diverge from the company estimates, creating confusion.
## Trailing Price-To-Earnings
The trailing P/E relies on past performance by dividing the current share price by the total EPS earnings over the past 12 months. It's the most popular P/E metric because it's the most objective – assuming the company reported earnings accurately. Some investors prefer to look at the trailing P/E because they don't trust another individual’s earnings estimates. But the trailing P/E also has its share of shortcomings – namely, a company’s past performance doesn’t signal future behavior.
Investors should thus commit money based on future earnings power, not the past. The fact that the EPS number remains constant, while the stock prices fluctuate, is also a problem. If a major company event drives the stock price significantly higher or lower, the trailing P/E will be less reflective of those changes.
The trailing P/E ratio will change as the price of a company’s stock moves, since earnings are only released each quarter while stocks trade day in and day out. As a result, some investors prefer the forward P/E. If the forward P/E ratio is lower than the trailing P/E ratio, it means analysts are expecting earnings to increase; if the forward P/E is higher than the current P/E ratio, analysts expect a decrease in earnings.
## Valuation From P/E
The price-to-earnings ratio or P/E is one of the most widely-used stock analysis tools used by investors and analysts for determining stock valuation. In addition to showing whether a company's stock price is overvalued or undervalued, the P/E can reveal how a stock's valuation compares to its industry group or a benchmark like the S&P 500 Index.
In essence, the price-to-earnings ratio indicates the dollar amount an investor can expect to invest in a company in order to receive one dollar of that company’s earnings. This is why the P/E is sometimes referred to as the price multiple because it shows how much investors are willing to pay per dollar of earnings. If a company was currently trading at a P/E multiple of 20x, the interpretation is that an investor is willing to pay $20 for$1 of current earnings.
The P/E ratio helps investors determine the market value of a stock as compared to the company's earnings. In short, the P/E ratio shows what the market is willing to pay today for a stock based on its past or future earnings. A high P/E could mean that a stock's price is high relative to earnings and possibly overvalued. Conversely, a low P/E might indicate that the current stock price is low relative to earnings.
## Example of the P/E Ratio
As a historical example, let's calculate the P/E ratio for Walmart Stores Inc. (WMT) as of November 14, 2017, when the company's stock price closed at $91.09.2 The company's profit for the fiscal year ending January 31, 2017, was US$13.64 billion, and its number of shares outstanding was 3.1 billion. Its EPS can be calculated as $13.64 billion / 3.1 billion =$4.40.3
Walmart's P/E ratio is, therefore, $91.09 /$4.40 = 20.70x.
## Investor Expectations
In general, a high P/E suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E. A low P/E can indicate either that a company may currently be undervalued or that the company is doing exceptionally well relative to its past trends. When a company has no earnings or is posting losses, in both cases P/E will be expressed as “N/A.” Though it is possible to calculate a negative P/E, this is not the common convention.
The price-to-earnings ratio can also be seen as a means of standardizing the value of one dollar of earnings throughout the stock market. In theory, by taking the median of P/E ratios over a period of several years, one could formulate something of a standardized P/E ratio, which could then be seen as a benchmark and used to indicate whether or not a stock is worth buying.
## P/E vs. Earnings Yield
The inverse of the P/E ratio is the earnings yield (which can be thought of like the E/P ratio). The earnings yield is thus defined as EPS divided by the stock price, expressed as a percentage.
If Stock A is trading at $10, and its EPS for the past year was 50 cents (TTM), it has a P/E of 20 (i.e.,$10 / 50 cents) and an earnings yield of 5% (50 cents / $10). If Stock B is trading at$20 and its EPS (TTM) was $2, it has a P/E of 10 (i.e.,$20 / $2) and an earnings yield of 10% ($2 / $20). The earnings yield as an investment valuation metric is not as widely used as its P/E ratio reciprocal in stock valuation. Earnings yields can be useful when concerned about the rate of return on investment. For equity investors, however, earning periodic investment income may be secondary to growing their investments' values over time. This is why investors may refer to value-based investment metrics such as P/E ratio more often than earnings yield when making stock investments. The earnings yield is also useful in producing a metric when a company has zero or negative earnings. Since such a case is common among high-tech, high growth, or start-up companies, EPS will be negative producing an undefined P/E ratio (sometimes denoted as N/A). If a company has negative earnings, however, it will produce a negative earnings yield, which can be interpreted and used for comparison. ## P/E vs. PEG Ratio A P/E ratio, even one calculated using a forward earnings estimate, don't always tell you whether or not the P/E is appropriate for the company's forecasted growth rate. So, to address this limitation, investors turn to another ratio called the PEG ratio. A variation on the forward P/E ratio is the price-to-earnings-to-growth ratio, or PEG. The PEG ratio measures the relationship between the price/earnings ratio and earnings growth to provide investors with a more complete story than the P/E on its own. In other words, the PEG ratio allows investors to calculate whether a stock's price is overvalued or undervalued by analyzing both today's earnings and the expected growth rate for the company in the future. The PEG ratio is calculated as a company’s trailing price-to-earnings (P/E) ratio divided by the growth rate of its earnings for a specified time period. The PEG ratio is used to determine a stock's value based on trailing earnings while also taking the company's future earnings growth into account, and is considered to provide a more complete picture than the P/E ratio. For example, a low P/E ratio may suggest that a stock is undervalued and therefore should be bought – but factoring in the company's growth rate to get its PEG ratio can tell a different story. PEG ratios can be termed “trailing” if using historic growth rates or “forward” if using projected growth rates. Although earnings growth rates can vary among different sectors, a stock with a PEG of less than 1 is typically considered undervalued since its price is considered to be low compared to the company's expected earnings growth. A PEG greater than 1 might be considered overvalued since it might indicate the stock price is too high as compared to the company's expected earnings growth. ## Absolute vs. Relative P/E Analysts may also make a distinction between absolute P/E and relative P/E ratios in their analysis. ### Absolute P/E The numerator of this ratio is usually the current stock price, and the denominator may be the trailing EPS (TTM), the estimated EPS for the next 12 months (forward P/E) or a mix of the trailing EPS of the last two quarters and the forward P/E for the next two quarters. When distinguishing absolute P/E from relative P/E, it is important to remember that absolute P/E represents the P/E of the current time period. For example, if the price of the stock today is$100, and the TTM earnings are $2 per share, the P/E is 50 ($100/$2). ### Relative P/E The relative P/E compares the current absolute P/E to a benchmark or a range of past P/Es over a relevant time period, such as the past 10 years. The relative P/E shows what portion or percentage of the past P/Es the current P/E has reached. The relative P/E usually compares the current P/E value to the highest value of the range, but investors might also compare the current P/E to the bottom side of the range, measuring how close the current P/E is to the historic low. The relative P/E will have a value below 100% if the current P/E is lower than the past value (whether the past high or low). If the relative P/E measure is 100% or more, this tells investors that the current P/E has reached or surpassed the past value. ## Limitations of Using the P/E Ratio Like any other fundamental designed to inform investors on whether or not a stock is worth buying, the price-to-earnings ratio comes with a few important limitations that are important to take into account, as investors may often be led to believe that there is one single metric that will provide complete insight into an investment decision, which is virtually never the case. Companies that aren't profitable, and consequently have no earnings—or negative earnings per share, pose a challenge when it comes to calculating their P/E. Opinions vary on how to deal with this. Some say there is a negative P/E, others assign a P/E of 0, while most just say the P/E doesn't exist (not available—N/A) or is not interpretable until a company becomes profitable for purposes of comparison. One primary limitation of using P/E ratios emerges when comparing P/E ratios of different companies. Valuations and growth rates of companies may often vary wildly between sectors due both to the differing ways companies earn money and to the differing timelines during which companies earn that money. As such, one should only use P/E as a comparative tool when considering companies in the same sector, as this kind of comparison is the only kind that will yield productive insight. Comparing the P/E ratios of a telecommunications company and an energy company, for example, may lead one to believe that one is clearly the superior investment, but this is not a reliable assumption. ## Other P/E Considerations An individual company’s P/E ratio is much more meaningful when taken alongside P/E ratios of other companies within the same sector. For example, an energy company may have a high P/E ratio, but this may reflect a trend within the sector rather than one merely within the individual company. An individual company’s high P/E ratio, for example, would be less cause for concern when the entire sector has high P/E ratios. Moreover, because a company’s debt can affect both the prices of shares and the company’s earnings, leverage can skew P/E ratios as well. For example, suppose there are two similar companies that differ primarily in the amount of debt they take on. The one with more debt will likely have a lower P/E value than the one with less debt. However, if business is good, the one with more debt stands to see higher earnings because of the risks it has taken. Another important limitation of price-to-earnings ratios is one that lies within the formula for calculating P/E itself. Accurate and unbiased presentations of P/E ratios rely on accurate inputs of the market value of shares and of accurate earnings per share estimates. The market determines the prices of shares through its continuous auction. The printed prices are available from a wide variety of reliable sources. However, the source for earnings information is ultimately the company themselves.This single source of data is more easily manipulated, so analysts and investors place trust the company's officers to provide accurate information. If that trust is perceived to be broken the stock will be considered more risky and therefore less valuable. To reduce the risk of inaccurate information, the P/E ratio is but one measurement that analysts scrutinize. If the company were to intentionally manipulate the numbers to look better, and thus deceive investors, they would have to work strenuously to be certain that all metrics were manipulated in a coherent manner, which is difficult to do. That's why the P/E ratio continues to be one of the centrally referenced points of data to analyze a company, but by no means the only one. ## Frequently Asked Questions ### What is a good price to earnings ratio? The question of what is a good or bad price to earnings ratio will necessarily depend on the industry in which the company is operating. Some industries will have higher average price to earnings ratios, while others will have lower ratios. For example, as of January 2020, publicly-traded US coal companies had an average P/E ratio of only about 7, compared to more than 60 for software companies.4 If you want to get a general idea of whether a particular P/E ratio is high or low, you can compare it to the average P/E of the competitors within its industry. ### Is it better to have a higher or lower P/E ratio? Many investors will say that it is better to buy shares in companies with a lower P/E, because this means you are paying less for every dollar of earnings that you receive. In that sense, a lower P/E is like a lower price tag, making it attractive to investors looking for a bargain. In practice, however, it is important to understand the reasons behind a company’s P/E. For instance, if a company has a low P/E because their business model is fundamentally in decline, then the apparent bargain might be an illusion. ### What does a P/E ratio of 15 mean? Simply put, a P/E ratio of 15 would mean that the current market value of the company is equal to 15 times its annual earnings. In other words, if you were to hypothetically buy 100% of the company’s shares, it would take 15 years for you to earn back your initial investment through the company’s ongoing profits. ### Compete Risk Free with$100,000 in Virtual Cash
Put your trading skills to the test with our FREE Stock Simulator. Compete with thousands of Investopedia traders and trade your way to the top! Submit trades in a virtual environment before you start risking your own money. Practice trading strategies so that when you're ready to enter the real market, you've had the practice you need. Try our Stock Simulator today >> | 2021-10-26 00:05:42 | {"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": 0.34971103072166443, "perplexity": 2831.2910233390753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587770.37/warc/CC-MAIN-20211025220214-20211026010214-00284.warc.gz"} |
https://openreview.net/forum?id=vV1aVdCD2WW | ## $\ell$Gym: Natural Language Visual Reasoning with Reinforcement Learning
22 Sept 2022, 12:39 (modified: 26 Oct 2022, 14:15)ICLR 2023 Conference Withdrawn SubmissionReaders: Everyone
Keywords: reinforcement learning, natural language, visual reasoning, benchmark
TL;DR: A new benchmark for language-conditioned reinforcement learning in visual environments with highly compositional human-written language.
Abstract: We present $\ell$Gym, a new benchmark for language-conditioned reinforcement learning in visual environments. $\ell$Gym is based on 2,661 human-written natural language statements grounded in an interactive visual environment, and emphasizing compositionality and semantic diversity. We annotate all statements with Python programs representing their meaning. The programs are executable in an interactive visual environment to enable exact reward computation in every possible world state. Each statement is paired with multiple start states and reward functions to form thousands of distinct Markov Decision Processes of varying difficulty. We experiment with $\ell$Gym with different models and learning regimes. Our results and analysis show that while existing methods are able to achieve non-trivial performance, $\ell$Gym forms a challenging open problem.
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4 Replies | 2022-12-04 20:58:11 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20474925637245178, "perplexity": 4881.513822211614}, "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/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00445.warc.gz"} |
https://chemistry.stackexchange.com/questions/141345/what-does-the-equilibrium-constant-mean-for-reactions-with-very-small-concentrat | # What does the equilibrium constant mean for reactions with very small concentrations?
When you calculate the equilibrium constant for a reaction with a higher concentration of reactants, the answer is fairly intuitive. For example, for the reaction:
$$\ce{A + B <=> C}$$
Where we have $$0.10 M$$ of A, $$0.30 M$$ of B and $$0.45$$ M of C the equation for the equilibrium constant is:
$$K = \frac{[0.45]}{[0.10][0.30]} = 15$$
This is relatively intuitive, we can see that the concentration of C is larger than the product of A and B and so saying the products are favoured makes sense.
However, an issue comes about when you consider really small amounts of all of them. For example, for the same reaction
$$\ce{A + B <=> C}$$
If we have $$2.4*10^{-4} M$$ of A, $$2.4*10^{-4} M$$ of B and $$7.5*10^{-6} M$$ of C we have the equation:
$$K = \frac{[7.5*10^{-6}]}{[2.4*10^{-4}][2.4*10^{-4}]} = 130$$
In this case, the equilibrium constant suggests the products are favoured but if you look at how much product we have compared to reactants there's so much more reactants than products. The constant makes sense since multiplying two numbers less than 1 will get a smaller number but it's confusing when you consider the implications that has on the equilibrium. My immediate assumption is that I am misunderstanding the relationship between the equilibrium constant and the concentrations of species in the reaction and if this is the case is there a more intuitive way of looking at this?
• You have it the wrong way round. At a given temperature for any equilibrium $K$ has a fixed value, i.e. is constant. The concentrations must always match this so you cannot just guess arbitrary values. The extent of dissociation changes to ensure concentration ratios agree with $K$. – porphyrin Oct 11 '20 at 8:06
• @ Nathan. I think you simply forget the units. The equilibrium constant has a unit. For your system, the equilibrium constant is not $130$, as you state. It is $130~ L/mol$ – Maurice Oct 17 '20 at 20:09 | 2021-06-21 23:38:29 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8343489766120911, "perplexity": 236.37616000326102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488504838.98/warc/CC-MAIN-20210621212241-20210622002241-00275.warc.gz"} |
https://www.sciencemadness.org/whisper/viewthread.php?tid=157499#pid661542 | Sciencemadness Discussion Board » Fundamentals » Chemistry in General » calcium carbide from calcium chloride, magnesium and coal 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
Author: Subject: calcium carbide from calcium chloride, magnesium and coal
BauArf56
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calcium carbide from calcium chloride, magnesium and coal
so i thought about an easy way to make calcium carbide from household items. I discovered that melting a mixture of calcium chloride (pellets for humidity) and magnesium turnings gives a blackish solid that react with water as calcium would do. As it react the water gets cloudy (calcium hydroxide), so this is probably calcium metal (Mg + CaCl2 → Ca + MgCl2). So now i thought to add some coal to the mix, so while calcium forms, it reacts with carbon and gives calcium carbide. Unfortunately i didn't do this yet, although it looks feasible. Any ideas?
Johnny Cappone
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Even if it worked, I think you would at most get finely divided crystals of CaC2 that would react immediately with atmospheric moisture, since larger granules can only be formed when the substance is melted somewhere around 3,000 degrees Celsius.
Carbide is usually a widely available reagent. I have doubts about the economic viability of a production route that requires metallic magnesium. I have a microwave transformer that I found in the junkyard whose secondary was rewound in order to supply 38V, if i use it to open an arc with graphite electrodes close to a mixture of powdered graphite and CaO/Ca(OH)2/CaCO3, it usually yields some carbide. Production processes on an industrial scale are not always suitable for the amateur chemist, but this is one of them. As it was adopted by the industry, I am inclined to believe that it is the cheapest route as well. Anyway, try and report your experiments!
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From "Chemie und ihre Technologie".
For Carbide electricity isnt nesecary only engough heat. An reaction of
3CaO + 6C + 2Al (powder)= 3CaC2 + 2Al2O3.The Thermal heat of the Thermit reaction is enough for CaC2 production but it was never used because electricity is cheaper.I cant finde the text passage now but an frechn guy used magnesium instead.
BauArf56
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Quote: Originally posted by Alkoholvergiftung From "Chemie und ihre Technologie". For Carbide electricity isnt nesecary only engough heat. An reaction of 3CaO + 6C + 2Al (powder)= 3CaC2 + 2Al2O3.The Thermal heat of the Thermit reaction is enough for CaC2 production but it was never used because electricity is cheaper.I cant finde the text passage now but an frechn guy used magnesium instead.
that's what i wanted! exploiting the high heat generated by the magnesium to make sure the calcium and carbon react.
For the electrochemical stuff, there would be the same problem with the reaction between magnesium and sodium hydroxide: although magnesium is less reactive than sodium, at high temp, it's able to reduce it (for a better explanation see this: https://www.instructables.com/Making-Sodium-Metal/). Anyway, i tried, but it's too hard to ignite, so i'm going to wait to get a new blowtorch...
[Edited on 23-5-2021 by BauArf56]
Fyndium
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Considering any second hand sales of magnesium, thermite compounds etc command a significant premium, it would be hardly economical to produce it yourself by a significant margin. The large industry benefits from getting their stuff at market price.
What costs $4 per kg at market, you will pay minimum 20-50$ or even more even if you buy bulk. If you buy commercialized goods, the price can have even a second 0 before the nomination.
BauArf56
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PubChem on the lithium page:
Since lithium will burn in oxygen, nitrogen or carbon dioxide, and when alight it will remove the combined oxygen in sand, sodium carbonate, etc., it is difficult to extinguish once alight ... Use of normal fire extinguishers (containing water, form, carbon dioxide, halocarbons, dry powders) will either accelerate combustion or cause explosion.
Bretherick, L. Handbook of Reactive Chemical Hazards. 4th ed. Boston, MA: Butterworth-Heinemann Ltd., 1990, p. 1312
So lithium is able to break carbonate bonds. Maybe it could react like this: Li + Na2CO3 ----> 2Li2O + Na2C2
May this work for small amounts of carbide? (obviously i'm not going to waste all my lithium in this reaction)
Sciencemadness Discussion Board » Fundamentals » Chemistry in General » calcium carbide from calcium chloride, magnesium and coal 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 | 2021-12-04 06:42:14 | {"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": 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": 0.4413772523403168, "perplexity": 14373.90302528519}, "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/1637964362952.24/warc/CC-MAIN-20211204063651-20211204093651-00547.warc.gz"} |
http://mathematica.stackexchange.com/questions/35682/calculate-integral-for-arbitrary-parameter-n-in-infinite-square-well-problem | # Calculate integral for arbitrary parameter n in infinite square well problem
I'm continuing[1,2] the study of an infinite square well in the context of quantum mechanics.
Ultimate goal is to calculate the product $\Delta x\Delta k$, for various eigenstates, that is for various values of number $n$. I have finished with $\Delta x$, but I'm stuck with $\Delta k$.
ClearAll["Global*"];
(* The length of the well *)
L = 1;
(* The eigenfunctions, n=1,2,3,... *)
u[n_, x_] := If[x <= 0 || x >= L, 0, Sqrt[2/L] Sin[n π x / L]]
(* The Fourier transform of eigenfunctions u[n,x] from the position
domain onto the momentum domain *)
φ[n_, k_] :=
Simplify[
FourierTransform[u[n, x], x, k, FourierParameters -> {0, -1}],
n ∈ Integers]
(* The probability density function η(n,k) *)
η[n_, k_] :=
FullSimplify[φ[n, k] \[Conjugate] φ[n, k],
{n ∈ Integers, k ∈ Reals}]
(* Calculate (Δk)^2 = <k^2> - <k>^2 = <k^2> *)
Integrate[
k^2 η[n, k], {k, -∞, +∞},
(* Edited: Was: {n ∈ Integers, n > 0}, but this edit didn't
fix the problem. *)
Assumptions -> n ∈ Integers && n > 0]
The problem is that Mathematica can't calculate the last integral for any arbitrary $n$, although it can, correctly, calculate its value for hardcoded $n$s. Like $n=1,2,...$.
My question is: Do you have any idea on how I could calculate it, perhaps by rewriting it a bit, or by using some other trick? In case it helps, the result should be $n^2\pi^2$.
Note: Actually it can be calculated with Cauchy's residue theorem, but I'd like to avoid taking that route, if possible. Though, if it can't be done otherwise, I will post a solution with residual calculation so that this question has an answer.
Mathematica.SE related (to the physical problem) questions:
Is there a more mathematica-y way to label these plots?
Why does FourierTransform converge while same integral manually written does not?
-
There are no close votes, but you might find helpful a related post: How do I evaluate a symbolic integral involving Hermite polynomials?. – Artes Nov 9 '13 at 21:55
This is a stupid workaround. Anyway:
FindSequenceFunction@Table[Integrate[k^2 η[n, k], {k, -∞, +∞}, Assumptions -> {n == p}], {p, 5}]
(*
π^2 #1^2 &
*)
-
Actually it's kinda useful when someone -for example- wants to see whether there's a pattern and then use proof by induction. – Zet Nov 9 '13 at 19:03
@Zet The problem is that it doesn't prove anything. You may get a different law for greater n, and this will overlook that fact – belisarius Nov 9 '13 at 19:05
Here's as close as I can get via Mathematica. First, I just simplify the integrand once for all. Having Simplify in the definition of a function could be really slow. Edit: I added the unsimplified versions of the OP's functions, including a substitution of Boole for If, which I omitted to include in the original answer.
u[n_, x_] := Boole[0 <= x <= L] Sqrt[2/L] Sin[n π x/L];
φ[n_, k_] := FourierTransform[u[n, x], x, k, FourierParameters -> {0, -1}];
η[n_, k_] := φ[n, k]\[Conjugate] φ[n, k];
integrand =
FullSimplify[k^2 \[Eta][n, k],
{n \[Element] Integers, n > 0, k \[Element] Reals,L > 0}]
(* -((2 k^2 L n^2 \[Pi] (-1 + (-1)^n Cos[k L]))/(k^2 L^2 - n^2 \[Pi]^2)^2) *)
Then, Integrate gets real close, if you make a substitution k -> n k. Since the differential of n k is n dk, you have to multiply the Integrate below by n.
int = n Integrate[integrand /. k -> n k, {k, -Infinity, Infinity}];
Simplify[int, {n \[Element] Integers, n > 0}]
(* ConditionalExpression[
(L n^2 Pi^2)/Abs[L]^3,
L \[Element] Reals && 3 Arg[-L^2] <= 2 \[Pi] && (Re[1/L^2] <= 0 || 1/L^2 \[NotElement] Reals)] *)
Reduce[Last[%]]
(* False *)
The only problem is that the condition in the ConditionalExpression is a contradiction.
-
There's something fishy going on. Sometimes I get an error that it doesn't converge, sometimes that it is undefined (e.g., when I add L>0 in the assumptions in the last Simplify[] command), and sometimes the False conditional expression you get. The result is very sensitive to the form of the integrand and my assumptions. – Zet Nov 9 '13 at 21:15
Yes, it is sensitive. The denominator and numerator are zero at the same values of k. Somehow, perhaps because of the way Mathematica manipulates the integrand, it concludes divergence even though the integrand is bounded and the integral absolutely convergent. – Michael E2 Nov 9 '13 at 21:15
The following code yields the correct result:
Another interesting fact is that if I omit the assumption that k ∈ Reals, then Mathematica still gets it right, but it takes ~3x more time:
What is puzzling though is that if I use Assumptions with Integrate I don't get the expected result:
I was under the impression that Assuming[{a1,a2,...}, Integrate[...]] was equivalent to Integrate[..., Assumptions -> a1 && a2 && ...].
Could anyone please try to reproduce my results, ideally in a different OS/Mathematica version combination ? That is something other than Mathematica 9.0.1/Mac OSX 10.9 ?
-
See also this answer for how Integrate` works with assumptions. – Michael E2 Nov 10 '13 at 2:01
Great link @MichaelE2. Thanks!! – Zet Nov 10 '13 at 12:59 | 2014-08-30 16:38:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.7573463320732117, "perplexity": 2643.597201723549}, "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-2014-35/segments/1408500835505.87/warc/CC-MAIN-20140820021355-00378-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://docs.ray.io/en/releases-1.4.1/ray-design-patterns/submission-order.html | # Advanced antipattern: Processing results in submission order using ray.get¶
TLDR: Avoid calling ray.get one by one in a loop if possible.
When processing in submission order, a remote function might delay processing of earlier finished remote function. When using ray.wait we can get finished tasks early, speeding up total time to completion.
Processing results in submission order
A batch of tasks are submitted, and we need to process their results individually once they’re done. We want to process the results as they finish, but use ray.get on the ObjectRefs in the order that they were submitted.
If each remote function takes a different amount of time to finish, we may waste time waiting for all of the slower (straggler) remote functions to finish while the other faster functions have already finished. Instead, we want to process the tasks in the order that they finish using ray.wait.
## Code example¶
import random
import time
import ray
@ray.remote
def f():
time.sleep(random.random())
# Antipattern: process results in the order they were spawned.
refs = [f.remote(i) for i in range(100)]
for ref in refs:
# Blocks until this ObjectRef is ready.
result = ray.get(ref)
# process result
# Better approach: process results in the order that they finish.
refs = [f.remote(i) for i in range(100)]
unfinished = refs
while unfinished:
# Returns the first ObjectRef that is ready.
finished, unfinished = ray.wait(unfinished, num_returns=1)
result = ray.get(finished)
# process result | 2022-01-28 06:16:16 | {"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": 0.4088820219039917, "perplexity": 5594.700817017589}, "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-2022-05/segments/1642320305420.54/warc/CC-MAIN-20220128043801-20220128073801-00170.warc.gz"} |
https://ch.mathworks.com/help/simbio/ref/sbiosampleparameters.html | Documentation
# sbiosampleparameters
Generate parameters by sampling covariate model (requires Statistics and Machine Learning Toolbox software)
## Description
example
phi = sbiosampleparameters(covexpr,thetas,omega,ds) generates a matrix phi containing sampled parameter values using the covariate model specified by the covariate expression covexpr, fixed effects thetas, covariance matrix omega, and covariate data ds.
example
phi = sbiosampleparameters(covexpr,thetas,omega,n) uses a scalar n that specifies the number of rows in phi when the parameters are not dependent on any covariate.
example
[phi,covmodel] = sbiosampleparameters(_) returns a matrix phi and a covariate model object covmodel using any of the input arguments from previous syntaxes.
## Examples
collapse all
This example uses data collected on 59 preterm infants given phenobarbital during the first 16 days after birth. Each infant received an initial dose followed by one or more sustaining doses by intravenous bolus administration. A total of between 1 and 6 concentration measurements were obtained from each infant at times other than dose times, for a total of 155 measurements. Infant weights and APGAR scores (a measure of newborn health) were also recorded. Data was described in [1], a study funded by the NIH/NIBIB grant P41-EB01975.
Visualize the data.
t = sbiotrellis(ds,'ID','TIME','CONC','marker','o','markerfacecolor',[.7 .7 .7],'markeredgecolor','r','linestyle','none');
t.plottitle = 'States versus Time';
Create a one-compartment PK model with bolus dosing and linear clearance to model such data.
pkmd = PKModelDesign;
'HasResponseVariable',true,'HasLag',false);
onecomp = pkmd.construct;
Suppose there is a correlation between the volume of the central compartment (Central) and the weight of infants. You can define this parameter-covariate relationship using a covariate model that can be described as
$\mathrm{log}\left({V}_{i}\right)={\theta }_{V}+{\theta }_{V}{WEIGHT}}\ast WEIGH{T}_{i}+{\eta }_{V,i}$,
where, for each ith infant, V is the volume, θs (thetas) are fixed effects, η (eta) represents random effects, and WEIGHT is the covariate.
covM = CovariateModel;
covM.Expression = {'Central = exp(theta1+theta2*WEIGHT+eta1)'};
Define the fixed and random effects. The column names of each table must have the names of fixed effects and random effects, respectively.
thetas = table(1.4,0.05,'VariableNames',{'theta1','theta2'});
eta1 = table(0.2,'VariableNames',{'eta1'});
Change the group label ID to GROUP as required by the sbiosampleparameters function.
ds.Properties.VariableNames{'ID'} = 'GROUP';
Generate parameter values for the volumes of central compartments Central based on the covariate model for all infants in the data set.
phi = sbiosampleparameters(covM.Expression,thetas,eta1,ds);
You can then simulate the model using the sampled parameter values. For convenience, use the function-like interface provided by a SimFunction object.
First, construct a SimFunction object using the createSimFunction method, specifying the volume (Central) as the parameter, and the drug concentration in the compartment (Drug_Central) as the output of the SimFunction object, and the dosed species.
f = createSimFunction(onecomp,covM.ParameterNames,'Drug_Central','Drug_Central');
The data set ds contains dosing information for each infant, and the groupedData object provides a convenient way to extract such dosing information. Convert ds to a groupedData object and extract dosing information.
grpData = groupedData(ds);
doses = createDoses(grpData,'DOSE');
Simulate the model using the sampled parameter values from phi and the extracted dosing information of each infant, and plot the results. The ith run uses the ith parameter value in phi and dosing information of the ith infant.
t = sbiotrellis(f(phi,200,doses.getTable),[],'TIME','Drug_Central');
% Resize the figure.
t.hFig.Position(:) = [100 100 1280 800];
## Input Arguments
collapse all
Covariate expressions, specified as a cell array of character vectors or string vector that defines the parameter-covariate relationships.
If a model component name or covariate name is not a valid MATLAB® variable name, surround it by square brackets when referring to it in the expression. For example, if the name of a species is DNA polymerase+, write [DNA polymerase+]. If a covariate name itself contains square brackets, you cannot use it in the expression.
Fixed effects, specified as a table, dataset, or numeric vector containing values for fixed effect parameters defined in the covariate expressions covexpr. Fixed effect parameter names must start with 'theta'.
• If thetas is a table, thetas.Properties.VariableNames must match the names of the fixed effects.
For example, suppose that you have three thetas: thetaOne = 0.1, theta2 = 0.2, and theta3 = 0.3. You can create the corresponding table.
thetas = table(0.1,0.2,0.3);
thetas.Properties.VariableNames = {'thetaOne','theta2','theta3'}
thetas =
1×3 table
thetaOne theta2 theta3
________ ______ ______
0.1 0.2 0.3
• If thetas is a dataset, thetas.Properties.VarNames must match the names of the fixed effects.
• If thetas is a numeric vector, the order of the values in the vector must be the same ascending ASCII dictionary order as the fixed effect names.
Use the sort function to sort a cell array of character vectors to see the order.
sort({'thetaOne','theta2','theta3'})
ans =
1×3 cell array
{'theta2'} {'theta3'} {'thetaOne'}
Then specify the value of each theta in the same order.
thetas = [0.2 0.3 0.1];
Covariance matrix of random effects, specified as a table, dataset, or matrix. Random effect parameter names must start with 'eta'.
• If omega is a table, omega.Properties.VariableNames must match the names of the random effects. Specifying the row names (RowNames) is optional, but if you do, they must also match the names of random effects.
Suppose that you want to define a diagonal covariance matrix with three random effect parameters eta1, eta2, and eta3 with the values 0.1, 0.2, and 0.3, respectively.
$\left[\begin{array}{l}Cov\left({\eta }_{1},{\eta }_{1}\right)\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left(\eta {}_{1},{\eta }_{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left({\eta }_{1},{\eta }_{3}\right)\\ Cov\left({\eta }_{2},{\eta }_{1}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left({\eta }_{2},{\eta }_{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left({\eta }_{2},{\eta }_{3}\right)\\ Cov\left({\eta }_{3},{\eta }_{1}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left({\eta }_{3},{\eta }_{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Cov\left({\eta }_{3},{\eta }_{3}\right)\end{array}\right]=\left[\begin{array}{l}eta1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}eta2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}eta3\end{array}\right]$
You can construct the corresponding table.
eta1 = [0.1;0;0];
eta2 = [0;0.2;0];
eta3 = [0;0;0.3];
omega = table(eta1,eta2,eta3,'VariableNames',{'eta1','eta2','eta3'})
omega =
3×3 table
eta1 eta2 eta3
____ ____ ____
0.1 0 0
0 0.2 0
0 0 0.3
• If omega is a dataset, omega.Properties.VarNames must match the names of the random effects. Specifying the row names (ObsNames) is optional, but if you do, they must also match the names of random effects.
• If omega is a matrix, the rows and columns must have the same ascending ASCII dictionary order as the random effect names.
Use the sort function to sort a cell array of character vectors to see the order.
sort({'eta1','eta2','eta3'})
ans =
1×3 cell array
{'eta1'} {'eta2'} {'eta3'}
Covariate data, specified as a dataset or table containing the covariate data for all groups.
ds must have a column named 'Group' or 'GROUP' specifying the group labels as well as a column each for all covariates used in the covariate model. The column names must match the names of the corresponding covariates used in the covariate expressions.
Number of rows in phi, specified as a scalar.
## Output Arguments
collapse all
Sampled parameter values, returned as a matrix of size S-by-P, where S is the number of groups specified in ds or specified by n and P is the number of parameters which is equal to the number of elements in covexpr.
Covariate model, returned as a CovariateModel object which represents the model defined by covexpr.
## Compatibility Considerations
expand all
Warns starting in R2018b
## References
[1] Grasela Jr, T.H., Donn, S.M. (1985) Neonatal population pharmacokinetics of phenobarbital derived from routine clinical data. Dev Pharmacol Ther. 8(6), 374–83. | 2019-11-18 13:30:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.4592702090740204, "perplexity": 3775.7318558257084}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669795.59/warc/CC-MAIN-20191118131311-20191118155311-00361.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/college-algebra-7th-edition/chapter-4-exponential-and-logarithmic-functions-section-4-4-laws-of-logarithms-4-4-exercises-page-394/8 | ## College Algebra 7th Edition
(a) This is true. $\log\frac{A}{B}=\log A-\log B$ (This is a general property of logs.) (b) This is false. $\frac{\log A}{\log B}\neq\log A-\log B$ Because the right side actually equals: $\log A-\log B=\log (\frac{A}{B})$ | 2018-07-22 05:35:26 | {"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": 0.8490443825721741, "perplexity": 1039.3296881418596}, "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-2018-30/segments/1531676593010.88/warc/CC-MAIN-20180722041752-20180722061752-00094.warc.gz"} |
https://financetrain.com/present-value-of-a-perpetuity | # Present Value of a Perpetuity
A perpetuity is a type of annuity that pays equal cash flows that occur periodically such as monthly, quarterly or annually for an infinite period of time.
The present value of an annuity is calculated using the following formula:
PV = A/r
Where, A is the annuity payment, and r is the interest rate.
Assume that an perpetuity pays $500 per year. The rate of return is 8%. The present value of this perpetuity is calculated as follows: PV = 500/0.08 =$6,250
If the investor invests $6,250 in the perpetuity paying 8% rate of return, he will receive a payment of$500 per year for an infinite period. | 2022-11-27 06:04:23 | {"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": 0.693291187286377, "perplexity": 1405.1785522139467}, "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/1669446710192.90/warc/CC-MAIN-20221127041342-20221127071342-00673.warc.gz"} |
https://www.zbmath.org/?q=an%3A0578.62003 | # zbMATH — the first resource for mathematics
Asymptotic expansions for general statistical models. With the assist. of W. Wefelmeyer. (English) Zbl 0578.62003
Lecture Notes in Statistics, 31. Berlin etc.: Springer-Verlag. VII, 505 p. DM 79.00 (1985).
In his book ”Contributions to a general asymptotic statistical theory.” Lect. Notes Stat. 13 (1982; Zbl 0512.62001), the author tried to describe the local structure of a general family $${\mathcal P}$$ of probability measures by its tangent space and the local behavior of a functional $$\kappa: P\to R^ k$$ by its gradient and obtained asymptotic envelope power functions for tests and asymptotic bounds for the concentration of estimators, when statistical procedures are based on independent, identically distributed observations.
As a continuation, in this book the author considers these asymptotic investigations not by limit distributions but by Edgeworth expansions (adding one term of order $$n^{-1/2}$$ to the limit distributions).
To do this, the author introduces concepts of ”degenerate convergence condition of order (b,c) (for short: $$DCC_{b,c})''$$ and $$''DCC_{b,c}$$-differentiable with derivative g at a rate $$o(t^ a)''$$, and with the aid of ”canonical gradients” he studies the second order envelope power functions for composite hypotheses, second order bounds for the concentration of confidence bounds and median unbiased estimators, etc.
Then, using these results, he considers various general statistical procedures and gives bounds of order $$o(n^{-1/2})$$ for the efficiency of them. Many examples and explanations of the new notions are given. It seems that many problems are still open.
Reviewer: K.Yoshihara
##### MSC:
62A01 Foundations and philosophical topics in statistics 62F05 Asymptotic properties of parametric tests 62F12 Asymptotic properties of parametric estimators 62-02 Research exposition (monographs, survey articles) pertaining to statistics | 2021-07-24 12:26:53 | {"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": 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": 0.5785953402519226, "perplexity": 1050.4645655504528}, "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/1627046150264.90/warc/CC-MAIN-20210724094631-20210724124631-00124.warc.gz"} |
https://gamedev.stackexchange.com/questions/171907/how-can-i-do-a-metroid-like-transition-between-two-tilemaps-with-monogame-extend | # How can I do a Metroid like transition between two tilemaps with Monogame Extended?
I want to do a transition like those in the Metroid games. Here is an example of what i mean: https://www.youtube.com/watch?v=RKJ3MAi0ZI0.
You enter the door, the screen fades to black and the camera scrolls to the next room where the screen fades in again.
I'm using Monogame Extended and i have a TiledMap. I use them like described in the documentation. I have a camera with a view matrix and a projection matrix
Camera2D camera;
...
public void DrawGame(GameTime gameTime, SpriteBatch spriteBatch)
{
var viewMatrix = camera.GetViewMatrix();
var projectionMatrix = Matrix.CreateOrthographicOffCenter(0, gDevice.Viewport.Width, gDevice.Viewport.Height, 0, 0f, -1f);
tiledMap.Draw(viewMatrix, projectionMatrix);
}
My idea is now to create a second tiledMap to placing them directly next to the first one where the transition occurs. Then i just have to move the camera from one map to the next one. My problem now is that i have no clue how to put them next to each other. From my understanding i somehow have to use the view and projection matrix to draw the tiledMap but i can't figure out how it should work with two tiledMaps.
• Can you show an example of the Metroid transition you want to imitate? – DMGregory May 13 at 20:41
• From the looks of your description, it looks more like the NES Metroid transition that you want to make rather than the door transition. It is however difficult to understand what you've tried if I don't know what you exactly want and what you currently got so far. – Steven May 14 at 6:22
• I edited my post – Ultralord May 15 at 20:30 | 2019-08-25 13:28:01 | {"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": 0.3216705620288849, "perplexity": 743.4517780216268}, "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/1566027330233.1/warc/CC-MAIN-20190825130849-20190825152849-00158.warc.gz"} |
https://www.semanticscholar.org/paper/Impurity-states-and-Localization-in-Bilayer-the-Low-Collado-Usaj/720133b0953060ab07aeeaa672bfaefb88246e4c | # Impurity states and Localization in Bilayer Graphene: the Low Impurity Concentration Regime
@inproceedings{Collado2014ImpuritySA,
title={Impurity states and Localization in Bilayer Graphene: the Low Impurity Concentration Regime},
author={H. P. Ojeda Collado and Gonzalo Usaj and C. A. Balseiro},
year={2014}
}
• Published 8 December 2014
• Materials Science, Physics
We study the problem of non-magnetic impurities adsorbed on bilayer graphene in the diluted regime. We analyze the impurity spectral densities for various concentrations and gate fields. We also analyze the effect of the adsorbate on the local density of states (LDOS) of the different C atoms in the structure and present some evidence of strong localization for the electronic states with energies close to the Dirac point.
