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http://soleadea.org/cfa-level-1/money-weighted-return-vs-time-weighted-return | # Choosing Between Money-Weighted Return & Time-Weighted Return In Your CFA Exam
CFA Level 1 / Quantitative Methods / Money-weighted VS Time-weighted return
return on an investment: HPR, MWRR, TWRR
When you put your money to work, sooner or later you simply want to know how successful the investment has turned out to be. To evaluate the investment merits, you need to measure the rate of return and assess the performance. The money-weighted rate of return measure and the time-weighted rate of return measure will come in handy then.
## Holding Period Return (HPR)
You must know that portfolio measurement is not as easy a concept as it may seem and especially in your CFA level 1 exam it might turn out to be quite challenging or even problematic.
The basic concept connected with the return on an investment portfolio is the holding period return (HPR):
$HPR=\frac{P_1+D_1}{P_0}-1$
Where:
• $HPR$ - holding period return
• $P_1$ – price at the end of the period
• $P_0$ – price at the beginning of the period
• $D_1$ – dividend at the end of the period
So, HPR is the total return earned by an investor over a single period and it includes all cash flows occurring at the end of the period.
HPR is a valuable tool when you want to calculate the rate of return on an investment over one period assuming that no additions or withdrawals of money occur meanwhile. However, how should we compute the rate of return on the portfolio over many periods if there are cash inflows and outflows?
In such a case, we have two alternative measurement tools at our disposal, that is:
• the money-weighted rate of return, and
• the time-weighted rate of return.
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## CFA Level 1 Exam: Money-Weighted Rate of Return
The money-weighted rate of return is simply an internal rate of return (IRR). However, we use the term internal rate of return in the context of capital budgeting. In portfolio management, this measure is called money-weighted rate of return. How was this term coined? Well, the money-weighted return accounts for the value of cash flows in given periods. So, logically the values of particular cash flows affect the value of the money-weighted rate of return.
The money-weighted rate of return is calculated through equating discounted cash inflows to discounted cash outflows, but in the exam we’ll use Cash Flow and IRR worksheets.
### Money-Weighted Return – Example
An investor purchased 10 shares of a technological company for USD 50 each. After a year, he received a per-share dividend of USD 7 and sold 4 shares for USD 55 each. The investor decided not to reinvest the dividends. At the end of the second year, he sold the remaining 6 shares for USD 60 each. What is the money-weighted rate of return on this 2-year investment?
Using a calculator we can compute this rate of return. We're going to use the Cash Flow and IRR spreadsheets.
We open the Cash Flow worksheet and enter the following:
• [2ND] [CE|C] (to clear the worksheet from previous data)
• CF0=500 [+/-] [Enter] (there is a minus sign because we bought 10 shares for USD 50 each = it’s an outflow)
• C01=290 [Enter] (because after one year we received a dividend in the amount of USD 7 for each of 10 shares and we sold 4 shares for USD 55 each)
• C02=360 [Enter] (because at the end of the second year we sold 6 shares for USD 60 each)
Finally, we press the IRR button, press the CPT button and get the result. The money-weighted rate of return is 18.67% annually.
#### Exam tip
You should develop a habit of clearing worksheets. Otherwise, previous data may mess up with your future computations.
## CFA Level 1 Exam: Time-Weighted Rate of Return
The time-weighted rate of return differs from the money-weighted rate of return as it does not depend on the value of particular cash flows. The time-weighted rate of return is a geometric mean return over the whole investment period:
$TWRR=\sqrt[T]{(1+HPR_1)\times(1+HPR_2)\times{...}\times(1+HPR_T)}-1$
$TWRR=\\\sqrt[T]{(1+HPR_1)\times{...}\times(1+HPR_T)}\\-1$
Where:
• $TWRR$ - time-weighted rate return
• $HPR$ – holding period return
### Time-Weighted Return – Example
An investor purchased 10 shares of a technological company for USD 50 each. After a year, he received a per-share dividend of USD 7 and sold 4 shares for USD 55 each. The investor decided not to reinvest the dividends. At the end of the second year, he sold the remaining 6 shares for USD 60 each. What is the time-weighted rate of return on this 2-year investment?
$TWRR=\sqrt{(1+HPR_1)\times(1+HPR_2)}-1$
$TWRR=\\\sqrt{(1+HPR_1)\times1+HPR_2)}\\-1$
To calculate the time-weighted rate of return, we need to compute holding period returns for Year 1 and Year 2 (to be more precise we need to find 1+HPR1 and 1+HPR2):
$1+HPR_1=\frac{55+7}{50}=1.24$,
because the share price at the end of the first year is USD 55 and the dividend amounts to USD 7.
$1+HPR_2=\frac{60}{55}=1.0909$,
because the share price at the beginning of the second year is USD 55 and at the end of the second year amounts to USD 60.
So, the time-weighted rate of return amounts to:
$TWRR=\sqrt{1.24\times1.0909}-1=16.31\%$
$TWRR=\\\sqrt{1.24\times1.0909}\\-1=16.31\%$
#### Exam tip
When calculating the time-weighted rate of return you don't need to know the number of shares that were bought/sold (this data is redundant). You only need to know the prices of stock in different periods and the value of dividends.
As you can see the money-weighted rate is higher than the time-weighted rate of return. Is that a rule? Nope. Depending on numbers it could be the other way round.
Please also note that the time-weighted rate of return gives the same weights to different periods, whereas the money-weighted rate of return gives different weights to different periods. In this particular example with the money-weighted rate of return, a higher weight will be given to the first period as the beginning value in the first year was USD 500 and in the second year it was only USD 290. Since HPR for the first year is higher than for the second year and the money-weighted rate of return gives a higher weight to the first period, the money-weighted rate of return must be higher than the time-weighted rate of return.
### Performance of Portfolios: Application
In the previous paragraph we learnt that the money-weighted rate of return gives different weights to different periods, while the time-weighted rate of return gives the same weights to different periods. Therefore, we should conclude that if an investment manager has full control of the timing of cash inflows and outflows, we should use the money-weighted rate of return. It would be appropriate in the case of our private investments when it is us who decides what amounts we invest in every period. When the portfolio manager has little influence on the invested amounts, we should apply the time-weighted rate of return. This is the case with a fund manager, who doesn’t have control over how much the clients add to or withdraw from the fund.
## CFA level 1 Exam Takeaway: MWRR vs TWRR
• The money-weighted rate of return is an internal rate of return (IRR).
• The time-weighted rate of return is a geometric mean return over the whole investment period.
• You should remember to clear calculator worksheets before doing any computations.
• To calculate the money-weighted return we use the CF and IRR worksheets (in your calculator remember to enter a minus sign in case of outflows).
• When calculating the time-weighted return instead of calculating HPRs for consecutive periods it would have been faster if we computed 1 plus HPRs for these periods.
• When calculating the time-weighted rate of return you don't need to know the number of shares that were bought/sold. You only need to know the prices of stock in different periods and the value of dividends.
• Generally the money weighted and the time-weighted return are not equal. Exception: If throughout the investment period the investor does not sell any shares, nor does he purchase any new ones, and all received dividends are reinvested, then the money-weighted rate of return is equal to the time-weighted rate of return.
• An investment manager has a full control of the timing of cash flows use money-weighted return for manager's evaluation; otherwise time-weighted return is preferable. | 2018-11-12 22:49:10 | {"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": 0.4280907213687897, "perplexity": 1396.00071865693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741151.56/warc/CC-MAIN-20181112215517-20181113001517-00424.warc.gz"} |
https://www.authorea.com/users/16716/articles/18248/_show_article | ROUGH DRAFT authorea.com/18248
Abstract
# ABSTRACT
DNA methylation is an epigenetic mark that plays an inadequately understood role in gene regulation, particularly in non-model species. Because it can be influenced by the environment and potentially transferred to subsequent generations, DNA methylation may contribute to the ability of organisms to acclimatize and adapt to environmental change. We evaluated the distribution of gene body methylation in reef-building corals, a group of organisms facing significant environmental challenges. Gene body methylation in six species of corals was inferred from in silico transcriptome analysis of CpG O/E, an estimate of germline DNA methylation that is highly correlated with patterns of methylation enrichment. Consistent with what has been documented in most other invertebrates, all corals exhibited bimodal distributions of germline methylation suggestive of distinct fractions of genes with high and low levels of methylation. The hypermethylated fractions were enriched with genes with housekeeping functions, while genes with inducible functions were highly represented in the hypomethylated fractions. In three of the coral species, we found that genes differentially expressed in response to thermal stress and ocean acidification exhibited significantly lower levels of methylation. These results support a link between gene body hypomethylation and transcriptional plasticity that may point to a role of DNA methylation in the response of corals to environmental change.
# Introduction
As human influence on the planet expands, many organisms must acclimatize and adapt to rapid environmental change. Phenotypic plasticity facilitates a more rapid response to environmental change than is possible through natural selection, and will likely be critical to the persistence of many species (Charmantier ), (Chevin 2010). Phenotypic change often involves modifications in gene expression. Epigenetic mechanisms, involving alterations to the genome that do not affect the underlying DNA sequence, are increasingly recognized as some of the principal mediators of gene expression (Duncan 2014). The most researched and best understood epigenetic process is DNA methylation, which most commonly involves the addition of a methyl group to a cytosine in a CpG dinucleotide pair. The role of DNA methylation is best understood in mammals, where methylation in promoter regions has a repressive effect on gene expression (Jones 2001). In plants and invertebrates, methylation of gene bodies prevails, and is thought to be the ancestral pattern (Zemach 2010). Gene body methylation appears to have a range of functions, including regulating alternative splicing, repressing intragenic promoter activity, and reducing the efficiency of transcriptional elongation (Duncan 2014). Methylation of gene bodies also varies according to gene function, and studies on invertebrates indicate that highly conserved genes with housekeeping functions tend to be more heavily methylated than those with inducible functions (Roberts 2012), (Sarda 2012), (Dixon 2014), (Gavery 2014). This has led to speculation that gene body methylation may promote predictable expression of essential genes for basic biological processes, while an absence of methylation could allow for stochastic transcriptional opportunities in genes involved in phenotypic plasticity (Roberts 2012), (Dixon 2014), (Gavery 2014).
Direct relationships between DNA methylation and phenotypic plasticity are increasingly being established. Some examples include caste structure in honeybees and ants (Kucharski 2008), (Bonasio ), expression of the agouti gene in mice ((Wolff )), and the influence of prenatal maternal mood on newborn stress levels in humans (Oberlander ). In many cases, changes in methylation patterns can be attributed to external cues such as temperature, stress, or nutrition. A prime example is the honeybee Apis mellifera, where larval consumption of royal jelly induces changes in methylation that ultimately determine the developmental fate of an individual into a queen or a worker (Kucharski 2008). Thus, DNA methylation has been established as a key link between environment and phenotype.
Reef-building corals, the organisms that form the trophic and structural foundation of coral reef ecosystems, are known to display a significant degree of phenotypic plasticity (Todd ), (Forsman 2009), (Granados-Cifuentes ). As long-lived, sessile organisms, corals are thought to be particularly reliant on phenotypic plasticity to cope with environmental heterogeneity, because they must be able to withstand whatever nature imposes on them over long periods of time (Bruno 1997). As phenotypically flexible as they may be, corals’ longevity and immobility may also contribute to their vulnerability in a changing environment. Reef corals worldwide are experiencing severe declines due to a variety of anthropogenic effects, including climate change, ocean acidification, and a host of local stressors (Hoegh-Guldberg 2007). This has raised doubt concerning the ability of corals to survive coming decades. Yet there are also signs that, at least in some cases, corals possess sufficient resiliency to overcome their numerous challenges (Palumbi 2014). Recent studies on gene expression variation, for example, support the view that phenotypic plasticity in corals is robust and may provide resilience in the face of ocean warming (Barshis 2013), (Granados-Cifuentes ), (Palumbi 2014). However, the underlying basis of gene expression variation, and indeed phenotypic plasticity, remain largely unknown.
Evaluation of epigenetic processes therefore represents a logical next step in understanding coral gene expression and phenotypic variation. While recent annotation of the Acropora digitifera genome revealed a broad repertoire of genes involved in DNA methylation and other epigenetic processes (Dunlap 2013), to date, only one study has investigated possible roles of epigenetic processes in corals (Dixon 2014). Germline DNA methylation patterns in the transcriptome of Acropora millepora corroborated findings reported in studies of other invertebrate species (Dixon 2014). Most interestingly, genes that were differentially expressed in response to a common garden transplantation experiment were among the genes exhibiting lower levels of germline methylation (Dixon 2014). This finding further supports studies on invertebrates showing that hypomethylated genes tend to be those with inducible functions (Gavery 2010), (Sarda 2012).
Coral gene expression studies continue to expand, providing rich datasets to further probe the relationship between DNA methylation and gene function. In this study, we performed a comprehensive evaluation of germline methylation patterns in reef corals by examining the transcriptomes of six scleractinian coral species. Germline methylation levels in these data were inferred based on the hypermutability of methylated cytosines, which leads to a reduction in CpG dinucleotides over evolutionary time (Sved ). These data were then matched with gene ontology information, permitting evaluation of methylation patterns associated with broad categories of biological processes. Lastly, in three of the six species, we evaluated germline methylation patterns in genes involved in response to thermal stress and ocean acidification.
# Methods
Transcriptome data sources
The transcriptomes of six scleractinian coral species were evaluated to determine germline methylation patterns in relation to gene function and activity. Species examined included Acropora hyacinthus, A. millepora, A. palmata, Pocillopora damicornis, Porites astreoides, and Stylophora pistillata (Table 1). These transcriptomes reflect a range of life history stages. Some transcriptomes were developed from life history stages that had not yet been infected with symbiotic dinoflagellates (Symbiodinium spp.), while others used bioinformatic techniques to filter out putative Symbiodinium sequences. However, two of the transcriptomes (P. damicornis and S. pistillata) were developed from adult corals and did not remove putative symbiont sequences. We therefore applied a filtering step to these transcriptomes by comparing them to Symbiodinium clade A and B transcriptomes from (Bayer 2012) using blastn (version 2.2.29). An evalue threshold of of 10-5 was used for these queries, and all matched sequences were removed from further analyses. Details of blastn query and filtering are provided (https://github.com/jldimond/Coral-CpG-ratio-MS)
Table 1. Transcriptomes used in this study
Organism Life history stage Method No. Contigs Reference
Acropora hyacinthus Adult Illumina 33,496 Barshis et al. (2013)(Barshis 2013)
Acropora millepora Embryo to adult 454 & Illumina 52,963 Moya et al. (2012)(Moya 2012)
Acropora palmata Larval 454 88,020 Polato et al. (2011)(Polato 2011)
Pocillopora damicornis Adult Illumina 72,890 Vidal-Dupiol et al. (2013)(Vidal-Dupiol 2013)
Porites astreoides Adult 454 30,740 Kenkel et al. (2013)(Kenkel 2013)
Stylophora pistillata Adult 454 15,052 Karako-Lampert et al. (2014)(Karako-Lampert 2014)
Illumina is Company - 454 is platform, several types of Illumina platforms.
Differentially expressed gene datasets
In addition to analyzing whole transcriptomes, we also examined genes differentially expressed in response to environmental stressors for the three acroporid species. For A. hyacinthus and A. millepora these gene sets were derived from the same studies that developed the reference transcriptomes ((Barshis 2013), (Moya 2012)), and for A. palmata differentially expressed genes sets were reported in (Polato 2013)(Polato Nicholas R.; Baums 2013) (Table 2).
Table 2. Differentially expressed gene sets examined
Organism Life history stage Method No. Contigs Environmental factor Reference
Acropora hyacinthus Adult sequencing 484 Thermal stress Barshis et al. (2013)(Barshis 2013)
Acropora millepora Juvenile sequencing 234 Ocean acidification Moya et al. (2012)(Moya 2012)
Acropora palmata Larval microarray 2002 Thermal stress Polato et al. (2013)(Polato 2013)(Polato Nicholas R.; Baums 2013)
Annotation
To maintain consistency in comparing datasets, all transcriptomes and differentially expressed gene sets were compared to the UniProt/Swiss-Prot protein database (version 2/17/2015) using Blastx (version 2.2.29) with an evalue threshold of 10-5. To further annotate genes with functional categories, corresponding Gene Ontology Slim (GOSlim) biological process were identified. Details of annotation are provided in jupyter notebooks in accompanying repository (https://github.com/jldimond/Coral-CpG-ratio-MS).
Predicted germline methylation
Germline methylation levels were inferred based on the hypermutability of methylated cytosines, which tend towards conversion to thymines over evolutionary time. This results in a reduction in CpG dinucleotides, meaning that heavily methylated genomic regions are associated with reduced numbers of CpGs. Thus, methylation patterns that have been inherited through the germline over evolutionary time can be estimated using the ratio of observed to expected CpG, known as CpG O/E. Germline DNA methylation estimated by analysis of CpG O/E is highly correlated with direct assays of methylation ((Suzuki 2007), (Sarda 2012), (Gavery 2013)). CpG O/E was defined as:
CpG O/E = (number of CpG / number of C x number of G) x (l2/l-1)
where l is the number of nucleotides in the contig.
Only annotated sequences were used for calculation of CpG O/E to increase the likelihood that sequences were oriented in the 5' to 3' direction. For subsequent analyses, we set minimum and maximum limits for CpG O/E at 0.001 and 1.5, respectively, to exclude outliers. Details of germline methylation prediction methods are provided in jupyter notebooks (https://github.com/jldimond/Coral-CpG-ratio-MS/)
Statistical analyses
Transcriptome CpG O/E patterns were fitted with the normalmixEM function in the mixtools package in the R statistical platform. Mixture models were evaluated against the null single component model by comparison of log-likelihood statistics. High and low CpG O/E components were delineated in mixture models using the intersection point of component density curves. For each GOSlim term, enrichment in the high and low CpG O/E components identified in mixture | 2016-10-22 09:00:25 | {"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": 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": 0.6021935343742371, "perplexity": 11753.96551323308}, "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-2016-44/segments/1476988718866.34/warc/CC-MAIN-20161020183838-00370-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/42965/why-is-the-boiling-point-of-ch3cooh-higher-than-that-of-c2h5oh | # Why is the boiling point of CH3COOH higher than that of C2H5OH?
Why is the boiling point of $\ce{CH3COOH}$ higher than that of $\ce{C2H5OH}$ ? Both are polar molecules held by hydrogen bond.
• CH3COOH is more polar. Dec 30 '15 at 8:32
• – user7951
Dec 30 '15 at 12:58
Yes you are correct. The molecules are polar in nature and are bound by intermolecular hydrogen bonding. But you must pay attention to the extent of polarization in both the molecules. Consider the alcohol. Here the carbon bearing the $\ce{-OH}$ group is the only polarizing group present. But for $\ce{CH3COOH}$ , the carbonyl carbon is polarized by an $\ce{-OH}$ group as well an $\ce{=O}$ group attached to it, thus increasing its effective polarization more than the alcohol. Therefore $\ce{CH3COOH}$ has greater boiling point. | 2021-11-29 06:03:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.584674060344696, "perplexity": 730.8286131610031}, "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/1637964358688.35/warc/CC-MAIN-20211129044311-20211129074311-00115.warc.gz"} |
https://dml.cz/handle/10338.dmlcz/119662 | # Article
Full entry | PDF (0.3 MB)
Keywords:
non-standard growth; vector case; local minimizers; interior regularity; problems of higher order
Summary:
We consider local minimizers $u : \Bbb R^2\supset \Omega \to \Bbb R^N$ of variational integrals like $\int_\Omega [(1+|\partial_1 u|^{2})^{p/2}+(1+|\partial_2 u|^{2})^{q/2}]\,dx$ or its degenerate variant $\int_\Omega [|\partial_1 u|^p+|\partial_2 u|^q]\,dx$ with exponents $2\leq p < q < \infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. {\bf 16} (2003), 177--186. We prove interior $C^{1,\alpha}$- respectively $C^{1}$-regularity of $u$ under the condition that $q < 2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. {\bf 31} (2006), 349--362.
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[Ma2] Marcellini P.: Regularity and existence of solutions of elliptic equations with $(p,q)$-growth conditions. J. Differential Equations 90 (1991), 1-30. MR 1094446 | Zbl 0724.35043
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[UU] Ural'tseva N.N., Urdaletova A.B.: Boundedness of gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad. Univ. Mat. Mekh. Astronom no. 4 (1983), 50-56 (in Russian); English translation: Vestnik Leningrad. Univ. Math 16 (1984), 263-270. MR 0725829
Partner of | 2017-12-18 03:30:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8773170113563538, "perplexity": 2338.9712282533596}, "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-2017-51/segments/1512948604248.93/warc/CC-MAIN-20171218025050-20171218051050-00749.warc.gz"} |
https://www.sarthaks.com/1178746/find-the-particular-solution-of-the-differential-equation-dy-dx-1-xy-given-that-0-when-x | # Find the particular solution of the differential equation dy/dx = 1 + x + y + xy, given that y = 0 when x = 1.
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Find the particular solution of the differential equation
$\frac{dy}{dx}$ = 1 + x + y + xy, given that y = 0 when x = 1.
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Given:
$\frac{dy}{dx}$ = (1 + x)(1 + y)
⇒ $\frac{dy}{1+y}$ = (1 + x)dx
⇒ log |y + 1| = (x + $\frac{x^2}2$ + c)
⇒ Now, for y = 0 and x = 1,
We have,
⇒ 0 = 1 + $\frac{1}2$ + c
⇒ c = - $\frac{3}2$
⇒ log |y + 1| = $\frac{x^2}2$ + x - $\frac{3}2$ | 2022-11-27 19:15:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5305464863777161, "perplexity": 1004.8372477704077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00793.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-2-linear-and-quadratic-functions-section-2-5-inequalities-involving-quadratic-functions-2-5-assess-your-understanding-page-163/6 | ## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
a) $(-\infty, -3 ) \cup (1, \infty)$ b) $[-3,1]$
a) We observe from the graph the interval where the function $f$ value is below the value of $g$. We get the interval: $(-\infty, -3 ) \cup (1, \infty)$ (b) We reverse the interval in (a) to get: $[-3,1]$ | 2021-10-28 11:29:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.821352481842041, "perplexity": 719.4354375938962}, "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/1634323588284.71/warc/CC-MAIN-20211028100619-20211028130619-00519.warc.gz"} |
http://faculty.franklin.uga.edu/petridis/content/publications | # Publications
My arXiv public author identifier is petridis_g_1.
I have a Google Scholar page.
A slightly longer account of my research can be found in the corresponding page.
Journal publications in reverse chronological order of completion.
13. Products of differences over arbitrary finite fields (with Brendan Murphy), 2017. arxiv
12. New results on sum-product type growth over fields (with Brendan Murphy, Oliver Roche-Newton, Misha Rudnev and Ilya D. Shkredov), 2017. arxiv
11. Pinned algebraic distances determined by Cartesian products in $\mathbb{F}_p^2$, Proceedings of the American Mathematical Society, volume 145, issue 11, pp. 4639-4645, 2017. arXiv | Journal
10. Bounds on trilinear and quadrilinear exponential sums (with Igor Shparlinski), accepted in Journal d'Analyse Mathematique, 2016. arXiv
9. A point-line incidence identity in finite fields, and applications (with Brendan Murphy), Moscow Journal of Combinatorics and Number Theory, volume 6, issue 1, pp. 64-95, 2016. Journal | arXiv
8. On the number of dot products determined by a large set and one of its translates in finite fields, Online Journal of Analytic Combinatorics, issue 12, #4, 2017. Journal | arXiv
7. Tiling sets and spectral sets over finite fields (REU project run together with Alex Iosevich and Jonathan Pakianathan), Journal of Functional Analysis, volume 273, issue 8, pp. 2547-2577, 2017. Journal | arXiv
6. The cardinality of sumsets: different summands (with Brendan Murphy and Eyvi Palsson), Acta Arithmetica, volume 167, issue 4, pp 375-395, 2015. Journal | arXiv
5. Matchings in random biregular bipartite graphs (with Guillem Perarnau), The Electronic Journal of Combinatorics, volume 20, issue 01, p60, 2013. Journal | arXiv
4. New proofs of Plünnecke-type estimates for product sets in groups, Combinatorica, volume 32, issue 06, pp. 721-733, 2012. Journal | arXiv
3. The $L^1$-norm of exponential sums in $\mathbb{Z}^d$, Mathematical Proceedings of the Cambridge Philosophical Society, volume 154, issue 03, pp. 381-392, 2013. Journal | arXiv
2. Upper bounds on the cardinality of higher sumsets, Acta Arithmetica, volume 158, issue 04, pp. 299-319, 2013. Journal | arXiv
1. Plünnecke's inequality, Combinatorics, Probability and Computing, volume 20, issue 06, pp. 921-938, 2011. Journal | arXiv
________
Peer reviewed expository articles in the proceedings of Mel Nathanson's legendary Combinatorial and Additive Number Theory yearly conference in Manhattan.
II. A second wave of expanders over finite fields, (with Brendan Murphy) to appear in Combinatorial and Additive Number Theory: CANT 2015 and 2016, 2017. arxiv
I. The Plünnecke-Ruzsa inequality: an overview, in: Combinatorial and Additive Number Theory: CANT 2011 and 2012, 229-241, 2014. Author's Copy | Proceedings
________
Preprints that did not make it to print.
Collinear triples and quadruples for Cartesian products in $\mathbb{F}_p^2$ (absorbed in [13] above). arXiv
Products of differences in prime order finite fields (absorbed in the second expository article above; see also [12] and [13]). arXiv | 2018-08-15 05:26:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5902892351150513, "perplexity": 2116.1327906467855}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-34/segments/1534221209884.38/warc/CC-MAIN-20180815043905-20180815063905-00312.warc.gz"} |
https://corumysysuvap.sunshinesteaming.com/dynamics-of-pulmonary-gas-exchange-and-heart-rate-changes-at-start-and-end-of-exercise-book-12936nd.php | Last edited by Zuluzshura
Saturday, August 1, 2020 | History
2 edition of Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. found in the catalog.
Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
# Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
## by Dag Linnarsson
• 195 Want to read
• 18 Currently reading
Published by Karolinska Institutet, Faculty of Medicine in Stockholm .
Written in English
Edition Notes
ID Numbers Series Acta physiologica Scandinavica -- supplementum 415 Open Library OL17335117M ISBN 10 9172220821
HIDROGINÁSTICA – Exercícios em suspensão, 82 Saltos, 85 Abdominais, 88 Membros Superiores, 89 Capítulo 4 ELAbORANDO umA. Respostas cardiorespiratórias em exercícios de hidroginástica executados. in the water and on land with and without the X walk’n tone exercise belt. 15 Minute Pool Exercise Routine For Rapid Weight Loss Exercicios Na Piscina, Hidro, Exercícios. Girandola, R.N., and F. I. Katch, Effects of physical conditioning in changes in exercise and recovery oxygen uptake and efficiency during constant load ergometer exercise. Med Sci. Sports 5: Linnarson, D. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. Acta. Phys. Scand,
Studies of the limited exercise capacity in patients with COPD have not assessed the recovery phase, although the phenomenon of increased oxygen uptake after exercise has been thoroughly investigated in normal subjects. Therefore, we compared the recovery of gas exchange variables and HR after maximal cycle ergometry in 16 patients with varying severities of airflow obstruction and ten aged. Attainment of a steady state for oxygen uptake () during constant work rate exercise has been reported to take longer for patients with chronic heart failure (CHF) compared with normal. The steady state is also delayed in normal subjects during high-intensity exercise compared with moderate exercise, however, and the delay correlates with the degree of associated lactic acidosis.
Start studying Pulmonary Ventilation & Gas Exchange. Learn vocabulary, terms, and more with flashcards, games, and other study tools. - rate of gas movement into or out of the lungs per minute. small amount of venous blood returns from certain tissues and bypasses lungs for gas exchange on way to L heart. Effect of interbreath fluctuations on characterizing exercise gas exchange kinetics. J Appl Physiol –, Link | ISI Google Scholar; Linnarsson D. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. .
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### Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise by Dag Linnarsson Download PDF EPUB FB2
Acta Physiol Scand Suppl. ; Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
Linnarsson by: Corpus ID: Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. @article{LinnarssonDynamicsOP, title={Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.}, author={Dag Linnarsson}, journal={Acta physiologica Scandinavica.
Linnarsson D. () Dynamics of pulmonary gas exchange and heart rate at start and end of exercise. Acta physiol. Scand. Miyamoto Y., Nakazono Y., Hiura T.
and Abe Y. () Cardiorespiratory dynamics during sinusoidal and im- pulse exercise in man. Jap. Physiol. 33, Cited by: 4. Six Standardbred horses were used to evaluate the time course of pulmonary gas exchange, ventilation, heart rate (HR) and acid base balance during different intensities of constant-load treadmill exercise.
Horses were exercised at approximately 50%, 75% and % maximum oxygen uptake ($$\dot V_{{\text{O}}_2 }$$ max) for 5 min and measurements taken every 30 s throughout by: Linnarsson D () Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
Acta Physiol Scand [Suppl] – Google Scholar. Murphy PC, Cuervo LA, Hughson RL () Comparison of ramp and step exercise protocols during hypoxic exercise in by: Linnarson D () Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
Acta Physiol Scand [Suppl] –68 Google Scholar. The rate of increase of pulmonary oxygen uptake (V ˙ o 2) at exercise onset depends on the rates of increase of cardiac output and tissue O 2 extraction (V ˙ o 2 = cardiac output X arterial-venous O 2 difference). As early as1 it was reported that the dynamic response of V ˙ o 2 to exercise was slowed in the presence of circulatory disease.
More recently, it has been recognized. This chapter discusses the arterial blood gases, heart rate, and gas exchange during rest and exercise in men saturated at a simulated seawater depth of ft.
In a study described in the chapter, physiological measurements were obtained from each of three healthy male volunteers during alternating periods of exercise and rest. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise.
Acta Physiologica Scandinavica 1 – 8 Whipp, BJ, Ward, SA, Lamarra, N, Davis, JA and Wasserman, K (). Cardiopulmonary exercise testing (CPX) measures a broader range of variables related to cardiorespiratory function, including expiratory ventilation (V̇ e) and pulmonary gas exchange (oxygen uptake [V̇ o 2] and carbon dioxide output [V̇ co 2]), along with the ECG and blood pressure, with the goal of quantitatively linking metabolic.
Linnarsson D. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. Acta Physiol Scand Suppl. ; – Miyamoto Y, Hiura T, Tamura T, Nakamura T, Higuchi J, Mikami T. Dynamics of cardiac, respiratory, and metabolic function in.
Cardio-Pulmonary Changes during Exercise – Heart rate increases linearly with the severity and duration of exercise. Heart rate is slightly increased even before the onset of exercise due to influence of cerebral cortex on the medullary cardiac centre.
The maximal heart rate achieved is determined by the age of the subject. Also called. Linnarsson D. Dynamics of Pulmonary Gas Exchange and Heart Rate Changes at Start and End of Exercise.
Acta Physiol Scand. ; – [Google Scholar] Lo LW, Vinogradov SA, Koch CJ, Wilson DF. A new, water soluble, phosphor for oxygen measurements in vivo. Adv Exp Med Biol. ; – [Google Scholar]. Linnarsson D. Dynamics of pulmonary gas exchange and heart rate changes at the start and end of exercise.
Acta Physiol Scand 1–68, Mahler M. First order kinetics of muscle oxygen consumption, and equivalent proportionality between Q˙ O2 and phosphorylcreatine level.
Implications for the control of respiration. J Gen Physiol Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. Effects of aquatic resistance training on neuromuscular performance in healthy women. The aim of the present study was to analyse the heart rate and the oxygen uptake in hydrogymnastics exercises performed with and without the use of resistive equipment.
To elucidate the role of factors other than the nervous system in heart rate (f c) control during exercise, the kinetics off c and plasma catecholamine concentrations were studied in ten heart transplant recipients during and after min cycle ergometer exercise at 50 W.
Thef c did not increase at the beginning of the exercise for about 60 s. 3 Metra M, Raccagni D, Carini G, Orzan F, Papa A, Nodari S, Cody RJ, Dejours P.
Ventilatory and arterial blood gas changes during exercise in heart failure. In: Exercise Gas Exchange in Heart Disease. Wasserman K, ed. Armonk, NY: Futura Publishing Co; Google Scholar; 4 Kobayashi T, Itoh H, Kato K.
The role of increased dead space. output (ho,), O2 uptake (vo2), and heart rate were comprised of an abrupt increase at exercise onset, followed by a slower rise to the new steady state (t = 48, 43, 31, and 33 s.
Linnarsson D. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. Acta Physiol Scand ; Suppl.: 1– CAS Google Scholar 6. Whipp BJ. Dynamics of pulmonary gas exchange. Circulation ; 76 Suppl.
VI: 18– Google Scholar. Linnarsson D. Dynamics of pulmonary gas exchange and heart rate changes at start and end of exercise. Acta Physiol Scand Suppl. ; – Linton RA, Band DM. The effect of potassium on carotid chemoreceptor activity and ventilation in the cat.
Respir Physiol. Jan; 59 (1)– Miller JP, Cunningham DJ, Lloyd BB, Young JM. A symptom-limited incremental exercise test with the measurement of expired gas is widely performed for evaluating exercise capacity in cardiac patients and for stratifying patients with heart fail-ure.
1 2 3 Among indexes of cardiopulmonary exercise testing, the peak oxygen uptake (V ˙ o 2) measured at maximal exercise has been considered a “gold standard” because it reflects maximal.
Gas exchange, ventilation and heart rate were measured with each breath, as described below. The separate transition responses for Vo: and "o~ were time aligned to the start of exercise and averaged for each subject.
Measurement of gas exchange.Linnarsson D (). Dynamics of pulmonary gas exchange and the heart rate changes at start and end of exercise. Acta Physiologica Scandinavica, (Suppl): [ Links ] Ribeiro TF, Gomes VR, Moura MAS et al. (). Estudo do limiar de anaerobiose em mulheres sedentárias durante esforço físico . | 2021-01-23 08:51:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4271370470523834, "perplexity": 12541.812520099214}, "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/1610703536556.58/warc/CC-MAIN-20210123063713-20210123093713-00169.warc.gz"} |
https://www.dan-moss.com/hypothesis-testing/ | Much of the statistics we end up meeting at university involves inference about some parameter of the model, wherein we try and figure out what process generated our data. This might pop up in a so-called "regression" problem where we seek to establish a relationship between some dependent variable (or response/output/label) $$Y$$ and some independent variables (or covariates/inputs/features) $$X=(X_1,\dots,X_p)$$. Then we might be able to make statements like "We are 95% confident that the average relationship between $$X$$ and $$Y$$ looks like this."
Two very much related tasks in statistics are prediction and (hypothesis) testing. In the former case, we might ask "If these are my covariates, what is the probability that my response will look like this?" In the latter case, we might ask questions such as, "Do I think that this covariate matters or not?" and it is this setting which I would like to discuss today.
Null, alternative and errors
When we formulate a testing problem, we typically compare a null hypothesis $$H_0$$ against an alternative hypothesis $$H_1$$. In this formulation, we intuitively think of $$H_0$$ as the thing we would initially presume, and $$H_1$$ as an alternative which we perhaps think less likely from the outset, or which we don't want as our starting assumption for some reason. In a sense, $$H_0$$ and $$H_1$$ are just different labels for the two hypotheses, but the usual framework for conducting tests introduces the kind of asymmetry I'm describing, which I will come back to later. For example in a court case where we have presumption of innocence until proven guilty, we would certainly set $$H_0$$ to be "innocent" and $$H_1$$ to be "guilty."
Each hypothesis will provide some (set of) possible process(es) by which the data we see was generated. This data will have a different distribution depending on whether $$H_0$$ is true. For example, we might that under $$H_0$$ it is pretty unlikely that a defendant's DNA was found on someone's coat, while that might be a pretty likely outcome under $$H_1$$. Thus the evidence in this case, or the data more generally, will have associated distributions $$P_0$$ and $$P_1$$ modelling the likelihood of different scenarios under the hypotheses $$H_0$$ and $$H_1$$ respectively.
Since we will use the data to guide our decision on $$H_0$$ and $$H_1$$, this decision is subject to (at least) the same randomness as is found in the data, and so it makes sense to talk about the probability of deciding on one hypothesis or the other. Since the distribution of the data will depend on which hypothesis is true, we can specifically talk about each of these probabilities under each of the distributions associated to $$H_0$$ or $$H_1$$. Mathematically speaking, we can write $$P_j(H_i)$$ to be "probability of accepting $$H_i$$ when $$H_j$$ is true." where $$i,j\in\{0,1\}$$ (meaning the subscripts $$i$$ and $$j$$ take values in the set "0 or 1," which indexes our hypotheses).
Then Type I and Type II errors correspond to particular instances of these probabilities where $$i$$ and $$j$$ do not match, in which case we accepted the wrong hypothesis. Type I error is $$P_0(H_1)$$ which is the probability of falsely rejecting $$H_0$$ (meaning that our evidence was gathered in a situation where the defendant was innocent, but it led us to a guilty verdict). Type II error is $$P_1(H_0)$$, the probability of falsely accepting $$H_0$$. Note that under both $$P_1$$ and $$P_0$$ we would end up either picking $$H_0$$ or $$H_1$$ (this framework doesn't provide a mechanism to reject both, and often rejecting both wouldn't make sense) and so this entails that $$P_0(H_0)+P_0(H_1)=1$$ and $$P_1(H_0)+P_1(H_1)=1$$ . This does not mean that $$P_0(H_0)+P_1(H_0)=1$$ (or for $$H_1$$) however (for instance if we are in a rigged court that always finds the defendant guilty regardless of what actually happened).
Size, power and asymmetry in testing
The Type I error $$P_0(H_1)$$ is often called the size of the test and denoted with the letter $$\alpha$$. The probability $$P_1(H_1)=1-P_1(H_0)$$ is called the power of the test and denoted $$\beta$$, with the fact that this is equal to one minus the Type II error being a consequence of what we discussed above. These definitions already nod towards the asymmetry we will see later.
We should note that we do have a certain amount of freedom over controlling these errors. As a silly example, we could imagine a situation where we always accept the null hypothesis. This would make the Type I error go to zero at the expense of the Type II error going to 100%. The question of having a good test is not just about controlling one error or the other - we just saw that this would be trivial - but rather about having a "good" control of the errors together.
To make our notation precise, we introduce a "test" $$\psi$$ which is a function of the data (i.e. it depends only on things we can observe, not on which hypothesis is actually true which we can't see) which spits out either a 0 or 1. We would then choose $$H_0$$ if it's 0 and $$H_1$$ if it's 1. Then the problem of hypothesis testing is the problem of choosing a suitable $$\psi$$. So my silly example above would involve always having $$\psi=0$$ all the time, which we didn't think was a very good idea. In this framework, $$P_i(H_j)=P_i(\psi=j)$$. This is just a way of making explicit the mechanism $$\psi$$ by which we decide which hypothesis to choose, then if we want to compare mechanisms we have a notation where we can distinguish between one test $$\psi$$ and a different test $$\psi'$$.
The question then remains as to a suitable criterion to decide how to do our hypothesis testing, which amounts to a suitable criterion to choose $$\psi$$. This is where the asymmetry tends to come in, although there is no reason it really has to. We might imagine that a sensible criterion would which tells us whether to take $$H_0$$ or $$H_1$$ such that $$\psi$$ minimises the sum of the Type I and Type II errors. Mathematically, that would mean
$P_1(\psi=0)+P_0(\psi=1) = \min_{\psi'}\{P_1(\psi'=0)+P_0(\psi'=1)\}.$
However, it is more standard (at least when the topic is first introduced in a statistics course) to first control the Type I error (or size) of the test $$\psi$$ to a desired level, and then choose which of those has the lowest Type II error (or equivalently, the highest power). This would mean first specifying some threshold level $$\alpha$$ (which tends to be put at 5%, but this is by no means always the right value) and then choosing, among all tests which have size at most $$\alpha$$ (which would include my silly test), the test which has the highest power. Mathematically, this amounts to choosing a test $$\psi$$ for which
$P_1(\psi=0) = \min_{\psi':P_0(\psi'=1)\leq\alpha}\{P_1(\psi'=0)\}.$
There is a bit of a subtlety here in that I allow tests to have size less than alpha as well as being equal to alpha. Intuitively, size being lower is better, so if it turns out my most powerful test happens to have a size lower than $$\alpha$$ then that's the best one. However, beyond the size threshold $$\alpha$$, we don't allow more tradeoff between $$\alpha$$ and $$\beta$$, for which the optimisation problem would be
$P_1(\psi=0) = \min_{\psi':P_0(\psi'=1)\leq\alpha}\{P_1(\psi'=0)+P_0(\psi'=1)\}.$
This would allow us to pick a $$\psi$$ where we get a size a fair bit less than $$\alpha$$ in return for a minor reduction in power, and maybe this is a reasonable criteria to use as well.
It is these last two formulations (the former being the typical one) which introduces asymmetry into the hypotheses. This is because my silly test where I always accept $$H_0$$ would actually be preferable to a test in which both the Type I and Type II errors are sat at 6% (for $$\alpha=5\%)$$. In fact, it would even be better than one where the Type I error was 5.000001% and the Type II error was zero - neither one would be a viable solution. Sometimes we want this asymmetry, but it is certainly not clear that we always do.
Remark on "optimal" tests (+Exercise for reader)
I may be wrong, but I think that the solution to the optimisation problem without tradeoff (the first asymmetric one) is always optimised with size being equal to alpha, assuming we have access to a external random number generator. If we have more "size budget" remaining, we could always add further randomisation to our test which allows us to use it up to get increased power. For instance, imagine $$\alpha=0.05$$ but my proposed optimiser had $$\alpha=0.04$$. I could generate a random number between 1 and 1000 independently of all the other data, and then if I get 1000 I just reject $$H_0$$ in favour of $$H_1$$, regardless of my data.
This sort of makes my test silly in that this bit doesn't use my data, but it has the effect of trading of Type I and Type II errors. If we pick a certain number $$n$$ instead of 1000 (maybe $$n$$ would need to be non-integer in which case a slightly more involved scheme might be necessary) then we could gain power until we reached the threshold $$\alpha$$. I am pretty sure $$n$$ wouldn't actually be 100 for this, I think it would be slightly less than that because 4% of the time we're rejecting anyway so the further randomness doesn't increase the probability in that case (so perhaps it needs to be 96?). I invite the reader to check what $$n$$ should be as an exercise and see if I'm right (which I may well not be).
Case study: Testing the mean of a normal distribution
I'd like to demonstrate these concepts with a discussion of a typical hypothesis testing problem faced by A Level maths students, where implicitly we use the asymmetric criterion to choose our test. We suppose that we have data gathered from a normal distribution with unit variance and unknown mean $$\theta$$ - that is, that $$X_1,\dots,X_n\sim N(\theta,1)$$. The null hypothesis will be $$H_0:\theta=0$$ (meaning that $$P_0$$ is "Normal distribution with mean zero, variance 1") and the alternative hypothesis will be $$H_1:\theta > 0$$.
As an aside, the reader may notice at this point that defining an "alternative distribution" $$P_1$$ is a bit hard if the mean could be "anything positive." This doesn't affect the definition of size, but when defining Type II error, or power, we have to adjust for the fact that we now have a whole collection $$\mathcal{P}_1$$ of candidate distributions which are valid under $$H_1$$, being "Normal distributions with mean greater than zero." We can define the Type II error to be the worst-case error over this family, and we can define a power function which tells us the power as we vary $$P_1$$ in the family $$\mathcal{P}_1$$. This formulation means that the corresponding notions of Type II errors and power are not related quite as straightforwardly as in the scenario we discussed, and that our corresponding optimality criteria is then based on both the worst case Type II error and this power function. However, the intuition is broadly the same as in the previous section, and so the reader will not lose anything by not worrying about these details, which I will not make precise.
I am going to make my life a bit easier and just consider the case $$n=1$$ (and write $$X=X_1$$ to demonstrate the concepts, although in practice you might be rather hesitant to make claims based on one data point. I will note this, however - regardless of sample size, the approach by which we control first size then optimize for power is no more "risky" for the null hypothesis (which you might have thought) - it's just less powerful. In other words, all the gain when you have more samples is realised in the power, and so you are always making valid statements concerning Type I errors.
Ok, so we have our data point $$X\sim N(\theta,1)$$ and we want to devise a test. The way I introduced these tests might have been a bit mysterious, but what it often boils down to is cooking up some kind of region, which we will call $$C$$, and then depending on whether or not $$X$$ lies in $$C$$ we will set $$\psi=1$$ (if it is) or $$\psi=0$$ (if it isn't) (and go for $$H_1$$ or $$H_0$$ accordingly). Sometimes $$C$$ is called the critical region for the test.
Remember that we need to make sure we're not rejecting the null falsely more than a proportion $$\alpha$$ of the time. Thus, under the null (in this case $$N(0,1)$$) we can't put more than $$\alpha$$ total probability. So something like this wouldn't work if $$\alpha=5\%$$
So we're looking for regions where the associated red area accounts for less than 5% of the total area under the blue curve. So something like this might be better:
Unfortunately, this test, while having an appropriate size, lacks power. It's not as bad as the "always accept $$H_0$$" test, as the values lying in the region $$C$$ specified here can arise from (a distribution associated to) the alternative hypothesis $$H_1$$, but it's still not very good.
It turns out that the most powerful choice of $$\psi$$ (in a suitable sense) is the one where we stuff the 5% of the total area on the far end of the positive axis. I've pictured that one below. Intuitively, these values are the ones which most strongly point towards positive values of the mean of $$X$$, and since these are the ones our alternative concerns, this procedure has good power. So our problem is solved by taking $$\psi$$ to be one when $$X$$ is in the red region, and zero otherwise. What region do you think would provide the worst power for this testing problem?
However, this is by no means the only way to think of extreme or critical values, rather it is simply the "optimal" way in a certain sense, described (roughly) by the minimisation problem formulated in the previous section.
Typically, questions might give students a value for $$X$$ from the get go (or work out a value for $$\bar{X}_n$$, the sample mean, when they have multiple observations) and are asked whether their result is significant at a certain threshold. I remember being asked to compute something like the "probability of seeing something this extreme," and this always bothered me. Because even under a normal distribution with mean zero, seeing a data point $$X\in(-0.000001,0.000001)$$ could certainly be considered "extreme." Really, what these questions are asking is to determine whether their result would put them in the critical region of a test of optimal power with a given size. So if our data satisfies $$X >1.645$$ and we are "optimally" testing (at the 5% level), we would reject the null. Students also often have to decide whether a test should be "one-tailed" (like the one above) or "two-tailed", and I think this is really a question about what the optimal test is in a suitable sense - even if both of them would define tests with the correct size.
Is this how we should be doing hypothesis testing?
I never much liked hypothesis testing in this formulation. I always found the asymmetry a bit contrived, and I felt that I didn't really have much flexibility in how this asymmetry was chosen. Perhaps I want to allow a bit of trade-off between size and power but weight one hypothesis more strongly - must I construct an ad-hoc optimality criteria in every instance?
It turns out that there is a fairly different framework for testing, called Bayesian hypothesis testing, which is really just a consequence of the Bayesian framework of doing statistics as a whole, which is fascinating and somehow hasn't yet made it into a blog post yet. I will certainly devote a whole post to the Bayesian framework in the not too distant future, but the way of testing in this framework is, from my perspective, much more natural in a lot of instances. It allows you to weight precisely your hypotheses prior to seeing your data, which gives you a neat way of incorporating asymmetry if it is desired, and it also provides a probabilistic framework in which to do these tests. So if this whole post felt a bit off, maybe you'll prefer my post on that subject. | 2023-02-06 22:00:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8071590662002563, "perplexity": 233.21063357126746}, "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/1674764500365.52/warc/CC-MAIN-20230206212647-20230207002647-00505.warc.gz"} |
https://codereview.stackexchange.com/questions/184203/f-operation-framework-railway-oriented-programming-task-async-lazy | # F# Operation Framework (Railway-Oriented Programming + Task/Async/Lazy)
I recently published the initial version of what is effectively my first open-source software package, a Railway-Oriented Programming Framework for F# that encapsulates Tasks, Async Workflows, and Lazy Computations (as well as any code written inside an operation {} computation as "Operations" and provides a library for working with Operations and their Results.
I have published the source for the project on GitHub, and I am looking for feedback on the design of the framework, the implementation of the OperationBuilder (effectively using Tasks behind the scenes) and any possible use-cases I have missed that could also be encapsulated as Operations. The github project has some documentation and examples in the readme, but the basic way the framework is used is as follows:
let readFile (fileName: string) =
operation {
}
This returns an Operation<string,exn>. An Operation is a type representing a computation, which can be either Completed, InProcess, Deferred, or Cancelled.
type Operation<'result,'event> =
| Completed of OperationResult<'result,'event>
| InProcess of InProcessOperation<'result,'event>
| Deferred of EventingLazy<Operation<'result,'event>>
| Cancelled of 'event list
Operations that are not completed or cancelled can be completed by calling Operation.wait or Operation.waitAsync. Multiple Operations can be executed concurrently by calling Operation.Parallel. Operation Results are either Success or Failure, and the Failure event type is assumed to be System.Exception unless a domain event is explicitly used in the Operation computation or in another Operation called by the computation.
type OperationResult<'result,'event> =
| Success of SuccessfulResult<'result,'event>
| Failure of 'event list
Operations and their Results can be used in other Operation computations, or acted upon by a set of functions provided in the framework, such as Operation.ok to test if an Operation was successful or Result.either which maps an OperationResult with one function if it was successful or another function if it failed.
Domain events are propagated from one Operation to another, such as in the following example:
type FileAccessEvents =
| FileOpenedSuccessfully
| FileNotFound of string
| UnhandledException of exn // This is returned automatically if an unhandled exception is thrown by an Operation
let getFile (fileName: string) =
operation {
let file = FileInfo fileName
return! if not file.Exists
then Result.failure [FileNotFound file.FullName]
else Result.success file
}
let openFile fileName =
operation {
let! file = getFile fileName
return! file.OpenText() |> Result.successWithEvents <| [FileOpenedSuccessfully]
}
Calling readFile with a valid file path returns an Operation that, when completed, contains the text from the file and the events [FileOpenedSuccessfully; FileReadSuccessfully].
When returning to a user interface or a non-ROP caller, the Operation can be passed to Operation.returnOrFail to either return the successful result or raise an exception based on the failure. | 2021-06-21 21:43:25 | {"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": 0.27420517802238464, "perplexity": 4681.838781211572}, "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-20210621212241-20210622002241-00503.warc.gz"} |
http://icpc.njust.edu.cn/Problem/Hdu/4049/ | # Tourism Planning
Time Limit: 2000/1000 MS (Java/Others)
Memory Limit: 32768/32768 K (Java/Others)
## Description
Several friends are planning to take tourism during the next holiday. They have selected some places to visit. They have decided which place to start their tourism and in which order to visit these places. However, anyone can leave halfway during the tourism and will never back to the tourism again if he or she is not interested in the following places. And anyone can choose not to attend the tourism if he or she is not interested in any of the places.