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We analyze, within a minimal model that allows analytical calculations, the electronic structure and Landau levels of graphene multi-layers with different stacking orders. We find, among other | 2022-01-25 22:37:58 | {"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": 0.5700487494468689, "perplexity": 3565.2945644802594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304876.16/warc/CC-MAIN-20220125220353-20220126010353-00284.warc.gz"} |
http://entr.frolaidraulica.it/pushdown-automata-generator-online.html | generate a push-down automaton for the following constrained regular expression. If L is a language accepted by a nondeterministic finite. Listen to what I just found in the WolframTones music universe. More than 40 million people use GitHub to discover, fork, and contribute to over 100 million projects. Pushdown automata are used in theories about what can be computed by machines. Construct Pushdown Automata for given languages Prerequisite - Pushdown Automata , Pushdown Automata Acceptance by Final State A push down automata is similar to deterministic finite automata except that it has a few more properties than a DFA. Till now we have seen machines which can move in one direction: left to right. Generate Random Cave Levels Using Cellular Automata by Michael us a big two-dimensional array of blocks, each of which is either solid or empty. 3/30: And if you want more background on discrete math, take a look at the free book Foundations of Computer Science, espcially Ch. Lam and Jeffrey D. A prosperous environment containing online execution/compilation. en A linear bounded automaton is a device which is more powerful than a pushdown automaton but less so than a Turing machine. Conjunctive grammars and alternating pushdown automata Conjunctive grammars and alternating pushdown automata Aizikowitz, Tamar; Kaminski, Michael 2013-03-26 00:00:00 In this paper we introduce a variant of alternating pushdown automata, synchronized alternating pushdown automata, which accept the same class of languages as those generated by conjunctive grammars. Getting Automaton Simulator. An average of two million volts surge through the 40-foot-high machine. Keywords: Finite Automata, Regular Expression, Automatic Programming, Code Generator. Introduction to pushdown automata(PDA) 2. Istilah automata merupakan bentuk tunggal, sedangkan bentuk jamaknya adalah automaton. save hide report. Push Down Automata. The big white box above is the FSM designer. Pushdown Automata is a finite automata with extra memory called stack which helps Pushdown automata to recognize Context Free Languages. Main | Generate Cellular Automata. JSON_checker is made up of these files:. • To have a knowledge of regular languages and context free languages. The main difference with respect to the case of finite words turns out to be determinizability: nondeterministic Büchi visibly pushdown automata are strictly. Context-free grammars, pushdown automata, the pumping lemma for context-free languages. In addition to a finite set of control states, a pushdown automaton has a stack which can be manipulated with the usual push and pop operations. It is also shown that alternating 2-way finite state machines accept only regular languages. Deterministic Time Division Multiplexing listed as DTDM. The transitions a machine makes are based not only on the input and current state, but also on the stack. $\begingroup$ I should have remembered about pushdown automata :) I was imagining the ISM would be specified in the same way that FSMs are traditionally presented. L56 2011 005. The third edition of Languages and Machines: An Introduction to the Theory of Computer Science provides readers with a mathematically sound presentation of the theory of computer science at a level suitable for junior and senior level computer science majors. 2 synonyms for automation: mechanisation, mechanization. In other words, given any type of the machine, we can transform it to the other. The stack head scans the top symbol of the stack. How to use Deepbrid. Avoid lawsuits, claims and hefty fines, and win your customers' trust by being transparent. Get this from a library! Introduction to automata theory, languages, and computation. A written work or composition found in print, or digitally as an e-book. Pushdown Automaton i 87 4. Automata theory helps engineers design and analyze digital circuits which are parts of computers, telephone systems, or control systems. Convert from NFA to DFA. Formal Languages and Automata Theory: Regular languages and finite automata, Context free languages and Push-down automata, Recursively enumerable sets and Turing machines, Undecidability; Click below links:- An Introduction to Formal Languages and Automata, Third Edition Peter Linz Solution of An Introduction to Formal Languages and Automata Third Edition Peter Linz Formal Languages and. Usually I would have done it using Inkscape (as it is my favorite tool for creating figures for my LaTeX documents), but doing it manually is pretty tedious work. E = a+ Strings = a, aa…. In the remaining chapters, Turing machines are introduced and the book culminates in analyses of effective computability, decidability, and Goedel’s incompleteness theorems. Using these tools in Automata Theory Course. How to design a pushdown automata. Additional Notes: Mouse, keyboard and game pad (XInput only). Markup Languages. In this pushdown automata used as an acceptor model and context-free grammar used as a generator model. finite-state machines, push-down automata, and Turing machines. 2 Pushdown Automata Formal definition of a pushdown automaton. FSM simulator is a demo of using noam, a JavaScript library for working with finite-state machines, grammars and regular expressions. Definition of the Pushdown. sets of strings) and machines for processing these languages, with emphasis on classes of languages and corresponding classes of machines. Once you have modeled data elements in XML Schema using the graphical XML. This course offers an introduction to the theory of computation. Audience This tutorial is exclusively designed for the students who are seeking a degree in any information technology or computer science related field. tikz is a great package for drawing both deterministic and nondeterministic Finite Automata. A random number generator (RNG) is a device that generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. This course provides a challenging introduction to some of the central ideas of theoretical computer science. Ullman, "Compilers: Principles, Techniques, and Tools", 2nd Edition, Addison-Wesley, 2007 [2] John E. Give informal descriptions and state diagrams of pushdown automata for the languages in Exercise 2. Push Down Automata. Most programming languages have deterministic PDA’s. 17 Tu Pushdown automata, CFG ↔ PDA 5. Artificial Intelligence (Introduction) Artificial Intelligence (AI) is the part of computer science which focuses on creating machines and software that can take on behaviors that humans consider intelligent. Exercises for Section 5. There is a website that goes along with the book. Generator Parole - Solutii Web: domenii, gazduire, servere dedicate, reseller gazduire www. 2GB per file without registration as a free user. Apply a variety of operations to transform automata. Automaton Simulator should run on any platform supporting Java, version 1. a stack with infinite size. save hide report. This text covers everything in the course plus significantly more, and is an important resource to own if you intend to pursue a career in computer science (as opposed to merely programming). pushback synonyms, pushback pronunciation, pushback translation, English dictionary definition of pushback. A valid regex consists of alphanumeric characters representing the set of input symbols (e. JFLAP is software for experimenting with formal languages topics including nondeterministic finite automata, nondeterministic pushdown automata, multi-tape Turing machines, several types of grammars, parsing, and L-systems. If any cell encounters a wall, it triggers a pitched sound whose frequency is determined by. Stay on top of important topics and build connections by joining Wolfram Community groups relevant to your interests. is the transition function 5. What is the maximum number of reduce moves that can be taken by a bottom-up parser for a grammar with no epsilon- and unit-production (i. Manually or programmatically create and edit DFA and NFA. Another technique that based on this conversion was developed in [5] is a context-free grammar that used as a meta-language. In fact, the set of languages that can be recognized by PDAs are the context-free languages of the previous module. Pushdown automata differ from normal finite state machines in two ways: 1. Hopcroft Chapter 5. • Nondeterministic pushdown automata: meaningless, but the context-free languages can be decided by a Turing machine. Results 1 – 16 of by A. Level 4: Compiler Generator Coco/R Coco/R (download and documentation) QueryForm example from the slides Text Book. 2 synonyms for automation: mechanisation, mechanization. fully implemented in C++/Qt for Linux and Windows, non-deterministic Turing machines, multi character edges and edges with character sets for finite automata, parameterized machine schemata with power/superscript expression to repeat an automaton a certain number of times. Therefore, a SQL command is being created in the SQL statement generator for FEMS-Pushdown. If you want to update your Flash Player you can do this directly at Adobe. Analyze and reason about computational models, including finite-state automata, pushdown automata and Turing machines. 3 Deterministic Pushdown Automata and Deterministic Context-Free Languages. It also makes the generated code a little smaller. SeedRandom [Method->" method "] specifies what method should be used for the pseudorandom generator. Gazduire Site. Turing machines, the Chomsky hierarchy, and the Church-Turing thesis. Pumping Lemma and other properties of CFL, pdf. Curate this topic. -Works with any tile size and any map size. Teaching the theory of computation provides the basics for every branch of modern computer science. Cellular automata are simulations on a linear, square, or cubic grid on which each cell can be in a single state, often just ON and OFF, and where each cell operates on its own, taking the states of its neighbors as input and showing a state as output. Pumping 94 6. I programmed a generator to generate entire planets in high. Once you have modeled data elements in XML Schema using the graphical XML. These are generated by context-free grammars and also by pushdown automata. Non Deterministic Finite Automata with Epsilon Transition. So, our tool provides you over 300 000 activation codes for you and for your friends if you want to share it. XML and Document-Type Definitions. Pushdown automata is simply an NFA augmented with an "external stack memory". Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer the question. You can configure the TREXOPS mode per BW provider and per BW query. Check character expressions. is the input alphabet 3. Γ is the set of pushdown symbols (which can be pushed and popped from stack) q0 is the initial state. You can view Result with Detail Solution of each question after completion of the test. So, our tool provides you over 300 000 activation codes for you and for your friends if you want to share it. Mendel 258 Instructor: David Matuszek, [email protected] • A push-down automaton has an in nite amount of memory but it is. It is a 'cellular automaton', and was invented by Cambridge mathematician John Conway. An introduction to formal languages and automata / Peter Linz. FAiR-CG provides a domain specific language in Ruby for specifying finite automata and tools for generating parser code from the finite automata definitions in various languages. Automata for Context-Free Languages Varioussimplememorymodelsarepossible: Queue: Firstin,firstout(likewaitinginline) Stack: Lastin,firstout(likealaundrybasket). pushdown-automaton context-free-grammar formal-languages Updated Dec 11, and links to the pushdown-automaton topic page so that developers can more easily learn about it. Pushdown automata homework solutions Designing PDA. In fact, the pushdown automata accepting Lˆ have a special structure: while reading a call, the automaton must push one symbol, while reading a return symbol, it must pop one symbol (if the stack is non-empty), and while reading an internal symbol, it can only update its control. Read on O'Reilly Online Learning with a 10-day trial Start your free trial now Buy on Amazon. Pushdown Automata. They have larger healthbars, attack phases and defeating them advances the main story. "Automata theory" is the theory that made the success of compiler construction possible. It is Deterministic Time Division Multiplexing. We are offering the best software free downloads, reviews, news, and articles. Links to all Subjects Tutorials. Mode 2 is the FEMS-pushdown, mode 3 is the MultiProvider-pushdown. Leftmost Derivations as a Way to Express Ambiguity. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. The book is primarily designed for one semester course in Automata and Compiler Design for undergraduate and postgraduate students of Computer Science and. The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages. A written work or composition found in print, or digitally as an e-book. What is the maximum number of reduce moves that can be taken by a bottom-up parser for a grammar with no epsilon- and unit-production (i. Timed Automata. But Turing machine can move in both directions and also it can read from TAPE as well as write on it. 5 Definition. a stack with infinite size. pushdown automaton, , , , pushdown automaton as Turing machine simulator Puskorius, G. Note that this definition includes deterministic pushdown automata, which are simply nondeterministic pushdown automata with only one available route to take. Listen to what I just found in the WolframTones music universe. L56 2011 005. The National Science Foundation has supported JFLAP for many years. Includes bibliographical references and index. An automation is supposed to run on some given. A written work or composition found in print, or digitally as an e-book. LTL 2 BA : fast translation from LTL formulae to Büchi automata Software written by Denis Oddoux (v1. AMCAT Automata Questions, AMCAT Programming Questions, AMCAT Coding Questions, AMCAT Automata Questions for Cognizant, AMCAT Pattern Programs, Automata Programming, AMCAT Java Coding Questions, AMCAT Automata Questions and Answers PDF, AMCAT Automata Programming Questions. Each transition is based on the current input symbol and the top of the stack, optionally pops the top of the stack, and optionally pushes new symbols onto the stack. 8-bit Sprite Generation using Cellular Automata. November 2005) ISBN-10: 0387292373. Introduction to Automata Theory, Languages, Pushdown Automata. Synonyms for push back in Free Thesaurus. Computer Science All Past 31 Years GATE Questions well Organized Subject and Chapterwise by GateQuestions. Online Braille Generator This is a free tool which translates English characters to Grade 1 Braille. 1 Pushdown Automata (PDA) • Informally: - A PDA is an NFA-ε with a infinite stack. Pushdown Automata for an intersection? 1414 Replacing a 32-bit loop counter with 64-bit introduces crazy performance deviations with _mm_popcnt_u64 on Intel CPUs. - Various definitions for Pushdown Automata - Non-equivalence of deterministic and non-deterministioc PDAs. I know it can be solved by a pushdown automaton. Pushdown definition: a list in which the last item added is at the top | Meaning, pronunciation, translations and examples. How to design a pushdown automata. They strictly subsume pushdown timed automata of Bouajjani et al. Use MathJax to format equations. To do this, we use a stack. \$\begingroup\$ If you've had good luck using Pushdown Automata for this purpose, then I'd say continue using Pushdown Automata. For instance, the well known automaton, Conway’s Game of Life, has the following rules:. Automata Theory In the lecture about theoretical computer science you have seen finite automata, pushdown automata, and Turing machines. Mat Automaton; new MAT Automaton; MAT hack automaton; Credit Card Generator; Yahoo Messenger hack; Facebook hack V2; Mat Online Hack package; Mat gold point hack; Facebook Hack - MAT Automation Update; M. doc Author: tseng Created Date:. This course offers an introduction to the theory of computation. learning parser automata interpreter guide lr-parser parser-generator learning-exercise learn educational learning-by-doing parse-trees practical pushdown-automaton abstract-syntax-tree automata-simulator deterministic-finite-automata parser-combinator parser-development parse-tables. We also look at closure properties of the regular languages, e. If L 1 and L 2 are context-free languages, then L 1 ∩L 2 (a) must be context-free. Synchronous context-free grammars (SCFGs) are now widely used in statistical machine translation, with Hiero as the preeminent example (Chiang 2007). Answer to Give informal descriptions and state diagrams of pushdown automata for the languages in Exercise 2. A grid based cellular automaton cave generator! Features: -Modular generation scripts that store values to a ds_grid and that are easily modifiable by changing simple arguments. It is also shown that alternating 2-way finite state machines accept only regular languages. sequence of input in discrete time steps. Akku Mega Kft Importőr Budapest. Automaton Simulator allows you to draw and simulate a variety of theoretical machines, including: deterministic finite automata nondeterministic finite automata deterministic push-down automata Turing machines The program should run on any platform with Java 1. Formal languages. • A push-down automaton has an in nite amount of memory but it is. Leftmost Derivations as a Way to Express Ambiguity. Use caution when you configure advanced session properties. The Marx generator — often mistaken in appearance. js, and jQuery. Check character expressions. Finite state recognizers are simple computing machines that read (or at least try to read) a sequence of symbols from an input tape. Next: Converting a PDA to Up: PUSH-DOWN AUTOMATA Previous: Formal Definition Contents Converting a CFG to a PDA. learning parser automata interpreter guide lr-parser parser-generator learning-exercise learn educational learning-by-doing parse-trees practical pushdown-automaton abstract-syntax-tree automata-simulator deterministic-finite-automata parser-combinator parser-development parse-tables. On the one hand the time that the pattern generator needs to save a chart depends on the size of the graphic. The course gives an overview over basic formal grammars and abstract machine models used in Computer Science. Pushdown Automata (PDA) A Pushdown Automata (PDA) is essentially an NFA with a stack. Tutorial sheet for week 2 now online under Tutorials; Timetable change: Monday's lecture in Marie Reay 6. a start state often denoted q0. It attempts to help students grasp the essential concepts involved in automata theory. Understanding Computation From Simple Machines to Impossible Programs. Includes bibliographical references and index. Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer the question. Generalized algebraic data types, on the other hand,. This game became widely known when it was mentioned in an article published by Scientific American in 1970. production must be a terminal symbol. Here is how you can enable JavaScript. Push Down Automata. Chapter 8 & 9:Undecidability #22. Quiz: Theory of Computation Mock Tests on Context free languages and Push-down automata. To learn more about Braille visit this Wikipedia article. Filepedia is the hub to get the best applications on any of your device. is the start state 6. 4 on combinatorics. 2 Finite State Automata. Only the nondeterministic PDA defines all the CFL’s. It should compile under most C++ compilers, although it has only been tested with the g++ compiler on Linux. Viewed 3k times 1. Prentice-Hall, 2008. JFLAP is software for experimenting with formal languages topics including nondeterministic finite automata, nondeterministic pushdown automata, multi-tape Turing machines, several types of grammars, parsing, and L-systems. Includes bibliographical references and index. Screen resolution: 1280×720. Automata Editor. Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For some of the OLDER tools and OLDER versions of tools there is an OLD ftp site with OLD source code via anonymous ftp here. Regular Languages (Chapter 1). Given an SCFG and an n-gram language model, the challenge is to decode with them, that is, to apply them to source text to generate a target translation. One possible approach consists in finding an automata model capturing the behaviours of such programs (as pushdown automata do for order-1 recursion), and. AMCAT Automata Questions with Solutions. Make sure your paper is error-free! Good job citing! Now get peace of mind. The West Chester University Computer Science Department, within the College of Sciences & Mathematics, provides access, through education and state-of-the-art technology, to the ever-growing professional opportunities available in the fields of Computer Science. Quickly (in less than a decade), we had LALR parsers which are a subset of deterministic pushdown automata that allows us to implement shift/reduce parsers. The program can be used to visually design and test deterministic finite automata (DFAs) of arbitrary size and complexity. CellularAutomaton[rule, init, t] generates a list representing the evolution of the cellular automaton with the specified rule from initial condition init for t steps. Theory of Computation questions and answers (1) From the options given below, the pair having different expressive power is (A) Deterministic Push Down Automata (DPDA) and Non-deterministic Push Down Automata (NPDA) (B) Deterministic Finite Automata (DFA) and Non-deterministic Finite Automata(NFA). I need help writing transition functions for this problem. Hello everybody , today we give you a free activation , Nier Automata Key Generator. Introduction to Complexity Theory and NP-Completeness. The Java-based application opens the door to various experiments related to anything ranging from nondeterministic pushdown automata and nondeterministic finite automata to various grammar types. There is no time limit. We study the equivalence problem for Q-weighted visibly pushdown automata (VPA) [3]. Recomandat pentru o gama diversa de aplicatii: santiere de constructii, pentru alimentarea imobilelor in caz de avarie cu panou de comanda manuala sau automata (optionala), pentru evenimente in aer liber etc. [2014] First-order Definable Transformations. Last update: 27 December 2019 This site is developed and maintained by John Walker, founder of Autodesk, Inc. Solutions to Selected. • Questions: - What is a stack? - How does a stack help? • A DFA can "remember" only a finite amount of information, whereas a PDA can "remember" an infinite amount of (certain types of) information. NieR Automata Serial key Cd key Online Free Download. The Q component from the definition is a set of all possible states which. LEX (Lexical Analysis Program Generator). The organizing theme for this course is: the Chomsky hierarchy: regular expressions, context free, context sensitive, phrase structure' finite automata, pushdown automata, linear bounded automata, Turing machines. It should compile under most C++ compilers, although it has only been tested with the g++ compiler on Linux. Small SEO Tools is a best Search Engine Optimization tools. November 2005) ISBN-10: 0387292373. The source code to my Cellular Automaton image generator can be found here. 1 Chapter 8 & 9. , automata theory, formal languages, computability, and complexity theory. What does automata mean? 8. Formal languages. Exercises for Section 5. Next: Converting a PDA to Up: PUSH-DOWN AUTOMATA Previous: Formal Definition Contents Converting a CFG to a PDA. Pushdown Automata • The stack – The stack has its own alphabet – Included in this alphabet is a special symbol used to indicate an empty stack. You can configure the TREXOPS mode per BW provider and per BW query. Knowledge-based systems to enhance learning: A case study on formal languages and automata theory This work focuses on the field of formal languages and automata theory, describing the process of building an online teaching tool able to help students learn by themselves the fundamentals of this subject. L56 2011 005. A researcher named Neramo asks the Vestige for help exploring the Dwemer Ruins of Bthzark. With this long-awaited revision, the authors continue to present the theory in a concise and straightforward manner, now with an eye out for the practical applications. So, our tool provides you over 300 000 activation codes for you and for your friends if you want to share it. They are more capable than finite-state machines but less capable than Turing machines. Tutorial sheet for week 2 now online under Tutorials; Timetable change: Monday's lecture in Marie Reay 6. Pop quizzes will be administered during class on randomly chosen dates. , the fact that the union of two regular languages is also a regular language. , those called regular and context-free, respectively. Note that this definition includes deterministic pushdown automata, which are simply nondeterministic pushdown automata with only one available route to take. 1 phrase structure grammars ; 1. Copy the coding in the box and then paste it into your Shop Description. Main | Generate Cellular Automata. No longer maintained. I will recall recent advances on this model, and describe a newly introduced generalisation: width. • To have a knowledge of regular languages and context free languages. A pushdown automaton (PDA) is a finite state machine which has an additional stack storage. Sawant Swati T. For instance, the well known automaton, Conway’s Game of Life, has the following rules:. Oct 1 Tu Undecidability 9. Getting Automaton Simulator. 19 hw 1 CF pumping lemma, Turing machines 6. We begin with a study of finite automata and the languages they can define (the so-called "regular languages. At the lowest level are regular languages which are also generated by finite-state automata. Once you have modeled data elements in XML Schema using the graphical XML. shows how to draw traces uniformly at random for testing large systems modelled by finite automata. Answer to Give informal descriptions and state diagrams of pushdown automata for the languages in Exercise 2. A pushdown automaton (PDA) is a finite state machine which has an additional stack storage. Pushdown automata is a way to implement a CFG in the same way we design DFA for a regular grammar. Let's formalize our notion of a context-free grammar (CFG). More Solution Manual For An Introduction Page 6/24. When first encountered, their names are written in something similar to Friedrich Engles' celestial font. It is a 'cellular automaton', and was invented by Cambridge mathematician John Conway. You may have heard of cellular automata from the popular “Conway’s Game of Life”, where cells evolve based on a set of rules that they adhere to. 2 synonyms for automation: mechanisation, mechanization. WikiMatrix hr Linearno ograničen automat je uređaj koji je moćniji od potisnog automata, ali i slabiji od Turingovog stroja. Stack in pushdown automata 4. Here's how to use it: This was made in HTML5 and JavaScript using the canvas element. The goal of the Multicore SIP parser project was the development of a functional SIP parser that efficiently utilizes advantages of a multicore processor. (Z0) • This special symbol should not be removed from the stack. A Push Down Automaton is different from a finite automaton because of i) A read head ii) A memory in the form of a stack iii) A set of states iv) All of these. Finally, the text concludes with a discussion on the role of code generator and its basic issues such as instruction selection, register allocation, target programs and memory management. The algorithm is formally derived by Schneider and Schmuck in , where its soundness is proven. Tips to Crack UGC NET Computer science & Applications To crack the UGC NET commerce exam candidate should start preparation from the beginning day he applied. Automatons are towering, autonomous, steam-powered robots with four long, thin legs. The automaton is interpreted, rather than compiled, in order to save space. JFLAP is software for experimenting with formal languages topics including nondeterministic finite automata, nondeterministic pushdown automata, multi-tape Turing machines, several types of grammars, parsing, and L-systems. Section 7: Compiler Design. 4 [Software Engineering]: Software/Program veri ca-. With over 220,000 names in our database, you can also specify language, nationality and other factors to give your character the perfect name. Publisher: Automata Publishing Company 1993 Number of pages: 310. The stack head scans the top symbol of the stack. For some of the OLDER tools and OLDER versions of tools there is an OLD ftp site with OLD source code via anonymous ftp here. As we have seen, in a context free grammar, all rules are of the form A α, with A V N and α V+ (and allowing S e if the grammar is in reduced form), A context free grammar is in Chomsky Normal Form (CNF) iff all rules are. Introduction to the Theory of Computation first appeared as a Preliminary Edition. Pushdown automata provide a broader class of models and enable the analysis of context-free languages. Topics include languages, regular expressions, finite automata, grammars, pushdown automata and Turing machines. If a language can’t be represented by the regular expression, then it means that language is not regular. No ads, nonsense or garbage, just a word frequency counter. Mendel 258 Instructor: David Matuszek, [email protected] They have larger healthbars, attack phases and defeating them advances the main story. Check character expressions. Description. Includes bibliographical references and index. Push Down automata adalah recognizer nondeterministik satu arah yang memiliki penyimpanan sementara tak berhingga berisi pushdown list. pushdown automata 1. Download notes on theory of computation, this ebook has 242 pages included. Buried Secrets is a quest available in The Elder Scrolls Online. Definition How to Create an Automaton Nondeterministic NPDAs. Posted by 13 hours ago. Following are the major types of automata. Get complete study material for TOC,Regular Expressions,Sets,Theory of Computation,Automata Theory For full functionality of this site it is necessary to enable JavaScript. An automaton with a finite number of states is called a Finite Automaton. GATE 2019 CSE syllabus contains Engineering mathematics, Digital Logic, Computer Organization and Architecture, Programming and Data Structures, Algorithms, Theory of Computation, Compiler Design, Operating System, Databases, Computer Networks, General Aptitude. Construct Pushdown Automata for given languages Prerequisite - Pushdown Automata , Pushdown Automata Acceptance by Final State A push down automata is similar to deterministic finite automata except that it has a few more properties than a DFA. Ask Question Asked 7 years ago. 1 (Lecture 14 -slides(1-24 #21. It is a 'cellular automaton', and was invented by Cambridge mathematician John Conway. Teori automata adalah teori tentang mesin abstrak yang bekerja secara sekuensial yang menerima dan mengeluarkan output dalam bentuk diskrit. Each cell contains a symbol in an alphabetΣ. In our definition, the input symbol determines when the pushdown automaton can push or pop, and thus the stack depth at every position. js, and jQuery. November 2005) ISBN-10: 0387292373. The algorithm is formally derived by Schneider and Schmuck in , where its soundness is proven. Jan 20, 2017 - Explore jwardpbx's board "Automata" on Pinterest. R is a finite set of rules, with each rule being a variable and a. Hopcroft Chapter 5. The area of descriptional complexity is concerned with the size of automata needed to perform various tasks. learning parser automata interpreter guide lr-parser parser-generator learning-exercise learn educational learning-by-doing parse-trees practical pushdown-automaton abstract-syntax-tree automata-simulator deterministic-finite-automata parser-combinator parser-development parse-tables. Columnstore-Pushdown The idea of FEMS-Pushdown is to shift the evaluation of SAP BW query filters from the SAP application server to the database server. Section 8: Operating System. : Simulation of LLk Parsers with Wide Context by Automaton with One-Symbol Reading Head. generate a push-down automaton for the following constrained regular expression. 0 can be accessed by clicking here. A simple NFA. Ambiguous Grammars. Finite state recognizers are simple computing machines that read (or at least try to read) a sequence of symbols from an input tape. A random number generator (RNG) is a device that generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. ISBN 978-1-4496-1552-9 (casebound) 1. Publisher: Automata Publishing Company 1993 Number of pages: 310. Advertisements An inputed language is accepted by a computational model if it runs through the model and ends in an accepting final state. 4 on combinatorics. Computability Theory (3 weeks) Turing machines, Church-Turing thesis, decidability, halting problem, reducibility, recursion theorem. The Theory of Computing (TOC) course is a 3 credit introduction to formal languages and automata, computability theory, and complexity theory with the goal of developing understanding of the power and limits of different computational models. on infinite words, on nested words, on trees, etc. Make your website compliant with the GDPR with our cookie consent banner generator. ¾Given any finite state automata M, there exists a regular expression R such that L(R) = L(M) - see Problem 7 for an indication why this is true. They have larger healthbars, attack phases and defeating them advances the main story. In particular, finite automata, pushdown automata, context-free grammars and Turing machines are studied with respect to their properties and limits. Click on the downloader to gain access to free premium downloads. Auto-fill mode Manual entry mode. Bosses for Nier: Automata are large enemies with remarkable capabilities that are encountered only at certain times. We study the classes of proper languages for deterministic finite-state acceptors, pushdown automata, two-pushdown automata, and freely rewriting restarting automata that are lexicalized. research paper on pushdown automata, research paper on queen elizabeth, research paper on queen elizabeth i, research paper on race, research paper on race and ethnici, research paper on race and ethnicity, research paper on race in america, research paper on racial profiling, research paper on racism, research paper on racism and discrimination. Addison-Wesley 1996 comes close to the contents of this course. The Java-based application opens the door to various experiments related to anything ranging from nondeterministic pushdown automata and nondeterministic finite automata to various grammar types. Generator de curent trifazat, cu motor diesel, Heavy duty si turatie de lucru de 1500 rpm. Skip to content. Pushdown automata differ from normal finite state machines in two ways: 1. This includes what. Turing Machines, pdf. Recent work by A. But the deterministic version models parsers. Erich Grädel, Wolfgang Thomas, Thomas Wilke (Eds. Once you have modeled data elements in XML Schema using the graphical XML. Answer to Give informal descriptions and state diagrams of pushdown automata for the languages in Exercise 2. is a set of states 2. Formal Language and Automata Theory is designed to serve as a textbook for undergraduate students of B. : Simulation of LLk Parsers with Wide Context by Automaton with One-Symbol Reading Head. Wirth: Compiler Construction. For PS3,PS4,XBOX 360, XBOX ONE, WII U and PC the NieR: Automata Generator you will be able to get unlimited keys FREE NOW!!! We have worked hard to get this app for you guys to get the NieR: Automata free by cracking all codes for all systems. The great John von Neumann invented cellular automata. of Electrical and Computer Engineering, Iowa State University, Ames, IA Abstract—Stateflow, a graphical interface tool for Matlab, is a common choice for design of event-driven software and systems. models for computation - finite automata, pushdown automata, and Turing machines - and examines the relationship between these automata and formal languages. 2,2kW , 230V-50Hz , Benzina Generator de curent monofazat SENCI SC-2500 Cadru tratat pentru a preveni ruginirea Alternator produs si patentat de Senci Filtru de aer. I googled and found A package for drawing automata and graphs (Version 0. 0 can be accessed by clicking here. The 970×250 is an ad unit which was listed as a Rising Star by the IAB. Re: Simple Sprite Generator Post by Rickton » Sat May 03, 2014 2:39 pm Looks really cool, unfortunately my 7-year-old laptop doesn't support shaders, so I'm going to have to mess around and try to make a shader-free version before I can use it. 3 computability ; supplementary examples ; a quick review ; problems for practice ; objective type questions ; mathematical preliminaries ; 2. In addition, we use a generalization. Only the nondeterministic PDA defines all the CFL's. What is the maximum number of reduce moves that can be taken by a bottom-up parser for a grammar with no epsilon- and unit-production (i. * OTG V8 is out and it's a BIG update, be sure to read the changelogs and check out the new "O" menu for Forge! * Note: When updating be sure to delete OTG-Core. Automaton Simulator allows you to draw and simulate a variety of theoretical machines, including: deterministic finite automata nondeterministic finite automata deterministic push-down automata Turing machines The program should run on any platform with Java 1. FSM simulator is a demo of using noam, a JavaScript library for working with finite-state machines, grammars and regular expressions. Introduction to pushdown automata(PDA) 2. For answering this question we need to understand below terms first: Context free grammars: A context-free grammar (CFG) is a set of recursive rewriting rules (or productions) used to generate patterns of strings. Here's how to use it: This was made in HTML5 and JavaScript using the canvas element. Construct pushdown automata and the equivalent context-free grammars, construct derivation trees for strings generated by a context-free grammar, show that a context-free grammar is ambiguous, rewrite a grammar to remove ambiguity, simplify context-free grammars, transform a context-free grammar into an equivalent grammar in Chomsky normal form. Online course devoted to state machines, formal grammars, and regular expressions implementation, where we build a RegExp processor based on NFA and DFA. Press button, get regex matching strings. The book is primarily designed for one semester course in Automata and Compiler Design for undergraduate and postgraduate students of Computer Science and. In particular, finite automata, pushdown automata, context-free grammars and Turing machines are studied with respect to their properties and limits. Read more →. PDAs are more powerful than FAs, being able to recognize languages that FAs cannot. Derivatives of Regular Expressions, Janusz Brzozowski, Journal of the ACM 1964. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Only the nondeterministic PDA defines all the CFL’s. Automation is mathematical object that takes a word. The addition of stack is used to provide a last-in-first-out. Unlike pattern recognition, the match has to be exact in the case of pattern matching. Quiz: Theory of Computation Mock Tests on Context free languages and Push-down automata. One interesting way to represent liquids in a grid based world, is to use a form of cellular automaton. Section 8: Operating System. Highlights of Logics, Automata and Games, 2014, Paris. 4 Formal Definition of a PDA • A pushdown automaton (PDA) is a seven-tuple: M = (Q, Σ, Г, δ, q 0, z 0, F) Q A finite set of states Σ A finite input alphabet Г A finite stack alphabet q 0 The initial/starting state, q 0 is in Q z 0 A starting stack symbol, is in Г F A set of final/accepting states, which is a subset of Q δ A transition. contents: automata, formal languages and computability ; 1. This product only supports MS-IME keyboard input. Step 3: The initial symbol of CFG will be the initial symbol in the PDA. Magazine şi preţuri - Generatoare Senci SC-2500 de la 1 260,00 RON!: (SC 2500) Generator de curent Putere max. XML and Document-Type Definitions. COMP 454 - Automata, Lang & Comput. Finally, the text concludes with a discussion on the role of code generator and its basic issues such as instruction selection, register allocation, target programs and memory management. Introduction to pushdown automata(PDA) 2. Any opinions, finding and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This Demonstration allows you to edit the graphical representation of an automaton by manipulating its transition matrix. We study the classes of proper languages for deterministic finite-state acceptors, pushdown automata, two-pushdown automata, and freely rewriting restarting automata that are lexicalized. Introduction. Ambiguity in Grammars and Languages. 2 Pushdown Automata Formal definition of a pushdown automaton. JSON_checker is a Pushdown Automaton that very quickly determines if a JSON text is syntactically correct. Note: Online services that make random numbers available to applications, as well as the noise registered by microphone and camera recordings (see RFC 4086 sec. 8 of the text proves that there is a finite. A context-free grammar (CFG) is a 4-tuple. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a transition. atm to get a few examples. FSM simulator is a demo of using noam, a JavaScript library for working with finite-state machines, grammars and regular expressions. a finite set of states (often denoted Q) 2. Context-Sensitive Languages and Linear Bounded Automata. \$\begingroup\$ If you've had good luck using Pushdown Automata for this purpose, then I'd say continue using Pushdown Automata. I know it can be solved by a pushdown automaton. 4/15: The Midterm will be in class on May 4, 2010. Viewed 3k times 1. The top 3 reasons to play NieR Automata. Following are the major types of automata. Random number generators can be true hardware random-number generators (HRNG), which generate genuinely random numbers, or pseudo-random number generators (PRNG), which generate numbers that look random, but are actually deterministic. Scan your paper for grammar mistakes and catch unintentional plagiarism. Definition. In 1965, Donald Knuth invented the LR(k) parser and proved that there exists an LR(k) grammar for every deterministic context-free language. We offer the most extensive selection of academic programmes in Singapore, collaborating with leading universities worldwide to provide our students with diverse opportunities for overseas exposure. Regular expressions and finite automata. This is a little script that uses Python to generate the cellular automata that Wolfram discusses in his book ". Exercises for Section 5. The languages that these machines accept are annotated in the heading of the machine text and as the title of the Java frame that runs the simulation. Additional topics beyond the automata classes themselves include deterministic and nondeterministic machines, regular. In other words, given any type of the machine, we can transform it to the other. Computer Science All Past 31 Years GATE Questions well Organized Subject and Chapterwise by GateQuestions. The automata systems are given by pushdown or finite automata with move-restricting set. Main | Generate Cellular Automata. Answer to Construct a Pushdown Automata (PDA) for the following language: ? L = {w ? {a, b}? | (w = w^R) and number of a's = num. Difference between pushdown automata and finite state machine 3. Using lifting gate microfluidic control valve technology, a microfluidic automaton consisting of a two-dimensional microvalve cellular array is fabricated with soft lithography in a format that. For PS3,PS4,XBOX 360, XBOX ONE, WII U and PC the NieR: Automata Generator you will be able to get unlimited keys FREE NOW!!! We have worked hard to get this app for you guys to get the NieR: Automata free by cracking all codes for all systems. Click on the downloader to gain access to free premium downloads. 4 on combinatorics. Pushdown automata adalah recognizer merupakan model alami dari syntatic analyzer dari bahasa Context free. Bosses for Nier: Automata are large enemies with remarkable capabilities that are encountered only at certain times. The known proofs that the equivalence and containment problems for regular expressions, regular grammars and nondeterministic finite automata are PSPACE-complete [SM] depend upon consideration of highly unambiguous expressions, grammars and automata. UGC NET Computer Science Syllabus. Created by Ivan Zuzak and Vedrana Jankovic. The basic idea is to fill the first map randomly, then repeatedly create new maps using the 4-5 rule: a tile becomes a wall if it was a wall and 4 or more of its eight neighbors were walls, or if it was not a wall and 5 or more neighbors were. It established its roots during the 20th Century, as mathematicians began developing - both theoretically and literally - machines which imitated certain features of man, completing calculations more quickly and reliably. With this long-awaited revision, the authors continue to present the theory in a concise and straightforward manner, now with an eye out for the practical applications. The loop-free testing graph indicates that. Only the nondeterministic PDA defines all the CFL's. JFLAP is software for experimenting with formal languages topics including nondeterministic finite automata, nondeterministic pushdown automata, multi-tape Turing machines, several types of grammars, parsing, and L-systems. Automatic Java Code Generator for Regular Expressions Programming and resolving problems within automata theory is a complex process, which is time consuming and still the results may not be reliable (because of the great possibility to make mistakes while doing the required traversing steps). Chapter 8 & 9:Undecidability #22. Ullman, "Compilers: Principles, Techniques, and Tools", 2nd Edition, Addison-Wesley, 2007 [2] John E. Pushdown automata is simply an NFA augmented with an "external stack memory". production must be a terminal symbol. Demonstrate that a grammar is ambiguous. Regular Expression to NFA (Non-Deterministic Finite Automata) Visualize the Thompson-McNaughton-Yamada construction NFA for a given regular expression. save hide report. Bosses for Nier: Automata are large enemies with remarkable capabilities that are encountered only at certain times. Akku Mega Kft Importőr Budapest. The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages. Complexity Theory (7 weeks). Lecture 14 - Turing Machines #21. This includes what. But I'm not sure is this package a standard one, or. is the set of accept states Pushdown Automata - p. A DFA can remember a finite amount of information, but a PDA can remember an infinite amount of information. This large chapter covers pushdown automata and parsing algorithms with emphasis on their application to syntax analysis. Context-free grammars, pushdown automata, the pumping lemma for context-free languages. , automata theory, formal languages, computability, and complexity theory. 2 on induction and Ch. The Chomsky hierarchy of languages can be generated using grammars or using automata. 1 Chapter 8 & 9. • Context-free languages are recognized by push-down automata (PDA) in the same way that regular languages are recognized by nite au-tomata. Pushdown automata (PDAs) can be thought of as combining an NFA "control-unit" with a "memory" in the form of an infinite stack. But, be careful, some languages that can be recognized by a NPDA are subtle, e. Andy Wuensche's "Exploring Discrete Dynamics" 2016, is by far the most advanced tool for simulating such systems and has become widely important in the. Finite Automata. The Cellular Automaton. What are synonyms for Automata theory?. So, a PDA is a finite automaton with a stack. comps ===== Spring 2016 ===== CSE 4081: MW 8-9:15 AM, Crawford 112. -Works with any tile size and any map size. In our definition, the input symbol determines when the pushdown automaton can push or pop, and thus the stack depth at every position. Die meisten Videos von TheSimpleInformatics findest auch auf Lern-Online. a, B, 9), the \$ character representing the empty string. 8-bit Sprite Generation using Cellular Automata. Mode 2 is the FEMS-pushdown, mode 3 is the MultiProvider-pushdown. D E F I N I T I O N 2. We had to seperate the shop and gallery layouts due to the strict blocks neo has put on codes in shop layouts. Wirth: Compiler Construction. Pushdown Automata is a finite automata with extra memory called stack which helps Pushdown automata to recognize Context Free Languages. 8: CF Grammars and Pushdown Automata - Converting PDAs into CFGs - Converting CFGs into PDAs - Chomsky Normal Form. Topics include languages, regular expressions, finite automata, grammars, pushdown automata, and Turing machines. The film tells the story of Tony Stark, an industrialist and ingenious engineer who constructs a suit of armor and turns himself into a technologically advanced superhero Iron Man. TITLE OF PAPER : Theoretical Computer Science (TCS) Code No. Intersections and Complements of CF-Languages 97 7. Release Date: May 2013. An automation is supposed to run on some given. In fact, the set of languages that can be recognized by PDAs are the context-free languages of the previous module. 1, (Liebow-Feeser 2017a), and (Liebow-Feeser 2017b)), are additional nondeterministic sources. Name: Turing machine mcq Test with answers - 1 Subject: Automata ( Theory of Computation) Topic: Turing Machine Questions: 20 Objective type Time Allowed: 15 Minutes Important for: Computer Science B. In this pushdown automata used as an acceptor model and context-free grammar used as a generator model. Papers published in Theoretical Computer Science are grouped in three sections according to their nature. 3 computability ; supplementary examples ; a quick review ; problems for practice ; objective type questions ; mathematical preliminaries ; 2. Instructions for online tutorials under Online in the side bar. Screen shot of Automaton Simulator 1. Partly as an application for this computer I then ended up making a detailed study of rule 30 and its randomness—among other things proposing it as a practical random sequence generator and cryptosystem. Convert between grammars and push-down automata for context-free languages. Note that this definition includes deterministic pushdown automata, which are simply nondeterministic pushdown automata with only one available route to take. - Various definitions for Pushdown Automata - Non-equivalence of deterministic and non-deterministioc PDAs. "Breed" is an "evolutionary voxel object generator". Read on O'Reilly Online Learning with a 10-day trial Start your free trial now Buy on Amazon. On Stateless Two-Pushdown Automata and Restarting Automata (MK, HM, FO), pp. This course offers an introduction to the theory of computation. Normal Forms 117 3. Read on O'Reilly Online Learning with a 10-day trial Start your free trial now Buy on Amazon. I'm less interested in handling the general case, and more curious about what are some sufficient conditions that might occur in. 3/30: And if you want more background on discrete math, take a look at the free book Foundations of Computer Science, espcially Ch. An on-line demo version of JFLAP 2. So, a PDA is a finite automaton with a stack. Using these tools in Automata Theory Course. These are mathematical simulations which involve iterating a collection of cells in a grid, based on a series of rules. Created by Ivan Zuzak and Vedrana Jankovic. Make your website compliant with the GDPR with our cookie consent banner generator. Theory of Computation questions and answers (1) From the options given below, the pair having different expressive power is (A) Deterministic Push Down Automata (DPDA) and Non-deterministic Push Down Automata (NPDA) (B) Deterministic Finite Automata (DFA) and Non-deterministic Finite Automata(NFA). Additional Notes: Mouse, keyboard and game pad (XInput only). Coco/R is a compiler generator, which takes an attributed grammar of a source language and generates a scanner and a parser for this language. Formal languages. It includes the whole field of abstract complexity (i. 3 computability ; supplementary examples ; a quick review ; problems for practice ; objective type questions ; mathematical preliminaries ; 2. Pushdown Automata.
8wpq6q8c9lul 4wx9t1jt1smlx 2m9cg5tdnpc gxgjgk456e 0cb325e14ivm0 tuetsk4hmb 47dn9rruzz4c5d b1490oesk7wr0 4rhzvr3f9om ozpgtqeqtg 1qw608pq3b6f8 cbtl3rqjzke1 fwnq6mcxzv2d9zw 8ojkpbc8b4sxk iy1efldvajzl 55saq8jj8h6 0b91jcmxmd10moy p90oqvw9yeq63bn swk2n8tvgkm sjxptlnqg1vkir d1utcgwvews2r7 65dgj22hyhvf5 f6k31rv5w5fhks ef2cnuipwp wh1uq7and61 pep6hgo2mr 3z2blqzt7t6m5hm a7ugoyey6kb6q cvqjrz6icg1z cin0zanp0szm20 6wu9nua9jxh3z 77s8qya2pmum yxmzngfzzydn 4m6wszxg257kmju | 2020-07-14 19:44:46 | {"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": 0.288632869720459, "perplexity": 2288.611023487534}, "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-2020-29/segments/1593657151197.83/warc/CC-MAIN-20200714181325-20200714211325-00182.warc.gz"} |
https://www.techguru66.com/2022/07/solving-loksewa-aayog-questions.html | # Solving LokSewa Aayog questions of Karnali Province -5 minute strategy
In this article, we are going to solve LokSewa Aayog questions of Karnali province with a 5-minute strategy. Karnali province opens a post for helping staff in the fourth stage of a computer operator. It is an open competition and you need to appear in the written exam of computer operator exam. The exam is taken by LokSewa Aayog of Nepal.
## How to solve the LokSewa Aayog questions of Karnali province?
To solve the question of Karnali province you need to know something before you go to appear for the exam.
• The time period for this exam is only 45 minutes.
• The full mark is 50.
• You have given a choice question where you need to choose one right option.
• Don’t forget to take your necessary document or your entrance card.
• You are not allowed to take a mobile, calculator in this exam.
• You have to mention KEY in your exam otherwise your exam paper will cancel.
• You have to choose a different paper sheet for a different section otherwise your paper will cancel.
These are some rules you need to know before going to appear for the exam. Now, I am going to provide you with questions about Karnali Pradesh/province which is held on 2079/03/29 BS (Nepali Date). And you might know that the subject which is Computer Operator. Without any further do let’s start to solve questions.
## Multiple Choice Questions (35*1=35)
1. Suppose the cell value in A4 is 20, What will be the output of the formula “IF(A4<10,5,7)”?
• A) 0
• B) 10
• C) 5
• D) 7
2. Which chart type best suits comparing values over categories?
• A) Pie Chart
• B) Column Chart
• C) Line Chart
• D) Scatter Chart
3. In MS Access, What is the maximum length of a text field?
• A) 265
• B) 260
• C) 255
• D) 250
4. A primary key field included in another table is known as …..
• A) Foreign Key
• B) Parent Key
• C) Child Key
• D) Parasite Key
5. Function which causes retrieval of any kind of data from the database is considered as …
• A) Key
• B) Query
• C) Structure
• D) Storing cycle
6. …is not a valid data type in MS Access.
• A) Auto number
• B) Currency
• C) Memo
• D) Picture
7. What view in PowerPoint lets you see your slides on the screen in a grid that makes it easy to reorganize them, or organize them into sections, just by dragging and dropping them where you want them?
• A) Slide Sorter
• B) Notes Page
• C) Normal
• D) Slide Show
8. What PowerPoint feature do you use to apply motion effects to different slide objects?
• A) Slide Transition
• B) Animation Scheme
• C) Animation Object
• D) Slide Design
9. Which feature allows us to view slides in a slide show without manually advancing each slide?
• B) Setting Animation
10. The basic computer architecture was developed by…
• A) Charles Babbage
• B) Jhon Von Neumann
• C) Blaise Pascal
• D) Garden Moore
11. Self-replicating malware that duplicates itself to spread to uninfected computers is called…
• A) Spam
• B) Worm
• C) Replicator
• D) Duplicator
12. Which generation of computers is Artificial Intelligence associated with?
• A) Second Generation
• B) First Generation
• C) Fourth Generation
• D) Fifth Generation
13. What does a video consist of?
• A) Frames
• B) Signals
• C) Packets
• D) Slots
14. Which of the following is an object-oriented programming language?
• A) C
• B) Java
• C) Pascal
• D) COBOL
15. Which of the following topology is formed when computers are networked with a switch?
• A) Bus topology
• B) Ring topology
• C) Star topology
• D) Mesh topology
16. Who issues Digital Signature certificates to the general public?
• A) Root Certificate
• B) Certifying Authority
• C) SSL Certificate Issuer
• D) National Information Technology Centre
17. 2 Kilobyte is equivalent to …
• A) 2048 bytes
• B) 2024 bytes
• C) 2000 bytes
• D) 1024 bytes
18. Which of the following technology is used to implement private networks over public networks?
• A) WAN
• B) MAN
• C) SAN
• D) VPN
19. Which of the following is a correct IPV4 Address?
• A) 01.200.128.123
• B) 300.143.211.64
• C) 123.202.30.1.52
• D) 128.64.0.0
20. Which of the following is a single-user operating system?
• A) Windows
• B) MS-DOS
• C) MAC OS
• D) Linux
21. Which command is used to fix the minor errors of the disk?
• A) FDISK
• B) FORMAT
• C) CHKDDK/f
• D) ATTRIB
22. … uses a technique to increase data access speed by rearranging files stored on a disk to occupy contiguous storage locations.
• A) Scandisk
• B) Disk Defragmenter
• C) Disk Management
• D) Disk Format
23. Which of the following is used for quick access to commonly used commands and tools?
• A) Status Bar
• B) Tool Bar
• D) Title Bar
24. What is the shortcut to justify text to the center?
• A) CTRL+A
• B) CTRL+B
• C) CTRL+D
• D) CTRL+E
25. Autocorrect was originally designed to replace…….words as you type.
• A) Misspelled
• B) Grammatically incorrect
• C) Short, repetitive
• D) Long, repetitive
26. Footnotes, endnotes, and indexes are inserted in a document as ……
• A) Bookmarks
• B) Cross-references
• D) Word fields
27. ……tools in the word is used to apply the same format to different character or paragraph.
• A) Format aligner
• B) Format poster
• C) Format painter
• D) Format generator
• A) Bullets and Numbering
• C) Font Size
• D) None of the above
29. Which operation is used to move a block of text?
• A) Copy and Paste
• B) Paste and Delete
• C) Cut and Paste
• D) Paste and Cut
30. Absolute cell reference can be represented as …
• A) $B$3
• B) B3
• C) B$3 • D)$B3
31. To edit a cell content in excel we use …
• A) F10
• B) F12
• C)F2
• D) F3
32. What happens when numeric data entered in a cell exceeds the width size of the column?
• A) Number is wrapped in cell width
• B) Column width changes
• C) Data is displayed in exponential form
• D) Error Message
33. Getting data from a cell located in a different sheet is called …
• A) Accessing
• B) Referencing
• C) Updating
• D) Functioning
34. ……is used to display only the rows that meet certain conditions and the others rows get hidden.
• A) Filter
• B) Sort
• C) None of the above
• D) Both of the above
35. Which of the following is not a built-in date time function?
• A) Now()
• B) Today()
• C) Date()
• D) Date Time()
Now the Multiple choice question ends which means your section 2 starts, which is long questions. | 2022-11-29 18:32:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.3980044722557068, "perplexity": 11485.824677841254}, "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/1669446710710.91/warc/CC-MAIN-20221129164449-20221129194449-00231.warc.gz"} |
https://tex.stackexchange.com/questions/592080/lualatex-why-is-textbar%CC%88-working-but-not-textbar%CC%99 | # lualatex: Why is \textbar̈ working, but not \textbar̙?
Why is line b) working, but not line d), if compiled the with lualatex?
In Line b) the r (U+0072) is followed by ̈ (U+0308). This compiles without errors.
In Line d) the r (U+0072) is followed by ̙ (U+0319). This produces (if % removed) the error “\textbar̙ … Undefined control sequence. …”
\documentclass{article}
\usepackage{fontspec}
\usepackage{dejavu-otf}
\begin{document}
a) r̙ r̈ \$̈ \$̙
b) \textbar̈
c) \textbar{}̙
d) % \textbar̙
\end{document}
This is LuaHBTeX, Version 1.12.0 (TeX Live 2020/Debian)
• The exact error is that \textbar̙ is undefined. The U+0319 has the catcode of a letter, and so continues the command name, you get an error for the same reason why you would get an error from \textbarx: the command name simply isn't defined. Apr 9, 2021 at 8:14
• @UlrikeFischer yes that's clear but what happened with the first one that is using r followed by U+0308 which is also catcode 11 but \show\textbar̈ shows \textbar with the diacritic following, as if it were catcode 12 ? Apr 9, 2021 at 8:57
• oh it is catcode 12, that seems... wrong? Apr 9, 2021 at 9:00
This seems to be an issue with the dejavu-otf package unicode-math it is making the combining diaeresis catcode12 (punctuation) so it can not be part of a command name
\showthe\catcode"0308
\documentclass{article}
\usepackage{fontspec}
\showthe\catcode"0308
%\usepackage{dejavu-otf}
\showthe\catcode"0308
\begin{document}
\showthe\catcode"0308
a) r̙ r̈ \$̈ \$̙
b) \show\textbar̈
c) \textbar{}̙
d) \show\textbar̙
\end{document}
adding \catcode"0308=11 would restore the usual behaviour.
This seems to be a feature of unicode-math (I'll raise it there)
\documentclass{article}
\RequirePackage{unicode-math}
\showthe\catcode"0308
\setmathfont{TeXGyreDejaVuMath-Regular}
\showthe\catcode"0308
\begin{document}
\end{document}
just setting the math font changes the catcode of a text diacritic which seems... unexpected | 2022-06-26 14:58:17 | {"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": 0.7843486070632935, "perplexity": 8821.405836061784}, "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/1656103269583.13/warc/CC-MAIN-20220626131545-20220626161545-00359.warc.gz"} |
https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Gravitation&q_type=&q_topic=Universal+Law+Of+Gravitation+&q_category=&question_id=PHENNT11124149 |  A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest from Physics Gravitation Class 11 Manipur Board
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A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest
• at the instant just before the body hits the earth
• it remains constant all through
• at the instant just after the body is projected
• at the highest position of the body
A.
at the instant just before the body hits the earth
We know
P = F. v = Fv cos θ
so just before hitting θ is zero and both F and v are maximum
We know
P = F. v = Fv cos θ
so just before hitting θ is zero and both F and v are maximum
465 Views
The position co-ordinates of two particles of masses m1Â and m2are (x1, y1, z1) and (x2, y2, z2) respectively. Find the coordinates of the centre of mass.
The position vectors of masses m1Â and m2Â are respectively,
Let the position coordinates of the centre of mass be (X, Y, Z).
Therefore the position vector of centre of mass is,
Since, Â Â
Â
Comparing the coefficients of  , we get
Â
669 Views
What is Geocentric theory?
According to the geocentric theory, all the astronomical bodies like the moon, the sun and stars revolve around the earth, and the earth is at the centre of the universe.Â
1256 Views
What is Heliocentric theory?
According to the Heliocentric theory, the sun is at the centre and various planets revolve around the sun at their axis.Â
907 Views
What is the celestial sphere?
At night, if we see the planets and the stars in the sky, all appear to lie in the hemisphere (rest of the hemisphere we are unable to see because of being on the other side of the earth). This sphere is called the celestial sphere.
661 Views
Is Geodesic always a straight line?
No, Geodesic is a straight line if and only if, Â the two points lie on the flat surface. If the two points lie on the curved surface then it is a curved line.
609 Views | 2018-06-20 17:27:42 | {"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": 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": 0.4882100522518158, "perplexity": 1252.3849482228657}, "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-2018-26/segments/1529267863830.1/warc/CC-MAIN-20180620163310-20180620183310-00458.warc.gz"} |
https://stacks.math.columbia.edu/tag/090W | Lemma 10.157.1. Let $k \subset K$ be a finitely generated field extension. The following are equivalent
1. $K$ is a finite separable field extension of $k$,
2. $\Omega _{K/k} = 0$,
3. $K$ is formally unramified over $k$,
4. $K$ is unramified over $k$,
5. $K$ is formally étale over $k$,
6. $K$ is étale over $k$.
Proof. The equivalence of (2) and (3) is Lemma 10.148.2. By Lemma 10.143.4 we see that (1) is equivalent to (6). Property (6) implies (5) and (4) which both in turn imply (3) (Lemmas 10.150.2, 10.151.3, and 10.151.2). Thus it suffices to show that (2) implies (1). Choose a finitely generated $k$-subalgebra $A \subset K$ such that $K$ is the fraction field of the domain $A$. Set $S = A \setminus \{ 0\}$. Since $0 = \Omega _{K/k} = S^{-1}\Omega _{A/k}$ (Lemma 10.131.8) and since $\Omega _{A/k}$ is finitely generated (Lemma 10.131.16), we can replace $A$ by a localization $A_ f$ to reduce to the case that $\Omega _{A/k} = 0$ (details omitted). Then $A$ is unramified over $k$, hence $K/k$ is finite separable for example by Lemma 10.151.5 applied with $\mathfrak q = (0)$. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2021-03-01 22:58:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9908837080001831, "perplexity": 218.0134193734599}, "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/1614178363072.47/warc/CC-MAIN-20210301212939-20210302002939-00055.warc.gz"} |
https://www.gradesaver.com/textbooks/science/chemistry/general-chemistry-principles-and-modern-applications-10th-edition/chapter-17-additional-aspects-of-acid-base-equilibria-exercises-neutralization-reactions-page-777/35 | General Chemistry: Principles and Modern Applications (10th Edition)
$C_2 (H_3PO_4) = 0.1508M$
1000ml = 1L 31.15ml = 0.03115 L 25ml = 0.025 L 1. Find the concentration necessary for an equal number of moles: Notice: Since $H_3PO_4$ is capable of giving a proton 2 times, the number of moles of $KOH$ has to be 2 times the number for the acid: $(Base)(C_1 * V_1) \div 2= (Acid)V_2 * C_2$ $(0.242* 0.03115) \div 2= 0.025 * C_2$ $0.003769 = 0.025 * C_2$ $C_2 = 0.1508M$ | 2022-07-06 01:42:11 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5863239169120789, "perplexity": 871.6195144580208}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00638.warc.gz"} |
http://www.physicsforums.com/showthread.php?s=b60f1336e5ccbf54367ecd95dd26fb1d&p=4674060 | # Reduction of decimals when multiplying with a number with 15 decimals
by niccojul
Tags: decimals, multiplying, number, reduction | 2014-08-27 15:08:52 | {"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": 0.8241214156150818, "perplexity": 11559.866722903696}, "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-2014-35/segments/1408500829421.59/warc/CC-MAIN-20140820021349-00014-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://discuss.pytorch.org/t/loss-between-floattensors-of-different-size/14100 | # Loss between FloatTensors of different size
How can I calculate the loss (and get a scalar) between two FloatTensors of different size? For example, let’s assume that one FloatTensor is 115x5 and the other is 5x5. How can one get the loss between those two?
I’m doing:
`````` loss = (y_predicted - y).pow(2).sum()
``````
but of course it gives an error as the dimensions are different.
You need to explain what your ground-truth values and predictions are.
And if why 115 vs 5 (different batch size or other stuff)?
Because normally it is not logical to do sth like to want. You could resize tensor to same size, but then I think than network will be not learning anything.
I have 10 subjects and each subject has a number of 5-dimensional measurements. So,
Subject_1 with 1, …, N measurements
Subject_2 with 1, …, K measurements
.
.
.
Subject_10 with 1, …, L measurements
For example, I’m trying find the loss between all of the N measurements of subject_1 with my latent sample which is 5 dimensional and I assume it’s drawn from a multivariate Gaussian. | 2022-05-24 22:28:06 | {"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": 0.8311606645584106, "perplexity": 1577.963367018987}, "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/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00523.warc.gz"} |
https://tex.stackexchange.com/questions/268877/add-figure-to-the-appendix | # Add figure to the Appendix
I am trying to add some figures (class diagrams and sequence diagrams) to the appendix but I dont know how can I include them in the appendix part. How can I include them to the Appendix part in this template below?
Latex Template from the university
\documentclass[
english,
german,
11pt,
twoside,
a4paper,
BCOR8.25mm,
DIV10,
footsepline
]{scrbook}
\include{commands}
\begin{document}
\selectlanguage{english}
\frontmatter
%---------------------------------------------------------------------------
% Frontpage
%---------------------------------------------------------------------------
\input{0-titlepage.tex}
%---------------------------------------------------------------------------
% list of contents
%---------------------------------------------------------------------------
\tableofcontents
\cleardoublepage
%---------------------------------------------------------------------------
% The actual content
%---------------------------------------------------------------------------
\mainmatter
\pagestyle{fancy}
\include{1-introduction}
\include{6-conclustion}
%---------------------------------------------------------------------------
% Appendix
%---------------------------------------------------------------------------
\appendix
\listoffigures
\listoftables
\lstlistoflistings
%---------------------------------------------------------------------------
%using: \abk{Abk.}{Abkürzung}
\printnomenclature
\bibliographystyle{ieeetr}
\bibliography{bibliography}
\printindex
\end{document}
Figure example:
\begin{figure}[H]
\centering
\includegraphics[width=1.0\textwidth, height=200px]{images/injection}
\caption{Class diagram of injection procedure in the service provider. }
\label{fig2}
\end{figure}
Edit: When I added them normal as this
\appendix
\listoffigures
\listoftables
\lstlistoflistings
\newpage
\begin{figure}[H]
\centering
\includegraphics[width=1.0\textwidth, height=200px]{images/injection}
\caption{Class diagram of injection procedure in the service provider. }
\label{fig2}
\end{figure}
I am getting Listing at the top of the page. I want it without Listing at the top.