Each place they visited will cost every person certain amount of money. And each person has a positive value for each place, representing his or her interest in this place. To make things more complicated, if two friends visited a place together, they will get a non negative bonus because they enjoyed each other’s companion. If more than two friends visited a place together, the total bonus will be the sum of each pair of friends’ bonuses.
Your task is to decide which people should take the tourism and when each of them should leave so that the sum of the interest plus the sum of the bonuses minus the total costs is the largest. If you can’t find a plan that have a result larger than 0, just tell them to STAY HOME.
## Input
There are several cases. Each case starts with a line containing two numbers N and M ( 1<=N<=10, 1<=M<=10). N is the number of friends and M is the number of places. The next line will contain M integers Pi (1<=i<=M) , 1<=Pi<=1000, representing how much it costs for one person to visit the ith place. Then N line follows, and each line contains M integers Vij (1<=i<=N, 1<=j<=M), 1<=Vij<=1000, representing how much the ith person is interested in the jth place. Then N line follows, and each line contains N integers Bij (1<=i<=N, 1<=j<=N), 0<=Bij<=1000, Bij=0 if i=j, Bij=Bji.
A case starting with 0 0 indicates the end of input and you needn’t give an output.
## Output
For each case, if you can arrange a plan lead to a positive result, output the result in one line, otherwise, output STAY HOME in one line.
## Sample Input
2 1
10
15
5
0 5
5 0
3 2
30 50
24 48
40 70
35 20
0 4 1
4 0 5
1 5 0
2 2
100 100
50 50
50 50
0 20
20 0
0 0
## Sample Output
5
41
STAY HOME
lcy
## Source
The 36th ACM/ICPC Asia Regional Beijing Sit | 2019-08-21 13:50:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25328657031059265, "perplexity": 1173.579088786229}, "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/1566027316021.66/warc/CC-MAIN-20190821131745-20190821153745-00000.warc.gz"} |
https://forum.allaboutcircuits.com/threads/identity-matrix.62172/ | # identity matrix
Discussion in 'Math' started by kokkie_d, Nov 18, 2011.
1. ### kokkie_d Thread Starter Active Member
Jan 12, 2009
72
0
Hi,
I have the following function:
$f(x) = e^{\underline{A}t}*\underline{A}^{-1}*\underline{A}*e^{\underline{A}T-t}$
Where $\underline{A}$ is A matrix.
Am I allowed to simplify $\underline{A}^{-1}*\underline{A}$ to an identity matrix?
I know a matrix times its inverse is an identity matrix but I am worried about the order to do the calculations in and if it then still is allowed.
Cheers
2. ### Georacer Moderator
Nov 25, 2009
5,151
1,268
If your function is exactly as you wrote it, then you can.
The inverse matrix of $A$, $A'$ is defined as:
$A \cdot A'=A' \cdot A=I$
kokkie_d likes this. | 2018-05-25 16:57:52 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 6, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8887166976928711, "perplexity": 1587.6350030244962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867140.87/warc/CC-MAIN-20180525160652-20180525180652-00300.warc.gz"} |
https://mathematica.stackexchange.com/questions/230013/pattern-matching-complex-numbers-as-a-bi | # Pattern-matching complex numbers as a + bi
I just started working with Mathematica and am toying with pattern matching. There may be something obvious I'm missing in this, but I can't figure it out by myself.
I want to write down a function that takes a complex number as arguments. So f[1 + 2 I] should be a valid input, as well as f[a + b I]. I want, however, to make my function parse this as two numbers of the form a + bi, getting a and b by pattern matching. I made several attempts similar to this:
f[a_ + b_ I] := NSolve[a^2 + b^2 == 1/2 (1 + z), z]
SetAttributes[f, HoldAll]
(I guess the NSolve doesn't matter in this case, but let it there in case it's part of the issue.)
This doesn't work as I planned. Any attempt to call it, like f[1 + 2 I], just echoes itself, but it does work fine when I call it with symbolic arguments, such as f[a + b I].
I guessed this should be due to some difference in the internal representation of symbolic expressions and complex numbers. Indeed, whenever I try to MatchQ[m + n I, a_ + b_ I], it says it's True. But when I try the sorts of MatchQ[Unevaluated[2 + 3 I], a_ + b_ I], it's False.
In trying to figure it out, I asked
FullForm[a + b I]
FullForm[Unevaluated[2 + 3 I]]
FullForm[a_ + b_ I]
and got
Plus[a,Times[Complex[0,1],b]]
Unevaluated[Plus[2,Times[3,\[ImaginaryI]]]]
Plus[Pattern[a,Blank[]],Times[Complex[0,1],Pattern[b,Blank[]]]]
My questions are:
1. Shouldn't the Plus[2,Times[3,\[ImaginaryI]]] match with Plus[Pattern[a,Blank[]],Times[Complex[0,1],Pattern[b,Blank[]]]]?
2. What is the difference between \[ImaginaryI] and Complex[0,1]? I know the first is a symbol as much as \[Alpha] is, and I guess me asking for Unevaluated is preventing it from being cast as a Complex[0,1]. Probably this would be needed for the matching, but I don't know a workaround.
3. Is there a better way to do what I'm attempting with my function?
Thanks!
• Why not simply use Re and Im to get the two parts? – Lukas Lang Sep 12 '20 at 16:32
• Yes, this is a good solution, of course. I should've put more contex to to my question, though: I'm actually doing this as an exercise to learn pattern matching, as I'm completely new to it. I'll edit this in, thanks! – ppln Sep 12 '20 at 16:54
f[a_. + b_ I | Complex[a_, b_]] := {a, b}
You need to catch both the unevaluated form for symbolic work and the evaluated Complex representation for numerical work. Note that the first a_. allows the real part to be omitted so that purely imaginary parts can be matched as well.
• Great, thanks! This works for all instances with an imaginary part, and it sets the real part to zero when I omit it. It doesn't, however, accept a pure real number as the argument. I then extended it as f[a_. + b_ I | Complex[a_, b_] | a_ + b_.] := {a, b}. I didn't simply use a_ as the last pattern because I wanted the imaginary part to be set to zero, and I also trusted that the matching will be tried from left to right and be short-circuited (so, e.g., f[2 + 3 I] matches with a_. + b_ I, and not with a_ + b_.). It worked for some test cases. Does this look OK? – ppln Sep 12 '20 at 17:21
• I don't think you can trust it to short-circuit left to right, but if it works for what you're doing then it's likely fine. If you are expecting arbitrarily mixed real and imaginary input, you are likely better off using Re and Im. – eyorble Sep 12 '20 at 17:41
• I hadn't noticed about the input ordering, thank you! So I'd have to mimick all those patterns changing a for b. And yes, Re and Im are much more straightforward for real use cases. This was just an exercise as I'm trying to get my head around pattern matching and substitution rules. – ppln Sep 19 '20 at 22:30 | 2021-03-06 08:08:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5690670609474182, "perplexity": 883.0590371136318}, "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/1614178374616.70/warc/CC-MAIN-20210306070129-20210306100129-00634.warc.gz"} |
http://dataprocessing.aixcape.org/Algorithms/Mode/index.html | # Mode¶
Mode algorithm corresponds to the most frequently occuring value of the data set.
Input Parameters
Parameter Type Constraint Description Remarks
$$Y$$ $$Y \in \mathbb R^N$$ $$N \in \mathbb{N}$$ Input data vector of length $$N$$ None
Output Parameters
Parameter Type Constraint Description Remarks
$$m$$ $$m \in \mathbb R$$ None Mode value of $$Y$$ None
Single Steps using the Algorithm | 2017-11-18 10:31:16 | {"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": 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": 0.6508345007896423, "perplexity": 6081.865475121812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934804724.3/warc/CC-MAIN-20171118094746-20171118114746-00254.warc.gz"} |
https://documen.tv/question/lim-4-4-1-2-2-question-related-to-limits-it-s-lim-arrow-to-4-try-to-answer-this-hardcor-24096437-54/ | ## lim x->4 [( x-4 / x^(1/2)-2 ) ] Question related to limits . It’s lim x arrow to 4 . Try to answer this hardcor
Question
lim x->4 [( x-4 / x^(1/2)-2 ) ]
Question related to limits .
It’s lim x arrow to 4 .
Try to answer this hardcore question.
in progress 0
6 months 2021-08-30T10:22:21+00:00 1 Answers 3 views 0 | 2022-09-29 05:38:01 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8028536438941956, "perplexity": 14987.873692430832}, "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-2022-40/segments/1664030335304.71/warc/CC-MAIN-20220929034214-20220929064214-00650.warc.gz"} |
http://edoc.mpg.de/display.epl?mode=doc&id=715996&col=61&grp=3972 | Institute: MPI für Physik Collection: YB 2016 Display Documents
ID: 715996.0, MPI für Physik / YB 2016
Measurements of Branching Fractions of $\\tau$ Lepton Decays with one or more $K^{0}_{S}$
Authors:
Date of Publication (YYYY-MM-DD):2014
Title of Journal:Physical Review D
Journal Abbrev.:Phys.Rev.D
Issue / Number:89
Start Page:072009
Audience:Not Specified
Intended Educational Use:No
Abstract / Description:We report measurements of branching fractions of $\\tau$ lepton decays to final states with a $K^{0}_{S}$ meson using a 669 fb$^{-1}$ data sample accumulated with the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. The inclusive branching fraction is measured to be $\\mathcal{B}(\\tau^{-} \\to K^{0}_{S}\\ X^{-} \\nu_{\\tau})=(9.15 \\pm 0.01 \\pm 0.15) \\times 10^{-3}$, where $X^{-}$ can be anything; the exclusive branching fractions are $\\mathcal{B}(\\tau^{-} \\to \\pi^{-} K^{0}_{S} \\nu_{\\tau}) = (4.16 \\pm 0.01 \\pm 0.08) \\times 10^{-3}$, $\\mathcal{B}(\\tau^{-} \\to K^{-} K^{0}_{S} \\nu_{\\tau}) = (7.40 \\pm 0.07 \\pm 0.27) \\times 10^{-4}$, $\\mathcal{B}(\\tau^{-} \\to \\pi^{-} K^{0}_{S} \\pi^{0} \\nu_{\\tau}) = (1.93 \\pm 0.02 \\pm 0.07) \\times 10^{-3}$, $\\mathcal{B}(\\tau^{-} \\to K^{-} K^{0}_{S} \\pi^{0} \\nu_{\\tau}) = (7.48 \\pm 0.10 \\pm 0.37)\\times 10^{-4}$, $\\mathcal{B}(\\tau^{-} \\to \\pi^{-} K^{0}_{S} K^{0}_{S} \\nu_{\\tau}) = (2.33 \\pm 0.03 \\pm 0.09) \\times 10^{-4}$, $\\mathcal{B}(\\tau^{-} \\to \\pi^{-} K^{0}_{S} K^{0}_{S} \\pi^{0} \\nu_{\\tau}) = (2.00 \\pm 0.22 \\pm 0.20) \\times 10^{-5}$, where the first uncertainty is statistical and the second is systematic. For each mode, the accuracy is improved over that of pre-$B$-factory measurements by a factor ranging from five to ten. In $\\tau^{-} \\to \\pi^{-} K^0_S K^0_S \\pi^{0} \\nu_{\\tau}$ decays, clear signals for the intermediate states $\\tau^{-} \\to \\pi^- f_{1}(1285)\\nu_{\\tau}$ and $\\tau^{-} \\to K^{*-}K^{0}_{S} \\pi^{0} \\nu_{\\tau}$ are observed.
Classification / Thesaurus:Belle II
Comment of the Author/Creator:17 pages, 18 figures.
External Publication Status:published
Document Type:Article
Communicated by:MPI für Physik
Affiliations:
Identifiers:
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The scope and number of records on eDoc is subject to the collection policies defined by each institute - see "info" button in the collection browse view. | 2020-08-14 12:28:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6519164443016052, "perplexity": 1434.5752095403734}, "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-2020-34/segments/1596439739211.34/warc/CC-MAIN-20200814100602-20200814130602-00218.warc.gz"} |
https://jordanleagle.com/category/new/ | YOUTUBE: A NEW BRANCH OF OUTREACH
Hi everyone!
I feel obligated to apologize for neglecting my personal blog while working on very new virtual outreach content. Hopefully you all understand the struggle! Between juggling research, preparing for my PhD defense (and thesis writing 🥵), post-doctoral grant applications, my personal blog, local volunteer work, etc, etc, and NOW this YouTube channel (‼️), my plate is pretty full 🥴.
The (On) Planet Nine YouTube channel‘s first ever season has commenced! So, I am accordingly putting all of my energy into these videos and sharing them. The first season is anticipated to end in February, and when that happens, I will re-arrange priorities to continue writing blog posts about ongoing research. I know there are a couple loose ends to be tied, and I also haven’t shared much regarding my current PhD research, so I’d love to do that more, too!
In the meantime, it would mean so much if you could watch/view/share our newest YouTube videos among your networks! Of course, I hope you enjoy the content and can think of one or more people who may also enjoy it 😄. Science is meant to be shared and rejoiced!
I do a little shameless plug of all of our links and latest available YouTube videos below! Lea’s episode two just came out TODAY and it features a little live music in the introduction ☺️.
Social media handle: @onplanetnine
NEW! Episode 2
Episode 1
Episode 1
Did you know we also feature guest interviews with other astronomers??
Next up is the incredible Dr. Alex Filippenko crashing On Planet Nine and telling us about how he (and his team ofc) ended up discovering the acceleration of the expansion of The Universe! The full video releases November 27 (NEXT SATURDAY!) so watch out
Get yourself hype about Alex. He is seriously an amazing presenter and I guarantee he will make you giddy about science!! 🤪 Below is a Tedx Talk he did back in 2013. He has also been featured on The Universe (more than a dozen times, actually).
(A) SURPRISE!
I have been waiting way too long to share this exciting project with you all! About one year ago, a dear colleague and friend who was in the same research group as me at Clemson was preparing to graduate and was asking the group for help with her proposal for job applications. So, I read her proposal, and at the end I read that she has plans to start a YOUTUBE CHANNEL for educational outreach. If you know me then you know I messaged her with my comments on her proposal and said I WOULD LOVE TO HELP WITH THE YOUTUBE CHANNEL IDEA.
A few months later we meet up, and it’s FIVE (5!!) of us that show up to a meeting, all with the same Clemson advisor (Marco must be proud 🥲). We had all read her proposal and said LET’S GO!🗣 📣 🤣
Over the next year, we prepared what this channel will be about, and man, it is no joke making videos and content of all kinds — it is a TON of work 🥵. Mainly because we all also have our day jobs, so juggling all of this has been an exciting challenge.
Without further ado, please go check out the channel we are building! We aim to share our passion for science with anyone and everyone because, as Nuria points out in the trailer below, it’s contagious! 😉
We plan to officially roll out our first season starting November 6 so be sure to follow/subscribe to your preferred platform (or all of them 😁 ) to make sure you get the latest news in the cosmos! 🌠
We aim to be active as an outreach group on our Youtube channel, webpage, and social media platforms, and I provide them all below! Go follow/subscribe/interact with us to help us grow ☺️
GREATER BOSTON LIVING
Noah and I have been living in the greater Boston area for a little over 1 ½ years now! This September first has marked our third time of moving around in that same time frame 😅. Don’t even ask how many other times we’ve moved around before even coming up here (it’s a ton, we have it just about down to a science now…). We initially moved into a super cute Cambridge neighborhood and lived there with all three of our pets for about 10 months before we crossed the river (a whole two miles) into the Brighton village of Boston city. We just spent the last year being walking distance to the river and that has been relaxing, to say the least.
Images below are just a few from our first neighborhood we lived in in Cambridge. It is such a cute area!!😍
Cambridge
Moving up North in the dead of winter from Clemson, South Carolina was a pretty big change to say the least. Add into the mix a pandemic and you have two very confused Southerners. We tried our best to take advantage of the city, but hopefully you will understand our slow progress 🥲.
I’ve experienced a lot of weird stuff that was new for me — Northern accents, Northern attitudes, snow! Lots of snow, including shoveling my car out and learning why it’s best to not just wait for the snow to melt… Then, there’s triple the cost in everything, parallel parking, the list goes on.
I have to say – the jury is still out on whether I see myself ever choosing to remain in the city. I still really love the idea of rolling hills for miles, no neighbors, no hassle, and reasonably priced everything. Houses, produce, and even shoes, all become more affordable the farther South you go, generally, haha.
Though there are a lot of great things about the city, too! For one, there really is always something to do. I would definitely have moments where I felt sort of stuck in Clemson whenever I was antsy to do something fun or adventurous. I felt like my only option was to go hiking (and I didn’t even go that often!). Whenever I feel bored at home in Boston, I hop on the next bus into town! Well, it hasn’t always been that easy, with the pandemic and everything, but during that brief hopeful moment this summer as vaccination rates and COVID-19 positivity rates switched spots, it started to feel something like a city life 😌
I share what things we have done so far that we really enjoyed below!
If you click the images you should see captions 🙂
Boston: the Freedom Trail
• The Freedom Trail
• Highly recommend for spring to summer months. Probably fine to do year round tbh, just stay warm ☃️
• Begins at the Welcome Center of the Public City Garden in Boston Common.
• It is ~2-3 miles long, and it takes you through all different parts of the city
• You follow a visible red brick line to every landmark! Don’t worry they are easy to pick out ☺️
Boston: BikeShare
• BikeRide with BlueBike Share!
• A 30 minute bike ride costs $3.00 and there’s dozens of drop off points all over the city. • You can pretty much follow the river to get to anywhere in the city by bike! Boston/Cambridge: Kayak the Charles River • Kayaking or canoeing along the Charles River • You can do a 2-3 hour kayaking trip for as cheap as$25 per person at any one of the popular Charles River Kayak Kiosks along the river banks!
• Canoeing is even better priced and all of the kiosks offer a range of boating equipment. You can choose from kayaks, canoes, paddleboards, and more (maybe?). The kiosks are located at several river access points so you can choose to paddle through the quieter suburbs toward Newton or dive right into the opening into the Atlantic Ocean near downtown.
That’s basically it for affordable things to do in Boston 😂 but I list below a few other places that you might find enjoyable to explore, too:
Cambridge: Fresh Pond
• Fresh Pond Reservation
• Lots of trails, parks, and nature in the middle of Cambridge.
• Harvard Football Stadium and the Recreational Area there
• Lots of open fields, tracks, and free recreational stuff. It’s really cool actually it features an outdoor skate park and even an ice skating rink in the winter.
Boston/Cambridge: the Charles River
• Any recreational spots along the river
• There is always some cool recreational spots along the river. Where we live now, there are two huge kid parks, two of them are water parks! I’m pretty sure it’s free, too. Just this past weekend I saw the water parks looking like Water Country, USA, with lifeguards and everything, just like in my backyard, it felt like!
• You just never know what you will find in the city when you explore. At the river near my house, there’s a regular drum circle that plays in the courtyard every weekend. There’s even Herter park, a full on amphitheater that hosts free shows weekly during the summer (yes, even right now!!).
• One time I even stumbled upon an artist painting the river early one morning on my run 🥺 it was such a whimsical moment running up among a Bob Ross Bob Rossing it right there in the open. Though I feel bad for the painter hearing me stomping, grunting, panting as I run by. Sorry, dude!
Cambridge: Harvard College Observatory
Oh, yeah! My work! The Harvard & Smithsonian Center for Astrophysics! Why, you might ask?! Well because of the super historical Harvard College Observatory, of course. How can you forget!
Boston: SoWa Open Market
And most recently we went to the SoWa open market in South End for the first time and it was lovely! There’s other more tourist-y things that we did (or tried to do) this summer that we’ve been wanting to do and I’d definitely recommend including the aquarium, whale watching (weather actually cancelled our whale watch event 😔), festivals like SoWa open market which are happening all times of the year all over the area, and visiting the many beautiful and immaculate college campuses that decorate the city.
Images are all my own and from April — August 2021.
OUR DEEPEST EXPLORATION OF THE UNIVERSE IN X-RAYS
eROSITA is an X-ray space telescope that was launched on July 13, 2019 by an international collaboration, mainly funded by Germany and Russia. The space telescope took its first ever X-ray image three months after orbiting the Earth the following October and has already released some of the first data collected in the first months of operation as well as a schedule confirming the official first data release by December 2022. Most recently, the Astronomy and Astrophysics peer-reviewed science journal has released a special issue including ~35 publications that analyze new eROSITA data. Given the exciting first light and the already big discoveries the telescope has made including the largest supernova remnant ever discovered in X-rays, I thought it would be appropriate to highlight a little bit more about the telescope on my blog! 😄
The eROSITA telescope flies aboard a large satellite: the Spektrum-Röntgen-Gamma (SRG) space satellite. Along with the primary instrument, eROSITA (extended ROentgen Survey with an Imaging Telescope Array), on the SRG is the Russian ART-XC instrument which can probe higher energy X-rays than eROSITA.
As you have probably guessed, this is an X-ray imaging space telescope. It turns out that the Earth’s atmosphere actually absorbs incoming X-rays (see image below).
This is precisely why all astrophysical X-ray instruments are deployed in space including eROSITA.
eROSITA is made up of seven identical and strategically aligned X-ray Mirror Assemblies (MAs) that are situated on an optical bench. Underneath this is the rest of the supporting structure (see the schematic view below), which includes connecting the MAs to the camera assemblies (CAs), i.e. the mirrors will deflect incoming X-rays from its surface in very tiny incident angles that then focus the incoming X-rays onto the cameras (called the grazing incidence angle and is a common practice for designing sensitive X-ray instruments).
The X-ray “baffles” are used to prevent X-ray photons that are outside of the field of view from contaminating the image being taken at that time. This is particularly important when you need to observe an object that may have bright X-ray sources nearby that can contaminate the X-ray measurements.
The telescope (not SRG, the observatory it is deployed on right now) itself is 1.9 meters wide and 3.2 meters high. For my American readers that is about 6 by 10 feet! 😀 The completed instrument weighs in at a whopping 808 kg or 1781 pounds!
The Field Of View (FOV) of the full instrument (including all seven cameras) is about 1 degree in diameter. To give you an idea of what portion of the sky eROSITA can see at any given time, the full moon is about 1/2 a degree in the night sky, so eROSITA is able to see an area in the sky that is 2 times larger than the full moon.
This FOV is considerably larger than both of the previously most sensitive X-ray space telescopes, Chandra and XMM-Newton. Further, eROSITA will operate optimally for a specific energy range of X-ray photons. You will almost always see X-ray astronomy use kiloelectron volts to describe the X-ray energies,
$1 \text{keV} = 1,000 \text{eV} = 1.6 \times 10^{-12} \text{Joules}$
eROSITA, along with Chandra, XMM-Newton, and several other currently operating (and retired) X-ray instruments, can detect X-ray photons between 0.2 keV and 10 keV (see image below, but don’t freak out 😉)
The above plot is showing the field of view averaged effective area in cm squared as a function of energy. You can think of this as the sensitivity of the instrument as a function of energy. Each line corresponds to a different instrument: eROSITA’s seven modules in solid red, Chandra’s ACIS-I setup in green dot-dashed, another Chandra instrument called HRC-I in purple dashed, XMM-Newton’s 3 cameras with the thin filter on, and the previously retired ROSAT PSCPC instrument.
You can see that eROSITA is just about the most sensitive instrument from energies ~0.5keV to ~2keV which is often referred to as the soft X-ray range which just indicates the lower energy range of the X-ray band. Above 2 keV, the sensitivity of eROSITA drops off at a similar rate as the Chandra instruments, while XMM-Newton wins the sensitivity competition at energies greater than about 2keV. With its large field of view in comparison to Chandra and XMM-Newton, eROSITA will make (and has already demonstrated) significant discoveries to X-ray astronomy.
What separates eROSITA from other current missions like Chandra, in addition to its large field of view and sensitivity, is its angular and energy resolution and most of all — the way it will take data. Chandra and XMM-Newton X-ray telescopes are pointing missions. This means the telescope has to position itself for specific observations in varying parts of the sky. The time gets “shared” among thousands of researchers who request for telescope observations every year. eROSITA, on the other hand, is an all-sky survey.
It is the first ever X-ray instrument to survey the entire sky from 0.2-10keV in astronomy HISTORY!
ROSAT was also an all-sky survey, but it only imaged soft X-ray photons, so it didn’t detect X-ray photons with energy more than 2.4keV. ROSAT also had a similar field of view of 2 degrees, but by inspecting the above effective area (i.e. sensitivity) plot, we can see that eROSITA will be a much deeper sky survey, by about 4 times!
To visualize this difference, here is a ROSAT view of the Vela supernova remnant (if you are familiar with my work you have seen the ROSAT image before) in the left panel below compared to the Vela SNR image from eROSITA on the right. I’m unable to find more details about the eROSITA image, but I’m guessing that the colors indicate three energy bands: red is likely the softest of X-rays < 0.6 keV, green is probably “medium” X-rays from 0.6 – 1-ish keV, and blue is likely 1-2.3 keV energies. If this assumption is correct, most of the Vela SNR is dominated by soft and medium X-rays (which is indeed the case, see the ROSAT image on the left lol!). We can also see the smaller overlapping supernova remnant Puppis A is bright in this X-ray range in both images, but that “hard” (higher-energy, see eROSITA image) X-rays dominate the observed emission. Additionally, one can easily spot the Vela central pulsar (lots of hard X-rays there in blue, too!) in the near-center of the eROSITA image, and a third supernova remnant in the lower left corner, visible by only a faint circular blue hue. Neither the central pulsar nor the lower-left supernova remnant is resolved in the ROSAT image. Note: do I see the third supernova remnant’s central compact object in the eROSITA image?!
First light for theeROSITA telescope occurred in mid-October just months after launch
Moreover, eROSITA has already detected 10 times more sources than ROSAT which is about as many as have been discovered by all previous X-ray missions combined. Less than a year after launch, eROSITA has already completed its first all-sky survey, one of eight anticipated full sky surveys.
eROSITA’s first all-sky survey will be released in 2022 (well, the half that the Germans own), reporting already thousands of new sources, most being active galactic nuclei. One of the exciting discoveries includes the largest supernova remnant discovered in X-rays to date which has been nicknamed “Hoinga”. There are a lot of special surprises associated with Hoinga, including its high location with respect to the Galactic plane, an unusual location for supernova remnants to be found.
Hoinga is estimated to have a diameter of about 4.4 degrees. Vela SNR has a diameter of 8 degrees, but it was discovered first in radio, not X-ray.
To conclude, here is a super cool visual graphic about the SRG observatory where eROSITA operates.
I will definitely be on the lookout 👀 for the first data release, although that means I will have to learn (yet another) new software to clean and analyze the data….. 🥵 😅
A random side note
What I think is extra intriguing about this telescope is the collaboration between Germany and Russia (Just hear me out lol). The terms of the collaboration seem a little unusual. They have defined a German half of the X-ray sky as well as a Russia half of the X-ray sky. Essentially the Western hemisphere of the Galaxy (in Galactic coordinates) is owned by the Germans with unique scientific data exploitation rights and the Eastern hemisphere belongs to the Russians. They have decided to equally share the all-sky surveys, so I suppose the data that has been divided will include individual mission projects i.e. pointed observations for a particular object will have certain proprietary rights depending on its location in the sky. With that being said, only the German half of the sky has been scheduled a public release of data for 2022, and all of the Russian X-ray data and its release schedule is to be determined.
It will be very interesting to see how the data-sharing pans out with this particular method. To be fair, I’m not totally sure if this is a standard practice in international space efforts such as this, but I would be surprised if it is. | 2021-12-01 10:20:43 | {"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": 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": 0.21516191959381104, "perplexity": 2309.7551825019923}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00144.warc.gz"} |
https://www.udacity.com/wiki/ma008/glossary/lesson15 | # Glossary for Lesson 15: Quadratic Formula
### Plus or Minus (\pm)
The plus or minus sign \pm is used to indicate that the value to its can be either added or subtracted to the value to its left. For example, 8\pm 3 is a way of writing 8+3 or 8-3 using just one expression. To write this symbol with MathQuill, type \pm and then a space. | 2016-09-30 06:39:57 | {"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": 0.952058732509613, "perplexity": 1145.0625337304982}, "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-2016-40/segments/1474738662058.98/warc/CC-MAIN-20160924173742-00269-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an%3A0989.35073 | Slowly modulated two-pulse solutions in the Gray–Scott model. II: Geometric theory, bifurcations, and splitting dynamics.(English)Zbl 0989.35073
The presented paper continues the study of A. Doelman, W. Eckhaus and T. J. Kaper regarding the slowly modulated two pulse solutions in the one-dimensional Gray-Scott model.
The introduction is followed by Section 2, “Geometry of governing equations”. Divided in three sub-sections, “Dynamics on the slow manifold”, “The fast subsystem when $$\varepsilon=0$$” and “Persistent fast connections”, Section 2 presents the fundamental geometric properties of the studied system. The third section, made by “The right-moving pulse with slowly changing $$c(t)$$: Hooking up the slow and fast segments”, “The ODE for $$c(t)$$ in case Ia”, “Bounded domains and $$N$$-pulse solutions $$(N \neq 2)$$” and “The validity of the quasi-stationary approach” contains the construction of the basic slowly modulated two-pulse solution (case Ia), the consideration of the bounded interval and asymmetric cases and the establishment of the validity of the quasi-stationary approach. Section 4, “Geometric constructions of two-pulse solutions: cases Ib and IIa”, treats the cases Ib in “case Ib: $$\varepsilon\Delta/ \delta=O(1)$$, the bifurcation of traveling waves” and IIa in “case IIa: $$\delta/ \varepsilon=0(1)$$, a saddle-node bifurcation of two-pulse solutions”, respectively. The authors discuss the implications of the analytical results for the understanding of the self-replicating process in the section “The self-replication process” and they finish their study by relating it to the existing literature on self-replication.
MSC:
35K57 Reaction-diffusion equations 35B25 Singular perturbations in context of PDEs 35B32 Bifurcations in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 92E20 Classical flows, reactions, etc. in chemistry
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http://fermionlattice.wikidot.com/dipole-trap | Dipole Trap
# Introduction
After magnetic transport we wish to transfer into a crossed optical dipole trap ('crossed ODT' or 'CDT') for further cooling before loading into the optical lattice. For more information about the theory of optical dipole traps see: Optical dipole traps for neutral atoms
The dipole trapping beam is generated by a fiber amplifier and a narrow linewidth source. The source is The Rock from NP Photonics (datasheet), and the fiber amplifier is a 40W 1.0 μm Single Frequency PM Fiber Amplifier from Nufern (datasheet).
After the amplifier, we use two AOMs to diffract out two power controlled beams in order to form a crossed dipole trap. These diffracted beams are then brought to the upper level of the experiment via two periscopes, shaped with telescopes and prisms, and then focused into the science chamber.
# The Rock
Turn On
The control box for The Rock has three switches for relative intensity noise suppression (RIN) On/Off, Laser On/Off, and constant current (ACC) versus constant power (ACP). There are two LEDs to indicate the presence of a 2nd cavity mode (red) and temperature stability (green). The sequence for turning on The Rock is:
1. - Laser is off from the previous day, switches should be set to: RIN Off, Laser Off, ACC
2. - Switch Laser to On and wait for the grating temperatures to stabilize. After ~20 minutes the green LED will light, and hopefully the red LED will be off
3. - Switch to RIN On to stabilize power (used to also switch to APC, but this was seen to add noise - now leave on ACC)
The output of The Rock is ~150mW. More details can be found here: The Rock
# Fiber Amp
Turn On
The key on the front of the fiber amplifier has 3 stages as pictured:
The amplifier is controlled via computer using USB. The recommended start procedure is:
1. - Key in position 0
2. - Couple an input signal of 50-200mW which is polarized along the slow axis of a PM fiber (some of this light will be transmitted through the amplifier and out through the output isolator, even when the power is off)
3. - Turn on water cooling at T=23+/-3 degrees Celsius
4. - Open NuFern_SFA_Rack_v1.0
5. - Turn key to position 1 and pull out emergency stop button (if needed)
6. - Turn key to position 2, GUI changes from 'Not Ready' to 'Ready'
7. - Press 'Enable' on GUI, preamplifier is now on and ~800mW of power should be observed from the output isolator
8. - Increase amplifier power from 0-70% using GUI
9. - Nufern output power is now ~45W
# AOMs
The AOMs used for the two crossed ODT beams are the Gooch and Housego I-FS080-2S2G-3-LV1. The two AOMs are labelled Dipole 1 and Dipole 2, and are driven by two Crystal Technologies 80MHz drivers (fixed frequency) controlled by two ALPS boxes for power control. The Dipole 1 and Dipole 2 AOMs diffract into the +1st and -1st order, respectively, giving frequency shifts of +80MHz and -80MHz. The layout of these AOMs is shown below:
(old picture) The dipole beams are now fiber coupled. The layout is the same as above, but instead of periscopes, each beam goes though a 2X expanding telescope and then a fiber coupler.
On the upper level, we use a PBS and pickoff to tidy the polarization and monitor the beam power. The beam then passes through a 2:1 reducing telescope (using fused silica lenses for high power performance), as well as a prism pair to make the beams elliptical. We would like the focus of the beams at the atoms to be roughly 60$\mu m$ in the vertical direction and 170$\mu m$ in the horizontal direction. The positions of the telescopes and prisms were decided by using Gaussian Beam Propagation, trying to maximally overlap the foci of both axes of the elliptical beams. Details of the layout can be found in the following .ppt file: Elliptical Beams | 2017-06-23 01:45:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.45855090022087097, "perplexity": 3171.6605283593644}, "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-2017-26/segments/1498128319943.55/warc/CC-MAIN-20170623012730-20170623032730-00709.warc.gz"} |
https://solvedlib.com/n/8-zo-3-2-1-acchk-quot-assure-j-20-x-con-4c-01-t-1-f-1-vs,16185984 | 5 answers
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##### An Ac series circuit built capac_tor using (C-12 pF), resistor along 300 Q, an inductor (L-800 mH); and with Vend operates at power source that supplies maximum voltage of AV max-220 f-60 Hz. Find the angular frequency in radians current, sec find the impedance, TS phase angle between the voltage and reaches current, and average power for the circuit Which is maximum first in the circuit; the current or the voltage? You must box each answer for credit: (30 pts )
An Ac series circuit built capac_tor using (C-12 pF), resistor along 300 Q, an inductor (L-800 mH); and with Vend operates at power source that supplies maximum voltage of AV max-220 f-60 Hz. Find the angular frequency in radians current, sec find the impedance, TS phase angle between the voltage an...
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https://math.stackexchange.com/questions/608764/how-many-ways-can-targets-be-broken-if-only-the-target-on-a-bottom-of-a-row-can/608771 | # how many ways can targets be broken if only the target on a bottom of a row can be broken?
Twelve clay targets (identical in shape) are arranged in four hanging columns. There are four red targets in the first column, three white ones in the second column, two green targets in the third column and three blue ones in the fourth column. To join her college drill team, Deborah must break all 12 of these targets (using her pistol and only 12 bullets) and in so doing must always break existing target at the bottom of a column. Under these conditions, in how many different orders can Deborah shoot down (and break) the 12 targets?
The answer key gives $\frac{12!}{4!3!2!3!}$ but I don’t see why? She definitely has 4 options to choose from the first shot, 4 options to choose from the second shot, but then she could have broken the two in the third column and I’m not sure where to go from here. I don’t see how this question is a matter of counting permutations and removing repetitions?
There are $12!$ different sequences of targets. The restriction that she can only shoot the bottom target means that she has only one option of each color. Thus the number of ways she can shoot the targets is the number of possible sequences of $4$ red, $3$ white, $2$ green and $3$ blue targets, where targets of the same color are indistinguishable. Thus the number of sequences of targets she could shoot is $$\frac{\text{# of sequences of targets}}{\text{# of sequences of red}\times\cdots\times \text{# of sequences of blue}}=\frac{12!}{4!\times 3!\times 2!\times 4!}$$
The sequence used by Deborah is completely described by a word of length $12$ obtained by arranging the letters in the word RRRRWWWGGBBB. | 2019-05-23 02:42:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5332489013671875, "perplexity": 318.37536298994684}, "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-2019-22/segments/1558232257002.33/warc/CC-MAIN-20190523023545-20190523045545-00301.warc.gz"} |
https://www.encyclopediaofmath.org/index.php/Neyman%E2%80%93Pearson_lemma | # Neyman-Pearson lemma
(Redirected from Neyman–Pearson lemma)
A lemma asserting that in the problem of statistically testing a simple hypothesis $H_0$ against a simple alternative $H_1$ the likelihood-ratio test is a most-powerful test among all statistical tests having one and the same given significance level. It was proved by J. Neyman and E.S. Pearson [1]. It is often called the fundamental lemma of mathematical statistics. See also Statistical hypotheses, verification of. | 2018-11-20 17:58:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874059557914734, "perplexity": 1155.7961635807214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039746528.84/warc/CC-MAIN-20181120171153-20181120193153-00495.warc.gz"} |
https://chemistry.stackexchange.com/questions/35174/synthesizing-2-methyl-2-butanol | # Synthesizing 2-methyl-2-butanol
I have recently been trying to come up with a method to synthesize 2-methyl-2-butanol or tert-amyl alcohol. I wish to avoid complex/expensive steps since as a home chemist/student, I don't have access to much money or equipment.
What I have done so far: I noticed that methyl ethyl ketone has pretty much the correct structure, minus a methyl group.
I then realized this meant I could use a methylating agent to get the correct molecule. What I needed was to break the $\ce{C=O}$ bond from the carbonyl group and add a methyl group.
Long story short, I found that either Grignard reagents or something like methyl lithium would do it since it would donate an electron pair to the carbon, and form a bond with the methyl. I would then have the 2-methyl-2-butanol as a conjugate base with $\ce{Li}$ ions present. I could then react them out with $\ce{HCl}$ to precipitate out the $\ce{Li}$ salt and add a hydrogen to the negatively charged oxygen.
My problem: I don't know if this would work at all. I didn't find any documentation. Also, methyl lithium is very expensive and difficult to make so I can't use it.
Does anyone have any other route suggestions? Would it be possible to make methyl sodium and if so, would it work the same? Is it possible to make Grignard reagents easily at home?
I'm a hobby chemist, so please be detailed with how you predicted the organic reaction.
Also, I don't necessarily care about what chemicals I use for the reaction, so if anyone can suggest a different starting chemical, I'd be glad to hear it. I was even thinking of using an alkene group of some kind and methylating the $\ce{C=C}$ bond, but I don't know for sure, as I'm pretty new to organic chem.
• Yes it would work excellently in theory with either Grignard reagents or alkyllithiums. I can't say about in practice although both reagents can be dangerous to handle so this is something to think about. However, if you're only interested in the product then buying it is the easiest option. – bon Aug 18 '15 at 15:37
• Grignards are not the kind of thing that should be worked with without a hood. – jerepierre Aug 18 '15 at 16:31
• Acid-catalyzed hydration of 2-methyl-2-butene should also work. – Jannis Andreska Mar 26 '16 at 18:57
Isopropyl lithium and ethyl bromide would be my go-to if I were to try to prepare this stuff (which I will call TAA). This requires a solid fume hood, and the resulting mixture of salt, TAA, and isopropyl lithium solvent (usually n-hexane) should be easy to separate. The hexane can be stripped from the TAA via fractional distillation, with more distillation passes resulting in a more pure product. TAA boils at $\pu{102 ^\circ C}$ (Sigma Aldrich), and n-hexane boils at around $\pu{70 ^\circ C}$ - this $\pu{30 ^\circ C}$ gap in BP passes the rule-of-thumb "can I separate with distillation?" question.
All that said, even pricey Sigma Aldrich can get you 99% for $115 a liter. I am aware that TAA is an experimental recreational drug, and a possible replacement for ethanol. I advise caution in sourcing this material for that purpose, as the impurities present are likely to be more toxic than TAA itself. I believe the industrial synth is from 2-methyl-2-butene (2m2b) with the addition of water (this would be cheapest); TAA prepared this way can be purified further by refluxing with an open condenser, enabling any unreacted alkene to evaporate (Boiling point =$\pu{39 ^\circ C}\$). Acetone and ethyl Grignard would also form TAA, but with much more hassle and hazard, so the remnant ethyl chloride and acetone from such a synthesis are not likely impurities in any cheap sample. | 2019-06-19 05:00:50 | {"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": 0.6183634996414185, "perplexity": 1763.042239495928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998913.66/warc/CC-MAIN-20190619043625-20190619065625-00547.warc.gz"} |
http://old.xwiki.org/xwiki/bin/view/ReleaseNotes/ReleaseNotesXWikiEnterprise15?viewer=changes&rev1=10.1&rev2=11.1 | Changes for page ReleaseNotesXWikiEnterprise15
<
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edited by Jean-Vincent Drean
on 2008/07/24
>
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... ... @@ -7,9 +7,11 @@ 7 7 8 8 The focus for the 1.5 release was on improving the stability, performance and administration. 9 9 10 +#warning("Since XWiki Enterprise 1.5 the Administration is distributed as an application. You can download it from http://code.xwiki.org/xwiki/bin/view/Applications/AdministrationApplication.") 11 + 10 10 1.1 Changes since XWiki Enterprise 1.4 11 11 12 -1.1.1 1 Platform Improvements 14 +1.1.1 1. Platform Improvements 13 13 14 14 * Mail sender plugin now supports SMTP authentication 15 15 * Better PDF export support ... ... @@ -21,7 +21,7 @@ 21 21 * XWiki class property names are now translatable 22 22 * Ability to change just one (or several) object properties 23 23 24 -1.1.1 2 Administration Overhaul 26 +1.1.1 2. Administration Overhaul 25 25 26 26 | 2023-03-24 03:19:29 | {"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": 0.8666393756866455, "perplexity": 14136.18522867218}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945242.64/warc/CC-MAIN-20230324020038-20230324050038-00797.warc.gz"} |
https://cxc.cfa.harvard.edu/ciao4.13/dictionary/backscal.html | ## BACKSCAL
The backscale value records the ratio of the data extraction region area to the total detector area for a PHA spectrum file. If the background scaling is independent of channel, it is written to the BACKSCAL header keyword; otherwise, it is recorded in a BACKSCAL column.
In this example, the extraction region is 6283.19 square pixels. Dividing by the total ACIS detector area (8192x8192), yields the backscale value:
6283.19/(8192*8192) = 9.36267e-05
which is the value recorded in the file header:
unix% dmkeypar simple.pi BACKSCAL echo+
9.3626757073098e-05 | 2022-11-30 14:08:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.819593608379364, "perplexity": 7387.183966365428}, "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/1669446710764.12/warc/CC-MAIN-20221130124353-20221130154353-00268.warc.gz"} |
https://codepsu18-intermediate.kattis.com/problems/codepsu18.isstravellogs | CodePSU 2018 - Intermediate
#### Start
2018-03-18 19:00 CET
## CodePSU 2018 - Intermediate
#### End
2018-03-18 23:00 CET
The end is near!
Contest is over.
Not yet started.
Contest is starting in -211 days 5:40:52
4:00:00
0:00:00
# Problem EISS Travel Logs
The Interplanetary Space Station, located at the midpoint of Mercury and Jupiter, has a registry of all space flights to and from its ports. Each flight is identified by a six-digit ID number. The first digit of that ID number signifies the origin or destination country of those flights. The last digit of that ID number signifies the level of difficulty of the flight.
Earth’s Department of Data Delivery needs these ID numbers to be easily classifiable and accessible based on these two digits. Specifically, the department needs to know if these two digits are the same in certain cases and, if so, how many occurrences of these two digits there are.
## Input
The input consists of $N$ test cases ($1 \leq N \leq 100$), each test case consisting of a line. The first line of input contains the single integer N, indicating how many lines will follow the first line. Each line after the first line contains a string of L ID numbers ($1 \leq N \leq 100$). There will be a single space separating each ID number on all lines. Each ID number is entirely numeric and the length of each ID number is always $6$ digits.
## Output
For each N test cases of ID numbers in the input, output one line, containing the number of occurrences of unique matches of the first and last digits of every possible first/last number combination separated by commas, in order from $00$ to $99$. Cases where there are zero occurrences should be excluded. The number of lines of output should correspond to the number of test cases $N$.
Sample Input 1 Sample Output 1
4
012340 092470 589199 012345
557840 543640 251805 232835
214497 580409 001279 531281 900388
413989 772246 625761 930964 767396 944064
2 1 1
2 2
1 1 1 1 1
1 1 2 2 | 2018-10-15 23:40:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23696161806583405, "perplexity": 815.5256066509146}, "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/1539583509958.44/warc/CC-MAIN-20181015225726-20181016011226-00509.warc.gz"} |
https://www.edaboard.com/threads/please-help-me-with-designing-a-bus-bridge-in-cpld.53848/ | ### Welcome to EDAboard.com
#### Welcome to our site! EDAboard.com is an international Electronics Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.
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#### buenos
bus bridge in CPLD
Hi!
I have to design a system bus interface and an address decoder, for the ADI's Blackfin processor in a CPLD. it has a 133MHz system bus. I have to connect a 133MHz SDRAM, and some low speed peripherals (USB/ethernet controllers, max 12/50MHz). The DSP will be the bus master. The USB-IF can be temporally a bus master, but it is not necessary. I would like to use a xilinx Coolrunner CPLD.
--Is it possible to make a 133MHz bus interface logic in a cpld? Is a Coolrunner fast enough for this? (the datasheet said: 5.7 ns pin-to-pin delays, and max 250MHz)
As I know, the 250MHz test system consists of one gate, and nothing else. But my IF will be more difficoult. So I think it will work slower. I do not wait an exact speed, only an estimated.
--What should do the bus-IF logic?
I can think it very simply (a clock prescaler, and some gates for the READY signal), but there are very difficoult bus-IF implementations, with dual port memories, and other... Like the OPB/PLB bridge for the Microblaze processor. Why need they that level of difficulity?
--Can I connect the SDRAM directly to the system-bus? (with address decoded-CS-signal from the CPLD) The processor ha a universal interface for DRAM, SRAM and other, and it has an integrated SDRAM controller.
--How can I know the required size of the CPLD?
--Will the CPLD address decoder decrease the system performance, and SDRAM access speed? Through an increased clk to CS (clk to address) SKEW. What should I do?
#### buenos
bus bridge in CPLD
has anybody designed any bus-bridge, or high-speed bus?
#### mc&fpga
##### Member level 1
Re: bus bridge in CPLD
as isee ,you want to communicate sdram and other bus like usb,...
as you know you need a sequncer for controlling the refresh timiing of sdram and the another sequncer for controlling the bus timming like usb or ethernet,
you will have problem to use coolrunner and other cpld for thier low macrocell,
and you need memory to control asynchrounce interface,so cpld and coolrunner aren't good device.
#### buenos
bus bridge in CPLD
I have a ADSP-BF533 processor, with internal SDRAM controller. The proc has one single bus inerface, for all the external peripherals, and mem. It has all the signals for everything.
As I see now in the UG, the ext bus has select signals, for SDRAM, and other 4 for peripherals. it is fine!
In this case, could I use a CPLD? With a fast separated 3state buffer? as an asinchronous memory-like device, with ready signal.
So, the CPLD (or whatever else) has to do the bus bridging only. In the proc side, there is the proc, and mem, and in the other side, there are the slow peripherals, in a slow 10-15MHz bus.
Do I need a bus bridge? Or, can I connect them directly? In many DSP boards, there is a CPLD, with some bus-function. What should they do in the buses?
What If I will have to design (in the far future, not now) a system without a this-like good ext-bu-IF? Will I have to produce these select signals? with what? Is it possible in this speed?
#### mc&fpga
##### Member level 1
Re: bus bridge in CPLD
ya,the dsp processor like ti c6000 and youranalog device processor like adsp have some peripheral to control the sdram or ddr bus,and they are cheap way to connect the memory and other bus(like ethernet), ya in some boards only for i/o buffering and implement other bus controlling like usb they use cpld (and for low power system they use collrunner),and cpld permit us in designingt to control the i/o port.as you said you don't need them for contrlloing sdram (cause adsp do that).so you don't need them in fast speed.
#### buenos
Re: bus bridge in CPLD
thanx again.
and what about the bridging between the 133MHz and the 15MHz bus? Is enough a fast bus-switch (select) and a clock prescaler? ready-signal logic? or do I need some temporal memory-buffering? or not?
If I would use without bus-sw, when the dsp takes back the select from the low-sp.per., it may go out from the bus slowly. (bus crash) And they need slower clk.
What do you think in that:
select signal
----------------------------
I I
I system bus V (en) slow bus
[DSP]---------------------[bus-sw]----------------------
I I I I
[sdram] [other hs] [lowsp.per.1] [lowsp2]
I hope it looks visible.
as I see it don't look like, how I draw.
I can not draw it, because the forum-system does not allow multiple spaces in the text.
another trying:
Code:
select signal
------------------------------------
I I
I system bus V (en) slow bus
[DSP]-----------------------------[bus-sw]---------------------------------------
I I I I
[sdram] [other hs] [lowsp.per.1] [lowsp2]
#### mc&fpga
##### Member level 1
Re: bus bridge in CPLD
as i see from your note,
you want and you can control sdram in speed of 133 mhz,and interface the data to
another bus,so you need an asynchrounse memory in your dsp processor or dma protochol in your dsp processor to control the speed of i/o bus in low speed,
#### buenos
bus bridge in CPLD
is it possible that the processor can access to the sdram at high speed, and on the same bus, it can access to the perif at low speed?
It has async memory control signals, like the ready signal. I saw that. only by waiting some cycles, as it do an access to the perif?
do not needed any external logic? buffers, switches?
Status
Not open for further replies. | 2021-12-03 14:33:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18707114458084106, "perplexity": 10273.576435585332}, "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/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00034.warc.gz"} |
http://runge.math.smu.edu/Courses/Math6370_Spring17/ | # Math 4370/6370 – Parallel Scientific Computing, Spring 2017¶
Instructor: Daniel R. Reynolds
Lectures: M/W/F 2-2:50, 156 Dallas Hall
Office Hours:
• 139 Clements Hall
• M/W/F 9-10, or by appointment (email to arrange)
Course Pages:
News (most recent at top):
• This term we will be using Piazza for class discussion. The system is designed to help you get assistance quickly and efficiently from classmates and myself. Rather than emailing questions directly to me, I encourage you to post your questions on Piazza.
• At the bookstore the text costs $80 new,$60 used or $180 digital (less for rentals); online it can be found for as little as$46 new or \$62 used (as of 11/19/2015). | 2020-06-05 09:32:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.2166382223367691, "perplexity": 12324.625052432595}, "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-2020-24/segments/1590348496026.74/warc/CC-MAIN-20200605080742-20200605110742-00566.warc.gz"} |
https://chat.stackexchange.com/transcript/71/2019/6/20 | 12:04 AM
::recalls Aristotle saying something similar::
@ACuriousMind yes it is
5 hours later…
4:45 AM
@skullpatrol When he was 22. But he was ROTC, so he went to OCS and off to the Navy.
5:26 AM
cool
5:42 AM
@JohnRennie just came out of Magdalene may ball
I love may balls
They are beautiful
@JakeRose you sound suspiciously high :-)
Just oxytocin, or might alcohol be involved as well? :-)
2 hours later…
7:58 AM
mornin
8:35 AM
@MoreAnonymous are positions in condensed matter physics easier to get?