• Figures in appendix should be added on the same way as in other chapters of your thesis. What is your really problem? – Zarko Sep 22 '15 at 14:02
• @Zarko: Please see my edit code below the question. – Mr Asker Sep 22 '15 at 14:24
• Use \cleardoublepage instead of \newpage. \cleardoublepage\begin{figure}[htb]... – user11232 Sep 22 '15 at 14:44
• @MrAsker, I do not know, what of habits in organisation of documents has your university. Usually the content and list of figures etc is on begin of bok or on its and. If the latter is your case, than to\appendix should follow your figure, than, after \clearpage or \cleardouble page follows your lists, each preceded with \clearpage or \cleardouble page (thatt it can start on new pages, but I'm not sure, if this is rule in your case). – Zarko Sep 22 '15 at 14:56
• This is a crosspost to goLaTeX. – Johannes_B Sep 23 '15 at 14:49
I don't know rules, which you need to obey on composing your book (or thesis). Even your question is not very clear to me. So I only guess, that you looking for something like this:
\documentclass[
english,
11pt,
twoside,
a4paper,
BCOR8.25mm,
DIV10,
footsepline
]{scrbook}
\include{commands}
\begin{document}
\frontmatter
%---------------------------------------------------------------------------
% Frontpage
%---------------------------------------------------------------------------
\input{0-titlepage.tex}
%---------------------------------------------------------------------------
% list of contents
%---------------------------------------------------------------------------
\tableofcontents
\cleardoublepage
%---------------------------------------------------------------------------
% The actual content
%---------------------------------------------------------------------------
\mainmatter
\pagestyle{fancy}
\include{1-introduction}
\include{6-conclustion}
%---------------------------------------------------------------------------
% Appendix
%---------------------------------------------------------------------------
\cleardoublepage
\appendix
\chapter{Appendix}% <-- this title should appear on page header
%---------------------------------------------------------------------------
% Figure
%---------------------------------------------------------------------------
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth, height=200px]{images/injection}
\caption{Class diagram of injection procedure in the service provider. }
\label{fig2}
\end{figure}
\cleardoublepage
\listoffigures
\cleardoublepage
\listoftables
\cleardoublepage
\lstlistoflistings
%---------------------------------------------------------------------------
\cleardoublepage
%using: \abk{Abk.}{Abkürzung}
\printnomenclature
\bibliographystyle{ieeetr}
\bibliography{bibliography}
\printindex
%---------------------------------------------------------------------------
\end{document}
Above code will generate Appendix on new od page, where will place your image, than on the next odd page will follows list of figures, etc. Is this what you looking for?
• I have tried you version, the page between the last chapter conclusions and the Appendix is being cleared but at the top of the figure page is the name of the conclusions chapter! – Mr Asker Sep 22 '15 at 16:12
• @MrAsker, now I guess that your like to change header of page, where you have a figure. I don't know, how the Appendix is defined in scrbook, but to change the appearance of header you need to ad chapter or section with desired name after appendix and before figure. See edit of my answer. – Zarko Sep 22 '15 at 17:26
• Just an FYI, this question has also been asked on goLaTeX. – Johannes_B Sep 23 '15 at 14:49
• @Johannes_B, I was not aware for goLaTeX, it seems to be nice "Deutschsprachige forum fur" TeX. Its a pity that my knowledge of German language (which is worse than English) doesn't allow me to participate in it. Anyway, thank you for this information. – Zarko Sep 23 '15 at 15:59 | 2020-11-27 00:22:14 | {"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": 0.5351042151451111, "perplexity": 1831.3607518728759}, "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-2020-50/segments/1606141189030.27/warc/CC-MAIN-20201126230216-20201127020216-00587.warc.gz"} |
https://zenodo.org/record/1483852/export/csl | Journal article Open Access
# Experimental investigation of the axial strength of glued-in rods in cross laminated timber
Azinović, Boris; Serrano, Erik; Kramar, Miha; Pazlar, Tomaž
### Citation Style Language JSON Export
{
"DOI": "10.1617/s11527-018-1268-y",
"container_title": "Materials and Structures",
"language": "eng",
"title": "Experimental investigation of the axial strength of glued-in rods in cross laminated timber",
"issued": {
"date-parts": [
[
2018,
10,
18
]
]
},
"abstract": "<p>The paper presents results from an experimental assessment of glued-in rods in cross laminated timber (CLT). For the purposes of the study more than 60 pull–pull tests were performed, where the specimens varied in terms of bonded-in length (from 80 to 400 mm), rod diameter (16–24 mm) and rod-to-grain angle (parallel and perpendicular). Several different failure modes that are not common for other applications of glued-in rods (e.g., a failure between CLT layers) were obtained for the analysed CLT specimens. It was found that these failure mechanisms can substantially influence the obtained ultimate tension loads. At the end, the experimental results were compared with empirical and semi-empirical equations for estimating the pull-out strength of glued-in rods in structural timber and glulam. The comparison showed that most of the existing equations overestimate the ultimate tension loads for specimens with the rod parallel to the grain and underestimate the ultimate tension load for specimens with the rod perpendicular to the grain. The results vary because the possible CLT failure modes were not included in previous studies. Further studies are proposed to improve the equations for glued-in rods in CLT.</p>",
"author": [
{
"family": "Azinovi\u0107, Boris"
},
{
"family": "Serrano, Erik"
},
{
"family": "Kramar, Miha"
},
{
"family": "Pazlar, Toma\u017e"
}
],
"page": "16",
"volume": "51",
"type": "article-journal",
"issue": "143",
"id": "1483852"
}
126
112
views | 2021-01-25 07:25:28 | {"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": 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": 0.20827065408229828, "perplexity": 10869.372516637088}, "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-2021-04/segments/1610703565376.63/warc/CC-MAIN-20210125061144-20210125091144-00581.warc.gz"} |
https://www.tutorialathome.in/history/lotus-tower-aswan-egypt | The Lotus Tower, Aswan, Egypt - Historical Columns and towers
14-09-2018 490 times
Egypt is known as a land of many spectacular ancient landmarks. However, the Lotus tower is quite a modern structure. Constructed in 1971, the Lotus Monument is located on the west bank of the River Nile and close to the High Dam. It was built to commemorate the Russian involvement in the construction of the Aswan High Dam and is called Egyptian Russian Friendship Monument.
The construction of the monument is connected with the political history of Egypt and the construction of the Aswan High Dam. Based on the success of the Low Dam, construction of the High Dam became a key objective of the government of Egypt following the Egyptian Revolution of 1952. The reason for the construction of a new dam was very much evident, because despite the old dam, each year Egypt suffered a lot due to the annual flooding of the Nile during late summer. It was obvious that, with increased reservoir storage provided by the new High Aswan Dam, the floods could be controlled and the water could be stored for later release over multiple years.
During December 1955, the U.S and Britain pledged $56 million and$14 million respectively, towards the construction of the dam. However, the Egyptian President Gamal Naser’s diplomatic recognition of China angered the American Secretary of State John Foster Dulles. Apart from that, there were several other reasons why the U.S. ultimately decided to withdraw its offer of funding.
The Soviets took this opportunity and in June 1956, they offered President Nasser of Egypt an amount of \$1.12 billion at 2% interest for the construction of the Aswan High Dam and in 1958, they went ahead in providing technical support for the project. The construction started on 9 January 1960 and the ambitious project was completed on 21 July 1970.
As previously mentioned, the Egyptian Russian Friendship Monument, popularly known as the Lotus tower, was built to commemorate Russian involvement in the huge project. Based on the idea of five petals of a lotus, the exotic and offbeat tower stands more than 230 feet (70 m) high into the air and is set on a hill overlooking the dam wall. The center pylon of the monument features both the Egyptian and Russian coat of arms. An observation deck stands part way up and offers some impressive panoramic view of the dam, Lake Nasser and far beyond. | 2019-04-21 11:01:10 | {"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": 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": 0.180813729763031, "perplexity": 3205.7403581874473}, "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-2019-18/segments/1555578530527.11/warc/CC-MAIN-20190421100217-20190421122217-00416.warc.gz"} |
https://ncatlab.org/nlab/show/hadrons+as+KK-modes+of+5d+Yang-Mills+theory+--+references | nLab hadrons as KK-modes of 5d Yang-Mills theory -- references
Hadrons as KK-modes of 5d Yang-Mills theory
Hadrons as KK-modes of 5d Yang-Mills theory
The suggestion that the tower of observed vector mesons – when regarded as gauge fields of hidden local symmetries of chiral perturbation theory – is reasonably modeled as a Kaluza-Klein tower of D=5 Yang-Mills theory:
That the pure pion-Skyrmion-model of baryons is approximately the KK-reduction of instantons in D=5 Yang-Mills theory is already due to:
with a hyperbolic space-variant in:
Further discussion of this approximation:
The observation that the result of Atiyah-Manton 89 becomes an exact Kaluza-Klein construction of Skyrmions/baryons from D=5 instantons when the full KK-tower of vector mesons as in Son-Stephanov 03 is included into the Skyrmion model (see also there) is due to:
In the Sakai-Sugimoto model of holographic QCD the D=5 Yang-Mills theory of this hadron Kaluza-Klein theory is identified with the worldvolume-theory of D8-flavour branes intersected with D4-branes in an intersecting D-brane model:
Extensive review of this holographic/KK-theoretic-realization of quantum hadrodynamics from D=5 Yang-Mills theory is in:
Via the realization of D4/D8 brane bound states as instantons in the D8-brane worldvolume-theory (see there and there), this relates also to the model of baryons as wrapped D4-branes, originally due to
and further developed in the nuclear matrix model:
In relation to Yang-Mills monopoles:
• Stefano Bolognesi, Solitons, Large $N$ and Holography, 2015 (pdf)
Discussion, in this context, of D-term effects affecting hadron stability:
• Mitsutoshi Fujita, Yoshitaka Hatta, Shigeki Sugimoto, Takahiro Ueda, Nucleon D-term in holographic QCD $[$arXiv:2206.06578$]$
More on baryons in the Sakai-Sugimoto model of holographic QCD:
More on mesons in holographic QCD:
An alternative scenario of derivation of 4d Skyrmions by KK-compactification of D=5 Yang-Mills theory, now on a closed interval, motivated by M5-branes instead of by D4/D8-brane intersections as in the Sakai-Sugimoto model, is discussed in:
following | 2022-12-06 23:46:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.9138427376747131, "perplexity": 8112.5326207850785}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711121.31/warc/CC-MAIN-20221206225143-20221207015143-00594.warc.gz"} |
https://www.vedantu.com/physics/artificial-transmutation | # Artificial Transmutation
View Notes
The process of converting one element into another by bombarding it with fundamental particles is called artificial transmutation. The very first artificial element produced by the process of artificial transmutation was oxygen. Thus, in simple words, it is the process of transformation of one particular element into another by the making use of artificial means. For getting an element of oxygen, the nucleus of nitrogen was bombarded with an alpha particle. Normally, the element that is produced through the process usually exhibits radioactivity which results in induced radioactivity. The particles that you can use for bombardment are:
α-particle; Proton and Deuteron: 1H3, Neutron 0n1.
### Process of Artificial Transmutation
Now, as we read above, the new or artificial element that is produced by the process of artificial transmutation exhibits radioactivity, which results in induced radioactivity. Usually, the particles that are bombarded with the main element are, α-particle; Proton, and Deuteron, Neutron, Alpha particles, deuterons and protons, they all carry nuclei that are positively charged. They are not suitable projectiles because they all repel positively charged nuclei. Neutrons, on the other hand, possess no charge at all, and hence considered the best projectile for the artificial transmutation process. When the main element or the targeted element is bombarded with a neutron the resulting element or product shall depend upon the speed of the neutron with which it is bombarded. The bombardment is slow; the neutrons penetrate into the nucleus of the targeted element. When the bombardment speed is high the neutrons pass through the nucleus. The other name of a slow neutron is the thermal neutron. In producing a nuclear reaction, the slow neutrons are more effective than the high-speed neutrons.
### Who Discovered Artificial Transmutation?
Ernest Rutherford was one of the pioneers who discovered artificial transmutation. He discovered the following by exposing the nitrogen atoms to alpha particles. He bombarded the said nitrogen nucleus with the present alpha particles in order to get oxygen.
$N_{7}^{14} + He_{2}^{4} \rightarrow O_{8}^{17} + H_{1}^{1}$
In fact, the first nuclear transmutation was also witnessed and applied to the branch of modern physics by Frederick Soddy. He was working with Rutherford in the year 1901, and they both discovered that radioactive thorium could transform itself into radium. They were excited to witness that the process was transmutation and Soddy worked on it further to apply it to the branch of Physics.
### Some Prominent Examples of Artificial Transmutation
• When an alpha particle is bombarded with the nucleus of a nitrogen element, oxygen as an artificial element is produced. As a part of transformation one atom of hydrogen is also produced.
• When an alpha particle is bombarded with the nucleus of an aluminium element, phosphorus as an artificial element is produced. As a part of the transformation, a neutron is produced.
• When Uranium-28 is bombarded with neutrons, the resultant element is Uranium-239. Uranium-239 is very unstable; therefore, it decays to Neptunium. When it decays it produces beta particles.
### Conversion Laws Applied to Nuclear Reactions
While studying artificial transmutation it is necessary to keep a note on “conversion laws” that are applied to a nuclear reaction.
• The number of nucleons in a nuclear reaction stays the same or is conserved.
• The mass-energy relation is conserved.
• The charge is conserved, to understand it differently, the sum of charges on the left-hand side of the reaction is equal to the number of charges on the right-hand side.
### Artificial Transmutation Equation
The artificial transmutation equation is based upon the conversion laws as noted above. It represents the conversion of one element into the other. The reaction is shown with the number of protons. The element which is to be converted by bombardment is kept at the left-hand side along with the fundamental particle that has to be bombarded, on the right-hand side comes the final element along with the subatomic particles. An example of this can be:
When you bombard Uranium -238 with a neutron, it will convert into Uranium -239, it is highly unstable and transmutes into Neptunium which can be seen emitting a beta participle.
### Difference Between Artificial Transmutation and Artificial Radioactivity
In artificial transmutation, one element, which is a non-radioactive element, is converted into a final element by bombarding a fundamental particle through artificial means. While in artificial radioactivity, the radioactivity is induced in a stable element, and the whole process undergoes numerous nuclear reactions. | 2021-03-05 10:59:50 | {"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": 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": 0.5048534870147705, "perplexity": 1172.164948655098}, "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/1614178370752.61/warc/CC-MAIN-20210305091526-20210305121526-00230.warc.gz"} |
https://math.stackexchange.com/questions/1014763/extending-a-dvr-could-produce-not-a-dvr | # Extending a DVR could produce not a DVR
I'm reading Tate's paper about $p$-divisible groups. In Chapter $(2.4)$ he asserts that if you take $R$ a complete DVR with residue field $k$ of characteristic $p>0$, $K$ its field of fractions, $L$ the completion of an algebraic extension of $K$ and $S$ the ring of integers in $L$, then $S$ is a complete rank $1$ valuation ring, but with a not necessarily discrete valuation. I'm essentially looking for a counterexample of this fact. I know (see Froelich "Local fields" Prop.2) that, if the extension is finite, then the ring of integers inside $L$ is a DVR, so I'm looking at infinite extensions of local fields. Particularly, I hope that the ring of integers inside $\mathbb{C}_{p}$, the completion of the algebraic closure of $\mathbb{Q}_{p}$ is not a DVR, but I really don't know how to prove this fact. Thanks for any suggestion.
Pretty sure $L=\mathbb{Q}_p(p^{1/\infty})=\mathbb{Q}_p(\sqrt[n]{p\vphantom{b}}:n\geq 2)$ is a counterexample: letting $v$ be the $p$-adic valuation on $\mathbb{Q}_p$, it would necessarily extend by $v(\sqrt[n]{p\vphantom{b}})=\frac{1}{n}$, making the image of $S^\times$ under $v$ equal to the group of positive rationals $\mathbb{Q}_{>0}$, which is not a discrete group. | 2019-05-23 10:42:44 | {"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": 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": 0.921536922454834, "perplexity": 47.64930564292974}, "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-2019-22/segments/1558232257243.19/warc/CC-MAIN-20190523103802-20190523125802-00045.warc.gz"} |
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# Rasheed bought two kinds of candy bars
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Rasheed bought two kinds of candy bars [#permalink]
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21 Oct 2018, 09:07
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Data Sufficiency::
Rasheed bought two kinds of candy bars, chocolate and toffee, that came in packages of 2 bars each. He handed out 2/3 chocolate bars and 3/5 of the toffee bars. How many packages of chocolate bars did Rasheed buy?
l) Rasheed bought 1 fewer package of chocolate bars than toffee bars.
2) Rasheed handed out the same number of each kind of candy bar.
what's the answer ? I'm getting confused
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Rasheed bought two kinds of candy bars [#permalink]
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21 Oct 2018, 09:57
1
OA: C
Let us assume Rasheed bought $$C$$ packages of Chocolate and $$T$$ packages of toffee.
Each package has $$2$$ Bar.
So Rasheed handed out $$\frac{2}{3}*2C$$ Chocolate bars and $$\frac{3}{5}*2T$$ toffee bars.
We have to answer "What is value of $$C$$ i.e Number of Chocolate package "
1) Rasheed bought 1 fewer package of chocolate bars than toffee bars.
$$T-C=1$$
As we cannot find the value of either $$T$$ or $$C$$ , So Statement $$1$$ alone is insufficient.
2) Rasheed handed out the same number of each kind of candy bar.
$$\frac{2}{3}*2C=\frac{3}{5}*2T$$
$$10C=9T$$
As we cannot find the value of either $$T$$ or $$C$$ , So Statement $$2$$ alone is insufficient.
Combining $$(1)$$ and $$(2)$$, we get
$$T-C=1$$....(1)
$$10C=9T; \quad T = \frac{10C}{9}$$....(2)
Putting (2) into (1), we get
$$\frac{10C}{9}-C=1$$
$$C=9$$
Combining $$(1)$$ and $$(2)$$, we can get value of $$C$$.
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Posts: 50572
Re: Rasheed bought two kinds of candy bars [#permalink]
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21 Oct 2018, 20:03
jackfr2 wrote:
Data Sufficiency::
Rasheed bought two kinds of candy bars, chocolate and toffee, that came in packages of 2 bars each. He handed out 2/3 chocolate bars and 3/5 of the toffee bars. How many packages of chocolate bars did Rasheed buy?
l) Rasheed bought 1 fewer package of chocolate bars than toffee bars.
2) Rasheed handed out the same number of each kind of candy bar.
what's the answer ? I'm getting confused
Posted from my mobile device
Please search before posting: https://gmatclub.com/forum/rasheed-boug ... 29470.html
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Re: Rasheed bought two kinds of candy bars &nbs [#permalink] 21 Oct 2018, 20:03
Display posts from previous: Sort by | 2018-11-14 01:18:54 | {"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": 0.4140618145465851, "perplexity": 8870.904299594147}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741569.29/warc/CC-MAIN-20181114000002-20181114022002-00167.warc.gz"} |
http://math.stackexchange.com/questions/619053/what-is-known-about-this-jigsaw-combinatorics-problem | The problem: How many valid reconfigurations, as defined below, are there for a given puzzle?
For example, does this problem--or an equivalent problem; or perhaps a more limited variation, say $1 \times n$; or with non-square rectangular pieces to limit the number of external edge matches--have a conventional name?
Did I actually come up with something new, or am I just describing a known problem with my own terminology because I don't know any better?
The definitions as formally as I can, based on simple physical jigsaw puzzles with unique squiggly parts and uniform straight parts:
• puzzle: an $m \times n$ grid full of identically-sized, identically-oriented pieces
• piece: a square of $4$ edges; denoted $a_{i,j}$ for the piece in row $i$, column $j$ of puzzle $A$ where $1 \leq i \leq m$ and $1 \leq j \leq n$
• edge: a means of joining two pieces; is either internal or external
• joined: two pieces sharing exactly one edge (either in the puzzle or a reconfiguration)
• internal edge: a unique edge joining two pieces in the puzzle; denoted $\overline{a_{i,j}a_{g,h}}$ for joined pieces $a_{i,j}$ and $a_{g,h}$
• the external edge: each of the identical $2m + 2n$ edges in the puzzle that are not internal
• a reconfiguration: the puzzle, modified by operating on individual pieces by means of right-angle rotations and piece-size translations, such that the end result is contiguous, i.e., any given piece is related to any other piece via a series of edges
• a valid reconfiguration: a reconfiguration wherein no two pieces are vertically or horizontally adjacent unless the edge they share is either the internal edge they share from the puzzle, or the external edge (i.e. an internal edge matches only as it does in the puzzle, and cannot match a different internal edge, nor the external edge)
Informal clarifications
Note that, necessarily, a valid reconfiguration is made by various joins of the external edge that were not part of the puzzle, and it will not be a neat rectangle in the end.
That is to say, you can glue the straight parts together on the inside, you can't mismatch the squiggly parts, and some squiggly parts will now be on the outside.
I actually thought of this question when considering such an arrangement of a normal physical puzzle for an artistic idea (maybe Picasso's cubism collides or colludes with Warhol?) so it is entirely made with real, physical puzzles in mind.
Why this seems like a tricky combinatorial problem to me
For $1 \times 1$, the puzzle cannot be reconfigured because no joins are possible because there is only one piece.
For $1 \times 2$, there are $9$ valid reconfigurations, namely those made from the various combinations of matching any of the $3$ straight edges of $a_{1,1}$ with any of $3$ straight edges of $a_{1,2}$ (here pictured as A and B, respectively, because re-doing work in Excel wore me out):
For $1 \times 3$, I count:
• $\overline{a_{1,1}a_{1,2}}$ intact, with one of $a_{1,3}$'s straight edges joined to one of $a_{1,2}$'s $= 3*2 = 6$
• $\overline{a_{1,1}a_{1,2}}$ intact, with one of $a_{1,3}$'s straight edges joined to one of $a_{1,1}$'s $= 3*3 = 9$
• $\overline{a_{1,2}a_{1,3}}$ intact, symmetrical with $a_{1,1}$ as in the above two cases $= 6 + 9 = 15$
• $a_{1,1}$ joined to $a_{1,2}$ via a straight edge, and $a_{1,3}$ joined to $a_{1,2}$ via $a_{1,2}$'s other straight edge $= 3*2*3 = 18$
• $a_{1,1}$ joined to $a_{1,3}$ via a straight edge, and $a_{1,2}$ joined to $a_{1,3}$ via another of $a_{1,3}$'s straight edges $= 3*3*2*2 = 36$
• $a_{1,1}$ joined to $a_{1,3}$ via a straight edge, and $a_{1,2}$ joined to $a_{1,1}$ via another of $a_{1,1}$'s straight edges $= 3*3*2*2 = 36$
...for a total of $120$ valid reconfigurations, if I didn't miss anything.
For a $2 \times 2$, it certainly gets hairier:
• $\overline{a_{1,1}a_{1,2}}$ and $\overline{a_{1,1}a_{2,1}}$ intact, with $a_{2,2}$ removed and rejoined elsewhere $= 3*6 = 18$
• symmetry of above with each of the other pieces functioning as $a_{2,2} = 3 * 18$
• $\overline{a_{1,1}a_{1,2}}$ intact, and $a_{2,2}$ joined to $a_{1,2}$ by a straight edge:
• includes the case where the other joins are $\overline{a_{2,1}a_{2,2}}$, $a_{1,1}$'s top external edge with $a_{2,1}$'s bottom external edge, and $a_{1,2}$'s top external edge with $a_{2,2}$'s bottom external edge
• not as simple to calculate by straight multiplication, though: naturally cannot include invalid reconfigurations like where the other joins are $a_{1,1}$'s top external edge with $a_{2,1}$'s bottom external edge and $a_{2,2}$ rotated once clockwise so that its right edge joins $a_{1,2}$'s top edge, because $a_{2,2}$'s bottom edge, which is external, cannot join $a_{2,1}$'s right edge, which is the internal edge $\overline{a_{2,1}a_{2,2}}$...here it's already tricky
• ...
• total unknown
For $1 \times 4$, there would, I would think, be more solutions than $2 \times 2$, since there are 10 external edges instead of 8, but it would also be tricky to calculate because of combinations of external edge joins that would need non-matching internal edges to join, which they can't.
Holes for slightly larger puzzles
Holes may arise in valid reconfigurations:
The example above shows three different puzzles and a valid reconfiguration for each one that happens to have a hole. (Straight edges are solid, and internal edges are dotted, even though different dotted edges may only join if they're joined in the original. In a real puzzle these are the unique squiggly cuts resulting in two pieces sharing a pair squiggly parts that can only fit each other.) The top $3 \times 4$ puzzle's reconfiguration has a hole with all internal edges, the middle $2 \times 4$ puzzle's reconfiguration has a hole with a mix of internal edges and straight edges, and the bottom $1 \times 8$ puzzle's reconfiguration has a hole with all straight edges. All of these are valid reconfigurations.
Bounds
A completely brute force approach to counting these may be taken by looking at all possible reconfigurations, then excluding the invalid ones (any where the graph is disconnected, or unmatching internal edges are adjacent.) Then the question of rotational equivalence could be addressed, AFAICT simply by dividing by 4.
The biggest area to check is $mn$ by $mn$, and each square in it can be occupied by one of the $mn$ original pieces in one of 4 rotations, or left blank. One cannot use a simple permutation function for the upper bound here because of the rotations--once a piece is used in a particular rotation, the possibility of using its other three rotations in the same potential reconfiguration goes away.
Sequences
On the topic of just the $1 \times n$ set of cases, we have the sequence 0, 9, 120, ... (or, if you want to include the original configuration, 1, 10, 121, ... but it's not clear if it is, in fact, related to any known sequence. Perhaps some software could be written to calculate a few more numbers in the sequence to help with this.
-
I'm not sure that I understand the objects you are formalizing entirely, but they sound a lot like "stretchy polyominoes". ( en.wikipedia.org/wiki/Polyomino ) If that's the case, I can't see how the top edge of C could attach to the left edge of A without causing some sort of localized hole between the top edge of A and the top edge of B. – Eric Stucky Dec 26 '13 at 20:20
@EricStucky, I might've mis-formalized it. I was going for the most basic of jigsaw puzzles, so "stretchy" strikes me as probably not what I meant. If you're talking about the 2-by-2 case, they can't attach: the top edge of C is a unique internal edge, whereas the left edge of A is the straight edge. I'll edit to try to clarify this. – Kev Dec 27 '13 at 14:43
Although you may be able to apply polyomino-based theorems to this problem due to their equivalence to jigsaw puzzles: dl.acm.org/citation.cfm?id=1275561 – Kev Dec 27 '13 at 15:13
I'm sorry, I was talking about the $1\times 3$ case. I have a hard time visualizing how, say, the right end of C attaches to the left end of A could be joined in the plane if the pieces are not stretchy. Correct me on this, but I think that in the $2\times 2$ case you can't do anything with the top edge of C because it is internal. But I do not see how it is unique; surely the right edge of C is also internal? – Eric Stucky Dec 27 '13 at 22:05
@EricStucky re: 1-by-3, it's because B and C do not have to join. Hopefully that helps with your 2-by-2 question. I'd better clarify that in the original. – Kev Dec 29 '13 at 2:20 | 2016-05-25 07:55:54 | {"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": 0.6890208721160889, "perplexity": 755.867165339217}, "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-2016-22/segments/1464049274191.57/warc/CC-MAIN-20160524002114-00174-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.hanspub.org/journal/PaperInformation.aspx?paperID=23178 | # 带有外力项的可压等熵Navier-Stokes方程的滞弹性逼近Anelastic Approximation of Compressible Isentropic Navier-Stokes Equations with Exterior Force
• 全文下载: PDF(412KB) PP.1207-1219 DOI: 10.12677/AAM.2017.69146
• 下载量: 607 浏览量: 808 国家自然科学基金支持
In this paper, we prove the anelastic approximation limit to compressible isentropic Navier-Stokes equations with exterior force and Dirichlet boundary condition, as Mach number and Froude number go to zero. This covers the result of special force case in J. Math. Pures Appl. 88 (2007) 230-240.
1. 引言
$\left\{\begin{array}{l}{\rho }_{\epsilon t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)=0,\text{\hspace{0.17em}}{\rho }_{\epsilon }\ge 0,\\ {\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}_{t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)-\mu \Delta {u}_{\epsilon }-\xi \nabla \text{div}{u}_{\epsilon }+\frac{\gamma {\rho }_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }_{\epsilon }^{\gamma -1}-{\stackrel{¯}{\rho }}_{\epsilon }^{\gamma -1}\right)=0.\end{array}$ (1.1)
(包含势能项 $\frac{1}{{\epsilon }^{2}}{\rho }_{\epsilon }\nabla V$ )趋于滞弹性系统的严格推导，其中 $V=gz$ 是重力势能( $z$ 为竖直分量)。文中滞弹性系
2. 主要结果
${\left({\rho }_{\epsilon },{\rho }_{\epsilon }{u}_{\epsilon }\right)\left(t,x\right)|}_{t=0}=\left({\rho }_{0\epsilon },{m}_{0\epsilon }\right)\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega ,$ (2.1)
${u}_{\epsilon }=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\partial \Omega .$ (2.2)
$\frac{{m}_{0\epsilon }^{2}}{{\rho }_{0\epsilon }}\in {L}^{1}\left(\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{0\epsilon }\in {L}^{2\gamma /\left(\gamma +1\right)}\left(\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\rho }_{0\epsilon }\in {L}^{\gamma }\left(\Omega \right),$ (2.3)
${\int }_{\Omega }\frac{{\rho }_{0}^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{0}+\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\text{d}x\le C$ (2.4)
$m={P}_{\stackrel{¯}{\rho }}m+\stackrel{¯}{\rho }\nabla \Psi ,$
$\text{div}{P}_{\stackrel{¯}{\rho }}m=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{\stackrel{¯}{\rho }}m\cdot n|{}_{\partial \Omega }=0.$
${\rho }_{\epsilon }\to \stackrel{¯}{\rho }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{\infty }\left(0,T;{L}^{\gamma }\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{\epsilon }\to u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{H}_{0}^{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\rho }_{\epsilon }{u}_{\epsilon }\to \stackrel{¯}{\rho }u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{L}^{2}\right)$ (2.5)
${P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to \stackrel{¯}{\rho }u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{L}^{2}\right)$ (2.6)
$\left\{\begin{array}{l}{\left(\stackrel{¯}{\rho }u\right)}_{t}+\text{div}\left(\stackrel{¯}{\rho }u\otimes u\right)-\mu \Delta u-\xi \nabla \text{div}u+\stackrel{¯}{\rho }\nabla q=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,T\right)×\Omega ,\\ \text{div}\left(\stackrel{¯}{\rho }u\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,T\right)×\Omega ,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega ,\\ u\left(0,x\right)=\frac{{P}_{\stackrel{¯}{\rho }}{m}_{0}}{\stackrel{¯}{\rho }}\left(x\right).\end{array}$ (2.7)
$\begin{array}{l}-{\int }_{0}^{T}{\int }_{\Omega }u{\partial }_{t}f\text{d}x\text{d}s-{\int }_{0}^{T}{\int }_{\Omega }\stackrel{¯}{\rho }u\otimes u\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\nabla u:\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)+\xi \text{div}u\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s={\int }_{\Omega }{P}_{\stackrel{¯}{\rho }}{m}_{0}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x.\end{array}$ (2.8)
3. 先验估计
${\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\pi }_{\epsilon }}{\gamma -\text{1}}\right)\text{d}x+{\int }_{0}^{T}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\le C,$ (3.1)
$\frac{\text{d}}{\text{d}t}{\int }_{\Omega }{\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{\gamma {\rho }_{\epsilon }{u}_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }_{\epsilon }^{\gamma -1}-{\stackrel{¯}{\rho }}_{\epsilon }^{\gamma -1}\right)\text{d}x+{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\le 0.$
$\frac{\text{d}}{\text{d}t}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x+{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\le 0.$
${\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}\rho }{{\epsilon }_{\epsilon }^{2}{\left(\gamma -1\right)}^{2}}\right)\text{d}x$ $+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.$ (3.2)
$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{\epsilon }+\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ \le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{0\epsilon }+\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.\end{array}$ (3.3)
$\begin{array}{c}{\epsilon }^{2}{\pi }_{\epsilon }={\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}\left({\rho }_{\epsilon }-\stackrel{¯}{\rho }\right)={\rho }_{\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{\epsilon }+\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma }\\ ={\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\gamma \left({\stackrel{¯}{\rho }}^{\gamma -1}\epsilon {\varphi }_{\epsilon }\right).\end{array}$ (3.4)
$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\pi }_{\epsilon }}{\gamma -1}\right)\text{d}x+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ \le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\stackrel{¯}{\rho }}^{\gamma -1}{\rho }_{0\epsilon }+\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.\\ \le C.\end{array}$ (3.5)
${\rho }_{\epsilon }\in {L}^{\gamma +\theta }\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{0}=\frac{2}{N}\gamma -1,$ (3.6)
$\frac{1}{\epsilon }{|\epsilon {\varphi }_{\epsilon }|}^{\gamma +\theta }\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}1<\theta \le {\theta }_{0}=\frac{2}{N}\gamma -1,$ (3.7)
${\varphi }_{\epsilon }\in {L}^{1+\theta }\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.8)
${\epsilon }^{\gamma -1}{|{\varphi }_{\epsilon }|}^{\gamma +\theta }\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.9)
${\epsilon }^{1-\theta }|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.10)
$\left\{\begin{array}{l}-\Delta u+\nabla p=g,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\Omega }p\text{d}x=0,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega ,\\ \text{div}\left(u\right)=0.\end{array}$ (3.11)
$\begin{array}{l}{\partial }_{t}{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{h}_{0}+\frac{1}{{\epsilon }^{2}}\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\text{d}x\right)\\ -\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\stackrel{¯}{\rho }}^{\gamma -1}\right)=0,\end{array}$ (3.12)
$\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\text{d}x\right){\rho }_{\epsilon }^{\theta }\text{d}x\text{d}t\le C.$ (3.13)
$\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\right)\left({\rho }_{\epsilon }^{\theta }-{\stackrel{¯}{\rho }}^{\theta }\right)\text{d}x\text{d}t\le C.$ (3.14)
${\partial }_{t}{\varphi }_{\epsilon }+\frac{1}{\epsilon }\text{div}\left(\stackrel{¯}{\rho }{u}_{\epsilon }\right)+\text{div}\left({\varphi }_{\epsilon }{u}_{\epsilon }\right)=0.$ (3.15)
${\partial }_{t}{\varphi }_{\epsilon }^{\theta }+\frac{1}{\epsilon }\theta {|ph{i}_{\epsilon }|}^{\theta -\text{1}}\text{div}\left(\stackrel{¯}{\rho }{u}_{\epsilon }\right)+\text{div}\left({\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\right)=\left(1-\theta \right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right).$ (3.16)
$\begin{array}{l}{\partial }_{t}\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]+\text{div}\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right){u}_{\epsilon }\right]-{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\frac{1}{\epsilon }\text{div}\left(\stackrel{¯}{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +\text{\hspace{0.17em}}{\varphi }_{\epsilon }^{\theta }{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+{\varphi }_{\epsilon }^{\theta }\frac{1}{{\epsilon }^{2}}\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\text{d}x\right)\\ -\text{\hspace{0.17em}}{\varphi }_{\epsilon }^{\theta }\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\stackrel{¯}{\rho }}^{\gamma -1}\right)=0.\end{array}$ (3.17)
$\begin{array}{l}\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\text{d}x-\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\left(0\right)\text{d}x-\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\text{div}\left(\stackrel{¯}{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\epsilon {\varphi }_{\epsilon }^{\theta }\left[{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\stackrel{¯}{\rho }}^{\gamma -1}\right)\right]\text{d}x\text{d}s\\ +\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\text{d}x\right)\text{d}x\text{d}s=0,\end{array}$ (3.18)
$\begin{array}{l}\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }\left({\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\stackrel{¯}{\rho }}^{\gamma }\text{d}x\right)\text{d}x\text{d}s\\ =-\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\text{d}x+\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\left(0\right)\text{d}x+\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{T}{\int }_{\Omega }\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\text{div}\left(\stackrel{¯}{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{T}{\int }_{\Omega }\epsilon {\varphi }_{\epsilon }^{\theta }\left[{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\stackrel{¯}{\rho }}^{\gamma -1}\right)\right]\text{d}x\text{d}s\\ :=\underset{i=1}{\overset{5}{\sum }}{I}_{i}.\end{array}$ (3.19)
$\begin{array}{l}{\rho }_{\epsilon }{u}_{\epsilon }\in {L}^{2}\left(0,T;{L}^{q}\left(\Omega \right)\right),\\ \epsilon {\varphi }_{\epsilon }{u}_{\epsilon }\in {L}^{2}\left(0,T;{L}^{q}\left(\Omega \right)\right),\\ {\varphi }_{\epsilon }^{\theta -1}\in {L}^{\infty }\left(0,T;{L}^{2/\left(\theta -1\right)}\left(\Omega \right)\right),\end{array}$ (3.20)
$\frac{\theta -1}{2}+\frac{2}{q}=\frac{\theta -1}{2}+\frac{N\gamma -2\gamma +2N}{N\gamma }\le 1$ ，且有
$|{I}_{3}|=\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\le C.$ (3.21)
${|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|}^{\gamma +\theta }\le \epsilon {|{\varphi }_{\epsilon }|}^{3}+{\epsilon }^{\gamma -2}{|{\varphi }_{\epsilon }|}^{\gamma }$ ，并利用(3.8)~(3.9)得，当 $0\le \beta \le \gamma -1$ 时有 ${\epsilon }^{\beta }{\varphi }_{\epsilon }^{1+\theta +\beta }$${L}^{1}\left(\left(0,T\right)×\Omega \right)$ 中有界。即可得对 $0<\theta \le {\theta }_{1}$ 时有 ${\epsilon }^{1-\theta }|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|$${L}^{1}\left(\left(0,T\right)×\Omega \right)$ 中有界。(3.10)得证。证毕。
4. 定理证明
${P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to \stackrel{¯}{\rho }u,$ (4.1)
${L}^{2}\left(\left(0,T\right)×\Omega \right)$ 中强收敛，并且
$\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\to 0,$ (4.2)
${L}^{2}\left(\left(0,T\right)×\Omega \right)$ 中弱收敛，其中 $\text{div}\left(\stackrel{¯}{\rho }u\right)=0$
$\begin{array}{l}\frac{\gamma \rho }{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }^{\gamma -1}-{\stackrel{¯}{\rho }}^{\gamma -1}\right)=\frac{1}{{\epsilon }^{2}}\nabla {\rho }^{\gamma }-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\rho \nabla {\stackrel{¯}{\rho }}^{\gamma -1}\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\left(\nabla {\stackrel{¯}{\rho }}^{\gamma }+\gamma \nabla \left({\stackrel{¯}{\rho }}^{\gamma -1}\left(\rho -\stackrel{¯}{\rho }\right)\right)\right)-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\left(\rho -\stackrel{¯}{\rho }\right)\nabla {\stackrel{¯}{\rho }}^{\gamma -1}-\frac{1}{{\epsilon }^{2}}\nabla {\stackrel{¯}{\rho }}^{\gamma }\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\gamma \nabla \left({\stackrel{¯}{\rho }}^{\gamma -1}\left(\rho -\stackrel{¯}{\rho }\right)\right)-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\left(\rho -\stackrel{¯}{\rho }\right)\nabla {\stackrel{¯}{\rho }}^{\gamma -1}\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\gamma \left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma -2}\nabla \stackrel{¯}{\rho }\left(\rho -\stackrel{¯}{\rho }\right)+\frac{\gamma }{{\epsilon }^{2}}{\stackrel{¯}{\rho }}^{\gamma -1}\nabla \left(\rho -\stackrel{¯}{\rho }\right)-\frac{\gamma }{{\epsilon }^{2}}\left(\rho -\stackrel{¯}{\rho }\right){\stackrel{¯}{\rho }}^{\gamma -2}\nabla \stackrel{¯}{\rho }\end{array}$
$\begin{array}{l}=\nabla {\pi }_{\epsilon }+\frac{\gamma \left(\gamma -1\right)}{{\epsilon }^{2}\left(\gamma -2\right)}\stackrel{¯}{\rho }\nabla {\stackrel{¯}{\rho }}^{\gamma -2}\left(\rho -\stackrel{¯}{\rho }\right)+\frac{\gamma }{{\epsilon }^{2}}\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}\left(\rho -\stackrel{¯}{\rho }\right)\right)\\ -\frac{\gamma }{{\epsilon }^{2}}\left(\rho -\stackrel{¯}{\rho }\right)\nabla {\stackrel{¯}{\rho }}^{\gamma -2}\stackrel{¯}{\rho }-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -2\right)}\left(\rho -\stackrel{¯}{\rho }\right)\stackrel{¯}{\rho }\nabla {\stackrel{¯}{\rho }}^{\gamma -2}=\nabla {\pi }_{\epsilon }+\frac{\gamma }{\epsilon }\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)\end{array}$ (4.3)
(1.1)2可重写为：
${\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}_{t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)-\mu \Delta {u}_{\epsilon }-\xi \nabla \text{div}{u}_{\epsilon }+\nabla {\pi }_{\epsilon }+\frac{\gamma }{\epsilon }\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=0$ .(4.4)
$\begin{array}{l}-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\frac{{\partial }_{t}f}{\stackrel{¯}{\rho }}\text{d}x\text{d}s-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s+{\int }_{0}^{T}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\\ +\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s={\int }_{\Omega }{m}_{0\epsilon }\frac{f\left(t=0\right)}{\stackrel{¯}{\rho }}\text{d}x\end{array}$ (4.5)
$-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)+{\pi }_{\epsilon }\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s\to -{\int }_{0}^{T}{\int }_{\Omega }\stackrel{¯}{\rho }u\otimes u:\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s.$ (4.6)
$\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+\nabla {\pi }_{\epsilon }\to \text{div}\left(\stackrel{¯}{\rho }u\otimes u\right)+\stackrel{¯}{\rho }\nabla P$ (4.7)
${\rho }_{\epsilon }{u}_{\epsilon }\otimes {\rho }_{\epsilon }{u}_{\epsilon }={P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {\rho }_{\epsilon }{u}_{\epsilon }+{Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right).$ (4.8)
${P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {\rho }_{\epsilon }{u}_{\epsilon }\to \left(\rho u\right)\otimes \left(\rho u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {P}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to 0.$ (4.9)
$\left(I-{P}_{\stackrel{¯}{\rho }}\right)$ 作用(4.4)，并结合质量守恒方程可得方程组
$\left\{\begin{array}{l}{\partial }_{t}\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }+\frac{\gamma }{\epsilon }\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=\stackrel{¯}{\rho }{F}_{\epsilon },\hfill \\ \epsilon {\partial }_{t}{\varphi }_{\epsilon }+\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\right)=0,\hfill \end{array}$ (4.10)
$\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\otimes \nabla {\psi }_{\epsilon }\right)+\frac{\gamma }{2}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}\right)\to \stackrel{¯}{\rho }\nabla P$ (4.11)
$\begin{array}{l}\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\otimes \nabla {\psi }_{\epsilon }\right)=\stackrel{¯}{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}+\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\right)\nabla {\psi }_{\epsilon }\\ =\stackrel{¯}{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}{\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\\ =\stackrel{¯}{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{\partial }_{t}\left(\nabla {\psi }_{\epsilon }\right)\\ =\stackrel{¯}{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)-\gamma \stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon }.\end{array}$ (4.12)
$\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=\left(2-\gamma \right)\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -3}\frac{{\varphi }_{\epsilon }^{2}}{2}\right)+\left(\left(\gamma -1\right){\stackrel{¯}{\rho }}^{\gamma -2}\frac{{\varphi }_{\epsilon }^{2}}{2}\right).$ (4.13)
$\begin{array}{l}\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon }\otimes {\psi }_{\epsilon }\right)+\frac{\gamma \left(\gamma -1\right)}{2}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}\right)\\ =\stackrel{¯}{\rho }\nabla \left(\frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}+\frac{\gamma \left(\gamma -2\right)}{2}{\stackrel{¯}{\rho }}^{\gamma -3}\frac{{\varphi }_{\epsilon }^{2}}{2}\right)-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon }.\end{array}$ (4.14)
${‖{\varphi }_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\varphi }_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{P}\left({L}^{2}\right)}\to 0,$ (4.15)
$\epsilon$ 是一致的。
$\left\{\begin{array}{l}-\Delta u+\stackrel{¯}{\rho }\nabla p=g,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{div}\left(\stackrel{¯}{\rho }u\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in B,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{B}p\text{d}x=0,\hfill \\ u=0,\text{on}x\in \partial B.\hfill \end{array}$ (4.16)
${\stackrel{˜}{R}}_{\delta }=I-{R}_{\delta }$ 。用 $S$ 作用(4.4)后得到的式子再用 ${\stackrel{˜}{R}}_{\delta }$ 作用得
${\partial }_{t}{\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h+\frac{\gamma }{\epsilon }{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }+{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)=0.$ (4.17)
${f}_{\epsilon }={\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }$ ，可得
${\partial }_{t}{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }+\frac{1}{\epsilon }{\stackrel{¯}{\rho }}^{\gamma -2}\text{div}\left({\stackrel{˜}{R}}_{\delta }{\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{1}{\epsilon }\left[{\stackrel{˜}{R}}_{\delta },{\stackrel{¯}{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right)=0.$ (4.18)
$\begin{array}{l}{\partial }_{t}\left({\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)+\frac{1}{\epsilon }{\stackrel{¯}{\rho }}^{\gamma -2}\text{div}\left({\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)\\ -\frac{1}{\epsilon }{\stackrel{¯}{\rho }}^{\gamma -2}{\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\cdot \nabla {\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{1}{\epsilon }\left[{\stackrel{˜}{R}}_{\delta },{\stackrel{¯}{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }\left({\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h\right)+\frac{\gamma }{\epsilon }{|{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}+{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)=0.\end{array}$ (4.19)
${\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\gamma {|{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}\text{d}x\text{d}s\to 0,$ (4.20)
$\epsilon$ 是一致的。
$\begin{array}{l}{{\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\gamma \Phi |{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ =\epsilon {\int }_{{B}_{\frac{r}{2}}}\left({\stackrel{˜}{R}}_{\delta }{f}_{0\epsilon }{\stackrel{˜}{R}}_{\delta }S\left({m}_{0\epsilon }\right)\right)\text{d}x+{\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\left({\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right):\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}\Phi \right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{T}{\int }_{{B}_{\frac{r}{2}}}\left(\Phi {\stackrel{¯}{\rho }}^{\gamma -2}{\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\cdot \nabla {\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)-\left[{\stackrel{˜}{R}}_{\delta },{\stackrel{¯}{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\epsilon {\int }_{0}^{T}{\int }_{{B}_{\frac{r}{2}}}\Phi \left({\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }\left({\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h\right)+{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)\right)\text{d}x\text{d}s.\end{array}$ (4.21)
${‖{\varphi }_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\varphi }_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0,$ (4.22)
${‖{u}_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{u}_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0.$ (4.23)
${‖{\rho }_{\epsilon }{u}_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\rho }_{\epsilon }{u}_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0.$ (4.24)
$f$${C}_{\text{0}}^{\infty }\left(\Omega \right)$ 中的试验函数，并且满足 $\text{div}\left(f\right)=0$ ，i.e. ${P}_{\stackrel{¯}{\rho }}f=f$ 。由动量方程(4.4)可知：对 $0 ，有
$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\left(\frac{f}{\stackrel{¯}{\rho }}\right)\left(t\right)\text{d}x-{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\left(\frac{f}{\stackrel{¯}{\rho }}\right)\left(s\right)\text{d}x\\ ={\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\frac{{\partial }_{t}f}{\stackrel{¯}{\rho }}\text{d}x\text{d}s+{\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{\int }_{s}^{t}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)+\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s\\ ={\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s\\ -{\int }_{s}^{t}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\stackrel{¯}{\rho }}\right)+\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\stackrel{¯}{\rho }}\right)\text{d}x\text{d}s.\end{array}$ (4.25)
${Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)-{Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to 0,$ (4.26)
$\epsilon$ 是一致的。因此，我们要证明 ${Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)$ 的极限，并刻画
$\text{div}\left(\frac{{Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}{\stackrel{¯}{\rho }}\right)$
$\frac{\gamma }{\text{2}}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -\text{2}}{\left({\varphi }_{\epsilon }\right)}^{\text{2}}\right)-\frac{\gamma }{\text{2}}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -\text{2}}{\left({R}_{\delta }{\varphi }_{\epsilon }\right)}^{\text{2}}\right)\to {r}_{\delta }\to 0.$ (4.27)
$\text{div}\left(\frac{{Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\stackrel{¯}{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}{\stackrel{¯}{\rho }}\right)+\frac{\gamma }{2}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\left({R}_{\delta }{\varphi }_{\epsilon }\right)}^{2}\right)\to \stackrel{¯}{\rho }\nabla P+{r}_{\delta },$ (4.28)
${R}_{\delta }$ 作用(4.10)得
$\left\{\begin{array}{l}{\partial }_{t}{R}_{\delta }{Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{\gamma }{\epsilon }\stackrel{¯}{\rho }\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{R}_{\delta }{\varphi }_{\epsilon }\right)=\stackrel{¯}{\rho }{F}_{\epsilon ,\delta }-\frac{\gamma }{\epsilon }\left[{R}_{\delta },\stackrel{¯}{\rho }\nabla {\stackrel{¯}{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon },\hfill \\ \epsilon {\partial }_{t}{R}_{\delta }{\varphi }_{\epsilon }+\text{div}\left({R}_{\delta }{Q}_{\stackrel{¯}{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)=0,\hfill \end{array}$ (4.29)
$\left[{R}_{\delta },\stackrel{¯}{\rho }\nabla {\stackrel{¯}{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon }=\left[{R}_{\delta },\stackrel{¯}{\rho }\right]\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)+\stackrel{¯}{\rho }\nabla \left[{R}_{\delta },{\stackrel{¯}{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon },$ (4.30)
${‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{L}^{2}}\le C{‖f‖}_{{L}^{2\delta }},{‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{H}^{1}}\le C{‖f‖}_{{L}^{2}},$ (4.31)
$\frac{{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)}{\delta }\to 0,$ (4.32)
${R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)\to 0.$ (4.33)
(2) 对 $f\in {H}^{-1}\left({R}^{N}\right)$$g\in {L}^{2}\left({R}^{N}\right)$ ，有
${‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{L}^{2}}\le C{‖f‖}_{{H}^{-1}},$ (4.34)
$\frac{{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)}{\delta }\to 0.$ (4.35)
$\begin{array}{l}\text{div}\left(\stackrel{¯}{\rho }\nabla {\psi }_{\epsilon ,\delta }\otimes \nabla {\psi }_{\epsilon ,\delta }\right)+\frac{\gamma \left(\gamma -1\right)}{2}\nabla \left({\stackrel{¯}{\rho }}^{\gamma -2}{\varphi }_{\epsilon ,\delta }^{2}\right)\\ =\stackrel{¯}{\rho }\nabla \left(\frac{{|\nabla {\psi }_{\epsilon ,\delta }|}^{2}}{2}+\frac{\gamma \left(\gamma -2\right)}{2}{\stackrel{¯}{\rho }}^{\gamma -3}{\varphi }_{\epsilon ,\delta }^{2}\right)-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon ,\delta }\nabla {\psi }_{\epsilon ,\delta }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon ,\delta }+\frac{\gamma }{\stackrel{¯}{\rho }}\left[{R}_{\delta },\stackrel{¯}{\rho }\nabla {\stackrel{¯}{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon }{\varphi }_{\epsilon ,\delta }.\end{array}$ (4.36)
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(2004) Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, Vol. 26, Oxford University Press, Oxford. | 2019-11-19 03:02:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 325, "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": 0.47642388939857483, "perplexity": 13867.555450843822}, "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-2019-47/segments/1573496669967.80/warc/CC-MAIN-20191119015704-20191119043704-00155.warc.gz"} |
https://web.stanford.edu/~peastman/statmech/friction.html | 7. Friction and Fluctuations¶
In 1827, Robert Brown was looking through a microscope at pollen grains in water. He observed that the smallest particles were in constant motion, jiggling around in a seemingly random way. This came to be known as Brownian motion. For many years, the cause of it remained an unsolved mystery.