@bolbteppa are these theses funded positions for applications? If funding is still uncertain, then even if one applies for one of the theses and gets acceptance of the principal investigator, whether they can work with that thesis still depends on whether funding is available.
9:17 AM
@JohnRennie I am drug free if we don’t include caffeine and ethanol ;) never seen the value tbh
@JakeRose I did security at a few May Balls, but I never went as a paying guest. They aren't really my thing. I have no romance in my soul :-)
9:31 AM
@CaptainBohemian depends where I guess ... but I knew a prof in durham university how thought I'd b good at it because he liked my insightful questions I'd ask ... Compared to that I couldn;t get anywhere for quantum gravity ... So in my experience yes
but to be fair networking can play a massive role in these kind of things
9:44 AM
Hm
Trying to find a timelike curve in my dumb spacetime is trickier than expected
@Slereah since ur a boss in GR ... Is there an explicit calculation showing the equivalence principle for stress energy tensors? (the reason I ask for this is instead of the usual line-elements is that fields cannot be thought like this)
Who told you I was a boss in GR
I am awful
really ?? I thought u were good at it ... L
(I presumed myself)
Any hints on how to approach this problem I posed?
10:01 AM
@MoreAnonymous what do you mean by the equivalence principle for stress energy tensors?
Equivalence princple = It is a physical principle in which given Einstein's thought experiment setup: One cannot distinguish if the lift is under the influence of gravity or acceleration using any experiment within the lift
The equivalence principle is just the statement that the four acceleration is the sum of a coordinate term and the curvature term.
I wanna somehow show a relation between the stress energy tensors of both cases
The four acceleration is:
$$A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha{}_{\mu\nu}U^\mu U^\nu \tag{1}$$
But neither of the two terms on the right are tensors so both are coordinate dependent.
That is either term can be made zero by an appropriate choise of coordinates.
yea ... but I'm not sure if u can apply this for a field ...
10:05 AM
@MoreAnonymous Well the coordinate term is zero in the rest frame of the observer and the curvature term is zero in the Fermi normal coordinates. So I guess you need to write your SE tensor in both coordinates and compare the two versions.
Alright ... I think it's best to just ask the question on PSE ... I think it's not a dumb question?
sigh ... Im really rusty with GR
@JohnRennie That's almost an answer to this question, but it might be too technical for the OP.
Working on a more rigorous proof for that dumb metric
hahaha
Any bookk recommendations for someone who wants to revise GR? like I'd prefer it to complete yet fast paced ...
@PM2Ring I'll have a look ...
10:18 AM
@JohnRennie Thanks. I added a related question, but I don't know if it's close enough to be a dupe target.
I am thinking about liquid spacetime
@MoreAnonymous if you want fast paced try Relativity: the special and general theory by Einstein
Worse than chaotic spacetimes where the metric varies chaotically from event to event, the metric itself changes in such a way such that it behaves like a fluid
Ugh ... He should put a warning out on his Boltzman brains
@skullpatrol I was hoping for something with more exercises (plenty of examples would b ideal) and more mathematical (something a physicist can still digest though)
And more modern as well
(Skimmed past the book)
10:46 AM
The worst thing about being a Boltzman brain? Boltzman zombies!
0
I have been on physics stack exchange for about four years. Some of my answers I am proud of some of them not so much. There is one particular answer where I did my best at the time but I lacked the knowledge to understand what OP was asking. Some people however liked the answer and nobody else r...
11:34 AM
@JohnRennie isn't this just the covariant derivative of the 4-velocity of an object? I didn't think of it can be an illustration of equivalence principle.
@MoreAnonymous I think that the gravitational energy-momentum is not a tensor but a pseudotensor is just an illustration of equivalence principle.
Feel free to answer here
0
Is there an explicit calculation showing the equivalence principle* for stress energy tensors? (the reason I ask for this is instead of the usual line-elements is that fields cannot be thought like this). $*$ =It is a physical principle in which given Einstein's thought experiment setup: One c...
@CaptainBohemian
@MoreAnonymous is writing to researchers who work in topics similar to my MSc thesis one kind of network? but usually they would just reply me that have no funding.
@MoreAnonymous I don't understand what "equivalence principle for stress-energy tensors" is supposed to mean
well knowing ppl helps ... Like the prof in durham was my lecturer for stat mech. ..
@ACuriousMind I mean if we are in einsitein's life the stress energy tensor of the lift should have some kind of relationship with the stress energy tensor of the life in a graviataional field
What's the "stress-energy tensor of the lift"?
11:49 AM
well we know kinda a line element since it's accelerating ... right? and then I can go EFE and find the stress energy tensor ... though I wanted to know if u could do something similar for fields ?
a pseudotensor is coordinate related and you may choose a coordinate to make it vanish.
I mean you know
Energy itself is coordinate related
it's not that weird that it would be
Particularly when involving gravity, you need to be careful to define what you mean by "stress-energy tensor".
Are you deriving it from an action? Are you defining it operationally through 4-momentum fluxes? Is it a tensor or a pseudo-tensor? Etc...
@CaptainBohemian the equivalence principle is the fact that by choosing appropriate coordinates you can make the Christoffel symbols zero or the coordinate acceleration zero. The former is what Newton would call gravity and the latter acceleration. So the two are interchangeable.
Also there's the whole stress energy tensor v. canonical stress energy tensor v. Beliphantes tensor business
11:53 AM
but usual energy is a tensor, which follows usual transformation law when you make coordinate transformation, and if it vanishes in a coordinate system, it vanishes in all coordinate systems.
Well no
Energy isn't a tensor
and you can indeed have no energy in a frame and energy in another frame
Just consider the case of classical mechanics
rest frame v. Galilean boost
Although things are more complex with GR involved certaily
gravitational energy-momentum is a pseudotensor which follows some physics principles but don't follow the usual coordinate transformation rules for tensors.
true
Since it is basically derived from Christoffel symbols
The basic idea being to have $\partial_t (T_M + T_g) = 0$
Since you have $\nabla T_M = 0$ you can sort of work out how it's derived
@ACuriousMind Im not really well versed in the technicalities .. I was thinking about the following: (forget about fields for the moment) .. If I think in terms of line elements ... The line element of the lift in presence of acceleration enables me to get a stress energy tensor ... Now the measuring apparatus is inside lift is also inside the lift undergoing acceleration ...
So using these I should be able to make a statement about the possible measurments such as energy difference, etc. On the other had I have the lift in the presense of a graviations field ...
Now re-enter fields ... Is it possible to do the above analysis in terms of fields (since they have a different starting point (no-line element))
?
I don't understand what you mean by "the line element of the lift", either
"Line element" is usually just another name for the metric
12:10 PM
It's also a section of the projective vector bundle
though I doubt that it is here
@MoreAnonymous but suppose you're in space near the Earth, then the stress-energy tensor at your position is zero.
wait i think u guys re right and Im confusing myself
You can't determine the SE tensor by measuring proper acceleration. You have to do some curvature measurement then you can get the SE from the Ricci tensor.
but can't u gain the chirstoffel symbols by knowing the proper accelreation
you may if you know the acceleration of a geodesic congruence, I think
Since knowing the Christoffel symbol at a point won't help you much
since you need to know the derivatives too
And knowing a geodesic congruence is basically what curvature measurement is, anyway
just the spread and shear and twist of the congruence
12:18 PM
alright ... I kinda get ur point
sigh ,.. I really need to revise GR
getting outta touch
any book recommendation?
44
This list is extensive, but not exhaustive. I am aware that there are more standard GR books out there such as Hartle and Schutz, but I don’t think these are worth mentioning. Books with stars are, in my opinion, “must have” books. (I) denotes introductory, (IA) denotes advanced introductory, i.e...
thannks
Speaking of GR, nice work on physics.stackexchange.com/a/487103/123208 John. Thanks! It could make a useful dupe target, but I guess we need the OP to clarify the question a little more.
12:40 PM
@Slereah doing god's work
I need to update that list
Also we need to make a list of worst books for GR
hahaa ... any popular books out there which should join the list :P
" worst books for GR"
@Slereah sachs-wu
@Slereah @ACuriousMind please yeet this into the shadow realm
I will not have my friend assaulted
You traitor
Sachs-Wu isn't that bad
Certainly not the best introduction
Hm
what's a really awful one
Besse certainly isn't great as far as notation goes
12:51 PM
wow!
This book claims that matter, energy and space are the same thing, with the particles we observe being 'knots' of space. This book is self-published, after having been rejected by a number of publishers. I would like to say that Deepak Chopra would turn down this author.
besse is a good book
savage
If I am paid enough I am willing to write a GR book entirely in Penrose notation
If I do GR for my PhD I'll write a GR book one day
I actually met Penrose once in Nottingham
12:52 PM
Any specific topic?
the book or the PhD
Ironically I don't think Penrose ever made any extensive use of penrose notation
it was just an idea he threw around
@RyanUnger Both I'd suppose
haha ... thats funny
I think there's a real lack of a book that addresses a good cross-section of GR topics that mathematicians could find interesting
Choquet-Bruhat talks about a lot of them but the book has so many issues
@MoreAnonymous That's the type of book I would write, except I wouldn't try to get it published, I would just self-publish for laughs to see if anyone actually tried to understand the word-salad. It's always seemed really alluring to me to try to write a nonsense "theory" just to get a bunch of parody out of my system.
12:54 PM
what would a mathematician find interesting
AFAIK none of the minimal surfaces stuff has made it into books
@JMac ... ur actions would seem synonymous with evil
Minimal surface stuff is as far as I know more used for the study of extended objects than GR proper
although I don't actually know
I am trying to write a general thing on extended objects currently
@MoreAnonymous Just Poe's law in action. It's basically impossible to tell who is serious or not online
although right now I have a much nobler goal
which is to make up a really stupid spacetime
The dumbest
There are errors I'm aware, didn't fix all the computations
But it's the law : every counterexample must use $\sin(1/x)$ somewhere
3
extended objects will be here soon, inch'allah : samuel-lereah.com/articles/Physics/extended-objects
12:59 PM
Anyone read this book
?
Seems ambitious
I have not
got it from the least voted answer in book recommendation post
it's hard to know what the worst GR book is because it's probably some crank book that nobody's ever heard of rly
it might b a worthwhile goal: "find the worst GR book"
:P)
I don't think I have a book that is really bad
1:04 PM
I feel like now upvoting that post ... So someone can actually tell me how much baloney is in that book
I wonder if it would fall under the ethical use of upvote
Now, on a more serious note (some QM) ... how does the program where they have wavefunction of the universe . How do they go about deifning a measurment
?
Jesus takes the measurement
I was actually being serious here ...
as thus
noooooo
I mean like how they overcome this kind of objection
2
Background Let's say I have a Hamiltonian $\hat H$ (assume the Schrodinger equation) and it be in an arbitrary eigen-energy state: $$\hat H_{\text{system}} | m \rangle = E_m |m \rangle$$ And I want to measure the momentum of the system without perturbing the Hamiltonian. We know this will...
not a clue
1:09 PM
@Slereah I notice u have a stash of memes available instanteous
-ly
I use the ancient art of google image search
theres a meme with lubos on it as well on google search top result
?
no, I made that one
it is the official logo of PSE
I remain unconvinced that QM is not just due to physicists being bad at experiments
Or at least so I would like to think
Feel free to try to prove it wrong
1:11 PM
oh I thought those google algorithm was really amazing for a moment
@RyanUnger if u have a seperate way to resolve the conundrum in my post ... please answer :)
There is kind of a Thing where it's kind of hard to do experiments wrt a model without some assumptions
Because you don't actually know the initial conditions properly
and measuring them is impossible without some assumptions beforehand
like how you have to assume that distances are roughly euclidian over a short enough time when you're doing the light clock thing
since the metric could be arbitrarily complicated in the way
@MoreAnonymous Sorry, but I don't understand that question, either. "energy cost of the measurement" is not a standard term, as glS points out. For energy to be conserved, of course changes in the expectation value of energy of the measured system result in changes of the expectation value of energy of the measurement apparatus. If you do not consider the measurement apparatus, measurement is never a unitary time evolution, hence will never conserve conserved quantities.
So can I conclude the: the measurement only makes sense if there is more than one system?
It's that Epistemology Thing where a theory's interaction with the real world has to include the definition of the measurement
otherwise it's not worth a damn
@MoreAnonymous Sure, a measurement needs at least two systems: A measured system and a measurement apparatus.
But you don't need to do the confusing computation you do there at all to conclude that - that's what measurement means
1:18 PM
The how does the wavefunction of the universe make sense
?>
@MoreAnonymous The "wavefunction of the universe" is never measured.
does it ever make contact with experiment
?
Only subsystems of the universe are measured, with one subsystem being the measured system and another being the measurement apparatus
So ur bsaically sayin I'm right ... I wasnt sure ... since the measurement was kinda a taboo in physics class
A long time ago, a friend of mine made a joke
Where he said that, if asked that ifthe universe is closed, then what is inside
you should answer Azathoth
1:20 PM
Whether you believe that there is a "wavefunction of the universe" or not is a matter of your chosen quantum interpretation, in particular whether it believes in true collapse or not. The formalism really doesn't care about "the universe", it just describes quantum systems.
and it is still what I do to this day
@ACuriousMind this one is a bit more involved but since u agree with the prevoius post .. I;d love ur thoughts?
1
Summary and Motivation "The below idea is about making a mathematical statement on system $2$ which induces a measurement on system $1$ while $1+2$ obeys unitary evolution." Basically, I'm modelling the measurement (occurring at time $t$) as an interaction and that I have some constraints based...
1:36 PM
It follows the idea of a subsystem performing a measuremnt on another subsytem but the whole system following Unitary evolution
1:50 PM
@MoreAnonymous That reminds me of Wigner's Friend; sorry, I should have mentioned that the other day.
@PM2Ring ... thanks .. To be fair I dont go into the case where $\tilde \epsilon_\pm \to 0$ so i sidestep the discontinous
@Slereah someone one needs to merge Wald, HE, and Straumann
They all have boring bits that can be cut out
Like do we really need to see the computation of geodesics
@RyanUnger Depends I suppose
Gotta target your audience
I’m going for big brain
I'm guessing most people who read HE know the geodesic formula
2:00 PM
Straumann has way too much stuff in the appendix
Just need to fix sign conventions really
I can feel his energy around here
Does he teach
Unclear
I’ll check when the course reg opens in the fall
There’s very little overlap between physics and math at Princeton
Witten's resume is quite to the point
But I guess that when you're Ed Witten
you don't need to put bells and whistles on it
2:04 PM
I don’t know how many courses I can take in the fall
Have to prepare for the generals
he is listed as "Visiting Lecturer"
He’s not a professor at Princeton
He’s faculty at the IAS
just go find him in the hall and yell at him that string theory is a sham
hahhaha
tnf Einstein said the same for religion
@Slereah I wouldn’t be the first to do that
Problem is he’s so smart that it’s impossible to argue with him
2:16 PM
@RyanUnger Next time this happens plz use ur cellphone to show footage of the person being intellectually pawned
I think we should have a big brother show where ppl of different camps come stay in the same house and discuss physics
though if u wanted to reach a more general audience replace camps with religion
Sorry @RyanUnger
you don't have the $\mathbb{PHENOTYPE}$
2:36 PM
@PM2Ring did anyone take the
Sorry accidently pressed eneter
but did anyone do the math behind Wigner's friend?
2:57 PM
nevermind just saw the wiki link
@CaptainBohemian "This page presents the thesis proposals submitted to the ED, irrespective of whether or not they are the subject of an application to the funding competition."
@MoreAnonymous I'm not sure the equivalence principle question makes sense
k . .. I sadly do not get Wigner's friend's paradox ... Like how does the friend measure the system without an interaction hamiltonian? If there is an interaction hamiltonaian the tensor product comes under question
@bolbteppa yea it doesnt i have to delete it
thanks for remindin me
3:12 PM
'Thus the properties of the motion in a non-inertial system are the same as those in an inertial system in the presence of a gravitational field. In other words, a non-inertial reference system is equivalent to a certain gravitational field. This is called the principle of equivalence.' Not sure what it means to apply this to the stress-energy tensor which is basically the derivative of the action with respect to the metric...
yea .. i was confused about some stuff ...
But I was thinking that the stress energy of both them must be related in some way ...
On a related note are there any fields which can be expressed in tensor notation but do not respect the equivalence princple?
@MoreAnonymous I don't know what it's supposed to mean for a field to "respect the equivalence principle".
Any field can be "expressed in tensor notation"
Hi, first time in chat :]
Like if I have stress energy tensor of lets say the EM field between a capcitor in a lift
accelrating
@Gyromagnetic Welcome!
3:23 PM
Please help me out with this: physics.stackexchange.com/questions/487094/… Im sure its trivial <3, but I want someone that knows their QM to say its right
then would it behave the same when in gravitational field?
Thanx ;)
Can I construct a field that does not behave like tis is the ("related note question")
@Gyromagnetic You need not guess the answer there and think this is a "conceptual" question - you can actually straightforwardly compute the answer! Write down the time evolution for spinning particles in a constant magnetic field, then put in those two states as the initial conditions and see how differently they evolve!
I actually envy those who have good physical intuition and can tell u stuff about a physical system without calculating it
3:27 PM
You get that intuition by having done these calculations so often that you no longer need to actually do them to make a stab at the result
3
@MoreAnonymous You still seem to think the "equivalence principle" is somehow a property of specific fields or something. It's not. It's a property of the theory of general relativity.
I really need to revise my GR (im muddled in the head right now)
Kind of embarrassing to admit but I dunno if I could even do it... I mainly build experiments, been years since I actually evolved a free particle hahaha... I guess I'll open my dusty Landau & Lifshitz and try to figure it out haha
The point of the equivalence principle is just that the motion of a body in a non-inertial system is indistinguishable (locally) from the motion of a body in an inertial system in a potential where all bodies move the same independently of their masses provided the initial conditions are the same, i.e. a gravitational field.
A tensor transformation represents going to a different coordinate system in the same gravitational field, so if your tensor transformation in a gravitational field is not a change of coordinates from one system to another using the line element of the GR field it's not respecting the equivalence principle I think
got it! (i think)
Any experts on Wigner's friend now?
"Like how does the friend measure the system without an interaction hamiltonian? If there is an interaction hamiltonaian the tensor product comes under question"
requoting myself
@Gyromagnetic By the way, I'd agree that the phase difference is what you write. But there isn't really any way to back up that claim without actually doing the computation, so we end up with at least one of us doing it either way ;)
1 hour later…
4:45 PM
weez
5:26 PM
hmmm
correct number of m's
good man
3 hours later…
8:10 PM
@JMac Walsby's at it again: physics.stackexchange.com/a/487166/123208 At least this time the OP is knowledgeable enough to see instantly that that answer is baloney.
3 hours later…
10:54 PM
Am I being super dense or maybe my brain is dead ... But I can;t get what he's saying
There is no energy cost here like you have defined above. See, in classical example, when your interacation doesn't matter, you still get different values for every measurements as such you can do what you have done here for classical case and still argue that this formula is "energy cost" but it isn't! We clearly know that in classical case there is nothing such as adding energy to the system and etc. If you write something similar for classical case, it would only mean error of each data relative to average, i am suggesting that same logic applies for quantum case, for that formula of course — Paradoxy 8 mins ago
2
Background Let's say I have a Hamiltonian $\hat H$ (assume the Schrodinger equation) and it be in an arbitrary eigen-energy state: $$\hat H_{\text{system}} | m \rangle = E_m |m \rangle$$ And I want to measure the momentum of the system without perturbing the Hamiltonian. We know this will...
11:32 PM
@PM2Ring I noticed this gem today physics.stackexchange.com/questions/487049/… and decided to look at his self published works on amazon, exact same type of rambling. It's like the gish gallop of physics questions, just so much nonsense that it takes longer to explain whats wrong than it does to come up with the rambling. | 2019-09-16 01:47:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.700900137424469, "perplexity": 1104.8269790150305}, "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/1568514572439.21/warc/CC-MAIN-20190915235555-20190916021555-00125.warc.gz"} |
https://www.jobilize.com/trigonometry/test/identifying-horizontal-asymptotes-of-rational-functions-by-openstax | # 5.6 Rational functions (Page 5/16)
Page 5 / 16
Find the vertical asymptotes and removable discontinuities of the graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{{x}^{2}-25}{{x}^{3}-6{x}^{2}+5x}.$
Removable discontinuity at $\text{\hspace{0.17em}}x=5.\text{\hspace{0.17em}}$ Vertical asymptotes:
## Identifying horizontal asymptotes of rational functions
While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the function that is the ratio of the leading terms.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator>degree of the numerator, there is a horizontal asymptote at $\text{\hspace{0.17em}}y=0.$
In this case, the end behavior is $\text{\hspace{0.17em}}f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}.\text{\hspace{0.17em}}$ This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function $\text{\hspace{0.17em}}g\left(x\right)=\frac{4}{x},\text{\hspace{0.17em}}$ and the outputs will approach zero, resulting in a horizontal asymptote at $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] . Note that this graph crosses the horizontal asymptote.
Case 2: If the degree of the denominator<degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is $\text{\hspace{0.17em}}f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x.\text{\hspace{0.17em}}$ This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function $\text{\hspace{0.17em}}g\left(x\right)=3x.\text{\hspace{0.17em}}$ As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of $\text{\hspace{0.17em}}g\left(x\right)=3x\text{\hspace{0.17em}}$ looks like a diagonal line, and since $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ will behave similarly to $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ it will approach a line close to $\text{\hspace{0.17em}}y=3x.\text{\hspace{0.17em}}$ This line is a slant asymptote.
To find the equation of the slant asymptote, divide $\text{\hspace{0.17em}}\frac{3{x}^{2}-2x+1}{x-1}.\text{\hspace{0.17em}}$ The quotient is $\text{\hspace{0.17em}}3x+1,\text{\hspace{0.17em}}$ and the remainder is 2. The slant asymptote is the graph of the line $\text{\hspace{0.17em}}g\left(x\right)=3x+1.\text{\hspace{0.17em}}$ See [link] .
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at $\text{\hspace{0.17em}}y=\frac{{a}_{n}}{{b}_{n}},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}_{n}\text{\hspace{0.17em}}$ are the leading coefficients of $\text{\hspace{0.17em}}p\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}q\left(x\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0.$
In this case, the end behavior is $\text{\hspace{0.17em}}f\left(x\right)\approx \frac{3{x}^{2}}{{x}^{2}}=3.\text{\hspace{0.17em}}$ This tells us that as the inputs grow large, this function will behave like the function $\text{\hspace{0.17em}}g\left(x\right)=3,\text{\hspace{0.17em}}$ which is a horizontal line. As $\text{\hspace{0.17em}}x\to ±\infty ,f\left(x\right)\to 3,\text{\hspace{0.17em}}$ resulting in a horizontal asymptote at $\text{\hspace{0.17em}}y=3.\text{\hspace{0.17em}}$ See [link] . Note that this graph crosses the horizontal asymptote.
Notice that, while the graph of a rational function will never cross a vertical asymptote , the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions | 2019-08-23 08:46:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 27, "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": 0.8927876353263855, "perplexity": 503.73008622299324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318243.40/warc/CC-MAIN-20190823083811-20190823105811-00081.warc.gz"} |
http://maths.ccnu.edu.cn/info/1045/24924.htm | Statistical Principles and Deep Modeling
Speaker: 刘传海 DateTime: 2020年12月17日（周四）上午10:30-11:30 Brief Introduction to Speaker: 刘传海，美国普渡大学统计系教授。武汉大学概率统计硕士，1987年、哈佛大学统计学硕士，1990年、哈佛大学统计学博士，1994年。主要研究兴趣包含：贝叶斯、统计推断的计算方法、数据分析的计算机语言和环境、缺失数据和多重插补、多重比较和时间序列等等。获得的奖项/荣誉有美国统计协会会员（2007年）、当选为国际统计学会成员（2006年）、杰出统计应用论文(《美国统计协会杂志》，2000年）、2000年弗兰克·威尔科克森奖、1998年贝尔实验室总裁银奖、1994年哈佛大学杰出教学研究员等等。 Place: 腾讯会议（会议号请联系左国新老师索取） Abstract: While the development of machine learning methods has dominated the recent research in the area of computer-intensive data analysis, one can imagine that future high-quality of research appears to be also in need of more principled ways of data analysis. In this talk, we will introduce the two obvious but fundamental principles, namely, the {\it Validity} principle and the {\it Efficiency} principle. In their book entitled {\it Inferential Models --- Reasoning with Uncertainty}, Ryan Martin and Chuanhai Liu argued for these two principles in the context of making reliable and efficient inference based on postulated models. With a brief review of the two principles for statistical inference, we discuss their implications in a scenario of model building. Implementation of the two principles for model building will be illustrated with an experimental method, which we call {\it Deep Modeling}, for analyzing the famous MNIST dataset. | 2021-03-04 06:23:04 | {"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": 0.843376100063324, "perplexity": 1198.6325536411218}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178368608.66/warc/CC-MAIN-20210304051942-20210304081942-00293.warc.gz"} |
https://gottwurfelt.com/2020/09/ | Would the 2016 US presidential election result be different if the Electoral College were bigger?
No.
A quick way to see this: Trump won 306 electoral votes, Clinton 232. (There were faithless electors, but we can ignore them for the purposes of this question.). Trump won 30 states, Clinton 21. For the purposes of this post, DC is a state, which it probably also should be in reality.
Each state has as many electoral votes as it has senators and representatives combined. (DC gets 3.). Imagine breaking up the electoral votes in each state into two classes: the “senatorial” electoral votes (two per state), and the “representational” electoral votes (the other ones). Then Trump won the “senatorial” electoral votes by 60 to 42, leaving a 246-190 margin among the “representational” votes.
In 2000, on the other hand, Bush won 271 electoral votes in 30 states, and Gore 267 in 21 states; the “representational” votes were therefore split as 211 for Bush, 225 for Gore. If there were, say, twice as many Representatives (870 instead of 435 – and let’s keep the math simple and say DC gets 6 electoral votes in this scenario), and every state had twice as many electoral votes, then Bush would have had about (60 + 211 × 2) = 482 EV, and Gore (42 + 225 × 2) = 490 EV. Also in this world Nate Silver’s web site is named nineseventytwo.com (which is available, as of this writing). In fact, Bush would have won with any number of representatives less than 491; Gore with more than 655; and in between, the lead swings back and forth, according to a 2003 analysis by Michael G. Neubauer and Joel Zeitlin.
In short: 2000 was the way it was because small states are overrepresented in the Electoral College; 2016 was the way it was because the Electoral College is winner-take-all.
Balancing the centrifuge – a Riddler puzzle
From the riddler:
Quoc’s lab has a microcentrifuge, a piece of equipment that can separate components of a liquid by spinning around very rapidly. Liquid samples are pipetted into small tubes, which are then placed in one of the microcentrifuge’s 12 slots evenly spaced in a circle.
For the microcentrifuge to work properly, each tube must hold the same amount of liquid. Also, importantly, the center of mass of the samples must be at the very center of the circle — otherwise, the microcentrifuge will not be balanced and may break.
Quoc notices that there is no way to place exactly one tube in the microcentrifuge so that it will be balanced, but he can place two tubes (e.g., in slots 1 and 7).
Now Quoc needs to spin exactly seven samples. In which slots (numbered 1 through 12, as in the diagram above) should he place them so that the centrifuge will be balanced? Extra credit: Assuming the 12 slots are distinct, how many different balanced arrangements of seven samples are there?
https://fivethirtyeight.com/features/can-you-break-a-very-expensive-centrifuge/
I’ve seen this one before, but I’ll take it on as a programming challenge.
I’ll say that hole k is at coordinates $(\sin 2\pi k/12, \cos 2\pi k/12)$. This is nonstandard, but the result looks like a clock, so it’s easier to visualize:
Now, there are ${12 \choose 7} = 792$ ways that we could pick 7 out of the 12 holes. To generate a list of these I can use the following R function:
combinations = function(n, k){
if (k == 0 & n == 0){return(list())} else {
if (k == 0){return(list(c()))} else {
if (n == k){return(list(1:n))} else {
without_n = combinations(n-1, k)
with_n = lapply(combinations(n-1, k-1), function(x){c(x, n)})
return(c(without_n, with_n))
}
}
}
}
To generate the subsets of [n] of size k, I just take:
• the subsets that don’t contain n (which are therefore subsets of [n-1] of size k), and
• the subsets that do contain n (which are therefore subsets of [n-1] of size k-1, with n adjoined)
(I learned this a good two decades ago, from section 3.6 of Herb Wilf’s notes East Side, West Side.)
Then iterate over the subsets to find their centers of mass:
our_comb = combinations(12, 7)
com_x = rep(NA, length(our_comb))
com_y = rep(NA, length(our_comb))
for (i in 1:length(our_comb)){
com_x[i] = mean(x[our_comb[[i]]])
com_y[i] = mean(y[our_comb[[i]]])
}
So our_comb runs through the combinations, and com_x and com_y are the x- and y-coordinates of the center of mass. Plotting the centers of mass gives we get an attractive symmetric pattern. Here I’ve plotted transparent points, so the darker points are points that arise from more distinct combinations. The outermost points represent the most imbalanced centrifuges – for example the one closest to the top represents loading slots 9, 10, 11, 12, 1, 2, and 3. The main question is: how many points are on top of each other at the very center.
I can extract those subsets from the matrix our_comb and build them into a matrix for looking at. There’s a small problem in that we used floating-point arithmetic, so the coordinates don’t come out exactly at zero, but it’s enough to ake the ones that are “close enough”:
epsilon = 10^(-6)
balanced_indices = which(abs(com_x) < epsilon & abs(com_y) < epsilon)
balanced_combs = t(matrix(unlist(our_comb[balanced_indices]), nrow = k))
and the matrix balanced_combs, which has one row for each combination that balances the centrifuge, is
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 1 2 3 6 7 9 10
[2,] 1 2 4 5 8 9 10
[3,] 1 2 5 6 7 10 11
[4,] 2 3 4 7 8 10 11
[5,] 2 3 5 6 9 10 11
[6,] 1 2 5 6 8 9 12
[7,] 1 3 4 7 8 9 12
[8,] 1 4 5 6 9 10 12
[9,] 1 4 5 7 8 11 12
[10,] 2 3 6 7 8 11 12
[11,] 3 4 5 8 9 11 12
[12,] 3 4 6 7 10 11 12
(The order here is lexicographic order, but you have to read the rows backwards.)
If you can make sense of this pile of numbers without making some pictures, good for you! But I am human, though a mathematician, so let’s plot. Here big black dots represent the loaded spots, and small blue dots represent the non-loaded spots.
(And yes, it’s in base R graphics. It’s been a while since I’ve used those – at my day job it’s all ggplot all the time.)
par(mfrow = c(4, 3), mar = c(1, 1, 1, 1))
for (i in 1:nrow(balanced_combs)){
full = balanced_combs[i,]
empty = setdiff(1:n, balanced_combs[i,])
plot(x[full], y[full], pch = 19,cex = 2,
asp = 1, axes = FALSE, xlab = '', ylab = '',
xlim = c(-1.25, 1.25), ylim = c(-1.25, 1.25))
points(x[empty], y[empty], pch = 19, cex = 1, col = 'lightblue')
}
So there are 12 ways to load the centrifuge… but we can clearly see that all of them are really just rotations of a single pattern. To me the visually most convenient representative of that pattern is the third one in the third row, which has loadings at 12, 1, 4, 5, 7, 8, and 11.
We can play a little game of connect-the-dots to see why this pattern is balanced. Connect 12, 4, and 8 to form an equilateral triangle. Connect 1 and 7; connect 5 and 11. Like this:
Each of those three subsets has its center of mass at the center of the circle, so the whole arrangement does too.
That decomposition turns out to be the key to the general problem of which centrifuge sizes can be loaded with which number of tubes and remain balanced. Stay tuned.
Rainbow cats considered harmful
Partner: you’re stressed! fill out this!
Me: I can’t do that, because rainbow color maps are considered harmful. This is especially true because if we’re going to use one, shouldn’t it be green for “go” and red for “stop”.
Partner: just shut up and color.
(The original version of this meme didn’t have the colors, so you can make whatever palette you want, as in this example.)
Twenty-nine
Evelyn Lamb has a delightful page-a-day calendar. Today (yes, a few days late) I learned that TWENTY NINE is the only word in English that is written with a number of straight-line strokes equal to its value. (This is in a sans serif font; in particular I is one stroke, not three.)
English is surprisingly rich in numbers that have all straight-line strokes. In the Latin alphabet the letters that are all straight lines are A, E, F, H, I, K, L, M, N, T, V, W, X, Y, Z. That leads to the following words that are all straight-lined and appear in numbers, and the number of straight lines that make them up:
– having more strokes than their value: FIVE (10), NINE (11), ELEVEN (19), TWELVE (18), FIFTEEN (20), NINETEEN (23)
– having less strokes than their value: TEN (9), TWENTY (18), FIFTY (12), NINETY (16)
Any word that equals its own value must combine elements from both these lists, and it’s not hard to see that TWENTY NINE is the only one that works. The full list of straight-line numbers in English is: 5, 9, 10, 11, 12, 15, 19, 25, 29, 55, 59, 95, 99. (All the larger numbers include “HUNDRED” or “THOUSAND” or something ending in “-ION”, so we can stop there.)
Evelyn suggests this as part of how to memorize the largest known prime (it’s a Mersenne prime, and she suggests doing it in binary so every bit is 1, so the hard part is remembering where you are).
It’s hard to even find straight-line numbers in other languages, because a lot of the alphabet is missing. They include:
• German ZWEI (2), ZEHN (10), ELF (11)
• Dutch EEN (1), TWEE (2), ZEVEN (7), TIEN (10), ELF (11) (edited 9/3: also ZWAALF (12))
• Norwegian has a lot: EN (1), FEM (5), ATTE (8), NI (9), TI (10), ELLEVE (11), FEMTEN (15), ATTEN (18), NITTEN (19), FEMTI (50), ATTI (80), NITTI (90) (and also 55 = FEMTIFEM, 58, 59, 85, 88, 89, 95, 98, 99)
• As does Danish: EN, FEM, NI, TI, ELLEVE, FEMTEN, ATTEN, NITTEN, with the same meanings as in Norwegian, but then the bigger numbers are formed irregularly.
• Swedish has less: EN (1), FEM (5), ATTA (8), ELVA (11) – the numbers are very similar to Norwegian but the “-teen” ending is “-ton”, not “-ti” like Norwegian.
• Spanish VEINTE (20), MIL (1000), and MIL VEINTE (1020)
• Italian VENTI (20), MILLE (1000), and MILLEVENTI (1020)
• Portuguese VINTE (20), MIL (1000) and MIL E VINTE (1020)
• French MILLE (1000), but not VINGT (20)
ELVA (Swedish for 11) is the only other one I could find that also has the self-referential property, and the Chinese numerals , , if you want to stray from the alphabetic world. (Edited 9/3: also Dutch TIEN = 10, which I inexplicably missed before.)
(This post was edited to add the list of numbers and to clarify that ELVA is not the only straight-line number outside of English, but the only one I could find with this self-referential property.) | 2022-01-18 06:54:46 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43703579902648926, "perplexity": 1652.532134800347}, "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-2022-05/segments/1642320300805.79/warc/CC-MAIN-20220118062411-20220118092411-00220.warc.gz"} |
https://mastodon.technology/@cj/102577563089305266 | $\int_{-\infty}^{+\infty}g(x)e^{-2\pi i f x}dx$
@cj Is it a hint for when you push the next apcore commit? :D
@21stio I wish! But I have a long flight tomorrow so expect some commits landing in a few days. :)
@cj Exciting, I'm looking forward :)
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon! | 2020-02-24 15:07:10 | {"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": 0.32025349140167236, "perplexity": 7429.656496496016}, "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/1581875145960.92/warc/CC-MAIN-20200224132646-20200224162646-00177.warc.gz"} |
https://softwarerecs.stackexchange.com/questions/1601/converting-big-list-of-questions-in-tex-into-single-question-view | # Converting Big List of Questions in TeX into Single Question View
I have questions in this format in tex files
Systemic venous congestion caused by
\begin{enumerate}[(a)]
\item heart problems,
\item thrombosis of the portal vein,
\item la la la lorem
\end{enumerate}a
... 777 questions
Systemic venous congestion caused by
\begin{enumerate}[(a)]
\item heart problems,
\item thrombosis of the portal vein,
\item la la la lorem
\end{enumerate}
I would like to make a simple app that asks me these questions randomly one at a time and finally returns rate. iPad and iPhone apps would be nice - probably HTML version is enough seen can be seen on the browser. Very simple GUI is enough.
Which software would you use for this?
• I doubt that you'll find something that handles everything from the LaTeX source to the prompting and scorekeeping. How much fiddling are you prepared to do to, say, convert that TeX source into a bunch of text or HTML files, that could be input to a flash card app? – Gilles 'SO- stop being evil' Feb 24 '14 at 20:39
• @Gilles The conversion of TeX into HTML should be straightforward. – Léo Léopold Hertz 준영 Feb 24 '14 at 20:42
• @Masi True, but you won't find a tool that does both the conversion AND the question asking. Maybe it is better to focus on the questioning and scorekeeping part, and take care of the conversion to the required format afterwards – Bernhard Feb 25 '14 at 6:42
• I think you need to break this up so that the recommendation here is for a flash-card like program with the options you need and some sort of import, this figure out how to convert your tex data to the input format required. I doubt asking for a flash card program that understands tex is going to get you anywhere. – Caleb Feb 25 '14 at 9:54
% -*- coding-system:utf-8 -*- \documentclass[12pt]{article} \usepackage[utf8]{inputenc} \newenvironment{note}{\paragraph{NOTE:}}{} \newenvironment{field}{\paragraph{field:}}{} \begin{document} \begin{note} \begin{field} Is it true that $$1 + 1 = 0$$? \end{field} \begin{field} Yes and no. The equation \begin{equation*} 1+1 = 0 \end{equation*} holds only in characteristic two. \end{field} \end{note} \end{document} | 2020-08-03 20:49:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7587226033210754, "perplexity": 3185.1630700282235}, "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-2020-34/segments/1596439735833.83/warc/CC-MAIN-20200803195435-20200803225435-00294.warc.gz"} |
http://websrv.cs.umt.edu/isis/index.php?title=Category_1:_Whole_Ice_Sheet&oldid=4402 | Category 1: Whole Ice Sheet
SeaRISE began with commitments from leaders of six whole ice sheet models. A number of models have been added strengthening the multi-model ensemble approach. The original models (along with the lead institutions or modeler) were: 3D
• CCSM (Community Climate System Model; Los Alamos)
• PISM (Parallel Ice Sheet Model; University of Alaska)
• UMISM (University of Maine Ice Sheet Model; University of Maine)
• PSU (Penn State University; Penn State University)
• GLAM (GLimmer with Advanced Mechanics; Los Alamos and University of Bristol)
• SICOPOLIS (SImulation COde for POLythermal Ice Sheets; Hokkaido University)
2D
• Parizek Flowline (Penn State University)
• GLAM (Los Alamos and University of Bristol)
Table 1 presents some specific characteristics for each of these models for comparisons.
Models (and modelers) added and not yet with their characteristics included in Table 1 are:
• ELMER (Hakime Seddik)
• Goddard (Weili Wang)
• GRISLI (Catherine Ritz)
• PISM+ (Maria Martin)
Contents
Model Tests
Testing models against analytic solutions is a valuable means of model verification. For the research effort described in this document, such verification will be bypassed because the results sought from the individual models are the deviations in ice sheet volume over the next 100-200 years from a control run of the same model. It is less essential to have agreement of absolute behaviors among models. Even so, many of the above models have already been verified through model intercomparison studies such as EISMINT (I and II), MISMIP and ISMIP-HOM. Such heritage adds credibility to the results of the experiment set.
A principal advantage sought by using multiple models is the power of ensemble studies and is a well-accepted method of detecting less reliable results. Many of the above models derive from GLIMMER and share some common numerical components, so model-to-model independence is not as large as the sheer number of models might suggest. Nevertheless, there are enough differences between even similarly constructed models to make the ensemble methodology worthwhile as an evaluation criterion. Large excursions of one model’s results from the ensemble mean will help identify model components that must be examined carefully and will figure into the derivation of confidence in the associated ice-sheet response.
Set-Up and Initialization
Surface and bed geometries, ice thickness, precipitation and near-surface air temperature, along with other datasets, are available for both Greenland and Antarctica through the Community Ice Sheet Model (CISM) project. Most of the whole ice sheet models use a similar grid size, so spatial interpolations of these data sets should create only small variations between the geometric initializations of different models.
Initial internal ice temperature fields are missing and are frequently generated through the procedure of “spin-up”. To guide the spinning up of a model, past near-surface air temperature and sea level are available in the Model Initialization section. Often spin-up spans many glacial-interglacial cycles in order to diminish numerical artifacts from initialization and to “set” its internal temperatures in accord with a long history of variable external temperature conditions (i.e., temperature diffusion within the ice column is very slow, and current ice temperatures reflect past climates). Alternatively, assimilation procedures can be used to force the model to match currently observed fields (e.g., velocity, bed and surface topography).
Spin-up is a very computationally demanding process, and it is unlikely that models other than the whole ice sheet models with a shallow-ice-approximation (SIA) balance of stresses will chose to complete the process. Recognizing this, data sets based on a long (150,000-year) spin-up process will be provided as a reference by CISM efforts. These reference spin-ups can then be used as an initial condition for models that must utilize some other spin-up procedure due to differences in stress balance or other model features. Additionally, in regions where interferometrically derived velocity data is not available, such as the upper reaches of the drainage basins investigated by regional models, output from reference spin-ups can be used for kinematic boundary conditions.
One requirement for this effort is for the model to be devoid of non-physical transients in the future behavior of the ice sheet at t0, i.e. the present day, so that control and future climate experiment runs can be made without needing to consider these non-physical transients. It is also important to keep in mind that the primary time horizon of interest in this effort is 100-200 years, with secondary interest extending out to as long as 500 years.
A second requirement for model spin-up is that data provided by CISM efforts are used for whatever spin-up process is utilized. This consists of modern day fields for surface, mean annual accumulation, InSAR surface velocity, and temperature. If lapse rates, sea level records, or ice core data are used as part of a spin-up process, values consistent with those in references provided in the Data and Model Initialization sections should be used. Measurements of velocity over large parts of both ice sheets are available and balance velocities can be used, in some cases, to fill gaps.
Similarly, basal conditions, subglacial hydrology and other internal or boundary fields may have to be generated by individual models, unless models are so similar in their parameterization that it makes sense to specify these for all models. Again, there are reference spin-up results from SIA models to initialize the process.
Our initial target for the prescribed state of balance at t0 is equilibrium, i.e., no net or local rate of volume change, even though this is known to be incorrect. It is a vexing modeling problem to initialize to a spatial field of non-zero elevation changes at t0, even though these changes are becoming well determined from satellite altimetry for most of the ice sheets. Many parameters could contribute to elevation change, some in a non-linear way, creating a highly unconstrained situation. This initial equilibrium condition might be relaxed in specific areas known to be changing rapidly to prevent a blatantly incorrect initial state.
The primary goal in set-up, spin-up and initialization is that each model has a minimal amount of non-physical transients at t0 and that it be a close approximation of the current geometric and dynamic state of either ice sheet. The degree to which it deviates from any other model is of lesser concern than the fact that its own deviations of future climate experiments from its own control run accurately capture predictions of physical changes in ice sheet mass.
Initialization data sets were frozen in October 2009 for the purposes of producing control runs. However, further discussion among SeaRISE participants have made it apparent that there is value in allowing parallel "developmental" data sets that either incorporate new observations and/or that improve model simulations. Developmental data sets may replace some of the original data sets and modelers are free to replace previous control and experiment runs with improved runs.
Control Run
The control run of each model is the reference against which all climate change experiments with that model will be compared. A reasonable choice for a control run is a continuation of the present climate run for 200 to 500 years into the future. All forcing fields such as temperature, precipitation and basal conditions (if these are prescribed) can be held fixed to their t0 values. In cases where the t0 state is not an equilibrium state, the control run will contain a prediction of additional ice mass changes. A “control-run” ice mass changes will be subtracted from ice mass changes resulting from changed-climate experiments to isolate the change that comes from experiment forcing.
Two control runs have been agreed upon:
• "Constant Climate Control (CC)” is, as it sounds, a run beginning at present and running for 200 (or 500) years holding the climate constant to the present climate.
• “4th Assessment Climate Control Run (AR4)” starts with the same present day condition, but the climate is modified according to anomalies from the present climate based on anomalies of the 4th Assessment model from a constant (t0) climate, up to year 2100. Beyond 2100, the year 2100 climate will persist to the end of the run (200 or 500 years).
For guidelines on how to submit your control run, see the Output Format section.
Experiments
Useful and feasible experiments are currently being fleshed out by the SeaRISE group. Agreed upon experiments are described below in sufficient detail for modelers to run the experiments and supply results to Sophie. (Please follow the submission requirements described in the Output Format section.)
The wiki format allows discussion of the experiments if they are either insufficiently clear or not feasible. Feedback is welcomed.
Greenland
Initial Experiment - E1 - Increased Basal Lubrication
The purpose of the initial Greenland experiment is to get a feel for the results of increased flux across the grounding line. Because most of Greenland's outlet glaciers lack an ice shelf large enough to be resolved in whole ice sheet models, the experiment forces an increase in modeled outlet glacier speed by doubling sliding speed everywhere. This condition has subtleties (next) but there are enough "adjustable knobs" on the basal relations in ice sheet models so as to make a version of this condition implementable.
Models vary in how sliding speed is calculated, and it is left up to the modeler to determine how to impose the doubling of sliding speed condition. The sliding speed and the basal shear stress are, generally, model results, not model inputs. Thus the implementation of this sliding speed doubling will likely involve adjustment of a lubrication or friction factor. An example mechanism is to halve the friction coefficient $C$ if sliding follows a linear relation: $\tau_b = C u$.
This is straightforward to understand and implement if the sliding velocity is a pointwise (local) function of basal shear stress, and vice-versa, and the basal shear stress is determined directly by the geometry (e.g. is the basal value of the driving stress). Such is true in the SIA. In the SIA case, therefore, a halving of $C$ for some patch of the ice base becomes a doubling of sliding speed in the same region directly. Dynamically, this doubling is sustained until the geometry (thus driving stress) undergoes significant change.
In models with longitudinal (membrane) stresses, however, sliding speed is not such a local function of ice sheet geometry. In most models we can still make an approximately-local assumption about the nature of sliding, and still replace a doubling of sliding speed condition by a halving of coefficient condition. With such a coefficient change there must be the understanding that the sliding speed will only actually double in the interior of large-ish patches where the basal shear stress was already fairly high. (At the extreme case we could replace $\tau_b = C u$ by $\tau_b = 0$, which multiplies $C$ by zero instead of 1/2, giving no basal resistance as in an ice shelf. But we would not expect the sliding velocity to be infinite because, as in ice shelves, membrane stresses connect to distant ice with high basal resistance.) If $C$ goes down by half in small patches, or even to zero, the velocity might not increase much at all because the ice flow is held steady by higher-basal-resistance neighboring columns.
A further condition is to require the basal ice to be at the pressure-melting point. This is likely already a condition for a non-zero sliding speed in most (all?) models. Thermodynamic feedbacks (i.e. not just geometry changes) may lead to the sliding speed initially doubling in response to a sliding coefficient change and then dropping again as basal strength increases; witness the century-scale response of Kamb Ice Stream (Kamb = C). It is preferred (by Bindschadler) that this doubling of sliding speed condition be dynamic, possibly spreading to formerly bed-frozen areas as the ice sheet evolves. Not all models may be able to respond so dynamically.
Antarctica
Initial Experiment - E1 - Increased Ice Shelf Melting
The initial experiment in Antarctica deals with increasing flux across the grounding line by manipulating the ice shelves. In experiments E1a, E1b, and E1c, a uniform sub-ice-shelf melting rate of 2, 20 and 200 meters per year (of ice equivalent) is applied, respectively. Ice shelves are treated differently in different models. If a whole ice sheet model lacks ice shelves, stopping at the grounding line, the above melt rates can be imposed at the grounding line. It is not certain that this prescription works for all models.
Discussion of Future Climate Experiments
The experiments described below are NOTIONAL ONLY and are ONLY FOR DISCUSSION . They are intended to give examples of how initial quantitative assessments of how large the ice sheet contribution to sea level could be generated. The experiments are discussed separately for Greenland and Antarctica.
Greenland
These experiments are of two types: those that addresses the role of surface meltwater on subglacial lubrication, and those that addresses the role of imposed changes at the margins of major outlet glaciers.
When the future climate experiment requires the forcings resulting from an IPCC scenario, these fields will be produced from the ensemble (ideally weighted in some manner) of results from all the GCMs used in the IPCC-Fourth Assessment Report.
Surface Melting
The correlation of surface meltwater production and ice flow has led to inferences that this meltwater penetrates to the bed and lubricates the ice-bed interface, reducing resistive stresses and/or decreasing bed normal stresses, much like mountain glaciers. The quantitative impact on overall ice dynamics is an active area of research, so the possible contribution of this effect on ice sheet mass loss in warmer future climates deserves careful examination. Three experiments are suggested aimed at examining this sensitivity. Details of their implementation have yet to be agreed upon.
No lubricative effect of meltwater: The most extreme IPCC climate scenario (A1F1: temperature rise of 4.0 °C with a likely range of 2.4 to 6.4 °C) could be used to force the ice sheets for 100 years into the future. Beyond 100 years, the final climate state wouldbe sustained by repeating the final year. The surface meltwater that is produced is deemed to have no impact on ice flow and is deposited directly into the ocean. Dynamic changes in ice flow will result primarily from changes in ice-sheet geometry driven by surface mass balance changes. Models that have actively evolving margins, especially at calving ice fronts will be encouraged to run this experiment both with the active evolution components “on” and, separately, with them “off”.
Meltwater penetrates vertically and lubricates bed: This experiment would impose the same forcing as above, but now the surface water is prescribed to reach the bed causing subglacial lubrication. The water is assumed to penetrate vertically, accessing the bed immediately below the production area. This water then leaves the ice sheet without further influence on ice flow. Previously frozen bed areas that become wetted could support basal sliding, creating changes in the ice flow and causing changes in ice-sheet geometry.
Horizontal propagation of subglacial water: This experiment also has the same climatic forcing as above, but horizontal water transport is included. For those models that include a subglacial water balance component, the water can move within the ice-bed interface, pool in lakes, influence effective pressure and basal lubrication, and move to other areas where it extends its influence on ice flow before exiting the subglacial system. Treatment of the water’s effect on lubrication will depend on the model. Here the challenge is a hydrological model that relates to surface meltwater input to basal sliding and seasonal acceleration.
Marginal Changes
Many deep outlet glaciers at Greenland’s perimeter are experiencing dramatic acceleration, increasing the present rate of ice loss. These changes appear to resemble drastic retreat of tidewater glaciers, a phenomenon known to lead to sustained and rapid retreat of calving glacier termini, and both flow acceleration and ice thinning, each propagating upstream.