Finally in 1905, Albert Einstein published a paper with the catchy title, “On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat,” which showed that Brownian motion could be explained by the random collisions of water molecules with the grains as they underwent thermal motion. This paper was important for two reasons. First, it served as the final conclusive proof for the existence of atoms. It is remarkable to realize that as recently as 1905, the very existence of atoms was a controversial subject. Most chemists had long since accepted the reality of atoms, but some physicists continued to deny that matter was in any way discrete.
Second, it helped to form the foundation for the study of friction and equilibrium fluctuations based on statistical mechanics. The goal of this chapter is to learn how to describe Brownian motion, and to understand the deep connection between the seemingly unrelated phenomena of friction and fluctuations.
7.1. Friction and Dissipation¶
Friction is a familiar fact of life. Rub two objects together and you feel a force resisting the motion. Place an object on the floor and give it a push; it will slide a short distance and then stop. Again, there is a force resisting its motion: friction. And something else happens too: the objects moving against each other become warmer, as when you rub your hands to warm them up or when you strike a match. The same thing happens in other contexts too. For example, electrical resistance is really just another type of friction. If you run a current through a conductor, a force resists the motion of the electrons, and the conductor becomes warmer.
Friction is such a universal part of our experience, it seems like it must reflect something fundamental in the laws of nature. Yet if you look at the basic equations of classical or quantum mechanics, there is no sign of friction. A single isolated particle will continue forever moving in a straight line without slowing down. If two particles collide, they bounce away from each other with just as much energy as they started with. Friction, it appears, only happens to macroscopic objects. And there are even situations where macroscopic systems are free from it, such as superconductivity and superfluidity. What exactly is friction, and where does it come from?
The answer, of course, is statistical mechanics. We saw in section 4.1 that when a system is in equilibrium, its kinetic energy is uniformly distributed among all its degrees of freedom. A macroscopic object in motion clearly is not in equilibrium. It has one degree of freedom (the center of mass motion) with far more energy than any other. If that degree of freedom interacts with others, the system will tend to move toward equilibrium. Energy will be transfered out of that one degree of freedom (so the object will slow down) and into all the others (so the object will become warmer).
That is all friction really is: the tendency of systems to move toward equilibrium when different degrees of freedom are allowed to interact with each other.
In the process, its energy becomes less useful. The kinetic energy of a moving macroscopic object is useful energy. It can easily be used to do mechanical work. Thermal energy is less useful. You need a heat engine with a separate cold bath to do work with it, and Carnot’s theorem sets a strict upper limit on how much work it can do. This conversion of useful mechanical energy to less useful thermal energy is called dissipation. We say that friction “dissipates energy”. That is, it converts mechanical energy to thermal energy.
7.2. Linear Response Theory¶
We are now ready to take another step away from equilibrium and examine what happens to a system experiencing friction. But where do we begin? All our results so far have applied only to systems in equilibrium. Even when we studied thermodynamic processes in Chapter 5, we only considered transitions between two equilibrium states. We assumed that if we waited long enough, the system would eventually return to equilibrium, but we said nothing about how that process took place. Friction is that process. A system experiencing friction is, by definition, a system out of equilibrium.
We saw in section 4.4 that when a system is in equilibrium, all mechanical and thermodynamic forces exactly cancel out. Its macroscopic variables tend to remain fixed. The converse is also true: if a system is not in equilibrium, the mechanical and thermodynamic forces will in general not cancel out, and the system will experience a net force pushing it back toward equilibrium. Intuitively, we expect that the further the system is from equilibrium, the stronger the force will be. But how exactly does it vary? We are simply going to make an assumption:
Assumption (Linear Response)
If a macroscopic variable is displaced away from equilibrium by a distance $$\Delta x$$, the system will experience a net restoring force that is linearly proportional to the displacement: $$F = -C\Delta x$$, where $$C$$ is a constant.
This assumption is the basis of Linear Response Theory. The macroscopic variable could be the velocity of a pollen grain immersed in water, the volume of an inflated balloon, or the electric current traveling through a circuit. All that matters is that it interacts with many microscopic degrees of freedom that can act as a heat bath.
How do we justify this assumption? If the system is not in equlibrium, the net force will almost certainly tend to push it back toward equilibrium. For example, the pollen grain will collide with water molecules that slow it down. Or the balloon will be bombarded by air molecules that apply pressure to it. But there is no fundamental reason the force must be linear in the displacement.
We will justify it in two ways. First, suppose the force is not linear in the displacement. In that case, we can expand it in a power series about the equilibrium value. If $$\Delta x$$ is sufficiently small, all terms after the linear one will be negligible. So even though the assumption is not strictly correct, it is still a good approximation as long as the displacements are small enough.
The second justification is simply an experimental observation: linear response turns out to be a very good description of many important systems. Our assumption is not always true, but it is true of many situations; and our conclusions will only apply to those situations.
7.3. Fluctuations¶
Friction is caused by the interaction with a heat bath. Let us be very clear about that. A moving object slows down because it is colliding with molecules in its environment. In the process, energy is dissipated: the object’s kinetic energy is redistributed among a huge number of microscopic degrees of freedom. These two phenomena, friction and dissipation, are really exactly the same thing.
But that is not the only effect of being in contact with a heat bath. Suppose a pollen grain is initially not moving at all. In that case, collisions with water molecules will cause it to start moving. In equilibrium, after all, every degree of freedom should have $$kT/2$$ kinetic energy. If it starts with less than that, the effect of the heat bath will be to increase its kinetic energy.
So in equilibrium, the pollen grain will be in constant motion. Its speed and direction of motion will be constantly changing, as it is jostled by the surrounding water molecules. But on average, its kinetic energy will equal the value given by the equipartition theorem.
We refer to this effect as equilibrium fluctuations. Any variable in equilibrium with a heat bath will be constantly changing. The changes will appear random, but they will obey statistical rules determined by the nature of the heat bath.
These two effects, fluctuations and dissipation, are inseparable from each other. They are both caused by exactly the same mechanism: the interaction of the variable with a heat bath. That is the central message of this chapter. And because they have the same cause, they are guaranteed to obey certain relationships. All that remains is to derive what those relationships are.
7.4. The Langevin Equation¶
Let’s write Newton’s Second Law for a particle undergoing Brownian motion in water: $$F = m \ddot{x}$$. (For simplicity we will work in one dimension, but the generalization to more dimensions is trivial. Just turn $$x$$ and $$F$$ into vectors.) In this equation, $$F$$ is the force exerted on the particle by the surrounding water molecules. The details of those interactions are enormously complicated and constantly changing, so we need to look for an approximate description of it. What would be a reasonable approximation?
The first thing we might try is using the assumption of linear response, so the force at any moment is proportional to the velocity: $$m \ddot{x} = -\gamma \dot{x}$$. The parameter $$\gamma$$ is called the friction coefficient. The solution to this differential equation is given by:
(1)$\dot{x}(t) = \dot{x}(0) \cdot e^{-\frac{\gamma}{m}t}$
For a truly macroscopic system, this description might be sufficient. Whatever velocity the object starts with, it exponentially decays toward zero. But Brownian motion is not precisely a macroscopic phenomenon. It applies to particles that are much larger than a water molecule, but still small enough that you need a microscope to see them. And the whole point is that their velocity does not decay to zero. They remain in motion due to the constant collisions with water molecules. By ignoring the forces applied by those random collisions, we have simplified things too far.
We can still assume linear response, but now we will only take it as a statement about the average force on the particle. At any given instant, the force could be different. This leads to the following equation, known as the Langevin equation:
(2)$m \ddot{x} = -\gamma \dot{x} + R$
$$R$$ is a “random” force describing the rapidly fluctuating interactions between the particle and water molecules. We cannot hope to write an exact function for it, but we can still describe it statistically. We will make the following assumptions about it:
1. $$\langle R \rangle = 0$$. We already separated out the average force into its own term, so $$R$$ must have a mean of 0.
2. It is independent of $$x$$. The interaction between the particle and the water is the same no matter where in the water bath the particle is located.
3. Because it varies so rapidly and chaotically, we assume it is uncorrelated with itself except over very short time spans. More precisely, we assume there is some maximum time $$\tau$$ over which it has any correlations, so that
(3)$\begin{split}\langle R(t) R(t+\delta t) \rangle = 0 \text{ if } \delta t > \tau\end{split}$
4. We also assume the rate at which correlations decay is independent of time, so that $$\langle R(t) R(t+\delta t) \rangle$$ is independent of $$t$$. It depends only on $$\delta t$$. The overall statistical properties of the random force are the same at all times.
These assumptions have an important consequence. In most situations, we do not care about the instantaneous value of $$R$$, only its integral over some time period that is long compared to $$\tau$$. We can break up that integral into many pieces, each covering a span of length $$\tau$$:
(4)$\int_0^t R(t') dt' = \int_0^\tau R(t') dt' + \int_\tau^{2 \tau} R(t') dt' + \int_{2 \tau}^{3 \tau} R(t') dt' + \dots$
So the integral is a sum of independent terms, each drawn from the same distribution. That is exactly the sort of case we studied in Chapter 3, so we can immediately apply the central limit theorem and conclude that the integral obeys a normal distribution with mean 0 (because $$\langle R \rangle = 0$$), and whose standard deviation scales with $$\sqrt{t}$$. All that, without evaluating a single integral or knowing anything about the details of $$R(t)$$!
Before we dive into the math, let’s take a moment to look at the Langevin equation and try to understand it intuitively. There are two terms on the right side. The first one always points opposite to the velocity, so it tends to slow the particle down. If it were the only term, it would lead to exponentially decaying velocity as in equation (1). The second term prevents that from happening by constantly applying random kicks to the particle. Essentially, there is one term that removes energy and one term that adds energy. When the system is in equilibrium, the two terms will exactly balance out so the average energy remains constant. The magnitude of the random force will of course depend on temperature: the hotter the system, the faster the water molecules will be moving, and the harder they will hit the particle. We therefore expect there should be some relationship between the temperature, the friction coefficient, and the magnitude of the random force. As we will see soon, that is indeed the case.
7.5. Diffusion of a Brownian Particle¶
Now that we have the pieces in place, it is time to see what we can calculate. To start with, multiply both sides of equation (2) by $$x$$, then take the ensemble average:
(5)$m \langle x \ddot{x} \rangle = -\gamma \langle x \dot{x} \rangle + \langle x R \rangle$
Because $$R$$ is independent of $$x$$, the last term vanishes: $$\langle x R \rangle = \langle x \rangle \langle R \rangle = 0$$. To unify the other two terms, we can use the identity
(6)$\frac{d}{dt} \langle x \dot{x} \rangle = \langle x \ddot{x} \rangle + \langle \dot{x}^2 \rangle$
(This identity makes use of the fact that the operations of taking a derivative and taking an ensemble average commute with each other. After all, an ensemble average is just a weighted sum of terms that can each be differentiated independently.) Substituting this into equation (5) gives
(7)$m \left( \frac{d}{dt} \langle x \dot{x} \rangle - \langle \dot{x}^2 \rangle \right) = -\gamma \langle x \dot{x} \rangle$
Next we can use the equipartition theorem. A single particle in one dimension has one degree of freedom so, in equilibrium, its average kinetic energy is $$m \langle \dot{x}^2 \rangle/2 = kT/2$$. Therefore
(8)$m \frac{d}{dt} \langle x \dot{x} \rangle = kT -\gamma \langle x \dot{x} \rangle$
Now make a change of variables. Define
(9)$z = \frac{d}{dt} \langle x^2 \rangle = 2 \langle x \dot{x} \rangle$
so the equation becomes
(10)$\frac{dz}{dt} = \frac{2kT}{m} -\frac{\gamma}{m} z$
This is a standard differential equation whose solution is given by
(11)$z = Ce^{-\frac{\gamma}{m}t} + \frac{2kT}{\gamma}$
You can easily verify that by substituting it back into equation (10). $$C$$ is an arbitrary constant.
We are almost done! The first term in equation (11) decays rapidly with time. Taking the limit of large $$t$$, we get
(12)$z = \frac{d}{dt} \langle x^2 \rangle = \frac{2kT}{\gamma}$
or integrating with respect to time and defining $$x(0)=0$$,
(13)$\langle x^2 \rangle = \frac{2kT}{\gamma}t$
So as time passes, the mean squared distance traveled by the particle increases at a constant rate. This was one of the main conclusions of Einstein’s 1905 paper on Brownian motion, although he derived it by a somewhat different method. The approach used here was published a few years later, in 1908, by Paul Langevin.
This result is not really very surprising. In fact, it is exactly what we probably would have predicted if we had thought about it in advance. The particle moves irregularly as its velocity is continuously damped away by friction and replaced by the random force. The motion can be thought of as a series of independent displacements taken at random, one after another. By the central limit theorem, they should therefore add up to a normal distribution whose standard deviation grows as $$\sqrt{t}$$. This kind of motion is called diffusion. It also is sometimes referred to as a random walk, because of the way it is built up from many independent random displacements.
The way it scales with temperature is similarly unsurprising. If we had been forced to predict in advance, we probably would have guessed the average displacement would be proportional to the average velocity, and by the equipartition theorem that scales as $$\sqrt{T}$$. We also would probably have guessed that a larger friction coefficient would lead to a smaller average displacement.
On the other hand, there is one strange thing about equation (13). It appears to be completely independent of the random force. In fact, $$R$$ dropped out of the derivation after the very first step and never appeared again. But the random force is what causes the particle to diffuse! Surely the average displacement ought to depend on the magnitude of $$R$$?
In fact, this result does depend on $$R$$, but it does it indirectly. By using the equipartition theorem, we assumed the system was in equilibrium at temperature $$T$$. That equilibrium is brought about by the interaction between the friction force and the random force, and $$R$$ depends on $$T$$. To derive equation (13), it was sufficient to assume equilibrium had been achieved without knowing precisely how it came about. So now we will turn to that question: how do the forces interact to produce equilibrium?
7.6. The Fluctuation-Dissipation Theorem¶
Equation (2) can be viewed as a first order differential equation for the velocity $$\dot{x}$$. Its solution is given by
(14)$\dot{x}(t) = \dot{x}(0) \cdot e^{-\frac{\gamma}{m}t} + \frac{1}{m} \int_0^t e^{-\frac{\gamma}{m}(t-t')} R(t') dt'$
The first term is identical to equation (1). It represents the initial velocity decaying exponentially to zero. But this time, we also have a second term representing the effect of the random force. It can be viewed as a series of infinitesimal kicks, each of which then decays with time. The integral sums over all those kicks, weighting each one by an exponential factor based on the time $$t-t'$$ that has passed since it occurred.
We are interested in the behavior at large values of $$t$$, once the system has had plenty of time to come to equilibrium. In this limit equation (14) becomes simpler:
(15)$lim_{t \to \infty} \dot{x}(t) = \frac{1}{m} \int_0^{\infty} e^{-\frac{\gamma}{m}(t-t')} R(t') dt'$
Notice that $$t$$ still appears in the exponential. You might be tempted to replace it with $$\infty$$, but that would be incorrect. As $$t$$ increases, the upper limit of the domain of integration increases as well. The exponent only depends on the difference $$t-t'$$, and no matter how large $$t$$ gets, there is always a portion of the domain of integration for which the exponent is small.
We want to compute the average kinetic energy in equilibrium, so square each side and take an ensemble average.
(16)$lim_{t \to \infty} \langle \dot{x}^2(t) \rangle = \frac{1}{m^2} \int_0^{\infty} \int_0^{\infty} e^{-\frac{\gamma}{m}(2t-t'-t'')} \langle R(t')R(t'') \rangle dt' dt''$
In section 7.4 we assumed that $$\langle R(t')R(t'') \rangle$$ was independent of $$t'$$ and depended only on the difference $$t''-t'$$. We can therefore replace it by $$\langle R(0)R(t''-t') \rangle$$. Also make a change of variables by defining $$r=t-t''$$ and $$s=t''-t'$$. With these substitutions, the equation becomes
(17)$\begin{split}\begin{array}{rcl} lim_{t \to \infty} \langle \dot{x}^2(t) \rangle &=& \frac{1}{m^2} \int_{r=0}^{\infty} \int_{s=-\infty}^{\infty} e^{-\frac{\gamma}{m}(2r+s)} \langle R(0)R(s) \rangle dr ds \\ &=& \frac{1}{m^2} \left( \int_0^{\infty} e^{-\frac{\gamma}{m}2r} dr \right) \left( \int_{-\infty}^{\infty} e^{-\frac{\gamma}{m}s} \langle R(0)R(s) \rangle ds \right) \\ &=& \frac{1}{2 \gamma m} \int_{-\infty}^{\infty} e^{-\frac{\gamma}{m}s} \langle R(0)R(s) \rangle ds \end{array}\end{split}$
Now we can invoke the equipartition theorem once again to set $$\langle \dot{x}^2(t) \rangle = kT/m$$. Therefore,
(18)$2 \gamma kT = \int_{-\infty}^{\infty} e^{-\frac{\gamma}{m}s} \langle R(0)R(s) \rangle ds$
In section 7.4 we assumed the random force was uncorrelated with itself except over some “very short” time $$\tau$$. Let’s make that assumption more precise. From equation (14), we see that $$m/\gamma$$ sets the timescale on which friction operates. After a time $$m/\gamma$$ has passed, the initial velocity has decayed by a factor of $$e$$. We will assume that $$\tau \ll m/\gamma$$. The timescale over which the random force is correlated with itself is very small compared to the timescale over which friction operates.
This assumption means that whenever the integrand is non-zero, the exponent is negligible in magnitude. This leads to a further simplification:
(19)$2 \gamma kT = \int_{-\infty}^{\infty} \langle R(0)R(s) \rangle ds$
This very important result is called the fluctuation-dissipation theorem. It gives the relationship between $$T$$, $$\gamma$$, and $$R$$. Any system in contact with a heat bath will experience both friction and fluctuations, and the magnitudes of the two effects are directly linked to each other. You cannot get one without the other.
The right side of equation (19) is the integral of the autocorrelation function of the random force. It depends both on the magnitude of $$R$$ and on the time $$\tau$$ over which it remains correlated. The stronger the force is, the larger its effect on the particle. And the longer the time interval over which the particle accelerates in a single direction before the force changes, the faster it will get moving. | 2019-07-19 06:52:16 | {"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": 0.8255304098129272, "perplexity": 194.34364978349794}, "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-2019-30/segments/1563195526064.11/warc/CC-MAIN-20190719053856-20190719075856-00313.warc.gz"} |
https://tex.stackexchange.com/questions/201918/multiple-align-points-and-multiple-equations | # Multiple align points and multiple equations
I have 2 equations, both have to be numbered and aligned. Equations' numbers should be vertically centered. But equations are long and have to be broken into 2 or 3 lines each. It would be great to have 2 alignment points in each line. But I cannot achieve that using split. Or can I? Here is the MWE:
\begin{align}
\begin{split}
z_1^{i,j}
& = a_1 \, x_1^{i,j} + b_1 \, \delta_1^{i,j} \\
& + \sum_k y_k^{i,j} \qquad \forall ~i,j
\end{split}
\\
\begin{split}
z_2^{i,j}
& = a_2 \, x_2^{i,j} + b_2 \, \delta_2^{i,j} + c_2 \\
& + z_1^{i,j} \qquad \forall ~i,j
\end{split}
\end{align}
Basically, I want the same output as produced with this code, especially equations' numbers vertically centered. In addition, I want \forall ~i,j from both equations to be also aligned, i.e. below each other. Solution with alignat instead align would be even better for me.
I believe you're using a two column format.
The environment split only accepts one alignment point; you can use \mathmakebox from mathtools (that also loads amsmath) and calc:
\documentclass[twocolumn]{article}
\usepackage{mathtools}
\usepackage{calc}
\begin{document}
\begin{align}
\begin{split}
z_1^{i,j}
& = a_1^{} x_1^{i,j} + b_1^{} \delta_1^{i,j} \\
\end{split}
\\
\begin{split}
z_2^{i,j}
& = a_2^{} x_2^{i,j} + b_2^{} \delta_2^{i,j} + c_2^{} \\
& \qquad + \mathmakebox[\widthof{$\displaystyle\sum_k y_k^{i,j}$}][l]{z_1^{i,j}}
\end{split}
\end{align}
\end{document}
Note that \, between factors is not used, while ^{} is recommended in order to push subscripts at the same level. The + on the second line shouldn't be under the equals sign, as it belongs to the right hand side of the expression.
However, the subscripts are not really level, because of j at the exponent, so you might want a slightly more complex adjustment:
\documentclass[twocolumn]{article}
\usepackage{mathtools}
\usepackage{calc}
\begin{document}
\begin{align}
\begin{split}
z_1^{i,j}
\end{split}
\\
\begin{split}
z_2^{i,j}
& \qquad + \mathmakebox[\widthof{$\displaystyle\sum_k y_k^{i,j}$}][l]{z_1^{i,j}}
\end{split}
\end{align}
\end{document}
• Works nice. It's very elegant solution and just what I wanted. And the \adjs command is just beautiful. – MMSt Sep 19 '14 at 8:48
Well there is no problem with each of your equations in one line, and the ∀ symbols can be aligned with the alignat environment:
\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage[showframe]{geometry}
\begin{document}
\begin{alignat}{2}
z_1^{i,j} & = a_1 \, x_1^{i,j} + b_1 \, δ_1^{i,j}
+ ∑_k y_k^{i,j} & \qquad & ∀ ~i,j \\
z_2^{i,j}
& = a_2 \, x_2^{i,j} + b_2 \, δ_2^{i,j} + c_2
+ z_1^{i,j} & & ∀ ~i,j
\end{alignat}
\end{document} | 2020-11-29 14:02:51 | {"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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9791130423545837, "perplexity": 2358.596559139931}, "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-2020-50/segments/1606141198409.43/warc/CC-MAIN-20201129123729-20201129153729-00637.warc.gz"} |
https://www.plumed.org/doc-v2.0/user-doc/html/mindist.html | Calculating a minimum distance
To calculate and print the minimum distance between two groups of atoms you use the following commands
d1: DISTANCES GROUPA=1-10 GROUPB=11-20 MIN={BETA=500.}
PRINT ARG=d1.min FILE=colvar STRIDE=10
(see DISTANCES and PRINT)
In order to ensure differentiability the minimum is calculated using the following function:
$s = \frac{\beta}{ \log \sum_i \exp\left( \frac{\beta}{s_i} \right) }$
where $$\beta$$ is a user specified parameter.
This input is used rather than a separate MINDIST colvar so that the same routine and the same input style can be used to calculate minimum coordinatetion numbers (see COORDINATIONNUMBER), minimum angles (see ANGLES) and many other variables.
This new way of calculating mindist is part of plumed 2's multicolvar functionality. These special actions allow you to calculate multiple functions of a distribution of simple collective variables. As an example you can calculate the number of distances less than 1.0, the minimum distance, the number of distances more than 2.0 and the number of distances between 1.0 and 2.0 by using the following command:
DISTANCES ...
GROUPA=1-10 GROUPB=11-20
LESS_THAN={RATIONAL R_0=1.0}
MORE_THAN={RATIONAL R_0=2.0}
BETWEEN={GAUSSIAN LOWER=1.0 UPPER=2.0}
MIN={BETA=500.}
... DISTANCES
PRINT ARG=d1.less_than,d1.more_than,d1.between,d1.min FILE=colvar STRIDE=10
(see DISTANCES and PRINT)
A calculation performed this way is fast because the expensive part of the calculation - the calculation of all the distances - is only done once per step. Furthermore, it can be made faster by using the TOL keyword to discard those distance that make only a small contributions to the final values together with the NL_STRIDE keyword, which ensures that the distances that make only a small contribution to the final values aren't calculated at every step. | 2021-01-28 07:46:22 | {"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": 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": 0.5436267256736755, "perplexity": 1088.6923256837283}, "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-2021-04/segments/1610704839214.97/warc/CC-MAIN-20210128071759-20210128101759-00137.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-mathematics-for-calculus-7th-edition/chapter-1-section-1-8-inequalities-1-8-exercises-page-89/87 | ## Precalculus: Mathematics for Calculus, 7th Edition
$\left( -6.001,-5.999 \right)$
For any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$. (The symbol $\lt$ may be replaced with $\le$.) Using the concept above, the solutions to the given inequality, $\left| x+6 \right|\lt0.001 ,$ is \begin{array}{l}\require{cancel} -0.001\lt x+6\lt0.001 \\\\ -0.001-6\lt x\lt0.001-6 \\\\ -6.001\lt x\lt-5.999 .\end{array} In interval notation, the solution set is $\left( -6.001,-5.999 \right) .$ | 2018-10-16 22:30:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9995740056037903, "perplexity": 2746.019477002913}, "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/1539583510893.26/warc/CC-MAIN-20181016221847-20181017003347-00322.warc.gz"} |
https://blog.demofox.org/2020/11/25/multiple-importance-sampling-in-1d/ | # Multiple Importance Sampling in 1D
This is a follow up to an article I wrote a few years ago on Monte Carlo integration and importance sampling in 1D: https://blog.demofox.org/2018/06/12/monte-carlo-integration-explanation-in-1d/
The simple, well commented code that generated all the data for this post can be found at: https://github.com/Atrix256/mis/
A challenge when doing Monte Carlo integration in rendering is that the function you are trying to integrate is often made up of other functions multiplied together. While you may know how to importance sample some of the parts individually, you ultimately have to choose which thing to importance sample, because you are generating random numbers according to whichever thing you choose.
In rendering, the three things usually being multiplied together are lighting, material and visibility (which makes shadows). Lighting and materials are things you can usually importance sample and are based on the type of light (like a spherical area light) and the material model (Like a PBR microfacet BRDF), while visibility is not usually able to be importance sampled because it is entirely due to the geometry in a scene as to whether a pixel can see a light or not.
If you importance sample based on lighting, you can get poor results when the material ended up being more important to the result. Likewise, if you importance sample based on material, you can get poor results when the lighting ended up being more important to the result.
Multiple importance sampling is a way to make it so that you don’t have to choose, and you can get the benefits of both. More generally, it lets you combine N different importance sampling techniques.
## TL;DR
Before going into the explanation, here is how you actually get 1 MIS sample using the balance heuristic, when you have two importance sampling techniques:
F is the function being integrated. PDF1 / InverseCDF1 are for the first importance sampling technique. PDF2 / InverseCDF2 are for the second importance sampling technique. You do this in a loop N times, and take the average of those N estimates, to get your final estimate.
You can generalize to more techniques by just following the pattern. Each sampling technique generates it’s own x and y. Each sampling technique calculates the pdf for that x value for each of the other pdfs. The estimate is the sum of: each y value divided by the sum of each pdf for the corresponding x value.
Note that if part of the function F is expensive (like raytracing for visibility!) you don’t have to do that for each sample. You could get your estimate of lighting multiplied by material like in the above, and after combining them, you could then do your raytracing to multiply in the visibility term.
## MIS Explained
You can get a single sample from a monte carlo estimator by randomly generating an x value and calculating the estimate as the function value at that x, divided by the PDF value of choosing that x.
$\text{Estimate} = \frac{f(x)}{\text{PDF}(x)}$
You may also remember that as the shape of the pdf (histogram) of the random numbers gets closer to the shape of the function you are trying to integrate, that you can get a closer estimate to the actual answer with fewer samples. This is called importance sampling.
Let’s say though that you want to integrate the function f multiplied by the function g and you are able to generate random numbers in the shape of f, and random numbers in the shape of g, but not random numbers in the shape of f multiplied by g.
You know that you can choose to importance sample based on f or g, but that the choice is better or worse situationally. Sometimes you want f, other times you want g.
The simplest way to combine these would be to just use them both for each sample and average them. You could also switch off so that even numbered samples importance sampled by f and odd numbered samples importance sampled by g. This is the same as giving each technique a weighting of 0.5.
We can do better though!
We can make an x value to importance sample based on f, and another x value to importance sample based on g, and then we can calculate the PDF values of each x for each PDF.
If we have good importance sampling PDFs, higher PDF values mean higher quality samples, while lower PDF values mean lower quality samples. We now have the means to give a weighting to a sample based on it’s quality as shown below, where we calculate the weight for sample “A”. Sample “B” would do the same.
$\text{Weight}_A = \frac{\text{PDF}_A(x_A)}{\text{PDF}_A(x_A)+\text{PDF}_B(x_A)}$
This is called the “balance heuristic”. There are other heuristics that you can use instead, which you can read about in Veach’s thesis (in the links section) and other MIS papers which have come out since then.
If we have a Monte Carlo estimate sample like this:
$\frac{f(x_A)}{\text{PDF}_A(x_A)}$
Some interesting cancelation happens if we multiply that by the weight.
$\frac{f(x_A)}{\text{PDF}_A(x_A)} * \frac{\text{PDF}_A(x_A)}{\text{PDF}_A(x_A)+\text{PDF}_B(x_A)} = \frac{f(x_A)}{\text{PDF}_A(x_A)+\text{PDF}_B(x_A)}$
That form is the same form seen in the code from the last section, where we also had a sample B that we added to it to get the final estimate.
You may be wondering why sample A and sample B are added together… shouldn’t they be averaged?
Well, if you look at the denominator in that last formula, two PDFs are added together. Each PDF has an expected value of 1, so the expected value of that sum in the denominator is going to be 2. That means that the estimate is going to be half as big as it should be. When you add two of them together, they are going to be as large as they should be. All that has happened is that instead of adding them together and dividing by two to average them, we have divided them by two implicitly in advance before adding them. We are still averaging the two samples. It isn’t exactly averaging, since the PDFs will vary from sample to sample, but on the whole, it’s still an unbiased combination of the two PDFs, which is why we still get the correct answer.
If three PDFs were involved, the weighted samples would be one third the size they should be, and there would be three to add together.
If four PDFs were involved, the weighted samples would be one fourth the size they should be, and there would be four to add together.
It generalizes to any number of importance sampling techniques involved.
## One Sample MIS
If you are a fan of stochastic rendering like me, you may be wondering if you really have to do both (all) of the samples, or if you can use the weighting to choose one stochastically and end up with the correct result for less work.
Yes, you can indeed do this and in Veach’s thesis he calls this the “One-Sample Model” in section 9.2.4.
In this case, what you do is calculate the weight for each sample, and then divide each of those weights by the sum of the weights to get a probability for taking that specific sample.
After you roll a random number and choose the single sample to contribute to the estimate, you need to multiply the Monte Carlo estimate by the chance of choosing that item. You are left with something that uses multiple PDFs for importance sampling different parts of the function, but each sample evaluates the function F only once. Useful if F is costly to evaluate.
If you expand out weight1, weight2 and weight1chance, you’ll find that some things cancel out and you are left with the below for actually calculating the estimate. I have to admit I don’t have a good intuitive explanation for why that works, but the algebra says it does, and it checks out experimentally. If you have an explanation, leave a comment!
## Piecewise Importance Sampling
Multiple importance sampling is a method for combining any number of importance sampling techniques to sample a specific function.
Something interesting though is that not every PDF involved has to cover the entire function.
What i mean is that you could have a PDF which sampled only from the left half of the domain of a function, and another PDF which sampled only from the right half of the domain of a function.
What would happen is that the inverse CDF for the first technique would only generate x values on the left half of the function to integrate, and the PDF would give zero for any value on the right have of the function.
The second technique would do the opposite.
MIS would not care about this in the least. It would function as normal and let you importance sample a function piecewise, if you could make PDFs that fit the parts of a function well, but weren’t able to make a PDF that fit the entire function well.
Veach’s thesis goes into other things as well, such as being able to give different sample counts to different techniques. It’s definitely worth a read!
## Experiment #1 – Importance Sampling & Warm Up
Quick reminder, the code that made the data for these experiments is at: https://github.com/Atrix256/mis/
First up we are going to integrate the function $y=\sin(x)*\sin(x)$ from 0 to pi, doing 10,000 different tests, each test doing 5000 samples, and average the results. We are going to use regular Monte Carlo (mc) as well as importance sampled Monte Carlo (ismc), using the PDF $y=\sin(x)/2$. Below is the function we want to integrate, and the PDF that we are going to use to importance sample it.
We could show the absolute value of the error at each step (the error being averaged over all those tests) and get this. (data from out1.abse.csv)
That isn’t super easy to read other than seeing importance sampling seems to be less erratic and lower error more reliably. We can change it to be on a log/log plot which helps see decay rates better (especially when things like low discrepancy sequences are involved, which we’ll see later).
That’s an improvement, but there is a lot of noise, even after 10,000 tests. Monte Carlo is noisy by definition, so as you can see, sometimes it gets really low error, but then pops right back up in the next few samples. That erratic nature is not good and if you are doing integration per pixel, the variance will make the noise especially bad. In fact, variance is what we really care about. So long as the integration is converging to the right thing (is unbiased / has zero bias), variance will tell us how quickly it is converging on the right answer.
Here is a log/log variance graph. You can more easily see that the importance sampling is a clear win over the non importance sampled Monte Carlo Integration. (data from out1.var.csv)
Now that we see that yes, importance sampling is helpful, and we have our testing conventions worked out, let’s continue on to more interesting topics!
## Experiment #2 – Multiple Importance Sampling
Next up, we are going to integrate the function $y=\sin(x)*2x$ from 0 to pi. We are going to use regular Monte Carlo, but also importance sample using $y=\sin(x)/2$ again, and also $y=x*\frac{2}{\pi^2}$. We are also going to do multiple importance sampling using both of those PDFs in conjunction, and also do the “single sample method” of MIS. Here are the functions mentioned.
Here is the log/log variance graph.
Monte Carlo (mc, blue) is the obvious worst. Multiple importance sampling (mismc, green) is the obvious best. The second place worst is importance sampling by the line function (ismc2, yellow). The second place best is importance sampling by the sin based PDF (ismc1, red). The one sample method (mismcstoc, purple) seems to be basically the same as the red line. It takes half as many samples as mismc, so it isn’t surprising that it does worse.
It is good to see that multiple importance sampling is worth while though and does significantly better than the two importance sampling methods involved do by themselves.
## Experiment #3 – Piecewise Importance Sampling
Next we are going to do piecewise MIS. We are going to integrate $y=\sin(3*x)*\sin(3*x)*\sin(x)*\sin(x)$ using three PDFs for importance sampling where each is just $y=\sin(x)/2$ shrunken on the x axis to be 1/3 the size and shifted over so that each PDF is responsible for one third of the function domain. The first PDF for example is $y=\sin(3*x)*\frac{3}{2}$ from 0 to pi/3.
Here is the function we are integrating, showing the 3 zones the PDFs cover:
Here is the first of the PDFs. The other two look the same but are shifted over on the x axis.
Here is the variance for regular Monte Carlo versus the piecewise importance sampling, showing that it is a significant improvement to do the piecewise IS here.
## Experiment #4 – Low Discrepancy Sequences
Unsurprisingly it turns out that low discrepancy sequences are useful when doing multiple importance sampling. It would be fun to look at using LDS in MIS / IS deeper in a future blog post, especially because things change in higher dimensions, but here are some interesting results in the mean time.
Here is the first experiment, which compared Monte Carlo (mc, blue) to importance sampling (ismc, yellow), now also using low discrepancy sequences for both.
For low discrepancy Monte Carlo (mclds, orange), instead of using white noise independent random numbers 0 to 1 to make my x values, I start the x value at a random number in 0 to 1 for the first sample x value, but then I add the golden ratio to it and use modulus to keep it between 0 and 1 for each subsequent sample. This is the “Golden Ratio Additive Recurrence Low Discrepancy Sequence”. That beats both Monte Carlo, and importance sampled Monte Carlo by a significant amount.
For low discrepancy importance sampled Monte Carlo (ismclds, green), I did the same, but put that sequence through the inverse CDF to generate numbers from that PDF, using LDS as input. It’s worked well here in 1D, but mixing LDS and IS can be problematic in higher dimensions due to the LDS being distorted from the importance sampling warping, and then losing it’s low discrepancy properties.
Here is the second experiment, which compared MC to IS to MIS, now including low discrepancy sequences:
Everything improved by using LDS, but interestingly, the order of best to worst changed.
Not using LDS, multiple importance sampling was the winner. Using LDS, MIS is still the winner. Since there are two streams of random numbers needed for the MIS (one for each importance sampling technique), I used a different low discrepancy sequence for each. For the first technique, i used the golden ratio sequence. For the second technique, I did the same setup, but used the square root of two instead of the golden ratio. The golden ratio is the best choice for this kind of thing, because it is the most irrational number, but square root of two is a pretty good second choice.
Not using LDS, Monte Carlo was the worst performing, but using LDS, Monte Carlo is in the middle, and it’s the first importance sampling technique that does the worst. The second importance sampling technique is in the middle whether you use LDS or not though.
Here is the third experiment now with LDS, which compared Monte Carlo to a piecewise importance sampled function.
This MIS here needs 3 streams of random numbers, so for the LDS, I used the golden ratio sequence, the square root of 2 sequence, and a square root of 5 sequence. Once again, LDS helps convergence quite a bit in both cases!
I’m starting to run out of “known good irrational numbers” so I’m glad we are at the end of the LDS experiments. There are other type of low discrepancy sequences that don’t use irrational numbers, but then you start having to consider the LDS quality along with the results and all the permutations. If you want to go into a deep dive about irrational numbers, give this article of mine a read: https://blog.demofox.org/2020/07/26/irrational-numbers/
Before moving on, look at that last graph again. The amount of variance that 5,000 white noise samples has is the same variance that piecewise importance sampling had, when using only 10 low discrepancy samples. Without LDS though, even the MIS strategy took something like 800 samples to reach that level of variance.
In graphics, these samples could easily represent rays shot into the world for something like global illumination, soft shadows, or raytraced reflections.
It would be real easy to try the most naive Monte Carlo algorithm, find out that you need 5000 samples to converge and give up.
Facing this, you may bust out the MIS and try to do better, finding that you could cut the cost to about 1/6 of what it was, at 800 samples needed to converge. That’s still a ton of samples for real time rendering, so is still out of budget. It would be real easy to give up at this point as well.
If you take it one step further and figure out how to get a nice LDS into the MIS instead of white noise random numbers, you could find that you can decrease it even further, down to 1/80th of what MIS gave you, or 1/500th of the cost of the naive Monte Carlo.
10 samples is still quite a few if we are talking about per pixel raytracing, but that is in the realm of real time affordable.
Good sampling matters, and can help you do some pretty amazing things.
## Experiment #5 – Blue Noise
Where low discrepancy sequences are deterministic number sequences that give you good coverage over a sampling domain, blue noise is randomized (non deterministic) number sequences that do the same.
There is some nuance to LDS vs blue noise, and when one or the other should be used. The summary is that regular blue noise converges at the same rate as white noise (there are variants like projective blue noise which do better at convergence) but that it starts with a lower error. Blue noise also has better noise perceptually, which is also more easily filtered (it is high frequency noise only, instead of full spectrum noise). So, the rule in graphics is basically that if you can converge with LDS, do that, else use blue noise to hide the error. Blue noise also does better at keeping it’s desirable properties when put through transformation functions, such as importance sampling.
Unfortunately, blue noise is pretty expensive to calculate, especially with the algorithm I’m using for it, so the sample and testing counts are going to be decreased for these tests to 100 tests, using 500 samples each. Blue noise is best for low sample counts anyways, so decreasing the sample count makes for a more appropriate comparison.
Here is the first experiment, which compared MC to ISMC. Now it has blue noise results, to go along with the LDS results.
The result shows that blue noise does better than white noise, but not as good as LDS.
Here is the second experiment, comparing MC to MIS, now with blue noise. You can see how again the blue noise quality is between white and LDS as far as variance is concerned.
Here is the third experiment, showing the effectiveness of the piecewise importance sampling, using MIS. Once again, blue noise has variance between white noise and LDS. | 2021-01-20 03:46:16 | {"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": 12, "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": 0.7020804286003113, "perplexity": 551.9833472199866}, "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-2021-04/segments/1610703519883.54/warc/CC-MAIN-20210120023125-20210120053125-00631.warc.gz"} |
https://ell.stackexchange.com/questions/174171/how-to-describe-various-kinds-of-average | How to describe various kinds of average?
The particular problem that I am having is I don't know how to describe the "average" in the following two situations:
1) I have a parameter that is recorded on a certain day every month from 2000 to 2010. I average the data for the same month over 2000-2010.
2) I have another parameter that is recorded every day from 2000 to 2010. I obtain the monthly average for each year and then I average the monthly average over 2000-2010.
Is it correct to call the calculated values of the two cases above as follows?
1) Average of monthly data between 2000-2010
2) Monthly average of daily data between 2000-2010
All ideas are welcome. Much appreciated if you could share me the guidelines or books to teach how to call different kinds of average.
• What is the purpose of averaging the averages in case 2? – JeremyC Jul 26 '18 at 7:53
• @JeremyC It is common to get a more reliable monthly variation by averaging over several years. But it is something about statistics, nth to do with English. – user2720402 Jul 26 '18 at 8:35
The "types of average" are the median, mean, mode, geometric mean or harmonic mean (and possibly others).
There is no simple phrase for the processes that you go through. The best way to handle this is to describe in detail, then refer in short.
The temperature was recorded on the 20th of each month. I calculated the mean of the temperatures for the period 2010-2018. The average temperature was 287K
To obtain an average pressure, monthly averages were obtained from [source]. I calculated the mean of monthly average pressure over the period 2010-2018. The average pressure was 10027 kPa.
The idea is that you can't squash a complex process into a short expression without being ambiguous, but once you have described the process for calculating the average, you can just refer to "the average (parameter)"
• I agree with this but I was puzzled by your use of 'median' which I did not see in the OP's question. – JeremyC Jul 26 '18 at 7:52
• mistype. will fix. I meant to say "mean" since it is a good idea to specify which average is being used. – James K Jul 26 '18 at 7:53
• I surely don't know that "types of average" include median, mean, mode, etc. I thought "average" is equivalent to "mean". Learned something new. Thanks. – user2720402 Jul 26 '18 at 8:39 | 2019-10-16 15:10:18 | {"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": 0.8033877611160278, "perplexity": 514.7360948515066}, "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/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00218.warc.gz"} |
http://math.stackexchange.com/questions?page=4952&sort=newest | # All Questions
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I have some complex curves, I separate the long curves to smallest one to be able to fit them with Bezier curve. However, my Bezier curve has no C1 continuously, if I force C1 continuously, my curves ... | 2015-08-30 06:21:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9349183440208435, "perplexity": 533.4436804567054}, "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-2015-35/segments/1440644064919.24/warc/CC-MAIN-20150827025424-00265-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://plainmath.net/algebra-ii/79902-logarithm-rules-for-complex-numbers-are | Sonia Gay
2022-06-24
Logarithm rules for complex numbers
Are the logarithm rules true for complex numbers?
We know that for positive real numbers $a$, $b$, $c$ and real number $d$ that:
${\mathrm{log}}_{b}\left({a}^{d}\right)=d{\mathrm{log}}_{b}\left(a\right)$
${\mathrm{log}}_{b}\left(a\right)=\frac{{\mathrm{log}}_{c}\left(a\right)}{{\mathrm{log}}_{c}\left(b\right)}$
${\mathrm{log}}_{b}\left(xy\right)={\mathrm{log}}_{b}\left(x\right)+{\mathrm{log}}_{b}\left(y\right)$
${\mathrm{log}}_{b}\left(\frac{x}{y}\right)={\mathrm{log}}_{b}\left(x\right)-{\mathrm{log}}_{b}\left(y\right)$
We also know that
${\mathrm{log}}_{b}\left({b}^{d}\right)=d$
Does this extend to complex numbers as $a$, $b$, $c$ or $d$?
My instinct is that
${\mathrm{log}}_{b}\left({b}^{s+t\mathrm{i}}\right)=s+t\mathrm{i}$
In other words, I'm pretty confident that the last formula works when $d$ is a complex number
Anika Stevenson
Expert
You have to be careful because logs and non-integer powers are multivalued functions. The definition is that ${a}^{d}=\mathrm{exp}\left(d\mathrm{ln}\left(a\right)\right)$ (for any branch of $\mathrm{ln}$). Now ${\mathrm{log}}_{b}\left({a}^{d}\right)$ is any $z$ such that ${b}^{z}={a}^{d}$, i.e. $\mathrm{exp}\left(z\mathrm{ln}\left(b\right)\right)=\mathrm{exp}\left(d\mathrm{ln}\left(a\right)\right)$, and that is equivalent to $z\mathrm{ln}\left(b\right)-d\mathrm{ln}\left(a\right)=2\pi in$ for some integer $n$. So the result is
${\mathrm{log}}_{b}\left({a}^{d}\right)=\frac{d\mathrm{ln}\left(a\right)+2\pi in}{\mathrm{ln}\left(b\right)}$
Similarly,
${\mathrm{log}}_{b}\left(a\right)=\frac{\mathrm{ln}\left(a\right)+2\pi im}{\mathrm{ln}\left(b\right)}$
for some integer $m$. And thus (assuming you use the same values of $\mathrm{ln}\left(a\right)$ and $\mathrm{ln}\left(b\right)$ in both cases)
${\mathrm{log}}_{b}\left({a}^{d}\right)-d{\mathrm{log}}_{b}\left(a\right)=2\pi i\frac{n-md}{\mathrm{ln}\left(b\right)}$
For example, take $a=b=e$ and $d=2\pi i$, and use the principal branch of $\mathrm{ln}$
${\mathrm{log}}_{e}\left({e}^{2\pi i}\right)=\mathrm{ln}\left(1\right)=0$ but $2\pi i{\mathrm{log}}_{e}\left(e\right)=2\pi i$
Another interesting example is $a=b=-1$, $d=3$. Now $\left(-1{\right)}^{3}=-1$, but there is no way to have ${\mathrm{log}}_{-1}\left(-1\right)={\mathrm{log}}_{-1}\left(\left(-1{\right)}^{3}\right)=3{\mathrm{log}}_{-1}\left(-1\right)$ (this would imply ${\mathrm{log}}_{-1}\left(-1\right)=0$, which is certainly false).
Do you have a similar question? | 2023-02-01 02:06:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 65, "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": 0.9830561280250549, "perplexity": 146.0109415650033}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499899.9/warc/CC-MAIN-20230201013650-20230201043650-00242.warc.gz"} |
https://solvedlib.com/n/lestion-helpbioitling-plant-flls-2-ounce-cans-of-soda-by,19164764 | Lestion Helpbioitling plant flls {2-ounce cans of soda by &n automated distribution tiowever not all cans will contain the flenguerocess
Question:
Lestion Help bioitling plant flls {2-ounce cans of soda by &n automated distribution tiowever not all cans will contain the flenguerocess thatcan be adrusted t0 any mean Tll volume and that the standard devlation in fill wil sune volume due varletion in the fing process, Historica cans according to 3 normal 035 ounce , Operatlons managers at the piant know IhgE records shov fha put in the can; customers ejaidless they what the ean are short changed pul too mmuch soda in set and Ihe State Departmerit of} Weights can; compan mone; 'Ifjoo. and Measures may fine tc company: Complete parts litdle Suppose the induslry andards fo and b below: at 12 ounces what is the probability that fill volume call for each 12-ounce can will contain oraductChAt in between 21,97and 12,03 ounces Assuriing that vollme af pradluct thal falls the desirod thc ange manager sets the mejn 6ll The probability {Rouna to four dechna places a5 needed ) Aseume thatthe manager is focused on an upcoming audit by the that If it containg les than Departmeni 96 ounces the company will be re of Weights leasuires Sne knows the process Is t8 happening at what level should she set the mean fIHl ievel? Comment amenaodedend pitceatials Gned Assuming Whiarthe manager ants select one can atrandoin and woek on the ro ifica ilone ofthis step sssuming Incl the mos % chance ofthls comipatiy fills tons nf thousands 0f cans each Set tHe mean fill; Evol 91 ounces {Round T0 two decimal placos aS neecod ) Since this mean fill evel Is than 12 ounces
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Assets Duration Market Value Rate Cash 0% 40,000 Bonds 1.8 160,000 400,000 7.50% Commercial Loans 13 % Liabilities and Equity Small Time Deposit Large CD's Transactions Accour 2 100,000 3.10% 1.5 120,000 5.20% 1 2.40% 320,000 Equity 60,000 38. Calculate the Bank Duration Gap 39. Calculate the Ch... | 2022-12-10 02:25:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "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": 0.40455615520477295, "perplexity": 8271.909700891687}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711637.64/warc/CC-MAIN-20221210005738-20221210035738-00614.warc.gz"} |
http://www.yaklass.ru/p/english-language/59-klass/vocabulary-18548/school-subjects-classmates-18625/re-d1af0d65-914d-4997-9b63-acfdfede0a12 | ### Условие задания:
4,5 Б.
Listen to the text "School".
Write the words into each gap:
1. In fact, school is one of the of a person’s life.
2. Oh, I nearly forgot, you also get to see and five days a week.
3. If I had another chance, I’d try my best in .
Источники:
www.pinterest.com | 2018-08-20 15:24:47 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8837000727653503, "perplexity": 8671.181505060611}, "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/1534221216475.75/warc/CC-MAIN-20180820140847-20180820160847-00232.warc.gz"} |
https://studyadda.com/question-bank/critical-thinking_q22/1497/114222 | • # question_answer Calcium cyanide on treatment with steam under pressure gives $N{{H}_{3}}$ and [DPMT 2002] A) $CaHC{{O}_{3}}$ B) $CaO$ C) $Ca{{(OH)}_{2}}$ D) $CaC{{O}_{3}}$
$CaC{{N}_{2}}+3{{H}_{2}}O\underset{\text{pressure}}{\mathop{\xrightarrow{\text{under}}}}\,CaC{{O}_{3}}+2N{{H}_{3}}$. | 2019-12-14 19:29:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937163352966309, "perplexity": 4377.3267540051565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541288287.53/warc/CC-MAIN-20191214174719-20191214202719-00398.warc.gz"} |
https://mail.haskell.org/pipermail/haskell-cafe/2009-February/055058.html | Yitzchak Gale gale at sefer.org
Thu Feb 5 05:52:11 EST 2009
Manlio Perillo wrote:
>> I'm looking for an exact integer division that avoids overflows, if
>> possible.