How to impose these margin-focused changes on whole ice sheet models is problematic. Grid resolution is often inadequate to capture the spatial details of narrow outlet glaciers, calving relationships and flow transitions at the grounding line are often ad hoc. Specifying discharge flux could be considered a possible approach, but so doing predetermines the sea level contribution, the primary predictive objective of this model exercise. The approach described below avoids this unwanted interdependency while making a simple prescription of marginal changes.
Single retreating glacier: The first case could be to force a retreat of the Jakobshavns Isbrae in western Greenland. This glacier has accelerated and retreated markedly in the past decade. It is wide enough that the main outlet can be resolved in most whole ice sheet models. The retreat will be prescribed by a time series of positions of the terminus (or grounding line) over the next 100 years. The retreat rate will continue the 10 km retreat in 5 years observed in the early 21st century. As the retreat proceeds beyond the present single ice-filled fjord, a more complex set of terminus positions will need to be specified. A modified experiment would be to half the retreat rate.
Due to grid size limitations, only the largest outlet glaciers can be resolved in whole ice sheet models. This is an example of where the regional models, with their higher spatial resolution and more complete physics, can provide a more realistic spatial and temporal prescription of margin changes. This is discussed more completely later. Half-dozen retreating glaciers Six deep outlet glaciers around Greenland will be selected and a Jakobshavns-level retreat imposed for each, also by a prescribed time series of terminus (or grounding-line) positions. A sub-case of halving the retreat rate could be an auxiliary experiment.
All tidewater glaciers: The most extreme case would be the imposing of drastic retreat on all the major tidewater glaciers around Greenland. Again, the half-speed retreat is a viable sub-case.
Calving Flux: Another method to force retreat and one that might be more amenable to the parameterizations of many models, would be to specify a calving flux in a variety of ways. A simple approach is to make the calving flux some fraction of the discharge flux. A fraction greater than unity will force retreat. The fraction could be prescribed as a time series to force accelerating retreat. Alternatively, the fraction could be tied to a physical parameter, such as water depth or height of ice thickness above flotation; and approach that expresses properties of earlier calving laws. Another alternative is to follow a recent paper on calving rate parameterization and make it dependent on the longitudinal stretching rate near the terminus.
Antarctica
Near-term climate change is impacting Antarctica within the Antarctic Peninsula and at the floating ice shelves through which most ice exits the ice sheet. The rugged topography and the glaciological complexity of the Antarctic Peninsula are beyond the abilities of most whole ice sheets to simulate. Based on the dramatic response of the feeding glaciers to the sudden removal (disintegration) of fringing ice shelves, a limiting scenario for the Peninsula is that all of its grounded ice will be removed this century through the eventual disintegration of its ice shelves. This will contribute a maximum of 34 cm to global sea level. The temporal schedule of this addition is beyond the capability of models to determine, at present.
Surface melting
Although there has not being strong evidence of the surface melting in Antarctica, except within the Antarctic Peninsula, there is no reason to assume that it would not play a role in a warming climate. These experiments would be similar to the surface melting experiments described for Greenland. If the initial surface melting experiment produced negligible surface water, the remaining surface melting experiments, where meltwater water lubricates the bed, might be abandoned.
The observations of grounded ice response to rapid ice shelf removal confirm that the ice shelf provides a significant longitudinal resistance to ice discharging across the grounding line. This interaction offers a convenient method to simulate the impact of ice shelf removal on Antarctic ice mass loss.
Marginal Changes
Remove all ice shelves currently thinning rapidly: The Pine Island Glacier ice shelf in West Antarctica and the Cook Ice Shelf in East Antarctica are observed to be thinning rapidly. This experiment would remove these ice shelves suddenly (either instantaneously or very rapidly) resulting in the hydrostatic equilibrium boundary condition being applied at the grounding line. Depending on the rate and spatial pattern of removal, there would be dramatic changes in the stress state at the margin that will result in large and rapid changes in ice flow and shape. Again, regional models could assist in providing more realism in the temporal pattern of change.
Remove all major Antarctic ice shelves: There are approximately 25 major outlet glaciers and ice streams in Antarctica that discharge more than half of the ice sheet’s annual flux to the ocean. Sudden removal, again guided by regional models, would explore a more radical scenario driven by the loss of multiple buttressing ice shelves.
Other
Jed Brown pointed the following out at the Summer 2009 CCSM/seaRISE meeting and it seemed too sensible an idea to leave off the wiki. At some level, all models solve the continuity equation
$\frac{\partial H}{\partial t} = -\nabla \cdot \left(\mathbf{u} H\right) + a$
to determine the thickness evolution. The details of how this equation is solved vary, but it is a starting point for reasoning about a class of experiments that appear to be straight forward to implement.
In SeaRISE experiments, arbitrarily increasing the flux at the grounding line is not useful because a primary goal of SeaRISE is to quantify the ice mass loss (i.e., the increase in grounding line flux) in a deterministic way using a dynamic ice sheet model forced by a specified environmental forcing condition. In SIA models, it is likely that the mass loss is attributed to either increased flow rates caused by a decrease in basal, lateral or longitudinal stresses) or increased melt. In SSA/SIA hybrid models, the additional flux of ice onto the shelf may be more problematic. Dave Pollard pointed out that it is probably better to impose the change by increasing the basal melt rate.
The difficulty arises in reconciling the various participating models. SIA models provide SeaRISE with a baseline, and are an important part of the exercise. However, prescription of experiments that are comparable across participating models can be very challenging. | 2020-10-21 14:04:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 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": 0.4517492353916168, "perplexity": 2071.9036301313713}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107876500.43/warc/CC-MAIN-20201021122208-20201021152208-00617.warc.gz"} |
https://physics.stackexchange.com/questions/412122/is-ether-wind-theory-completely-unverifiable | Is Ether wind theory completely unverifiable?
I recently learned that Ether wind theory and Special theory of Relativity predict same experimental outcomes and that there was no way to prove or disprove the existence of ether. And that Einstein gave a theory where ether was not necessary but still retained all the experimental outcomes.
I still can't get around this as in STR if I and you are in relative motion, I'll shrink for you and you'll shrink for me, but in Ether theory, if I am at rest w.r.t ether and you are moving, then you will shrink for me, but I will expand for you. So one of us will know for sure that which theory is correct. If this is true then how can Ether theory be identical with SRT, if this is false, how so?
Note: When I say Ether theory, I am considering Ether theory with Lorentz-FitzGerald space-time contraction, which is caused by "Ether wind"
• Thanks for responding, I tried no question tries to address this particular concept. – VARUN.N RAO Jun 16 '18 at 17:37
• My doubt is Ether wind theory and STR are said to be identical when it comes to predicting outcomes in an experiment, I just tried disprove it by giving a counter example – VARUN.N RAO Jun 16 '18 at 17:38
• The counter example being length contraction is mutual in STR unlike in Ethere Wind Theory, where one sees contraction and other sees expansion. – VARUN.N RAO Jun 16 '18 at 17:39
• I agree with Countto10 - it would be helpful if you identified the source of the statements in your 1st paragraph, if possible with a link to an accessible text. Also that you should look at Related questions to see if they solve your problem. – sammy gerbil Jun 16 '18 at 17:40
• Possible duplicate of How did we disprove aether wind? or About the Ether Theory acceptance – sammy gerbil Jun 16 '18 at 17:45
In the Lorentz Ether theory the same thing takes place, we see the other guy as being shorter and slower than you are, if all observers synchronize clocks in their reference frames by Einstein signalling method, a.k.a Einstein synchronization.
https://en.wikipedia.org/wiki/Einstein_synchronisation
That means, that moving observer is actually shorter and slower, than one that is at rest in the Ether. However, if moving observer synchronizes clocks in his frame by Einstein technique, he would measure, that clock a rest is ticking $\gamma$ times slower and measuring rod at rest is $\gamma$ times shorter than his own.
“Full simulation of SR by classical mechanics”. All effects and paradoxes of the SR are simulated in aquatic medium (universal frame), including time dilation, Lorentz transformation, relativistic velocity addition, twin paradox, mathematical formalism e.t.c. https://arxiv.org/abs/1201.1828
Chapter 3.5.5 - A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble https://sites.google.com/a/umn.edu/micheljanssen/home/papers
That makes these theories empirically equivalent, as the article explains: https://en.wikipedia.org/wiki/Lorentz_ether_theory
However, measuring technique by means of Einstein – synchronized clocks is not the only way to measure time dilation. Actual experiments have been conducted, and in these experiments researchers measure time dilation by means of the Transverse Doppler Effect.
https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment
Imagine that two observers rotate around a source a radiation in a circumference of very large diameter, let’s say many light years in diameter, so they are almost inertial. If they send a beam of light through the center from one point of the circumference to the opposite one, they would measure no time dilation, i.e. any Doppler shift.
http://iopscience.iop.org/article/10.1088/0370-1328/77/2/318/meta
Neither rotating ones, nor purely inertial ones that momentarily coincide with them would see any Doppler shift, since their clocks dilate at the same magnitude. | 2021-07-31 06:25:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5673239231109619, "perplexity": 546.3280257207277}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154053.17/warc/CC-MAIN-20210731043043-20210731073043-00127.warc.gz"} |
http://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-11-oscillations-and-waves-questions-page-320/3 | ## Physics: Principles with Applications (7th Edition)
If the other two variables are kept constant, one could double the amplitude A, quadruple the spring constant k, or reduce the mass to $\frac{1}{4}$ of its old value.
The maximum speed is given by equation 11-5a: $v_{max}=A\sqrt{\frac{k}{m}}$. This equation suggests various ways to double the maximum speed. | 2017-02-24 10:41:12 | {"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": 0.9179518818855286, "perplexity": 789.3136578554715}, "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-2017-09/segments/1487501171463.89/warc/CC-MAIN-20170219104611-00527-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://web2.0calc.com/questions/calculus-2-help_1 | +0
# Calculus 2 help
0
169
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+1904
Consider the curve defined by $$y = x + \frac{{x}^{2}}{2}$$ over the interval [0, 3].
a) Write (and simplify) a Riemann sum with n subintervals to approximate the area of the region between the curve and the x-asix over this interval. You may use either Right Hand Rule or Left Hand Rule. Note: your final expression should only involve the unknown n.
b) Take the linit of your expression as $$n \rightarrow ∞$$. Give a conclusion that clearly states what this limit represents as a definite integral.
Anyone who know how to answer this question and can give step by step instructions, I would really appreciate it. Thanks.
Feb 4, 2019
edited by gibsonj338 Feb 4, 2019
#1
+5788
+4
$$\text{using }n \text{ subintervals we get}\\ x_k = \dfrac{(3-0)k}{n} = \dfrac{3k}{n}\\ \Delta = x_1- x_0 = \dfrac 3 n$$
$$\text{the left hand Riemann sum is} \\ \dfrac 3 n\sum \limits_{k=0}^{n-1}~\dfrac{3k}{n} + \dfrac{9k^2}{2n^2} = \dfrac{9}{4 n^2}-\dfrac{45}{4 n}+9\\ \text{you need to show this last equality of course}$$
$$\text{Pretty clearly}\\ \lim \limits_{n\to \infty}~\dfrac{9}{4n^2}-\dfrac{45}{4n} + 9 = 9\\ \text{which is the value of the definite integral of }y \text{ over }[0,3]$$
.
Feb 5, 2019 | 2019-09-20 17:02:25 | {"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": 0.9835193753242493, "perplexity": 1608.270287403854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574050.69/warc/CC-MAIN-20190920155311-20190920181311-00328.warc.gz"} |
https://search.datacite.org/works/10.5281/zenodo.5748321 | ### Time-Dependent and Independent Quantum Entropy
Francesco R. Ruggeri
Entropy in quantum mechanics is often based on von Neuman’s prescription (1) which yields zero for a pure eigenstate. Some alternative approaches also yield this result (2). In other words, it seems this type of entropy deals with transitions between different energy eigenstates and excludes the intrinsic entropy of a pure energy eigenstate. There is, however, literature (3) dealing with both spatial and momentum entropy of a single energy eigenstate. In particular, Shannon’s entropy equation...
This data repository is not currently reporting usage information. For information on how your repository can submit usage information, please see our documentation. | 2022-05-22 17:14:30 | {"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": 0.8421563506126404, "perplexity": 1173.7549510296894}, "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-2022-21/segments/1652662545875.39/warc/CC-MAIN-20220522160113-20220522190113-00238.warc.gz"} |
https://tantalum.academickids.com/encyclopedia/index.php/Cartesian_product | # Cartesian product
In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y.
X Y = { (x, y) | xX and yY }
The Cartesian product is named after Ren Descartes whose formulation of analytic geometry gave rise to this concept.
For example, if set X is the 13-element set { A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2 } and set Y is the 4-element set {♠, ♥, ♦, ♣}, then the Cartesian product of those two sets is the 52-element set { (A, ♠), (K, ♠), ..., (2, ♠), (A, ♥), ..., (3, ♣), (2, ♣) }.
The Cartesian square of a set X is the Cartesian product X X. An example is the 2-dimensional plane R × R where R is the set of real numbers - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The binary Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn:
X1 ... Xn = { (x1, ...,xn) | x1 in X1 and ... and xn in Xn }
Indeed, it can be identified to (X1 ... Xn-1) Xn. It is a set of n-tuples.
An example of this is the Euclidean 3-space R × R × R, with R again the set of real numbers.
As an aid to its calculation, a table can be drawn up, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table by choosing the element of the set from the row and the column.
The Cartesian product can be introduced by the familiar calendar format:
• weeks as rows;
• weekdays as columns;
• a given day as a cell.
## Infinite products
The above definition is usually all that's needed for the most common mathematical applications. However, it is possible to define the Cartesian product over an arbitrary (possibly infinite) collection of sets. If I is any index set, and {X i | i in I} is a collection of sets indexed by I, then we define
[itex]\prod_{i \in I} X_i = \{ f : I \to \bigcup_{i \in I} X_i\ |\ (\forall i)(f(i) \in X_i)\}[itex]
i.e. the set of all functions defined on the index set such that the value of the function at a particular index i is an element of Xi. This coincides with the finite case, when I is a finite set, say {1, 2, ..., n}; any such function f defined on I is simply identified with the n-tuple (f(1), f(2), ..., f(n)). In the infinite case this can be thought of as an infinite-tuple. Conversely, an n-tuple can be viewed as a function on {1, 2, ..., n} that simply takes its value at i to be the ith position of the tuple.
One particular and familiar infinite case is when the index set is [itex]\mathbb N[itex], the natural numbers: this is just the set of all infinite sequences with the ith term in its corresponding set Xi. Once again, trusty old [itex]\mathbb R[itex] provides an example of this:
[itex]\prod_{n = 1}^\infty \mathbb R =\mathbb{R}^\omega= \mathbb R \times \mathbb R \times \ldots[itex]
is the collection of infinite sequences of real numbers, and it is easily visualized as a vector or tuple with an infinite number of components. Another special case (the above example also satisfies this) is when all the factors Xi involved in the product are the same, being like "cartesian exponentiation." Then the big union in the definition is just the set itself, and the other condition is trivially satisfied, so this is just the set of all functions from I to X.
Otherwise, the infinite cartesian product is less intuitive; though valuable in its applications to higher mathematics. In fact, asserting even whether or not the cartesian product is the empty set is one of the formulations of the axiom of choice.
• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy | 2021-10-18 04:33:12 | {"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": 0.9673680663108826, "perplexity": 437.5611775856028}, "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/1634323585196.73/warc/CC-MAIN-20211018031901-20211018061901-00406.warc.gz"} |
https://studydaddy.com/question/mth-157-week-1-dqs | # MTH 157 Week 1 DQs
This archive file of MTH 157 Week 1 Discussion Questions shows the solutions to the following problems:
DQ 1: Under what conditions might it be appropriate to use the mean? Median? Mode? When the median and mean are substantially different, what does that tell you about the data? What information does the range and standard deviation provide about the data?DQ 2: Choose two of the following types of displays: dot plots, stem and leaf plots, histographs, pictographs, line plots, or circle graph. In which situations is it appropriate to display data using each type you selected? When is it inappropriate to use the data displays you selected?
• Drwit
1 orders completed
$5.19 ANSWER Tutor has posted answer for$5.19. See answer's preview
*** 157 **** 1 *** | 2017-03-27 16:33:58 | {"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": 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": 0.27394238114356995, "perplexity": 1972.3749022351221}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189490.1/warc/CC-MAIN-20170322212949-00621-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.wyzant.com/resources/lessons/math/algebra/quadratic_formula | # Solve By Using the Quadratic Equation Lessons
The quadratic formula gives us an alternative to Completing the Square when we cannot factor an equation. People often find the Quadratic Formula method easier and more convenient because it does not require many operations on the equation being solved.
For solving an equation in the variable x, the Quadratic Formula is:
To find the solutions to an equation, we simply need to identify what a, b, and c are, then substitute them into this formula, and simplify.
## First Example (Two Solutions)
We begin applying the Quadratic Formula by putting the equation in the following form:
Where a, b, and c are constants
This means that each term in the equation must be on the left side, just like when we are factoring or Completing the Square. So we subtract
from each side.
Now for consistency, we will rearrange the terms so that they are in the same order: The x2 term first, the x term second, and the constant term last.
Now by comparing our equation with “ax2 + bx + c = 0″, we can see that a must equal 1, b must equal 1, and
c must equal .
Now that the values of a, b, and c have been determined, we may return to the quadratic formula and use substitution. (Remember to use parentheses when substituting to avoid problems with negative signs.)
We must now simplify this equation keeping the Order of Operations in mind. We begin by simplifying (1)2.
Next, we simplify multiplication. We see that 11 is equal to 15:
Now 1 and 15 are added, resulting in 16.
The square root of 16 is 4.
You may recall that the methods of solving by factoring and solving by completing the square required you to split each problem into multiple subproblems to obtain multiple solutions. Since we are again looking for more than one solution, we must split
this problem in two.
So far, we have been carrying the ± sign through the problem. Now, we will create two problems, one with a plus sign, and one with a minus sign.
and
Simplifying the first subproblem gives
Simplifying the second subproblem gives
We can now combine these two solutions into the solution to the original example problem:
## Second Example: One Repeated Solution
Examine the following problem.
We are using decimal numbers in place of fractions here, but this is a matter of choice. Using decimal numbers will not require us to use the Quadratic Formula any differently.
In this case, the equation is already in the correct format. Using “ax2 + bx + c = 0,” we must now determine a,
b, and c.
Now we can substitute into the Quadratic Formula:
Now we begin simplifying by replacing (22)2 with 484.
Simplifying multiplication, we see that -4(10)(12.1) is equal to -484, and 2(10) is equal to 20.
Notice that we cannot create two different subproblems because +0 and -0 are the same value. Therefore, we simply drop the ±0 from the problem.
Without the two subproblems, we will only have one solution.
## Imaginary and Complex Solutions
Consider the following problem.
We can identify that
Now, substituting this into the Quadratic Formula:
Simplifying the expression under the radical gives:
Notice that there is a negative number under the square root symbol. If you are familiar with imaginary numbers, you know that the square root of a negative number is an imaginary number, which will cause the solutions to this example to also be imaginary
or complex numbers.
In this lesson, we chose to expose you to this situation, but not to provide details on imaginary and complex solutions. We will add a lesson on this topic after we have introduced imaginary and complex numbers.
A more general form of algorithm for finding roots of quadratic equations by completing squares leads to the derivation of what in known as the Quadratic Formula. This is a general formula which can be used to solve for the roots of any quadratic equation.
then the roots of the equation can be found by completing the square as below:
This can be further simplified as follows
Putting everything under the
under one denominator results in
The above equation is known as the Quadratic Formula.
From this derivation, we can generalize a few equalities based on the formula.
For all real numbers b and c,
For all real numbers b and c,
For all real numbers a, b, and c where a does not equal 0,
## The Discriminant of the Quadratic Formula
The part of the quadratic formula under the radical sign is referred to as the discriminant. This is because this expression
is what determines if the quadratic equation whose roots we’re trying to find has real roots, imaginary (complex) roots, or has the same root repeated.
is important because this expression is under the square root sign. Remember that the square root of a number greater than zero (a positive number) is a real number, the square root of zero is zero and the
square root of a number less than zero (a negative number) is an imaginary or complex number. Thus the value of
says a lot about the nature of the roots of the quadratic equation.
1. If
i.e.
This quadratic equation is said to have one repeated root. For example:
looking at only
The above would indicate that the equation has one repeated root, and we already saw that it does indeed have one repeated root.
The roots of the equation are given by x = {-2,-2} which is the same root repeated.
2. If
then
is greater than zero and the quadratic equation whose roots we’re finding is said to have real roots. For example, if asked to find the roots of the given quadratic equation
looking at only
and 1 is greater than zero, we can conclude that the quadratic equation has real roots, which is proved by finding the roots of the equation using the quadratic formula.
and
Therefore the roots of the equation are given by x={-2,-1} which are both real numbers.
3. If
is less than zero and the quadratic equation whose roots we’re finding is said to have complex or imaginary roots. A
complex or imaginary number
is denoted by “i“, e.g. 4i is an imaginary number 4.
, look at only
to find the roots.
which is a negative indicating that the roots of the quadratic equation are imaginary. Knowing this, the roots can be found as follows:
is a negative number, we can find the roots by making the following adjustment to the quadratic formula:
which results in
which gives the roots
## Solving Using the Quadratic Formula Resources
Practice Problems / WorksheetPractice completing the square with 20 equations and solutions.
## Tutoring
Looking for someone to help you with algebra? At Wyzant, connect with algebra tutors and math tutors nearby. Prefer to meet online? Find online algebra tutors or online math tutors in a couple of clicks.
Scroll to Top | 2023-02-04 19:44:14 | {"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": 0.8386218547821045, "perplexity": 334.0960980261029}, "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/1674764500151.93/warc/CC-MAIN-20230204173912-20230204203912-00277.warc.gz"} |
http://hdrmx.medsbio.org/manual/build/html/classes/applications/NXmx.html | # 3.3.2.10. NXmx¶
Status:
application definition, extends NXobject
Description:
functional application definition for macromolecular crystallography
Symbols:
These symbols will be used below to coordinate datasets with the same shape. Most MX x-ray detectors will produce two-dimensional images. Some will produce three-dimensional images, using one of the indices to select a detector element.
dataRank: rank of the data field
np: number of scan points
i: number of detector pixels in the slowest direction
j: number of detector pixels in the second slowest direction
k: number of detector pixels in the third slowest direction
Groups cited:
NXattenuator, NXbeam, NXcollection, NXdata, NXdetector_group, NXdetector_module, NXdetector, NXentry, NXgeometry, NXinstrument, NXnote, NXsample, NXsource, NXtransformations
Structure:
(entry): (required) NXentry
title: (optional) NX_CHAR
start_time: (required) NX_DATE_TIME
ISO 8601 time/date of the first data point collected in UTC, using the Z suffix to avoid confusion with local time. Note that the time zone of the beamline should be provided in NXentry/NXinstrument/time_zone.
end_time: (optional) NX_DATE_TIME
ISO 8601 time/date of the last data point collected in UTC, using the Z suffix to avoid confusion with local time. Note that the time zone of the beamline should be provided in NXentry/NXinstrument/time_zone. This field should only be filled when the value is accurately observed. If the data collection aborts or otherwise prevents accurate recording of the end_time, this field should be omitted.
end_time_estimated: (required) NX_DATE_TIME
ISO 8601 time/date of the last data point collected in UTC, using the Z suffix to avoid confusion with local time. Note that the time zone of the beamline should be provided in NXentry/NXinstrument/time_zone. This field may be filled with a value estimated before an observed value is avilable.
definition: (required) NX_CHAR
NeXus NXDL schema to which this file conforms
Obligatory value: NXmx
(data): (required) NXdata
data[np, i, j, k]: (recommended) NX_NUMBER
For a dimension-2 detector, the rank of the data array will be 3. For a dimension-3 detector, the rank of the data array will be 4. This allows for the introduction of the frame number as the first index.
(sample): (required) NXsample
name: (required) NX_CHAR
Descriptive name of sample
depends_on: (required) NX_CHAR
This is a requirement to describe for any scan experiment.
The axis on which the sample position depends may be stored anywhere, but is normally stored in the NXtransformations group within the NXsample group.
If there is no goniometer, e.g. with a jet, depends_on should be set to ”.”
temperature: (optional) NX_CHAR {units=NX_TEMPERATURE}
(transformations): (required) NXtransformations
This is the recommended location for sample goniometer and other related axes.
This is a requirement to describe for any scan experiment. The reason it is optional is mainly to accommodate XFEL single shot exposures.
Use of the depends_on field and the NXtransformations group is strongly recommended. As noted above this should be an absolute requirement to have for any scan experiment.
The reason it is optional is mainly to accommodate XFEL single shot exposures.
(instrument): (required) NXinstrument
name: (required) NX_CHAR
Name of instrument
@short_name: (required) NX_CHAR
short name for instrument, perhaps the acronym
time_zone: (recommended) NX_DATE_TIME
ISO 8601 time_zone offset from UTC
(attenuator): (optional) NXattenuator
attenuator_transmission: (optional) NX_NUMBER {units=NX_UNITLESS}
(detector_group): (recommended) NXdetector_group
Optional logical grouping of detector elements.
Each detector element is represented as an NXdetector group with its own detector data array. Each detector data array may be further decomposed into array sections by use of NXdetector_module groups. The names are given in the group names field.
The groups are defined hierarchically, with names given in the group_names field, unique identifiing indices given in the field group_index, and the level in the hierarchy given in the group_parent field. For example if an x-ray detector, DET, consists of four elements in a rectangular array:
DTL DTR
DLL DLR
We could have:
group_names: ["DET", "DTL", "DTR", "DLL", "DLR"]
group_index: [1, 2, 3, 4, 5]
group_parent: [-1, 1, 1, 1, 1]
group_names: (required) NX_CHAR
An array of the names of the detector elements or hierarchical groupings of detector elements.
Specified in the base classes as comma separated list of names, but new code should use an array of names as quoted strings.
group_index[i]: (required) NX_INT
An array of unique indices for detector elements or groupings of detector elements.
Each element is a unique ID for the corresponding group named in the field group_names. The IDs are positive integers starting with 1.
group_parent[group_index]: (required) NX_INT
An array of the hierarchical levels of the parents of detector elements or groupings of detector elements.
A top-level element or grouping has parent level -1
(detector): (required) NXdetector
Normally the detector group will have the name detector. However, in the case of multiple detector elements, each element needs a uniquely named NXdetector group.
depends_on: (required) NX_CHAR
NeXus path to the detector positioner axis that most directly supports the detector.
data[np, i, j, k]: (recommended) NX_NUMBER
For a dimension-2 detector, the rank of the data array will be 3. For a dimension-3 detector, the rank of the data array will be 4. This allows for the introduction of the frame number as the first index.
description: (recommended) NX_CHAR
name/manufacturer/model/etc. information
time_per_channel: (optional) NX_CHAR {units=NX_TIME}
todo: define more clearly
distance: (recommended) NX_FLOAT {units=NX_LENGTH}
Distance from the sample to the beam center. Normally this value is for guidance only, the proper geometry can be found following the depends_on axis chain, But in appropriate cases where the dectector distance to the sample is observable independent of the axis chain, that may take precedence over the axis chain calculation.
distance_derived: (recommended) NX_BOOLEAN {units=NX_LENGTH}
Boolean to indicate if the distance is a derived, rather than a primary observation. If distance_derived true or is not specified, the distance is assumed to be derived from delector axis specifications.
count_time: (recommended) NX_NUMBER {units=NX_TIME}
Elapsed actual counting time
beam_center_derived: (optional) NX_BOOLEAN {units=NX_LENGTH}
Boolean to indicate if the distance is a derived, rather than a primary observation. If true or not provided, that value of beam_center_derived is assumed to be true
beam_center_x: (recommended) NX_FLOAT {units=NX_LENGTH}
This is the x position where the direct beam would hit the detector. This is a length and can be outside of the actual detector. The length can be in physical units or pixels as documented by the units attribute. Normally, this should be derived from the axis chain, but the direct specification may take precendence if it is not a derived quantity.
beam_center_y: (recommended) NX_FLOAT {units=NX_LENGTH}
This is the y position where the direct beam would hit the detector. This is a length and can be outside of the actual detector. The length can be in physical units or pixels as documented by the units attribute. Normally, this should be derived from the axis chain, but the direct specification may take precendence if it is not a derived quantity.
angular_calibration_applied: (optional) NX_BOOLEAN
True when the angular calibration has been applied in the electronics, false otherwise.
angular_calibration[i, j, k]: (optional) NX_FLOAT
Angular calibration data.
flatfield_applied: (optional) NX_BOOLEAN
True when the flat field correction has been applied in the electronics, false otherwise.
flatfield[i, j, k]: (optional) NX_FLOAT
Flat field correction data. If provided, it is recommended that is be compressed
flatfield_error[i, j, k]: (optional) NX_FLOAT
Errors of the flat field correction data. If provided, it is recommended that it be compressed
True when the pixel mask correction has been applied in the electronics, false otherwise.
The 32-bit pixel mask for the detector. Can be either one mask for the whole dataset (i.e. an array with indices i, j) or each frame can have its own mask (in which case it would be an array with indices np, i, j). Contains a bit field for each pixel to signal dead, blind or high or otherwise unwanted or undesirable pixels. They have the following meaning:
• bit 0: gap (pixel with no sensor)
• bit 2: under responding
• bit 3: over responding
• bit 4: noisy
• bit 5: -undefined-
• bit 6: pixel is part of a cluster of problematic pixels (bit set in addition to others)
• bit 7: -undefined-
• bit 8: user defined mask (e.g. around beamstop)
• bits 9-30: -undefined-
• bit 31: virtual pixel (corner pixel with interpolated value)
Normal data analysis software would not take pixels into account when a bit in (mask & 0x0000FFFF) is set. Tag bit in the upper two bytes would indicate special pixel properties that normally would not be a sole reason to reject the intensity value (unless lower bits are set.
If the full bit depths is not required, providing a mask with fewer bits is permissible.
If provided, it is recommended that it be compressed
countrate_correction_applied: (optional) NX_BOOLEAN
True when a count-rate correction has already been applied in the data recorded here, false otherwise.
How many bits the electronics record per pixel.
Time it takes to read the detector (typically milliseconds). This is important to know for time resolved experiments.
frame_time: (optional) NX_FLOAT {units=NX_TIME}
This is time for each frame. This is exposure_time + readout time.
gain_setting: (optional) NX_CHAR
The gain setting of the detector. This influences background.
saturation_value: (optional) NX_INT
The value at which the detector goes into saturation. Data above this value is known to be invalid.
sensor_material: (required) NX_CHAR
At times, radiation is not directly sensed by the detector. Rather, the detector might sense the output from some converter like a scintillator. This is the name of this converter material.
sensor_thickness: (required) NX_FLOAT {units=NX_LENGTH}
At times, radiation is not directly sensed by the detector. Rather, the detector might sense the output from some converter like a scintillator. This is the thickness of this converter material.
threshold_energy: (optional) NX_FLOAT {units=NX_ENERGY}
Single photon counter detectors can be adjusted for a certain energy range in which they work optimally. This is the energy setting for this.
type: (optional) NX_CHAR
Description of type such as scintillator, ccd, pixel, image plate, CMOS, ...
(transformations): (required) NXtransformations
Location for axes (transformations) to do with the detector
(collection): (optional) NXcollection
Suggested container for detailed non-standard detector information like corrections applied automatically or performance settings.
(detector_module): (required) NXdetector_module
Many detectors consist of multiple smaller modules that are operated in sync and store their data in a common dataset. To allow consistent parsing of the experimental geometry, this application definiton requires all detectors to define a detector module, even if there is only one.
This group specifies the hyperslab of data in the data array associated with the detector that contains the data for this module. If the module is associated with a full data array, rather than with a hyperslab within a larger array, then a single module should be defined, spanning the entire array.
data_origin: (required) NX_INT
A dimension-2 or dimension-3 field which gives the indices of the origin of the hyperslab of data for this module in the main area detector image in the parent NXdetector module.
The data_origin is 0-based.
The frame number dimension (np) is omitted. Thus the data_origin field for a dimension-2 dataset with indices (np, i, j) will be an array with indices (i, j), and for a dimension-3 dataset with indices (np, i, j, k) will be an array with indices (i, j, k).
The order of indices (i, j or i, j, k) is slow to fast.
data_size: (required) NX_INT
Two or three values for the size of the module in pixels in each direction. Dimensionality and order of indices is the same as for data_origin.
data_stride: (optional) NX_INT
Two or three values for the stride of the module in pixels in each direction. By default the stride is [1,1] or [1,1,1], and this is the most likely case. This optional field is included for completeness.
module_offset: (optional) NX_NUMBER {units=NX_LENGTH}
Offset of the module in regards to the origin of the detector in an arbitrary direction.
@transformation_type: (required) NX_CHAR
Obligatory value: translation
@vector: (required) NX_CHAR
@offset: (required) NX_CHAR
@depends_on: (required) NX_CHAR
fast_pixel_direction: (required) NX_NUMBER {units=NX_LENGTH}
Values along the direction of fastest varying pixel direction. The direction itself is given through the vector attribute
@transformation_type: (required) NX_CHAR
Obligatory value: translation
@vector: (required) NX_CHAR
@offset: (required) NX_CHAR
@depends_on: (required) NX_CHAR
slow_pixel_direction: (required) NX_NUMBER {units=NX_LENGTH}
Values along the direction of slow varying pixel direction. The direction itself is given through the vector attribute
@transformation_type: (required) NX_CHAR
Obligatory value: translation
@vector: (required) NX_CHAR
@offset: (required) NX_CHAR
@depends_on: (required) NX_CHAR
(beam): (required) NXbeam
incident_wavelength: (required) NX_FLOAT {units=NX_WAVELENGTH}
In the case of a monchromatic beam this is the scalar wavelength.
In the case of a polychromatic beam this is an array of the wavelengths with the relative weights in incident_wavelength_weight.
incident_wavelength_weight: (optional) NX_FLOAT
In the case of a polychromatic beam this is an array of the relative weights of the corresponding wavelengths in incident_wavelength.
The wavelength spread FWHM for the corresponding wavelength(s) in incident_wavelength.
flux: (optional) NX_FLOAT {units=NX_FLUX}
flux incident on beam plane area in photons per second per unit area
total_flux: (required) NX_FLOAT {units=NX_FREQUENCY}
flux incident on beam plane in photons per second
incident_beam_size[2]: (recommended) NX_FLOAT {units=NX_LENGTH}
Two-element array of FWHM (if Gaussian or Airy function) or diameters (if top hat) or widths (if rectangular) of beam in the order x, y
profile: (recommended) NX_CHAR
The beam profile, Gaussian, Airy function, top-hat or rectangular. The profile is given in the plane of incidence of the beam on the sample.
Any of these values: Gaussian | Airy | top-hat | rectangular
incident_polarisation_stokes[np, 4]: (recommended) NX_CHAR
incident_wavelength_spectrum: (optional) NXdata
(source): (required) NXsource
The neutron or x-ray storage ring/facility.
distance: (optional) NX_FLOAT {units=NX_LENGTH}
Effective distance from sample Distance as seen by radiation from sample. This number should be negative to signify that it is upstream of the sample.
name: (required) NX_CHAR
Name of source
@short_name: (optional) NX_CHAR
short name for source, perhaps the acronym
type: (optional) NX_CHAR
type of radiation source (pick one from the enumerated list and spell exactly)
Any of these values:
• Spallation Neutron Source
• Pulsed Reactor Neutron Source
• Reactor Neutron Source
• Synchrotron X-ray Source
• Pulsed Muon Source
• Rotating Anode X-ray
• Fixed Tube X-ray
• UV Laser
• Free-Electron Laser
• Optical Laser
• Ion Source
• UV Plasma Source
probe: (optional) NX_CHAR
type of radiation probe (pick one from the enumerated list and spell exactly)
Any of these values:
• neutron
• x-ray
• muon
• electron
• ultraviolet
• visible light
• positron
• proton
power: (optional) NX_FLOAT {units=NX_POWER}
Source power
emittance_x: (optional) NX_FLOAT {units=NX_EMITTANCE}
Source emittance (nm-rad) in X (horizontal) direction.
emittance_y: (optional) NX_FLOAT {units=NX_EMITTANCE}
Source emittance (nm-rad) in Y (horizontal) direction.
sigma_x: (optional) NX_FLOAT {units=NX_LENGTH}
particle beam size in x
sigma_y: (optional) NX_FLOAT {units=NX_LENGTH}
particle beam size in y
flux: (optional) NX_FLOAT {units=NX_FLUX}
Source intensity/area (example: s-1 cm-2)
energy: (optional) NX_FLOAT {units=NX_ENERGY}
Source energy. For storage rings, this would be the particle beam energy. For X-ray tubes, this would be the excitation voltage.
current: (optional) NX_FLOAT {units=NX_CURRENT}
Accelerator, X-ray tube, or storage ring current
voltage: (optional) NX_FLOAT {units=NX_VOLTAGE}
Accelerator voltage
frequency: (optional) NX_FLOAT {units=NX_FREQUENCY}
Frequency of pulsed source
period: (optional) NX_FLOAT {units=NX_PERIOD}
Period of pulsed source
target_material: (optional) NX_CHAR
Pulsed source target material
Any of these values:
• Ta
• W
• depleted_U
• enriched_U
• Hg
• Pb
• C
number_of_bunches: (optional) NX_INT
For storage rings, the number of bunches in use.
bunch_length: (optional) NX_FLOAT {units=NX_TIME}
For storage rings, temporal length of the bunch
bunch_distance: (optional) NX_FLOAT {units=NX_TIME}
For storage rings, time between bunches
pulse_width: (optional) NX_FLOAT {units=NX_TIME}
temporal width of source pulse
mode: (optional) NX_CHAR
source operating mode
Any of these values:
• Single Bunch: for storage rings
• Multi Bunch: for storage rings
top_up: (optional) NX_BOOLEAN
Is the synchrotron operating in top_up mode?
last_fill: (optional) NX_NUMBER {units=NX_CURRENT}
For storage rings, the current at the end of the most recent injection.
@time: (required) NX_DATE_TIME
date and time of the most recent injection.
notes: (optional) NXnote
any source/facility related messages/events that occurred during the experiment
bunch_pattern: (optional) NXdata
For storage rings, description of the bunch pattern. This is useful to describe irregular bunch patterns.
title: (required) NX_CHAR
name of the bunch pattern
pulse_shape: (optional) NXdata
source pulse shape
geometry: (optional) NXgeometry
“Engineering” location of source
distribution: (optional) NXdata
The wavelength or energy distribution of the source
NXDL Source:
https://github.com/nexusformat/definitions/blob/master/applications/NXmx.nxdl.xml | 2022-01-25 17:32:16 | {"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": 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": 0.7735090851783752, "perplexity": 7992.087892372804}, "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/1642320304859.70/warc/CC-MAIN-20220125160159-20220125190159-00001.warc.gz"} |
https://itectec.com/matlab/matlab-how-to-check-to-see-if-a-specific-package-is-installed-during-runtime/ | # MATLAB: How to check to see if a specific package is installed during runtime
packages runtime
My personal computer has the Parallel Computing Toolbox, but my work computer does not. How can I check during runtime if the computer I am running a program on has that package installed? Thanks in advance!
ver('distcomp') %this tells you whether it is _installed_license('test','Distrib_Computing_Toolbox') %this tells you whether it is _licensed_ | 2021-05-18 21:19:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.2469320297241211, "perplexity": 2001.4841879673797}, "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-2021-21/segments/1620243991514.63/warc/CC-MAIN-20210518191530-20210518221530-00582.warc.gz"} |
https://stats.stackexchange.com/questions/linked/118 | 35k views
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### Why is the variance squared, and does it mean the same thing as the standard deviation? [duplicate]
I am still confused, despite similar questions being asked, about the difference between the variance and standard deviation in statistics. Why is the variance squared? | 2019-10-15 02:01:56 | {"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": 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": 0.9109438061714172, "perplexity": 878.459163889302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986655735.13/warc/CC-MAIN-20191015005905-20191015033405-00148.warc.gz"} |
http://itknowledgeexchange.techtarget.com/itanswers/automate-tsql-query-output/ | ## Automate TSQL Query Output
175 pts.
Tags:
SQL
SQL Query
T-SQL
I want to have a stored procedure run on a schedule that automatically creates a database output file as a result and saves it to a specific location on my network. Can someone please tell me how to do this?
Software/Hardware used:
SQL Server 2005
Thanks. We'll let you know when a new response is added.
One option would be to create a stored procedure using xp_cmdshell (which is disabled by default) to call bcp, and schedule a task to execute it.
Something like this:
CREATE PROCEDURE yourProcedure AS
EXEC master..xp_cmdshell 'bcp "SELECT * FROM yourDB.dbo.yourTable" queryout "c:\text.txt" -c -T'
GO
——————– | 2016-02-14 02:52:07 | {"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": 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": 0.17245683073997498, "perplexity": 5455.423851312855}, "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-2016-07/segments/1454701168076.20/warc/CC-MAIN-20160205193928-00027-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/228273/using-another-entrys-date-field-with-biblatex | Using another entry's date field with biblatex
I am trying to use the date field of an entry in a bib file for another entry. The idea is that I am using another source to date the entry (which is itself undated) and I want to avoid duplicating data.
Biblatex provides a simple way to use a field from another entry in a bib file with the \entrydata{entry_key}{\thefield{field}} command.
Although it works well for fields like title and although using the \printdate command also works if used for example in a title field (\entrydata{entry_key}{\prindate}), using \entrydata{entry_key}{\thefield{date}} in a date field doesn't work.
I am guessing it has to do with the nature of the date field which is not a simple string, but after having empirically tried every command I could think of in the biblatex manual, nothing works.
Additionally, the date field I want to use will come in YYYY-MM-DD form: I'd like the full date to be retained for sorting purposes but only the year to be printed.
Any help would be much appreciated.
MWE:
\documentclass{article}
\usepackage{biblatex}
\usepackage{filecontents}
\begin{filecontents}{sources.bib}
@article{testart1,
title = {First Article Title},
journaltitle = {Something Times},
date = {1964-02-01},
}
@article{testart2,
title = {\entrydata{testart1}{\thefield{title}}},% using the title of testart1 entry, working
journaltitle = {Another Times},
date = {1975-05-10},
}
@video{testvid,
title = {A Film Title},
editor = {Doe, John},
date = {},% should use date field from testart1 entry, not working with \thefield{date}
}
\end{filecontents}
\begin{document}
Lorem.
\nocite{*}
\printbibliography
\end{document}
• Welcome to TeX.SX! Please help us to help you and add a minimal working example (MWE) that illustrates your problem. It will be much easier for us to reproduce your situation and find out what the issue is when we see compilable code, starting with \documentclass{...} and ending with \end{document}. – user31729 Feb 15 '15 at 16:31
• I'm not entirely sure whether I understood what you want to do. It seems much easier to just give the date than use this convoluted method. Plus, I don't think \entrydata was intended to be used in the .bib file. The fact that it works for titles and such is due to biblatex (or rather Biber) copying these fields verbosely and pasting them to the bibliography where the commands are expanded; the date field, however is actually consumed by Biber at a a point where the expression \entrydata{entry_key}{\thefield{date}} is in a way utterly meaningless and cannot be expanded. – moewe Feb 15 '15 at 16:46
• Without an actually use-case or an MWE it is hard to say, but there might be ways using data inheritance or source-mapping to do what you want. – moewe Feb 15 '15 at 16:50
• Thanks for your answer and for pointing that I might be on the wrong track with \entrydata. The idea might sound weird but it actually makes sense. Of course, for a single entry, just writing the correct date in the date field would be easier but I need to do this on a larger scale. I've a got a large number of entries that have no precise dates (just YYYY, no MM-DD) and that I need to date more finely, based on when they were first discussed in print. Think for example of movies that would need to be sorted by actual release dates, not just years. – syntax Feb 15 '15 at 17:28
• @syntax - an MWE isn't absolutely mandatory. However, among other things, providing an MWE tends to dramatically raise the odds that one or more readers will understand what exactly you're trying to achieve. Of course, you can always hope that a biblatex-savvy clairvoyant will come along... – Mico Feb 15 '15 at 17:42
Found a simple solution:
To use the date field of another bib entry, one just needs to add a crossref field to the destination entry containing the key of another entry (source entry).
Using crossref has a side effect, it populates the destination entry with unnecessary fields from the source entry if those fields are not already defined (a common occurence if the source and destination entries are of different types). This is easily prevented by using \DeclareDataInheritance. For example:
\DeclareDataInheritance{article}{music}{% source entry type / destination entry type
\inherit{date}{date}% field we want to use
\noinherit{entrysubtype}% fields we don't want to use
\noinherit{author}
\noinherit{title}
\noinherit{subtitle}
\noinherit{journaltitle}
\noinherit{pages}
} | 2019-11-18 03:02:48 | {"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": 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": 0.7011311650276184, "perplexity": 1341.20415937353}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669431.13/warc/CC-MAIN-20191118030116-20191118054116-00278.warc.gz"} |
https://stats.stackexchange.com/questions/183474/odds-made-simple | I am having some trouble in understanding odds, and I would like just a basic explanation for how to interpret them.
I have found various posts related to odds but most of them are more complex than what I am trying to understand. Here is an example of how I am interpreting odds: if the odds of an event happening are 3 to 1, then the event will happen 3 times for every 1 time that it doesn't happen. I do not know if this interpretation would be correct. So, any guidance and more examples on interpreting odds would be greatly appreciated.
• That is correct. Nov 25 '15 at 4:15
On another thread there is a much broader answer by @gung that also deals with related technical issues such as the odds ratio, but I am going to stick to the topic at hand: how to interpret odds, and particularly the formulation "$a$ to $b$". As a beginner's question, it's worth thinking how "odds" are expressed in everyday speech (especially in betting parlance) as well as what odds mean to a statistician, because discrepancies between the two are problematic for learners.
For odds as expressed by a statistician, your contention is correct. Suppose a bag contains four tokens, of which three are $\color{aquamarine}{\text{aquamarine}}$ and one is $\color{brown}{\text{brown}}$, and one token is selected at random. The probability that the selected token is aquamarine is 3 out of 4, i.e. $\frac{3}{4}$, often read "3 in 4". With equally likely outcomes, the odds for aquamarine would be calculated as the number of favourable outcomes (3) divided by the number of unfavourable outcomes (1), which is $\frac{3}{1} = 3$, often read as $\color{aquamarine}{\text{three}}\text{ to }\color{brown}{\text{one}}$ or just as the number "three". More generally, you could take the fraction of "favourable outcomes over unfavourable outcomes" and cancel (divide) both numerator and denominator by the total number of outcomes, to obtain "the probability of a favourable outcome over the probability of an unfavourable outcome", from which a little algebra gives:
$$\text{Odds} = \frac{p}{1-p} \implies p = \frac{\text{Odds}}{1 + \text{Odds}}$$
Odds as expressed by a bookmaker are usually quoted either as "odds against" or "odds on", and which way they are written seems to be a common cause of confusion. In so-called British odds, fractional odds or traditional odds, the odds for aquamarine would be written "3/1 on" or "3-1 on", read as $\color{aquamarine}{\text{three}}\text{ to }\color{brown}{\text{one}}\text{ on}$.* For a gambler, the fact these are "odds on" indicates that a stake of £3 on aquamarine would return £1 profit if successful (they actually receive £4, of which £3 is simply the return of the original stake) whereas a failed bet results in the loss of the £3 stake. We can see these are "fair odds" because the gambler has three chances of gaining £1 and one chance of losing £3, so on average there is no expected gain or loss. So far, so little discrepancy: "odds on" are simply the "odds in favour" preferred by statisticians.
For events with a 50% probability, such as heads on a coin toss — two equally likely outcomes of success or failure — a statistician would say the odds are "one to one", $\frac{1}{1}$ or simply $1$ whereas a fair bookmaker would give fractional odds of 1/1 (read as "evens"). So no problems here either; however, when the probability falls below 50%, we see the bookmaker resumes quoting the larger number in the ratio before the smaller one.
Consider a race in which all four horses (let's say Foinavon, Gregalach, Mon Mome and Tipperary Tim) are equally likely to win: then in terms of probabilities, we would say each had a "1 in 4" or 0.25 chance of victory. What would the fair odds be for a bet on, say, Foinavon? There is only one favourable outcome (victory for F) versus three unfavourable outcomes (victory for G, M or T), so a statistician would describe the odds as "1 to 3", or numerically as $\frac{1}{3}$. However, a bookmaker using British odds would see the odds as "3 to 1 against", and write them as simply "3/1" or 3-1" (both read "three to one"; the "against" is implicit and goes unspoken). For a gambler, "odds against" means a stake of £1 would return £3 profit if successful (they will actually receive £4, but £1 of this is the return of the original stake) whereas if unsuccessful they lose the £1 stake. The gambler has three chances of losing £1 and one chance of gaining £3, so again there is zero expected profit/loss and the odds are fair. Sadly, "odds against" (the usual form of odds) does not correspond to a statistician's "odds in favour".
Each horse in our hypothetical race achieved fame by once winning the Grand National at odds of 100/1: since these were high ("long") odds against, they were "long shots" considered extremely unlikely to win, and their backers were handsomely rewarded with £100 profit per pound wagered. If we pretend that the bookmakers' odds were the fair odds (which would ignore the bookmaker's overround, or "vig"), it was felt that there were 100 ways the horse could lose for each way the horse could win, so the implied probability of success was $\frac{1}{101}$. On the contrary, if a statistician claimed an event had odds of "100 to 1", that is a claim the event is very likely (with a probability of $\frac{100}{101}$).
If any layperson in your audience comes from a country where fractional odds are used by bookmakers, and regularly quoted in the media (e.g. "Jeremy Corbyn set to beat 100-1 odds to become leader of UK's Labour party", The Guardian, 11 September 2015; "11 million to one: Quadruplet calves born in South Australia", Sydney Morning Herald, 30 July 2015) then quoting odds in the form "$a$ to $b$" is almost certain to cause confusion.
I've seen people try this, perhaps in the belief that "the general public is more familiar with odds than probabilities", but statisticians wise to the bookmaker's overround, and who have therefore never placed a bet in their lives, may be caught by surprise that the popular conception of odds is "the wrong way round". If this confusion is felt to outweigh the advantages of the "$a$ to $b$" formulation (particularly that it makes clear odds express a ratio of favourable to unfavourable) then it might be better to express "statistical odds" as a single number, to distinguish them from a bookmaker's fractional odds. Before presenting statistical odds to such an audience, I would at least make them aware of the following points:
• A statistician's odds correspond to a bookmaker's "odds on". If you are used to "odds against", a statistician's odds may seem "the wrong way round". For example, "10 to 1" indicates a very likely event, and "1000 to 1" an extremely likely one!
• A statistician need not put the higher number first, so odds like "2 to 3" can be used to indicate "2 chances of success to 3 chances of failure" (i.e. after many trials, the ratio of successes to failures should be around 2:3 and hence the probability of success is $\frac{2}{5}$).
• While bookmakers prefer to give odds as a ratio of whole numbers,** statisticians will often simplify their odds into the form "something to one", even if this introduces a decimal (e.g. "5 to 2" becomes "2.5 to 1").