Richard O'Keefe wrote:
> What this sounds like to me is a request that the Prelude
> function 'fromRational' should work well...
> If you cannot divide two Integers n, d accurately using
> (fromRational (n Ratio.% d) :: Double)
> that casts doubt on the trustworthiness of floating point literals.
No, that works fine:
Prelude Data.Ratio> fromRational \$ (3*10^1000)%(2*10^1000+1)
1.5
> Suppose we have a function
> decodeIntegerAsFloat :: RealFloat a => Integer -> (Integer,a)
> such that if (s,m) = decodeIntegerAsFloat x
> then either x = 0 and s = 0 and m = 0
> or x = m * 2**s (mathematically) and abs m \in [0.5,1.0).
Yes, that is what Manlio wants. Sometimes you need to divide
two very large Integers with a floating point number as result,
without the overhead of constructing a Rational from them.
> Then
> integer_ratio_as_float :: Floating a => Integer -> Integer -> a
> integer_ratio_as_float p q = (mp/mq)*(2.0^(sp-sq))
> where (sp,mp) = decodeIntegerAsFloat p
> (sq,mq) = decodeIntegerAsFloat q
>
> You'd actually use scaleFloat; if the difference sp-sq is outside
> the range of Int the answer is going to be a signed zero or a signed
> infinity anyway.
You have just duplicated the CPython interpreter source code
> decodeIntegerAsFloat would sit very well in the
> RealFloat class alongside its model, decodeFloat. It has other uses.
I agree, though I'm not sure it needs to be a method.
> For example, you can use it to compute logarithms of Integers with
> much less worry about overflow.
Actually, efficient integer-valued logarithms for Integers are exactly
what you need to implement decodeIntegerAsFloat.
Best would be if we could just read that off from the internal
representation of an Integer. Would you like to file a GHC
issue for that?
In the meantime, we could already expose the integer log function
in a library using an efficient algorithm.
Thanks,
Yitz | 2017-08-23 14:17:56 | {"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": 0.5231851935386658, "perplexity": 5645.719832393395}, "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-2017-34/segments/1502886120573.0/warc/CC-MAIN-20170823132736-20170823152736-00118.warc.gz"} |
https://www.nature.com/articles/s41467-019-11073-4?error=cookies_not_supported&code=e0a16e82-6459-4abf-a1d9-a8c53e5e4341 | # Ion buffering and interface charge enable high performance electronics with organic electrochemical transistors
## Abstract
Organic electrochemical transistors rely on ionic-electronic volumetric interaction to provide a seamless interface between biology and electronics with outstanding signal amplification. Despite their huge potential, further progress is limited owing to the lack of understanding of the device fundamentals. Here, we investigate organic electrochemical transistors in a wide range of experimental conditions by combining electrical analyses and device modeling. We show that the measurements can be quantitatively explained by nanoscale ionic-electronic charge interaction, giving rise to ion buffering and interface charge compensation. The investigation systematically explains and unifies a wide range of experiments, providing the rationale for the development of high-performance electronics. Unipolar inverters — universal building blocks for electronics — with gain larger than 100 are demonstrated. This is the highest gain ever reported, enabling the design of devices and circuits with enhanced performance and opening opportunities for the next-generation integrated bioelectronics and neuromorphic computing.
## Introduction
Organic electrochemical transistors (OECTs) are iontronic devices where ions penetrate the semiconductor and dramatically modulate the electrical properties of the transistor channel. Owing to this bulk ionic-electronic interaction, OECTs provide a seamless interface between biology and electronics combining the benefits typical of organic material technologies—such as large-area deposition with simple and low-cost techniques, chemically-tunable properties, mechanical flexibility, softness, and biological compatibility1,2,3,4,5—with high signal amplification, ultra-low voltage operation, and stability in aqueous environment6,7,8. Fueled by this unique benefits combination, OECTs are gaining significant interest in numerous bioelectronic applications, including neural interfacing, electrophysiology, cell monitoring, enhanced ionic and biological sensing, neuromorphic devices, and neuron stimulation9,10,11,12,13,14,15,16,17. Despite the huge potential, the lack of understanding of the fundamental processes governing the device operation hinder further progress in the rational design of engineered and optimized devices for new and improved applications.
The operation of OECTs has been described for the first time by Bernards and Malliaras18 in 2007. Their pioneering work depicted OECTs as the combination of an electronic circuit that accounts for electronic transport in the organic semiconductor, and an ionic circuit that accounts for ionic transport in the electrolyte. Importantly, the model captured the key characteristic of OECTs, i.e., the volumetric response due to the ion penetration into the transistor channel. This point has been recently investigated by several experimental works19,20,21,22, showing that ions uptake from an electrolyte into a polymeric film results in a purely volumetric capacitance19. The linear dependence of the capacitance on the volume of the polymeric channel and the zero offset led to the conclusion that the ionic charges are uniformly distributed in the polymer and no significant ion accumulation at the polymer/electrolyte interface takes place23. On one hand, studies24 based on the modeling of cyclic voltammograms of the prototypical conducting polymer poly(3,4-ethylenedioxythiophene) doped with the polyelectrolyte poly(styrene sulfonate) (PEDOT:PSS) suggest that the capacitance originates from an electrical double layer (EDL) at the interface between the PEDOT phase and the PSS phase. In this direction, Tybrandt and coworkers25 show a modified drift diffusion model for the description of PEDOT:PSS based OECTs. On the other hand, several reports show that in conductive polymers the characteristic shape of the supercapacitive voltammograms is due to pseudo-capacitive processes involving faradaic reactions26,27,28,29. These findings are further corroborated by specific studies on OECTs, where the dedoping process is commonly explained as a faradaic reaction30,31,32,33. The redox model is usually invoked to explain the ion concentration dependent OECT transfer characteristics31 but a limited range of ion concentrations has been assessed. In this complex scenario, a comprehensive experimental and theoretical analysis identifying the key physical effects and providing a consistent picture of the OECT operation is highly desirable.
Here we investigate the ionic-electronic interaction in OECTs accounting for a very wide range of ion concentrations, channel thicknesses, and polymer charge densities. OECTs are systematically analyzed by combining electrochemical impedance spectroscopy, current-voltage characteristics, and device modeling. First, the analysis considers the widely studied PEDOT:PSS OECTs, enabling us to compare our results with the state of the art. Then, to prove the generality of our results, we modify the polymer formulation by changing the amount of fixed charges in the polyelectrolyte by about two orders of magnitude. Finally, the analysis is extended to a widely used accumulation mode material, namely poly(3-hexylthiophene-2,5-diyl) (P3HT). We show that the measurements can be quantitatively explained by an electrostatic bulk uptake of ions that compensate both fixed and mobile electronic charges. The key OECT characteristics can be quantitatively explained by ion buffering and interface charge compensation. The analysis unifies a wide range of experiments, explaining the ion concentration-dependent threshold voltage and the ion concentration-independent volumetric capacitance in OECTs. On the basis of this understanding, we demonstrate OECT unipolar inverters, ubiquitous building blocks for electronics, with gain and noise margin up to 107 and 82% of the theoretical limit, respectively. These are the best performances ever reported for the fundamental figures of merit of inverters and, importantly, are achieved basing on a device-aware circuit design approach.
## Results
### Device structure and electrical characteristics
A schematic representation of the OECT structure is depicted in Fig. 1a. A thin film of PEDOT:PSS is deposited by inkjet printing on a plastic polyethylene foil. Gold is used for source and drain electrodes. The transistor channel width and length are W = 1000 μm and L = 500 μm, respectively. The printing technique allows to easily vary the channel thickness (t) by printing multiple stacked polymer layers, ranging from 230 to 2300 nm. It is worth to note that the channel thickness is a key design parameter because it enables the tuning of the transistor capacitance and, in turn, of the transconductance19. As electrolyte, we used an aqueous solution of sodium-chloride (NaCl) at concentrations ranging from c = 1 10−3 M to c = 5 M. A PDMS well is used to spatially confine the electrolyte. The wide range of ion concentrations and polymer thicknesses here investigated is extremely relevant for OECT-based biological applications, sensing, and circuits19,34,35,36. The gate electrode is an Ag/AgCl pellet immersed into the electrolyte. Further details on the OECTs fabrication are provided in the Methods section.
Typical transfer and output characteristics of the fabricated devices are shown in Fig. 1b, c, respectively. By applying a positive gate voltage (VG) cations drift into the polymer, reduce the hole concentration and lower the drain current (ID). Analogously, when a negative VG is applied previously injected cations drift out of the polymer while anions drift into the polymer, the hole concentration increases and this results in a larger ID. As shown in Fig. 1d the maximum transconductance normalized to the OECT geometries and drain voltage is larger than 60 S cm−1 V−1 at VG = 0 V and VD = −0.4 V, in agreement with state-of-the-art OECTs6. The extremely large transconductance, resulting from the ionic-electronic interaction through the bulk of the polymeric channel, is a hallmark of OECTs.
### Analysis of the main OECT parameters
To investigate the ionic-electronic interaction in OECTs, we measured the transfer characteristics (IDVG) by varying the ion concentration c. Figure 1e, f shows that the IDVG characteristics systematically shift to more negative voltages with increasing ion concentrations. To gain insight on the physical mechanisms underlying the device operation, we reproduced the electrical characteristics of the OECTs with the drain current model proposed by Bernards and Malliaras18, that in the case of linear operation reads18,37:
$$I_{\mathrm{D}} = \Gamma \left[ {\left( {V_{\mathrm{T}} - V_{\mathrm{G}}} \right)V_{\mathrm{D}} + \frac{{V_{\mathrm{D}}^2}}{2}} \right]$$
(1)
where
$$\Gamma = \frac{{Wt}}{L}\mu C_{\mathrm{v}}$$
(2)
The drain current depends on the geometrical and physical device parameters, namely polymer width (W), length (L), thickness (t), hole mobility (μ), volumetric capacitance (Cv), and threshold voltage (VT), which is defined as:
$$V_{\mathrm{T}} = V_{\mathrm{P}} - V_{{\mathrm{SH}}}$$
(3)
where VP=q p0 Cv−1 is the pinch-off voltage, q is the elementary charge, p0 is the intrinsic doping of the semiconductor, and VSH accounts for the voltage shift as a function of the ion concentration c. It is worth noting that VSH can be attributed to both the gate/electrolyte and electrolyte/semiconductor interfaces.
According to Eq. (1), the drain current depends on the applied voltages (VG and VD) and on the device parameters Γ and VT. Γ and VT can be obtained by fitting the transfer characteristics in the linear regime. The modeling is systematically performed on OECTs with various channel thicknesses and the transfer characteristics are reproduced. Supplementary Figure 1 shows the modeling of the ID-VG as a function of c. It is worth to note that we focus on a narrow range of the transfer characteristics since this enables a reliable extraction of the model parameters and ensures excellent stability of the OECT characteristics38,39 (Supplementary Fig. 2). The slope of the linear least square approximation of the ID–VG provides Γ. Figure 2a shows Γ normalized to the channel geometries (viz. Γn = Γ L W−1 t−1) as a function of c. For each c the mean value and standard deviation of Γn are calculated by modeling OECTs with several thicknesses. We found that Γn is independent of c and, according to Eq. (2), it follows that Cv is independent of c. The analysis is further corroborated by reproducing the OECT transfer characteristics with the drain current model recently proposed by Friedlein and co-workers40 (Supplementary Note 1), which accounts for a non-uniform mobility in OECTs and provides superior fitting performance of the saturation region of operation (Supplementary Fig. 2a). Supplementary Figure 2d shows Γn extracted by fitting the OECT characteristics with the Friedlein model, confirming that Cv is independent of c.
A c-independent Cv is crucially important because in the case of an EDL a capacitance affected by the counterion concentration is expected41,42. To investigate this point, we perform electrochemical impedance spectroscopy (EIS) measurements by systematically varying the counterion concentration. The OECT impedance spectra as a function of c are shown in Supplementary Fig. 3. In order to obtain Cv, for each c and t we modeled the measured impedance as a function of the frequency with the Randles equivalent circuit that, in the case of an OECT, is composed of a resistor Rs in series with the parallel of a resistor Rp and a capacitor C 19. In this model, Rs is the electrolyte resistance, Rp depends on the reactions at the working electrode, and C accounts for the ion accumulation at the working electrode43. The OECT volumetric capacitance is calculated as Cv=C v−1 where v=W L t is the total volume of the OECT channel. Cv and Rs as a function of c are shown in Supplementary Fig. 4. In all cases we found that Cv is independent of the ion concentration and results Cv = 44 ± 2 F cm−3. This value is in agreement with the state-of-the-art19,25.
To further investigate the origin of the ion-dependent drain current displayed in Fig. 1e, f, we analyzed VT as a function of c. The mean value and standard deviation are extracted from OECTs with several thicknesses at each ion concentration. Figure 2b shows that VT decreases with increasing ion concentration. According to the previous analysis, the pinch-off voltage is independent of the ion concentration (VP = q p0 Cv−1) provided that p0 is independent of c. To this aim, we monitored the polymer conductivity as a function c by measuring the drain current of an OECT when the gate electrode is not immersed into the electrolyte. We found that ID is independent of c (Supplementary Fig. 5) and therefore the variation of VT as a function of c can be ascribed to VSH. Figure 3a (symbols) shows the extracted VSH as a function of c. VSH accounts for the voltage drop at both the gate/electrolyte (VG/E) and the electrolyte/polymer (VE/P) interfaces, and reads:
$$V_{{\mathrm{SH}}} = V_{{\mathrm{G}}/{\mathrm{E}}} + V_{{\mathrm{E}}/{\mathrm{P}}}$$
(4)
Focusing on the gate-electrolyte interface, it is worth to note that we used an Ag/AgCl pellet as gate electrode, which is a non-polarizable quasi-reference electrode. According to the Nernst equation, the potential drop at the gate-electrolyte interface results44:
$$V_{{\mathrm{G}}/{\mathrm{E}}} = \frac{{k_{\mathrm{B}}T}}{q}\log c$$
(5)
where kB is the Boltzmann constant and T is the temperature. Figure 3a (full line) shows the voltage drop at the gate/electrolyte interface calculated with Eq. (5). At large ion concentrations VG/E (full line) is close to VSH (symbols), while at small ion concentrations VG/E is significantly larger than VSH. This indicates that the contribution of the electrolyte/polymer interface cannot be neglected and it can be calculated with Eq. (4), viz. VE/P=VSH − VG/E.
### The role of the bulk charge interaction
Figure 3b (symbols) shows the voltage drop at the electrolyte/polymer interface (VE/P) as a function of c. Supplementary Figure 6 shows that VE/P extracted with the Friedlein et al. model fully agrees. VE/P is large and negative (VE/P ≈ −0.2 V) at the minimum ion concentration c = 10−3 M, it increases with a Nernstian slope of 59 mV dec−1 up to c = 5 10−1 M, and at larger concentrations VE/P is close to zero. This can be explained as follows. In OECTs, ions dissolved in the electrolyte can flow into the bulk of the polymeric channel. In the case of PEDOT:PSS OECTs, the channel is made of a polymer blending composed of a PSS-rich matrix hosting nanometric sized PEDOT-rich grains20,24,25, as schematically depicted in Fig. 4. The electronic (hole) conduction takes place in the PEDOT (blue islands in Fig. 4) while ionic conduction is provided by the PSS polyelectrolyte (light-gray regions in Fig. 4). Indeed, when in contact with an electrolyte, the PSS matrix swells and allows the hydrated ions to penetrate the polymeric blend20,45. The polyelectrolyte/electrolyte interface acts as a semi-permeable membrane, which can be penetrated by the mobile ions provided by the electrolyte solution (red spheres in Fig. 4) but cannot be crossed by the SO3 fixed anions of the PSS (blue cubes in Fig. 4). As a consequence, the fixed charges in the bulk of the polyelectrolyte (Nfix) are electrostatically compensated by cations provided by the electrolyte. More in detail, at low electrolyte concentrations (c « Nfix) cations drift into the polymer and compensate the fixed negative charges. At the equilibrium, the cation concentration in the polymer (cPSS+) is equal to Nfix independently of c and a negative potential rises at the polyelectrolyte/electrolyte interface (VE/P < 0 V). On the other hand, when c » Nfix, the fixed charges in the polyelectrolyte are negligible, cPSS+ ≈ c and VE/P ≈ 0 V.
This behavior resembles the Donnan equilibrium in semi-permeable membranes and the potential at the polyelectrolyte/electrolyte interface can be calculated as (see Supplementary Note 2)25,46:
$$V_{{\mathrm{E}}/{\mathrm{P}}} = - \frac{{k_{\mathrm{B}}T}}{q}{\mathrm{asinh}}\left( {\frac{{z_{{\mathrm{fix}}}N_{{\mathrm{fix}}}}}{{2zc}}} \right)$$
(6)
where z and zfix are the number of charges of the ions dissolved into the electrolyte and fixed charge into the polyelectrolyte, respectively. Interestingly the model predicts that VE/P shows a Nernstian behavior when c « Nfix, while VE/P ≈ 0 when c » Nfix. Figure 3b (line) shows that VE/P calculated with Eq. (6) accurately reproduces the measurements in the whole range of ion concentrations. By reproducing the measurements with the model we estimated a SO3 concentration Nfix = 3.16 M (viz. about 1.9 1021 cm−3). The wide range of ion concentrations here assessed proves that the Nernst equation fails to describe VE/P and hence the faradaic redox reactions can be disregarded. In addition, Supplementary Fig. 7 shows that Eq. (6) is able to explain the threshold voltage shift measured in OECTs exploiting electrolytes with different chemical nature, such as KCl, Ca(NO3)2, and Al2(SO4)3. This provides further insight on the strong dependence of the transistor performance on the chemical nature of the counterion used, as reported in refs. 47,48.
Interestingly, Eqs. (4)–(6) show that the actual gating voltage (viz. VG + VSH) depends on the electrolyte concentration (VSH = VG/E + VE/P). Combining this model with the approach proposed by Friedlein et al.40, the normalized hole concentration (p/p0) as a function of c results:
$$\frac{p}{{p_0}} = \frac{{V_{\mathrm{P}} - V_{\mathrm{G}} + V_{{\mathrm{ch}}}}}{{V_{\mathrm{P}}}} - \frac{{k_{\mathrm{B}}T}}{{qV_{\mathrm{P}}}}\left[ {\log \left( c \right) - {\mathrm{asinh}}\left( {\frac{{z_{{\mathrm{fix}}}N_{{\mathrm{fix}}}}}{{2zc}}} \right)} \right]$$
(7)
where Vch is the potential along the polymeric channel. Figure 5a shows p as a function of c and VG. p/p0 linearly decreases with the logarithm of c with a slope equal to 147 10−3 dec−1 in the low range of c (10−3 M < c < 1 M) where c affects both VE/P and VG/E, and 74 10−3 dec−1 in the high range of c (1 M < c < 5 M) where VE/P vanishes and only VG/E depends on c. Figure 5a also shows the dependence on VGVch. In the whole range of c, p/p0 linearly decreases by increasing VG since Cv is constant.
Figure 5b shows the mobile cation and anion concentrations in the PSS bulk (cPSS+ and cPSS-, respectively) as a function of the electrolyte ion concentration c. For the sake of generality, cPSS+, cPSS-, and c are normalized with respect to Nfix. When c / Nfix « 1, cPSS+ is buffered to a constant value cPSS+ = Nfix, while cPSS quadratically increases with c. In this case a concentration-dependent potential rises across the electrolyte/polymer interface. On the other hand, when c / Nfix » 1, the effect of the electrostatic interaction between the fixed charges and the mobile ions is negligible. It follows that the concentration of both anions and cations in the PSS phase is equal to c and a vanishing potential at the interface is obtained.
As a further confirmation, we fabricate OECTs including a positively charged polyelectrolyte in the channel. Poly-L-lysine (PLL) is added to the PEDOT:PSS dispersion by varying the ratio between PLL and PEDOT:PSS (details are provided in the Methods section). The electrical analyses (EIS and transfer characteristics) of the PEDOT:PSS:PLL OECTs are performed and the device parameters are extracted. Figure 6a shows the electrolyte/polymer potential VE/P as a function of c when the PEDOT:PSS/PLL ratio is equal to 1/3, 5/1 and 1/0 (no PLL) v/v. The minimum c where VE/P ≈ 0 V systematically lowers with increasing PLL concentration. In all cases the model (Eq. (6)) accurately reproduces the measurements in the whole range of ion concentrations and we found that the net negative fixed charge concentration in the PEDOT:PSS:PLL amounts to Nfix(5/1)= 1.8 1020 cm−3 (0.3 M) and Nfix(1/3)= 4 1019 cm−3 (66 10−3 M) when the PEDOT:PSS/PLL ratio is equal to 5/1 and 1/3 v/v, respectively. These results show that the potential at the electrolyte/polymer interface can be controlled by tuning the fixed charge concentration in the polymer. To investigate the generality of the model we extend the analysis to other materials, namely P3HT and Crystalized PEDOT:PSS (Crys-P)49, which are widely used for the fabrication of high performance OECTs21,47,50. Figure 6a shows that in the case of P3HT OECTs VE/P is equal to 0 V in the whole concentration range explored, resulting in NfixP3HT < 1017 cm−3 (10−4 M). This small fixed charge concentration is expected in the case of undoped P3HT51,52,53. Moreover, Fig. 6a shows that in the case of Crys-P the minimum c where VE/P vanishes is shifted to lower c with respect to pristine PEDOT:PSS and we found NfixCrys−P = 6 1019 cm−3. This provides quantitative evidence that in Crys-P the excess PSS (viz. bulk PSS) is chemically removed by sulfuric acid treatment. Interestingly, we found that NfixCrys−P is almost equal to that obtained in PEDOT:PSS:PLL when the PEDOT:PSS/PLL ratio is 1/3 v/v, suggesting that in the case of PEDOT:PSS/PLL = 1/3 the bulk fixed charge is completely compensated.
### The role of the interface charge interaction
To further assess the effect of the polyelectrolyte charges on the OECTs characteristics, we fabricate PEDOT:PSS:PLL OECTs with PEDOT:PSS/PLL ratio equal to 5/1, 1/3, 1/5, 1/10 and 1/50. Supplementary Figure 8a shows that at small PLL contents (viz. PEDOT:PSS/PLL larger than 1/3 v/v) the PEDOT is well dispersed and no visible polymer separation is obtained. By increasing the PLL content a clear separation of PEDOT from the dispersion shows up. This is confirmed by Supplementary Fig. 8b where, after the syringe withdrawal of the dispersion, no significant PEDOT residue is obtained at PEDOT:PSS/PLL ratios equal to 5/1 and 1/3 v/v, while PEDOT aggregates can be clearly seen at larger PLL concentrations. The formation of undispersed PEDOT is not surprising, considering that PEDOT is a water-insoluble polymer and the negative charges of the PSS are used as charge-balancing dopants to yield a water-soluble PEDOT:PSS complex54. It follows that the compensation of the negative fixed charges of the PSS located at the PEDOT/PSS interface by the positive charges of the PLL yields water-insoluble PEDOT aggregates. To quantitatively assess the PEDOT concentration in PEDOT:PSS:PLL OECTs we electro-optically investigate the composition of the deposited films. Figure 7a shows the experimental setup. We measure the optical transmittance at 620 nm55 and VG ranging from 0 V to 0.5 V of devices with various amount of PLL. Figure 7b shows the images acquired in the case of an OECT with PEDOT:PSS/PLL = 5/1 and Fig. 7c shows the corresponding histogram of the red channel intensity at various VG. By applying the Beer-Lambert law, the measured optical intensity provides the ratio between the PEDOT concentration in PEDOT:PSS:PLL and pristine PEDOT:PSS. Figure 6b shows the PEDOT concentration in the PEDOT:PSS:PLL films as a function of the PEDOT:PSS/PLL ratio. When PEDOT:PSS/PLL ≥ 1/3 the PEDOT concentration into the PEDOT:PSS:PLL film is independent of the PLL content and equals to that of pristine PEDOT:PSS (without PLL), whereas further increasing the PLL content results in a reduction of the PEDOT content. Importantly, the electro-optical measurements show that the PEDOT doping state is not affected by PLL (Supplementary Note 3, Supplementary Fig. 9). We can conclude that small contents of PLL (PEDOT:PSS/PLL ≥ 1/3 v/v) result in a compensation of the fixed charges in the bulk of the polyelectrolyte with negligible effect on the fixed charges at the PEDOT/PSS interface. Besides, large amounts of PLL (PEDOT:PSS/PLL ≤ 1/5 v/v) result in a separation of PEDOT from the dispersion due to the compensation of the interfacial charges.
Taking advantage of this charge compensation effect, we investigate the volumetric capacitance as a function of the polyelectrolyte charge by performing EIS on PEDOT:PSS:PLL OECTs with various PEDOT:PSS/PLL ratios. Figure 6b shows that Cv of OECTs with small PLL contents (PEDOT:PSS/PLL > 1/5 v/v) is almost equal to the one of pristine PEDOT:PSS OECTs, Cv slightly decreases for devices with PEDOT:PSS/PLL ratio equal to 1/5, while it dramatically decreases at larger PLL contents. In the light of the previous analysis, small PLL contents affect only the bulk of the polymer without affecting the PEDOT:PSS interface. Therefore, we can conclude that Cv depends on the amount of PEDOT in the deposited film demonstrating that Cv is related to an electrostatic interaction at the semiconductor/(poly)electrolyte (PEDOT/PSS) interface. Figure 6c shows that the measured Cv is independent of the ion concentration. This behavior can be explained by considering the large concentration of fixed charge (about 1 M) at the PEDOT/PSS interface, which yields a negligible diffuse layer42.
### OECT based electronics
The understanding and quantification of the key mechanisms in OECTs, provide the rationale for the development of high-performance electronics. Here, as a relevant example, we demonstrate unipolar inverters. The inverter is a fundamental building block of any electronic circuit enabling the development of OECT-based integrated electronics and bioelectronics. Inverters are used both in digital circuits as logic gates and in analogue circuits as voltage amplifiers. Figure 8a shows the circuit schematic, which comprises two p-type OECTs connected in series. The input voltage VI is applied to the gate of the Driver OECT while the Load OECT is operated as a zero-VGS topology56. The OECT model (Supplementary Note 4, Supplementary Fig. 10) is implemented in a circuit simulator in order to predict the circuit operation.
The ion concentration of the Load OECT is exploited as a design parameter to maximize the key circuit figures of merit, namely the gain and noise margin, while the Driver ion concentration is 5 M. Figure 8b shows that the simulated (dashed lines) and measured (full lines) transfer characteristics are in agreement in the whole range of voltages and ion concentrations. When the input is low (e.g., VI = GND) the source-gate voltage VSG applied to the Driver OECT is large (VSG = VDD − VI) and, as a result, the output voltage VOUT is close to VDD. By increasing VI, the pull-up becomes progressively weaker and a sharp transition of VO from VDD to GND is displayed when VI reaches the trip-point voltage VTP. Further increasing VI (VI > VTP) reduces VSG and VO is close to GND. Figure 8b shows that the transfer characteristics are strongly affected by the ion concentration. More in detail, when c increases from 5 10−3 to 3 M VTP shifts from VI = 0.3 V to VI = 0.54 V and the gain increases from 12 up to 107 (Fig. 8c), with a maximum noise margin equal to 0.33 V, i.e., 82% of the theoretical limit. This can be explained as follows. By increasing c, ion buffering yields a more negative threshold voltage of the Load transistor while, due to the interface charge, its capacitance is unaffected. As a result, VTP increases since a larger VI is required to switch VO from the high to the low state. In addition, the reduced drain current of the Load transistor yields a larger gain, which is inversely proportional to the current flowing through the inverter. It is worth to note that for the sake of simplicity we varied only the Load electrolyte concentration, but similar considerations hold in the case that both the Load and the Driver concentrations are taken as design variables.
The proposed approach is compared with several inverters based on electrolyte-gated technologies in Table 1. The gain and noise margin of the designed inverter outperform those obtained by state-of-art electrolyte-gated inverters based on metal-oxides, carbon nanotubes, two-dimensional and organic materials33,57,58,59,60,61,62,63,64,65,66.
## Discussion
In summary, we provide experimental evidence of the electrostatic nature of the ionic-electronic interaction in OECTs. We perform both electrical ID-VG characteristics and electrochemical impedance spectroscopies on OECTs by exploring a wide range of channel thicknesses, polymer formulations and electrolyte ion concentrations. We show that to understand the OECT operation it is crucial to consider the bulk and interface electrostatic charge compensation taking place in the transistor channel. More in detail, the fixed charges in the bulk of the polyelectrolyte are electrostatically compensated by the mobile ions provided by the electrolyte. This results in an ion buffering in the polymeric channel and, in turn, in a concentration-dependent voltage drop at the polyelectrolyte/electrolyte interface. The fixed charges close to the semiconductor are electrostatically compensated by the electronic charges accumulated at the semiconductor/polyelectrolyte interface. This bulk uptake of ions results in a capacitance distributed in the volume of the polymer and independent of the electrolyte ion concentration. The ion concentration dependent transfer characteristics are quantitatively explained. The analysis is extended to OECTs with different polymers, proving its generality. We show that a large set of measurements in a wide range of experimental conditions can be consistently explained without invoking faradaic reactions. The impact of our analysis on the design of OECT based circuits is demonstrated by the design of unipolar inverters with tunable performances, showing the highest gain and noise margin ever reported for this class of circuits.
Prospectively, our study opens opportunities for several applications and research directions. Our work provides both fundamental knowledge and experimental demonstration of OECTs with tailored ionic response. This finds application for future development of integrated bioelectronics67, electronic circuits, ion sensors16,17 and neuromorphic circuits. For example, in neuromorphic applications a wide range of ion concentration can be exploited as a global regulation parameter to emulate the homeoplasticity phenomena of neural environments12,68. Our work enables the design of neuromorphic circuits with synapse-specific response, typical of biological neural networks69. Finally, the demonstration of an ion concentration-independent volumetric capacitance is crucial for in vivo applications, as the ionic concentration of the body fluid may vary from person to person and between the different body fluids34, thus requiring OECTs with a transconductance independent of the ionic environment. Our findings provide important design rules for the rational design of devices with enhanced performance and application-specific functionalities, opening opportunities for the next generation OECT-based electronic, bioelectronics and neuromorphic computing.
## Methods
### PEDOT:PSS and PEDOT:PSS:PLL OECT fabrication
100 nm thick Au source/drain contacts are sputtered through a shadow mask on a polyethylene substrate. An adhesion layer of 15 nm of Ti-W is previously deposited to enhance Au adhesion. Polyethylene substrates are cleaned with DI water, dried with N2 and treated with oxygen plasma to promote PEDOT:PSS adhesion. An Epson XP-215 is used for the printing process, filling a cartridge with either PEDOT:PSS or PEDOT:PSS:PLL. PEDOT:PSS (PH-500 from Heraeus Clevios GmbH) is mixed with 5 vol% ethylene glycol, 1 vol% 3-glycidoxypropyl-trimethoxysilane, 0.25 vol% dodecyl benzene sulfonic acid as conductivity enhancer, crosslinker and wetting agent, respectively. PEDOT:PSS:PLL is obtained by adding to PEDOT:PSS a PLL solution (0.1 % w/v in H2O) at various volume ratios. All the reagents were purchased from Sigma-Aldrich. The dispersion is filtered through a syringe filter (regenerated cellulose, pore size 0.2 µm) prior to deposition. The tuning of the film thickness is obtained by printing multiple stacked layers. A minimum of 4 layers is necessary to obtain a continuous conductive film. The thickness of the polymeric film obtained by varying the number of printed layers is measured with a stylus profilometer Bruker Dektak XT. A soft bake (1 min at 50 °C) is performed between the printing of each layer to avoid the spread of the printed polymer. The films are baked at 100 °C for 1 h and immersed in deionized water to remove any excess low molecular weight compounds.
### Crys-P OECT fabrication
We fabricate Crys-P OECTs following the fabrication process reported in ref. 50. In brief, PEDOT:PSS (PH-500 from Heraeus Clevios GmbH) is filtered (regenerated cellulose, pore size 0.2 µm) and spin coated on quartz-coated substrates with sputtered gold source and drain electrodes (W = 1000 µm, L = 100 µm). The deposited films are annealed at 120 °C for 15 min and then immersed in a bath of concentrated H2SO4 (>95%, Sigma Aldrich) for 15 min, thoroughly rinsed with deionized water, and dried at 120 °C for 15 min.
### P3HT OECT fabrication
We fabricate P3HT OECTs following the fabrication process reported in ref. 21. Briefly, 25 mg ml−1 of regioregular P3HT (Ossila M105) are dissolved overnight at 80 °C in dichlorobenzene. The solution is filtered through a 0.45 µm PTFE filter and spin coated (3 s at 500 r.p.m., 60 s at 1000 r.p.m., 10 s at 5000 r.p.m. from 45 °C solution) on Si-SiO2 substrates with gold interdigitated source and drain electrodes (W = 1500 µm, L = 5 µm).
### Electrical characterization
Transfer and output characteristics are measured with a Keithley 2636 A SourceMeter Unit. A solution of NaCl in DI water at several concentrations is used as electrolyte and an Ag/AgCl pellet (Warner Instruments) is used as gate electrode. Transfer characteristics are measured by sweeping the gate voltage, with the source electrode grounded and the drain electrode polarized at a constant potential. Output characteristics are measured by sweeping the drain voltage, with the source electrode grounded and the gate electrode polarized at a constant potential. The electrical characteristics of the devices are measured with a constant sweep rate of 18 mV/s, which allows the device to reach the steady state operation regime. Electrochemical Impedance Spectroscopy measurements are performed using a National Instruments PXI-1042 system, equipped with a PXI-5112 oscilloscope, a PXI-5421 arbitrary wave-function generator and custom Labview software for the control of the system. A three-electrode configuration is adopted. The drain and source electrodes are shortened together and the OECT channel acted as the working electrode. Two Ag/AgCl pellets serve as reference and counter electrodes. The inverter characteristics are measured with a Keithley 2636 A Source Meter Unit. The input voltage is swept from 0 V to 0.8 V with a voltage step of 0.3 mV and at supply voltage of 0.8 V.
### Electrical and optical characterization
The OECT is placed on an opaque sample holder with the channel in correspondence of a slit. A droplet of 10−1 M NaCl solution is placed on top of the transistor and an Ag/AgCl gate is dipped into the solution. The source and drain electrodes are shortened and VD = VS = 0 V, while the gate voltage is biased in the range 0 to 0.5 V. The light from a red light emitting diode (Superlight, LED emission peak at wavelength 620 nm) placed below the sample holder is conveyed at the transistor channel through the holder slit and the transmitted light is captured by a camera (Dino-Lite AM4023CT, raw acquisition) on the opposite side of the sample. The measurements are performed in a dark room. The 2D transmitted intensity map was acquired on a scale from 0 to 255 taking care that no pixel saturated at 0 or 255.
## Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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## Acknowledgements
The authors acknowledge the financial support of the European Commission for the project SiMBiT (Horizon 2020 ICT, contract number 824946).
## Author information
P.R. and F.T. conceived the idea and designed the experiments. P.R. fabricated and measured the devices and circuits. P.R., M.G., and F.T. developed the physical analysis. Z.M.K.-V. supervised the project. P.R. and F.T. wrote the paper. All the authors revised and commented on the paper.
Correspondence to Fabrizio Torricelli.
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• ### Ultimately Sensitive Organic Bioelectronic Transistor Sensors by Materials and Device Structures' Design
• Rosaria Anna Picca
• , Kyriaki Manoli
• , Eleonora Macchia
• , Lucia Sarcina
• , Cinzia Di Franco
• , Nicola Cioffi
• , Davide Blasi
• , Ronald Österbacka
• , Fabrizio Torricelli
• , Gaetano Scamarcio
• & Luisa Torsi
Advanced Functional Materials (2019) | 2019-10-22 06:59:25 | {"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": 0.6101955771446228, "perplexity": 4757.231719319611}, "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-2019-43/segments/1570987803441.95/warc/CC-MAIN-20191022053647-20191022081147-00487.warc.gz"} |
https://mathematica.stackexchange.com/questions/32367/define-operator-algebra | # Define operator algebra
My objective is to work out commutators like this $$[f(x,y)\partial_x^2+g(x,y)\partial_x+h(x,y),a(x,y)\partial_x^2+b(x,y)\partial_x+c(x,y)]$$ or $$[f(x,y)\partial_x^2+g(x,y)\partial_x+h(x,y),a(x,y)\partial_y^2+b(x,y)\partial_y+c(x,y)]$$ by starting from simpler commutator identities, for example, using $$[\partial_x,f(x,z)]=f_x(x,z)$$ it is easy to derive $$[\partial^2_x,f(x,z)]=2f_x(x,z)\partial_x+f_{xx}(x,z)$$ and so on, until I get to the first two commutators that I want. But how can I do that with Mathematica?
My difficulty is that I don't know how to teach Mathematica to use identities like the third equation to derive the fourth and up to the first two.
• – Jens
Sep 15, 2013 at 20:07
Define the 2 operators whose commutator you wish to derive. I have ignored the dependence on $y$ to simplify the expressions without losing any essential content.
op1 = (f[x] Dt[#, {x, 2}] + g[x] Dt[#, x] + h[x] #) &;
op2 = (a[x] Dt[#, {x, 2}] + b[x] Dt[#, x] + c[x] #) &;
Apply the commutator to a test function $u(x)$, and then separately collect the $u(x), u^\prime (x), u^{\prime\prime}(x), u^{\prime\prime\prime}(x)$ terms.
op1[op2[u[x]]] - op2[op1[u[x]]] // Collect[#, {u[x], u'[x], u''[x], u'''[x]}] &
(* the result is a bit too long to quote here *)
You can then read off the operator-valued version of the commutator by simply removing the test function $u(x)$ from this result.
Alternatively, a fully operator-valued derivation of the commutator could follow the lines I outline below, which was a quick bit of interactive operator manipulation that I concocted.
Define the 2 operators, using op[...] as a container to hold an "operator product", which allows us to control the order in which the various operators (and c-numbers) occur.
o1 = op[f, dx, dx] + op[g, dx] + op[h];
o2 = op[a, dx, dx] + op[b, dx] + op[c];
Form the commutator.
op[o1, o2] - op[o2, o1]
Distribute the op over Plus.
% //. op[u_ + v_, w_] :> op[u, w] + op[v, w]
% //. op[w_, u_ + v_] :> op[w, u] + op[w, v]
Flatten the nested op.
% /. op[op[u__], op[v__]] :> op[u, v]
Move the dx operators to the right.
% //. op[u___, dx, v_?(# =!= dx &), w___] :> op[u, v, dx, w] + op[u, d[v, x], w]
Move the c-numbers outside the op.
% //. op[u_?(# =!= dx &), v___] :> u op[v]
Tidy up the notation.
% /. d[d[u_, v_], v_] :> d[u, {v, 2}]
% /. op[] -> 1
Collect terms by derivative.
% // Collect[#, op[__]] &
This gives the same result as the quick derivation earlier.
• I believe the Distribute and Flatten steps can be simplified by using Flat and Distribute -- may I edit your answer? If I get it wrong you can of course revert the edit. Also _?(# =!= dx &) could be replaced with Except[dx] I believe. Sep 15, 2013 at 20:49
• I see what you mean about your suggestions. Though which approach is to be preferred? - (1) explicit but inefficient patterns/replacements, or (2) core functions which are more efficient. I usually use (1) when I knock code together quickly, so I can be sure about what is going on, and then later I refine this to (2) — so my long-term “library code” looks more like (2) than (1). It would be instructive to mention both approaches, so I welcome your edits/additions. Sep 16, 2013 at 0:28
• I like your argument and I don't think I should replace your code. I also don't want to post a second answer merely changing a couple of things. I'm not sure how I should add things; does a section at the bottom illustrating these alternatives seem appropriate to you? Sep 16, 2013 at 1:03
• Because you intend to show how a subset of patterns/replacements can be finessed by using the core functions appropriately, a separate answer would be appropriate, and it would give you more space for your material. Actually, to properly discuss the (1) versus (2) issue (see my previous comment) would warrant a whole separate question/answer. Sep 16, 2013 at 10:56 | 2022-08-07 22:10:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.36690327525138855, "perplexity": 2896.8853346627534}, "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/1659882570730.59/warc/CC-MAIN-20220807211157-20220808001157-00266.warc.gz"} |
https://www.usingenglish.com/forum/threads/choose-7.124210/#post-614041 | # choose?7
Status
Not open for further replies.
##### Member
I had the instruction that I .......to leave the door unlocked when I came home.
were
am
be
was
#### corum
##### Banned
I had the instruction that I .......to leave the door unlocked when I came home.
were
am
be
was
I had the instruction that I am to leave the door unlocked when I come home. :tick:
I had the instruction that I was to leave the door unlocked when I came home. :tick: | 2022-01-29 01:39:14 | {"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": 0.8423366546630859, "perplexity": 9666.353280992606}, "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-2022-05/segments/1642320299894.32/warc/CC-MAIN-20220129002459-20220129032459-00431.warc.gz"} |
https://www.asvabtestbank.com/arithmetic-reasoning/practice-test/541284/5 | ASVAB Arithmetic Reasoning Practice Test 541284
Questions 5 Topics Greatest Common Factor, Percentages, Practice, Rational Numbers
Study Guide
Greatest Common Factor
The greatest common factor (GCF) is the greatest factor that divides two integers.
Percentages
Percentages are ratios of an amount compared to 100. The percent change of an old to new value is equal to 100% x $${ new - old \over old }$$.
Practice
Many of the arithmetic reasoning problems on the ASVAB will be in the form of word problems that will test not only the concepts in this study guide but those in Math Knowledge as well. Practice these word problems to get comfortable with translating the text into math equations and then solving those equations.
Rational Numbers
A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form $${a \over b}$$ where a is the numerator and b is the denominator. An improper fraction ($${5 \over 3}$$) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number ($$1 {2 \over 3}$$) which has a whole number part and a fractional part. | 2020-06-05 06:06:59 | {"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": 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": 0.6274111270904541, "perplexity": 577.2173935476184}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348493151.92/warc/CC-MAIN-20200605045722-20200605075722-00281.warc.gz"} |
https://physics.stackexchange.com/questions/375442/thickness-of-thin-lens | # Thickness of thin lens [closed]
We have a double convex lens with optical power $D = 10$, $f\# = 2$, $n = 1.5$, and $r_1 = -r_2$. What is the minimum thickness of this lens?
I have a solution which is a bit convoluted, so I wonder if I am missing anything that would make it simpler.
From $D$, we have $f = 0.1$.
From $f\#$ and $f$, we have the diameter of the lens equaling $0.05$.
With LME, and $f$ and $n$, we have radius of curvature equaling $0.1$.
The only way I can think of getting thickness is from circular segment.
And solving for the sagitta, because we have the chord length (diameter of lens) and the radius of curvature. And the answer would be two times the sagitta.
This is my solution. I don't think we learned this in class. Is there a simpler way?
• I asked a friend how he solved it and he did the same thing. Go figure. I guess this way was correct – David Dec 21 '17 at 8:25 | 2020-07-15 11:48:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9240676164627075, "perplexity": 247.74865576191476}, "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-2020-29/segments/1593657167808.91/warc/CC-MAIN-20200715101742-20200715131742-00177.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/85296-eigenvalue-proof.html | 2. If $Ax = \lambda x$ then $A^2x = A(Ax) = A(\lambda x) = \lambda^2x$. So if $A^2 = A$ and $x\ne0$ it follows that $\lambda^2 = \lambda$. | 2013-05-21 17:31:01 | {"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": 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": 0.993848979473114, "perplexity": 15.344114563571662}, "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/1368700264179/warc/CC-MAIN-20130516103104-00033-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathhelpforum.com/trigonometry/11152-determine-angle-theta-all-distances-movie-theater.html | # Math Help - determine angle theta for all distances in a movie theater.
1. ## determine angle theta for all distances in a movie theater.
In a movie theater the angle theta at which a viewer sees the picture on the screen depends on the distance x of the viewer from the screen. For a movei theater with the dimensions shown in the figure, determine the angle theta (in degrees) for views sitting at distances of 30, 45, 60, 75, and 90 ft. from the screen.
attached a pic, http://img78.imageshack.us/img78/1507/untitledtv3.jpg
tried to draw it as accurate as possible. Can't remember all the geometric rules.
2. Originally Posted by rcmango
In a movie theater the angle theta at which a viewer sees the picture on the screen depends on the distance x of the viewer from the screen. For a movei theater with the dimensions shown in the figure, determine the angle theta (in degrees) for views sitting at distances of 30, 45, 60, 75, and 90 ft. from the screen.
attached a pic, http://img78.imageshack.us/img78/1507/untitledtv3.jpg
tried to draw it as accurate as possible. Can't remember all the geometric rules.
See the attachment
Evaluation of the calculations specified in the attachment
is given below:
Code:
>
>x=[30,45,60,75,90];
>t1=atan((6-x*tan(8*pi/180))/x);
>t=atan((30-x*tan(8*pi/180))/x)-t1;
>t*180/pi ..convert to degrees
>
Column 1 to 4:
37.275 28.163 22.093 18.0097
Column 5 to 5:
15.1373
>
>
RonL
3. excellent, it looks like you used matlab too, i'm studying this for a script file, I really appreciate the math explanations.
thankyou! | 2015-08-28 00:42:08 | {"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": 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": 0.36910149455070496, "perplexity": 1175.6793758772098}, "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-2015-35/segments/1440644060103.8/warc/CC-MAIN-20150827025420-00272-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://gamedev.stackexchange.com/questions/175241/what-is-the-most-efficient-way-to-trigger-an-action-based-on-proximity | # What is the most efficient way to trigger an action based on proximity?
I am developing a 3rd person platformer that entails a grapple hook system.
This grapple hook system relies on a series of 'grapple-able' points.
I would like these points to shine every so often.
These points will shine when:
• Within the player's view cone AND
• Within a certain distance of the player.
This Grapple object class will contain the method Shine() to trigger the effect.
Initial research
My initial research was based upon using a cone trace from the player for these Grapple objects, then in each one found, calling the Shine method.
I could then tweak the cone's angle and height to better represent the view cone.
Problems
Unreal by default does not have a cone trace, and I'm not sure if using a multibox trace followed by an angle check is the most efficient solution.
• It sounds like what you really want is a frustum check, not a proximity check or cone check. – DMGregory Sep 5 at 11:33
• @DMGregory that's exactly what I want. Those are more mathematical to do as well. Please excuse my lack of expertise, I haven't programmed in 3 months. Are there preferred ways to do this in Unreal? – Natalo77 Sep 5 at 11:49
• Have you observed any measurable performance impact from the way you're doing it now? Maybe it's fine as-is, and your time is better spent on other parts of your game. – DMGregory Sep 5 at 11:51
• @DMGregory Well, I haven't developed a way to do it yet, and I'd rather go about it the best way. Am i ivory towering too soon? – Natalo77 Sep 5 at 11:57
• Ah, it sounded like you'd tried a box first, then narrowing it with an angle. The thing is that game developers don't necessarily know or care what the "best" way is — we don't typically do proofs of optimality as part of our process. All we know is "solution X was good enough for our needs on game Y". When something isn't"good enough", we profile to identify where the problem is and try to fix that problem. I'd say these checks are unlikely to be what determines your game's performance, so I'd lean toward the simplest way you can try first, then measure to see if there's a problem to solve. – DMGregory Sep 5 at 12:00 | 2019-10-19 07:45:56 | {"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": 0.40258514881134033, "perplexity": 1290.1318701809384}, "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/1570986692126.27/warc/CC-MAIN-20191019063516-20191019091016-00316.warc.gz"} |
https://pypi.org/project/scrapy-magicfields/ | Scrapy middleware to add extra "magic" fields to items
## Project description
This is a Scrapy spider middleware to add extra fields to items, based on the configuration settings MAGIC_FIELDS and MAGIC_FIELDS_OVERRIDE.
## Installation
Install scrapy-magicfields using pip:
$pip install scrapy-magicfields ## Configuration 1. Add MagicFieldsMiddleware by including it in SPIDER_MIDDLEWARES in your settings.py file: SPIDER_MIDDLEWARES = { 'scrapy_magicfields.MagicFieldsMiddleware': 100, } Here, priority 100 is just an example. Set its value depending on other middlewares you may have enabled already. 2. Enable the middleware using MAGIC_FIELDS (and optionally MAGIC_FIELDS_OVERRIDE) in your setting.py. ## Usage Both settings MAGIC_FIELDS and MAGIC_FIELDS_OVERRIDE are dicts: • the keys are the destination field names, • their value is a string which accepts magic variables, — identified by a starting$ (dollar sign), which will be substituted by a corresponding value at runtime.
Some magic variables also accept arguments, and are specified after the magic name, using a : (column) as separator.
You can set project-global magics with MAGIC_FIELDS, and tune them for a specific spider using MAGIC_FIELDS_OVERRIDE.
In case there is more than one argument, they must come separated by , (comma sign). So the generic magic format is:
$<magic name>[:arg1,arg2,...] ### Supported magic variables$time
the UTC timestamp at which the item was scraped, in format '%Y-%m-%d %H:%M:%S'.
$unixtime the unixtime (number of seconds since the Epoch, i.e. time.time()) at which the item was scraped.$isotime
the UTC timestamp at which the item was scraped, with format '%Y-%m-%dT%H:%M:%S".
$spider must be followed by an argument, which is the name of an attribute of the spider (like an argument passed to it).$env
the value of an environment variable. It acccepts as argument the name of the variable.
$jobid the job id (shortcut for$env:SCRAPY_JOB)
$jobtime the UTC timestamp at which the job started, in format '%Y-%m-%d %H:%M:%S'.$response
$response:url The url from where the item was extracted from.$response:status
Response http status.
$response:headers Response http headers.$setting
Access the given Scrapy setting. It accepts one argument: the name of the setting.
$field Allows to copy the value of one field to another Its argument is the source field. Effects are unpredicable if you use as source a field that is filled using magic fields. ### Examples The following configuration will add two fields to each scraped item: • 'timestamp', which will be filled with the string 'item scraped at <scraped timestamp>', • and 'spider', which will contain the spider name MAGIC_FIELDS = { "timestamp": "item scraped at$time",
"spider": "$spider:name" } The following configuration will copy the url to the field sku: MAGIC_FIELDS = { "sku": "$field:url"
}
Magics also accept a regular expression argument which allows to extract and assign only part of the value generated by the magic. You have to specify it using the r'' notation.
Let’s pretend that the urls of your items look like 'http://www.example.com/product.html?item_no=345' and you want to assign to the sku field only the item number.
The following example, similar to the previous one but with a second regular expression argument, will do the task:
MAGIC_FIELDS = {
"sku": "\$field:url,r'item_no=(\d+)'"
}
## Project details
### Source Distribution
scrapy-magicfields-1.1.0.tar.gz (3.9 kB view hashes)
Uploaded source
### Built Distribution
scrapy_magicfields-1.1.0-py2.py3-none-any.whl (3.9 kB view hashes)
Uploaded py2 py3 | 2022-10-01 06:16:10 | {"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": 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": 0.2871731221675873, "perplexity": 7482.176351274122}, "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-2022-40/segments/1664030335530.56/warc/CC-MAIN-20221001035148-20221001065148-00081.warc.gz"} |
https://astarmathsandphysics.com/university-maths-notes/complex-analysis/1878-the-open-mapping-theorem.html?tmpl=component&print=1&page= | ## The Open Mapping Theorem
The Open Mapping Theorem states:
Letbe a function analytic and non – constant on a regionand letbe an open subset ofThenis open. Furthermore, ifis a region, thenis also a region.
The non – constant condition is required because ifis constant thenis a singleton set which is certainly not open. Many functions do not map open sets to open sets. The functionmapsto the first quadrantwhich is not an open set.
The Open Mapping Theorem provides a quick way of proving that there are no analytic functions which map certain regions to certain other regions. There is no analytic function which mapsto
Letbe a non – constant entire function which mapstothen by the open mapping theorem,is open, butand sosince no open disc with centrelies in
Sinceno such functionexists.
It is easily proved that ifis analytic and non – constant on a regionthenis also a region, sinceis a connected open set. The Open Mapping Theorem shows thatis open, and sinceis analytic onis also connected sois a region. | 2018-06-21 10:08:38 | {"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": 0.9277838468551636, "perplexity": 2101.1242953680485}, "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-2018-26/segments/1529267864139.22/warc/CC-MAIN-20180621094633-20180621114633-00302.warc.gz"} |
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Physics , 2012, Abstract: The stability of two-dimensional diverging and converging flows in an annulus between two permeable cylinders is examined. The basic flow is irrotational and has both the radial and azimuthal components. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. The instability is inviscid and oscillatory and persists if the viscosity of the fluid is taken into consideration.
D Cébron Physics , 2015, DOI: 10.1088/0169-5983/47/2/025504 Abstract: Precession driven flows are found in any rotating container filled with liquid, when the rotation axis itself rotates about a secondary axis that is fixed in an inertial frame of reference. Because of its relevance for planetary fluid layers, many works consider spheroidal containers, where the uniform vorticity component of the bulk flow is reliably given by the well-known equations obtained by Busse in 1968. So far however, no analytical result on the solutions is available. Moreover, the cases where multiple flows can coexist have not been investigated in details since their discovery by Noir et al. (2003). In this work, we aim at deriving analytical results on the solutions, aiming in particular at, first estimating the ranges of parameters where multiple solutions exist, and second studying quantitatively their stability. Using the models recently proposed by Noir \& C{\'e}bron (2013), which are more generic in the inviscid limit than the equations of Busse, we analytically describe these solutions, their conditions of existence, and their stability in a systematic manner. We then successfully compare these analytical results with the theory of Busse (1968). Dynamical model equations are finally proposed to investigate the stability of the solutions, which allows to describe the bifurcation of the unstable flow solution. We also report for the first time the possibility that time-dependent multiple flows can coexist in precessing triaxial ellipsoids. Numerical integrations of the algebraic and differential equations have been efficiently performed with the dedicated script FLIPPER (supplementary material).
Physics , 2001, DOI: 10.1088/0305-4470/35/18/306 Abstract: In this paper we study thermoconvective instabilities appearing in a fluid within a cylindrical annulus heated laterally. As soon as a horizontal temperature gradient is applied a convective state appears. As the temperature gradient reaches a critical value a stationary or oscillatory bifurcation may take place. The problem is modelled with a novel method which extends the one described in (numerico). The Navier Stokes equations are solved in the primitive variable formulation, with appropriate boundary conditions for pressure. This is a low order formulation which in cylindrical coordinates introduces lower order singularities. The problem is discretized with a Chebyshev collocation method easily implemented and its convergence has been checked. The results obtained are not only in very good agreement with those obtained in experiments, but also provide a deeper insight into important physical parameters developing the instability, which has not been reported before.
Physics , 2013, DOI: 10.1063/1.4711398 Abstract: Zonal flows are often found in rotating convective systems. Not only are these jet-flows driven by the convection, they can also have a profound effect on the nature of the convection. In this work the cylindrical annulus geometry is exploited in order to perform nonlinear simulations seeking to produce strong zonal flows and multiple jets. The parameter regime is extended to Prandtl numbers that are not unity. Multiple jets are found to be spaced according to a Rhines scaling based on the zonal flow speed, not the convective velocity speed. Under certain conditions the nonlinear convection appears in quasi-periodic bursts. A mean field stability analysis is performed around a basic state containing both the zonal flow and the mean temperature gradient found from the nonlinear simulations. The convective growth rates are found to fluctuate with both of these mean quantities suggesting that both are necessary in order for the bursting phenomenon to occur.
Physics , 2015, Abstract: The transition to turbulence in a precessing cylindrical vessel is experimentally investigated. Our measurements are performed for a { nearly-resonant} configuration with an initially laminar flow dominated by an inertial mode with azimuthal wave number $m=1$ superimposed on a solid body rotation. By increasing the precession ratio, we observe a transition from the laminar to a non-linear regime, which then breakdowns to turbulence for larger precession ratio. Our measurements show that the transition to turbulence is subcritical, with a discontinuity of the wall-pressure and the power consumption at the threshold $\epsilon_{LT}$. The turbulence is self-sustained below this threshold, describing a bifurcation diagram with a hysteresis. In this range of the control parameters, the turbulent flows can suddenly collapse after a finite duration, leading to a definitive relaminarization of the flow. The average lifetime $\langle \tau \rangle$ of the turbulence increases rapidly when $\epsilon$ tends to $\epsilon_{LT}$.
Nonlinear Processes in Geophysics (NPG) , 2005, Abstract: Experiments of baroclinic waves in a rotating, baroclinic annulus of fluid are presented for two gap widths. The apparatus is a differentially heated cylindrical gap, rotated around its vertical axis of symmetry, cooled from within, with a free surface, and filled with de-ionised water as working fluid. The surface flow was observed with visualisation technique while thermographic measurements gave a detailed understanding of the temperature distribution and its time-dependent behaviour. We focus in particular on transitions between different flow regimes. Using a wide gap, the first transition from axisymmetric flow to the regular wave regime was characterised by complex flows. The transition to irregular flows was smooth, where a coexistence of the large-scale jet-stream and small-scale vortices was observed. Furthermore, temperature measurements showed a repetitive separation of cold vortices from the inner wall. Experiments using a narrow gap showed no complex flows but strong hysteresis in the steady wave regime, with up to five different azimuthal wave modes as potential steady and stable solutions.
Physics , 2009, Abstract: The two-dimensional flow of viscous incompressible fluid in the domain between two concentric circles is investigated numerically. To solve the problem, the low-order Galerkin models are used. When the inner circle rotates fast enough, two axially asymmetric flow regimes are observed. Both regimes are the stationary flows precessing in azimuthal direction. First flow represents the region of concentrated vorticity. Another flow is the jet-like structure similar to one discovered earlier in Vladimirov's experiments.
Physics , 2008, DOI: 10.1088/0031-8949/2008/T132/014029 Abstract: We present numerical simulations of circular Couette flow in axisymmetric and fully three-dimensional geometry of a cylindrical annulus inspired by Princeton MRI liquid gallium experiment. The incompressible Navier-Stokes equations are solved with the spectral element code Nek5000 incorporating realistic horizontal boundary conditions of differentially rotating rings. We investigate the effect of changing rotation rates (Reynolds number) and of the horizontal boundary conditions on flow structure, Ekman circulation and associated transport of angular momentum through the onset of unsteadiness and three-dimensionality. A mechanism for the explanation of the dependence of the Ekman flows and circulation on horizontal boundary conditions is proposed.
Mathematics , 2012, Abstract: We consider the fully-developed flow of an incompressible Newtonian fluid in a cylindrical vessel with elliptical cross-section (both an ellipse and the annulus between two confocal ellipses). In particular, we address an inverse problem, namely to compute the velocity field associated with a given, time-periodic flow rate. This is motivated by the fact that flow rate is the main physical quantity which can be actually measured in many practical situations. We propose a novel numerical strategy, which is nonetheless grounded on several analytical relations. The proposed method leads to the solution of some simple ordinary differential systems. It holds promise to be more amenable to implementation than previous approaches, which are substantially based on the challenging computation of Mathieu functions. Some numerical results are reported, based on measured data for human blood flow in the internal carotid artery, and cerebrospinal fluid (CSF) flow in the upper cervical region of the human spine. As expected, computational efficiency is the main asset of our solution: a speed-up factor over 10^3 was obtained, compared to more elaborate numerical approaches. The main goal of the present study is to provide an improved source of initial/boundary data for more ambitious numerical approaches, as well as a benchmark solution for pulsatile flows in elliptical sections with given flow rate. The proposed method can be effectively applied to bio-fluid dynamics investigations (possibly addressing key aspects of relevant diseases), to biomedical applications (including targeted drug delivery and energy harvesting for implantable devices), up to longer-term medical microrobotics applications.
Physics , 2011, Abstract: Several galaxy clusters are known to present multiple and misaligned pairs of cavities seen in X-rays, as well as twisted kiloparsec-scale jets at radio wavelengths. It suggests that the AGN precessing jets play a role in the formation of the misaligned bubbles. Also, X-ray spectra reveal that typically these systems are also able to supress cooling flows, predicted theoretically. The absence of cooling flows in galaxy clusters has been a mistery for many years since numerical simulations and analytical studies suggest that AGN jets are highly energetic, but are unable to redistribute it at all directions. We performed 3D hydrodynamical simulations of the interaction between a precessing AGN jet and the warm intracluster medium plasma, which dynamics is coupled to a NFW dark matter gravitational potential. Radiative cooling has been taken into account and the cooling flow problem was studied. We found that precession is responsible for multiple pairs of bubbles, as observed. The misaligned bubbles rise up to scales of tens of kiloparsecs, where the thermal energy released by the jets are redistributed. After $\sim 150$ Myrs, the temperature of the gas within the cavities is kept of order of $\sim 10^7$ K, while the denser plasma of the intracluster medium at the central regions reaches $T \sim 10^5$ K. The existence of multiple bubbles, at diferent directions, result in an integrated temperature along the line of sight much larger than the simulations of non-precessing jets. This result is in agreement with the observations. The simulations reveal that the cooling flows cessed $\sim 50 - 70$ Myr after the AGN jets are started.
Page 1 /100 Display every page 5 10 20 Item | 2019-11-18 13:16:14 | {"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": 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": 0.6737358570098877, "perplexity": 889.9763215535045}, "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-2019-47/segments/1573496669795.59/warc/CC-MAIN-20191118131311-20191118155311-00059.warc.gz"} |
https://www.physicsforums.com/threads/figures-in-latex.385533/ | # Figures in Latex
## Homework Statement
Is there a way i can attatch matlab graphs onto the Latex file?
## The Attempt at a Solution
I copied and pasted it onto paint then saved it as .jpg but i was wondering if theres a way i could attatch the original graph without copying and pasting it onto paint.
Thank you
LCKurtz
Homework Helper
Gold Member
I don't use Matlab. Can it export its graphs as encapsulated PostScript files? It it can, then you can do it like this:
\usepackage{graphicx}
\begin{center}
\includegraphics[width=2.5in,height=2.5in]{yourgraph.eps}
\end{center}
I don't use Matlab. Can it export its graphs as encapsulated PostScript files? It it can, then you can do it like this:
\usepackage{graphicx}
\begin{center}
\includegraphics[width=2.5in,height=2.5in]{yourgraph.eps}
\end{center}
Yes, it can
## Homework Statement
Is there a way i can attatch matlab graphs onto the Latex file?
## The Attempt at a Solution
I copied and pasted it onto paint then saved it as .jpg but i was wondering if theres a way i could attatch the original graph without copying and pasting it onto paint.
Thank you
Sara, here's the magic:
MATLAB > SVG > PDF > Latex
1) You need to download "Scalable Vector Graphics (SVG) Export of Figures" from MATLAB Central.
a. Go to
Code:
http://www.mathworks.com/matlabcentral/fileexchange/7401-scalable-vector-graphics-svg-export-of-figures
b. Extract zip file into the matlab folder you will use as a current directory (i.e. where your M-files are located)
c. Simply type "plot2svg" in command window or M-file, a GUI will pop up and ask you where to save the figure as an SVG file. Of course, you need a plot( ) command just before this step to generate a figure!
Note: svgs do not have resolution constraints, like true pdfs.
2) You need inkscape
b. Open the svg file from matlab in inkscape
c. shift+ctrl+d: opens document properties, you might want to click fit page to selection - this removes any extra space surrounding the figure.
d. Add any annotations as you wish.
e. shift+ctrl+s: save as... pdf via Cairo
3) Latex
Example of scaled figure (33% of the initial size and lossless!).
Preamble:
Code:
\usepackage{graphics}
Body:
Code:
\begin{figure}[!h]
\centerline{
{\scalebox{0.33}{\includegraphics{figure.pdf}}
}
\caption{Any caption.}
\label{fig:figlabel}
\end{figure}
You may wanna look into the subfloat package for having subfigures (a) (b) (c) ... etc within one plot. Example:
Preamble:
Code:
\usepackage{graphics}
\usepackage{subfig}
Body:
Code:
\begin{figure}[!h]
\centerline{
\subfloat[Vertical cutlines.]%
{\scalebox{0.33}{\includegraphics{NMOS_cutlines.pdf}} \label{fig:nmos1athena2dcut}}
\subfloat[Doping profile for cutline \# 1.]%
{\scalebox{0.33}{\includegraphics{NMOS_cutlines1.pdf}} \label{fig:nmos1athena2dcut1}}
\vspace{0.5cm}
}
\centerline{
\subfloat[Doping profile for cutline \# 2.]%
{\scalebox{0.33}{\includegraphics{NMOS_cutlines2.pdf}} \label{fig:nmos1athena2dcut2}}
\subfloat[Doping profile for cutline \# 3.]
{\scalebox{0.33}{\includegraphics{NMOS_cutlines3.pdf}} \label{fig:nmos1athena2dcut3}}
\vspace{0.5cm}
}
\caption{NMOSFET doping profile at critical locations.}
\label{fig:nmos2}
\end{figure}
The code looks ugly, but the final result is amazing!
I hope this helps.
Last edited: | 2021-04-21 14:01:31 | {"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": 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": 0.552810788154602, "perplexity": 6003.089607753943}, "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/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00052.warc.gz"} |
http://mathhelpforum.com/new-users/281599-finding-payout-ratio-finance.html | ## Finding Payout Ratio - Finance
Fin-Tech, a finance and technology firm, floated its shares today. The IPO prospectus notesthat Fin-Tech does not plan to pay any dividend in the foreseeable future and reported annualearnings of $4 per share (this is the earning as of today). Fin-Tech’s Return on Equity is 15%and expected to stay the same forever. Investors agree that the appropriate discount rate is10% and that in 4 years the firm will start distributing a dividend keeping the payout ratioconstant after that.Fin-Tech’s share price closed at$52.56. This trading price is the consensus valuation amonginvestors and analysts.
1. What is the payout ratio on and after year 4 implied by investors’ valuation?
2. What is the implied PVGO?
EPS= 4
ROE= 15%
Share price = 52.56
Discount rate = 10%
Growth rate (not sure) = 0?
The formula is P(t=4)= D(t=5)/(r-g). ROE is 15%, and g = 0% (not sure about this one) so D/P = 15%
Now we know that at t=0, Actual P is 52.56 per share. This is the NPV at t=0. We have to equate this the NPV of the future Price P at time (t=4). We can do this using the formula NPV (t=0) = P(t=4)/(1+i)^4 where i = annual discount factor. But we know P(t=4) = D(t=5)/0.15 from the formula above. Substituting into the NPV give us: 52.56 = [D(t=5)/0.15]/[1+0.1]^4. Then, D(t=5)/P(t=4) is the payout ratio in year 4 and after.
Is this correct? I wasn't very sure about how to approach this question, so any help would be greatly appreciated! | 2019-08-22 10:09:29 | {"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": 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": 0.5018618106842041, "perplexity": 2844.3655427275107}, "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/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00010.warc.gz"} |
https://zbmath.org/?q=an:0832.28011 | # zbMATH — the first resource for mathematics
Topological spaces admitting a unique fractal structure. (English) Zbl 0832.28011
Summary: Each homeomorphism from the $$n$$-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, “fractal structure” is not a metric but a topological phenomenon.
##### MSC:
28A80 Fractals 54E35 Metric spaces, metrizability
Full Text: | 2021-10-21 02:09:19 | {"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": 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": 0.7597071528434753, "perplexity": 1111.6646923874468}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00190.warc.gz"} |
http://www.faqoverflow.com/unix/12075.html | # FAQ overflow
#### QUESTION
I have a server log that outputs a specific line of text into its log file when the server is up. I want to execute a command once the server is up, and hence do something like the following:
tail -f /path/to/serverLog | grep "server is up" ...(now, e.g., wget on server)?
What is the best way to do this?
A simple way would be awk.
tail -f /path/to/serverLog | awk '/Printer is on fire!/ { system("shutdown -h now") }
/new USB high speed/ { system("echo New USB" | mail admin")'
And yes, both of those are real message from a kernel log. Perl might be a little more elegant to use for this and can also replace the need for tail. If using perl, it will look something like this:
open(my $fd, "<", "/path/to/serverLog") or die "Can't open log"; while(1) { if(eof$dataFd) {
sleep 1;
$fd->clearerr; next; } my$line = <$fd>; chomp($line);
if($line =~ /Printer is on fire!/) { system("shutdown -h now"); } elsif($line =~ /new USB high speed/) {
system("echo New USB" | mail admin");
}
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http://aventurasturisticas.com/hiha06z/orthocentre-of-a-triangle-properties-6d2632 | The orthocentre of triangle properties are as follows: If a given triangle is the Acute triangle the orthocenter lies inside the triangle. 5pm !! Why can't we build a huge stationary optical telescope inside a depression similar to the FAST? The centroid is the centre point of the object. Thanks for contributing an answer to Mathematics Stack Exchange! Orthocentre: where the triangle’s three altitudes intersects. Different triangles like an equilateral triangle, isosceles triangle, scalene triangle, etc will have different altitudes. does not have an angle greater than or equal to a right angle). パンの耳? does not have an angle greater than or equal to a right angle). Please take a look on the following question: Does the orthocenter have any special properties? Government censors HTTPS traffic to our website. How did 耳 end up meaning edge/crust? The orthocenter properties of a triangle depend on the type of a triangle. The orthocenter of a triangle is the point of intersection of any two of three altitudes of a triangle (the third altitude must intersect at the same spot). Let's learn these one by one. Free classes & tests. Altitudes are the perpendicular drawn from the vertex to the sides. The circumcenter, centroid, and orthocenter are also important points of a triangle. The orthocentre of triangle properties are as follows: If a given triangle is the Acute triangle the orthocenter lies inside the triangle. This location gives the incenter an interesting property: The incenter is equally far away from the triangle’s three sides. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle. Orthocenter - The orthocenter lies at the intersection of the altitudes. Find the point in a triangle, that is closest to the triangle's 3 points. In this class ,Abhinay sharma will discuss Orthocentre, incentre & circumcentre in triangle. The orthocenter can also be considered as a point of concurrency for the supporting lines of the altitudes of … In the applet below, point O is the orthocenter of the triangle. ... theorem on the line segments connecting the point of intersection of the heights with the vertices of an acute-angled triangle. If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. Find the slopes of the altitudes for those two sides. The centroid is an important property of a triangle. If the orthocentre of the triangle is the origin, then the third vertex is. Adjust the figure above and create a triangle where the orthocenter is outside the triangle. Then we have to calculate the slopes of altitudes of the triangle. Look at Euler line or Euler circle, and these are just examples. SSC Exams. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. The x-coordinate of the incentre of the triangle that has the coordinates of mid-points of its sides as (0, 1), (1, 1) and (1, 0) is. Since the triangle has three vertices, we have three altitudes in the triangle. Besides this, the Orthocenter has several other properties related to circumcenter, incenter, and area of a triangle. Answer: The Orthocenter of a triangle is used to identify the type of a triangle. Finally by solving any two altitude equations, we can get the orthocenter of the triangle. As far as triangle is concerned, It is one of the most important ‘points’. While solving one of Brilliant problems I came across an interesting property of an orthocentre which I have not thought of before, so I decided to share it with Brilliant community. Dealing with orthocenters, be on high alert, since we're dealing with coordinate graphing, algebra, and geometry, all … An altitude of a triangle is a line passing through the vertex of a triangle such that it is perpendicular to the opposite side of the vertex. View solution. 2. If one angle is a right angle, the orthocenter coincides with the vertex of the right angle. The point-slope formula is given as. If a given triangle is the Obtuse triangle the orthocenter lies outside the triangle. Here AD, BE and CF are the altitudes drawn on the sides BC, AC and AB respectively, all these three altitudes intersect at a point O. Orthocentre distance to triangle vertices as a function of triangle angles and side lengths. Find the orthocenter of the triangle with the given vertices: CBSE Class 9 Maths Number Systems Formulas, CBSE Class 9 Maths Surface Areas and Volumes Formulas, Important Four Marks Questions for CBSE Class 10 Maths, Important 3 Marks Question For CBSE Class 10 Maths, Vedantu For example: Does the orthocenter have any similar property? And there are litterally hundreds of special points. How about the symmedian center or the nine-point center? 1. The triangle is one of the most basic geometric shapes. Aren't the Bitcoin receive addresses the public keys? So these two are going to be congruent to each other. If the Orthocenter of a triangle lies in the center of a triangle then the triangle is an acute triangle. How likely it is that a nobleman of the eighteenth century would give written instructions to his maids? Orthocenter of a Triangle || GeoGebra || Mr. Binod Pandey#Orthocenter #GeoGebra #MrBinodPandey Use MathJax to format equations. GRE question bank. The ORTHOCENTER of a triangle is the point of concurrency of the LINES THAT CONTAIN the triangle's 3 ALTITUDES. Some even say it's a sin to spend too much time looking for such properties. Which instrument of the Bards correspond to which Bard college? We know that, for a triangle with the circumcenter at the origin, the sum of the vertices coincides with the orthocenter. Orthocenter as Circumcenter Orthocenter Properties. And so we can say that O is the orthocentre of a triangle ABC. Hypothetically, why can't we wrap copper wires around car axles and turn them into electromagnets to help charge the batteries? What is the Galois group of one ultrapower over another ultrapower? The orthocenter is known to fall outside the triangle if the triangle is obtuse. For example, due to the mirror property the orthic triangle solves Fagnano's Problem. The orthocenter is the point of concurrency of the three altitudes of a triangle. In other words, the point of concurrency of the bisector of the sides of a triangle is called the circumcenter. The altitude of a triangle is a segment from a vertex of the triangle to the opposite side (or to the extension of opposite side if necessary). “The orthocenter of a triangle is the point at which the three altitudes of the triangle meet.” We will explore some properties of the orthocenter from the following problem. For an obtuse triangle, it lies outside of the triangle. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. The three altitudes intersect in a single point, called the orthocenter of the triangle. If the Orthocenter of a triangle lies outside the triangle then the triangle is an obtuse triangle. How can I disable OneNote from starting automatically? Activity 6 Objective: To find Incentre, Circumcentre and Orthocentre by paper folding. Just as a review, the orthocenter is the point where the three altitudes of a triangle intersect, and the centroid is a point where the three medians. We know that, for a triangle with the circumcenter at the origin, the sum of the vertices coincides with the orthocenter. Centroid - The centroid, or a triangle's center of gravity point, is located where all three medians intersect. Circumcenter. Login. The points symmetric to the orthocenter have the following property. GRE Coordinate Geometry sample question. Some even say it's a sin to spend too much time looking for such properties. If a given triangle is the Obtuse triangle the orthocenter lies outside the triangle. To calculate the perpendicular slope we have, Perpendicular Slope of Line = - (1/slope of a line). Take isogonal conjugate of orthocenter and you get the circumcenter of that triangle. Then over here, on this inner triangle, our original triangle, the side that's between the orange and the blue side is going to be congruent to the side between the orange and the blue side on that triangle. If a given triangle is the right-angled triangle the orthocenter lies on the triangle. Since a triangle has three vertices, it also has three altitudes. The circumcenter is also the centre of the circumcircle of that triangle and it can be either inside or outside the triangle. So not only is this the orthocenter in the centroid, it is also the circumcenter of this triangle right over here. ), B( 3,0) and C(0,4) then Find the Orthocenter of the Triangle. Orthocenter of a Triangle In geometry, we learn about different shapes and figures. :-). There are numerous properties in the triangle, many involving the orthocenter. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. To download free study materials like NCERT Solutions, Revision Notes, Sample Papers and Board … Pro Lite, Vedantu 1. For an acute triangle, it lies inside the triangle. Isaiah 5:14 - Sheol/Hell personified as a woman? Sorry!, This page is not available for now to bookmark. Finding Orthocenter of the Triangle with Coordinates : In this section, we will see some examples on finding the orthcenter of the triangle with vertices of the triangle. What did Asimov find embarrassing about "Marooned Off Vesta”? The orthocenter of a triangle is the point where all three of its altitudes intersect. If the Orthocenter of a triangle lies on the triangle then the triangle is a right-angled triangle. Here $$\text{OA = OB = OC}$$, these are the radii of the circle. The point-slope formula is given as, Now, the slope of side YZ with Y( 3, -1) and Z(4, 2), Solving equation 1 and 2 we get, the values of, thus , we get the coordinates of Orthocenter as ( -4 , 10/3). Altitudes are the perpendicular drawn from the vertex to the sides. Orthocenter Formula - Learn how to calculate the orthocenter of a triangle by using orthocenter formula prepared by expert teachers at Vedantu.com. Workarounds? The slope of XY with X ( 5, 3) and Y(3, -1). The points symmetric to the point of intersection of the heights of a triangle with respect to the middles of the sides lie on the circumscribed circle and coincide with the points diametrically opposite the corresponding vertices (i.e. These altitudes intersect each other at point O. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. Hardness of a problem which is the sum of two NP-Hard problems. The circumcenter of a triangle is defined as the point where the perpendicular bisectorsof the sides of that particular triangle intersects. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. 4. The foot of an altitude also has interesting properties. When constructing the orthocenter or triangle T, the 3 feet of the altitudes can be connected to form what is called the orthic triangle, t.When T is acute, the orthocenter is the incenter of the incircle of t while the vertices of T are the excenters of the excircles of t.When the triangle is obtuse then the roles of the vertex of the obtuse angle and the orthocenter are reversed. The orthocenter of a triangle varies according to the triangles. How to Calculate Orthocenter of a Triangle : Let us calculate the slopes of the sides of the given triangle. The point in which the three medians of the triangle intersect is known as the centroid of a triangle. The circumcenter is the center of the circle defined by three points. How to compute the circumcentre and orthocentre of a right triangle if the equation of one of its sides is known. For every three points on a line, does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid? It has several important properties and relations with other parts of the triangle, including its circumcenter, incenter, area, and more. Properties of the incenter. If a given triangle is the right-angled triangle the orthocenter lies on the triangle. The properties of the points symmetric to the orthocenter. But with that out of the way, we've kind of marked up everything that we can assume, given that this is an orthocenter and a center-- although there are other things, other properties of … Consider a triangle ABC in which the altitudes are drawn from the vertex to the opposite side of the vertex such that it forms a right angle with the side. Angle-side-angle congruency. Main & Advanced Repeaters, Vedantu Orthocenter of a Triangle (Definition, How to Find, Video, & Examples) The orthocenter of a triangle, or the intersection of the triangle's altitudes, is not something that comes up in casual conversation. Why don't video conferencing web applications ask permission for screen sharing? Show that the orthocenter must coincide with one of the vertices of triangle ABC. The orthocenter of an acute triangle lies inside the triangle. And this point O is said to be the orthocenter of the triangle ABC. Oo; orthocentre, orthocenter • a point where the three altitudes of a triangle meet which may lie inside or outside the triangle. Can we get rid of all illnesses by a year of Total Extreme Quarantine? 2. For right-angled triangle, it lies on the triangle. In triangle ABC AD, BE, CF are the altitudes drawn on the sides BC, AC and AB respectively. The orthocenter of a triangle is the point of intersection of the heights of the triangle. “The orthocenter of a triangle is the point at which the three altitudes of the triangle meet.” We will explore some properties of the orthocenter from the following problem. The orthocenter of a triangle can be calculated as follows: Step 1: Let us calculate the slopes of the sides of the given triangle. Orthocentre, incentre & circumcentre in triangle -ABHINAYMATHS. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Sum of the angle in a triangle is 180 degree. ChemDraw: how to change the default aromatic ring style for drawing from SMILES. Construction of a triangle given some special points ($O,H,I$). Pro Subscription, JEE The orthocenter properties of a triangle depend on the type of a triangle. Step 4: Finally by solving any two altitude equations, we can get the orthocenter of the triangle. Asking for help, clarification, or responding to other answers. 7mathswithrichabhardwaj.blogspot.in 8. Is there a book about the history of linear programming? There are numerous properties in the triangle, many involving the orthocenter. Orthocentre 8mathswithrichabhardwaj.blogspot.in 9. Step 1 Step 3: Then by using the point-slope form, calculate the equation for the altitudes with their respective coordinates. This is Corollary 3 of Ceva's theorem. The orthocenter of a triangle is the intersection of the triangle's three altitudes. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. When the position of an Orthocenter of a triangle is given. Example: Find the Orthocenter of the Triangle with the Given Vertices: O is the Orthocenter of altitudes drawn from X, Y and Z. Orthocenter. Definition of Orthocenter : The altitudes of a triangle are concurrent and the point of concurrence is called the orthocentre of the triangle.The orthocentre is denoted by O. Making statements based on opinion; back them up with references or personal experience. It only takes a minute to sign up. See Orthocenter of a triangle. The various properties of the orthocenter are: 1. Then a Google search should work, and sites like Mathworld or Wikipedia and their sources might help. It is denoted by P(X, Y). Altitudes as Cevians. * The three heights (altitudes) of a triangle intersect at one point (are concurrent at a point), called the orthocentre of the triangle. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. The height and circumscribed circle. So these two-- we have an angle, a side, and an angle. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle. Then by using the point-slope form, calculate the equation for the altitudes with their respective coordinates. Example 2: If the Coordinates of the Vertices of Triangle ABC are A(0,0), B( 3,0) and C(0,4) then Find the Orthocenter of the Triangle. Wizako offers online GRE courses for GRE Quant and GRE Verbal @ https://online.wizako.com and GRE coaching in Chennai. ... Properties of triangle. First of all, let’s review the definition of the orthocenter of a triangle. Here you can see we have AB on the Y- axis and AC passes through point zero, which shows that triangle is a right angled triangle. Н is an orthocenter of a triangle Proof of the theorem on the point of intersection of the heights of a triangle As, depending upon the type of a triangle, the heights can be arranged in a different way, let us consider the proof for each of the triangle types. Example 2: If the Coordinates of the Vertices of Triangle ABC are A(0,0. It is one of the points that lie on Euler Line in a triangle. Construct the Orthocenter H. The product of the parts into which the orthocenter divides an altitude is the equivalent for all 3 perpendiculars. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The vertices of the triangle are A(0,0), B( 3,0) and C( 0,4). If a given triangle is the Obtuse triangle the orthocenter lies outside the triangle. A fascinating application of Steiner's theorem for trapezium: geometric constructions using straightedge alone Center of the incircle: ... Constructing the Orthocenter of a Triangle. Are there explainbility approaches in optimization? properties of triangle 1. MathJax reference. If the triangle is obtuse, it will be outside. Orthocentre is the point of intersection of altitudes from each vertex of the triangle. Steps Involved in Finding Orthocenter of a Triangle : Find the equations of two line segments forming sides of the triangle. Move the white vertices of the triangle around and then use your observations to answer the questions below the applet. To learn more, see our tips on writing great answers. A geometrical figure is a predefined shape with certain properties specifically defined for that particular shape. In this case, the orthocenter lies in the vertical pair of the obtuse angle: It's thus clear that it also falls outside the circumcircle. To make this happen the altitude lines have to be extended so they cross. The circumcenter of a triangle is the center of a circle which circumscribes the triangle.. Equation of altitude through Z(4, 2) is perpendicular to XY. Given triangle ABC. 3. The orthocenter of a triangle is the point of intersection of all the three altitudes drawn from the vertices of a triangle to the opposite sides. Triangles have three vertices so these three altitudes are drawn will intersect at a certain point and that point is said to be the orthocenter of the respective triangle. No other point has this quality. Each of the commonly known triangle centers I know has some sort of special property. And there are litterally hundreds of special points. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. It turns out that all three altitudes always intersect at the same point - the so-called orthocenter of the triangle. Orthocenter is the intersection point of the altitudes drawn from the vertices of the triangle to the opposite sides. This will occur inside acute triangles, outside obtuse triangles, and for right triangles, it will occur at the midpoint of the hypotenuse. Find the orthocenter of the triangle with the given vertices: Answer: in a triangle a point of intersection of all the three altitudes is said to be orthocenter. So do you mean properties which are not directly geometric? The incenter is the center of the inscribed circle. That opposite side is called as base. EXAMPLE: Statement 1 . Properties of parallelogram. In any given triangle the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides is called the Orthocenter of a triangle. 1mathswithrichabhardwaj.blogspot.in Therefore, orthocenter lies on the triangle I.e Orthocenter is ( 0,0). 2. The centroid is the gravitational center of an object. So I have a triangle over here, and we're going to assume that it's orthocenter and centroid are the same point. Other triangle … In this case, the orthocenter lies in the vertical pair of the obtuse angle: It's thus clear that … It is an important central point of a triangle and thus helps in studying different properties of a triangle with respect to sides, vertices, other … The orthocenter properties of a triangle depend on the type of a triangle. The orthocenter is not always inside the triangle. Centroid Definition. math.stackexchange.com/questions/2321816/…, Gergonne Point of a triangle coinciding with other triangle centers. Let us discuss the definition of centroid, formula, properties and centroid for different geometric shapes in detail. Given triangle ABC. In this drawing of the Avengers, who's the guy on the right? Incenters, like centroids, are always inside their triangles.The above figure shows two triangles with their incenters and inscribed circles, or incircles (circles drawn inside the triangles so the circles barely touc… Pro Lite, NEET The orthocenter is known to fall outside the triangle if the triangle is obtuse. When constructing the orthocenter or triangle T, the 3 feet of the altitudes can be connected to form what is called the orthic triangle, t. When T is acute, the orthocenter is the incenter of the incircle of t while the vertices of T are the excenters of the excircles of t. When the triangle is obtuse then the roles of the vertex of the obtuse angle and the orthocenter are reversed. Repeaters, Vedantu You find a triangle’s incenter at the intersection of the triangle’s three angle bisectors. Step 2: Then we have to calculate the slopes of altitudes of the triangle. The orthocentre of triangle properties are as follows: If a given triangle is the Acute triangle the orthocenter lies inside the triangle. In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Expectations from a violin teacher towards an adult learner. In the case of an equilateral triangle, all four of the above centers occur at the same point. Nine-point circle - proof using plane geometry, An identity associated with the centroid of a triangle. Hindi Practice & Strategy. Now to bookmark specifically defined for that particular shape scalene triangle, isosceles triangle scalene. 6 Objective: to find Incentre, circumcentre orthocentre of a triangle properties orthocentre of triangle ABC can say that O is acute... What did Asimov find embarrassing about Marooned Off Vesta ” sources might.! Of that triangle of all, let ’ s three sides linear programming an object coincide with one of triangle! Geometric shapes in detail orthocentre distance to triangle vertices as a function of triangle properties as. This drawing of the eighteenth century would give written instructions to his maids a single point, located! Circumscribes the triangle ’ s review the definition of the triangle I know has some sort of special property a! Commonly known triangle centers I know has some sort of special property angle bisectors acute-angled... Orthocenter lies inside the triangle location gives the incenter is equally far away from the vertices with! Intersect at the origin, the point of a triangle, that is closest the. So we can say that O is the obtuse triangle the orthocenter of the bisector of altitudes. Triangle around and then use your observations to answer the questions below the.. The so-called orthocenter of an acute-angled triangle privacy policy and cookie policy also important points of triangle! That all three medians intersect special points ( $O, H, I$ ) 6. Geometrical figure is a predefined shape with certain properties specifically defined for that particular triangle intersects properties the! The public keys area of a triangle depend on the line segments connecting the point in triangle! This location gives the incenter is also the circumcenter at the origin then..., is located where all three of its sides is known to outside! ( $O, H, I$ ) solves Fagnano 's Problem ( 3,0 ) and C ( ). Most basic geometric shapes in detail various properties of the lines that CONTAIN the.! As triangle is a right-angled triangle the orthocenter points that lie on Euler line Euler... Following question: does the orthocenter of the Bards correspond to which Bard college other of. Applet below, point O is said to be congruent to each other any level and professionals related... Optical telescope inside a depression similar to the triangle this drawing of the triangle 's 3 points a is. Angle ) mathematics Stack Exchange Inc ; user contributions licensed under cc by-sa Stack Exchange Inc ; contributions! Point where the perpendicular drawn from the vertex to the sides of a triangle into your RSS reader public?! Work, and sites like Mathworld or Wikipedia and their sources might help, and area of a is! Geometrical figure is a right-angled triangle be either inside or outside the triangle this page is not available for to. Constructing the orthocenter properties of the lines that CONTAIN the triangle intersect is known right angle the... Activity 6 Objective: to find Incentre, circumcentre and orthocentre by paper folding single point, called orthocenter... A single point, is located where all three altitudes of the points that lie on Euler in! Triangle over here us calculate the slopes of altitudes of a triangle coinciding with other parts of the BC! Figure is a question and answer site for people studying math at level... Can we get rid of all illnesses by a year of Total Extreme Quarantine always. The circle triangle around and then use your observations to answer the questions below the applet given special... Follows: if the orthocenter lies inside the triangle a nobleman of the triangle! This RSS feed, copy and paste this URL into your RSS reader 5, 3 ) C... Is located where all three of its sides is known to fall outside the triangle ’ s altitudes... Construct the orthocenter in the case of an equilateral triangle, that is closest to FAST! In which the orthocenter lies on the sides 's the guy on the triangle intersect is known the..., isosceles triangle, including its circumcenter, incenter, and more to compute the circumcentre orthocentre... Centroid is the obtuse triangle is equally far away from the triangle is the where!, Y ) is given 's orthocenter and centroid for different geometric shapes shapes in detail, )... 3 perpendiculars then a Google search should work, and these are examples! Predefined shape with certain properties specifically defined for that particular shape the of! Triangle with the centroid is the gravitational center of a triangle from each vertex of the circle instrument the. Offers online GRE courses for GRE Quant and GRE Verbal @ https: //online.wizako.com and GRE in. Slope we have three altitudes of the triangle, etc will have different altitudes segments connecting the of... Triangle properties are as follows: if a given triangle is obtuse type of a triangle acute... Get the orthocenter of the inscribed circle be, CF are the radii of triangle! Take a look on the type of a triangle is the point of concurrency of the eighteenth century give! Ca n't we wrap copper wires around car axles and turn them into electromagnets to help the. Triangle then the triangle and these are just examples centroid is an important property of a triangle video! A sin to spend too much time looking for such properties points of a triangle is called circumcenter! Is concerned, it is denoted by P ( X, Y.. } \ ), these are the perpendicular drawn from the vertices coincides the... And orthocentre by paper folding = - ( 1/slope of a triangle the slopes of the other three,... That CONTAIN the triangle if and only if the orthocenter lies at the right angle altitude also has three,... Lines have to calculate the equation for the altitudes drawn on the type of a is... Copper wires around car axles and turn them into electromagnets to help charge the batteries orthocenter as circumcenter if triangle. You mean properties which are not directly geometric incenter an interesting property: incenter! The following property 's incircle - the centroid of a triangle is the acute triangle the orthocenter on. Property of a triangle is the sum of the sides shape with certain specifically... Is used to identify the type of a line ) of concurrency of the triangle to the?... Have any similar property the orthocentre of a triangle properties triangle is acute ( i.e the public keys 5! O is the obtuse triangle the orthocenter have any similar property angle ) bisectors. Triangle to the orthocenter of an acute-angled triangle equal to a right angle ) sum of the basic! To XY on Euler line or Euler circle, and area of a triangle over here guy!!, this page is not available for now to bookmark for drawing from SMILES other parts of other. Oa = OB = OC } \ ), B ( 3,0 ) Y! To XY AB respectively directly geometric by a year of Total Extreme?! To triangle vertices as a function of triangle ABC two sides OC } \ ), B ( 3,0 and! Be the orthocenter the orthic triangle solves Fagnano 's Problem triangle angles and side lengths O! Finally by solving any two altitude equations, we can say that O is origin... Properties of a triangle lies inside the triangle 2021 Stack Exchange Inc ; user contributions under. Euler line in a triangle has three altitudes of the points symmetric to opposite... Extended so they cross = OB = OC } \ ), B ( )... Right angle ) etc will have different altitudes this RSS feed, copy paste! Radii of the vertices of the triangle triangle angles and side lengths questions below applet... Take isogonal conjugate of orthocenter and you get the orthocenter lies outside the triangle take conjugate. A single point, called the orthocenter of the orthocenter is concerned, it lies inside the then. 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Inside or outside the triangle identity associated with the orthocenter of the triangle as... Location gives the incenter is the center of gravity point, called circumcenter. As a function of triangle properties are as follows: if a given triangle is.... All illnesses by a year of Total Extreme Quarantine of Total Extreme Quarantine circumcentre orthocentre! The obtuse triangle the orthocenter properties of a triangle is the point of of... Slopes of the triangle is defined as the centroid is an obtuse triangle orthocenter... And create a triangle coinciding with other parts of the triangle the various of. Fit inside the triangle 's 3 points out that all three medians intersect ‘ points ’ shortly for online. You get the orthocenter have any similar property show that the orthocenter lies outside of the triangle an orthocentric,! | 2023-03-28 02:19:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4839959740638733, "perplexity": 496.1618902193341}, "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/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00278.warc.gz"} |
https://www.lil-help.com/questions/249603/give-a-specific-example-of-how-the-introduction-of-a-gene-into-a-bacterium-can-change-the-phenotype-of-the-bacterium-also-explain-the-specific-role-of-the-protein-expressed | Give a specific example of how the introduction of a gene into a bacterium can change the phenotype of the bacterium. Also explain the specific role of the protein expressed
# Give a specific example of how the introduction of a gene into a bacterium can change the phenotype of the bacterium. Also explain the specific role of the protein expressed
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Give a specific example of how the introduction of a gene into a bacterium can change the phenotype of the bacterium. Also explain the specific role of the protein expressed by the gene in changing the bacteriums phenotype. You have isolated one of the genes for producing once of the blood-clotting proteins needed by some hemophiliacs. Briefly describe how you could create bacteria that would produce this protein. ANSWER:
Part 1: Examples
# Suppose a strain ofE. colithat has been made competent to allow it to incorporate and produce a plasmid containing...
Give a
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Asked: 12 months ago | 2018-01-17 11:03:56 | {"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": 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": 0.3434447944164276, "perplexity": 3942.0916071925267}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886895.18/warc/CC-MAIN-20180117102533-20180117122533-00176.warc.gz"} |
https://stats.libretexts.org/Courses/Fresno_City_College/Book%3A_Business_Statistics_Customized_(OpenStax)/13%3A_Linear_Regression_and_Correlation/13.04%3A_Linear_Equations | # 13.4: Linear Equations
Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:
$y=a+b x\nonumber$
where $$a$$ and $$b$$ are constant numbers.
The variable $$\bf x$$ is the independent variable, and $$\bf y$$ is the dependent variable. Another way to think about this equation is a statement of cause and effect. The $$X$$ variable is the cause and the $$Y$$ variable is the hypothesized effect. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
Example $$\PageIndex{1}$$
The following examples are linear equations.
$$y=3+2x$$
$$y=–0.01+1.2x$$
The graph of a linear equation of the form $$y = a + bx$$ is a straight line. Any line that is not vertical can be described by this equation
Example $$\PageIndex{2}$$
Graph the equation $$y = –1 + 2x$$.
Exercise $$\PageIndex{2}$$
Is the following an example of a linear equation? Why or why not?
Example $$\PageIndex{3}$$
Aaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a$31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.
Find the equation that expresses the total cost in terms of the number of hours required to complete the job.
Solution 13.3
Let $$x$$ = the number of hours it takes to get the job done.
Let $$y$$ = the total cost to the customer.
The $31.50 is a fixed cost. If it takes $$x$$ hours to complete the job, then (32)($$x$$) is the cost of the word processing only. The total cost is: $$y = 31.50 + 32x$$ ## Slope and Y-Intercept of a Linear Equation For the linear equation $$y = a + bx$$, $$b$$ = slope and $$a = y$$-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the $$y$$-intercept is the $$y$$ coordinate of the point $$(0, a)$$ where the line crosses the y-axis. From calculus the slope is the first derivative of the function. For a linear function the slope is $$dy / dx = b$$ where we can read the mathematical expression as "the change in y (dy) that results from a change in $$x (dx) = b * dx$$". Example $$\PageIndex{4}$$ Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of$25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $$y = 25 + 15x$$. What are the independent and dependent variables? What is the y-intercept and what is the slope? Interpret them using complete sentences. Answer Solution 13.4 The independent variable ($$x$$) is the number of hours Svetlana tutors each session. The dependent variable ($$y$$) is the amount, in dollars, Svetlana earns for each session. The y-intercept is $$25 (a = 25$$). At the start of the tutoring session, Svetlana charges a one-time fee of$25 (this is when $$x= 0$$). The slope is $$15 (b = 15)$$. For each session, Svetlana earns \$15 for each hour she tutors.
This page titled 13.4: Linear Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2023-02-09 10:20:01 | {"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.822006106376648, "perplexity": 528.789305827089}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00835.warc.gz"} |
https://physics.stackexchange.com/questions/277968/what-is-precision | # What is precision?
I’ve learnt that accuracy is how close to the true value a measurement is (i.e. low discrepancy between the experimental and theoretical values). Precision, on the other hand, is the reproducibility of experimental results—high precision means that the data points are closely located. However, precision can also refer to how many significant figures a measurement has.
Are those two definitions of precision the same thing? What exactly is precision?
• Although not a precise duplicate I would suggest that @LubošMotl 's answer explains the difference between accuracy and precision very well? It is also worth reading some of the other answers. physics.stackexchange.com/q/126895 – Farcher Sep 3 '16 at 8:10
• @Farcher I think my original question was a little ambiguous and I’ve made a slight change. Hopefully it now reflects what I’m asking—not the difference between accuracy and precision, but the definition of precision. – lightweaver Sep 3 '16 at 8:17
• In the @LubošMotl answer he writes "Before 2008, people would agree that "precision" refers to the typical difference between individual measurements, and the precision is good if a "statistical error" is low or if the measurements are producing "many significant figures" for the result." and that probably is still a good way of explaining what precision is? – Farcher Sep 3 '16 at 8:23
• @Farcher So many significant figures = low difference between individual measurements? Why is that? – lightweaver Sep 3 '16 at 9:23
• in radar measurement $precision = \frac {1}{standard \: deviation}$ – hyportnex Sep 3 '16 at 15:24
Think of data points in just 1 dimension: you can plot their distribution with values on an axis and frequencies on the other axis. If the points are closely located, their dispersion spans a small region on the value axis. Therefore you need more significant figures to express the region in which the data are located. You can think of an higher precision, or a smaller dispersion, as a deeper insight in determining the location parameter (the mean value) and the dispersion parameter (standard deviation) of the distribution.
Those two definitions relate to the same concept but in two different frames: structure of data (they are closely located) and their rapresentation (how many significant figures you need to express the parameters of the distribution).
I would say that precision is the resolution of a measurement (eg how many decimal places a measurement has) and accuracy is how true that number is traceable to an official standard.
So you can buy a cheap digital thermometer that will read out to 0.01deg but will be 1deg wrong in the absolute value if you put it on eg melting ice.
• So how does that relate to precision’s definition as the closeness of measurement values? – lightweaver Sep 3 '16 at 13:26
• this kind of usage, that is equating accuracy with resolution, has created immense amount of confusion for the word $resolution$ could be given another very good and useful meaning, namely the ability to distinguish two equal strength "sources" (of information) in noise. See, Rayleigh's use of the notion of "diffraction limited resolution" of two point sources. – hyportnex Sep 3 '16 at 15:29
• @hyportnex - I meant the exact opposite. Precision = resolution not accuracy. If I say c=3E8 m/s that is accurate (ie correct) but not precise. If I say it is 300000000 m/s that is precise but inacurate ie wrong – Martin Beckett Sep 3 '16 at 18:36
in my opinion precision tells us to what resolution or limit the quantity is measured, since the number of significant figures implies resolution in measurement, we say that more number of significant figures implies higher is the precision.consider an example, i m taking measurement of a rod and i have got two sets of results 1. 5.5cm and 5.6cm , 2. 5.0005cm and 5.0006cm, since second set of measurement contain mesurements with 5 number of significant figures hence measurements in second set are highly precise. hope you got what i m trying to say | 2019-10-22 14:24:54 | {"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": 0.6713360548019409, "perplexity": 857.5056701431862}, "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/1570987822098.86/warc/CC-MAIN-20191022132135-20191022155635-00120.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-1-section-1-7-absolute-value-equations-and-inequalities-1-7-exercises-page-120/104 | ## Intermediate Algebra (12th Edition)
Published by Pearson
# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 104
#### Answer
$x=\dfrac{1}{3}$
#### Work Step by Step
The given inequality, $|6x-2|=0 ,$ is equivalent to \begin{array}{l}\require{cancel} 6x-2=0 .\end{array} Using the properties of equality, the solution to the equation above is \begin{array}{l}\require{cancel} 6x=2 \\\\ x=\dfrac{2}{6} \\\\ x=\dfrac{\cancel{2}}{\cancel{6}^3} \\\\ x=\dfrac{1}{3} .\end{array}
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-04-26 17:21:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9576593637466431, "perplexity": 5498.090068989649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948426.82/warc/CC-MAIN-20180426164149-20180426184149-00520.warc.gz"} |
http://mathhelpforum.com/algebra/33715-quadratic-formula.html | 1. ## Quadratic Formula
(a) Use the quadratic formula to solve the distance formula
for in terms of the other quantities. (The quadratic formula gives you two solutions, enter the larger one first.)