• A statistician may leave off the "to one" and quote a single number (e.g. odds of 3.5 mean "3.5 to 1", or "7 to 2", so the long-run ratio of successes to failures is expected to be 7:2, from which the probability of success can easily be seen to be $\frac{7}{9}$).
• On this scale, odds of zero represent an impossibility; odds between 0 and 1 indicate a less-than-evens chance; odds of 1 show a 50% chance; odds above 1 indicate the event is more likely than not; a certain event would have infinite odds.
Mathematically, we have
$$\text{Odds}_\text{statistician} = \text{Odds on}_\text{British}; \quad \text{Odds}_\text{statistician} = \frac{1}{\text{Odds against}_\text{British}}$$
Even this may not be sufficient to avoid confusion. Decimal odds, also known as continental odds or European odds, have become more prevalent in an era of online gambling, especially for in-play betting and betting exchanges where fractional odds are unwieldy for displaying small, rapid changes in implied probabilities. European odds quote the payout per unit staked, including the return of the stake. For the aquamarine bet, a winning £3 stake makes a profit of £1, so each £1 staked would make a profit around £0.33 (in a payout of £1.33). The European odds for aquamarine are therefore about $1.33$. For the coin toss, a gambler staking £1 receives a payout of £2 (if successful) or £0 otherwise, so the European odds are $2.00$. For a £1 bet on Foinavon, a gambler has a winning payout of £4, so the European odds are $4.00$. You may have noticed that the European odds are the reciprocal of the implied probability of success: for the odds to be fair on a £1 stake, the expected payout (which is the probability of success multiplied by the winning payout) must equal the £1 wagered, so the winning payout must be the reciprocal of the probability. Since $\text{Odds}_\text{European} = \frac{1}{p}$ we find
$$\text{Odds}_\text{statistician} = \frac{p}{1-p} = \frac{1}{p^{-1}-1} = \frac{1}{\text{Odds}_\text{European}-1}$$
We might also have deduced this from noting $\text{Odds}_\text{European} = \text{Odds against}_\text{British} + 1$ (because of European odds including the return of the stake in the payout).
European odds have several advantages to the gambler. Comparing two fractional odds (try 8/15 versus 4/7) involves greater feats of mental arithmetic than comparing two decimals. Small changes to the implied probability work "smoothly" for a decimal whereas the form of a fraction may have to completely change as a different denominator is required. Calculating the payout from a win is as simple as multiplying the stake by the European odds (e.g. a winning stake of €300 at European odds of $1.50$ receives a payout of €450, of which €150 is profit). The reciprocal relationship with implied probability is especially useful for spotting "value bets": if a gambler believes the true probability of success on a bet at European odds of $6.00$ is greater than the bet's implied probability of $\frac{1}{6}$, the bet is good value and the gambler's expected profit is positive.
However, it's harder for a statistician to explain mathematical odds to a layperson accustomed to European odds! Like British "odds against", higher European odds indicate an event that's deemed less likely ($1.00$ for a certainty, $2.00$ for an even chance, $\infty$ for an impossibility). Even worse, the numbers are not simply the "wrong way round" but completely misleading: the entire concept of a ratio of favourable and unfavourable outcomes has been lost.
This key conceptual ratio is retained in the moneyline system used in US sports betting, even though it looks more complex at first sight. Positive figures indicate profit (excludes return of stake) on a winning $\$100$stake, essentially the same idea as "odds against". A figure of +300 indicates$\$300$ of profit on a $\$100$stake, equivalent to "3/1 [against]" in the British system or "1 to 3" for a statistician (the Foinavon bet). Negative figures indicate the required stake to win a profit of$\$100$, equivalent to "odds on". A figure of -300 shows a $\$300$stake makes$\$100$ profit, which is "3/1 on" in the British system or "3 to 1" for a statistician (the aquamarine bet).
$$\text{Odds}_\text{statistician} = \begin{cases} \frac{|\text{Moneyline}|}{100} & \text{if Moneyline} < 0 \\[5pt] \frac{100}{\text{Moneyline}} &\text{if Moneyline} > 0 \end{cases}$$
I appreciate much of this answer has concerned betting and pay-offs rather than statistics, but I've found the everyday usage of "odds" differs so markedly from the statistician's technical definition, that a thorough comparison might address some confusion (both of non-technical gamblers, and non-gambling statisticians). There are, of course, deep historical and philosophical links between betting and statistics. The problem of points concerned the fair division of the prize pot in an interrupted gambling game, and had generated discussion since medieval times. When Antoine Gombaud, chevalier de Méré posed a version of the problem in 1654, the subsequent correspondence of Blaise Pascal and Pierre de Fermat on the issue laid the foundations of probability theory. More recently, Frank Ramsey (in the 1920s) and Bruno de Finetti (in the 1930s) examined the coherence of wagers (related to the gambling phenomenon of a Dutch book) as a justification of Bayesian probability: if an agent's subjective probabilities or degrees of belief do not obey the axioms of probability, then they are incoherent and a Dutch book can be made against the agent, exposing them to a certain loss. The Stanford Encyclopedia of Philosophy has an article on the "Dutch Book argument".
($*$) I've deliberately oversimplified here for pedagogical purposes. In fact bookmakers are not consistent on this point: these odds may well be written "1/3" (signifying "one to three against"), though this may still be read aloud as "three to one on"! However, while a bookmaker might write the smaller number first in an odds against bet, they will never frame an odds on bet in this way: "1/3 on" would theoretically be the same as "3/1 [against]", but in practice would always be quoted in the latter form.
($**$) As an aside, bookmakers do not always cancel these whole numbers to their lowest terms: "6/4" is often advertised ("ear'ole"), so perhaps bookmakers believe a £6 profit on a £4 stake is more psychologically enticing than the prospect of £3 profit on a £2 stake. I have heard it argued, though the truth I know not, that "100/30" survives because "10 to 3" could be mistaken for the time of a race. Hong Kong odds are fractional odds (against) cancelled down to a single number, so "5/2 against" becomes 2.5; the profit from a winning bet (excluding return of the stake) is then the Hong Kong odds multiplied by the stake. Hong Kong odds below one indicate a greater than 50% chance; they are the reciprocal of statistical odds.
• When I was 14 and studied statistics as a separate subject at high school for the first time, my textbook examined carefully gambling odds and payoffs versus probabilities and "statistical odds": in retrospect, the level of detail was rather disturbing :) Completists may mourn the absence of Caughoo's controversial 1947 Grand National victory, the only other 100/1 winner, but in keeping with the original question I wanted to compare "1 to 3" and "3 to 1", leaving no room for Caughoo in the lineup. Nov 27 '15 at 14:13
• I'm not sure is gung's answer is really "much broader" right now ;)
– Tim
Nov 27 '15 at 15:43
• A much more in-depth answer than I thought it would be. +1 Dec 2 '15 at 19:26 | 2021-11-28 07:53:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6672122478485107, "perplexity": 1806.594013458421}, "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/1637964358480.10/warc/CC-MAIN-20211128073830-20211128103830-00121.warc.gz"} |
https://math.stackexchange.com/questions/1946104/what-is-the-natural-natural-symbol | # What is the "natural" ($\natural$) symbol?
This paper talks about "$M^\natural$-concave functions". The paper defines what an M$^\natural$-concave function is, but, I would like to know specifically what does the $\natural$ symbol mean? Why is it called "natural" and where else in mathematics is it used?
• Sep 29 '16 at 5:55
• This post on TeX.SE implies that the OP was wishing to use it as a boundary connected sum in analogy to how $\sum$ is used for ordinary summation. Looking at other texts about boundary connected sums however, I see $\#$ or some other variation being used instead. Sep 29 '16 at 6:33
• Should we read for example M natural? May 27 '17 at 8:30
• It's the natural sign from music, to undo a sharp or flat set by the key signature. Unsure how it's used in math. Here's the Unicode for the symbol: fileformat.info/info/unicode/char/266e/index.htm Jul 4 '19 at 3:44 | 2021-12-01 04:22:19 | {"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": 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": 0.7964741587638855, "perplexity": 849.84515545788}, "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/1637964359082.78/warc/CC-MAIN-20211201022332-20211201052332-00131.warc.gz"} |
https://chat.stackexchange.com/transcript/36/2021/4/10 | 12:00 AM
We're using the definition of: every neighborhood of a point $a$ contains both points in $S$ and points not in $S$
I use frontier, not boundary. Because students get insanely confused when we get to boundaries of manifolds with boundary.
heh
I vividly remember having my head spining in circles when I dealt with the boundary definition copper is talking about the first time
@HereToRelax I've heard Grand Tour isn't very cool
or whatever it's called
@copper For submanifolds with or without boundary, everything is boundary points. So frontier is just way less confusing.
I am not without reason for my madness.
12:07 AM
@TedShifrin understandable.
@dc3rd using that definition you can see that every point is in exactly one of the interior of $S$, the frontier of $S$ or the exterior of $S$. Since the first & last are open, the remainder must be closed.
visually i see what is happening, in terms of shrinking the size of $\epsilon$ for all neighbourhoods around all my boundary points will give me convergent sequences. Just the solid logic of explaining it is what is missing from me.
if I were to use that definition.
Do we explicitly use these ideas in the upcoming sections @TedShifrin ? I ask because when I did multivariable previously I didn't see these ideas pop up explicitly too much. THey kinda just sat in the background except for defining the usual differentiation and integration ideas, but we didin't use frontier points, limit points, closures, etc too explicitly. I've always wanted to see it in action
12:23 AM
This is not a calculus course. This is a multivariable analysis course. If you study the proofs, they will be used. You will be doing proofs with compactness and integrability ...
Not to mention inverse/implicit function theorems ...
THe course I did was a "middle ground" it wasn't all calculus and wasn't full analysis. We used Folland's "Advanced Calculus"
it was the course for the "inbetweeners" who were not full math specialists and also not math minors.
If you learned that course, you should know most of mine. Hmmm .... Or it could have been taught watered. Down.
"learned" is a stretch..............memorized definitions, theorems, and proofs?.......yes.
but that was my experience because of the weak foundations. But I do recognize all the content you're teaching
also they really didn't do any of the geometry the way you do.
This comes back to the ability to "teach" that you have talked about
1:04 AM
$$\Demarois$$
1:19 AM
apparently someone even tried to post an answer on a sock puppet account to prevent the bounty and got owned
Legend depends on one's universe, evidently
I thought it was funny :'(
I guess I didn't pay enough attention.
1 hour later…
2:33 AM
Perfect timing if you are bored @TedShifrin . I either just finished 9a, or I mixed the results up. What do you think:
i) True/ False: all interior points of $\bar{S}$ are points of $S$?
Claim: False.
Pf: Let $\bar{S} = [a,b]$. Then $b \in S^{f}$. Take $\delta$ nbhds for $b \in S$. If we take $\frac{\delta}{2}$ nbhds we can have $b \in \bar{S}$.
ii) True/ False: If $S$ is open, then all interior points of $\bar{S}$ are points of $S$
Claim: True.
pf: Suppose towards contradiction that there exists interior points $a \in \bar{S}$ such that $a \notin S$. Then $a \in S^{F}$. This means f
I don't follow a at all. b is wrong.
back to the drawing board...
For a) what I wanted to argue was that the frontier points are also interior points of $\bar{S}$ specifically.
@robjohn yes, then ...?
@dc3rd check definitioms?
So whatever $\delta$ I have dictating the nbhd around a frontier point of $S$, there exists a smaller $\delta$, this one I said $\frac{\delta}{2}$ that can be chosen to make every nbhd of a frontier point with respect to $\bar{S}$ be fully contained in $\bar{S}$
2:47 AM
Nope.
Alright. I'll go back and read the definitions and proabably check in with you tomorrow about it if you're not around tonight.
The definition of frontier point says nope.
> Findings of these studies suggest that the risk of SARS-CoV-2 infection via the fomite transmission route is low, and generally less than 1 in 10,000, which means that each contact with a contaminated surface has less than a 1 in 10,000 chance of causing an infection 7, 8, 9.
is that a "yes", sir?
2:51 AM
I don't know any new news, and I have no idea what 7,8,9 is there.
sorry, those are the study references
[7], [8], [9]; I guess I'll have to read them
As I say, for months it's been known that it's aerosol transmission that is the grave danger — hence the importance of masks. Hand-washing continues to be a precaution.
"1 in 10,000 after contact" struck me as odd
Surfaces are not a big deal.
Why odd?
3:05 AM
Just unraveling the definition a bit more I see what you're getting at with respect to i). To explain it informally. If I take an interior point $a \in \bar{S}$ then there exists a nbhd fully contained in $\bar{S}$. Equivalently it means $a \in S$ or $a \in S^{F}$. We are only concerned with the $S^{F}$ case. If $a \in S^{F}$ then by definition every nbhd of $a$ will contian points not in $S$, negating the definition of an interior point
....... So then that means i) is true.....just have to write it out
3:24 AM
No, it's still not correct. You're not quite catching the subtlety yet. Think about more examples.
Alright will do
The subtlety yo're hinting at has to do with the interior of the set of frontier points I gather.
Not quite.
hmmmm....interesting.
I don't actually know what you mean by that.
Well I was thinking about the set of frontier points exclusively and disregarded the point of $S$ because I thought it was trivial.
3:39 AM
That seems correct, but you went the wrong way down the road.
@dc3rd Pick a point $x$. Then exactly one of the following three things is true: 1. there is an open set containing $x$ that is entirely in $S$. 2. there is an open set containing $x$ that is entirely in $S^c$. 3. neither 1 nor 2 hold.
Now note that if 3. holds, then no open set containing $x$ can be a subset of $S$ or a subset of $S^c$. In other words, any open set containing $x$ must contain points of both $S$ and $S^c$.
Sorry, I've been saving that since before I started dinner.
The issue is to imagine different scenarios with where frontier points are located. Let's not give it away.
@geocalc33 there?
The room is surprisingly lively despite its dearth of tools. I think that is due to you @Ted.
3:56 AM
And whom are you calling tools?
@copper.hat yeah, who da tools ?
:-)
We all can be tools of the prolific spread of mathematics I guess
Tool is also slang.
Perhaps dated slang.
nah it's still relevant. prbably has greater emphasis now because of its age as a term
4:01 AM
As slang, it's rather derogatory. Whence my original question.
Well, if it's slang it could mean 1 of 10 things, and 80% of us will fall into a tool category
Therefore, I think it can be replaced with "people"
Whence my original question.
i just wanted to use the word dearth.
LOL ... or death
dear'th
4:04 AM
a few years ago my daughter gave me a present of "vader's little princess"
Dearth of t-word, could mean stuff
But who cares, time's little
a Haiku I composed
I might have some questions in just a bit. I'm working through this #-theory book at the moment.
how did hash become number?
lol
LOL, another victim of the primes
@zacts burton?
4:06 AM
@LeonhardEuler I'm looking at that and another text too.
# has always been number!
@zacts what's the q?
[#] looks like a rubik's cube to me
Math is indeed an infinite-dimensional rubik's cube phenomenon
Just as is coding
@StudySmarterNotHarder well, I'm kind of stuck with some of the problems. maybe my first question could be what's a good study strategy for these texts? Also, might analysis be a useful way for me to get started before these texts, like with Tso's Analysis for example.
4:08 AM
i think i have only seen it used for number in the us, at least a few decades ago.
@zacts what part do you first get stuck on in the book?
@copper.hat what is your etymological meaning of #?
the interweb is homogenising.
@StudySmarterNotHarder I'll link a particular problem
Seems also like you need to know some Python from that page
4:09 AM
@StudySmarterNotHarder no python is needed for the actual text of an illustrated theory
that's like extra supplement for the text
Oh, cool, either way I could teach Python 3.x / SymPy coding if that's what you wanted
i could easily have forgotten, but i believe it was used to indicate a number as a label.
as opposed to having some order. similar to a modern SKU
#:
$\# A$ is an alternative to $|A|$ when $A$ is a set
that is where the cs usage came form
4:11 AM
@copper.hat ah cool
Ok, let me link a question from this text by burton
it is called an octothorpe
Prove the following facts concerning triangular numbers: (a) A number is triangular iff it is of the form n(n+1)/2 for some n>=1.
Another way to phrase that is they want you to prove that $T(n) = \dfrac{n(n+1)}{2}$
The information given is: "This led the ancient Greeks to call a number /triangular/ if it is the sum of consecutive integers, beginning with 1."
4:14 AM
is that a definition?
where $T$ is the sequence of triangular numbers
It's actually a very well-tought proof, what are you having trouble with? It's an obvious candidate for inductive-style proof because $n + 1$ occurs directly within it and that's how induction proceeds.
Prove it first for $n = 1$
I think my confusion was with the definition for /triangular/.
$1(1 + 1)/2 = 1$, done
Oh, cool
triangular means "The sum of consecutive integers, beginning with 1"
so if I can prove that n(n+1)/2 is the sum of consecutive integers, beginning with 1,, I have proven it's triangular, right?
iff that is
I guess because you can form a "pyramid" with 1 object at the top, 2 objects on the row below that, 3 on the row below it and so on, and the sum of all row counts is equal to the area approximately (as $n$ grows) of a triangle.
4:17 AM
Show $T(1) = 1$ and $T(n+1) = n+1+T(n)$.
If part is proof the formula $T(n) = \dfrac{n (n+1)}{2}$, and only if would be simply by definition I think
The integer n is a triangular number iff 8n+1 is a perfect square.
I also am looking at this Analysis I text by Tso.
Ok either I'm thinking too hard or I'm on to something. But I've been sitting here thinking about the statement of
"either $a \in \bar{S}$ or $a \in \bar{S}^{c}$ or $a \in \bar{S}^{F}$"........should I unravel the idea from there?
For that you're asked to solve: $8 T(n) + 1 = m^2$ for $m$, for any given $n$. And the converse part is for all $m$ such that $m^2 = 8 k + 1$, we have that $k = T(n)$ for some $n$.
but let me post a cool looking question from the other #-theory (pardon my use of #) book.
4:23 AM
Nice, let's hear it
*see it
@dc3rd i'm not sure what you are trying to do.
Either $a \in A$ or $a \notin A$ for all objects $a$ in the universe of thought, for any given set $A$.
try a few examples. what is the frontier of $\mathbb{R}$?
Draw a spiral to demonstrate that 100 = 10 + 2(9) + 2(8) + 2(7) + 2(6) + 2(5) + 2(4) + 2(3) +2(2) + 2(1).
What does the spiral look like?
4:26 AM
I don't know.
What's the general pattern they are implying?
it's got to be stacked somehow
like stacked dots
Then I would try it with a smaller number than 100 maybe
ok
yeah
25 maybe
4:27 AM
that's kind of what is going on in the ch for sure
I don't think this would be too difficult now that you mention it.
let's see if I can sketch a pic for that.
25 = 5 + 2(4) + 2(3) + 2(2) + 2(1)
Look at each term, can you tell me the general formula for a any square $n^2$, $n \in \Bbb{N}$?
That's a good question @copper.hat ....I'm drawing it now and I'm asking what IS the frontier of $\mathbb{R}$..only thing that comes to mind is $-\infty$ and $+\infty$ and those don't mean much.
The frontier of $\Bbb{R}$ is the great wide open $\Bbb{C}$-plane, son!
J/k
@StudySmarterNotHarder n^2 = 2n(n+1)/2
wait
I was wondering if it was $\mathbb{C}$ or not.
4:31 AM
I was speaking poetically though
Lets stick with reals for now.
It is not the complex plane.
$n^2 = n + ?$
what points can be approached from the complement?
sorry just a sec..
Use \sum_{k = 1}^{n-1} ?$notation 4:33 AM if a point is$x \notin \mathbb{R}$....hmmmm....I'm truly stumped would we have to include higher dimensions? Noooooooo$\mathbb{R}$is the whole universe here. Well maybe I shouldn't search too hard because all I got is the empty set beyond$\mathbb{R}$exactly the complement is the empty set @zacts I'm not sure about what spiral it should be or look at it this way, for any$x$you can find a open set containing$x$that is a subset of$\mathbb{R}$. 4:36 AM but the empty set has no points in it, so how can I have a frontier point at "the end" of$\mathbb{R}$? if$x$was a frontier point then any open set containing$x$must contain points in$\mathbb{R}$and$\mathbb{R}^c$. Peas porridge hot, peas porridge cold however, there are no points in$\mathbb{R}^c$. ha.....which it cannot because there are no points in$\mathbb{R}^{c}$from which we conclude that the frontier of$\mathbb{R}$is .... 4:38 AM remember that the frontier is a set. ^ I think it's got to be something like this stacked dots idea ooooohhhh..............the set is empty exactly how about the rationals as a subset of the reals. what points$x$can be approached arbitrarily closely by rationals and irrationals? See the rationals I used as an example of a frontier set. 4:41 AM so what is the frontier? sorry I used the integers what real numbers can be approached arbitrarily closely by rationals? the integers would serve as a frontier set not sure how you're getting that. in a metric space, a point is in the frontier if (and only if) it can be approached arbitrarily closely from the set and its complement. the number${1 \over 2}$is a rational that is far away from the integers. @zacts I think you take a square, say$n^2 =25$and draw an actual square of 5 x 5 units Cross out the top row (5) and fit in$2(k)$for each$k \lt 5$in a spiral-like pattern 4:46 AM well to your question of what real numbers$x$can be approached by rationals. I would say the irrationals. @StudySmarterNotHarder would it be$n^{2} = 2(n-1)n/2$? oops that is not all$0$as well well, any rational$n^2 = n + \sum_{k = 1}^{n-1} 2k = n + 2 \sum_{k=1}^{n-1} k = n + 2 T(n-1)$4:47 AM$n^{2} = n + \frac{2(n-1)n}{2}$? I left in the 2 there to reflect the grouping in the original problem But the spiral they ask for is starting with an$n \times n$square. and then vice versa for the irrationals towards rationals as well. @dc3rd given any real$x$, and an open set$U$containing$x$. Does$U$contain rationals? 4:49 AM @StudySmarterNotHarder oh, so I'm drawing a spiral within the square? like draw a line within the dots kind of idea? But the spiral for case$n$has in it the spiral for case$n - 1$which has in it the spiral for case$n - 2$, and so on down to$1$. I would use grid squares Where each grid box is a unit in area Yes it does. Does$U$contain irrationals? Definitely does So what is the boundary of the rationals? 4:52 AM Was just going to type it too.....the boundary is the reals correct. good. 551133 552233 552233 554444 554444 what is the boundary of the irrationals? @zacts where the labeling just reflects what it index it corresponds to sorry i meant frontier. old habits die hard. 4:56 AM boundary is the rationals. Nope. lol....ted with the RKO out of nowhere...... what real numbers can be approached arbitrarily closely by irrationals? i'm going to ask the same questions again... Don't worry: I'll disappear as quickly as I appeared. @StudySmarterNotHarder is the spiral formed from the actual dots themselves within the grid? 4:58 AM The spiral is centered approximately at$1 \ 1$You then double 2 to get:$1 \ 1 \\ 2 \ 2 \\ 2 \ 2$That always gives you a height or width that is$3, 4, 5, \dots$The next is 3 so: 1 1 3 3 2 2 3 3 2 2 3 3 The point is that it's always of the form$k(k-1)$real numbers that can be approached by irrationals are the irrationals that is a different answer to the same question when we were looking for the boundary of the rationals. can 2 be approached by irrationals? yes it can so, what real numbers can be approached arbitrarily closely by irrationals? the integers but then the rationals contain the integers..... 5:05 AM lets back up a bit pick any real$x$. Pick any open set$U$containing$x$. Does$U$contain irrationals? was just doing that too.... but yes$U$contains irrationals Does$U$contain rationals? @StudySmarterNotHarder thanks. I'm still trying to visualize this, but that looks like a cool solution. Yes$U$contains rationals. so the reals would serve as a boundary for the irrationals... given that the irrationals are viewed as a subset of the reals.....very important 5:12 AM correct. note that the boundary of the rationals and the boundary of its complement (the irrationals) is the same. @TedShifrin Poof! i mean frontier @copper.hat depending on the book you use the subtlety in this is keeping note of what is a subset of what. I understand what you mean by boundary, no worries what is the frontier of$(-\infty,0]$as a subset of the reals? @robjohn Ted wants me to use frontier in current discussion so i am complying :-) 5:14 AM @StudySmarterNotHarder just to be clear, but was my solution to the$n^2$problem correct? @copper.hat Yes, master. @dc3rd It is important, since complements are involved. i am very obedient :-)$n^2 = n + 2[\frac{(n-1)n}{2}]$@zacts yes ok, thanks 5:16 AM @zacts$n^2=2\binom{n}{2}+\binom{n}{1}$the frontier would be$(0, \infty)
nope
what $x$s can be approached from inside and outside the set?
@dc3rd are you using the definition $\partial S=\overline{S}\cap\overline{S^C}$?
for any $y>0$ i can find some $r>0$ such that $(y-r,y+r) \subset (0, \infty)= (-\infty, 0]^c$.
no closures on them @robjohn
5:18 AM
@dc3rd then that would be empty
@StudySmarterNotHarder @robjohn thanks
@dc3rd loosely, for 'nice' sets, the frontier is the 'edge' of the set.
what is the edge of $(-\infty, 0]$ ? In the context of the reals?
@copper.hat that would seem to be what I wrote...
in this case then $0$ is the frontier
@zacts in your proof you have to state that $\sum_{k = 1}^{n-1}k = T(n-1)$
5:20 AM
@copper.hat $\partial S=\overline{S}\cap\overline{S^C}$
@robjohn would you mind explaining in a nutshell how that notation works?
@robjohn i think @dc3rd is using the defn that $x$ is in the frontier iff any open set containing $x$ contains points in the set and its complement.
@zacts which notation?
which is, of course, equivalent,
{ n \choose r }
5:21 AM
and the closure definition is one i prefer.
the bionomial coefficient notation parenthetical (n/2) (n/1)?
binom{n}{m}
with a slash
backslash
$binom{n}{m}$
verbatim the definition is: "$a \in \mathbb{R}^{n}$ is a frontier point of S if every neighbourhood of $a$ contains both points in $S$ and points not in $S$"
$\binom{n}{m}$
5:22 AM
@zacts right-click on the mathjax?
ok
@TedShifrin I have complex shapes of geometries which makes it hard to integrate over the area, I am thinking to use the theory to make things more analytical and reduce integrations. do not know though how things may work or succeed.
\binom{n}{m}
@robjohn no, can you conceptually explain how bionomial coefficients work in a nutshell, not how to write it in $\latex$.
ah, okay
5:23 AM
Although, I do apologize as I'm still learning LaTeX at the moment as well
I do have another venue for a practice pad too which I'll use.
@dc3rd in my example. pick 3 different classes of points. $x<0$ $x=0$ and $x>0$.
@zacts They are the coefficients of $(1+x)^n=\sum\limits_{k=0}^n\binom{n}{k}x^k$
for each class. ask if it can be 'approached' from in the set and from outside the set.
well I was typing something, but I'll save it. I think I'm going to conclude the same thing, but it is worth the practice.
@zacts They can also be computed in Pascal's Triangle
5:26 AM
@dc3rd you need to work examples involving intervals open closed half inifinite and become comfortable with the reasoning.
Can anyone describe a simply connected covering space of a 2-sphere with a circle intersects with two points?
@zacts also $\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots(n-k+1)}{k!}$
@dc3rd as i mentioned, in 'nice' cases the frontier is what you might expect the 'edge' of the set to be.
i should not have started with the rationals.
I don't know how to describe when a portion of circle is inside of sphere
a not so great circle?
5:28 AM
@zacts I am not sure what you mean by "how they work"
Got it. with regards to 'approached', I take it you mean that it can "eventually" get extremely close to the point?
@robjohn how would you think of that notation conceptually within the context of the $n^2$ problem, perhaps in terms of how things are grouped together? if that makes sense. I mean you wouldn't be thinking in terms of the factorial formula given above right?
@dc3rd well, more precisely, any open set contains points from the set and its complement.
@zacts No. I was simply considering exactly the formula you stated: $\frac n1+2\frac{n(n-1)}2$
but thinking of it as binomial coefficients allows it to be manipulated using things used to manipulate the binomial coefficients
such as the hockey-stick identity
Or Vandermonde's Identity
what would $\binom{n}{3}$ look like?
5:34 AM
@zacts $\frac{n(n-1)(n-2)}{3!}$
8 mins ago, by robjohn
@zacts also $\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots(n-k+1)}{k!}$
so, would $\binom{n}{4} = \frac{n(n-1)(n-2)(n-3)}{4!}$
ok
@zacts yes
in $\binom{n}{k}$ I only consider $k$ being a non-negative integer. $n$ can be any real (or complex even)
so you were directly notating my original solution to the $n^2$ problem in terms of binomial coefficient notation, which then allows you to use operations specific to binomial coefficients.
yes
cool, thanks.
5:37 AM
@zacts there is a nice interpretation as the number of paths to get from the top to a point in Pascal's triangle.
@copper.hat did you see my Paschal Triangle?
Mar 29 at 13:52, by robjohn
:-)
the number of stripes on each egg is important
@enthu I'd need to see an example to comment further.
very seasonably appropriate
5:40 AM
of course the eggs have tge powers in the stripes
what do the colors mean?
expand some binomial variables and you will see it
i can't figure out the coloUr mapping
oh wait are those easter eggs?
@zacts the number of colors is the value of the same place in Pascal's Triangle
@zacts yes
@copper.hat The colors are random, the number of colors is important
5:41 AM
@dc3rd what is the boundary of $\emptyset$ in the reals?
@robjohn i got the number part, i thought there was another map
he was pop quizzing me @robjohn
i'm quizzing dc3rd
sorry
part of the fun :-0
lol.....too late to delete the answer.
5:43 AM
note that the empty set is the complement of the reals in the reals.
I'm kind of looking for a text to latch onto this kind of idea.
in general the frontier of a set and its complement is the same.
I think that was actually the next question I have to do in the text
I'm also looking at this text, Analysis I by Tso.
@zacts my favorite text for discrete math is Concrete Mathematics
5:47 AM
I have that book too, and I'm looking at that, but I wonder if I'm ready for it.
this discrete type of mathematics definitely heavily interests me
i like flamboyant math.
i prefer short dense books.
@copper.hat (shhh!)
I think I was confused with the [p prime] notation.
Iverson's convention
@zacts The Iverson brackets?
yes
yeah
this was like a year ago I tried to dive into it a bit
I'm looking at the book now
5:51 AM
@zacts I use those in several of my answers
66 to be exact :-)
@robjohn hum... I wonder what text I might dive into at the moment.
the pleasures of counting
probably not what you are looking for, but a nice read.
hey chat, good evening
Would you say that understanding some real analysis would be a prerequisite for Concrete Mathematics, or could one just dive into the book? | 2021-05-15 08:01:54 | {"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": 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": 0.7579020857810974, "perplexity": 939.4077231942815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991378.48/warc/CC-MAIN-20210515070344-20210515100344-00546.warc.gz"} |
http://wikieducator.org/General_Search | # General Search
Assignment
Open your favourite search engine and search for websites about the:
Medici Family
Then check the material you have found for its declared license.
Attention: Often you will not find clear information regarding copyright.
To learn more about BOOLEAN search look at Search Engine Math). | 2017-11-24 20:29:52 | {"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": 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": 0.3324024975299835, "perplexity": 4642.403072795964}, "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-2017-47/segments/1510934808935.79/warc/CC-MAIN-20171124195442-20171124215442-00084.warc.gz"} |
https://proofwiki.org/wiki/Category:Complete_Elliptic_Integral_of_the_Third_Kind | # Category:Complete Elliptic Integral of the Third Kind
This category contains results about Complete Elliptic Integral of the Third Kind.
Definitions specific to this category can be found in Definitions/Complete Elliptic Integral of the Third Kind.
### Definition 1
$\ds \map \Pi {k, n} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} }$
is the complete elliptic integral of the third kind, and is a function of the variables:
$k$, defined on the interval $0 < k < 1$
$n \in \Z$
### Definition 2
$\ds \map \Pi {k, n} = \int \limits_0^1 \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
is the complete elliptic integral of the third kind, and is a function of the variables:
$k$, defined on the interval $0 < k < 1$
$n \in \Z$
## Pages in category "Complete Elliptic Integral of the Third Kind"
This category contains only the following page. | 2023-03-22 07:32:28 | {"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": 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": 0.8747658729553223, "perplexity": 538.4193475352121}, "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-2023-14/segments/1679296943750.71/warc/CC-MAIN-20230322051607-20230322081607-00152.warc.gz"} |
http://openstudy.com/updates/4f185f45e4b00328e4c515b4 | ## anonymous 5 years ago Simplify the expression below. √x^17
1. anonymous
2 goes in to 17 8 times with a remainder of one, so answer is $x^8\sqrt{x}$
2. anonymous
great! thank you :D
3. anonymous
yw
Find more explanations on OpenStudy | 2017-01-21 19:48:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7864992618560791, "perplexity": 6342.5191695871545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281202.94/warc/CC-MAIN-20170116095121-00203-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://cvgmt.sns.it/paper/1567/ | Weak and strong density results for the Dirichlet energy
created on 27 Nov 2002
modified on 06 Apr 2004
[BibTeX]
Published Paper
Inserted: 27 nov 2002
Last Updated: 6 apr 2004
Journal: Jour. Eur. Math. Soc.
Volume: 6
Pages: 95-117
Year: 2004
Abstract:
Let \,${{\cal Y}}$\, be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in \,$B^n\times{{\cal Y}}$\, with equibounded Dirichlet energies, \,$B^n$\, being the unit ball in \,${\mbox*R*}^n$. More precisely, weak limits of graphs of smooth maps \,$u_k:B^n\to{{\cal Y}}$\, with equibounded Dirichlet integral give rise to elements of the space \,${\mbox{\rm cart}}^{2,1}(B^n\times{{\cal Y}})$. In this paper we prove that every element \,$T$\, in \,${\mbox{\rm cart}}^{2,1}(B^n\times{{\cal Y}})$\, is the weak limit of a sequence of smooth graphs \,$\{u_k\}$\, with equibounded Dirichlet energies. Moreover, in dimension \,$n=2$, we show that the sequence \,$\{u_k\}$\, can be chosen in such a way that the energy of \,$u_k$\, converges to the energy of \,$T$. | 2017-11-18 19:47:39 | {"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": 0.9288966655731201, "perplexity": 1148.372353612558}, "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-2017-47/segments/1510934805023.14/warc/CC-MAIN-20171118190229-20171118210229-00797.warc.gz"} |
https://www.physicsforums.com/threads/entropy-maxwell-relations.273450/ | # Entropy - Maxwell Relations
1. Nov 20, 2008
### Master J
Hey guys.
Right, I have been studying the Maxwell thermodynaic relations. But I have come across entropy as
dS = (bS/bT)_P(dT) + (bS/bP)_T(dP)
where b is the partial differential symbol.
I dont understand where this comes from, which suggests S(T,P). I cant find a derivation of this.
Could you help?
2. Nov 20, 2008
### nasu
It's nothing really to derive. Physically, I mean.
For any function of two variables, the general formula for differentiation looks like this.
I mean, for f(x,y)
df=(bf/bx)dx+(bf/by)dy.
As for why S(p,T), you can use any two independent variables as "dependent variables". | 2016-10-26 00:37:19 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8450035452842712, "perplexity": 3361.605315323449}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720471.17/warc/CC-MAIN-20161020183840-00111-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/tags/two-column/new | # Tag Info
1
I'm not sure the multi-page table packages are helping here, you can simply force the column widths and break by hand: \documentclass[11pt]{article} \usepackage[left=0.25in,right=0.25in,top=0.25in,bottom=0.25in]{geometry} \geometry{letterpaper} \usepackage{helvet} \renewcommand{\familydefault}{\sfdefault} \usepackage{colortbl} \newenvironment{zz} ...
3
figures won't normally go to the end of the document unless you prevent them going anywhere else, but in anycase large spaces at page breaks are inevitable if you use H as the figure will not move relative to the surrounding text so if it needs to go over the page a gap is left, that's what [H] means. Your second example for example uses [hbt] the main ...
0
Not for KOMA article class, but for scrbook and scrreprt there is the \setpartpreamble command. In my version of the docs (2012-07-22), p.95: Parts and chapters in KOMA-Script can be started with a preamble. This is particularly useful when you are using a two column layout with the class option twocolumn, since the heading together with the ...
2
Package algorithm2e does not allow H in twocolumn mode, regardless if single or double float. A little excerpt from the package file: \if@twocolumn\@latex@error{[H] in two columns mode is not allowed for algorithms}\fi% TODO: SCREAM if H in two colums! Bad thing is, for some reason two captions in an algorithm environment don't work. The solution can ...
4
The class uses \LoadClass[fleqn]{article} to specify flush left equations, with no documented way of over-riding that. that is really the point of publisher specific classes, to remove flexibility to enforce a house style.
4
this just forces text page floats to the right, \clearpage still flushes all floats to the first available column. \documentclass[twocolumn]{article} \def\f{\begin{figure}\rule{1cm}{1cm}\caption{fff}\end{figure}} \def\t{one two three four five six sevn eight nine ten eleven} \def\tt{Red green \t. Blue yellow orange purple \t, \t, \t, \t.\par} \makeatletter ...
0
I suggest you not take any steps that end up reducing the font size used in the math expression relative to that used for the surrounding text. Instead, you might want to pursue the following approach: Don't use the \nolimits modifier after each \sum macro. Instead, encase the \sum{...} expressions in \smashoperator directives; this reduces the amount of ...
0
This is one possible solution. use of resizebox from graphicx package and parbox combo as shown below. \resizebox{0.48\textwidth}{!}{\parbox{\linewidth}{ math envrionment}} or {\tiny \begin{align*} ... \end{align*} environment} Code \documentclass[10pt,conference,letterpaper]{IEEEtran} \usepackage{amsmath,graphicx} \begin{document} as abcd dsfe ...
Top 50 recent answers are included | 2014-10-23 21:26:58 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8671055436134338, "perplexity": 4846.263265182637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413558067648.77/warc/CC-MAIN-20141017150107-00054-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://gateoverflow.in/14305/what-saying-candidate-super-which-proper-subset-also-super | +1 vote
255 views
Since candidate key is a minimal key and it is a proper subset of a super-key , then how is it that for a candidate key ,its proper subset is not a super key ?
| 255 views
A candidate key is the minimal superkey. That means if we try to remove some attributes from the candidate key (minimal super key) we will no more get a superkey as it was the minimal. Hence, any proper subset of the candidate key is not a superkey.
by Active (3.8k points)
selected
+1 vote | 2019-12-09 18:32:14 | {"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": 0.8254780173301697, "perplexity": 2062.251767061434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540521378.25/warc/CC-MAIN-20191209173528-20191209201528-00543.warc.gz"} |
http://jcreed.org/math/modal-encoding/ | $\def\imp{\Rightarrow} \def\celse{\ |\ } \def\prov{\vdash} \def\dns{{\downarrow}} \def\ups{{\uparrow}} \def\preq{\dashv\vdash} \def\sh{\#} \def\tensor{\land^+}$
Source Language
The source language is a straightforward polarized modal language. Its syntax goes like this:
Negative $N$ $::=$ $P \imp N \celse N \land N \celse \top \celse \Diamond P \celse a^- \celse \ups P$ Positive $P$ $::=$ $P \tensor P \celse P \lor P \celse 1 \celse 0 \celse \square N \celse a^+ \celse \dns N$
Target Language
The target language is just ordinary focused first order logic, nothing surprising. Pick one distinguished negative atomic formula $h(w, \phi, m)$ (pronounced 'here') with three arguments.
• $w$ is the Kripke modal world for the modal logic of $\square$ and $\Diamond$.
• $\phi$ can be seen as a modal world in the sense that intuitionistic logic is a modal logic over classical logic. We shunt conclusions into the context, (arguably a classical-logical sort of trick to pull) so we need $\phi$ to make sure only the 'fresh' ones can be used.
• $m$ is a parameter used to enforce the condition that $\Diamond$ can only be unpacked on the left when the conclusion is $\mathrm{poss}$. There is a distinguished first-order term $\star$ constant which may appear in the third argument of $h$.
Translation
Define the functions $N^w$ and $P^w$ by
$N$ $N^w$ $P \imp N$ $P^w \imp N^w$ $N_1 \land N_2$ $N_1^w \land N_2^w$ $\top$ $\top$ $\Diamond P$ $\forall \phi . \dns (\forall v \ge w . P^v \to h(v \phi \star)) \to h(w \phi \star)$ $a^-$ $\forall \phi m . a^+(w\phi) \to h(w \phi m)$ $\ups P$ $\forall \phi m . \dns (P^w \to h(w \phi m)) \to h(w \phi m)$
$P$ $P^w$ $P_1 \tensor P_2$ $P_1^w \tensor P_2^w$ $P_1 \lor P_2$ $P_1^w \lor P_2^w$ $1$ $1$ $0$ $0$ $\square N$ $\dns \forall v \ge w . N^v$ $a^+$ $a^+(w)$ $\dns N$ $\dns N^w$ | 2022-08-11 17:41:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.837100625038147, "perplexity": 302.8049308259097}, "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-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00210.warc.gz"} |
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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 4th 2012
added to pure type system in the Idea-section the statement
In other words a pure type system is
1. a system of natural deduction
2. with dependent types
3. and with the dependent product type formation rule.
and to the Related concepts-Section the paragraph
Adding to a pure type sytstem
1. rules for introduing inductive types
2. possibly a type of types hierarchy
makes it a calculus of inductive constructions.
Finally I added to the References-section a pointer to these slides
• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012
I think CiC is a particular pure type system; your phrasing sounds like it is a class of them.
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
Okay.
Can you maybe help me understand what these 8 different pure type systems in the cube are? I don’t understand the notation when it comes to discussion of that.
• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012
The notational names for those things like $\lambda_\to$ are ad hoc, I wouldn’t pay them too much mind.
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
What I don’t understand is what $(\ast,\Box)$ and $(\Box, \Box)$ and so on means.
• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012
They’re specifying the relevant set $R$ of “relations” as described in the “Definition” section.
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
Gee, I still don’t get it. Not only that I don’t understand the message, but I cannot even recognize a message.
Is “$\Box$” the name of some generic set? Could I write “$X$” instead of “$\Box$”? Why do we suddenly use geometric figures instead of variables? Is “$\ast$” also any set? It’s not meant to be singleton set?
Then: what does it mean to say that “$(*,\Box)$” is a relation? Which relation is this supposed to denote? On which set? And how am I supposed to think of it in words? Which role does that relation play?
Gee, I must be missing something really basic here. That feels weird.
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
Ah, I am making progress. $S$ is the set with one or two or three elements, right?
There are a bunch of curly brackets missing in the entry, could that be?
• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012
If putting curly brackets in would help, please do.
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
Okay, here I am, at past 2:00 in the morning, editing an entry that I couldn’t parse a minute ago. :-)
Right after where the triples appear, I am adding the folllowing remark. Give me a sanity check:
Remark. These relations will appear in the type formation rule for dependent product types below. They will say that for a type of sort $s_2$ depending on a type of sort $s_1$ its dependent product type has sort $s_3$.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
Does this here parse for you:
symbol actual value
$S =$ $\{\ast, \square\}$
$A =$ $\{(\ast : \square)\}$
$R =$ $\{(\ast, \ast), (\ast, \square), (\square, \ast), (\square, \square)\}$
?
I am now doing the other tables accordingly. Hope that’s right.
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
• (edited Oct 5th 2012)
Checking if I am following:
For the case of the PTS that is the CoC (the yes, yes, yes, yes-case :-) we have
• $\ast = Prop$
• $\Box = Type$
is that right?
• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012
Yes, I think all of that is right. Sorry, I didn’t realize how opaque the previous entry was! Your added remark after the definition of the “relations” is I think greatly helpful (I fixed a typo).
• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 5th 2012
Okay, thanks.
I have added a remark in a new subsection to make the special case of the CoC more explicit.
• CommentRowNumber15.
• CommentAuthorJonasFrey
• CommentTimeMay 1st 2015
Hi, I’ve been looking into pure type systems in the last week, and I also had a look on the nlab page. According to the nlab, a pure type system is a triple (S,A,R) of a set S of “sorts”, a set A of axioms, and a set R of “relations”.
However, I think the elements of R are called “rules”, not “relations” in the literature that I know. Is there a reason why they are called “relations” here or is it by mistake? If nobody objects I can change it.
• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeAug 5th 2015
Belatedly: thanks Jonas!
I added to pure type system a remark that they can be augmented with a cumulativity relation between sorts, and a reference.
Anonymous
• CommentRowNumber18.
• CommentAuthoratmacen
• CommentTimeApr 9th 2019
Changing the metavariable names in the idea section to use $b:B$ rather than $B:C$. The latter isn’t following any convention I’ve seen. | 2021-12-06 11:27:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 23, "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": 0.8138371109962463, "perplexity": 3667.637422434728}, "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/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00358.warc.gz"} |
https://physics.stackexchange.com/questions/230249/entropy-is-disorder/230270 | Entropy is…disorder?
As I read somewhere, it said that the universe is heading toward disorder a.k.a entropy increasing.
Now as far as I know from the second law of thermodynamics it states that entropy is indeed increasing and in the end, the entropy of the universe will be maximum, so everything will evolve toward thermodynamic equilibrium (e.g same temperature everywhere in the universe).
So my question is: isn't equilibrium order? Why is entropy called a measure of disorder if more entropy means more order?
• Why do you think equilibrium means ordered? – velut luna Jan 18 '16 at 12:33
• yes, it's a shortcut. More entropy is more "more or equal variousness" than "more or equal disorder". The first is factual, the second subjective – user46925 Jan 18 '16 at 12:43
• @igael To me, the second is less than subjective ... its meaningless. At least to me it is. I've never been able to make any sense of it. I like your phrase. – garyp Jan 18 '16 at 12:52
• @OP: Note that thermal and mechanical equilibrium are two different things. The latter one usually is associated with some (spatial) ordering – Bort Jan 18 '16 at 13:54
• Essential: Entropy Demystified: The Second Law Reduced to Plain Common Sense by Arieh Ben-Naim. – Martín-Blas Pérez Pinilla Jan 19 '16 at 7:33
8 Answers
What you are missing is the microscopic definition of entropy, once you know that, you will understand why people say that entropy is disorder.
Equilibrium as order
First, let's address your valid intuition that equilibrium as a form of order. Indeed, if everything is in thermal equilibrium, you just need to measure the temperature somewhere, and then you will know the temperature of everything. In our out of equilibrium, my body, my laptop, the room, outer space, all have different temperatures, and I need more information to know the state of everything, and I feel this is less "ordered" than the thermal equilibrium case.
What transpires is that less information needed corresponds to a higher degree of order. Well, let's keep that in mind for the next bit.
Entropy is microscopic disorder
In Physics, we know that the properties of macroscopic objects are determined by the motions of the particles that compose them. In particular, temperature of a gas is the disorganised jiggling of the atoms making it up.
As you increase the temperature, the atoms will move more and more erratically, and will have diverse speeds at any given time.
As you cool it, the particles will move slower and slower, until perhaps they freeze in place, forming a solid.
Which of the two - the still, regular lattice of the solid or the whizzing commotion of the particles that forms a gas - seems to you more disordered? Definitely the second. You know from thermodynamics that the gas has higher entropy than the solid. Indeed, there is a precise formula linking the macroscopic state variable $S$, entropy, and the microscopic conception of disorder I described.
Conclusion: the two ideas are reconcilable
In the projected "heat death" of the universe, everywhere there is constant temperature and density. In that sense, the universe is homogeneous and thus ordered. But microscopically - in the movements of the particles - that is the state in which there is the least order: no structure whatsoever, just a big soup of whizzing particles.
• Hmm...really interesting, I thought that in the "heat death" scenario the particles won't really move because energy can't be moved in the eventual case of thermal equilibrium. – griffinwish Jan 18 '16 at 14:40
• That's right: there is no coherent movement of energy from one region of space to another because everything is in thermal equilibrium. But at the microscopic level, there is movement. Remember that temperature is the jiggling of particles: no movement would mean absolute zero temperature. Individual particles themselves have no temperature: just position and velocity. They move around like billiard bald, colliding and exchanging energy. Thermal equilibrium means that the collisions don't have the effect of transporting energy around. – Andrea Jan 18 '16 at 15:12
• In fact, as the entropy increases constantly, the universe must have, started with surprisingly low entropy. The why of this is still an unresolved mystery. For the second question: Entropy is has a technical definition, and can be calculated precisely for various macroscopic configurations. Maybe you can start from here en.wikipedia.org/wiki/Entropy_(order_and_disorder) although this is definitely not the best resource. Penrose's talk linked by @rmhleo is also a very good place to get more. – Andrea Jan 18 '16 at 16:07
• @griffinwish Because the universe apparently started from a very ordered state. I don't think it's known why that happened. (Any state leads towards one with maximum entropy; we have less-than-maximum entropy so we must have started from a state with even less) – user253751 Jan 18 '16 at 20:18
• Exactly, and I finally think I got a better understanding of this. Disorder tends to increase because there are many more disordered states than ordered ones so it is more probable for a thing to be disordered, right? Just like ice cubes in a glass and liquid water, the latter having more entropy because water molecules are flying around inside and are not standing still like in ice cubes, so there are many more ways to arrange those molecules in a disordered state thus increasing the entropy. – griffinwish Jan 18 '16 at 20:47
I personally find the terms consistent. Think of the entropy as Boltzman proposes: $S=k \, \ln W$ Meaning high entropy states can be realized via many different configurations. Truly ordered state (assume you arrange a sculpture from atoms) can be realized via much smaller number of microscopic states. So again, equilibrium is not order - it is a mess.
• equilibrium is not order - it is a mess I might quote you for this some day. +1 – Steeven Jan 18 '16 at 14:00
• This is not true: a solid lattice is a highly ordered configuration, and is the most probable one as you cool down a gas, or liquid. Also different states of matter can coexist in certain trivial conditions and in a very stable status (i.e. water in triple point). – rmhleo Jan 18 '16 at 15:29
• @rmhleo: processes like crystallisation take place because $\Delta G= \Delta H - T\Delta S < 0$ and much lattice energy is released. – Gert Jan 18 '16 at 16:34
• @Steeven: I like Peter Atkins' "Everything always ends badly!". – Gert Jan 18 '16 at 16:35
• The trouble with using "disorder" is that it assumes a microscopic view. Equilibrium systems are complicated at the microscopic level, but simple at the macroscopic level. Indeed that is one of the ways that classical thermodynamics defines equilibrium "being adequately described by a small number of state variables" – dmckee Jan 18 '16 at 18:08
First of all as stated by Madan Ivan: equilibrium is not order. But you can get certain systems that are in a meta-stable "local" equilibrium (here meaning that you need some energy to move it from there), for example a crystal. These can be highly ordered.
Intuitively: if you smack the crystal with a hammer it breaks to pieces. This brings your closer to the global equilibrium. In the universe as a whole there is energy exchange between such subsystems and the second law of thermodynamics states that the overall order decreases by these processes.
So I think your problem is the two uses of the word equilibrium. Meta-stable equilibria can be order while the one that is used in the second law is the global minimum.
A comment on entropy in general: there isn't just one, there is a lot of them. In thermodynamics only there are 3 distinct ones. The names I use in the following are not official, since the literature mostly does not distinguish between them.
1. The Gibbs entropy: $$S_G = -k \sum_{N} \int d \tau_N p_N \log(p_N)$$ where the sum is over all the states of the system and $p_N$ is the probability of it. It turns out that this is a constant of the equations of motion.
2. The Boltzmann entropy: $$S_B = -k \sum_{1} \int d \tau_1 p_1 \log(p_1)$$ where $p_1$ is now the one particle distribution. This entropy is just wrong, but used a lot.
3. Experimental entropy: $$\Delta S_E = \int dQ/T$$ This is the one that increases.
It can be shown that both 1. and 3. are important quantities, but the second law applies to the 3. one.
References: Unfortunately I can only link to this http://www.oxfordmartin.ox.ac.uk/event/1348 which is where I got the information from.
• No wonder you found no references, because this is wrong. In particular the Gibbs entropy is equal to the thermodynamical entropy. The Gibbs entropy never decreases, and is not a "constant of the equations of motion", except at equilibrium of course. – ederag Jan 19 '16 at 16:30
• Jayne's article, which is referenced in the wikipedia article you posted, can be found here: bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf I quote from the abstract: "(5) the dynamical invariance of the Gibbs H gives a simple proof of the second law for arbitrary interparticle forces." So in fact he states (in the obscured form of H) that the Gibbs entropy is invariant. What he also shows though is that the experimental and the Gibbs entropy are equal FOR THE CANONICAL ENSEMBLE. For a general distribution he then obtains that S_E >= S_Gibbs, which is the second law. – Wolpertinger Jan 19 '16 at 19:44
• Thanks for the link. In this paper the Gibbs entropy is claimed to be constant only in a special transformation: "Now force the system to carry out an adiabatic change of state (i.e., one involving no heat flow to or from its environment), by applying some time-dependent term in the Hamiltonian (such as moving a piston or varying a magnetic field)." This transformation is both adiabatic and reversible (and thus isentropic), hence the Gibbs entropy, like the thermodynamic one, is constant. For other transformations the Gibbs entropy may vary (like the thermodynamic one). – ederag Jan 20 '16 at 19:43
• What you are saying now is absolutely correct. The Gibbs entropy of a SUBSYSTEM may vary. But the universe as a whole always undergoes an adiabatic change. About the reversible assumption: I think it is still safe to say that the Gibbs entropy is a constant of the full quantum mechanical equations of motion (i.e. Schrödinger equation), which doesn't seem to be proven in this paper though. Thermodynamically irreversible processes wash out quantum correlations, but are still microscopically reversible. I will try to find a reference for this. – Wolpertinger Jan 21 '16 at 10:35
• 1) The entropy of an isolated system (like the universe) may increase, even if the microscopic equations of motion are reversible. Do you really disagree with that ? 2) No, in general the Gibbs entropy is not a "constant of the full quantum mechanical equations of motion". Except in some cases, for instance an isolated system undergoing a reversible transformation. 3) Do you really assume that the universe is following a reversible transformation ? – ederag Jan 21 '16 at 17:34
Entropy is not disorder; it is a lack of information.
Consider the entropy formula $S = k_b \log \Omega$. Here, $\Omega$ is the number of microstates (sets of particle positions/momenta) corresponding to an observed macrostate (something macroscopic we can observe, like 'the gas has volume $V$ and pressure $P$). What this formula means is that the entropy is proportional to the amount of information we are missing -- the number of extra bits we would need to know, on top of knowing the macrostate, to full specify the microstate.
For example, consider heat transfer $Q$ and work $W$. Though both exchange energy, only the first increases entropy. That makes sense, because the only difference between heat transfer and work is that heat transfer is done in a disordered way. We don't know exactly how it happened, so our lack of information goes up.
Since heat transfer increases entropy, the maximum entropy is achieved at thermal equilibrium. At that point, we basically know nothing at all.
• but wouldn't thermal equilibrium mean that no heat transfer exists in the whole universe? So basically, wouldn't we know that there is nothing to know hence no entropy? – griffinwish Jan 18 '16 at 16:02
• You won't know exactly where all of the energy is. In thermal equilibrium, can you tell me where molecule #1375039 is and how fast it's moving? – knzhou Jan 18 '16 at 16:03
• @griffinwish However, note that this is unlikely to be a stable state - once you achieve maximum entropy, random actions will result in decreasing entropy. There's a neat little idea that that's what our universe is - a blob of low entropy that spontaneously formed in a maximum entropy "superuniverse". Energy and information are conserved, you just get local minima all the time - until the bubble of low entropy reaches maximum entropy again in a few (hundred) billion years. There's a lot of problems with that idea, but... it's pretty neat :) – Luaan Jan 18 '16 at 17:03
• wow..amazing theory. So after we reach maximum entropy we will create another blob of low entropy. Hah, that's interesting – griffinwish Jan 18 '16 at 20:49
No actually this is one perpetuating myth about entropy that even scientists themselves (and school curricula) propagate.
To answer this and dispel the myth, ask this simple question: disorder with respect to what exactly?
Why is a uniform gas disordered than a gas with two phases?
Of course a uniform gas has more (another) symmetry, in fact aquires the symmetries of the underlying environment. But so does the the two-phase gas, it has a certain symmetry (and not others) deriving from the underlying environmental process. So far so good. Where is the "disorder" exactly, and with respect to what and to whom is this a "disorder"? i think you get the point meant here.
Clearly there is a very subjective (to mention the least) concept of disorder used here which is not explained anywhere. Just stated as fact which is not.
Some take this further equating entropy with death vs life which is even more absurd. One can have a series of cages perfectly ordered, yet one will not have life in them.
Please consider this before you just accept anything thrown at you sounding scientific (while it is not)
PS
If you want the full scientific version of this answer check (especialy) the works of I. Prigogine on Entropy, Complex Dynamic Systems and Biological Systems. e.g "From Being to Becoming: Time and Complexity in the Physical Sciences"
Other schools of thermodynamics also have similar approaches and hard facts to consider. For a popular, yet somewhat thorough exposition check, for example: "The Arrow Of Time: A Voyage Through Science To Solve Time's Greatest Mystery"
To summarise:
entropy
1. is NOT disorder (mechanistic approach)
2. is NOT lack of information (bayesian/subjectivist approach),
3. is NOT contrary to evolution (inteligent design-approach)
4. is NOT simply a statistical effect (quantum-mechanical/statistical approach)
5. is NOT related solely to linear and (static) equilibrium processes, in fact entropy and (yes) the 2nd Law have been generalised (i would say simply clarified) for (dynamic) non-equilibrium / non-linear processes
abstract
In the scientific and engineering literature, the second law of thermodynamics is expressed in terms of the behavior of entropy in reversible and irreversible processes. According to the prevailing statistical mechanics interpretation the entropy is viewed as a nonphysical statistical attribute, a measure of either disorder in a system, or lack of information about the system, or erasure of information collected about the system, and a plethora of analytic expressions are proposed for the various measures. In this paper, we present two expositions of thermodynamics (both ’revolutionary’ in the sense of Thomas Kuhn with respect to conventional statistical mechanics and traditional expositions of thermodynamics) that apply to all systems (both macroscopic and microscopic, including single particle or single spin systems), and to all states (thermodynamic or stable equilibrium, nonequilibrium, and other states). .. Here entropy emerges as a microscopic nonstatistical property of matter.
Entropy is one of the most basic facts (and least understood, analysed) related directly to causality, the arrow of time, quantum-mechanics and evolution.
In fact most (if not all) time-reversible equations are wrong (ot at least crude approximations) rather than entropy and the time arrow itself.
To quote the cosmologist Arthur Eddington:
The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations - then so much the worse for Maxwell's equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
The references given above dispel all these misconceptions.
• This is the only valid answer. – SuchDoge Jul 10 '17 at 20:46
Entropy is a tricky concept and hard to understand. Personally I tend to avoid speaking of systems and phenomena in terms of entropy and/or temperature because they say very little of the dynamics, and I believe dynamical laws are the ones driving the universe.
When we hear that systems tend to increase entropy, we are saying there are dynamical laws driving them towards states of higher entropy. But this comes from our knowledge that for simple systems with elementary microscopic behavior (like ideal gases, or ideal liquids) when comparing two states of equilibrium, the one with higher entropy is more stable.
This might be misunderstood as an evidence that systems in general evolve by increasing entropy, which can be proven wrong. In fact the universe evolves in such a way that instead of tending to be homogeneous, is highly organised (galaxies, stars, planets, living beings).
My approach to this would be twofold: first microscopic dynamics is not elementary, which means that molecules have more degrees of freedom than we conceive when we tend to think only in terms of entropy to predict the behavior of the system. This is the same idea of Gibbs when he extended classical thermodynamics by allowing the number of molecules to change, which accounts for systems in which reactions may occur. But we can think of other types of "qualitative changes" (as I like to call them), as did Terrell Hill in his conception of Thermodynamics of Small Systems.
Secondly, I think we should not forget that the dynamics of evolution of physical system are fundamentally different from what we expect by saying that systems tend to increase their entropy, this is simply not verified, and in my opinion, misleading.
A final note in saying that Temperature, as Entropy, refers to equilibrated states and is also wrongly believed to behave the way energy does. But this is not the case: the dynamics of systems does not depend on temperature, but on the relative energies of the involved parties. Microscopically speaking, the collision dynamics depends on the relative energies or momenta, rather than on their average. Also in a non-equilibrated system, temperature (understood as mean kinetic energy) will largely fluctuate spatially before the whole system achieves the equilibrium.
PS: Sir Roger Penrose has very interesting arguments on the concept of Entropy and Universe evolution in this talk
• It is misleading to mention galaxies and living beings as examples of cases where entropy does not increase. Living being subsist only because they eat low entropy, organised, food and excrete high entropy matter. The second law states that the entropy of an isolated system always increases. A living being is emphatically not an isolated system: it is firmly embedded in the Earth. The Earth itself manages to keep a relatively stable entropy because it absorbs low entropy light from the Sun and emits high entropy infrared. Penrose himself points that out in his talk. – Andrea Jan 18 '16 at 15:25
• I think those examples show how the reality evolves in such a way that gives rise to highly ordered systems. This disproves the idea that growth of entropy is the direction of evolution of the systems. I agree that leaving systems is not a good example of physical one, since highly evolved organisms can actively act against physical processes. But again, this is not the point for mentioning them. Also a star and planets are more organised forms compared to matter being smeared homogeneously. – rmhleo Jan 18 '16 at 15:37
• Living beings are physical systems, and they do not act against physical processes. They are just not closed physical systems. It is true that their entropy does not increase, but that's only because they increase the entropy of the surroundings. – Andrea Jan 18 '16 at 15:56
• Stars and planets are more organised forms of matter, in the same sense that a stationary ball at the bottom of a well is more organised than one that is frantically bouncing around. But again, both processes involved the increase of entropy in the universe as a whole! The 2nd law is not incompatible with the creation of structure. – Andrea Jan 18 '16 at 15:56
• "I do not agree that all organised systems subsist by disorganising other systems." It depends on what you mean by systems. An homogeneous gas cloud that collapses under the gravitational attraction into a concentrated star and orbiting planets does so while increasing the entropy of the universe. Indeed the star is radiating heat all around, spreading countless photons in the universe. The radiation might not disrupt other "systems" ( as in other stars and planets, for example) but it is still increasing the entropy of the universe as a whole. There is no arguing with this. – Andrea Jan 18 '16 at 16:13
The entropy law can be (comically) reinterpreted like "equilibrium is a state of maximum possible disorder under given physical constraints". So... things keep getting worse until it's as bad as it can get. Intuitively, large entropy means that things look more or less the same (macroscopically) for many different microscopic realizations. When the system evolves, it's statistically easy to find yourself in one of the many high-entropy states, but very rarely you can randomly stumble upon an ordered state. Imagine trying to shake a box of coins: what's the probability that you'll get all tails? The equilibrium state (you keep shaking the box - simulation of thermal motion) will be somewhere around half tails half heads, plusminus the standard deviation, typical for this system (after binomial distribution). So... disorder. In other comparison, parents all over the world know that the room only gets messier and reaches a state of chaos (this being the equilibrium state). You must put in work to make it tidy again, and it doesn't stay that way for very long.
I'm giving a common sense illustration because the physics has already been covered by other posts. People keep saying entropy is a difficult concept to grasp, but that's only if you don't explain it right.
• So correct me if I'm mistaken, the entropy of the universe is increasing because particles are always moving towards a disordered state because there are many more disordered states than ordered ones so statistically it is more probable for a system to end up disordered than ordered. Another question: why when I exercise I increase the entropy of the universe? I get that I release more energy into the surroundings but what does that have to do with increasing the disorder? – griffinwish Jan 19 '16 at 11:21
• Yes, that's correct. Whatever you do that isn't reversible, increses the entropy of the universe. To do work, you take advantage of some order (temperature difference, chemical compounds in non-equilibrium [volatile] form, potential energy...) and make it disordered in the process. By doing work, you have a side effect of waste heat (heat is the most disordered, high-entropy form of energy that can't be used for work anymore). Heat engines, for instance, operate on temperature difference, producing work, but heat up the cold reservoir, ruining the temperature difference. – orion Jan 19 '16 at 15:48
Terms are conventions. With a point of view from the humans we are the order. Collecting something and order it in shells is order. But I agree with you that to order something needs energy and this led to misorder and this could be a possible convention too. But it is not.
protected by Qmechanic♦Jan 19 '16 at 10:54
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https://www.appropedia.org/Alternative_Transportation_Wheel | Project data
Authors Jake MayO. Rogerskaelon martinChase E. Bloom 2012 USD 50 Download Upload your project too!body.poncho-dark-mode .mw-parser-output .mw-ui-button{background:#303134;border-color:#3c4043;color:#bdc1c6}@media screen and (max-width:500px){.mw-parser-output .button .mw-ui-button{display:block;margin:.5em 0}}
The purpose of this page is to show off the Alternative Transportation Wheel that our ERE215 group built for Friends of the Dunes.
Our main goal of the transportation wheel is to get visitors of FOTD to think about the mode of transportation they used to arrive there and the corresponding carbon footprint that they have just produced.
## Problem Statement and Criteria
### Problem Statement
The objective is to create an interactive exhibit that will get visitors to think about their means of travel and the impact they are having on the environment.
### Criteria
Criteria Weight Description
Cost 6 Must be less than or equal to $400. Longevity 10 Must withstand a lifespan of 6 or more months at friends of the dunes. Aesthetics 9 Must fit in with the surrounding area, must stand out and grab visitor's attention. Safety 10 Must be usable by all ages and have no parts that can harm users. Educational Value 7 Teaches people about carbon footprints and gets them to think about their impact on the environment. Simplicity 9 Easy to use by all ages, easy to read for adults. Site Appropriate 10 Must fit in with FOTD and teach a lesson that's related to coastal ecosystems. Durability 10 Must be able to stand on its own without breaking in any way. ## Description of Final Design ### Design Cost Included is a graph showing the amount of hours we put into each individual part of the project. ### Implementation Cost Materials cost table Material Quantity Our Cost Subtotal Retail Cost Subtotal Bike Tire 1 Donated$50.00
Total $0$50.00
### Maintenance Cost
Maintenance of the Alternate Transportation Wheel is illustrated in the table below, in number of hours per month.
Maintenance cost table | 2022-08-13 21:34:06 | {"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": 0.366524338722229, "perplexity": 4379.622982020175}, "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/1659882571987.60/warc/CC-MAIN-20220813202507-20220813232507-00035.warc.gz"} |
http://mathhelpforum.com/calculus/33667-critical-numbers.html | # Math Help - Critical numbers
1. ## Critical numbers
Need a bit of help
Determine
a.) All the critical numbers
b.) all relative extrema
$y(x)=(x-1)e^{2x}$
I'm thinking my derivative is ...
$2*(x-1)*(1)*e^{2x}$
Multiple choice:
a.){ $\frac{1}{2}$,max}
b.){0,min}
c.) { $\frac{1}{2}$, min}
d.) none of these
I'm thinking it is $\frac{1}{2}$ min
2. Originally Posted by XIII13Thirteen
Need a bit of help
Determine
a.) All the critical numbers
b.) all relative extrema
$y(x)=(x-1)e^{2x}$
I'm thinking my derivative is ...
$2*(x-1)*(1)*e^{2x}$
Multiple choice:
a.){ $\frac{1}{2}$,max}
b.){0,min}
c.) { $\frac{1}{2}$, min}
d.) none of these
I'm thinking it is $\frac{1}{2}$ min
you derivative is wrong. you need the product rule here.
3. $(x-1)2e^{2x}+(1)e^{2x}$?
4. Hello,
Yes, it's it
$(x-1)2e^{2x}+(1)e^{2x}=e^{2x}(2x-2+1)=e^{2x}(2x-1)$
5. Originally Posted by XIII13Thirteen
$(x-1)2e^{2x}+(1)e^{2x}$?
very good. and Moo was kind enough to factorize it for us. now set Moo's factorized form to zero and solve for your critical points.
6. Originally Posted by Jhevon
very good. and Moo was kind enough to factorize it for us. now set Moo's factorized form to zero and solve for your critical points.
Yes, x=1/2 will zero out the equation. I do believe X>0 is a minimum? If so than c. Thanks guys | 2014-03-08 18:21:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807246685028076, "perplexity": 6792.75922426383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999657009/warc/CC-MAIN-20140305060737-00079-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/479433/combining-categorical-and-continuous-features-for-neural-networks | # Combining categorical and continuous features for neural networks
Is it OK to combine categorical and continuous features into the same vector for training deep neural networks? Say there is a categorical feature and continuous feature that I want to feed into a deep neural net at the same time. Is this the way to do it?
categorical feature (one-hot encoded) = [0,0,0,1,0]
continuous feature (number) = 8
final feature vector passed into neural network = categorical feature vector CONCATENATE continuous feature = [0,0,0,1,0,8]
Basically, the question is, is it OK to have a one-hot encoding and a continuous feature together in one feature vector?
• – Sycorax Aug 1 '20 at 15:31
• – Sycorax Aug 1 '20 at 15:31
Yes, that is one typical way of doing it. But, you need to standardize your features so that gradient descent doesn't suffer, and the regularization treats your weights equally. One way is to standardize the numerical features and then concatenate the one-hot vectors, and the other way is standardizing together. As far as I see, there is no consensus over the two.
Yes, this is absolutely standard. | 2021-04-20 13:36:24 | {"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": 0.4386289417743683, "perplexity": 957.2549346303276}, "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-2021-17/segments/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00426.warc.gz"} |
https://www.physicsforums.com/threads/residue-calculus.386039/ | # Homework Help: Residue Calculus
1. Mar 12, 2010
### kuahji
$$\int$$u^-B sin(u) du, 0<B<2 integrating from 0 to infinity. What is really throwing me off is the condition, I'm not sure why it's there or really what to do with it. Can I just solve this the same way I'd solve sin(x)/x?
2. Mar 12, 2010
### kuahji
Had I not something strange like u^B, I'd solve the problem as follows.
$$\frac{1}{u^B}$$(u-$$\frac{z^3}{6}$$+...)
I'd then look for C_-1, the residue. Then it's just a simple matter of 2pi(i)*res. But again, having u^B does not allow me to do this. There is a systematic formula I can apply to find the residue, but it does not allow me to eliminate i. | 2018-06-21 01:09:03 | {"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": 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": 0.8133964538574219, "perplexity": 662.9094107996291}, "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/1529267863980.55/warc/CC-MAIN-20180621001211-20180621021211-00120.warc.gz"} |
http://unapologetic.wordpress.com/2008/02/15/ | # The Unapologetic Mathematician
## The Fundamental Theorem of Calculus (all together now)
So we’ve seen two sides of the FToC: the first part, which says that given a continuous function $f:\left[a,b\right]\rightarrow\mathbb{R}$ we can integrate and differentiate to get our function back:
$\displaystyle\frac{d}{dx}\int\limits_a^xf(t)dt=f(x)$
and the second part, which says that given a differentiable function $F:\left[a,b\right]\rightarrow\mathbb{R}$ whose derivative is the continuous function $f$, we can integrate to get (part of) our function back again:
$\displaystyle F(b)-F(a)=\int\limits_a^bf(x)dx$
Now, we proved these two sides in very different ways, but it turns out that we can get from one to the other.
Let’s assume the first part holds. Then we take the function $F$ and define $f(x)=F'(x)$ as its derivative. The first part of the theorem tells us that we know a function whose derivative is $f$: the function defined by $G(x)=\int_a^xf(t)dt$. And we know that any two functions with the same derivative must differ by a constant! That is, there is some real number $C$ with $F(x)=G(x)+C$. Using this to evaluate $F(b)-F(a)$ we find:
$\displaystyle F(b)-F(a)=(G(b)+C)-(G(a)+C)=G(b)-G(a)=$
$\displaystyle\int\limits_a^bf(x)dx-\int\limits_a^af(x)dx=\int\limits_a^bf(x)dx$
Which gives us the second part of the theorem.
On the other hand, what if we assume the second part of the theorem holds? Then we start with a continuous function $f:\left[a,b\right]\rightarrow\mathbb{R}$. Given $x\in\left[a,b\right]$, the function is continuous on the subinterval $\left[a,x\right]$, and so the second part of the FToC says that $F(x)-F(a)=\int_a^xf(t)dt$. That is, the integral in the first part of the FToC differs by a constant ($F(a)$) from the function $F$ we assumed to be an antiderivative of $f$. Thus it must itself be an antiderivative of $f$.
So each half of the Fundamental Theorem implies the other, and we can prove either one first before immediately deriving the other.
February 15, 2008 Posted by | Analysis, Calculus | 125 Comments | 2013-12-10 12:01:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 22, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9840460419654846, "perplexity": 108.6164483899354}, "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/1386164018116/warc/CC-MAIN-20131204133338-00089-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/91837/how-to-apply-the-simplex-method-to-prove-that-the-following-problem-is-unbounded | # How to apply the simplex method to prove that the following problem is unbounded?
$\max 6t_1 + 4t_2$
$-t_1 + t_2 \leq 6$
$t_1 - t_2 \leq 1$
$t_1 - 2t_2 \leq 8$
$t_1, t_2 \geq 0$
Anyone?
-
Do you know how the simplex method works? Starting with $t_1 = t_2 = 0$, it shouldn't take more than a few iterations. – Mike Spivey Dec 15 '11 at 21:00
draw a picture of the inequalities to see they define an unbounded region – yoyo Dec 15 '11 at 21:21
In particular, note that if $t_1-t_2\leq 1$ and $t_2\geq 0$, then $t_1-2t_2\leq 1< 8$ so the condition $t_1-2t_2\leq 8$ is redundant. – Thomas Andrews Dec 15 '11 at 22:22
If you have been taught the relationship between the primal and dual forms of a linear program then you can take advantage of the fact that if the primal is unbounded then the dual is infeasible.
The dual constraints can be written as:
$-y_1 + y_2 +y_3 \ge 6$
$y_1-y_2-2y_3 \ge 4$
and
$y_1 \ge 0$, $y_2 \ge 0$ and $y_3 \ge 0$.
Adding up the first two inequalities we get:
$-y_3 \ge 10$ which is incompatible with the condition that $y_3 \ge 0$.
Thus, the dual constraints are infeasible which implies that the primal is unbounded.
-
Your argument is missing one small piece. Infeasibility of the dual implies that either the primal is unbounded or the primal is infeasible. It's easy to show the latter isn't the case, though; $t_1 = t_2 = 0$ is feasible for the primal. And then your argument goes through from there. – Mike Spivey Dec 17 '11 at 21:13 | 2016-05-01 12:25:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9649872779846191, "perplexity": 196.44789049767954}, "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-2016-18/segments/1461860115836.8/warc/CC-MAIN-20160428161515-00040-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/write-answer-directedgive-any-four-functional-groups-containing-oxygen-heteroatom-it-write-name-structural-formula-one-example-each-functional-groups-carbon-compounds_51830 | # Write Answer as Directed.Give Any Four Functional Groups Containing Oxygen as the Heteroatom in It. Write Name and Structural Formula of One Example Each. - Science and Technology 1
Give any four functional groups containing oxygen as the heteroatom in it. Write name and structural formula of one example each.
#### Solution
Four functional groups containing oxygen as the heteroatom in it are as follows:
1. Alcohols: The functional group, which is present in alcohol, is -OH. The IUPAC group suffix of alcohol is –ol.
Example: Ethanol
$\begin{array}{cc} |\phantom{........}|\phantom{....}\\ \ce{-C-OH-C-OH}\\ |\phantom{........}|\phantom{....}\\ \end{array}$
2. Aldehydes: The functional group, which is present in an aldehyde, is -CHO. The IUPAC group suffix of an aldehyde is –al.
Example: Formaldehyde
$\begin{array}{cc} \phantom{..}\ce{O}\\ |\phantom{...}||\phantom{..}\\ \ce{-C-C-H}\\ |\phantom{......} \end{array}$
3. Carboxylic acid: The functional group present in a carboxylic acid is -COOH. The IUPAC group suffix of a carboxylic acid is –oic acid.
Example: Acetic acid
$\begin{array}{cc} \phantom{.}\ce{O}\\ |\phantom{...}||\phantom{...}\\ \ce{-C-C-OH}\\ |\phantom{.......} \end{array}$
4. Ketones
Examples:- butanone
$\begin{array}{cc} \phantom{......}\ce{O}\\ \phantom{........}||\phantom{..}\\ \ce{CH3-CH2-C-CH3}\\ \end{array}$
#### Notes
$\begin{array}{cc} |\phantom{........}|\phantom{....}\\ \ce{-C-OH-C-OH}\\ |\phantom{........}|\phantom{....}\\ \end{array}$
Concept: Functional Groups in Carbon Compounds
Is there an error in this question or solution? | 2021-04-12 15:54:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3633922040462494, "perplexity": 12498.665034143736}, "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/1618038067870.12/warc/CC-MAIN-20210412144351-20210412174351-00346.warc.gz"} |
http://www.math.keio.ac.jp/~ishikawa/QLEJ/index115.html | Abstract (18.0: The reliability in psychological test)
Abstract: In this chapter, we will introduce the measurement theoretical approach to a problem of analyzing scores of tests for students. The obtained score is assumed to be the sum of a true value and a measurement error caused by the test, in which a student's score is subject to a systematic error (=noise) depending on his/her health or psychological condition at the test. In such cases, statistical measurements are convenient since these two errors (i.e., measurement error and systematic error) in measurement theory can be characterized in the different mathematical structures respectively. The goodness of a test is characterized by "reliability coefficient" such that \begin{align*} \mbox{ [reliability coefficient] }^2 = \frac{\mbox{[systematic error]}^2}{ \mbox{[systematic error]}^2+\mbox{[measurement error]}^2 } \end{align*} Now we have the following problem:
How do we calculate the "reliability coefficient"?
This will be answered in this chapter.
This chapter is extracted from the following.
$(\sharp):$ K. Kikuchi, S. Ishikawa, "Psychological tests in Measurement Theory," Far east journal of theoretical statistics,} 32(1) 81-99, (2010)
Again recall that, as mentioned in $\S$1.1, the main purpose of this book is to assert the following figure 1.1:
Fig.1.1: the location of "quantum language" in the world-views
This(particularly, ⑦--⑨) implies that quantum language has the following three aspects: $$\left\{\begin{array}{ll} \mbox{ ⑦ :the standard interpretation of quantum mechanics} \\ \mbox{ \qquad (i.e., the true colors of the Copenhagen interpretation) } \\ \\ \mbox{ ⑧ : the final goal of the dualistic idealism (Descartes=Kant philosophy) } \\ \\ \mbox{ ⑨ : theoretical statistics of the future } \end{array}\right.$$ | 2021-10-22 07:11:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8851101994514465, "perplexity": 1436.3290818926944}, "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-2021-43/segments/1634323585460.87/warc/CC-MAIN-20211022052742-20211022082742-00682.warc.gz"} |
http://clay6.com/qa/33264/-ch-3-3cmgcl-on-reaction-with-d-2o-produces | Browse Questions
# $(CH_3)_3CMgCl$ on reaction with $D_2O$ produces
$\begin{array}{1 1}(a)\;(CH_3)_3CD&(b)\;(CH_3)_3OD\\(c)\;(CD_3)_3CD&(d)\;(CD_3)_3OD\end{array}$
hence (a) is the correct answer. | 2017-06-26 14:02:53 | {"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": 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": 0.984485924243927, "perplexity": 5092.023504192496}, "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-2017-26/segments/1498128320763.95/warc/CC-MAIN-20170626133830-20170626153830-00680.warc.gz"} |
https://toontown.fandom.com/wiki/Megaphone | ## FANDOM
2,275 Pages
Needs Sound This article is in need of a sound file or more! Please help Toontown Wiki by inserting some!
Megaphone
Level 2 Toon-Up Gag
General information
Minimum Laff Heal: 15
Maximum Laff Heal: 18
Accuracy: Medium
Organic boost: 19 Laff Heal
Targets: All Toons
Minimum carry capacity: 5
Maximum carry capacity: 25
Experience points needed for next gag: 200
Lineage
Preceded by:
Feather
Succeeded by:
Lipstick
v • d • e
Megaphone is the level two Toon-Up gag. It succeeds the Feather but precedes the Lipstick.
## General
The Megaphone can be obtained after gaining twenty skill points in the Toon-Up track.
The Megaphone can heal all toons, excluding the user; the healing will be divided evenly among all toons. The minimum heal is fifteen for one toon, eight for two toons, and five for three toons. The maximum heal is eighteen for one toon, nine for two toons, and six for three toons. If the Megaphone is grown on a tree and is organic, the heal increases by one, totaling to a maximum of nineteen laff points healed.
In the event that the megaphone misses, it will still restore between "3" and "4" laff points depending on its skill points, and the laff is distributed amongst the other toons. This will result in a restoration of "3" to "4" laff points for one toon, "2" laff points for two toons, and "1" laff point for three toons.
At first, a toon can carry a minimum of five Megaphones. As the toon trains their Toon-Up track and obtains new gags, the toon will be able to carry more Megaphones. After obtaining the Juggling Balls, a maximum of twenty-five Megaphones can be carried at once.
## Skill points
Previous level Next level
20 200
Formula
In equation form: $\frac{X - Y}{(M - O)}$
• X represents the skill points for the next gag
• Y represents the starting point of this gag
• M represents the maximum laff heal
• O represents the original starting point
To determine the next increase in laff heal for a Megaphone, take the number of skill points required to obtain the next gag and divide it by the number of increase (the maximum laff heal - the original laff heal). You should therefore get an estimate.
The Megaphone equation:
$\frac{200-20}{18-14}=\frac{180}{4}=45$
For every 45 skill points, the Megaphone's healing attributes increase.
Laff heal Skill points
15 20
16 65
17 110
18 155
## Animation
1. The toon runs to the center of the cog battle.
2. A Megaphone is taken out from the toon's right pocket.
3. The toon says a joke through the Megaphone.
4. Other toons are healed by an amount of laff points depending on how funny the joke was (if it misses, it gives 2 points; if it hits when used on only one toon, it gives a maximum of 18 laff)
5. The toon runs back to the original position.
## Trading card
Gag
Does your megaphone sounds funny? Ours is downright hilarious! After a hard day of battling Cogs, did you ever wish you could cheer up your friends with a funny joke, but just can't think of one? Your troubles are over with MegO'Feeney's Mega-Funny Megaphones! Crammed full of dozens of the funniest jokes ever told in Toontown, your friends will be laffing themselves back to good health in no time!
## Jokes
What goes 'Ha Ha Ha Thud'?Someone laughing his head off. What goes TICK-TICK-TICK-WOOF?A watchdog! Why do male deer need braces?Because they have 'buck teeth'! Why is it hard for a ghost to tell a lie?Because you can see right through him. What did the ballerina do when she hurt her foot?She called the toe truck! What has one horn and gives milk?A milk truck! Why don't witches ride their brooms when they're angry?They don't want to fly off the handle! Why did the dolphin cross the ocean?To get to the other tide. What kind of mistakes do spooks make?Boo boos. Why did the chicken cross the playground?To get to the other slide! Where does a peacock go when he loses his tail?A retail store. Why didn't the skeleton cross the road?He didn't have the guts. Why wouldn't they let the butterfly into the dance?Because it was a moth ball. What's gray and squirts jam at you?A mouse eating a doughnut. What happened when 500 hares got loose on the main street?The police had to comb the area. What's the difference between a fish and a piano?You can tune a piano, but you can't tuna fish! What do people do in clock factories?They make faces all day. What do you call a blind dinosaur?An I-don't-think-he-saurus. If you drop a white hat into the Red Sea, what does it become?Wet. Why was Cinderella thrown off the basketball team?She ran away from the ball. Why was Cinderella such a bad player?She had a pumpkin for a coach. What two things can't you have for breakfast?Lunch and dinner. What do you give an elephant with big feet?Big shoes. Where do baby ghosts go during the day?Day-scare centers. What did Snow White say to the photographer?Some day my prints will come. What's Tarzan's favorite song?Jungle bells. What's green and loud?A froghorn. What's worse than raining cats and dogs?Hailing taxis. When is the vet busiest?When it's raining cats and dogs. What do you call a gorilla wearing ear-muffs?Anything you want, he can't hear you. Where would you weigh a whale?At a whale-weigh station. What travels around the world but stays in the corner?A stamp. What do you give a pig with a sore throat?Oinkment. What did the hat say to the scarf?You hang around while I go on a head. What's the best parting gift?A comb. What kind of cats like to go bowling?Alley cats. What did one eye say to the other?Between you and me, something smells. What's round, white and giggles?A tickled onion. What do you get when you cross Bambi with a ghost?Bamboo. Why do golfers take an extra pair of socks?In case they get a hole in one. What do you call a fly with no wings?A walk. Who did Frankenstein take to the prom?His ghoul friend. What lies on its back, one hundred feet in the air?A sleeping centipede. How do you keep a bull from charging?Take away his credit card. What do you call a chicken at the North Pole?Lost. What do you get if you cross a cat with a dog?An animal that chases itself. What did the digital watch say to the grandfather clock?Look dad, no hands. Where does Ariel the mermaid go to see movies?The dive-in. What do you call a mosquito with a tin suit?A bite in shining armor. What do giraffes have that no other animal has?Baby giraffes. Why did the man hit the clock?Because the clock struck first. Why did the apple go out with a fig?Because it couldn't find a date. What do you get when you cross a parrot with a monster?A creature that gets a cracker whenever it asks for one. Why didn't the monster make the football team?Because he threw like a ghoul! What do you get if you cross a Cocker Spaniel with a Poodle and a rooster?A cockapoodledoo! What goes dot-dot-dash-dash-squeak?Mouse code. Why aren't elephants allowed on beaches?They can't keep their trunks up. What is at the end of everything?The letter G. How do trains hear?Through the engineers. What does the winner of a marathon lose?His breath. Why did the pelican refuse to pay for his meal?His bill was too big. What has six eyes but cannot see?Three blind mice. What works only when it's fired?A rocket. Why wasn't there any food left after the monster party?Because everyone was a goblin! What bird can be heard at mealtimes?A swallow. What goes Oh, Oh, Oh?Santa walking backwards! What has green hair and runs through the forest?Moldy locks. Where do ghosts pick up their mail?At the ghost office. Why do dinosaurs have long necks?Because their feet smell. What do mermaids have on toast?Mermarlade. Why do elephants never forget?Because nobody ever tells them anything. What's in the middle of a jellyfish?A jellybutton. What do you call a very popular perfume?A best-smeller. Why can't you play jokes on snakes?Because you can never pull their legs. Why did the baker stop making donuts?He got sick of the hole business. Why do mummies make excellent spies?They're good at keeping things under wraps. How do you stop an elephant from going through the eye of a needle?Tie a knot in its tail. My friend thinks he's a rubber band.I told him to snap out of it. My sister thinks she's a pair of curtains.I told her to pull herself together! Did you hear about the dentist that married the manicurist?Within a month they were fighting tooth and nail. Why do hummingbirds hum?Because they don't know the words. Why did the baby turkey bolt down his food?Because he was a little gobbler. Where did the whale go when it was bankrupt?To the loan shark. How does a sick sheep feel?Baah-aahd. What's gray, weighs 10 pounds and squeaks?A mouse that needs to go on a diet. Why did the dog chase his tail?To make ends meet. Why do elephants wear running shoes?For jogging of course. Why are elephants big and gray?Because if they were small and yellow they'd be canaries. If athletes get tennis elbow what do astronauts get?Missile toe. Did you hear about the man who hated Santa?He suffered from Claustrophobia. Why did Donald sprinkle sugar on his pillow?Because he wanted to have sweet dreams. Why did Goofy take his comb to the dentist?Because it had lost all its teeth. Why did Goofy wear his shirt in the bath?Because the label said wash and wear. Why did the dirty chicken cross the road?For some fowl purpose. Why didn't the skeleton go to the party?He had no body to go with. Why did the burglar take a shower?To make a clean getaway. Why does a sheep have a woolly coat?Because he'd look silly in a plastic one. Why do potatoes argue all the time?They can't see eye to eye. Why did Pluto sleep with a banana peel?So he could slip out of bed in the morning. Why did the mouse wear brown sneakers?His white ones were in the wash. Why are false teeth like stars?They come out at night. Why are Saturday and Sunday so strong?Because the others are weekdays. Why did the archaeologist go bankrupt?Because his career was in ruins. What do you get if you cross the Atlantic on the Titanic?Very wet. What do you get if you cross a chicken with cement?A brick-layer. What do you get if you cross a dog with a phone?A golden receiver. What do you get if you cross an elephant with a shark?Swimming trunks with sharp teeth. What did the tablecloth say to the table?Don't move, I've got you covered. Did you hear about the time Goofy ate a candle?He wanted a light snack. What did the balloon say to the pin?Hi Buster. What did the big chimney say to the little chimney?You're too young to smoke. What did the carpet say to the floor?I got you covered. What did the necklace say to the hat?You go ahead, I'll hang around. What goes zzub-zzub?A bee flying backward. How do you communicate with a fish?Drop him a line. What do you call a dinosaur that's never late?A prontosaurus. What do you get if you cross a bear and a skunk?Winnie-the-phew. How do you clean a tuba?With a tuba toothpaste. What do frogs like to sit on?Toadstools. Why was the math book unhappy?It had too many problems. Why was the school clock punished?It tocked too much. What's a polygon?A dead parrot. What needs a bath and keeps crossing the street?A dirty double crosser. What do you get if you cross a camera with a crocodile?A snap shot. What do you get if you cross an elephant with a canary?A very messy cage. What do you get if you cross a jeweler with a plumber?A ring around the bathtub. What do you get if you cross an elephant with a crow?Lots of broken telephone poles. What do you get if you cross a plum with a tiger?A purple people eater. What's the best way to save water?Dilute it. What's a lazy shoe called?A loafer. What's green, noisy and dangerous?A thundering herd of cucumbers. What color is a shout?Yellow! What do you call a sick duck?A mallardy. What's worse then a giraffe with a sore throat?A centipede with athlete's foot. What goes ABC...slurp...DEF...slurp?Someone eating alphabet soup. What's green and jumps up and down?Lettuce at a dance. What's a cow after she gives birth?De-calf-inated. What do you get if you cross a cow and a camel?Lumpy milk shakes. What's white with black and red spots?A Dalmatian with measles. What's brown has four legs and a trunk?A mouse coming back from vacation. What does a skunk do when it's angry?It raises a stink. What's gray, weighs 200 pounds and says, Here Kitty, kitty?A 200 pound mouse. What's the best way to catch a squirrel?Climb a tree and act like a nut. What's the best way to catch a rabbit?Hide in a bush and make a noise like lettuce. What do you call a spider that just got married?A newly web. What do you call a duck that robs banks?A safe quacker. What's furry, meows and chases mice underwater?A catfish. What's a funny egg called?A practical yolker. What's green on the outside and yellow inside?A banana disguised as a cucumber. What did the elephant say to the lemon?Let's play squash. What weighs 4 tons, has a trunk and is bright red?An embarrassed elephant. What's gray, weighs 4 tons, and wears glass slippers?Cinderelephant. What's an elephant in a fridge called?A very tight squeeze. What did the elephant say to her naughty child?Tusk! Tusk! What did the peanut say to the elephant?Nothing -- Peanuts can't talk. What do elephants say when they bump into each other?Small world, isn't it? What did the cashier say to the register?I'm counting on you. What did the flea say to the other flea?Shall we walk or take the cat? What did the big hand say to the little hand?Got a minute. What does the sea say to the sand?Not much. It usually waves. What did the stocking say to the shoe?See you later, I gotta run. What did one tonsil say to the other tonsil?It must be spring, here comes a swallow. What did the soil say to the rain?Stop, or my name is mud. What did the puddle say to the rain?Drop in sometime. What did the bee say to the rose?Hi, bud. What did the appendix say to the kidney?The doctor's taking me out tonight. What did the window say to the venetian blinds?If it wasn't for you it'd be curtains for me. What did the doctor say to the sick orange?Are you peeling well? What do you get if you cross a chicken with a banjo?A self-plucking chicken. What do you get if you cross a hyena with a bouillon cube?An animal that makes a laughing stock of itself. What do you get if you cross a rabbit with a spider?A hare net. What do you get if you cross a germ with a comedian?Sick jokes. What do you get if you cross a hyena with a mynah bird?An animal that laughs at its own jokes. What do you get if you cross a railway engine with a stick of gum?A chew-chew train. What would you get if you crossed an elephant with a computer?A big know-it-all. What would you get if you crossed an elephant with a skunk?A big stinker. Why did Mickey Mouse take a trip to outer space?He wanted to find Pluto. What's wrong if you keep seeing talking animals?You're having Disney spells.
## Trivia
• This is the only gag where the toon speaks.
• On the trading card, the Megaphone icon looks similar to the Foghorn.
• Apparently, the Megaphone is just for aesthetic purposes, because when the toon is talking through it, the noise level does not rise.
• Other Disney characters are mentioned in the jokes.
• The Megaphone is also used in all Sound Gags and is used in conjunction with annoying, loud things (horns, whistles, etc.) with a Megaphone to deafen cogs.
• There is a spelling error in one of the Megaphone jokes: "What's worse then a giraffe with a sore throat?"
• There is also a spelling mistake in the trading card description: "Does your megaphone sounds funny?"
## In other languages
Language Name
French Mégaphone
Spanish ???
German Megafon
Brazilian Portuguese Megafone
Japanese ???
Community content is available under CC-BY-SA unless otherwise noted. | 2019-08-18 09:48:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20802068710327148, "perplexity": 6089.044788791227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313747.38/warc/CC-MAIN-20190818083417-20190818105417-00264.warc.gz"} |
https://brilliant.org/problems/moment-of-inertia-problem/ | # My First Brilliant Problem!
A uniform disk of radius $$R$$ and mass $$M$$ has a hole of diameter $$R$$ drilled out from it. The hole that is drilled out spans from the center of the disk to the edge of the disk.
$$R = 10 \text{ meters}, M = 5 \text{ kilograms}$$
The axis of rotation is through the center of the disk. What is the moment of inertia of the disk with the hole drilled out? Answer in $$\text{kg m}^2$$.
× | 2017-01-22 14:19:59 | {"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": 0.6987641453742981, "perplexity": 242.83865357478584}, "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-2017-04/segments/1484560281426.63/warc/CC-MAIN-20170116095121-00037-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_7%3A_Techniques_of_Integration/7.7%3A_L'H%C3%B4pital's_Rule/7.7E%3A_Exercises_for_L'H%C3%B4pital's_Rule |
# 7.7E: Exercises for L'Hôpital's Rule
## Terms and Concepts
1. List the different indeterminate forms described in this section.
The forms $$0⋅∞, ∞−∞, 1^∞, ∞^0$$, and $$0^0$$ are all considered indeterminate.
2. List similar looking forms that are not indeterminate.
Among others, the forms $$∞⋅∞, ∞+∞, -∞−∞, 0^∞, 1^0$$, and $$∞^∞$$ are not considered indeterminate, as these limits can be determined clearly.
3. T/F: l'Hôpital's Rule states that $$\frac{d}{dx} \left ( \frac{f(x)}{g(x)}\right ) = \frac{f'(x)}{g'(x)}$$.
False. L'Hôpital's Rule is a method for taking limits of rational functions in certain cases. It does not replace the Quotient Rule when taking the derivative of these rational functions.
4. Explain what the indeterminate form "$$1^{\infty}$$" means. Why is it indeterminate?
When a limit has the form "$$1^{\infty}$$", this means that the function in the base of the exponent is approaching $$1$$, while the function in the exponent is approaching $$\infty$$. It's indeterminate, since if the base function is approaching 1, but always is less than 1, then the limit could be 0, while if the base function were approaching 1, but always is greater than 1, the limit could be $$\infty$$. But since this uncertainty exists, the limit could, in fact, be anything.
5. Explain why limits of the form $$\infty - \infty$$ are indeterminate.
Limits with this form depend on the relative speed with which the two terms are approaching $$\infty$$. If the first term approaches $$\infty$$ faster than the second term, the limit would be $$\infty$$. If the second term apprproaches $$\infty$$ faster than the first term, the limit would be $$-\infty$$. But if they both appraoch $$\infty$$ at about the same rates, the limit could be anything!
6. Fill in the blanks" The Quotient Rule is applied to $$\frac{f(x)}{g(x)}$$ when taking its _____; l'Hôpital's Rule is applied when taking _______ of $$\frac{f(x)}{g(x)}$$ when the form is _______ or _______.
derivative; limits; $$\dfrac{0}{0}$$ or $$\dfrac{\pm\infty}{\pm\infty}$$
7. Create (but do not evaluate) a limit that initially has the form "$$\infty^0$$".
8. Create a function $$f(x)$$ such that $$\lim\limits_{x\to1}f(x)$$ initially has the form "$$0^0$$".
## Problems
For exercises 1 - 6, evaluate the limit.
1) Evaluate the limit $$\displaystyle \lim_{x→∞}\frac{e^x}{x}$$.
2) Evaluate the limit $$\displaystyle \lim_{x→∞}\frac{e^x}{x^k}$$.
$$\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞$$
3) Evaluate the limit $$\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}$$.
4) Evaluate the limit $$\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}$$.
$$\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}$$
5. Evaluate the limit $$\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}$$.
6. Evaluate the limit $$\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}$$.
$$\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}$$
For exercises 7 - 11, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
7) $$\displaystyle \lim_{x→0^+}x^2\ln x$$
8) $$\displaystyle \lim_{x→∞}x^{1/x}$$
Cannot apply directly; use logarithms
9) $$\displaystyle \lim_{x→0}x^{2/x}$$
10) $$\displaystyle \lim_{x→0}\frac{x^2}{1/x}$$
Cannot apply directly; rewrite as $$\displaystyle \lim_{x→0}x^3$$
11) $$\displaystyle \lim_{x→∞}\frac{e^x}{x}$$
For exercises 12 - 44, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
12) $$\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}$$
$$\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6$$
13) $$\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}$$
14) $$\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}$$
$$\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2$$
15) $$\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}$$
16) $$\displaystyle \lim_{x→π}\frac{x−π}{\sin x}$$
$$\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1$$
17) $$\displaystyle \lim_{x→1}\frac{x−1}{\sin x}$$
18) $$\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}$$
$$\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n$$
19) $$\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}$$
20) $$\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}$$
$$\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}$$
21) $$\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}$$
22) $$\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}$$
$$\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}$$
23) $$\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}$$
24) $$\displaystyle \lim_{x→1}\frac{x-1}{\ln x}$$
$$\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1$$
25) $$\displaystyle \lim_{x→0}\,(x+1)^{1/x}$$
26) $$\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}$$
$$\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}$$
27) $$\displaystyle \lim_{x→0^+}x^{2x}$$
28) $$\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)$$
$$\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1$$
29) $$\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}$$
30) $$\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)$$
$$\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0$$
31) $$\displaystyle \lim_{x→∞}(x−e^x)$$
32) $$\displaystyle \lim_{x→∞}x^2e^{−x}$$
$$\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0$$
33) $$\displaystyle \lim_{x\to 1^+} \left[\frac{1}{\ln x}-\frac{1}{1-x}\right]$$
34) $$\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right]$$
$$\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right] = \lim_{x\to 3^+} \frac{5-x^2-3x}{x^2-9} \quad = \quad -∞$$
35) $$\displaystyle \lim_{x\to \infty} \frac{\sqrt{2x^2-3}}{x+2}$$
Note:
L’Hôpital’s rule fails to help us find this limit, although the form seems appropriate. But you can evaluate this limit using techniques you learned earlier in calculus.
36) $$\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x}$$
$$\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x} \quad = \quad e^{4}$$
37) $$\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}$$
38) $$\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}$$
$$\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1$$
39) $$\displaystyle \lim_{x→π/4}(1−\tan x)\cot x$$
40) $$\displaystyle \lim_{x→∞}xe^{1/x}$$
$$\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞$$
41) $$\displaystyle \lim_{x→0}x^{1/\cos x}$$
42) $$\displaystyle \lim_{x→0^+}x^{1/x}$$
$$\displaystyle \lim_{x→0^+}x^{1/x} \quad = \quad 0$$
43) $$\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x$$
44) $$\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x$$
$$\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}$$
For exercises 45 - 54, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
45) [T] $$\displaystyle \lim_{x→0}\frac{e^x−1}{x}$$
46) [T] $$\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)$$
$$\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0$$
47) [T] $$\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}$$
48) [T] $$\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}$$
$$\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1$$
49) [T] $$\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}$$
50) [T] $$\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}$$
$$\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0$$
51) [T] $$\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)$$
52) [T] $$\displaystyle \lim_{x→0^+}\tan\left(x^x\right)$$
$$\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1$$
53) [T] $$\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}$$
54) [T] $$\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}$$
$$\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2$$ | 2021-04-10 19:32:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.977227509021759, "perplexity": 1026.5586983639062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038057476.6/warc/CC-MAIN-20210410181215-20210410211215-00600.warc.gz"} |
http://www.aimspress.com/article/10.3934/math.2020230/fulltext.html | AIMS Mathematics, 2020, 5(4): 3547-3555. doi: 10.3934/math.2020230
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On the Tame automorphisms of differential polynomial algebras
Department of Mathematics, Harran University, Şanlıurfa, Turkey
## Abstract Full Text(HTML) Figure/Table Related pages
Let $R\{x,y\}$ be the differential polynomial algebra in two differential indeterminates $x,y$ over a differential domain $R$ with a derivation operator $\delta$. In this paper, we study on automorphisms of the differential polynomial algebra $R\{x,y\}$ with one derivation operator. Using a method in group theory, we prove that the Tame subgroup of automorphism of $R\{x,y\}$ is the amalgamated free product of the Triangular and the Affine subgroups over their intersection.
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# References
1. M. Aschenbrenner, L. Van Den Dries, J. Van Der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, Princeton University Press, 2017.
2. A. G. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2. I & II, T. Am. Math. Soc., 160 (1971), 393-401; 171 (1972), 309-315.
3. P. M. Cohn, Subalgebras of free associative algebras, P. Lond. Math. Soc., 56 (1964), 618-632.
4. B. A. Duisengaliyeva, A. S. Naurazbekova, U. U. Umirbaev, Tame and wild automorphisms of differential polynomial algebras of rank 2, Fund. Appl. Math., 22 (2019), 101-114.