(b) Substitute and into the above formula and find
(c) Now check your answer to (a) is correct by plugging in the values for and from (b) and seeing if your formula gives
2. Originally Posted by lilikoipssn
(a) Use the quadratic formula to solve the distance formula
for in terms of the other quantities. (The quadratic formula gives you two solutions, enter the larger one first.)
(b) Substitute and into the above formula and find
(c) Now check your answer to (a) is correct by plugging in the values for and from (b) and seeing if your formula gives
the quadratic formula says, the solutions in x to the quadratic equation $ax^2 + bx + c = 0$ are given by:
$x = \frac {-b \pm \sqrt{b^2 - 4ac}}{2a}$
you want to solve: $\frac 12at^2 + ut - s = 0$ for t. just follow the formula
3. ## Ok
Originally Posted by lilikoipssn
(a) Use the quadratic formula to solve the distance formula
for in terms of the other quantities. (The quadratic formula gives you two solutions, enter the larger one first.)
(b) Substitute and into the above formula and find
(c) Now check your answer to (a) is correct by plugging in the values for and from (b) and seeing if your formula gives
Applying the quadratic formula we get $t=\frac{-u\pm\sqrt{u^2-4\cdot{\frac{1}{2}a}\cdot{-s}}}{2\cdot{\frac{1}{2}a}}$ | 2017-09-26 05:55:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "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": 0.8549788594245911, "perplexity": 397.4675514634798}, "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-2017-39/segments/1505818695066.99/warc/CC-MAIN-20170926051558-20170926071558-00598.warc.gz"} |
https://www.degruyter.com/view/j/forum.2018.30.issue-6/forum-2018-0130/forum-2018-0130.xml?rskey=tGbMal&result=8 | Show Summary Details
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# Forum Mathematicum
Managing Editor: Bruinier, Jan Hendrik
Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna
IMPACT FACTOR 2018: 0.867
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Volume 30, Issue 6
# Homogeneous Finsler spaces and the flag-wise positively curved condition
Ming Xu
/ Shaoqiang Deng
Published Online: 2018-08-07 | DOI: https://doi.org/10.1515/forum-2018-0130
## Abstract
In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space ${S}^{3}×ℝ$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on ${S}^{2}×{S}^{3}$ and ${S}^{6}×{S}^{7}$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on ${S}^{3}×{S}^{3}$, shedding some light on the long standing general Hopf conjecture.
MSC 2010: 22E46; 53C30
## References
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S. Bácsó, X. Cheng and Z. Shen, Curvature properties of $\left(\alpha ,\beta \right)$-metrics, Finsler Geometry (Sapporo 2005), Adv. Stud. Pure Math. 48, Mathematical Society of Japan, Tokyo (2007), 73–110. Google Scholar
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D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000. Google Scholar
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S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240 (2013), 194–226.
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Z. Hu and S. Deng, Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature, Math. Z. 270 (2012), no. 3–4, 989–1009.
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L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differential Geom. Appl. 40 (2015), 187–208.
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L. Huang and X. Mo, On the flag curvature of a class of Finsler metrics produced by the navigation problem, Pacific J. Math. 277 (2015), no. 1, 149–168.
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X. Mo and L. Hang, On curvature decreasing property of a class of navigation problems, Publ. Math. Debrecen 71 (2007), no. 1–2, 141–163. Google Scholar
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G. Randers, On an asymmetrical metric in the fourspace of general relativity, Phys. Rev. (2) 59 (1941), 195–199.
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M. Xu, Examples of flag-wise positively curved spaces, Differential Geom. Appl. 52 (2017), 42–50.
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M. Xu and S. Deng, Homogeneous $\left(\alpha ,\beta \right)$-spaces with positive flag curvature and vanishing S-curvature, Nonlinear Anal. 127 (2015), 45–54.
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M. Xu and S. Deng, Normal homogeneous Finsler spaces, Transform. Groups 22 (2017), no. 4, 1143–1183.
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M. Xu and S. Deng, Towards the classification of odd-dimensional homogeneous reversible Finsler spaces with positive flag curvature, Ann. Mat. Pura Appl. (4) 196 (2017), no. 4, 1459–1488.
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M. Xu, S. Deng, L. Huang and Z. Hu, Even-dimensional homogeneous Finsler spaces with positive flag curvature, Indiana Univ. Math. J. 66 (2017), no. 3, 949–972.
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M. Xu and J. A. Wolf, Killing vector fields of constant length on Riemannian normal homogeneous spaces, Transform. Groups 21 (2016), no. 3, 871–902.
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M. Xu and W. Ziller, Reversible homogeneous Finsler metrics with positive flag curvature, Forum Math. 29 (2017), no. 5, 1213–1226.
Published Online: 2018-08-07
Published in Print: 2018-11-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11771331
Award identifier / Grant number: 11671212
Award identifier / Grant number: 51535008
Funding Source: Natural Science Foundation of Beijing Municipality
Award identifier / Grant number: 1182006
Supported by NSFC (no. 11771331, 11671212, 51535008), Beijing Natural Science Foundation (no. 1182006) and the Fundamental Research Funds for the Central Universities.
Citation Information: Forum Mathematicum, Volume 30, Issue 6, Pages 1521–1537, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,
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© 2018 Walter de Gruyter GmbH, Berlin/Boston. | 2019-10-14 04:33:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4539862871170044, "perplexity": 3112.782557451012}, "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/1570986649035.4/warc/CC-MAIN-20191014025508-20191014052508-00310.warc.gz"} |
https://stacks.math.columbia.edu/tag/0GVF | Eilenberg swindle
[Eilenberg's lemma, Bass]
In [Bass] we find: “...is an elegant little swindle, observed several years ago by Eilenberg, and which might well have sprung from the brow of Barry Mazur.”
Proof.
$F \cong F \oplus F \oplus \ldots \cong P \oplus Q \oplus P \oplus Q \oplus \ldots \cong P \oplus F \oplus F \oplus \ldots \cong P \oplus F$
$\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2022-08-09 05:48:12 | {"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": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.7568103075027466, "perplexity": 3589.2982973185544}, "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/1659882570901.18/warc/CC-MAIN-20220809033952-20220809063952-00590.warc.gz"} |
https://nononsensebooks.com/qm/part3/ | Part 3 Bonus Material
No-Nonsense Quantum Mechanics
Exercises
How many dimensions do the following mathematical arenas for a system consisting of N free particles have:
• everyday space
• configuration space
• phase space
• Hilbert space?
• everyday space : 3.
• configuration space : 3N . The configuration space of a free particle is 3-dimensional. Thus, for N particles we glue N differnt 3-dimensional configuration spaces together and the resulting space is 3N-dimensional.
• phase space : 6N. The phase space of a free particle is 6-dimensional: 3 to specify the location and 3 to specify the momentum. Thus for N particles we get a 6N-dimensional phase space.
• Hilbert space : $\infty$.
Let's assume the configuration space of one object is a line and the configuration space of a second object is a circle. How does the total configuration space look like?
We have to glue a copy of the circle above each point of the line. What we end up with this way is a cylinder.
What's the difference between the Schrödinger picture and the Heisenberg picture?
Both pictures are formulation in Hilbert space but:
• In the Schrödinger picture, the states evolve in time while the operators do not change.
• In the Heisenberg pictures, the operators evolve in the and the states do not change.
Which formulation of Quantum Mechanics is the best one?
Objectively they are all equivalent and therefore equally good. However, of course, you are free to pick your personal favorite.
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Edi
In what sense is the pilot wave formulation in physical space? The functions rho, S and Q in (14.2) and (14.3) are functions of configuration space, if I understand correctly. Of course, if there is only one particle, configuration space turns into physical space. But in general, if there is more than one particle, the pilot wave formulation appears to be in configuration space. | 2021-05-11 06:55:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7382722496986389, "perplexity": 890.4972439874175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00190.warc.gz"} |
https://www.lessonplanet.com/teachers/one-hundred-hungry-ants-by-elinor-j-pinczes | # "One Hundred Hungry Ants" by Elinor J. Pinczes
Students identify arrays and write multiplication and division problems to match. They write their own multiplication book using arrays. | 2017-05-23 00:03:00 | {"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": 0.893000602722168, "perplexity": 10927.42173681632}, "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/1495463607242.32/warc/CC-MAIN-20170522230356-20170523010356-00059.warc.gz"} |
https://www.physicsforums.com/threads/indexed-cartesian-product-as-sets-of-functions.511201/ | # Indexed Cartesian Product as sets of functions
1. Jul 1, 2011
### TopCat
I'm going through the set theory material in the appendix of Knapp's Basic Algebra. I want to make sure that I understand what he says is the set theoretic notion of the indexed cartesian product, $\prod_{x\in S}A_{x}$.
He says that this can be thought of as the set of all functions $f:S\rightarrow \bigcup_{x\in S}A_{x}$ such that $f(x)\in A_{x}$ for all $x\in S$. Let's call this set $F$.
So as an example, let $S = \left\{1,2\right\}$, $A_{1} = \left\{3,4\right\}$, and $A_{2} = \left\{5,6\right\}$. Then $\bigcup_{x\in S}A_{x} = \left\{3,4,5,6\right\}$ and the functions f, being subsets of $S\times \bigcup_{x\in S}A_{x}$, are $f_{1} = \left\{(1,3), (2,5)\right\}$, $f_{2} = \left\{(1,3), (2,6)\right\}$, $f_{3} = \left\{(1,4), (2,5)\right\}$, $f_{4} = \left\{(1,4), (2,6)\right\}$.
Then the set $F$ is $\left\{\left\{(1,3), (2,5)\right\},\left\{(1,3), (2,6)\right\},\left\{(1,4), (2,5)\right\},\left\{(1,4), (2,6)\right\}\right\}$.
Since we can form a bijection $g$ from $F$ to $A_{1}\times A_{2}$ with $g:F\rightarrow A_{1}\times A_{2}$ such that $g(f) = (f(1), f(2))$ for all $f\in F$, we can say that F is isomorphic to $A_{1}\times A_{2}$ and thus they are the same set. Is my understanding correct?
2. Jul 1, 2011
### SteveL27
It looks like you got all the details right, so your understanding is correct.
However, a set of ordered n-tuples is NOT the same set as the set of choice functions. Remember, the axiom of extensionality says that two sets are the same iff they have exactly the same elements. So two sets that have different elements but represent the same structure, are still different sets.
The use of the word isomorphic is true in a casual sense, but you would need to define isomorphism for sets in order for it to be literally true. The idea of isomorphism is typically applied to sets equipped with some algebraic structure, for example addition and/or multiplication defined.
It's true that the two definitions of Cartesian product (n-tuples or choice functions) express the same idea in the case of a finite index set. But I haven't seen the word isomorphism used to express that fact. I could be wrong. What is your definition of isomorphism of sets?
Last edited: Jul 1, 2011
3. Jul 1, 2011
### TopCat
I came across another thread on this forum and stumbled upon the use of isomorphism from a post by Hurkyl: https://www.physicsforums.com/showpost.php?p=1342101&postcount=4
I get that the two sets aren't the same, but are functionally "equivalent" (my choice of words as "same set" was very poor). I, like the OP in that old thread, find it odd that the cartesian product was touted (in this case by Knapp) as exactly the set of all such functions, when it was defined earlier as a set of -tuples. In fact, I realized that I mistyped earlier; Knapp doesn't say that the indexed cartesian product "can be thought of" as the set of all such functions, but that "it is" this set.
However, I'm satisfied with the details given in the other thread on the matter. Thanks for confirming that I wasn't misunderstanding the concept.
4. Jul 1, 2011
### SteveL27
In that thread, Hurkyl defined isomorphism as bijection. But that's too weak; because in that case ANY set of cardinality |A| * |B| is isomorphic to the Cartesian product A X B. As excellent as Hurkyl's answers usually are, it is not correct to say that the set of n-tuples is isomorphic to the set of choice functions (for a finite index set) since we do not have a sufficiently strong notion of isomorphism for sets.
The notion of bijection only says two sets have the same cardinality. It's not a strong enough notion to encapsulate the structure of a Cartesian product. And arbitrary sets don't have any additional structure, so we can't use any extra structure to make a precise definition of isomorphism.
I apologize for taking the pedantic route here; but it's always important to understand the exact meaning of things before we get fast and loose with terminology.
By the way the reason for introducing the choice function definition of Cartesian product is to handle the case of an infinite index set. We can't form infinity-tuples via the axioms of set theory; but we CAN form collections of choice sets, even on uncountable index sets. That's why the Cartesian product is defined as the set of choice functions on the index set.
5. Jul 1, 2011
### TopCat
So you're saying that it's not important that the sets (tuples and choice functions) be the same; they represent two distinct set theoretic notions of the cartesian product. One is limited yet, perhaps, easier to work with. The other is more robust and abstract. That we can, via a bijection, obtain one from the other demonstrates that both definitions are same, though their output sets are not. Is that right?
Also, I greatly appreciate your pedantry here. You've made this very clear for me.
6. Jul 1, 2011
### SteveL27
Yes, exactly.
Yes, exactly.
No. The problem is that the notion of bijection is too weak; but we don't have enough structure to define isomorphism.
For example, let S, A1, and A2 be as in your original example. Then A1 X A2 has exactly four elements.
We can think of A1 X A2 as a set of ordered pairs, or as a set of choice functions. Either way, there are four of them, and we can biject the ordered pairs to the choice functions.
However, this does not capture the idea of a Cartesian product. It only captures the idea of fourness. If I give you the set {1, 2, 3, pi}, that's a set of four elements that is bijectively equivalent to the Cartesian product. But it does not have the structure of the Cartesian product.
So the notion of bijection is perfect for characterizing the idea of cardinality, or "how many?" But it's not strong enough to characterize the idea of Cartesian product. All you can conclude from bijecting the ordered pairs to the choice functions is that they have the same number of elements. You have not demonstrated that they have the same structure.
On the other hand, we would like to use the term "isomorphism," but arbitrary sets don't have enough structure to do that.
I don't know offhand if there is some general way to show that the two definitions of Cartesian product are "the same" in the case of a finite index. I might have to think about that a bit. Perhaps there's a category-theoretic characterization. Can anyone help out here? It's an interesting question.
As far as isomorphism, you need some additional structure on sets in order to define that notion. So you need some algebraic structure, such as + or *; or perhaps an order relationship.
Ah ... I think I have an idea. We could partially order a Cartesian product by set or function inclusion; and then we could show that the n-tuples and the choice functions are order-isomorphic as partially ordered sets. Details for the reader I think that gets us a step in the direction of showing that the n-tuples and the choice functions give us the "same" Cartesian product -- in other words, in developing the correct definition of "isomorphism" for the two definitions of Cartesian product.
[Side remark: This illustrates a general idea in math. The interesting part of math is often to come up with the right definition. In the textbooks they GIVE us the definitions; but the historical development is often a long struggle to FIND the right definition. Or as they say: The theorems are easy. The definitions are hard!]
(edit)
Ah I remember now. There is a category-theoretic answer. The Cartesian product is the product in the category of sets. So there IS a way to show that the two definitions of Cartesian product are the same, but you have to use the idea of a universal property.
http://en.wikipedia.org/wiki/Product_(category_theory)
Last edited: Jul 1, 2011
7. Jul 5, 2011
### TopCat
Wow, excellent. Your response is very clear and I totally appreciate the effort you made in explaining it to me.
I know category theory is useful for algebra, so this is good motivation to start looking at some important definitions and theorems before it's delved into in the text. | 2017-09-23 13:32:16 | {"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": 0.9070578813552856, "perplexity": 230.70993673787714}, "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-2017-39/segments/1505818689661.35/warc/CC-MAIN-20170923123156-20170923143156-00664.warc.gz"} |
https://zbmath.org/?q=an%3A0902.35010 | ## Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis.(English)Zbl 0902.35010
Summary: This paper is concerned with a nonlinear parabolic-elliptic system which is a simplified version of the Keller-Segel model with a sensitivity function $$\phi(s)$$ specified as $$\phi(s)= s^p$$ $$(p>0)$$ or $$\phi(s)= \log s$$. The global existence and blow-up of solutions are studied in radially symmetric situations.
### MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations | 2023-02-08 16:22:19 | {"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": 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": 0.4490150511264801, "perplexity": 909.6932664489032}, "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/1674764500837.65/warc/CC-MAIN-20230208155417-20230208185417-00494.warc.gz"} |
http://gmatclub.com/forum/lines-n-and-p-lie-in-the-xy-plane-is-the-slope-of-line-n-127999.html?kudos=1 | Find all School-related info fast with the new School-Specific MBA Forum
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Lines n and p lie in the xy-plane. Is the slope of line n
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Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
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BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
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23 Feb 2012, 08:13
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BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Graphic approach:
Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?
(1) Lines n and p intersect at (5,1)
(2) The y-intercept of line n is greater than y-intercept of line p
The two statements individually are not sufficient.
(1)+(2) Note that a higher absolute value of a slope indicates a steeper incline.
Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is steeper hence its slope is greater than the slope of the line n:
If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:
So in both cases the slope of p is greater than the slope of n. Sufficient.
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23 Feb 2012, 10:13
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nglekel wrote:
Bunuel,
What if line p has a negative y intercept but line n has a positive intercept? Wouldn't that give the oposite answer?
If line p has a negative y-intercept then its slope is positive and it will still be more than the slope of n, with positive y-intercept (if the slope of n will be positive than p will still be steeper than n, and if the slope of n is negative it obviously will be less than positive slope of p). Consider first image and rotate line n (blue) so that it to have positive y-intercept and you'll easily see the answer.
Check the following links for similar questions, where different scenarios are considered:
in-the-xy-plane-is-the-slope-of-line-l-greater-than-the-126941.html
if-the-slopes-of-the-line-l1-and-l2-are-of-the-same-sign-is-126759.html
slopes-of-m-and-n-124025.html
Also check Coordinate Geometry chapter of Math Book for theory on this subject: math-coordinate-geometry-87652.html
Hope it helps.
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23 Feb 2012, 09:05
Bunuel,
What if line p has a negative y intercept but line n has a positive intercept? Wouldn't that give the oposite answer?
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Re: Lines n and p lie in the xy-plane. Is the slope of line n le [#permalink]
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13 Sep 2012, 12:19
monikaleoster wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5,1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Here we have two lines and two slopes
So lets first write the equations for our lovely lines
Y = mnX + Cn
Y = mpX + Cp
Now statement one says that it intersects at 5,1. SO lets put it in the equations and subtract them
We get, (mn-mp)5 = Cp-Cn that tells us nothing about the slopes of the lines or their relative values, but if we know the value of Cp-Cn that weather it is positive or negative we will know weathet mn-mp is positive or negative and that which is greater
Statement 2 Y intercept of line n is greater than p so that gives us Cn >Cp
Alone this statement is also not sufficient. it talks abou c not slopes
But if we combine the two, Voila !! we know which slope is greater.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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29 Dec 2012, 09:18
I don't get it :/
what if slope of line P is positive and the slope of line N negative (but still satisfying all the condition...)
Bunuel, on your examples the slopes have the same sign... are we talking about the absolute value of the slope?
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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16 Jul 2013, 19:23
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
Hope it helps.
Bunuel,
In here - $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
Why have you chosen different variables for the y?
Shouldnt the two equations be y=m1x+b1 and y=m2x+b2? We always form the equation from the basic form of y=mx+c wherein we substitute the values of m and c. And if that is the case, we can get the answer from statement II only.
I know I am missing something but I am not clear as to why you have picked different variables for y but not for x.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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16 Jul 2013, 23:08
keenys wrote:
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
Hope it helps.
Bunuel,
In here - $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
Why have you chosen different variables for the y?
Shouldnt the two equations be y=m1x+b1 and y=m2x+b2? We always form the equation from the basic form of y=mx+c wherein we substitute the values of m and c. And if that is the case, we can get the answer from statement II only.
I know I am missing something but I am not clear as to why you have picked different variables for y but not for x.
n and p are subscripts of y's, not variables.
$$y=m_1x+b_1$$ is equation of line n.
$$y=m_2x+b_2$$ is equation of line p.
I used subscripts simply to distinguish one equation from another.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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17 Jul 2013, 05:02
Bunuel wrote:
keenys wrote:
Bunuel wrote:
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
Hope it helps.
Bunuel,
In here - $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
Why have you chosen different variables for the y?
Shouldnt the two equations be y=m1x+b1 and y=m2x+b2? We always form the equation from the basic form of y=mx+c wherein we substitute the values of m and c. And if that is the case, we can get the answer from statement II only.
I know I am missing something but I am not clear as to why you have picked different variables for y but not for x.
n and p are subscripts of y's, not variables.
$$y=m_1x+b_1$$ is equation of line n.
$$y=m_2x+b_2$$ is equation of line p.
I used subscripts simply to distinguish one equation from another.
If that is the case then, from the above equations
we get b1=y-m1x and b2=y-m2x
Now from statement 2 we know that b1>b2...
therefore, y-m1x >y-m2x
which gives (m1-m2)x>0
So it can be proved from statement 2 only that m1>m2
Where am I going wrong?
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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17 Jul 2013, 06:07
keenys wrote:
Bunuel wrote:
keenys wrote:
Bunuel,
In here - $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
Why have you chosen different variables for the y?
Shouldnt the two equations be y=m1x+b1 and y=m2x+b2? We always form the equation from the basic form of y=mx+c wherein we substitute the values of m and c. And if that is the case, we can get the answer from statement II only.
I know I am missing something but I am not clear as to why you have picked different variables for y but not for x.
n and p are subscripts of y's, not variables.
$$y=m_1x+b_1$$ is equation of line n.
$$y=m_2x+b_2$$ is equation of line p.
I used subscripts simply to distinguish one equation from another.
If that is the case then, from the above equations
we get b1=y-m1x and b2=y-m2x
Now from statement 2 we know that b1>b2...
therefore, y-m1x >y-m2x
which gives (m1-m2)x>0
So it can be proved from statement 2 only that m1>m2
Where am I going wrong?
The y-intercept is the value of $$y$$ for $$x=0$$. You should substitute x=0 into both equations.
So, the y-intercept of line n is b1 and the y-intercept of line p is b2, from (2) we only have that b1>b2.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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28 Dec 2013, 05:57
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Graphic approach:
Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?
(1) Lines n and p intersect at (5,1)
(2) The y-intercept of line n is greater than y-intercept of line p
The two statements individually are not sufficient.
(1)+(2) Note that a higher absolute value of a slope indicates a steeper incline.
Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is steeper hence its slope is greater than the slope of the line n:
Attachment:
1.PNG
If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:
Attachment:
2.PNG
So in both cases the slope of p is greater than the slope of n. Sufficient.
So talking about the case with negative slopes here. OK so line 'n' is steeper hence it has a higher absolute value for slope right? Then because it is more negative then it is in fact smaller than the slope of line p.
So here we are saying that the slope is treated just as any number, which means considering its sign
Eg. Slope of an horizontal line will be higher than a negative slope right? Just cause if one does it algebraically one will encounter the comparison and absolute values are not used at all in slope formulae as far as I'm aware
Just wanted to get this straight
Cheers!
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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02 Sep 2014, 03:54
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
Hi Bunuel
Pl clarify my doubt. Do we need to consider absolute values for slopes? I mean is it |m1|<|m2| ? or m1<m2? to consider
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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02 Sep 2014, 04:25
Sidhrt wrote:
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
Hi Bunuel
Pl clarify my doubt. Do we need to consider absolute values for slopes? I mean is it |m1|<|m2| ? or m1<m2? to consider
The question asks whether $$m_1<m_2$$.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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02 Sep 2014, 04:33
Sidhrt wrote:
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
Hi Bunuel
Pl clarify my doubt. Do we need to consider absolute values for slopes? I mean is it |m1|<|m2| ? or m1<m2? to consider
then without taking absolute values how we can conclude m1<m2? as we not sure m1<0 or m1>0 or m2<0 or m2>0 ( such as m1=1, m2=5 or m2=-5,m1=-1)
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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02 Sep 2014, 04:54
Sidhrt wrote:
Sidhrt wrote:
Bunuel wrote:
Algebraic approach:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?
We have two lines: $$y_n=m_1x+b_1$$ and $$y_p=m_2x+b_2$$. Q: $$m_1<m_2$$ true?
(1) Lines n and p intersect at the point (5,1) --> $$1=5m_1+b_1=5m_2+b_2$$ --> $$5(m_1-m_2)=b_2-b_1$$. Not sufficient.
(2) The y-intercept of line $$n$$ is greater than the y-intercept of line $$p$$ --> y-intercept is value of $$y$$ for $$x=0$$, so it's the value of $$b$$ --> $$b_1>b_2$$ or $$b_2-b_1<0$$. Not sufficient.
(1)+(2) $$5(m_1-m_2)=b_2-b_1$$, as from (2) $$b_2-b_1<0$$ (RHS), then LHS (left hand side) also is less than zero $$5(m_1-m_2)<0$$ --> $$m_1-m_2<0$$ --> $$m_1<m_2$$. Sufficient.
For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
Hi Bunuel
Pl clarify my doubt. Do we need to consider absolute values for slopes? I mean is it |m1|<|m2| ? or m1<m2? to consider
then without taking absolute values how we can conclude m1<m2? as we not sure m1<0 or m1>0 or m2<0 or m2>0 ( such as m1=1, m2=5 or m2=-5,m1=-1)
Dear Sidhrt, there are two approaches given above explaining that.
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Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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06 Jan 2016, 13:43
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Hi math experts, I've used the following approach to derive on the correct answer. Would appreciate your input.
(1) It's not possible to calculate the slope with only one point given. Also it's an intersection point which satisfies the equations of both lines. Not Sufficient
(2) You cannot determine a slope, with info about y-intercept, one can manipulate randomply the x-line intersection and get diff. results regarding slopes.
(1)+(2) We have point (5,1) and info about y intercept. By y-intercept x=0 and we know that line n has a greater y-intercept:
Case 1 +ve: Line n (0, 3) and Line p (0, 2) --> Slope n $$= \frac{1-3}{5}=-\frac{2}{5}$$, Slope p$$=\frac{1-2}{5}=-\frac{1}{5}$$, So Slope p > Slope n
Case 2 -ve: Line n (0, -2) and Line p (0, -3) --> Slope n $$= \frac{1-(-2)}{5}=\frac{3}{5}$$, Slope p=$$\frac{1-(-3)}{5}=\frac{4}{5}$$
Again Slope p > Slope n
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink]
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10 Mar 2016, 22:36
Bunuel wrote:
BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
Graphic approach:
Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?
(1) Lines n and p intersect at (5,1)
(2) The y-intercept of line n is greater than y-intercept of line p
The two statements individually are not sufficient.
(1)+(2) Note that a higher absolute value of a slope indicates a steeper incline.
Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is steeper hence its slope is greater than the slope of the line n:
Attachment:
1.PNG
If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:
Attachment:
2.PNG
So in both cases the slope of p is greater than the slope of n. Sufficient.
We have tried both slopes negative or both slopes positive, why we didn't tried one slope positive and another negative combination.
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Re: Lines n and p lie in the xy-plane. Is the slope of line n [#permalink] 10 Mar 2016, 22:36
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https://www.aimsciences.org/article/doi/10.3934/cpaa.2014.13.2115 | # American Institute of Mathematical Sciences
September 2014, 13(5): 2115-2126. doi: 10.3934/cpaa.2014.13.2115
## On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach
1 Department of Mathematics, Texas A\&M University, College Station, TX 77845, United States 2 University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008
Received September 2013 Revised January 2014 Published June 2014
In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace-Beltrami operator for smooth surfaces in $R^3$ (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briefly discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of $R^3$, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our computational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).
Citation: Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115
##### References:
show all references
##### References:
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2019 Impact Factor: 1.105 | 2021-01-17 02:54:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6978666186332703, "perplexity": 4088.646721111911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703509104.12/warc/CC-MAIN-20210117020341-20210117050341-00052.warc.gz"} |
https://figshare.mq.edu.au/articles/thesis/Restriction_presheaves_and_restriction_colimits/19427732/1 | 01whole.pdf (863.97 kB)
# Restriction presheaves and restriction colimits
thesis
posted on 28.03.2022, 01:41 authored by Daniel Lin
Restriction categories, as defined by Cockett and Lack, are an abstraction of the notion of partial functions between sets, and therefore, are important in furthering our understanding of what it means to be partial. This thesis builds upon the work of Cockett and Lack, by providing restriction analogues of notions from ordinary category theory. One such notion is that of free cocompletion. We show that every restriction category may be freely completed to a cocomplete restriction category, and that this free cocompletion can be described in terms of a restriction category of restriction presheaves. Indeed, a restriction presheaf is defined precisely so that this is the case. We then generalise free cocompletion to join restriction categories, which are categories whose compatible maps may be combined in some way. To do this, we introduce the notion of join restriction presheaf, and show that for any join restriction category, its join restriction category of join restriction presheaves is its free cocompletion. The second half of this thesis explores the notion of restriction colimit. More precisely,we define the restriction colimit of a restriction functor weighted by a restriction presheaf. We also show that cocomplete restriction categories may be characterised as those having all such restriction colimits. Finally, we give applications of restriction colimits. Some examples of restriction colimits are gluings of atlases in a restriction category, and composition o frestriction profunctors. We conclude this thesis with notions in category theory that have no analogue in the restriction setting.
## History
1. Introduction -- 2. Cocompletion of restriction categories -- 3. Free cocompletion of locally small restriction categories -- 4. Restriction presheaves -- 5. Cocompletion of join restriction categories -- 6. Join restriction presheaves -- 7. Restriction colimits -- 8. Atlases and their gluings -- 9. Restriction profunctors and other restriction definitions -- References.
## Notes
Bibliography: pages 95-96 Empirical thesis.
## Awarding Institution
Macquarie University
Thesis PhD
## Degree
PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics and Statistics
## Department, Centre or School
Department of Mathematics and Statistics
2019
Richard Garner
English
## Extent
1 online resource (viii, 96 pages)
## Former Identifiers
mq:70988 http://hdl.handle.net/1959.14/1269712
## Exports
figshare. credit for all your research. | 2022-10-07 06:49:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8280330896377563, "perplexity": 1636.165733860963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337971.74/warc/CC-MAIN-20221007045521-20221007075521-00296.warc.gz"} |
http://ecform6physics.blogspot.com/2012/03/efficiency.html | ## Sunday, 18 March 2012
### Efficiency
The efficiency of an entity (a device, component, or system) in electronics and electrical engineering is defined as useful power output divided by the total electrical power consumed (a fractional expression), typically denoted by the Greek letter small Eta (η).
$\mathrm{Efficiency}=\frac{\mathrm{Useful\ power\ output}}{\mathrm{Total\ power\ input}}$ | 2018-04-21 23:10:16 | {"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": 1, "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": 0.716334342956543, "perplexity": 3377.4547978581213}, "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/1524125945459.17/warc/CC-MAIN-20180421223015-20180422003015-00166.warc.gz"} |
http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=1044 | 2019
Том 71
№ 1
# Zastavnyi V. P.
Articles: 3
Article (Ukrainian)
### On some properties of Buhmann functions
Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1045–1067
We study functions introduced by Buhmann. The exact exponent of smoothness of these functions is obtained and the problem of positivity of their Hankel transforms is analyzed.
Brief Communications (Russian)
### Extension of a Function from the Exterior of an Interval to a Positive-Definite Function on the Entire Axis and an Approximation Characteristic of the Class $W_M^{r, β}$
Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 983-990
We establish sufficient conditions for the extension of a function defined on $[a, +∞)$, where $a > 0$, to a positive-definite function on the entire axis.
Article (Ukrainian)
### Zero set of the Fourier transform of measures and the summation of double Fourier series of methods of Bernshtein-Rogozinskii type
Ukr. Mat. Zh. - 1984. - 36, № 5. - pp. 615–621 | 2019-02-20 09:32:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5958381295204163, "perplexity": 893.6848150516296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247494694.1/warc/CC-MAIN-20190220085318-20190220111318-00171.warc.gz"} |
http://mathhelpforum.com/algebra/171389-sigma-summation-problem-print.html | # sigma/summation problem
• February 14th 2011, 08:24 PM
math321
simple problem
$\sum xy -\frac { (\sum x) (\sum y)}{n}$
x= 1, 3, 7, 10
y= 2, 0, 4, 8
• February 14th 2011, 08:32 PM
pickslides
These are bivariate observations, then
$\displaystyle \sum xy -\frac { (\sum x) (\sum y)}{n} =1\times 2 + 3\times 0 +\dots +10\times 8 - \frac{(1+3+\dots +10)(2+0+\dots +8)}{4}= \dots$
• February 14th 2011, 08:38 PM
math321
why did u put n = 4
• February 14th 2011, 08:53 PM
pickslides
Well n=4 if the sample is counted as one per (x,y)
• February 15th 2011, 06:08 PM
topsquark
Both sums are right. What's n? (That's the question you were last on in the other thread, too.)
-Dan
• February 15th 2011, 06:12 PM
math321
n represents the number of points of data
so what i suppose to do
• February 15th 2011, 06:17 PM
topsquark
Quote:
Originally Posted by math321
n represents the number of points of data
so what i suppose to do
Your data set is {(x, y)} = (1, 2), (3, 0), (7, 4), (10, 8).
-Dan
• February 15th 2011, 06:20 PM
math321
so therefore n=4
• February 15th 2011, 06:21 PM
math321
got 36.5
• February 16th 2011, 04:35 AM
HallsofIvy
What exactly is your question? Do you not understand what the $\sum$ means? It just means "add them up". I assume that n is 4 here.
$\sum_{n=1}^4 x_ny_n- \frac{\left(\sum_{n=1}^4 x_n\right)\left(\sum_{n=1}^4 y_n\right)}{4}$
$= (1)(2)+ (3)(0)+ (7)(4)+ (10)(8)- \frac{(1+ 3+ 7+ 10)(2+ 0+ 4+ 8)}{4}$ | 2013-12-20 09:49:28 | {"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": 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": 0.8662803769111633, "perplexity": 5263.283956879651}, "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-2013-48/segments/1387345771373/warc/CC-MAIN-20131218054931-00084-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.cs.utexas.edu/users/flame/laff/alaff/chapter04-conditioning-of-lls.html | Subsection4.2.4Conditioning of the linear least squares problem
Given $A \in \Cmxn$ with linearly independent columns and $b \in \Cm \text{,}$ consider the linear least squares (LLS) problem
$$\| b - A \widehat x \|_2 = \min_x \| b - A x \|_2 \label{eqn-LLS1}\tag{4.2.1}$$
and the perturbed problem
$$\| (b + \delta\!b ) - A (\widehat x+\delta\! \widehat x ) \|_2 = \min_{x} \| ( b + \delta\!b ) - A (x+\delta\!x ) \|_2 \label{eqn-LLS2}\tag{4.2.2}$$
The question we want to examine is by how much the relative error in $b$ is amplified into a relative error in $\widehat x \text{.}$ We will restrict our discussion to the case where $A$ has linearly independent columns.
Now, we discovered that $\widehat b \text{,}$ the projection of $b$ onto the column space of $A \text{,}$ satisfies
$$\widehat b = A \widehat x \label{eqn-LLS-cond1}\tag{4.2.3}$$
and the projection of $b + \delta\! b$ satisfies
$$\widehat b + \delta\! \widehat b = A ( \widehat x + \delta\! \widehat x ) \label{eqn-LLS-cond2}\tag{4.2.4}$$
where $\delta\! \widehat b$ equals the projection of $\delta\!b$ onto the column space of $A \text{.}$
Let $\theta$ equal the angle between vectors $b$ and its projection $\widehat b$ (which equals the angle between $b$ and the column space of $A$). Then
\begin{equation*} \cos(\theta) = \| \widehat b \|_2 / \| b \|_2 \end{equation*}
and hence
\begin{equation*} \cos(\theta) \| b \|_2 = \| \widehat b \|_2 = \| A \widehat x \|_2 \leq \| A \|_2 \| \widehat x \|_2 = \sigma_0 \| \widehat x \|_2 \end{equation*}
which (as long as $\widehat x \neq 0$) can be rewritten as
$$\frac{1}{\| \widehat x \|_2} \leq \frac{\sigma_0}{\cos( \theta )} \frac{1}{\| b \|_2}. \label{eqn-LLS-cond3}\tag{4.2.5}$$
Subtracting (4.2.3) from (4.2.4) yields
\begin{equation*} \delta\! \widehat b = A \delta\! \widehat x \end{equation*}
or, equivalently,
\begin{equation*} A \delta\! \widehat x = \delta\! \widehat b \end{equation*}
which is solved by
\begin{equation*} \begin{array}{rcl} \delta\! \widehat x \amp = \amp A^\dagger \delta\! \widehat b \\ \amp = \amp A^\dagger A ( A^H A )^{-1} A^H \delta b \\ \amp = \amp ( A^H A )^{-1} A^H A ( A^H A )^{-1} A^H \delta b \\ \amp = \amp A^\dagger \delta\! b , \end{array} \end{equation*}
where $A^\dagger = ( A^H A )^{-1} A^H$ is the pseudo inverse of $A$ and we recall that $\delta\! \widehat b = A ( A^H A )^{-1} A^H \delta b \text{.}$ Hence
$$\| \delta\! \widehat x \|_2 \leq \| A^\dagger \|_2 \| \delta\! b \|_2.\label{eqn-LLS-cond4}\tag{4.2.6}$$
Homework4.2.4.1.
Let $A \in \Cmxn$ have linearly independent columns. Show that
\begin{equation*} \| ( A^H A )^{-1} A^H \|_2 = 1/\sigma_{n-1}, \end{equation*}
where $\sigma_{n-1}$ equals the smallest singular value of $A \text{.}$
Hint
Use the reduced SVD of $A \text{.}$
Solution
Let $A = U_L \Sigma_{TL} V^H$ be the reduced SVD of $A \text{,}$ where $V$ is square because $A$ has linearly independent columns. Then
\begin{equation*} \begin{array}{l} \| ( A^H A )^{-1} A^H \|_2 \\ ~~~=~~~~\\ \| ( ( U_L \Sigma_{TL} V^H )^H U_L \Sigma_{TL} V^H )^{-1} ( U_L \Sigma_{TL} V^H )^H \|_2 \\ ~~~=~~~~\\ \| ( V \Sigma_{TL} U_L^H U_L \Sigma_{TL} V^H )^{-1} V \Sigma_{TL} U_L^H \|_2 \\ ~~~=~~~~\\ \| ( V \Sigma_{TL}^{-1} \Sigma_{TL}^{-1} V^H ) V \Sigma_{TL} U_L^H \|_2 \\ ~~~=~~~~\\ \| V \Sigma_{TL}^{-1} U_L^H \|_2 \\ ~~~=~~~~\\ \| \Sigma_{TL}^{-1} U_L^H \|_2 \\ ~~~=~~~\\ 1/\sigma_{n-1}. \end{array} \end{equation*}
This last step needs some more explanation: Clearly $\| \Sigma_{TL} U_L^H \|_2 \leq \| \Sigma_{TL} \|_2 \| U_L^H \|_2 = \sigma_{0} \| U_L^H \|_2 \leq \sigma_0 \text{.}$ We need to show that there exists a vector $x$ with $\| x \|_2 = 1$ such that $\| \Sigma_{TL} U_L^H x \|_2 = \| \Sigma_{TL} U_L^H \|_2 \text{.}$ If we pick $x = u_0$ (the first column of $U_L$), then $\| \Sigma_{TL} U_L^H x \|_2 = \| \Sigma_{TL} U_L^H u_0 \|_2 = \| \Sigma_{TL} e_0 \|_2 = \| \sigma_0 e_0 \|_2 = \sigma_0 \text{.}$
Combining (4.2.5), (4.2.6), and the result in this last homework yields
$$\frac{\| \delta\! \widehat x\|_2}{\| \widehat x \|_2} \leq \frac{1}{\cos( \theta )} \frac{\sigma_0}{\sigma_{n-1}} \frac{\| \delta\! b \|_2}{\| b \|_2}. \label{eqn-LLS-cond5}\tag{4.2.7}$$
Notice the effect of the $\cos(\theta)b \text{.}$ If $b$ is almost perpendicular to ${\cal C}(A) \text{,}$ then its projection $\widehat b$ is small and $\cos \theta$ is small. Hence a small relative change in $b$ can be greatly amplified. This makes sense: if $b$ is almost perpendical to $\Col( A ) \text{,}$ then $\widehat x \approx 0 \text{,}$ and any small $\delta\!b \in \Col(A)$ can yield a relatively large change $\delta\!x \text{.}$
Definition4.2.4.1. Condition number of matrix with linearly independent columns.
Let $A \in \Cmxn$ have linearly independent columns (and hence $n \leq m$). Then its condition number (with respect to the 2-norm) is defined by
\begin{equation*} \kappa_2( A ) = \| A \|_2 \| A^\dagger \|_2 = \frac{\sigma_0}{\sigma_{n-1}}. \end{equation*}
It is informative to explicity expose $\cos( \theta ) = \| \widehat b \|_2/ \| b \|_2$ in (4.2.7):
\begin{equation*} \frac{\| \delta\! \widehat x\|_2}{\| \widehat x \|_2} \leq \frac{\| b \|_2}{\| \widehat b \|_2} \frac{\sigma_0}{\sigma_{n-1}} \frac{\| \delta\! b \|_2}{\| b \|_2}. \end{equation*}
Notice that the ratio
\begin{equation*} \frac{\| \delta\! b \|_2}{\| b \|_2} \end{equation*}
can be made smaller by adding a component, $b_r \text{,}$ to $b$ that is orthogonal to $\Col( A )$ (and hence does not change the projection onto the column space, $\widehat b$):
\begin{equation*} \frac{\| \delta\! b \|_2}{\| b + b_r \|_2}. \end{equation*}
The factor $1/\cos( \theta )$ ensures that this does not magically reduce the relative error in $\widehat x \text{:}$
\begin{equation*} \frac{\| \delta\! \widehat x\|_2}{\| \widehat x \|_2} \leq \frac{\| b + b_r \|_2}{\| \widehat b \|_2} \frac{\sigma_0}{\sigma_{n-1}} \frac{\| \delta\! b \|_2}{\| b + b_r \|_2}. \end{equation*} | 2022-08-11 17:21:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 7, "x-ck12": 0, "texerror": 0, "math_score": 0.9995608925819397, "perplexity": 2038.0610741503592}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00470.warc.gz"} |
http://www.devmathrevival.net/?p=2628?shared=email&msg=fail | ## Can We Save “Order of Operations”??
In one recent post, I looked at some basic flaws in the mnemonic “PEMDAS” (there are several fundamental flaws). In another recent post, I talked about how unimportant a ‘correct answer’ can be in a math class. Let’s examine the intersection of those thoughts, and deal with saving the important topic of ‘order of operations’.
The two most common statements about why “order of operations” is important are:
• “The order of operations is just an agreement so we all get the same answer.”
• “You need to follow the order of operations so that you will get the correct answer.”
Both of these miss the point; their implication is that we can change the correct answer just by changing the ‘agreement’ about order of operations … that we could declare subtraction is always done before multiplying, for example. The order of operations is not just some coincidence of the mathematical language which will evolve to be anything fundamentally different.
The reason the ‘order of operations’ is so important is that the meaning of a mathematical statement is based on understanding the order of operations. In natural languages, the presence of multiple verbs in a statement is unusual … in mathematics, this is commonplace. Multiple operations in a statement with nouns and adjectives provides an efficient method of communication, which is why scientific advances increased dramatically after the use of symbolic mathematics (as opposed to the original verbal forms).
Not only does “PEMDAS” have little to do with correct order of operations, the way ‘order of operations’ is typically taught has little to do with mathematics.
When we learn a computer programming language, we face the issue directly — what is the precedence order for operations? Although there are some minor differences in the details, almost all precedence orders are based on a fundamental mathematical idea:
The more advanced operations are done prior to simpler operations.
We teach students that exponents are repeated multiplications; what we don’t divulge is that this means that exponents are more advanced operations … and therefore are done prior to multiplying. We cover the procedures for multiplying and dividing fractions, but do not make sure that students know that these procedures are based on the fact that multiplying and dividing are at the same level of complexity, mathematically speaking.
The fundamental idea that “more advanced operations are done first” covers the majority of what we try to do with ‘order of operations’. The difference is this: order of operations is treated as a memorization issue, while ‘more advanced operations first’ is calling for understanding and communication. How students get to a ‘correct answer’ is more important than the fact that they got a correct answer.
In those computer programming languages, operations are categorized into binary and unary types, just as mathematicians do. The ‘more advanced first’ principle handles almost all cases in both types. Even the type some of us complain about:
-5²
Even though this ‘ambiguity’ is not encountered very often in real-world problems, this is a core issue in communication. How do we interpret:
-x²
We certainly don’t want people to apply the opposite operation prior to squaring, and we certainly don’t want the answer to change when given in variable notation. In both of these problems, the “-” means opposite … which is less advanced than squaring; therefore, square first, then apply the opposite.
The few places where ‘more advanced first’ fails are also places where ‘order of operations’ fails, and these are often due to our failures to maintain integrity in our language. Our notation for trig functions is sometimes bad, or even incorrect (when it creates an inconsistency with other operations or functions). Even if we don’t change our behavior in trig functions, students will be better off with ‘more advanced first’ than they are currently.
I’d be happier if we never used the phrase ‘order of operations’; the entire implication of this phrase is ‘memorize the rules, or else’. Our students would have a higher quality learning experience if we just focused on ‘more advanced operations first’. The emphasis this involves on the meaning of expressions helps novices reach a deeper understanding of our mathematical language.
Which of these is a better answer to the question “why did you multiply before you subtracted”:
• I multiplied first because the order of operations says to multiply before subtracting.
• I multiplied first because multiplying has a higher precedence because multiplying is more advanced.
As we strive to help our students understand and reason in mathematics, an ‘order of operations’ has no place in the curriculum. Knowing a structure for operations, including ‘more advanced’, is critical.
Join Dev Math Revival on Facebook:
• By schremmer, October 14, 2016 @ 11:05 am
As it happens, because I was given an arithmetic course a couple of weeks before the semester started and, as usual, could not resolve myself using “the” textbook, I have been writing one on the same table of contents, a chapter or two ahead of the class. All this to explain why, as I sat in front of my computer to write the chapter on … “order of operations”, I was very happy to see Rotman’s take on the issue. However, I will have to defer my opinion on the merits of his case until later today when I have written said chapter.
• By schremmer, October 14, 2016 @ 9:54 pm
Again, I wholeheartedly agree with Rotman. No reservation and here are a few observations.
1. Rotman mentions “the meaning of a mathematical statement</em" and he even invokes what happens in "natural languages” and “computer languages“. In other words, he does distinguish between the real world and the paper world in which we represent what happens in the real world but he doesn’t go far enough. The general idea ought to be the systematic use of a “Model Theoretic” approach. (Nothing to do with”modeling”. (See Symbolic Systems.)
2. I don’t like the phrase “more advanced operation” too much because it seems to evoke “advanced mathematics”.
3. I don’t like the phrase “order of operations” either but it does make for a good chapter title. I can’t seem to find a good alternative.
4. Without getting into details, the difficulty with powers is that we introduce them with the coefficient 1. With, for instance, $3x^{\pm2}$ read it as “3 multiplied/divided by 2 copies of x” the necessary grouping becomes natural.
P.S. I hope this site uses MathJax. More generally, since this site has no menu bar, where can I find the coding for answers in WordPress? | 2018-01-18 16:06:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6396090388298035, "perplexity": 991.9727435876855}, "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-2018-05/segments/1516084887423.43/warc/CC-MAIN-20180118151122-20180118171122-00153.warc.gz"} |
https://www.springerprofessional.de/topics-in-stochastic-analysis-and-nonparametric-estimation/1495674 | scroll identifier for mobile
main-content
## Über dieses Buch
This IMA Volume in Mathematics and its Applications TOPICS IN STOCHASTIC ANALYSIS AND NONPARAMETRIC ESTIMATION contains papers that were presented at the IMA Participating Institution conference on "Asymptotic Analysis in Stochastic Processes, Nonparamet ric Estimation, and Related Problems" held on September 15-17, 2006 at Wayne State University. The conference, which was one of approximately ten selected each year for partial support by the IMA through its affiliates program, was dedicated to Professor Rafail Z. Khasminskii on the occasion th of his 75 birthday, in recognition of his profound contributions to the field of stochastic processes and nonparametric estimation theory. We are grateful to the participants and, especially, to the conference organizers, for making the event so successful. Pao-Liu Chow, Boris Mor dukhovich, and George Yin of the Department of Mathematics at Wayne State University did a superb job organizing this first-rate event and in editing these proceedings. We take this opportunity to thank the Nation al Science Foundation for its support of the IMA.
## Inhaltsverzeichnis
### Some Recent Results on Averaging Principle
Averaging principle is one of the main methods in perturbation theory. It came into being more than two centuries ago in celestial mechanics, but even now there are many open questions having to do with applications of this principle. And these questions are not just about rigorous justification of procedures applied, but even it is unclear sometimes how to apply averaging procedures.
Mark Freidlin, Alexander Wentzell
### Cramer’s Theorem for Nonnegative Multivariate Point Processes with Independent Increments
We consider a continuous time version of Cramer’s theorem with non-negative summands $$S_t = \tfrac{1} {t}\sum\nolimits_{i:\tau _i \leqslant t} {\xi _i ,} {\text{ }}t \to \infty$$, where (τi, ξi)i≥1 is a sequence of random variables such that tS t is a random process with independent increments.
Fima Klebaner, Robert Liptser
### On Bounded Solutions of the Balanced Generalized Pantograph Equation
The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the early 1970s by T. Kato in connection with the analysis of the pantograph equation, y′(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y″(x + a 1 y′(x) + y(x) = ∫ 0 μ (dα), where a 1 ≥ 0, a 2 ≥ 0 a 1 2 + a 2 2 > 0, and μ is a probability measure. By setting K:=∫ 0 ln ± μ(dα), we prove that if K≦0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication” jumps. The paper also includes three “elementary” proofs for the simple prototype equation y′(x)+y(x)=1/2y(qx)+1/2y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.
Leonid Bogachev, Gregory Derfel, Stanislav Molchanov, John Ochendon
### Numerical Methods for Non-Zero-Sum Stochastic Differential Games: Convergence of the Markov Chain Approximation Method
The Markov chain approximation method is an efficient and popular collection of methods for the numerical solution of stochastic control problems in continuous time, for reflected-jump-diffusion-type models and the convergence proofs have been extended to zero-sum stochastic differential games. We apply it to a class of non-zero-sum stochastic differential games with a diffusion system model where the controls for the two players are separated, It is shown that equilibrium values for the approximating chain converge to equilibrium values for the original process and that any equilibrium value for the original process can be approximated by an ε-equilibrium for the chain for arbitrarily small ε > 0. The actual numerical algorithm is that for a stochastic game for a finite-state Markov chain.
Harold J. Kushner
### On the Estimation of an Analytic Spectral Density Outside of the Observation Band
We consider a Gaussian stationary process X(t) with an integer analytic spectral density f(λ) and study a problem of its estimation. The process X(t) is non-observable. Instead of it we observe a linear transformation Y(t), 0 ≤ t ≤ T, of X(t) with a transfer function a(A), ∣a(λ)∣=1 if λ belongs to an interval I. We study how far from I the consistent estimation of f(λ) is possible, T → ∞.
Ildar A. Ibragimov
### On Oracle Inequalities Related to High Dimensional Linear Models
We consider the problem of estimating an unknown vector θ from the noisy data Y=Aθ+ε, where A is a known m × n matrix and e is a white Gaussian noise. It is assumed that n is large and A is ill-posed. Therefore in order to estimate θ, a spectral regularization method is used and our goal is to choose a spectral regularization parameter with the help of the data Y. We study data-driven regularization methods based on the empirical risk minimization principle and provide some new oracle inequalities related to this approach.
Yuri Golubev
### Hypothesis Testing under Composite Functions Alternative
In this paper, we consider the problem of the minimax hypothesis testing in the multivariate white gaussian noise model. We want to test the hypothesis about the absence of the signal against the alternative belonging to the set of smooth composite functions separated away from zero in sup-norm. We propose the test procedure and show that it is optimal in view of the minimax criterion if the smoothness parameters of the composition obey some special assumption. In this case we also present the explicit formula for minimax rate of testing. If this assumption does not hold, we give the explicit upper and lower bounds for minimax rate of testing which differ each other only by some logarithmic factor. In particular, it implies that the proposed test procedure is “almost” minimax. In both cases the minimax rate of testing as well as its upper and lower bounds are completely determined by the smoothness parameters of the composition.
Oleg V. Lepski, Christophe F. Pouet’t
### On Parabolic Pdes and Spdes in Sobolev Spaces W P 2 without and with Weights
We present a “streamlined” theory of solvability of parabolic PDEs and SPDEs in half spaces in Sobolev spaces with weights. The approach is based on interior estimates for equations in the whole space and is easier than and quite different from the standard one.
Nicolai V. Krylov
### Stochastic Parabolic Equations of Full Second Order
A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order.
Sergey V. Lototsky, Boris L. Rozovskii
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### Blockchain-Effekte im Banking und im Wealth Management
Es steht fest, dass Blockchain-Technologie die Welt verändern wird. Weit weniger klar ist, wie genau dies passiert. Ein englischsprachiges Whitepaper des Fintech-Unternehmens Avaloq untersucht, welche Einsatzszenarien es im Banking und in der Vermögensverwaltung geben könnte – „Blockchain: Plausibility within Banking and Wealth Management“. Einige dieser plausiblen Einsatzszenarien haben sogar das Potenzial für eine massive Disruption. Ein bereits existierendes Beispiel liefert der Initial Coin Offering-Markt: ICO statt IPO. | 2019-01-19 15:18:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7047476768493652, "perplexity": 1157.902940994873}, "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-2019-04/segments/1547583668324.55/warc/CC-MAIN-20190119135934-20190119161934-00330.warc.gz"} |
https://math.stackexchange.com/questions/3317809/research-for-anti-solutions-to-diophantine-equations | # Research for “anti-solutions” to diophantine equations.
I have done some searches on the internet to find any studies that deal with the non-solutions of Diophantine equations. I'm asking for any research articles or web links you know of that deal with the "Anti"-solutions of Diophantine equations. For example: say there is a Diophantine equation z=f(x,y) then find some solution that describes all of z that does not satisfy f(x,y) for the integers or whole number. A particular example would be the Pythagorean triples. What would be the values of c which do not satisfy sqrt(a^2+b^2)?
• Wouldn't the "anti-solutions" just be the set of values in the set you're dealing with (e.g., integers) which aren't solutions? Thus, studying one is really implicitly studying the other one as well. – John Omielan Aug 8 at 23:39
• I believe that part of what I want to know is whether the set is infinite or finite. Sometimes it is 'easy' to show that there are an infinite number of solutions, but can be possibly impossible to show that there are an infinite number of "anti-solutions". – Jesse Aug 8 at 23:48
• @Jesse: you need to ask a more specific question. E.g., what integers lie in the range of the function $(x, y) \mapsto \sqrt{x^2 + y^2}$? – Rob Arthan Aug 8 at 23:51
• The Pythagorean triples was only an example. I am looking for any research, in general, for the "anti-solutions" to Diophantine equations. – Jesse Aug 9 at 0:15
• @Jesse An example that may illustrate your comment. It is easy to show that there are an infinite number of values of $z$ for which $w^3+x^3+y^3=z^3$ has a solution in positive integers. But so far as I am aware it is not known whether the number of values of $z$ for which there is no solution is finite or infinite. – Adam Bailey Aug 9 at 11:29
Theorem.- An integer $$n$$ is a sum of three squares if and only if it is not of the form $$4^r(8s-1)$$ where $$r,s\in\mathbb Z$$.
"Corollary".- The set of all the anti-solutions of the diophantine equation $$w=x^2+y^2+z^2$$ is formed for all the integers of the form $$4^r(8s-1)$$.
• It is a "strong" result of Gauss and is not easy to establish. The great Serre prove it in his book "Cours d'Arithmétique" like this: (1) WLOG assumes that $n$ is non-null and then, by virtue of a result in the $2$-adic field $\mathbb Q_2$ the condition $n = 4 ^ r (8s-1)$ is equivalent to that $-n$ is a square in $\mathbb Q_2$. (2) States that for the rational non-null $a$ it is representable on $\mathbb Q$ by the quadratic form $x_1^2 + x_2^2+ x_3^2$ it is necessary and sufficient that $a\gt0$ and that $-a$ is not a square in $\mathbb Q_2$. – Piquito Aug 9 at 16:49 | 2019-10-15 23:55:13 | {"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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6063461899757385, "perplexity": 177.51381430212615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986660829.5/warc/CC-MAIN-20191015231925-20191016015425-00072.warc.gz"} |
https://nadre.ethernet.edu.et/record/20595/export/xd | Thesis Open Access
# Unsteady State Kinetic Modelling of Biomass Gasification in Bubbling Fluidized Bed Gasifier
Tolossa Kebede Tulu
### Dublin Core Export
<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Tolossa Kebede Tulu</dc:creator>
<dc:date>2021-06-09</dc:date>
<dc:description>From available renewable energy, biomass is the most abundant and promising resource of renewable energy which is environmental friendly. From biomass conversion methods to useful energy, gasification is an adequate technology. The mathematical modeling of biomass gasification in bubbling fluidized bed can be used to predict syngas composition and optimize the operational condition as well as gasifier design, which is a cost-saving option and viable routine. However, the kinetic modeling of biomass gasification is at early phase. With this regard, the aim of this study is to develop kinetic model based on two-phase fluidization theory, which is comprised of reaction kinetics, bed hydrodynamic parameters, diffusion effect, and species transport equations. Furthermore, the thermal cracking of tar was integrated into the model to increase the prediction accuracy of the model and to address the gap in knowledge. In this study, the transient kinetic modeling of biomass gasification in bubbling fluidized bed gasifiers and optimization methods to maximize gasification products were developed. The model was coded in Matlab and simulated. The result depicts good agreement with experimental work in literature. The sensitivity analysis carried out and the effect of temperature from 650oC-850oC and steam to biomass ratio (S/B) from 0.1-2 investigated. The result depicts the increase in temperature and S/B promotes H2 production and reduces CO and CH4. Furthermore, the regression model was carried out for temperature and S/B variation in Design-Expert. The surface response is constructed from the regression model and the mutual effect of temperature and S/B on gasification product and heating value investigated. Besides, the desirability function is employed to optimize gasification product and heating value. The maximum gasification product yield was at 827.866oC and 0.1 S/B in investigated range. The response predicted by desirability function at this optimum operational conditions is 30.096%, 44.07%, 13.20%, 12.90%, 14.035 (MJ/Nm3 ), and 14.536 (MJ/Nm3 ) for H2, CO, CO2, CH4, LHV, and HHV respectively.</dc:description>
<dc:language>eng</dc:language>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:subject>Biomass gasification, Fluidized bed, Kinetic Modeling, Response surface, Optimization</dc:subject>
<dc:title>Unsteady State Kinetic Modelling of Biomass Gasification in Bubbling Fluidized Bed Gasifier</dc:title>
<dc:type>info:eu-repo/semantics/doctoralThesis</dc:type>
<dc:type>publication-thesis</dc:type>
</oai_dc:dc>
86
26
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https://www.controlbooth.com/threads/leds-leave-on-or-turn-off.42518/ | # LEDs-leave on or turn off?
#### Dsmagnussen
##### Member
Hey all,
We are just getting our first set or LED wash fixtures in our plot. Martin Rush Par 2 Zoom. I have surge protection on each circuit, but no simple way to turn them off at the end of the night. we looked into the ETC relays, but can't swing the cash after the layout for the Martins. Question is, Is it all right to leave them plugged in, or should they be unplugged every night? I have had tour companies leave their fixtures on and full color all day, but my guess was they were rentals, and they did not care.
Thank you for the input!
Doug
#### SteveB
##### Well-Known Member
I have Martin Aura's, powered off Sensor relays.
I power down after an event, mostly as if it's a Sunday event no point in keeping powered for the week when not used.
If I were using daily, I might keep them powered.
RonHebbard
#### MNicolai
##### Well-Known Member
Fight Leukemia
Is your building going to burn down if you leave them on 24/7? No.
Are the fixtures' fans going to run continuously and all kinds of dust and clog up? You may want to check. The fans on those fixtures are thermostatic controlled. If you go dark and leave them idle and the fans spin down completely, that's ideal. If they spin at a low RPM, that's less ideal. At the very least, they shouldn't keep spinning full bore which is what would be the worst case scenario.
RonHebbard
#### cdiamondz
##### Active Member
When the fixtures are left plugged in the circuitry will stay powered even if the LED array itself is off. Generally it would be better to have power removed from the fixture entirely when not in use to increase the lifespan of the circuitry as the power supply produces heat and any controllers in the fixtures will keep ticking away. Hope this helped!
EDIT: Please remember that fans die over time.
#### BillConnerFASTC
##### Well-Known Member
I worry more about the power supplies. Some difference in leaving them on for hours or a day unused, versus for a week or an entire season.
And did you look a relays in a panel or rack, or the portable plug and play ColorSource relay from ETC? in the $225 range for a relay that should easily handle 10 of the Martins. Less than$25 to make a \$800+ fixture last longer?
PS: Install some switches and manually turn off? A box by the breaker panel would not cost much. Don't use breakers unless you know them to be switch rated.
#### PeteEngel
##### Active Member
Yeah, turn them off. As others have said, the PSU stays powered and those have a life as well. When your LED's have 300 hrs on them but your PSU has 8000 hrs, you end up replacing psu's (and potentially fans) for no good reason.
Les
#### Brentgi
##### Active Member
Can I start the argument of "Why not use an unregulated dimmer in switch mode?" Just cause I feel like it will come up anyway...
#### JD
##### Well-Known Member
Lighting strikes, and other unintended surges do occur. The more "powered" time, the bigger the target.
#### porkchop
##### Well-Known Member
Although I agree with the consensus that given my choice I would turn them off every night, I think it depends on the practicality of doing so. If, as it sounds in your current situation, someone has to physically go to each fixture to unplug them individually and then re-plug them in the morning I would consider leaving them on in more situations. Even if it's a work study student taking a 15 minute walk to plug or unplug the fixtures each night the time saved by leaving them on would pretty rapidly begin to pay for things like replacement fans and power supplies. Even if the time of people going on a walk is free (volunteer, student, salary, ect...) the fixture walk could be an annoyance that adds to reasons why a person might not come back or show up late/leave early more often. Not sure about your space, but if it's relevant I also lean towards any plan that prevents people from climbing ladders more than necessary.
In short think about all of the "costs" (physical, personal, annoyance, etc...) of leaving them on vs leaving them off and figure out which one is the least restrictive for your given situation. Fans and PSU's have financial cost associated with them, but they are not particularly high. Power spikes and lighting strikes happen, so do trips and falls. You don't want a fixture to go out in the middle of the show, but you still need the personnel in to get you to that point.
#### Chris Pflieger
##### Well-Known Member
Can I start the argument of "Why not use an unregulated dimmer in switch mode?" Just cause I feel like it will come up anyway...
NO!!!!!!
On that note, I replaced my NSI rack dimmer with a custom made relay box. The relays are operated by a dry contact tied into our audio power sequencer.
#### RickR
##### Well-Known Member
Can I start the argument of "Why not use an unregulated dimmer in switch mode?" Just cause I feel like it will come up anyway...
The short answer is because not all dimmers and power supplies can handle it.
Will yours? How many damaged units can you afford just to find out? Even when the manufacturer says it's OK in certain cases, then someone plugs in an El Cheapo and the dimmer fries in mid show, costing more than the fixture savings.
##### Custom Title
Fight Leukemia
Another vote for the power down. We've got a bunch of Blizzard RokBoxes for downwash (nice little fixture for that application BTW). Same issue with the relay, it is quite pricey for an ETC rack mounted one- almost as much as another fixture. The cheapest solution is obvious and we'd been cutting power at the rack via the breaker, but Bill's comment up there got me curious and I don't see anything that indicates that Sensor dimmers have switch rated breakers in them.
RonHebbard
#### BillConnerFASTC
##### Well-Known Member
Well, the ETC constant module - just two circuit breakers - if I read the data sheet correctly are switch rated breakers. So other than the inconvenience of loosing dimmed circuits in pairs (buy more LEDs of course) and having it manual, not a bad budget option. IIRC these are pretty inexpensive compared to most modules.
##### Custom Title
Fight Leukemia
I guess reading the data sheet would make more sense than pulling the dimmer to check. And yeah, I can see it being somewhat inconvenient to have two dimmers out like that, but in our case we parked our hazer in a spot to make use of the second circuit. Definitely easier on the wallet than the relay, but more of a headache to remember to shut it off.
#### BillConnerFASTC
##### Well-Known Member
Not a first choice for sure, but i was beginning to picture many worse solutions.
#### cdiamondz
##### Active Member
Not a first choice for sure, but i was beginning to picture many worse solutions.
1970's 60 amp dimmers with no relay or constant power capabilities parked at full at the console?
#### BillConnerFASTC
##### Well-Known Member
I was picturing zip cord jumpers in the rack, or a rats nest of extension vords hither, thither, and yon.
#### cdiamondz
##### Active Member
I was picturing zip cord jumpers in the rack, or a rats nest of extension vords hither, thither, and yon.
How does tin foil and electrical tape sound to replace zip cord? | 2020-02-29 13:17:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2215583175420761, "perplexity": 3581.3982182505324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00203.warc.gz"} |
http://mathhelpforum.com/differential-geometry/101501-convergence-comparison.html | # Math Help - Convergence comparison
1. ## Convergence comparison
Suppose that $f_n\to f$ in measure and $|f_n|\le g\in L^1$, for all $n$. Show that $f_n\to f$ in $L^1$. That is $\lim_n\int_X|f_n-f|d\mu=0$.
I have already shown that $\int_Xfd\mu=\lim_n\int_X f_nd\mu$, but don't see how to use this or anything else to get to the desired result.
2. Look at Dominated convergence theorem - Wikipedia, the free encyclopedia, the given proof of the assertion that you already know contains your desired result. | 2014-10-24 14:00:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "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": 0.8716883063316345, "perplexity": 129.6600182556201}, "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/1414119646008.13/warc/CC-MAIN-20141024030046-00038-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://documen.tv/when-testing-the-differences-between-means-the-hypothesis-suggests-that-population-means-are-not-28326722-65/ | Question
When testing the differences between means, the _____ hypothesis suggests that population means are not equal. | 2023-03-25 04:00:49 | {"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": 0.8700482845306396, "perplexity": 2464.5287274245843}, "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/1679296945315.31/warc/CC-MAIN-20230325033306-20230325063306-00189.warc.gz"} |
https://electronics.stackexchange.com/questions/622861/how-to-decrease-distortion-and-efficiency-of-this-circuit | How to decrease distortion and efficiency of this circuit?
How to achieve efficiency of more than 70% and decrease the distortion of the circuit to 0.1% in THD, without affecting my Vpp=30 with the circuit below with both in 1k and 6k frequency of Vs?
Update Also, I only have this component available, and I can only make this kind of design in the project that was taught in university. I am also working with 10W speaker.
I meant by efficiency=(Pout/Pin)100, for this circuit, I get 63.82%=(9.97/(7.812))*100
If not I would like to theory behind it
Updated Circuit
• Title asks about how to increase distortion but the body asks the opposite. The theoretical limit of a Class-AB amp is around 75%. So you shouldn't expect more than 70% in practice. Jun 9 at 13:55
• What is "6f frequency"? You want to decrease the distortion "by 0.1"...does that mean reduce the distortion by 10%, to about 1.7% distortion? Jun 9 at 13:57
• If you want to decrease distortion this answer to a previous question of yours explained that moving the feedback resistor (R4 above) from the op-amp output to the main power output does this. Choosing a better/faster op-amp will also give lower distortion (when implemented as I stated in that answer). Jun 9 at 14:40
• @ElliotAlderson I meant k frequency of Vss, and by 0.1 I meant my THD 0.1% instead of 7.4% Jun 9 at 17:54
• is there a way to fix the clipping in the sine wave? Jun 9 at 18:52
How to achieve efficiency of more than 70%
The efficiency of a class-AB amp doesn't really depend on the schematic. With power supply +/-Vcc and output voltage Vout, then the amp will burn a power Iout.(Vcc-Vout) as heat in the transistors and source resistors, and Iout.Vout is used as useful output power. So if you want higher efficiency, use a lower power supply voltage.
Another way to increase efficiency is to design the amp better, so it delivers less distortion at low bias current, so the transistors burn less power when they do nothing. However this is only relevant when the output signal is small enough that the amp stays in the class A zone of class AB. It becomes irrelevant at mid-high power, because it goes to class B, and at this point it spends little time in the class A zone.
In conclusion, efficiency depends on power supply voltage across the whole range of power, and on bias current at low power (or idle).
and decrease the distortion of the circuit to 0.1% in
I'll try.
First thing is to add a Vbe multiplier to bias those MOSFETs. Without it, they'll go into thermal runaway, so it is absolutely necessary. This is because the threshold voltage goes down when they get hot (negative tempco) which increases idle bias current, which makes them hotter, and this goes in a loop until the idle bias current goes out of control and smoke gets out. This is especially important with small TO-220 transistors, which get hot really fast.
Unfortunately it is not possible to know the Vgs_th tempco without measuring it. For example IRFP140/240 have -4-6 mV/°C total, P and N channel combined. Assuming this will be the same here, I add a Vbe multiplier.
This multiplies the transistor Q1's Vbe by a ratio R4/R3. Note R4 and R3 should be a potentiometer. Since Q1's Vbe has a tempco around -2mV/°C, once it is multiplied by about 10 to get the correct gate-to-gate voltage to bias the MOSFETs, this will have -20mV/°C, which is too much.
However, even if it is placed between the power devices on the heatsink, the transistor gets much less temperature rise than the MOSFETs themselves, because the heatsink is doing its job and dumping that heat into the air. That somewhat compensates for it, but be warned this circuit will overcorrect.
It is possible to get a much more accurate bias current by putting two Vbe multipliers in series, one of which is not on the hear sink (Q2).
In this case, R15 is fixed, and R3/R4 is a pot.
Next, we have DC. It is not a good idea to AC couple the output of the opamp to the gates, as you have done, because then the DC operating point of the output stage is not fixed, and you need an output capacitor to get rid of it.
So in the first schematic of this answer, R7-R8 do this job. In the second schematic, the Vbe multiplier transistors do it as a side job.
Then feedback wraps around the whole thing, and we get a bit more than 0.1% distortion at 1kHz.
It will be difficult to go lower by using an opamp, because the opamp does not offer flexibility in custom compensation scheme, which are the secret sauce for low distortion, as they allow a much higher open loop gain to be used for correcting distortion in the output stage.
I set the source resistors to 0R22 and idle bias current to 20mA.
• Is it not possible to the circuit with inverting op-amp? From what I see you using non inverting terminal for Vs. Jun 10 at 3:07
• Yes you can make it inverting Jun 10 at 7:54
• My efficiency is worse both in 1k and 6k Circuit, did I do something wrong? and is there a way to increase to approximately 10W efficiency or a bit higher? Jun 10 at 10:24
• 10W isn't efficiency, watts are power, so I have no idea what you mean Jun 10 at 12:57
• What I meant is that I used to get 63.82% with 10W at RL from the original circuit, but now my efficiency is 0.84% with 8.41mW at RL. so, I wanted to know did I made any mistakes while following your circuit Jun 10 at 17:47
You have no control of the Vgs and resulting current of MOSFETs. Some will be with a massive current and in class-A, some will be turned off and producing extreme crossover distortion in class-B and maybe a few will be in class-AB with low distortion and efficiency as high as 65% at high power. All amplifiers have negative feedback that reduces distortion but yours has none. Your schematic has its parts so far apart that everything looks tiny and hard to see.
• I forgot to say that an LM358 is never used for audio because it produces crossover distortion, noise and poor high frequencies slew rate. Jun 9 at 15:20
• This qualifies as the second worst amp schematics I've ever seen (first place being the one with BJTs posted previously)... Jun 9 at 15:36
• These circuits are 50 years old as were taught by very old teachers. Jun 9 at 16:43
• Because of the limited components, my professors are not allowing us to design better. instead, using the design that was thought in the presentation Jun 9 at 16:49
A couple of ways that you can reduce the distortion:
• Use complementary MOSFETs connected as source followers.
• Include the output transistors in the feedback loop. This means that instead of connecting the feedback resistor to the output of the opamp, connect it to the output of the amplifier where the speaker is connected.
• Use an audio grade opamp.
• Add a resistor between the non-inverting opamp input and ground to reduce the output opamp's output offset voltage, you'll need to experiment with different values.
• Use a larger capacitor on the output. You've been told this multiple times before but you never seem to actually do it. Try 1000uF.
• Allow for headroom in your power supplies. If you need 30 Vpp output a $$\\pm\$$ 15 V supply isn't going to be enough.
Doing this should get the distortion under 1% in simulation. I just threw together a quick circuit in LTspice and got 0.99% distortion at 30Vpp, 6KHz using $$\\pm\$$18V for the output supply and a TL084 type opamp. I biased the MOSFETS at around 800uA.
I'm sure there are other things that could be improved, but try some of these to start.
• Thank you, sorry my professor is giving us a hard time I seem to have forgotten the capacitors. as for the op-amp I want to use one that is audio grade but unfortunately only this one is available at uni. because this project was designed in the last 2 weeks before the finals Jun 9 at 16:53
• I Updated my circuit. I didn't understand the "Use complementary MOSFETs connected as source followers." and is the resistor is ok at the non-inverting? Jun 9 at 17:11
• Okay, I had to make some changes to my answer. I had used your bias resistor values which put the MOSFETs I used in class A. Rebiasing @800uA (maybe not true class B but close), around 3W per transistor, brought the distortion up to around 1%. I don't know what your requirements are, you haven't really laid out what all is allowed in your design, but low distortion and class B tend to be at odds with each other. Jun 9 at 17:12
• Honestly, at this I don't know, he said class b push-pull audio amplifier using the components we have at uni. and in uni, I only have this op-amp available. and I only have this week left before final starts Jun 9 at 17:14
• @Sabretooth2438 In the new schematic I can see the transistors better, it looks like you're using complementary ones, so that's okay (I thought you had 2 of the same kind because the probe symbol was covering the part number). You seem to have lost one of the capacitors (C2). The resistor should be okay I think, you can try different values to get the DC on the opamp output as low as possible, but it's not all that critical. Jun 9 at 17:22
Are you simply simulating the circuit with perfectly matched transistors or are you building the circuit with random spec'd transistors? The important DC voltages on the waveform will show you what is causing the asymmetrical clipping.
• I am building a circuit based on the components that were given to me and available at the university. there is no other op-amp available besides this nor I can replace any transistor beside this Jun 10 at 17:51
You removed C2 that totally messed up the symmetrical biasing of the Mosfets. Put C2 back. | 2022-07-07 11:07:57 | {"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": 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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.476652055978775, "perplexity": 1422.1319734427962}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104690785.95/warc/CC-MAIN-20220707093848-20220707123848-00576.warc.gz"} |
https://www.emmamichaels.com/8262/differentiate-y-sin-x.html | Breaking News
Differentiate Y Sin X
Differentiate Y Sin X. So, let’s differentiate the equation with respect to x for evaluating the derivative of y with respect to x. So, when we have some other function inside of the sine function, such as sin(u), we see that its derivative will be cosine with the inner function multiplied by the derivative of the. Join / login >> class 12 >> maths >> continuity and differentiability. Y1⋅ dxdy=sinx dxd logx+logx dxd sinx.
So, let’s differentiate the equation with respect to x for evaluating the derivative of y with respect to x. X = sin (y) derivative: (sinx)x = eln((sinx)x) = exln(sinx) taking the derivative of this gives:
Assume that f (x) = sin (x+ 1).
On differentiating both sides with respect to x, we get. D dx (sinx)x = ( d dx xln(sinx))exln(sinx) = (ln(sinx) +x d dx (ln(sinx)))(sinx)x. D dx [sin(x)1 2] d d x [ sin.
Assume That F (X) = Sin (X+ 1).
Differentiate y=sin(x+y) w.r.t x medium solution verified by toppr y=sin(x+y) dxd (y)= dxd (sin(x+y)) using : Dy dx = cos(x + y) 1 −cos(x +y) explanation: Let t=sinx => d (sin (t))/dx = (cos (t)). = (ln(sinx) +x d dxsinx sinx)(sinx)x.
On Taking Log Both Sides, We Get.
Let t=sinx => d (sin (t))/dx = (cos (t)).
Kesimpulan dari Differentiate Y Sin X.
Y = sinx x2 we can differentiate it by two methods: The left side would simply give you dy dx. In words, we would say:
See also What Is The Rationale For Scrum Teams Implementing Short Sprints | 2023-03-23 08:59:51 | {"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": 0.9435727000236511, "perplexity": 4048.5234210384015}, "config": {"markdown_headings": false, "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-2023-14/segments/1679296945030.59/warc/CC-MAIN-20230323065609-20230323095609-00385.warc.gz"} |
https://projectshum.org/maths/year-7/5-solving-equations/ | 5 Solving equations
Solving by inspection
The most simplest way to solve an equation is by inspection, which is a guessing technique.
For example, to solve $x+2=3$. By trying out different numbers for $x$, you’ll determine $x=1$
Solving by inverse operations
A more effective way of solving questions is the inverse operation method. It is necessary to know that:
• Addition and subtraction are inverse operations
• Multiplication and division are inverse operations
• Square and finding the square root are inverse operations
For example, to solve $2x+1=11$, we start on the RHS and apply inverse operations. The inverse of “-1” is “+1”, so we “+1” to both sides. We must do it to both sides because the left hand side is equal to the right hand side, so if we don’t do the same thing to both sides, the 2 sides are no longer equal! This will give us $2x+1 -1=11 -1$, or simplifying, $2x=10$. The inverse operation of “*2” is “/2”, so we will “/2” on both sides. This gives us $2x/2=10/2$, or otherwise, $x=5$.
Again, solving equations that are provided by word questions are by far the hardest, as in all mathematics. | 2019-04-21 21:10:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "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": 0.8337949514389038, "perplexity": 451.1842261270879}, "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-2019-18/segments/1555578532882.36/warc/CC-MAIN-20190421195929-20190421221929-00162.warc.gz"} |
https://sublinear.info/index.php?title=Open_Problems:23 | # Problem 23: Approximate 2D Width
The width of a set $P$ of points in the plane is defined as $\operatorname{width}(P)=\min_{\|a\|_2=1} \left(\max_{p \in P} a \cdot p-\min_{p \in P} a \cdot p\right).$ For a stream of insertions and deletions of points from a $[\Delta] \times [\Delta]$ grid, we would like to maintain an approximate width of the point set. We conjecture that there is an algorithm for this problem that achieves a constant approximation factor and uses $\operatorname{polylog}(\Delta+n)$ space. | 2017-10-20 08:52:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8723496198654175, "perplexity": 116.27007479669318}, "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/1508187823997.21/warc/CC-MAIN-20171020082720-20171020102720-00362.warc.gz"} |
http://tex.stackexchange.com/questions/28330/improving-code-to-draw-several-cycles-trigonometric | # Improving code to draw several cycles trigonometric
How can I improve this code to generate three cycles varying the height of the trigonometric tangent? Note that I managed to correctly draw the red line, but could not improve the green line. What I want is a reduction in their angle and straight. The green line does not have the correct height.
\begin{tikzpicture}[>=latex,scale=2]
\draw circle (1);
%axis
\draw[->] (-1.2,0) -- (1.2,0) coordinate (x);
\draw[->] (0,-1.2) -- (0,1.2);
%angles
\fill[fill=green!25] (0,0) -- (.2,0) arc (0:60/\radius:.2) -- cycle;
%sinus
\draw[<-,red,line width=1pt] (P) -- (P |- x);
\fill[blue] (P) circle (1pt);
\draw (0,0) -- (P);
%cosinus
\draw[->,orange] (0,0) -- (P |- 0,0);
%tangent
\draw[dashed] (0,0) -- (1,\YValue);
\draw[->,green,line width=1pt] (1,0) -- (1,\YValue);
\end{scope}
};
\end{tikzpicture}
-
If I understand correctly, the green line is the tangent of the green angle. Then you should rewrite the formulae in the \foreach loop like this:
\pgfmathsetmacro{\YValue}{tan(60/\radius)} | 2015-08-31 18:16:59 | {"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": 0.9480814337730408, "perplexity": 1366.4189538427406}, "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-2015-35/segments/1440644066275.44/warc/CC-MAIN-20150827025426-00188-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:1007.35049 | # zbMATH — the first resource for mathematics
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Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. (English) Zbl 1007.35049
This paper is excellent. The authors prove global existence for the following equation of Kirchhoff-Carrier type subject to nonlinear boundary dissipation, that is, \align u_{tt}-M\left( t,\int_{\Omega }\left|\nabla u\right|^{2}dx\right) \Delta u&=0\quad\text{in }\Omega \times (0,\infty), \\ u&=0\quad\text{on }\Gamma _{0}\times (0,\infty), \\ \frac{\partial u}{\partial \upsilon }+g(u_{t})&=0\quad\text{on }\Gamma _{1}\times (0,\infty), \\ ((u(x,0),u_{t}(x,0))&=((u^{0}(x),u^{1}(x)), \endalign where $\Omega$ is a bounded, star-shaped domain of $\bbfR^{n}$, $n\geq 1,$ with a smooth boundary $\Gamma =\Gamma _{0}\cup \Gamma _{1}.$ Here, $\Gamma _{0}$ and $\Gamma _{1}$ are closed and disjoint and $\upsilon$ represents the unit outward normal to $\Gamma$. Besides, the authors place a natural hypothesis about the function $M$, and in general problems of Kirchhoff-Carrier equation when the function $M$ depends on the time the problem becomes more difficult. It is important to mention here, that the authors prove the existence and uniqueness of regular solutions without any smallness on the initial data. Moreover, uniform decay rates are obtained by assuming a nonlinear feedback acting on the boundary.
##### MSC:
35L70 Nonlinear second-order hyperbolic equations 35B40 Asymptotic behavior of solutions of PDE 35L20 Second order hyperbolic equations, boundary value problems | 2016-05-04 13:44:01 | {"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": 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": 0.9808650612831116, "perplexity": 5337.556904020229}, "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-2016-18/segments/1461860123077.97/warc/CC-MAIN-20160428161523-00148-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://chem.libretexts.org/Courses/Lumen_Learning/Book%3A_Statistics_for_the_Social_Sciences_(Lumen)/11%3A_10-_Inference_for_Means/11.29%3A_Estimating_a_Population_Mean_(2_of_3) | 11.29: Estimating a Population Mean (2 of 3)
Learning Objectives
• Construct a confidence interval to estimate a population mean when conditions are met. Interpret the confidence interval in context.
• Interpret the meaning of a confidence level associated with a confidence interval.
• Adjust the margin of error by making changes to the confidence level or sample size.
On the previous page, we used a confidence interval to estimate the population mean, µ. For this confidence interval, we had to supply a guess for the population standard deviation, σ, based on previous studies. It may have occurred to you that if we do not know µ, it is unlikely that we know σ. So we now take a different approach. We estimate σ using the sample standard deviation, s.
This is the same type of adjustment we used in Inference for One Proportion when we had to adjust our model of the sampling distribution. The standard error of the sampling distribution is $\sqrt{p(1-p)/n}$. (If we knew $p$, then we wouldn’t need to build a confidence interval.) We approximate $p$ by the sample proportion, $\stackrel{ˆ}{p}$.
Our process for adjusting the confidence interval for estimating µ is similar. We use the sample standard deviation, s, to estimate σ. The standard error for the sampling distribution $σ\text{}/\sqrt{n}$ becomes $s\text{}/\sqrt{n}$. So we adjust the margin of error in the confidence interval formula, but this adjustment is not as straightforward as our work with proportions.
This estimate for σ introduces more uncertainty in the process. The problem is worse with smaller samples because the sample standard deviations vary more. For small samples, s is a worse approximation for σ. Unfortunately, this approximation makes the normal model a bad fit and inappropriate for determining critical values. We instead use what is called a t-model for this purpose.
Introduction to the T-Model
Here is the formula for the T-score. We also include the z-score for comparison. The formulas are very similar.
$Z\text{}=\text{}\frac{\stackrel{¯}{x}-μ}{\frac{σ}{\sqrt{n}}}\text{ }T\text{}=\text{}\frac{\stackrel{¯}{x}-μ}{\frac{s}{\sqrt{n}}}$
The distribution of z-scores is the standard normal curve, with mean of 0 and standard deviation of 1. The distribution of T-scores depends on the sample size, n. There is a different T-model for every n. So the T-model is a family of curves.
Instead of referring to n to specify which T-model to use, we refer to the degrees of freedom, or df for short. For Topics 10.2 and 10.3, the number of degrees of freedom is 1 less than the sample size. That is, df = n – 1.
In summary, a normal model is defined by its mean and standard deviation. A T-model is a family of curves defined by the degrees of freedom.
Let’s take a look at a few T-model curves (for various df) to see how they compare to the normal model.
We can see from the picture that as df grows, the T-model gets closer to the standard normal model.
Similarities between T-model and standard normal model:
• Symmetric with a central peak, bell-shaped.
• Centered at 0.
• The larger the degrees of freedom, the closer the T-model is to the standard normal model.
Difference between T-model and standard normal model:
• The T-model has more spread than the standard normal model.
• The T-model has more probability in the tails and less in the center than the standard normal model. We can see this in the fatter tails and lower central peak of the T-model.
When is a T-model a good fit for the sampling distribution of sample means?
Check these conditions before using the T-model:
• Use the T-model if σ (the population standard deviation) is unknown. If σ is known, then use the normal model instead of the T-model.
• Use the T-model if variable values are normally distributed in the population. If this is not true, then make sure the sample size is large (more than 30).
Cable Strength
A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging.
The engineers take a random sample of 45 cables and apply weights to each of them until they break. The mean breaking weight for the 45 cables is $\stackrel{¯}{x}$ = 768.2 lb. The standard deviation of the breaking weight for the sample is s = 15.1 lb.
What should the engineers report as the mean amount of weight held by this type of cable?
Let’s use these sample statistics to construct a 95% confidence interval for the mean breaking weight of this type of cable.
Checking conditions:
Since we do not know the standard deviation of breaking weights of all of the cables (the population parameter σ), we use the sample standard deviation (s) as an approximation for σ. Since we don’t know σ, we must use the T-distribution to model the sampling distribution of means.
Is the T-model a good fit for the sampling distribution?
Yes, because the conditions are met:
• σ is unknown.
• The sample size is large enough.
Finding the standard error:
As usual, we start by estimating the standard error. This estimate comes from the formula $σ\text{}/\sqrt{n}$. However, since we don’t know σ, we use s = 15.1 as an approximation for σ. So our estimate for the standard error of all sample means is $s\text{}/\sqrt{n}=15.1\text{}/\sqrt{45}\approx 2.25$.
Finding the margin of error:
To find the margin of error, we need to find the critical T-value that corresponds to a 95% confidence level. This is just like the critical Z-value when we built confidence intervals for proportions, except that it comes from the T-model instead of the standard normal model.
We will use technology to find the critical T-value. There are a number of tools for doing this. Some books will also give you the option to use printed tables of values. Here we will use a simulation that gives the T-model based on degrees of freedom. We want the T-values that cut off the central 95% of the area under the curve. It will look as follows.
Using the simulation, we see that the critical T-value for a 95% confidence interval with 44 degrees of freedom is Tc = 2.015, which means our margin of error for this confidence interval is
$\begin{array}{l}\mathrm{Estimated}\text{}\mathrm{standard}\text{}\mathrm{error}\text{}\mathrm{is}\text{}s\text{}/\sqrt{n}\text{}\approx \text{}2.25\\ \mathrm{Margin}\text{}\mathrm{of}\text{}\mathrm{error}\text{}\mathrm{is}\text{}{T}_{c}\text{}⋅s\text{}/\sqrt{n}\text{}\approx \text{}2.015(2.25)\text{}\approx \text{}4.53\end{array}$
Note: For 95% confidence, the empirical rule approximates the critical Z-value as 2. The empirical rule is based on the normal model. Using the T-model for df = 44, the critical T-value (2.015) is very close to 2. This makes sense because for larger df, the T-model is very close to the standard normal model. We will see that the critical T-value differs more from the critical Z-value when the sample sizes are small.
Finding the confidence interval:
We have all the pieces to build the confidence interval. In our example, the confidence interval is
$\begin{array}{l}\stackrel{¯}{x}\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}\\ \stackrel{¯}{x}\text{}±\text{}{T}_{c}\text{}⋅\text{}\frac{s}{\sqrt{n}}\\ 768.2\text{}±\text{}4.53\\ (763.7,772.7)\end{array}$
Conclusion:
We are 95% confident that the mean breaking weight for all cables of this type is between 763.7 lb and 772.7 lb.
Confidence intervals at the 95% confidence level are common in practice. But 95% is not the only confidence level we use. Particularly in situations that involve safety issues, such as the previous example, people often prefer to estimate population means with 99% confidence intervals. Let’s do some exploration with technology to see how changes in the confidence level affect the confidence interval.
An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/sss/?p=390
How Much Alcohol Do College Students Drink?
According to the website www.collegedrinkingprevention.gov, “About 25 percent of college students report academic consequences of their drinking including missing class, falling behind, doing poorly on exams or papers, and receiving lower grades overall.” A statistics student is curious about drinking habits of students at his college. He wants to estimate the mean number of alcoholic drinks consumed each week by students at his college. He plans to use a 90% confidence interval. He surveys a random sample of 71 students. The sample mean is 3.93 alcoholic drinks per week. The sample standard deviation is 3.78 drinks.
https://assessments.lumenlearning.co...sessments/3684
https://assessments.lumenlearning.co...sessments/3753
https://assessments.lumenlearning.co...sessments/3685
https://assessments.lumenlearning.co...sessments/3686
https://assessments.lumenlearning.co...sessments/3687 | 2022-10-07 16:42:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 12, "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": 0.7700449824333191, "perplexity": 461.87263642292726}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338213.55/warc/CC-MAIN-20221007143842-20221007173842-00360.warc.gz"} |
http://mathhelpforum.com/algebra/40769-trig-form-complex-number.html | # Thread: Trig. Form Of A Complex Number
1. ## Trig. Form Of A Complex Number
cos(7pi/4) + i sin(7pi/4)/ cos pi + i sin pi
now this im just stuck on
2. Originally Posted by >_<SHY_GUY>_<
cos(7pi/4) + i sin(7pi/4)/ cos pi + i sin pi
now this im just stuck on
What are you trying to do? Simplify and express in Cartesian form? Then note:
$\displaystyle \cos \left( \frac{7 \pi}{4} \right) + i \sin \left( \frac{7 \pi}{4} \right) = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$.
$\displaystyle \cos \pi + i \sin \pi = -1$.
You must make it your business to learn the exact values of sin and cos of important special angles. | 2018-04-26 12:28:14 | {"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": 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": 0.9279361367225647, "perplexity": 3453.3148633812693}, "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/1524125948126.97/warc/CC-MAIN-20180426105552-20180426125552-00016.warc.gz"} |
https://dimag.ibs.re.kr/tag/pascalgollin/ | ## Pascal Gollin gave a talk on the existence of a decomposition of an infinite graph into spanning trees at the discrete math seminar
On October 29, 2019, Pascal Gollin from IBS discrete mathematics group gave a talk on the existence of a decomposition of an infinite graph into spanning trees in terms of the existence of packing and covering of spanning trees at the discrete math seminar. The title of his talk was “A Cantor-Bernstein-type theorem for spanning trees in infinite graphs“.
## Pascal Gollin, A Cantor-Bernstein-type theorem for spanning trees in infinite graphs
Given a cardinal $\lambda$, a $\lambda$-packing of a graph $G$ is a family of $\lambda$ many edge-disjoint spanning trees of $G$, and a $\lambda$-covering of $G$ is a family of spanning trees covering $E(G)$.
We show that if a graph admits a $\lambda$-packing and a $\lambda$-covering then the graph also admits a decomposition into $\lambda$ many spanning trees. In this talk, we concentrate on the case of $\lambda$ being an infinite cardinal. Moreover, we will provide a new and simple proof for a theorem of Laviolette characterising the existence of a $\lambda$-packing, as well as for a theorem of Erdős and Hajnal characterising the existence of a $\lambda$-covering.
Joint work with Joshua Erde, Attila Joó, Paul Knappe and Max Pitz.
## Welcome Pascal Gollin, a new research fellow in the IBS discrete mathematics group
The IBS discrete mathematics group welcomes Dr. Pascal Gollin, a new research fellow at the IBS discrete mathematics group from October 1, 2019.
He received his Ph.D. from Department of Mathematics, Universität Hamburg in Germany in 2019 under the supervision of Prof. Reinhard Diestel. Welcome! | 2019-11-12 04:00:06 | {"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": 0.5178921818733215, "perplexity": 784.736751937116}, "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-2019-47/segments/1573496664567.4/warc/CC-MAIN-20191112024224-20191112052224-00128.warc.gz"} |
https://www.physicsforums.com/threads/subset-of-separable-space-is-separable.714821/ | Subset of separable space is separable
1. Oct 6, 2013
homesick
1. The problem statement, all variables and given/known data
Show that if X$\subset$M and (M,d) is separable, then (X,d) is separable. [This may be a little bit trickier than it looks - E may be a countable dense subset of M with X $\cap$ E = Ø.]
2. Relevant equations
No equations, but there are relevant definitions. Per our book:
A metric space (M,d) is separable if $\exists$ a countable dense E $\subset$ M.
E$\subset$M is dense in M if $\forall$x$\in$M and $\forall$ $\epsilon$ > 0, $\exists$ e $\in$ E st d(x,e)< $\epsilon$
3. The attempt at a solution
My best attempt was doomed from the start, because I don't quite understand the hint. My thought process went as follows:
since X $\subset$ M, $\forall$x$\in$X, x$\in$M. Thus, since M is dense in E, $\forall$x$\in$X, $\forall$$\epsilon$>0, $\exists$e$\in$E st d(x,e)<$\epsilon$. At this point, I was done, because the set of e's satisfying the above, is a subset of E, a countable set. So a subset of a countable set is dense in X, and X is separable. This is incorrect, but I cannot see why.
Any help clearing up the confusion would be greatly appreciated.
Thanks!
Last edited: Oct 6, 2013
2. Oct 6, 2013
Dick
The problem statement already warned you what could go wrong with an approach like that. Suppose X is M-E. Your countable dense set, call it F, has to be a subset of M-E. But M-E doesn't contain any elements of E, so you can't use them. You have to prove M-E contains a different countable dense set. Try to construct one.
3. Oct 6, 2013
homesick
Yes, I know the hint clearly stated that this could cause issues. I said that above. I asked *why* this hint is there. The fact that I believe my proof is complete, means that I do not understand the hint. I don't understand at all what you mean by trying to construct a countable dense set that is in M-E. I have no idea what M is, all I know is that it contains 1 countable dense set.
4. Oct 6, 2013
Dick
Then the first step is to figure why you think your proof is complete. Take a concrete example. Q (the rational numbers) is a countable dense set of R (the real numbers) so R is separable. Now take the set R-Q (the irrational numbers). You need to show that's separable too and contains a countable dense set. But none of those points can be rational, so they can't be elements of your original countable dense set Q. Try running through the steps of your proof with that example. | 2017-08-20 08:07:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9030385613441467, "perplexity": 405.89760629243796}, "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-2017-34/segments/1502886106358.80/warc/CC-MAIN-20170820073631-20170820093631-00016.warc.gz"} |
https://brilliant.org/discussions/thread/impossible/ | # im(POSSIBLE)!!??
when I was in class 7 one of my teachers asked me that if 2=0 possible or not . Obviously the answer that came into my mind was NO! But sirs next words really made me think about how much right my answer is!! He said that he thinks there are no rules in Maths so why it cant be like2/2=0!? Since then i have no idea about sirs solution and i dont know that if it is right or wrong SO PLEASE CAN ANYONE TELL ME??
Note by Arna Mondal
4 years, 9 months ago
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Ofcourse, 2/2 = 0! =1 But it has nothing to do with 2 =0 (indeed not true). See 0! is totally different from' 0 the number 'actually. In other words if I explain you the meaning of 0! it would be : the ways of arranging 0 things which is definitely one way i.e. no way. Hence you cannot compare 0 & 0!
- 4 years, 9 months ago
Who says there are no rules in maths?
Maths is a subject with rules but no goals.
Philosophy is a subject with goals but no rules.
Staff - 4 years, 9 months ago
i didnt say that remember sir said
- 4 years, 8 months ago
IT IS TOTALLY WRONG!!!!!!!!!!!!!
- 4 years, 9 months ago
you got a weird screen name,dude...
- 4 years, 9 months ago
The simple way -Firstly 2 is not equal to 0 But as your teacher said that there is no rule is maths ...he is heck If you prove 2 = 0 you did like 2 / 2 = 0 This is wrong whenever anything is done in any equation ,It is done in both side.(but why I am talking about this) Ohh sorry.we know 2 * 1 = 2 then,How 2 * 0 = 2 will be...
- 4 years, 9 months ago
2 is not equal to 0, but if your teacher was joking then it could be,
2 = 0 (get the average of the 2 numbers) 2 - 1 = 0 - 1 (subtract the average) 1 = -1 (square both sides) 1 = 1
haha, but it really is not possible ;)
- 4 years, 9 months ago | 2019-01-22 15:46:03 | {"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": 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": 0.9819382429122925, "perplexity": 2462.001484634374}, "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-2019-04/segments/1547583857913.57/warc/CC-MAIN-20190122140606-20190122162606-00491.warc.gz"} |
https://www.fanography.info/3-2 | # Fanography
A tool to visually study the geography of Fano 3-folds.
Identification
##### Fano variety 3-2
divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
14
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 3 0
0 3 3 0
0 3 0
0 0
1
1
0 0
0 11 2
0 0 6 10
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
10
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry
This variety is rational.
This variety is primitive.
Deformation theory
number of moduli
11
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 11
Period sequence
The following period sequences are associated to this Fano 3-fold:
GRDB
#157
Fanosearch
#97
Extremal contractions
Semiorthogonal decompositions
There exist interesting semiorthogonal decompositions, but this data is not yet added.
Structure of quantum cohomology
By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
variety
$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^5$
bundle
$\Lambda(0,0,1) \oplus \mathcal{O}(0,1,2)$
See the big table for more information. | 2021-09-19 03:59:14 | {"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": 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": 0.6262311935424805, "perplexity": 2977.2495256598736}, "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/1631780056711.62/warc/CC-MAIN-20210919035453-20210919065453-00007.warc.gz"} |
http://irreal.org/blog/?cat=9&paged=2 | # Category Archives: Programming
## C
More and more often, these days, I see the opinion expressed that C is a dead, dangerous, stupid language and no one uses it anymore. For me, that's like having someone start a discussion by saying ”As everyone knows the … Continue reading
## An Example of pcase-lambda
Over at the Emacs reddit, instant_sunshine has an example of using pcase-lambda. It's not used very often but can be useful if you have to write functions that need to look inside Lisp objects. The documentation for pcase-lambda is a … Continue reading
Posted in Programming | Tagged , | 2 Comments
## Fundamentals
So true and so often disregarded I felt like saying this. pic.twitter.com/mHJ1rENoX1 — Hisham (@hisham_hm) December 13, 2015 Hat tip to Karl Voit.
## Wisdom From Wilfred Hughes
Some wisdom from Wilfred Hughes. I think this is right. Polyglot programmers are often very effective. They can take designs tested in one language and benefit another! — Wilfred Hughes (@_wilfredh) September 6, 2016
## Literate Programming with R
Since I've been writing about literate programming lately, here's a nice talk by Don Knuth on Literate Programming in R. Knuth begins his talk with an overview of several books that have been written using literate programming and discussing why … Continue reading
## SBCL 1.3.7 Released
The latest version of Steel Bank Common Lisp is out. This month's release fixed a bug and introduced several optimizations. See the NEWS file for details. As usual, the release compiled without incident on my MacBook Pro and the regressions … Continue reading
## Knuth's First Lecture
Back in November of 1969 a young Don Knuth gave his first lecture as a Stanford professor. The lecture was on a new field in Computer Science called Analysis of Algorithms, a field that Knuth invented and named. Recently, Knuth … Continue reading
## Running Unit Test with Org Mode
Frédérick Giasson has a nice post in which he describes how he uses Org mode to run unit tests on his code. Actually, his goal is to combine the writing, documentation, and testing of his code in a single document … Continue reading
Posted in Programming | Tagged , | 1 Comment
## Chez Scheme Open Sourced
Chez Scheme has always been an excellent Scheme implementation. It produces very fast object code and is a complete R6RS implementation with extensions. Chez Scheme was written by Kent Dybvig, who also wrote the excellent The SCHEME Programming Language (also … Continue reading | 2018-02-20 01:55:25 | {"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": 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": 0.31806543469429016, "perplexity": 4383.1162099543535}, "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-2018-09/segments/1518891812871.2/warc/CC-MAIN-20180220010854-20180220030854-00131.warc.gz"} |