5. G. Gallo, B. Mishra, F. Ollivier, Some constructions in rings of differential polynomials, In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, Berlin, 1991, 171-182.
6. H. W. E. Jung, Uber ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.
7. I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957.
8. E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
9. W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, 3 (1953), 33-41.
10. L. Makar-Limanov, The automorphisms of the free algebra with two generators, Funct. Anal. Appl., 4 (1970), 262-263.
11. M. Nagata, On the Automorphism Group of k[x,y], Kinokuniya, Tokio, 1972.
12. J. F. Ritt, Differential Algebra, American Mathematical Society, New York, 1950.
13. I. P. Shestakov, U. U. Umirbaev, Tame and wild automorphisms of rings of polynomials in three variables, J. Am. Math. Soc., 17 (2004), 197-227.
14. W. Sit, The Ritt-Kolchin Theory for Differential Polynomials, World Scientific, Singapore, 2001.
15. U. U. Umirbaev, The Anick automorphism of free associative algebras, J. Reine Angew. Math., 605 (2007), 165-178.
16. A. Van den Essen, Polynomial Automorphisms: and the Jacobian Conjecture, Birkhauser, 2012. | 2020-09-24 15:51:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8884148597717285, "perplexity": 1879.4604465710224}, "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-2020-40/segments/1600400219221.53/warc/CC-MAIN-20200924132241-20200924162241-00156.warc.gz"} |
http://atom-packages.directory/package/atom-latex/ | # Atom Packages Directory
a package directory for a text editor of the 21st Century
# Latex
Install with:
apm install atom-latex
# Atom-LaTeX package
Atom-LaTeX is an extension for Atom.io, aiming to provide all-in-one features and utilities for latex typesetting with Atom.
## Update
Recently (Mar. 29th, 2017 now) I switched back to Visual Studio Code for most coding and typesetting tasks due to reasons 1, 2, 3, and more. Atom-LaTeX will still be maintained, though new feature development is in stale. I would recommend you to have a check on VS Code as well as LaTeX Workshop, a counterpart of Atom-LaTeX to VS Code. Thank you all for your long time support!
## Features
Atom-LaTeX is currently under active development. More features coming soon. Some features have screenshots/screencasts available here. Have a check!
• Compile LaTeX with BibTeX
• Preview PDF with build-in viewer
• Parse LaTeX compiling log
• Autocomplete
• Syntax highlight
• Direct and reverse SyncTeX
If you figured out some features neat but not included, create an issue!
## Why another LaTeX package?
Unification provides seamless experience. Aiming to make it work and work perfectly.
## Requirements
• LaTeX distribution in system PATH. For example, TeX Live.
• MiKTeX does not ship with SyncTeX, but basic build and preview and non-SyncTeX related features work fine.
• Set LaTeX root file.
## Installation
Installing Atom-LaTeX is simple. You can find it in the atom.io package registry, or simply run apm install atom-latex in command line.
For cutting edge features or changes, you can check out this repository to the Atom package folder: - Windows %USERPROFILE%\.atom\packages - Mac/Linux \$HOME/.atom/packages
## Usage
All commands can be invoked from Package-Atom-LaTeX menu or by command palette. Alternatively, keybinds are provided. Each command is invoked if the two key combinations are pressed sequentially.
For reverse SyncTeX from PDF to LaTeX, use ctrl+Mouse Left Click in the PDF viewer to reveal the line in editor.
Mac OS users can use command key as a replacement of ctrl.
Command Default Keybind Function
atom-latex:build ctrl+L ctrl+B Build LaTeX file.
atom-latex:build-here ctrl+L ctrl+H Build LaTeX using active text editor file if possible.
atom-latex:clean ctrl+L ctrl+C Clean LaTeX auxillary files.
atom-latex:preview ctrl+L ctrl+P Preview generated PDF file with in-browser viewer.
atom-latex:kill ctrl+L ctrl+K Terminate current LaTeX building process.
atom-latex:synctex ctrl+L ctrl+S Direct SyncTeX from the current cursor position.
atom-latex:toggle-panel ctrl+L ctrl+L Toggle Atom-LaTeX panel display.
## Project-based Configuration
Atom currently does not provide per-project configuration. Atom-LaTeX uses a .latexcfg file under the root directory of LaTeX project to partially control its behavior. Following is a complete example of its content. { "root" : "\path\to\root\file.tex", "toolchain" : "%TEX %ARG %DOC", "latex_ext": [".tikz", ".Rnw"] } If a key is set, the config will overwrite the global one in atom settings.
## How To
### Set LaTeX root file LaTeX root file is essential to Atom-LaTeX. Building, preview, autocompletion, and more features rely on its proper configuration. You can select to manually set the file by clicking the home icon on the control bar, or let Atom-LaTeX automatically find it given proper project structures: { "root" : "\path\to\root\file.tex" }
1. Create a .latexcfg file at the root directory of your project. The file should contain a json object with root key set to the root file. An example: { "root" : "\path\to\root\file.tex" }
2. Add a magic comment % !TEX root = \path\to\root\file.tex to all of your LaTeX source file. The path can be absolute or relative.
3. Open the root file and use Build Here command. Alternatively, use Build LaTeX from active editor menu item.
4. If all previous checks fail to find a root file, Atom-LaTeX will iterate through all LaTeX files in the root directory.
You can choose one or multiple methods stated above to set the root file.
### Set per-project LaTeX toolchain
LaTeX toolchain can be controlled by either atom configuration or .latexcfg file under root directory. If LaTeX projects need special toolchain, one can add a toolchain key to this file. An example: { "toolchain" : "%TEX %ARG %DOC" } This example will only use the defined compiler in atom configuration to build the project. Alternatively, you can also have this example that provides the same functionality: { "toolchain" : "pdflatex -synctex=1 -interaction=nonstopmode -file-line-error -pdf %DOC" }
### Support non-tex files
Atom-LaTeX has limited support to LaTeX source files with a non-.tex extension. To consider such files as valid LaTeX documents, one can add a latex_ext key to the .latexcfg local configuration file. An example: { "latex_ext": [".tikz", ".Rnw"] } Note that the value must be a JSON array, even when there is only one alternative file extension. For toolchain settings of non-tex files, @ashthespy gave a very good summary here.
### Enable spell check
• Open setting panel of build-in package spell-check.
• Add text.tex.latex to the Grammars edit box.
## Contributing
Keywords: latex, compile, preview, synctex, highlight Suggest keywords | 2018-12-13 21:55:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8437576293945312, "perplexity": 14350.515874874945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825112.63/warc/CC-MAIN-20181213215347-20181214000847-00152.warc.gz"} |
http://clay6.com/qa/46373/calculate-the-mass-of-a-non-volatile-solute-molar-mass-40-g-mol-which-shoul | # Calculate the mass of a non-volatile solute (molar mass 40 g mol$^{−1}$) which should be dissolved in 114 g octane to reduce its vapour pressure to 80%.
8g
Hence (A) is the correct answer. | 2018-02-25 05:59:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5889893770217896, "perplexity": 1484.3681207182074}, "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/1518891816138.91/warc/CC-MAIN-20180225051024-20180225071024-00467.warc.gz"} |
https://support.bioconductor.org/p/53418/ | Question: Repeated Measures mRNA expression analysis I
0
6.0 years ago by
United States
Charles Determan Jr140 wrote:
Greetings,
I need to analyze data collected from an RNA-seq experiment. This consists of comparing two groups (control vs. treatment) and repeated sampling (1 hour, 2 hours, 3 hours). If this were a univariate problem I know I would use a 2-way rmANOVA analysis but this is RNA-seq and I have thousands of variables. I am very familiar with multiple packages for RNA differential expression analysis (e.g. DESeq2, edgeR, limma, etc.) but I have been unable to figure out what the most appropriate way to analyze such data in this circumstance. The closest answer I can find within the DESeq2 and edgeR manuals (limma is somewhat confusing to me) is to place to main treatment of interest at the end of the design formula, for example:
design(dds) <- formula(~ time + treatment)
Is this what is considered the appropriate way to address repeated measures in mRNA expression experiments? Any thoughts are appreciated.
Regards,
--
Charles Determan
Integrated Biosciences PhD Candidate
University of Minnesota
limma edger deseq2 • 2.8k views
modified 4.5 years ago by Gordon Smyth37k • written 6.0 years ago by Charles Determan Jr140
Charles,
I am looking to do a similar analysis. What was the final method you ended up using to do your repeated measures analysis?
Cheers,
Nate
This question was continued and answered on a later thread, see: Repeated Measures mRNA expression analysis II
Answer: Repeated Measures mRNA expression analysis
1
6.0 years ago by
Michael Love24k
United States
Michael Love24k wrote:
Hi Charles, On Jun 24, 2013, at 10:08 PM, Charles Determan Jr <deter088 at="" umn.edu=""> wrote: > > design(dds) <- formula(~ time + treatment) > > Is this what is considered the appropriate way to address repeated measures > in mRNA expression experiments? Any thoughts are appreciated. > Yes, this is the correct design formula for DESeq2 for estimating and testing the effect of treatment over all time points. We use by default a Wald test, and the likelihood ratio test is also implemented (see vignette). This is then a similar approach to calling anova.glm() on a glm fit for a single gene. Best, Mike
Answer: Repeated Measures mRNA expression analysis
1
6.0 years ago by
Gordon Smyth37k
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Gordon Smyth37k wrote:
Dear Charles,
The term "repeated measures" describes a situation in which repeated measurements are made on the same biological unit. Hence the repeated measurements are correlated. It is not clear from the brief information you give whether this is the case, or whether the different time points derive from independent biological samples. The model you give might or might not be correct, depending on the experimental units and the hypotheses that you plan to test. For most experiments it is not the right approach, for reasons that I have pointed out elsewhere:
https://www.stat.math.ethz.ch/pipermail/bioconductor/2013-June/053297.html
Best wishes
Gordon
Gordon,
I apologize for not being more definitive with my description. Your initial definition is my intention, consecutive measurements on the same biological units. I will look over the comments in the link you provided. Thank you for your insight, I appreciate any further thoughts you may have.
Regards,
Charles
ADD REPLYlink modified 4.5 years ago by Gordon Smyth37k • written 6.0 years ago by Charles Determan Jr140
Charles,
Are there only 2 biological units in your experiment? (One for treatment and one for control?) Or do you have multiple biological units in each group? Surely it must be the latter but, if so, your model does not take this into account.
What questions do you want to test?
Best
Gordon
To help clarify further here is a dataframe of the design.
subject group times
1 1 Treated 0hr
2 2 Treated 0hr
3 3 Control 0hr
4 4 Treated 0hr
5 5 Control 0hr
6 6 Control 0hr
7 1 Treated 1hr
8 2 Treated 1hr
9 3 Control 1hr
...
17 5 Control 2hr
18 6 Control 2hr
My thought process has been as follows:
In the edgeR userguide there is the treatment combination example
> targets
Sample Treat Time
1 Sample1 Placebo 0h
2 Sample2 Placebo 0h
3 Sample3 Placebo 1h
4 Sample4 Placebo 1h
5 Sample5 Placebo 2h
6 Sample6 Placebo 2h
7 Sample1 Drug 0h
8 Sample2 Drug 0h
9 Sample3 Drug 1h
10 Sample4 Drug 1h
11 Sample5 Drug 2h
12 Sample6 Drug 2h
which combines the groups to produce a single group (ex. Drug.1, Placebo.1, Drug.2, etc)
This seems potentially appropriate but this appears to assume independence between samples whereas my data consists of what you could call 'true repeated measures' on the same sample. This seems to draw on the paired samples and blocked examples. These proceed by having the 'subject' as a factor as well, for example:
design <- model.matrix(~Subject+Treatment)
This leads me to guess that a combination of these techniques is required. Perhaps merging the times and group factors in my dataset (see above) as 'newgroup' (e.g. Control.0, Control.1, Treatment.0, etc). Then create the model formula:
design <- model.matrix(~Subject+newgroup)
Does this seem appropriate or am I way off base and over thinking this? Thanks for any suggestions.
Regards,
Charles
ADD REPLYlink modified 4.5 years ago by Gordon Smyth37k • written 6.0 years ago by Charles Determan Jr140
Answer: Repeated Measures mRNA expression analysis
0
4.5 years ago by
Gordon Smyth37k
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Gordon Smyth37k wrote:
This question was continued and answered on a later thread, see: Repeated Measures mRNA expression analysis II | 2019-06-19 14:59:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4012843668460846, "perplexity": 5090.156938665929}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999000.76/warc/CC-MAIN-20190619143832-20190619165832-00060.warc.gz"} |
https://johncarlosbaez.wordpress.com/page/2/ | ## Applied Category Theory 2018
12 September, 2017
There will be a conference on applied category theory!
Applied Category Theory (ACT 2018). School 23–27 April 2018 and workshop 30 April–4 May 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).
The plenary speakers will be:
• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Kathryn Hess (EPFL)
• David Spivak (MIT)
There will be a lot more to say as this progresses, but for now let me just quote from the conference website:
Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.
Towards an Integrative Science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.
While the workshop will host discussions on a wide range of applications of category theory, there will be four special tracks on exciting new developments in the field:
1. Dynamical systems and networks
2. Systems biology
3. Cognition and AI
4. Causality
Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.
Sincerely,
The organizers
Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)
Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry.
This site originated from discussions at the Computational Category Theory Workshop at NIST on Sept. 28-29, 2015. It serves to collect and disseminate research, resources, and tools for the development of applied category theory, and hosts a blog for those involved in its study.
### The proposal: Towards an Integrative Science
Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].
A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].
In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.
In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].
In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].
[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.
[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.
[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.
[4] John C. Baez and Mike STay. Physics, topology, logic and computation: a Rosetta stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.
[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.
[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945–1957, 2011.
[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127–135, 2003.
[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.
[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.
[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991.
[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121–134, 1997.
[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317–341, 1958.
[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.
[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.
## Postdoc in Applied Category Theory
8 September, 2017
guest post by Spencer Breiner
### One Year Postdoc Position at Carnegie Mellon/NIST
We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF’s Systems Science program. The position will be managed through Carnegie Mellon University (PI: Eswaran Subrahmanian), but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, Maryland outside of Washington, DC.
The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such “distributed energy resources” including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid.
This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st.
## Complex Adaptive Systems (Part 5)
4 September, 2017
When we design a complex system, we often start with a rough outline and fill in details later, one piece at a time. And if the system is supposed to be adaptive, these details may need to changed as the system is actually being used!
The use of operads should make this easier. One reason is that an operad typically has more than one algebra.
Remember from Part 3: an operad has operations, which are abstract ways of sticking things together. An algebra makes these operations concrete: it specifies some sets of actual things, and how the operations in the operad get implemented as actual ways to stick these things together.
So, an operad $O$ can have one algebra in which things are described in a bare-bones, simplified way, and another algebra in which things are described in more detail. Indeed it will typically have many algebras, corresponding to many levels of detail, but let’s just think about two for a minute.
When we have a ‘less detailed’ algebra $A$ and a ‘more detailed’ algebra $A',$ they will typically be related by a map
$f : A' \to A$
which ‘forgets the extra details’. This map should be a ‘homomorphism’ of algebras, but I’ll postpone the definition of that concept.
What we often want to do, when designing a system, is not forget extra detail, but rather add extra detail to some rough specification. There is not always a systematic way to do this. If there is, then we may have a homomorphism
$g : A \to A'$
going back the other way. This is wonderful, because it lets us automate the process of filling in the details. But we can’t always count on being able to do this—especially not if we want an optimal or even acceptable result. So, often we may have to start with an element of $A$ and search for elements of $A'$ that are mapped to it by $f : A' \to A.$
Let me give some examples. I’ll take the operad that I described last time, and describe some of its algebras, and homomorphisms between these.
I’ll start with an algebra that has very little detail: its elements will be simple graphs. As the name suggests, these are among the simplest possible ways of thinking about networks. They just look like this:
Then I’ll give an algebra with more detail, where the vertices of our simple graphs are points in the plane. There’s nothing special about the plane: we could replace the plane by any other set, and get another algebra of our operad. For example, we could use the set of points on the surface of the Caribbean Sea, the blue stuff in the rectangle here:
That’s what we might use in a search and rescue operation. The points could represent boats, and the edges could represent communication channels.
Then I’ll give an algebra with even more detail, where two points connected by an edge can’t be too far apart. This would be good for range-limited communication channels.
Then I’ll give an algebra with still more detail, where the locations of the points are functions of time. Now our boats are moving around!
Okay, here we go.
The operad from last time was called $O_G.$ Here $G$ is the network model of simple graphs. The best way to picture an operation of $O_G$ is as a way of sticking together a list of simple graphs to get a new simple graph.
For example, an operation
$f \in O_G(3,4,2;9)$
is a way of sticking together a simple graph with 3 vertices, one with 4 vertices and one with 2 vertices to get one with 9 vertices. Here’s a picture of such an operation:
Note that this operation is itself a simple graph. An operation in $O_G(3,4,2;9)$ is just a simple graph with 9 vertices, where we have labelled the vertices from 1 to 9.
This operad comes with a very obvious algebra $A$ where the operations do just what I suggested. In this algebra, an element of $A(t)$ is a simple graph with $t$ vertices, listed in order. Here $t$ is any natural number, which I’m calling ‘t’ for ‘type’.
We also need to say how the operations in $O_G$ act on these sets $A(t).$ If we take simple graphs in $A(3), A(4),$ and $A(2)$:
we can use our operation $f$ to stick them together and get this:
But we can also make up a more interesting algebra of $O_G.$ Let’s call this algebra $A'.$ We’ll let an element of $A'(t)$ be a simple graph with $t$ vertices, listed in order, which are points in the plane.
My previous pictures can be reused to show how operations in $O_G$ act on this new algebra $A'.$ The only difference is that now we tread the vertices literally as points in the plane! Before you should have been imagining them as abstract points not living anywhere; now they have locations.
Now let’s make up an even more detailed algebra $A''.$
What if our communication channels are ‘range-limited’? For example, what if two boats can’t communicate if they are more than 100 kilometers apart?
Then we can let an element of $A''(t)$ be a simple graph with $t$ vertices in the plane such that no two vertices connected by an edge have distance > 100.
Now the operations of our operad $O_G$ act in a more interesting way. If we have an operation, and we apply it to elements of our algebra, it ‘tries’ to put in new edges as it did before, but it ‘fails’ for any edge that would have length > 100. In other words, we just leave out any edges that would be too long.
It took me a while to figure this out. At first I thought the result of the operation would need to be undefined whenever we tried to create an edge that violated the length constraint. But in fact it acts in a perfectly well-defined way: we just don’t put in edges that would be too long!
This is good. This means that if you tell two boats to set up a communication channel, and they’re too far apart, you don’t get the ‘blue screen of death’: your setup doesn’t crash and burn. Instead, you just get a polite warning—‘communication channel not established’—and you can proceed.
The nontrivial part is to check that if we do this, we really get an algebra of our operad! There are some laws that must hold in any algebra. But since I haven’t yet described those laws, I won’t check them here. You’ll have to wait for our paper to come out.
Let’s do one more algebra today. For lack of creativity I’ll call it $A'''.$ Now an element of $A'''(t)$ is a time-dependent graph in the plane with $t$ vertices, listed in order. Namely, the positions of the vertices depend on time, and the presence or absence of an edge between two vertices can also depend on time. Furthermore, let’s impose the requirement that any two vertices can only connected by an edge at times when their distance is ≤ 100.
When I say ‘functions of time’ here, what ‘time’? We can model time by some interval $[T_1, T_2].$ But if you don’t like that, you can change it.
This algebra $A'''$ works more or less like $A''.$ The operations of $O_G$ try to create edges, but these edges only ‘take’ at times when the vertices they connect have distance ≤ 100.
There’s something here you might not like. Our operations can only try to create edges ‘for all times’… and succeed at times when the vertices are close enough. We can’t try to set up a communication channel for a limited amount of time.
But fear not: this is just a limitation in our chosen network model, ‘simple graphs’. With a fancier network model, we’d get a fancier operad, with fancier operations. Right now I’m trying to keep the operad simple (pun not intended), and show you a variety of different algebras.
And you might expect, we have algebra homomorphisms going from more detailed algebras to less detailed ones:
$f_T : A''' \to A'', \quad h : A' \to A$
The homomorphism $h$ takes a simple graph in the plane and forgets the location of its vertices. The homomorphism $f_T$ depends on a choice of time $T \in [T_1, T_2].$ For any time $T,$ it takes a time-dependent graph in the plane and evaluates it at that time, getting a graph in the plane (which obeys the distance constraints, since the time-dependent graph obeyed those constraints at any time).
We do not have a homomorphism $g: A'' \to A'$ that takes a simple graph in the plane obeying our distance constraints and forgets about those constraints. There’s a map $g$ sending elements of $A''$ to elements of $A'$ in this way. But it’s not an algebra homomorphism! The problem is that first trying to connect two graphs with an edge and then applying $g$ may give a different result than first applying $g$ and then connecting two graphs with an edge.
In short: a single operad has many algebras, which we can use to describe our desired system at different levels of detail. Algebra homomorphisms relate these different levels of detail.
Next time I’ll look at some more interesting algebras of the same operad. For example, there’s one that describes a system of interacting mobile agents, which move around in some specific way, determined by their location and the locations of the agents they’re communicating with.
Even this is just the tip of the iceberg—that is, still a rather low level of detail. We can also introduce stochasticity (that is, randomness). And to go even further, we could switch to a more sophisticated operad, based on a fancier ‘network model’.
But not today.
## Voyager 1
3 September, 2017
Launched 40 years ago, the Voyagers are our longest-lived and most distant spacecraft. Voyager 2 has reached the edge of the heliosphere, the realm where the solar wind and the Sun’s magnetic field live. Voyager 1 has already left the heliosphere and entered interstellar space! A new movie, The Farthest, celebrates the Voyagers’ journey toward the stars:
What has Voyager 1 been doing lately? I’ll skip its amazing exploration of the Solar System….
### Leaving the realm of planets
On February 14, 1990, Voyager 1 took the first ever ‘family portrait’ of the Solar System as seen from outside. This includes the famous image of planet Earth known as the Pale Blue Dot:
Soon afterwards, its cameras were deactivated to conserve power and computer resources. The camera software has been removed from the spacecraft, so it would now be hard to get it working again. And here on Earth, the software for reading these images is no longer available!
On February 17, 1998, Voyager 1 reached a distance of 69 AU from the Sun — 69 times farther from the Sun than we are. At that moment it overtook Pioneer 10 as the most distant spacecraft from Earth! Traveling at about 17 kilometers per second, it was moving away from the Sun faster than any other spacecraft. It still is.
That’s 520 million kilometers per year — hard to comprehend. I find it easier to think about this way: it’s 3.6 AU per year. That’s really fast… but not if you’re trying to reach other stars. It will take 20,000 years just to go one light-year.
### Termination shock
As Voyager 1 headed for interstellar space, its instruments continued to study the Solar System. Scientists at the Johns Hopkins University said that Voyager 1 entered the termination shock in February 2003. This is a bit like a ‘sonic boom’, but in reverse: it’s the place where the solar wind drops to below the speed of sound. Yes, sound can move through the solar wind, but only sound with extremely long wavelengths — nothing you can hear.
Some other scientists expressed doubt about this, and the issue wasn’t resolved until other data became available, since Voyager 1’s solar-wind detector had stopped working in 1990. This failure meant that termination shock detection had to be inferred from the other instruments on board. We now think that Voyager 1 reached the termination shock on December 15, 2004 — at a distance of 94 AU from the Sun.
### Heliosheath
In May 2005, a NASA press release said that Voyager 1 had reached the
heliosheath
. This is a bubble of stagnant solar wind, moving below the speed of sound. It’s outside the termination shock but inside the heliopause, where the interstelllar wind crashes against the solar wind.
On March 31, 2006, amateur radio operators in Germany tracked and received radio waves from Voyager 1 using a 20-meter dish. They
checked their data against data from the Deep Space Network station in Madrid, Spain and yes — it matched. This was the first amateur tracking of Voyager 1!
On December 13, 2010, the the Low Energy Charged Particle device
aboard Voyager 1 showed that it passed the point where the solar wind flows away from the Sun. At this point the solar wind seems to turn sideways, due to the push of the interstellar wind. On this date, the spacecraft was approximately 17.3 billion kilometers from the Sun, or 116 AU.
In March 2011, Voyager 1 was commanded to change its orientation to measure the sideways motion of the solar wind. How? I don’t know. Its solar wind detector was broken.
But anyway, a test roll done in February had confirmed the spacecraft’s ability to maneuver and reorient itself. So, in March it rotated 70 degrees counterclockwise with respect to Earth to detect the solar wind. This was the first time the spacecraft had done any major maneuvering since the family portrait photograph of the planets was taken in 1990.
After the first roll the spacecraft had no problem in reorienting itself with Alpha Centauri, Voyager 1’s guide star, and it resumed sending transmissions back to Earth.
On December 1, 2011, it was announced that Voyager 1 had detected the first Lyman-alpha radiation originating from the Milky Way galaxy. Lyman-alpha radiation had previously been detected from other galaxies, but because of interference from the Sun, the radiation from the Milky Way was not detectable.
Puzzle: What the heck is Lyman-alpha radiation?
On December 5, 2011, Voyager 1 saw that the Solar System’s magnetic field had doubled in strength, basically because it was getting compressed by the pressure of the interstellar wind. Energetic particles originating in the Solar System declined by nearly half, while the detection of high-energy electrons from outside increased 100-fold.
### Heliopause and beyond
In June 2012, NASA announced that the probe was detecting even more charged particles from interstellar space. This meant that it was getting close to the heliopause: the place where the gas of interstellar space crashes into the solar wind.
Voyager 1 actually crossed the heliopause in August 2012, although it took another year to confirm this. It was 121 AU from the Sun.
What’s next?
In about 300 years Voyager 1 will reach the Oort cloud, the region of frozen comets. It will take 30,000 years to pass through the Oort cloud. Though it is not heading towards any particular star, in about 40,000 years it will pass within 1.6 light-years of the star Gliese 445.
NASA says:
The Voyagers are destined — perhaps eternally —
to wander the Milky Way.
That’s an exaggeration. The Milky Way will not last forever. In just 3.85 billion years, before our Sun becomes a red giant, the Andromeda galaxy will collide with the Milky Way. In just 100 trillion years, all the stars in the Milky Way will burn out. And in just 10 quintillion years, the Milky Way will have disintegrated, with all the dead stars either falling into black holes or being flung off into intergalactic space.
But still: the Voyagers’ journeys are just beginning. Let’s wish them a happy 40th birthday!
My story here is adapted from this Wikipedia article:
• Wikipedia, Voyager 1.
• NASA, NASA and iconic museum honor Voyager spacecraft 40th anniversary, August 30, 2017.
## Complex Adaptive System Design (Part 4)
22 August, 2017
Last time I introduced typed operads. A typed operad has a bunch of operations for putting together things of various types and getting new things of various types. This is a very general idea! But in the CASCADE project we’re interested in something more specific: networks. So we want operads whose operations are ways to put together networks and get new networks.
That’s what our team came up with: John Foley of Metron, my graduate students Blake Pollard and Joseph Moeller, and myself. We’re writing a couple of papers on this, and I’ll let you know when they’re ready. These blog articles are kind of sneak preview—and a gentle introduction, where you can ask questions.
For example: I’m talking a lot about networks. But what is a ‘network’, exactly?
There are many kinds. At the crudest level, we can model a network as a simple graph, which is something like this:
There are some restrictions on what counts as a simple graph. If the vertices are agents of some sort and the edges are communication channels, these restrictions imply:
• We allow at most one channel between any pair of agents, since there’s at most one edge between any two vertices of our graph.
• The channels do not have a favored direction, since there are no arrows on the edges of our graph.
• We don’t allow a channel from an agent to itself, since an edge can’t start and end at the same vertex.
For other purposes we may want to drop some or all of these restrictions. There is an appalling diversity of options! We might want to allow multiple channels between a pair of agents. For this we could use multigraphs. We might want to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. For this we could use directed graphs. And so on.
We will also want to consider graphs with colored vertices, to specify different types of agents—or colored edges, to specify different types of channels. Even more complicated variants are likely to become important as we proceed.
To avoid sinking into a mire of special cases, we need the full power of modern mathematics. Instead of separately studying all these various kinds of networks, we need a unified notion that subsumes all of them.
To do this, the Metron team came up with something called a ‘network model’. There is a network model for simple graphs, a network model for multigraphs, a network model for directed graphs, a network model for directed graphs with 3 colors of vertex and 15 colors of edge, and more.
You should think of a network model as a kind of network. Not a specific network, just a kind of network.
Our team proved that for each network model $G$ there is an operad $O_G$ whose operations describe how to put together networks of that kind. We call such operads ‘network operads’.
I want to make all this precise, but today let me just show you one example. Let’s take $G$ to be the network model for simple graphs, and look at the network operad $O_G.$ I won’t tell you what kind of thing $G$ is yet! But I’ll tell you about the operad $O_G$.
Types. Remember from last time that an operad has a set of ‘types’. For $O_G$ this is the set of natural numbers, $\mathbb{N}.$ The reason is that a simple graph can have any number of vertices.
Operations. Remember that an operad has sets of ‘operations’. In our case we have a set of operations $O_G(t_1,\dots,t_n ; t)$ for each choice of $t_1,\dots,t_n, t \in \mathbb{N}.$
An operation $f \in O_G(t_1,\dots,t_n; t)$ is a way of taking a simple graph with $t_1$ vertices, a simple graph with $t_2$ vertices,… and so on, and sticking them together, perhaps adding new edges, to get a simple graph with
$t = t_1 + \cdots + t_n$
vertices.
Let me show you an operation
$f \in O_G(3,4,2;9)$
This will be a way of taking three simple graphs—one with 3 vertices, one with 4, and one with 2—and sticking them together, perhaps adding edges, to get one with 9 vertices.
Here’s what $f$ looks like:
It’s a simple graph with vertices numbered from 1 to 9, with the vertices in bunches: {1,2,3}, {4,5,6,7} and {8,9}. It could be any such graph. This one happens to have an edge from 3 to 6 and an edge from 1 to 2.
Here’s how we can actually use our operation. Say we have three simple graphs like this:
Then we can use our operation to stick them together and get this:
Notice that we added a new edge from 3 to 6, connecting two of our three simple graphs. We also added an edge from 1 to 2… but this had no effect, since there was already an edge there! The reason is that simple graphs have at most one edge between vertices.
But what if we didn’t already have an edge from 1 to 2? What if we applied our operation $f$ to the following simple graphs?
Well, now we’d get this:
This time adding the edge from 1 to 2 had an effect, since there wasn’t already an edge there!
In short, we can use this operad to stick together simple graphs, but also to add new edges within the simple graphs we’re sticking together!
When I’m telling you how we ‘actually use’ our operad to stick together graphs, I’m secretly describing an algebra of our operad. Remember, an operad describes ways of sticking together things together, but an ‘algebra’ of the operad gives a particular specification of these things and describes how we stick them together.
Our operad $O_G$ has lots of interesting algebras, but I’ve just shown you the simplest one. More precisely:
Things. Remember from last time that for each type, an algebra specifies a set of things of that type. In this example our types are natural numbers, and for each natural number $t \in \mathbb{N}$ I’m letting the set of things $A(t)$ consist of all simple graphs with vertices $\{1, \dots, t\}.$
Action. Remember that our operad $O_G$ should have an action on $A$, meaning a bunch of maps
$\alpha : O_G(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)$
I just described how this works in some examples. Some rules should hold… and they do.
To make sure you understand, try these puzzles:
Puzzle 1. In the example I just explained, what is the set $O_G(t_1,\dots,t_n ; t)$ if $t \ne t_1 + \cdots + t_n?$
Puzzle 2. In this example, how many elements does $O_G(1,1;2)$ have?
Puzzle 3. In this example, how many elements does $O_G(1,2;3)$ have?
Puzzle 4. In this example, how many elements does $O_G(1,1,1;3)$ have?
Puzzle 5. In the particular algebra $A$ that I explained, how many elements does $A(3)$ have?
Next time I’ll describe some more interesting algebras of this operad $O_G.$ These let us describe networks of mobile agents with range-limited communication channels!
## Complex Adaptive System Design (Part 3)
17 August, 2017
It’s been a long time since I’ve blogged about the Complex Adaptive System Composition and Design Environment or CASCADE project run by John Paschkewitz. For a reminder, read these:
Complex adaptive system design (part 1), Azimuth, 2 October 2016.
Complex adaptive system design (part 2), Azimuth, 18 October 2016.
A lot has happened since then, and I want to explain it.
I’m working with Metron Scientific Solutions to develop new techniques for designing complex networks.
The particular problem we began cutting our teeth on is a search and rescue mission where a bunch of boats, planes and drones have to locate and save people who fall overboard during a boat race in the Caribbean Sea. Subsequently the Metron team expanded the scope to other search and rescue tasks. But the real goal is to develop very generally applicable new ideas on designing and ‘tasking’ networks of mobile agents—that is, designing these networks and telling the agents what to do.
We’re using the mathematics of ‘operads’, in part because Spivak’s work on operads has drawn a lot of attention and raised a lot of hopes:
An operad is a bunch of operations for sticking together smaller things to create bigger ones—I’ll explain this in detail later, but that’s the core idea. Spivak described some specific operads called ‘operads of wiring diagrams’ and illustrated some of their potential applications. But when we got going on our project, we wound up using a different class of operads, which I’ll call ‘network operads’.
Here’s our dream, which we’re still trying to make into a reality:
Network operads should make it easy to build a big network from smaller ones and have every agent know what to do. You should be able to ‘slap together’ a network, throwing in more agents and more links between them, and automatically have it do something reasonable. This should be more flexible than an approach where you need to know ahead of time exactly how many agents you have, and how they’re connected, before you can tell them what to do.
You don’t want a network to malfunction horribly because you forgot to hook it up correctly. You want to focus your attention on optimizing the network, not getting it to work at all. And you want everything to work so smoothly that it’s easy for the network to adapt to changing conditions.
To achieve this we’re using network operads, which are certain special ‘typed operads’. So before getting into the details of our approach, I should say a bit about typed operads. And I think that will be enough for today’s post: I don’t want to overwhelm you with too much information at once.
In general, a ‘typed operad’ describes ways of sticking together things of various types to get new things of various types. An ‘algebra’ of the operad gives a particular specification of these things and the results of sticking them together. For now I’ll skip the full definition of a typed operad and only highlight the most important features. A typed operad $O$ has:
• a set $T$ of types.
• collections of operations $O(t_1,...,t_n ; t)$ where $t_i, t \in T$. Here $t_1, \dots, t_n$ are the types of the inputs, while $t$ is the type of the output.
• ways to compose operations. Given an operation
$f \in O(t_1,\dots,t_n ;t)$ and $n$ operations
$g_1 \in O(t_{11},\dots,t_{1 k_1}; t_1),\dots, g_n \in O(t_{n1},\dots,t_{n k_n};t_n)$
we can compose them to get
$f \circ (g_1,\dots,g_n) \in O(t_{11}, \dots, t_{nk_n};t)$
These must obey some rules.
But if you haven’t seen operads before, you’re probably reeling in horror—so I need to rush in and save you by showing you the all-important pictures that help explain what’s going on!
First of all, you should visualize an operation $f \in O(t_1, \dots, t_n; t)$ as a little gizmo like this:
It has $n$ inputs at top and one output at bottom. Each input, and the output, has a ‘type’ taken from the set $T.$ So, for example, if you operation takes two real numbers, adds them and spits out the closest integer, both input types would be ‘real’, while the output type would be ‘integer’.
The main thing we do with operations is compose them. Given an an operation $f \in O(t_1,\dots,t_n ;t),$ we can compose it with $n$ operations
$g_1 \in O(t_{11},\dots,t_{1 k_1}; t_1), \quad \dots, \quad g_n \in O(t_{n1},\dots,t_{n k_n};t_n)$
by feeding their outputs into the inputs of $f,$ like this:
The result is an operation we call
$f \circ (g_1, \dots, g_n)$
Note that the input types of $f$ have to match the output types of the $g_i$ for this to work! This is the whole point of types: they forbid us from composing operations in ways that don’t make sense.
This avoids certain stupid mistakes. For example, you can take the square root of a positive number, but you may not want to take the square root of a negative number, and you definitely don’t want to take the square root of a hamburger. While you can land a plane on an airstrip, you probably don’t want to land a plane on a person.
The operations in an operad are quite abstract: they aren’t really operating on anything. To render them concrete, we need another idea: operads have ‘algebras’.
An algebra $A$ of the operad $O$ specifies a set of things of each type $t \in T$ such that the operations of $O$ act on these sets. A bit more precisely, an algebra consists of:
• for each type $t \in T,$ a set $A(t)$ of things of type $t$
• an action of $O$ on $A,$ that is, a collection of maps
$\alpha : O(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)$
obeying some rules.
In other words, an algebra turns each operation $f \in O(t_1,...,t_n ; t)$ into a function that eats things of types $t_1, \dots, t_n$ and spits out a thing of type $t.$
When we get to designing systems with operads, the fact that the same operad can have many algebras will be useful. Our operad will have operations describing abstractly how to hook up networks to form larger networks. An algebra will give a specific implementation of these operations. We can use one algebra that’s fairly fine-grained and detailed about what the operations actually do, and another that’s less detailed. There will then be a map between from the first algebra to the second, called an ‘algebra homomorphism’, that forgets some fine-grained details.
There’s a lot more to say—all this is just the mathematical equivalent of clearing my throat before a speech—but I’ll stop here for now.
And as I do—since it also takes me time to stop talking—I should make it clear yet again that I haven’t even given the full definition of typed operads and their algebras! Besides the laws I didn’t write down, there’s other stuff I omitted. Most notably, there’s a way to permute the inputs of an operation in an operad, and operads have identity operations, one for each type.
To see the full definition of an ‘untyped’ operad, which is really an operad with just one type, go here:
They just call it an ‘operad’. Note that they first explain ‘non-symmetric operads’, where you can’t permute the inputs of operations, and then explain operads, where you can.
If you’re mathematically sophisticated, you can easily guess the laws obeyed by a typed operad just by looking at this article and inserting the missing types. You can also see the laws written down in Spivak’s paper, but with some different terminology: he calls types ‘objects’, he calls operations ‘morphisms’, and he calls typed operads ‘symmetric colored operads’—or once he gets going, just ‘operads’.
You can also see the definition of a typed operad in Section 2.1 here:
• Donald Yau, Operads of wiring diagrams.
What I would call a typed operad with $S$ as its set of types, he calls an ‘$S$-colored operad’.
I guess it’s already evident, but I’ll warn you that the terminology in this subject varies quite a lot from author to author: for example, a certain community calls typed operads ‘symmetric multicategories’. This is annoying at first but once you understand the subject it’s as ignorable as the fact that mathematicians have many different accents. The main thing to remember is that operads come in four main flavors, since they can either be typed or untyped, and they can either let you permute inputs or not. I’ll always be working with typed operads where you can permute inputs.
Finally, I’ll say that while the definition of operad looks lengthy and cumbersome at first, it becomes lean and elegant if you use more category theory.
Next time I’ll give you an example of an operad: the simplest ‘network
## Norbert Blum on P versus NP
15 August, 2017
There’s a new paper on the arXiv that claims to solve a hard problem:
• Norbert Blum, A solution of the P versus NP problem.
Most papers that claim to solve hard math problems are wrong: that’s why these problems are considered hard. But these papers can still be fun to look at, at least if they’re not obviously wrong. It’s fun to hope that maybe today humanity has found another beautiful grain of truth.
I’m not an expert on the P versus NP problem, so I have no opinion on this paper. So don’t get excited: wait calmly by your radio until you hear from someone who actually works on this stuff.
I found the first paragraph interesting, though. Here it is, together with some highly non-expert commentary. Beware: everything I say could be wrong!
Understanding the power of negations is one of the most challenging problems in complexity theory. With respect to monotone Boolean functions, Razborov [12] was the first who could shown that the gain, if using negations, can be super-polynomial in comparision to monotone Boolean networks. Tardos [16] has improved this to exponential.
I guess a ‘Boolean network’ is like a machine where you feed in a string of bits and it computes new bits using the logical operations ‘and’, ‘or’ and ‘not’. If you leave out ‘not’ the Boolean network is monotone, since then making more inputs equal to 1, or ‘true’, is bound to make more of the output bits 1 as well. Blum is saying that including ‘not’ makes some computations vastly more efficient… but that this stuff is hard to understand.
For the characteristic function of an NP-complete problem like the clique function, it is widely believed that negations cannot help enough to improve the Boolean complexity from exponential to polynomial.
A bunch of nodes in a graph are a clique if each of these nodes is connected by an edge to every other. Determining whether a graph with $n$ vertices has a clique with more than $k$ nodes is a famous problem: the clique decision problem.
For example, here’s a brute-force search for a clique with at least 4 nodes:
The clique decision problem is NP-complete. This means that if you can solve it with a Boolean network whose complexity grows like some polynomial in n, then P = NP. But if you can’t, then P ≠ NP.
(Don’t ask me what the complexity of a Boolean network is; I can guess but I could get it wrong.)
I guess Blum is hinting that the best monotone Boolean network for solving the clique decision problem has a complexity that’s exponential in $n.$ And then he’s saying it’s widely believed that not gates can’t reduce the complexity to a polynomial.
Since the computation of an one-tape Turing machine can be simulated by a non-monotone Boolean network of size at most the square of the number of steps [15, Ch. 3.9], a superpolynomial lower bound for the non-monotone network complexity of such a function would imply P ≠ NP.
Now he’s saying what I said earlier: if you show it’s impossible to solve the clique decision problem with any Boolean network whose complexity grows like some polynomial in n, then you’ve shown P ≠ NP. This is how Blum intends to prove P ≠ NP.
For the monotone complexity of such a function, exponential lower bounds are known [11, 3, 1, 10, 6, 8, 4, 2, 7].
Should you trust someone who claims they’ve proved P ≠ NP, but can’t manage to get their references listed in increasing order?
But until now, no one could prove a non-linear lower bound for the nonmonotone complexity of any Boolean function in NP.
That’s a great example of how helpless we are: we’ve got all these problems whose complexity should grow faster than any polynomial, and we can’t even prove their complexity grows faster than linear. Sad!
An obvious attempt to get a super-polynomial lower bound for the non-monotone complexity of the clique function could be the extension of the method which has led to the proof of an exponential lower bound of its monotone complexity. This is the so-called “method of approximation” developed by Razborov [11].
I don’t know about this. All I know is that Razborov and Rudich proved a whole bunch of strategies for proving P ≠ NP can’t possibly work. These strategies are called ‘natural proofs’. Here are some friendly blog articles on their result:
• Timothy Gowers, How not to prove that P is not equal to NP, 3 October 2013.
• Timothy Gowers, Razborov and Rudich’s natural proofs argument, 7 October 2013.
From these I get the impression that what Blum calls ‘Boolean networks’ may be what other people call ‘Boolean circuits’. But I could be wrong!
Continuing:
Razborov [13] has shown that his approximation method cannot be used to prove better than quadratic lower bounds for the non-monotone complexity of a Boolean function.
So, this method is unable to prove some NP problem can’t be solved in polynomial time and thus prove P ≠ NP. Bummer!
But Razborov uses a very strong distance measure in his proof for the inability of the approximation method. As elaborated in [5], one can use the approximation method with a weaker distance measure to prove a super-polynomial lower bound for the non-monotone complexity of a Boolean function.
This reference [5] is to another paper by Blum. And in the end, he claims to use similar methods to prove that the complexity of any Boolean network that solves the clique decision problem must grow faster than a polynomial.
So, if you’re trying to check his proof that P ≠ NP, you should probably start by checking that other paper!
The picture below, by Behnam Esfahbod on Wikicommons, shows the two possible scenarios. The one at left is the one Norbert Blum claims to have shown we’re in. | 2017-11-18 21:38:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 120, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5901637077331543, "perplexity": 736.7470057881292}, "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-2017-47/segments/1510934805049.34/warc/CC-MAIN-20171118210145-20171118230145-00471.warc.gz"} |
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Calling all IT Pros from Canada and Australia.. we need your help! Support our site by taking a quick sponsored surveyand win a chance at a $50 Amazon gift card. Click here to get started! # Are Pack Files (PAK, ZIP, WAD, etc) Worth It? Old topic! Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic. 20 replies to this topic ### #1Cornstalks Crossbones+ - Reputation: 7003 Like 0Likes Like Posted 31 March 2012 - 10:06 PM Here's a question I've been mulling over. Are pack files worth using in a game? I've looked into using PhysicsFS, but I'm not liking its global state so much. I don't need any compression in my pack files, as just about everything will already be compressed (png for images, vorbis for audio, vp8 for video, protobuf binary objects for units/objects/maps, etc), and I'm more concerned about read/write times. If I used a pack file, I'd just need a format with random access (like zip). Here are the pros and cons as I see it of having an uncompressed pack file: Pros • (Slightly) harder for users to muck with • It's only one file (it's kinda nice having things grouped in one file) Cons • (Slightly) harder for me to work with • Increased save times when modifying the file (which the game won't do, but I will in my editor, so it's a con for me, though users won't experience this) ??? • Faster read times? (I've heard it can help not thrash the hard drive so much, but is this really much of a concern today on modern operating systems, and does it really help a significant amount?) Does anyone have much experience with the pros/cons of using pack files? Are there any significant pros to using pack files, and are there any significant cons to just using the normal file system? [ I was ninja'd 71 times before I stopped counting a long time ago ] [ f.k.a. MikeTacular ] [ My Blog ] [ SWFer: Gaplessly looped MP3s in your Flash games ] Sponsor: ### #2Hodgman Moderators - Reputation: 42108 Like 8Likes Like Posted 31 March 2012 - 11:37 PM (Slightly) harder for me to work with Increased save times when modifying the file (which the game won't do, but I will in my editor, so it's a con for me, though users won't experience this) In my engine, I only use packs/archives for retail/shipping builds. In development builds, each asset is stored in a separate file in the data directory. When making a shipping build, the data directory is "zipped" into an archive, and the code is compiled to use a different asset-loading class. This lets me have the benefits of archives on the end-user's machine, while having the benefit of easy content iteration during development. IMO, I find working with "files" much harder than working with assets. Yes, in development my assets are stored as files, so when I say Load("foo.texture"), that does turn into CreateFile("data/foo.texture", GENERIC_READ, FILE_SHARE_READ, 0, OPEN_EXISTING, FILE_FLAG_OVERLAPPED, 0)... However, because I never think of these assets as "files", I'm free to change the behind the scenes behavior. Maybe I'll look in the OS's file system first, and then in a patch archive, and then in the shipping archive. Maybe I'll pre-load a chunk of an archive that contains several assets at once, then when the user asks for any of those assets, I've already got the data sitting there, etc... Also, because I don't think of them as files, I don't author them as files. I never copy files into the data directory, and I never use file->save as to create any of the data files. Instead, we have a content directory, which does contain files, and a data-compiler, which scans the content directory for modifications, compiles any modified files, and writes them into the data directory. This means that for example, if I want to change my textures to use a different DXT compression algorithm, or I decide that materials should be embedded into level files, then I change the data-compiler's rules, and the data directory can be recompiled. The association between an asset name (e.g. "foo.texture") and the content file path (e.g. "d:\myProject\content\foo.png") aren't hard-coded; our compilation routines are written in C#, the build steps described in Lua, and regular expressions are used to find a suitable file to use as input when building an asset. e.g. The following Lua script tells the asset-compiler that: * If the asset "foo.geo" is required, then use the GeometryBuilder plugin (a C# class) and load "temp/foo.daebin" as input. * If "temp/foo.daebin" is required, then use the DaeParser plugin and search the content directory recursively for "foo.dae" (a COLLADA model). local DaeParser = Builder("DaeParser") Rule(DaeParser, "temp/(.*).daebin", "$1.dae")
local GeometryBuilder = Builder("GeometryBuilder")
Rule(GeometryBuilder, "data/(.*).geo", "temp/$1.daebin") These kinds of data compilers can also be used to ensure that only the data that's actually used by the game ends up in the data directory. Instead of compiling every file inside the content directory, we only compile the required files. We start off with the hard-coded asset-names used in the game's source code (ideally this number is quite small), then we find the linked assets (e.g. a material links a texture and a shader), and repeat until we've got a full dependency tree. Another neat feature you can add to a system like this is asset-refreshing -- the data-compiler is already scanning for file modifications to rebuild new data, so when it re-builds a file it can check if the game is currently running, and if so, send a message to the game instructing it to reload the modified asset. In the industry, every company I've worked for the past 6 years has used some kind of automated asset pipeline like this, and I just can't imagine going back to manually placing files in the game's data directory -- to me, it seems like a lot more of a hassle Faster read times? (I've heard it can help not thrash the hard drive so much, but is this really much of a concern today on modern operating systems, and does it really help a significant amount?) Assuming a non-SSD drive, it can give a significant reduction in loading times. With 1000 seperate files, the OS can keep each one defragmented, so that each individual file load can be done without wasteful seek periods, however, the OS doesn't know the order in which you want to load all of those files, so you'll pay a seek penalty in-between each file and likely won't benefit from automatic pre-caching. If you pack all the files end-to-end, in the order in which you want to load them, you can spend more time reading and less time seeking. As for modern OS's helping out, either way, make sure you're using the OS's native file system API (e.g. on windows, CreateFile/ReadFileEx instead of fopen/fread). By using these API's you can take advantage of modern features like threadless background loading (DMA), file (pre)caching or memory mapping. ### #3Cornstalks Crossbones+ - Reputation: 7003 Like 0Likes Like Posted 01 April 2012 - 08:56 AM Wow, thanks a ton for the great insights Hodgman! I'm definitely looking at implementing a data compiler like that. The auto-asset-refresh sounds *really* nice. That, and it lets the artists maintain their normal workflow when it comes to updating assets. And I think I'll do what you do: use pack files in the release builds and the filesystem for development builds. Abstracting the data storage and using a swappable loading class would be nice. [ I was ninja'd 71 times before I stopped counting a long time ago ] [ f.k.a. MikeTacular ] [ My Blog ] [ SWFer: Gaplessly looped MP3s in your Flash games ] ### #4Madhed Crossbones+ - Reputation: 3705 Like 3Likes Like Posted 01 April 2012 - 09:10 AM After a long search i found the paper that i read long ago: http://wassenberg.dreamhosters.com/articles/study_thesis.pdf It's by Jan Wassenberg of Wildfire games (0 A.D). He describes how packing of assets resulted in a massive reduction of load times. ### #5frob Moderators - Reputation: 31253 Like 2Likes Like Posted 01 April 2012 - 01:54 PM Properly packed, you can reduce load times. That is by far the biggest compelling reason. Ideally packed you have a small pointer table up front followed by all the data that gets memory-mapped and copied into place as fast as the OS streams it in. However, do that wrong and it will be SLOWER than a traditional load. Profile and proceed with careful measurements. Making it harder for end users to reverse engineer is perhaps the most invalid reason. If that is your motivation then stop. Properly packed you can have independent resource bundles that can be worked and replaced as individual components. A great example of this is The Sims where you can download tiny packs of clothes, people, home lots, and more. People generate custom content all the time and upload their hair models, body models, clothing models, the associated textures and whatnot, all in their own little bundle. Many comprehensive systems will use dual-load systems, first checking the packaged resources and then checking the file system for updated resources. That enables you to make changes without rebuilding all the packages. Even better systems will watch the file system and automatically update when changes are detected. This is extremely useful when there are external tools, such as string editors, tuning editors, and various resource editors so you can see your changes immediately in game. Check out my book, Game Development with Unity, aimed at beginners who want to build fun games fast. Also check out my personal website at bryanwagstaff.com, where I occasionally write about assorted stuff. ### #6Krohm Crossbones+ - Reputation: 4083 Like 0Likes Like Posted 02 April 2012 - 07:00 AM I'm very interested by this. I initially started by referring to resources (possibly the same thing as "asset names") however I had a few collisions here and there and I later switched to using file names directly. I didn't like this and I don't like it now, I want to go back to asset names in the future however I am still unsure on how to deal with naming collisions and in general provide a fine degree of flexibility. Perhaps it would be just better to give better naming conventions? Suggestions on rules about resource->file mappings? ### #7Cornstalks Crossbones+ - Reputation: 7003 Like 0Likes Like Posted 02 April 2012 - 07:22 AM Making it harder for end users to reverse engineer is perhaps the most invalid reason. If that is your motivation then stop. It's not. My primary goal is load times (though I wanted to confirm that was still an issue, as the last time I considered this topic was years and years ago). The bundles idea is a cool concept I hadn't thought of. While I don't plan on my current game being very moddable, it's definitely something I'd like to do if I make a more moddable game. Keep the good input flowing! This has all helped me a lot. [ I was ninja'd 71 times before I stopped counting a long time ago ] [ f.k.a. MikeTacular ] [ My Blog ] [ SWFer: Gaplessly looped MP3s in your Flash games ] ### #8samoth Crossbones+ - Reputation: 6328 Like 2Likes Like Posted 02 April 2012 - 08:57 AM @Madhed: In respect of the generally very interesting paper by Jan Wassenberg, one should note that it contains a lot of very useful information for some cases, and a lot of consideration in general. If one develops for a console or considers streaming data from CD, the paper hits the spot 100%. Some of the techniques described (e.g. duplicating blocks) are big win when you read from a medium where seeking is the end of the world (such as a DVD), or when you can't afford clobbering some RAM. On the other hand, if one targets a typical Windows desktop PC with "normal" present time hardware, almost all of the claims and assumptions are debatable or wrong (that was already the case in 2006 when the paper was written). What is indisputably right is that it's generally a good idea to have one (or few) big files rather than a thousand small ones. Other than that, one needs to be very careful about which assumptions are true for the platform one develops on. On a typical dektop machine which typically has half a gigabyte or a gigabyte of unused memory (often rather 2-4 GiB nowadays, or more), you absolutely do not want to bypass the file cache. If speed (and latency, and worst case behaviour) is of any concern, you also absolutely do not want to use overlapped IO. Overlapped IO rivals memory mapping in raw disk throughput if the file cache is disabled and if no pages are in cache. This is cool if you want to stream in data that you've never seen and that you don't expect to use again. It totally sucks otherwise, because the data is gone forever once you don't use it any more. With memory mapping, you pull the pages from the cache the next time you use the data. Even with some seeks in between (if only part of a large file is in the cache), pulling the data from the cache is no slower and usually faster (much to my surprise -- this is counterintuitive, but I've spend some considerable time on benchmarking that). Ironically, overlapped IO runs at about 50% of the speed of synchronous IO, if it is allowed to use the cache (this is, other than under e.g. Linux, actually possible under Windows). Pulling data from the cache into the working set synchronously peaks at around 2 GiB/s on my system (this is surprisingly slow for "doing nothing", a memcpy at worst, but it beats anything else by an order of magnitude). Asynchronous IO will silently, undetectably, unreliably, and differently between operating systems and versions, and depending on user configuration, revert to synchronous operation. Also, if anything "unexpected" happens, queueing an overlapped request can suddenly block for 20 or 40 milliseconds or more (so much for threadless IO, which means your render thread stalls during that time). This is not singular to Windows, Linux has the exact same problem. If the command queue is full or some other obscure limit (that you don't know about and that you cannot query!) is hit, your io_submit blocks. Surprise, you're dead. What you ideally want is to memory map the entire data file and prefault as much of it as you can linearly at application start (from a worker thread). If you, like me, own a "normal, inexpensive" 3-4 year old harddisk, you can observe that this will suck a 200 MiB data file into RAM in 2 seconds, with few or no seeks at all. If you, like me, also have a SSD, you can verify that the same thing will happen in well under a second. Either way, it's fast and straightforward. If your users, like pretty much everyone, have half a gigabyte of unused memory, the actual read later will be "zero time" without ever accessing the disk. This is admittedly the best case, not the worst case. But the good news is that the worst case is no worse than otherwise. The best (and average) case, on the other hand, is much better. ### #9Madhed Crossbones+ - Reputation: 3705 Like 0Likes Like Posted 02 April 2012 - 09:01 AM @samoth fair point. I just wanted to point out the paper since that was the first thing that sprung to my mind when reading the thread title. I haven't acually implemented or verified the results but found the paper interesting enough to share. Cheers ### #10ill Members - Reputation: 320 Like 1Likes Like Posted 02 April 2012 - 05:37 PM I use PhysFS myself and it works great I think. It allows you to not use an archive, but instead mount an actual folder. This means in development you can still be using PhysFS and be working with the resources on disk directly, and then create an archive and switch to using the archive by mounting the .pak file or whatever. PhysFS has really nice FileIO functions too. I find it's also pretty easy to write a little batch script or shell script that creates the archive from a folder in one click if you add something and want to see changed results. ### #11Hodgman Moderators - Reputation: 42108 Like 3Likes Like Posted 03 April 2012 - 10:58 PM I initially started by referring to resources (possibly the same thing as "asset names") however I had a few collisions here and there and I later switched to using file names directly. I didn't like this and I don't like it now, I want to go back to asset names in the future however I am still unsure on how to deal with naming collisions and in general provide a fine degree of flexibility. Perhaps it would be just better to give better naming conventions? Suggestions on rules about resource->file mappings? My build tool scans the entire content directory and builds a map of filenames to paths. If the same filename appears at multiple paths (e.g. content/lvl1/foo.png, content/lvl2/foo.png), then a boolean is set in the map, indicating that this name->path mapping is conflicted. When evaluating build rules, this table is used to locate input files on disk. If an entry from the table is used that has it's conflict flag set, then the tool spits out an error (describing the two paths) and refuses to build your data. This is similar to bad code spitting out assertion failures and refusing to run. Because my particular data-compiler is designed to always be running, it listens to changes to the content directory, and if you create a duplicate file, I can pop up one of those annoying bubbles from the system tray, letting you know you've just made a potential conflict before you even try to build your data. Regarding naming conventions, I can somewhat enforce these by specifying them in my build rules. For example, if I wanted to disallow "foo.texture" and enforce "foo_type.texture", where "type" is some kind of abbreviation, I can set only rules that contain "_type". Let's say one of my "types" is "colour+alpha", and that I want the artists want to author colour and alpha seperately. Rule(TexCombiner, "temp/(.*)_ca.tga", {"$1_c.tga", "$1_a.tga"}) Rule(TexCompiler, "data/(.*)_ca.texture", "temp/$1_ca.tga" )
Then, if someone sets up a material to link to "foo_ca.texture", the data compiler follows this sequence:
Build: data/foo_ca.texture
*Matches rule: "data/foo_ca.texture", inputs are "temp/foo_ca.tga"
**Build: temp/foo_ca.tga
***Matches rule: "temp/(.*)_ca.tga", inputs are "foo_c.tga" and "foo_a.tga"
***Search content map for "foo_c.tga" and "foo_a.tga"
***Run plugin: TexCombiner, inputs {"content/bar/foo_c.tga", "content/bar/foo_c.tga"}, output "temp/foo_ca.tga"
*Run plugin: TexCompiler, input "temp/foo_ca.tga", output "data/foo_ca.texture"
Whereas, if someone sets up a material to link to something like "foo.texture", which doesn't follow the convention, the data compiler follows this sequence:
Build: data/foo.texture
*Error: No rule matches "data/foo.texture"
### #12Krohm Crossbones+ - Reputation: 4083
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Posted 04 April 2012 - 01:01 AM
Thank you very much, I think I understand the basic principles. I am currently doing something similar to the example you describe. It now appears reasonable not allowing conflicts to happen is better than fixing them.
### #13Ashaman73 Crossbones+ - Reputation: 11954
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Posted 04 April 2012 - 01:34 AM
I use a system similar to java classpath/jars (or PhysFS ?). In fact I have a virtual filesystem with multiple layers of archives or directories which could be mounted. Important is the fact, that there're layers of archives. I.e.
data-archive:
/data/texture/tex1.png (Version 1.0)
/data/texture/tex2.png (Version 1.0)
/data/scripts/script1.lua (Version 1.0)
patch-archive:
/data/texture/tex1.png (Version 1.1)
/data/scripts/script1.lua (Version 1.1)
directory:
/data/texture/tex1.png (Version 1.2)
When I mount the archives/directories in the order data->patch->directory, I got the final virtual filesystem:
/data/texture/tex1.png (Version 1.2)
/data/texture/tex2.png (Version 1.0)
/data/scripts/script1.lua (Version 1.1)
This comes in really handy when delivering patches or exchanging single files for debugging purpose (atleast for a hobby dev ).
Ashaman
### #14markr Crossbones+ - Reputation: 1692
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Posted 16 April 2012 - 06:12 AM
Yes, they typically are.
The main motivations are quicker load times and more convenience of distribution.
Suppose you have 10,000 files. This is a fairly low number of individual objects, even for a game without much content.
On a desktop OS, your on-access AV program must scan every file. This is usually very time consuming.
You'll probably distribute your game as an archive (e.g. zip) anyway. So it doesn't make any difference. Your packer may be less efficient than zip. or more, but it doesn't really matter.
The overhead of having large numbers of files in the OS filesystem is quite significant, particularly when you remember that EVERYONE has on-access AV scanners!
Development convenience can be provided by having dev-builds search the filesystem first (and the resources.zip second) for files.
---
There is no security / reverse engineering benefit, because it is just as easy for a cracker to modify your big zip file as it would be if they were individual files. If you want to discourage casual reverse-engineering (or graphics ripping etc), then rename your .zip file to .zpi or something
### #15Antheus Members - Reputation: 2405
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Posted 16 April 2012 - 09:29 AM
Thousands of files in one directory will simply kill Windows-based machine, regardless of whether NTFS or FAT, SSD or regular, AV or not.
I don't know the reason, but this has been pathological worst case, perhaps due to CreateFile() or something.
At minimum, put all those files into an uncompressed zip and it will improve access times dramatically, despite same amount of data.
### #16Ashaman73 Crossbones+ - Reputation: 11954
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Posted 17 April 2012 - 06:00 AM
An other point is, that many third party resources (textures,sounds,models etc.) are coming with a license which forces you to deliver the resources in a protected way. A resource file(!=simple zip) is atleast a basic protection.
Ashaman
### #17Cornstalks Crossbones+ - Reputation: 7003
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Posted 17 April 2012 - 09:06 AM
An other point is, that many third party resources (textures,sounds,models etc.) are coming with a license which forces you to deliver the resources in a protected way. A resource file(!=simple zip) is atleast a basic protection.
Ha, no it's not. Any game that gains popularity is not protected (I mean really... Spore, anyone?), and any game that doesn't gain popularity isn't worth overcomplicating in the name of unnecessary and ineffective protection. And I highly doubt media licenses would seriously force me to deliver them in a "protected" pack file (I could be wrong, but I'd be surprised)... The reason I listed "(Slightly) harder for users to muck with" as a pro is not so much because it would stop Average Joe from replacing a texture (because if Pro-Hacker Henry hacks the file anyway, all he needs to do is release a program and Average Joe can now do everything Pro-Hacker Henry can do), but because I think some modders get a kick out of reverse engineering things and in a way it helps develop a modding community for the game, which (if done correctly) I think can be a good thing. [edit: Hodgman has pointed out that this has come across as quite arrogant; please read my post below, as that was not my intention (and realize I am not talking about the legal issues of fulfilling a contract here; I'm talking more about what frob said above)]
Anyway, sorry, I'm not trying to start a holy war here. You've all brought up some great points. I hadn't thought about anti-virus programs, and I didn't realize Windows struggled with lots of files in one folder (I'd probably categorize them in subfolders anyway, but it's good to know).
[ I was ninja'd 71 times before I stopped counting a long time ago ] [ f.k.a. MikeTacular ] [ My Blog ] [ SWFer: Gaplessly looped MP3s in your Flash games ]
### #18L. Spiro Crossbones+ - Reputation: 21326
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Posted 17 April 2012 - 10:22 AM
Sorry to whomever voted down Cornstalks but I had to undo your vote.
I am the author of MHS (Memory Hacking Software) and I have a very informed view on this topic.
Hackers can’t really be prevented. Instead for what we anti-cheat specialists aim is to simply minimize the spread of cheats, and Cornstalks was basically trying to illustrate this.
I have models that I use as test material for my own engine which I got from a site, but in the back of my mind I know for a fact that they ripped that content out of a Final Fantasy game illegally. The model is a raw hack of data.
Fine. I am still going to use that data as test material so I can gauge the progress of my engine. Some of my data I know to be illegally ripped from Halo as well but I was not the one who ripped it.
But this is peanuts compared to some of the things I myself have ripped from games, which include entire levels, not just individual models.
If you were to proclaim that you had any form of unhackable resource in any product you made I would simply laugh. The digital age + protection? Give me a break.
No we can’t stop hackers. But you really don’t realize how effective the small stuff is.
As mentioned by Cornstalks, the “slightly harder for users to muck with” situation is actually extremely effective when combating hackers. I have first-hand experience talking with hackers of all levels and I can personally confirm that hackers without much skill give up very easily.
In my own engine I have a custom compression system that acts as a deterrent for hackers. Why?
Only a little has changed from the standard libraries, but in order to handle that change you still have to rewrite the entire decompression system.
Even if some people realize that, very few of them are willing to actually do it. “Meh, I will just hack something else,” is how most will reply.
They end up creating more basic-level hacks and then keeping them for themselves. Why? Because there is no prestige in releasing a hack that everyone else can make.
The benefits in deterring the basic and obvious cheats are actually quite huge and very very frequently underestimated.
L. Spiro
### #19Hodgman Moderators - Reputation: 42108
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Posted 17 April 2012 - 09:18 PM
Sorry to whomever voted down Cornstalks but I had to undo your vote.
I downvoted it because of this arrogant dismissal of the reality that lawyers don't understand technology. I can paraphrase it as "Ashaman73, I've no experience with such legal requirements so I'll declare that they don't exist".
An other point is, that many third party resources (textures,sounds,models etc.) are coming with a license which forces you to deliver the resources in a protected way. A resource file(!=simple zip) is atleast a basic protection.
Ha, no it's not. Any game that gains popularity is not protected (I mean really... Spore, anyone?), and any game that doesn't gain popularity isn't worth overcomplicating in the name of unnecessary and ineffective protection. And I highly doubt media licenses would seriously force me to deliver them in a "protected" pack file (I could be am wrong, but I'd be surprised)...
The point was that if you're legally obliged to use 'protected' files then you're forced to jump through this hoop. Yes, your 'protected' files can easily be opened, but that doesn't change the fact that if there's a legal requirement to use 'protected' files, then you may have to. And yes, such legal hoops do exist and are an important detail in the real world.
For example, anyone can rip a copy-protected DVD easily, but the fact that the DVD has weak anti-copying measures means that you've crossed a particular legal line in the sand, which makes the lawyers job much easier when trying to prosecute pirates. Even though this copy-protection is useless in stopping copies from being made, publishers use it anyway as it becomes a legal weapon (the data was "protected" and you "broke" that protection).
### #20Cornstalks Crossbones+ - Reputation: 7003
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Posted 17 April 2012 - 10:25 PM
Sorry to whomever voted down Cornstalks but I had to undo your vote.
I downvoted it because of this arrogant dismissal of the reality that lawyers don't understand technology.
An other point is, that many third party resources (textures,sounds,models etc.) are coming with a license which forces you to deliver the resources in a protected way. A resource file(!=simple zip) is atleast a basic protection.
Ha, no it's not. Any game that gains popularity is not protected (I mean really... Spore, anyone?), and any game that doesn't gain popularity isn't worth overcomplicating in the name of unnecessary and ineffective protection. And I highly doubt media licenses would seriously force me to deliver them in a "protected" pack file (I could be wrong, but I'd be surprised)...
The point was that if you're legally obliged to use 'protected' files then you're forced to jump through this hoop. Yes, your 'protected' files can easily be opened, but that doesn't change the fact that if there's a legal requirement to use 'protected' files, then you may have to. And yes, such legal hoops do exist and are an important detail in the real world.
For example, anyone can rip a copy-protected DVD easily, but the fact that the DVD has weak anti-copying measures means that you've crossed a particular legal line in the sand, which makes the lawyers job much easier when trying to prosecute pirates. Even though this copy-protection is useless in stopping copies from being made, publishers use it anyway as it becomes a legal weapon (the data was "protected" and you "broke" that protection).
I will apologize for the arrogance, as I honestly didn't intend for it to come across as arrogant as I suppose it has (when I read "protection" I immediately thought of Spore, which has always been a funny example of epic failure to me, hence the "ha" part). I am indeed sorry for that.
I will say I am surprised that it sounds like (from Hodgman and Ashaman) it's not uncommon for contracts to require things to be packed and protected... I understand that of course if a contract states that, that is what needs to be done. I was not disagreeing with that. My point was more in line with frob's point that "Making it harder for end users to reverse engineer is perhaps the most invalid reason." Sure, it can prevent newbs from messing around with things, but I'd use a simple encryption/obfuscation scheme for that if that was my goal (which could be implemented on top of the pack file or raw files on the file system) (I see "packing things into a file" and "encrypting/obfuscating them" as two different problems with two different goals, though they can be used in combination).
[ I was ninja'd 71 times before I stopped counting a long time ago ] [ f.k.a. MikeTacular ] [ My Blog ] [ SWFer: Gaplessly looped MP3s in your Flash games ]
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PARTNERS | 2015-08-30 10:04:46 | {"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": 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": 0.20986308157444, "perplexity": 2823.5554949425}, "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/1440644065241.14/warc/CC-MAIN-20150827025425-00266-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.techwhiff.com/learn/prove-that-a-line-passing-through-the-midpoints/381799 | # Prove that: A line passing through the midpoints of two sides of a triangle is parallel...
###### Question:
Prove that:
A line passing through the midpoints of two sides of a triangle is parallel to and half the length of the third side of the triangle.
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##### How do you integrate int xsin2x by integration by parts method?
How do you integrate int xsin2x by integration by parts method?... | 2022-12-02 18:43:12 | {"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": 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": 0.5458337664604187, "perplexity": 1830.4491734602918}, "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/1669446710916.40/warc/CC-MAIN-20221202183117-20221202213117-00579.warc.gz"} |
https://socratic.org/questions/is-it-possible-to-find-the-derivative-of-3x-x-2-4-without-using-the-quotient-rul | # Is it possible to find the derivative of 3x/(x^2+4) without using the quotient rule?
$y = 3 x {\left({x}^{2} + 4\right)}^{-} 1$ so that you could use the Product and Chain Rule:
$y ' = 3 {\left({x}^{2} + 4\right)}^{-} 1 - 3 x {\left({x}^{2} + 4\right)}^{-} 2 \left(2 x\right) =$
$= 3 {\left({x}^{2} + 4\right)}^{-} 1 \left[1 - 2 {x}^{2} {\left({x}^{2} + 4\right)}^{-} 1\right] =$
$= \frac{3}{{x}^{2} + 4} \left[1 - 2 {x}^{2} / \left({x}^{2} + 4\right)\right] =$
$= \frac{3}{{x}^{2} + 4} ^ 2 \left[{x}^{2} + 4 - 2 {x}^{2}\right] =$
$= \frac{3}{{x}^{2} + 4} ^ 2 \left[4 - {x}^{2}\right]$ | 2021-01-27 08:17:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6976404786109924, "perplexity": 215.66792157003826}, "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/1610704821253.82/warc/CC-MAIN-20210127055122-20210127085122-00618.warc.gz"} |
https://www.bartleby.com/solution-answer/chapter-15-problem-5e-precalculus-mathematics-for-calculus-6th-edition-6th-edition/9780840068071/8ff6bb16-c2af-11e8-9bb5-0ece094302b6 | The blanks in the statement “The equation ( x + 1 ) 2 − 5 ( x + 1 ) + 6 = 0 is of ____ type. To solve the equation, we set W = _ _ _ _ _ . The resulting quadratic equation is _____”.
BuyFind
Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind
Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
Solutions
Chapter 1.5, Problem 5E
To determine
Expert Solution
Answer to Problem 5E
The complete statement is “The equation (x+1)25(x+1)+6=0 is of quadratic_ type. To solve the equation, we set W=x+1_. The resulting quadratic equation is W25W+6=0_.”
Explanation of Solution
The given equation is (x+1)25(x+1)+6=0.
If the expression x+1 in (x+1)25(x+1)+6=0 is considered as a variable, then the resulting equation will be of the form ax2+bx+c=0 where a=1,b=5and c=6.
Thus, the given equation is a quadratic type equation.
To solve the equation, substitute W for x+1. Then, the equation becomes as follows.
(x+1)25(x+1)+6=0W25W+6=0
Thus, the complete statement is “The equation (x+1)25(x+1)+6=0 is of quadratic_ type. To solve the equation, we set W=x+1_. The resulting quadratic equation is W25W+6=0_.”
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Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers! | 2021-09-16 22:42:23 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.82309490442276, "perplexity": 2751.205934735521}, "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-2021-39/segments/1631780053759.24/warc/CC-MAIN-20210916204111-20210916234111-00027.warc.gz"} |
https://proofwiki.org/wiki/Diophantus_of_Alexandria/Arithmetica/Book_1/Problem_17 | # Diophantus of Alexandria/Arithmetica/Book 1/Problem 17
## Problem
The sums of $4$ numbers, omitting each of the numbers in turn, are $22$, $24$, $27$ and $20$ respectively
What are the numbers?
## Solution
$\ds a$ $=$ $\ds 9$ $\ds b$ $=$ $\ds 7$ $\ds c$ $=$ $\ds 4$ $\ds d$ $=$ $\ds 11$
## Proof 1
Let $x$ be the sum of all $4$ numbers.
Then the numbers individually are:
$\ds a$ $=$ $\ds x - 22$ $\ds b$ $=$ $\ds x - 24$ $\ds c$ $=$ $\ds x - 27$ $\ds d$ $=$ $\ds x - 20$ $\ds \leadsto \ \$ $\ds a + b + c + d = x$ $=$ $\ds 4 x - \paren {22 + 24 + 27 + 20}$ $\ds \leadsto \ \$ $\ds 3 x$ $=$ $\ds 93$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds 31$
The result follows.
$\blacksquare$
## Proof 2
Let $a$, $b$, $c$ and $d$ be the numbers requested.
Then we have:
$\text {(1)}: \quad$ $\ds b + c + d$ $=$ $\ds 22$ $\text {(2)}: \quad$ $\ds a + c + d$ $=$ $\ds 24$ $\text {(3)}: \quad$ $\ds a + b + d$ $=$ $\ds 27$ $\text {(4)}: \quad$ $\ds a + b + c$ $=$ $\ds 20$ $\ds \leadsto \ \$ $\ds a - b$ $=$ $\ds 2$ $(2) - (1)$ $\ds b - c$ $=$ $\ds 3$ $(3) - (2)$ $\ds d - c$ $=$ $\ds 7$ $(4) - (3)$
We use the method of false position.
Set $c = 1$.
Then:
$\ds b$ $=$ $\ds 1 + 3 = 4$ $\ds d$ $=$ $\ds 1 + 7 = 8$ $\ds a$ $=$ $\ds 4 + 2 = 6$ $\ds \leadsto \ \$ $\ds b + c + d$ $=$ $\ds 13$ $\ds a + c + d$ $=$ $\ds 15$ $\ds a + b + d$ $=$ $\ds 18$ $\ds a + b + c$ $=$ $\ds 11$
These sums are all $9$ less than what they should be.
So we have to add $3$ to each number, to make:
$\ds a$ $=$ $\ds 9$ $\ds b$ $=$ $\ds 7$ $\ds c$ $=$ $\ds 4$ $\ds d$ $=$ $\ds 11$
$\blacksquare$ | 2021-12-08 10:23:44 | {"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": 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": 0.8623297214508057, "perplexity": 53.31970579346717}, "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/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00090.warc.gz"} |
http://eli5.readthedocs.io/en/latest/autodocs/permutation_importance.html | # eli5.permutation_importance¶
A module for computing feature importances by measuring how score decreases when a feature is not available. It contains basic building blocks; there is a full-featured sklearn-compatible implementation in PermutationImportance.
A similar method is described in Breiman, “Random Forests”, Machine Learning, 45(1), 5-32, 2001 (available online at https://www.stat.berkeley.edu/%7Ebreiman/randomforest2001.pdf), with an application to random forests. It is known in literature as “Mean Decrease Accuracy (MDA)” or “permutation importance”.
get_score_importances(score_func, X, y, n_iter=5, columns_to_shuffle=None, random_state=None)[source]
Return (base_score, score_decreases) tuple with the base score and score decreases when a feature is not available.
base_score is score_func(X, y); score_decreases is a list of length n_iter with feature importance arrays (each array is of shape n_features); feature importances are computed as score decrease when a feature is not available.
n_iter iterations of the basic algorithm is done, each iteration starting from a different random seed.
If you just want feature importances, you can take a mean of the result:
import numpy as np
from eli5.permutation_importance import get_scores_importances
base_score, score_decreases = get_score_importances(score_func, X, y)
feature_importances = np.mean(score_decreases, axis=0)
iter_shuffled(X, columns_to_shuffle=None, pre_shuffle=False, random_state=None)[source]
Return an iterator of X matrices which have one or more columns shuffled. After each iteration yielded matrix is mutated inplace, so if you want to use multiple of them at the same time, make copies.
columns_to_shuffle is a sequence of column numbers to shuffle. By default, all columns are shuffled once, i.e. columns_to_shuffle is range(X.shape[1]).
If pre_shuffle is True, a copy of X is shuffled once, and then result takes shuffled columns from this copy. If it is False, columns are shuffled on fly. pre_shuffle = True can be faster if there is a lot of columns, or if columns are used multiple times. | 2017-10-22 13:48:56 | {"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": 0.34553247690200806, "perplexity": 3576.778747369766}, "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/1508187825264.94/warc/CC-MAIN-20171022132026-20171022152026-00116.warc.gz"} |
http://demon-software.com/public_html/support/htmlug/ug-node9.html | ## How to Use deMon2k
The deMon2k program implements DFT in the Kohn-Sham formulation. It uses the linear combination of Gaussian type orbital (LCGTO) method. In this framework, the Kohn-Sham orbitals are expanded in an atomic orbital basis:
(1)
Here denotes an atomic orbital (built from contracted Gaussian basis functions) and the corresponding molecular orbital coefficient. With this expansion, the electronic density is:
(2)
is an element of the (closed-shell, also called non-spin-polarized in the DFT literature) density matrix defined as:
(3)
Using the LCGTO expansions for the Kohn-Sham orbitals (1.1) and the electronic density (1.2), the Kohn-Sham self-consistent field (SCF) energy expression [49] can be expressed as:
(4)
The total energy is the sum of and the nuclear repulsion energy, which can be calculated analytically. In (1.4), are elements of the core Hamiltonian matrix. They are built from the kinetic and nuclear attraction energy operators of the electrons and describe the distribution of an independent electron in the nuclear framework. The second term in (1.4) is the Coulomb repulsion energy of the electrons. In the short-hand notation for the four-center electron repulsion integrals (ERIs) the symbol represents the two-electron Coulomb operator and separates functions of electron 1 from those of electron 2. In contrast to Hartree-Fock theory, the calculations of the Coulomb and exchange energies are separate in Kohn-Sham DFT. Calculation of the exchange-correlation energy requires numerical integration. In deMon2k, the scaling of straight-forward calculation of the Coulomb repulsion energy is avoided by introducing an auxiliary function density [50]. This approximated density is expanded in primitive Hermite Gaussians which are centered on the atoms [51,52]:
(5)
The primitive Hermite Gaussian auxiliary functions are grouped in auxiliary function sets that share the same exponent [53,54]. For this reason, they usually are denoted as s, p, d etc. auxiliary function sets. With the LCGTO expansion for and we obtain the following approximate SCF energy:
(6)
Therefore, only three-center electron repulsion integrals are necessary for the SCF and energy calculation in deMon2k. This represents the density fitting Kohn-Sham method available in deMon2k. It is activated by the keyword VXCTYPE BASIS (see Section 4.2.1 for more details about the VXCTYPE keyword). However, by default (VXCTYPE AUXIS), the approximated density is also used for the calculation of the exchange-correlation energy:
(7)
This is the auxiliary density functional theory (ADFT) energy expression. For more details on ADFT, see the reviews [55,56,57,58,59]. Typically, the optimized ADFT structure parameters are indistinguishable from their full DFT counterparts even for weakly bound systems (here the use of the GEN-A2* auxiliary function set is recommended; see Section 4.3.3 and Appendix A). For binding energies, ADFT and Kohn-Sham results typically deviate by less than 1 kcal/mol if GEN-A2* or larger auxiliary function sets are used. Thus, the differences between ADFT and Kohn-Sham DFT geometries and bond energies are usually in the range of the accuracy of the underlying approximate exchange-correlation functional. Because of the considerable savings in computational time, we suggest to use ADFT for all studies including frequency analysis and property calculations. The VXCTYPE BASIS option Eq. (1.6) should be employed only if direct comparison with four-center DFT calculations is required. It should be noted that the default setting for the auxiliary functions is GEN-A2, independent of which energy expression is used (see Section 4.3.3). For all theoretical models available in deMon2k, VXCTYPE AUXIS results can be used as a restart guess (GUESS RESTART; see Section 4.5.5) for VXCTYPE BASIS calculations.
The most frequently encountered problem in DFT calculations is the failure to achieve SCF convergence. Usually this is caused by the small energy gap between the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbital. In deMon2k, the DIIS procedure (Section 4.5.8) is activated by default. For a small HOMO-LUMO gap, DIIS may be counterproductive and should be switched off. There are several options available in deMon2k to achieve SCF convergence. Most important are modifications of the choice of the starting GUESS (Section 4.5.5) and the MIXING (Section 4.5.6) of the old and new (auxiliary) densities as well as enlargement of the HOMO-LUMO gap by the level-SHIFT (Section 4.5.7) procedure. If a static level-shift is employed it is advisable to check the orbital energies and occupations at the HOMO-LUMO gap by use of the PRINT keyword (Section 4.12.2). Other relevant keywords to alter or achieve SCF convergence are MOEXCHANGE (Section 4.4.3), FIXMOS (Section 4.4.5) and SMEAR (Section 4.4.6). For atomic calculations, the CONFIGURE keyword (Section 4.4.7) should be used in order to ensure SCF convergence.
In deMon2k 5.0 the calculation of Hartree-Fock energies by the variational fitting of the Coulomb and Fock potentials is also available. The corresponding SCF energy has the form [43]:
(8)
Note that the same auxiliary function sets are used for the Coulomb and Fock potential fitting. As a result, the approximated Hartree-Fock energy, Eq. (1.8), is self-interaction free. To obtain a computationally efficient methodology the Fock potential fitting is performed with localized molecular orbitals [61]. This yields a computationally efficient and very accurate approximate Hartree-Fock energy expression that only requires three-center ERIs. Deviations with respect to four center ERIs total energies are below 1 kcal/mol if GEN-A2* auxiliary function sets are used. With this development hybrid functionals such as B3LYP [62,63], PBE0 [64,65] and M06-2X [66] are now available in deMon2k [67].
For QM/MM calculations in deMon2k 5.0 the following energy expression is used [44]:
(9)
The QM energy, , can be calculated with any of the above discussed SCF energy expressions given in Eqs. (1.6) to (1.8) or corresponding hybrid functional expressions. In all cases the core Hamiltonian matrix elements, , are augmented in order to take into account the electrostatic embedding of the QM system by the MM region:
(10)
In Eq. (1.10) denotes original core Hamilton matrix elements of the QM system and denotes the atomic charges of the MM atoms . The general form of the nuclear attraction type operator. , is given by:
(11)
This general definition permits immediately the inclusion of MM atoms with higher point moments. Note that Eq. (1.10) is also used for pure electrostatic embedding [68] with the EMBED keyword (see 4.2.6). In both cases asymptotic expansions for the long-range nuclear attraction type integrals are implemented in order to improve computational efficiency [69]. Another part of the QM energy in Eq. (1.9) is the MM augmented nuclear repulsion energy,
which can be calculated analytically from the structure of the QM/MM system. Because the so-defined QM energy contains all quantum mechanical terms plus the electrostatic embedding from the MM region the Kohn-Sham or Hartree-Fock matrix elements can be defined as partial derivatives of this energy with respect to density matrix elements.
The second term in Eq. (1.9) contains the mechanical interaction energy between the QM and MM regions. It is expressed in the form of a Lennard-Jones potential:
(12)
The are combinations of the van der Waals radii of QM atom and MM atom . By default these radii are taken from the MM force field. The parameter defines the depth of the Lennard-Jones potential. As for the van der Waals radii it is also taken from the MM force field. Therefore, an MM atom type has to be assigned to each QM atom in the input. This is done with the QM/MM keyword (Section 4.2.4).
The last term in Eq. (1.9) is the MM energy. In deMon2k 5.0 it can contain the following terms:
(13)
The first four terms in Eq. (1.14) denote bond stretching, angle bending, dihedral torsion and Urey-Bradley energy terms. Their calculation requires molecular connectivity information that is usually given in the input along with the geometrical definition of the MM atoms under the GEOMETRY keyword (see 4.1.1). As an alternative, the automatic generation of molecular connectivity information on the basis of the distances between MM atoms is also available. The last two terms in Eq. (1.14) represent van der Waals and point-charge interaction energies between the MM atoms. The force fields for MM and QM/MM calculations available in deMon2k are OPLS-AA [45] and AMBER [70]. They are selected by the FORCEFIELD keyword (see 4.2.3) and read from the FFDS (force field dataset) file. For these MM and QM/MM calculations all deMon2k functionalities, such as geometry optimization, transition state finding, molecular dynamics, frequency analysis etc., are available. Also property calculations for the QM system in QM/MM calculations are possible [71].
Besides the internal MM capability, deMon2k can also be externally interfaced with force fields. To this end a standard interface output for CHARMM [72] can be activated with the QM/MM keyword [44,46,73].
By default, the ERIs are calculated in each SCF cycle (direct SCF) using recurrence relations for near-field ERIs [20,51] and double asymptotic expansions [74] for far-field ERIs. This approach minimizes the random access memory (RAM) demand of deMon2k. If sufficient RAM is available the code performance can be improved by the MIXED option of the ERIS keyword (see 4.5.4). The RAM usage of deMon2k can be monitored by PRINT RAM (see Section 4.12.2 for more details). It also should be noted that, for larger systems, the linear algebra steps in deMon2k may become a bottleneck. With the keywords MATDIA and MATINV (see 4.11.2 and 4.11.3) alternative diagonalizers and matrix inversion techniques can be selected.
Several optimization and transition state search algorithms are implemented in deMon2k. For structure optimization, the default setting is the Levenberg-Marquardt restricted step method in delocalized internal redundant coordinates. This method has excellent convergence behavior and is very robust. However, it requires an iterative back transformation of the coordinates. Thus, to reach tight structure convergence, it may be necessary to switch to Cartesian coordinates at the end of the optimization (see 4.6.1). For ultimate accuracy, this might be combined with a Hessian calculation in each optimization step (UPDATE EXACT; Section 4.6.5). If effective core potentials (ECPs), Section 4.3.4, or model core potentials (MCPs), Section 4.3.5, are used, care must be taken regarding the accuracy of the gradients. Here it may be necessary to tighten the numerical integration threshold with the GRID keyword (see 4.3.6). Usually a FINE grid will be sufficient. The same holds for weak and nonbonded interactions. For the local transition state search, we recommend starting the optimization from a calculated Hessian (see 4.6.5) or restarting it from a frequency analysis (the Hessian from the frequency analysis is then used in the first optimization step). If a SADDLE point interpolation (Section 4.6.2) is to be performed, the starting points must be local minima, i.e. reactants and products. All optimizations and interpolations can be restarted with the deMon.new and deMon.mem files. These must be copied into the new input file deMon.inp and the corresponding restart file deMon.rst. The new input file may be modified and extended but the molecular geometry definitions must be left untouched in order to guarantee a successful restart run.
Born-Oppenheimer molecular dynamics (BOMD) simulations are initialized by the DYNAMICS keyword (see 4.7.1). In these calculations a trajectory file deMon.trj is created which can be large! For compatibility reasons, the trajectory file is written in ASCII (Note that *********************** are used as separations in this file). It should not be modified. The data from the trajectory file can be used to restart BOMD runs or to analyze them (Sections 4.7.2 and 4.7.3). Because BOMD runs may take weeks, we recommend that regular snapshots of the deMon working directory be produced from which restarts are possible. During such a copy the trajectory, deMon.trj, and new input file, deMon.new, must be unchanged. With these files, a restart run is possible just as in the case of structure optimizations, i.e. the deMon.new must be copied into the new input file deMon.inp. If requested, the restart file can also be used, e.g. for a restart density (GUESS RESTART; see 4.5.5). However, this is not mandatory.
Usually the default settings of deMon2k are sufficient for standard calculations. However, if extended basis sets are used or higher accuracy is required, it may be necessary to adjust the accuracy and performance settings of the code. This is achieved by the keywords GRID, SCFTYPE and ERIS (see 4.3.6, 4.5.1 and 4.5.4) for the electronic structure calculation, the keyword OPTIMIZATION (see 4.6.1) for the structure optimization and the keywords MATDIA and MATINV (see 4.11.2 and 4.11.3) for the linear algebra parts of the code. The keywords WEIGHTING, QUADRATURE and CFPINTEGRATION control the accuracy settings for the numerical integration (see 4.11.5, 4.11.6 and 4.11.7). The keyword DAVIDSON (4.11.4) controls the iterative diagonalization in time-dependent DFT calculations. In general, modification of the standard settings may alter the performance and accuracy of the code quite substantially. Therefore, such modifications should be tested carefully before being used for production runs. | 2020-01-25 06:10:52 | {"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": 0.8005803227424622, "perplexity": 1555.551636642153}, "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-2020-05/segments/1579251669967.70/warc/CC-MAIN-20200125041318-20200125070318-00539.warc.gz"} |
http://mathhelpforum.com/algebra/11087-find-b.html | # Math Help - Find A And B
1. ## Find A And B
Find A and B so that 1 / [(4n-3)(4n+1)] = A / (4n-3) + B / (4n+1).
Thanks for your time, efforts, and help!
2. Originally Posted by Mr_Green
Find A and B so that 1 / [(4n-3)(4n+1)] = A / (4n-3) + B / (4n+1).
You have,
$\frac{1}{(4n-3)(4n+1)}=\frac{A}{4n-3}+\frac{B}{4n+1}$
Multiply through by,
$(4n-3)(4n+1)$,
$1=A(4n+1)+B(4n-3)$
$1=n(4A+4B)+(A-3B)$
We have,
$4A+4B=0$
$A-3B=1$
Solve, to get,
$A=1/4,B=-1/4$
3. Originally Posted by Mr_Green
Find A and B so that 1 / [(4n-3)(4n+1)] = A / (4n-3) + B / (4n+1).
Thanks for your time, efforts, and help!
Here is one way.
1 /[(4n-3)(4n+1)] = A/(4n-3) +B/(4n+1)
Multiply both sides by (4n-3)(4n+1)
1 = A(4n+1) +B(4n-3) ---------------(i)
So that B is eliminated, make (4n-3) equal to zero.
4n -3 = 0
n = 3/4
So, when n=3/4, in (i),
1 = A(4(3/4) +1) +B(4(3/4) -3)
1 = A(3+1) +B(3-3)
1 = 4A
So that A is eliminated, make (4n+1) equal to zero.
4n +1 = 0
n = -1/4
So, when n = -1/4, in (i),
1 = A(4(-1/4) +1) +B(4(-1/4) -3)
1 = A(-1 +1) +B(-1 -3)
1 = -4B
4. Hello, Mr_Green!
Another method . . .
Find $A$ and $B$ so that: . $\frac{1}{(4n-3)(4n+1)} \:= \:\frac{A}{4n-3} + \frac{B}{4n+1}$
Multiply through by $(4n-3)(4n+1)\!:\;\;1 \;=\;(4n+1)A + (4n-3)B$
Let $n = \frac{3}{4}\!:\;\;1 \:=\:\left[4\left(\frac{3}{4}\right) + 1\right]A + \left[4\left(\frac{3}{4}\right) - 3\right] B$
. . $1 \:=\:4\!\cdot\!A + 0\!\cdot\!B\quad\Rightarrow\quad\boxed{ A\,=\,\frac{1}{4}}$
Let $n = -\frac{1}{4}\!:\;\;1\:=\:\left[4\left(-\frac{1}{4}\right) + 1\right]A + \left[4\left(-\frac{1}{4}\right) - 3\right]B$
. . $1 \:=\:0\!\cdot\!A - 4\!\cdot\!B\quad\Rightarrow\quad\boxed{ B = -\frac{1}{4}}$ | 2014-11-27 23:30:12 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8504337668418884, "perplexity": 9384.571151280816}, "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-2014-49/segments/1416931009292.37/warc/CC-MAIN-20141125155649-00153-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://archive-ouverte.unige.ch/unige:84914 | Title
# Scanning of the supersymmetry breaking scale and the gravitino mass in supergravity
Authors
Farakos, Fotis
Published in Journal of High Energy Physics. 2016, vol. 1606, p. 120-140
Collection Open Access - SCOAP3
Abstract We consider the minimal three-form N $$\mathcal{N}$$ = 1 supergravity coupled to nilpotent three-form chiral superfields. The supersymmetry breaking is sourced by the three-forms of the chiral multiplets, while the value of the gravitino mass is controlled by the three-form of the supergravity multiplet. The three-forms can nucleate membranes which scan both the supersymmetry breaking scale and the gravitino mass. The peculiar supergravity feature that the cosmological constant is the sum of a posictive contribution from the super-symmetry breaking scale and a negative contribution from the gravitino mass makes the cosmological constant jump. This can lead to a phenomenologically allowed small value of the cosmological constant even though the supersymmetry breaking scale and the gravitino mass are dynamically large.
Keywords P-branesSupergravity ModelsSupersymmetry Breaking
Identifiers
arXiv: 1605.07631
Full text
Structures
Citation
(ISO format)
FARAKOS, Fotis et al. Scanning of the supersymmetry breaking scale and the gravitino mass in supergravity. In: Journal of High Energy Physics, 2016, vol. 1606, p. 120-140. https://archive-ouverte.unige.ch/unige:84914
145 hits | 2018-11-20 17:22:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7266358137130737, "perplexity": 1538.6772717429403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039746528.84/warc/CC-MAIN-20181120171153-20181120193153-00389.warc.gz"} |
https://github.com/scy/forscript/blob/3ca67b520a260178bee2570937ca4faa9cf5f037/thesis.nw | # scy/forscript
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\documentclass[abstracton]{scrartcl} \usepackage[english]{babel} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{relsize} \usepackage{lmodern} \PreloadUnicodePage{0} \usepackage[pdfborder={0 0 0}]{hyperref} \usepackage{noweb} \def\cmd{\textsc} \def\pack{\emph} \def\term{\emph} \def\param{\texttt} \def\str{\texttt} \def\bs{\textbackslash} \def\key{\texttt} \newcommand\ctrl[1]{\key{\char\^#1}} \titlehead{University of Mannheim\\Laboratory for Dependable Distributed Systems} \subject{\vspace{3cm}Bachelor Thesis} \title{Design~and~Implementation~of a~Forensic~Documentation~Tool for~Interactive~Command\mbox{-}line~Sessions} \author{Tim Weber} \date{February 23, 2010} \publishers{ \vspace{5cm} \begin{tabular}{ll} Primary examiner: & Prof. Dr. Felix C. Freiling \\ Secondary examiner: & Dipl.-Inf. Andreas Dewald \\ Supervisor: & Prof. Dr. Felix C. Freiling \end{tabular} } \begin{document} \maketitle \begin{center}Bachelorstudiengang Software- und Internettechnologie\end{center} \thispagestyle{empty} \pagebreak \pagenumbering{roman} \setcounter{page}{1} {\phantom{.}\vspace{5cm}} \abstract{ In computer forensics, it is important to document examination of a computer system with as much detail as possible. Many experts use the software \cmd{script} to record their whole terminal session while analyzing the target system. This thesis shows why \cmd{script}’s features are not sufficient for documentation that is to be used in court. A new system, \cmd{forscript}, providing additional capabilities and mechanisms will be designed and developed in this thesis. } \pagebreak \tableofcontents \pagebreak \section*{Acknowledgements} First of all, I would like to thank Prof. Dr. Freiling for the opportunity to write a thesis about this interesting subject and for supporting me during the process of writing. I would also like to thank Andreas Dewald for being available as secondary examiner and Prof. Dr.-Ing. Effelsberg for postponing my thesis deadline. Thanks to Alexander Brock for testing and fuzzing \cmd{forscript} as well as proofreading the thesis. Michael Stapelberg, thank you for testing \cmd{forscript} and for giving me some hints about Unix system calls and why \cmd{script} does some things the way it does. Many thanks go to the free software community: The people who created C, GCC, Git, \LaTeX, Linux, make, noweb, script and Vim, but especially those who create comprehensive documentation. I would like to explicitly mention the BSD \emph{termios(4)} manual page as an example of how good documentation should look like. Thanks to my father for giving me the time to study at my own pace, and thanks to the hacker community for inspiring me every day. Finally, I would like to thank Nathalie for the love, support and understanding before, during and after this thesis. \pagebreak \pagenumbering{arabic} \section{Introduction}\label{intro} \subsection{Background: Computer Forensics} Computer forensics is a branch of forensic science.~\cite{casey} In the digital age we live in, an increasing number of crimes is performed using or at least aided by digital devices and computer systems. To analyze the evidence that may be present on these devices, specially trained experts are required. Having knowledge about the technology behind the systems, these forensic investigators are able to search for evidence without destroying traces, modifying or even accidentally inserting misleading data. Principles and techniques of computer forensics are, among others, employed to \begin{itemize} \item analyze computers, mobile phones and other electronic devices a suspected criminal has used, \item recover data after a hardware or software failure, \item gain information during or after attacks or break-in attempts on a computer system. \end{itemize} \subsubsection*{Documentation of Terminal Sessions} A forensic investigator has to keep a detailed record of his or her actions while analyzing a system. That way, in case of dispute about a piece of evidence, another forensic investigator can review the steps that led to certain conclusions. This \term{forensic log} improves the credibility of the investigator and protects a possible defendant from false accusations. Additionally, the investigator protects himself from forgetting how the evidence was found and what additional details (which probably seemed to be not important at that time) were present. The protocol consists of, depending on the type of analysis, notes on paper, images, videos and data files on the investigator’s computer. For example, to perform a \term{static analysis} of a suspect’s computer’s hard disk drive, i.e. searching the drive for suspicious data without modifying it, an investigator normally uses his computer, which is equipped with a software that records every action the investigator performs. Often a Unix-based operating system like Linux or Mac OS~X and command-line based software (also called \term{CLI software} for its command-line user interface) is used to perform such an analysis, for example \cmd{dd} to create a snapshot of the suspect’s hard drive, \cmd{sha1sum} to verify its integrity and other tools like \cmd{foremost} to find evidence in the snapshot. All interaction with the forensic software takes place in a text-based interface; the investigator uses his keyboard to perform commands, his workstation responds by displaying\footnote{also called “printing”, even though the output appears on the screen, not on paper} text and data. A text-based interface cannot display graphics or use the mouse\footnote{Using the mouse is possible via several extensions, but mouse commands are simply translated to special control characters and can be read by the application just like any other keyboard input.}. In principle, CLI sessions can be documented quite easily by creating a piece of software that records everything typed on the keyboard and everything sent to the screen. The \cmd{script} utility is often used to accomplish this; however, it has several limitations described in section~\ref{scriptissues} which greatly limit its usefulness as a forensic tool. \subsection{Tasks} Several tasks have to be solved in this bachelor thesis: \begin{itemize} \item Analyze \cmd{script} with regard to weaknesses concerning its usage as a forensic tool. \item Describe \cmd{script}’s output format and its disadvantages. \item Describe in detail an output format suitable for forensic usage. \item Implement a software for Linux that is used like \cmd{script}, but creates output in the new forensic output format. In order to minimize the requirements a target system has to meet to be able to run the software, it has to be implemented in the \term{C} programming language. \item Document the software according to the methods of \term{literate programming}. \end{itemize} \term{Literate programming}~\cite{literate} is a technique invented by Donald~E. Knuth, the author of the \TeX{} typesetting system. Instead of writing more or less commented source code, it propagates writing a continuous text with embedded code fragments. These do not necessarily appear in the order they are executed, but where they are didactically useful. The software \cmd{noweb}~\cite{noweb} is used to generate the layouted thesis as well as the final program’s source code out of a single file. \subsection{Results} It is apparent that \cmd{script} is not suited for forensic usage, especially because it does not record the user’s input and data about the environment it is running in. A successor, \cmd{forscript}, has been designed and developed in this thesis. Its output format is portable, extensible and contains detailed information about the environment. The disadvantages of \cmd{script} are eliminated. Following the paradigm of literate programming, this thesis is \cmd{forscript} and vice versa. \subsection{Outlook on the Thesis} Section~\ref{intro}, which you are currently reading, contains the introduction into the topic of computer forensics. It explains why detailed documentation of forensic analyses is an important task, what a command-line interface is, which subjects will be presented in this thesis and also provides an overview of the tasks and results. In section~\ref{script}, one of the most popular tools for recording interactive terminal sessions, \cmd{script}, will be presented and the format of the files it generates will be described. Afterwards, several issues regarding its usage as a forensic tool are presented, leading to the conclusion that it should be replaced with a more suitable software. This new software called \cmd{forscript} will be drafted in section~\ref{design}, focusing on its file format and the resulting properties. The invocation syntax of \cmd{forscript}, which is based on that of \cmd{script}, and the differences in behavior compared to \cmd{script} is also described. Section~\ref{implementation}, by far the longest section, contains a detailed step-by-step description of \cmd{forscript}’s source code. It describes how to write \cmd{forscript}’s data format, parsing the command line, what a pseudo terminal is and how to create one to access the input and output streams of an application, how to deal with subprocesses and signals and other things. The resulting application will be evaluated in section~\ref{evaluation}, which includes an example transcript file and a description of \cmd{forscript}’s known limitations. Finally, section~\ref{summary} summarizes the work that has been done. It talks about the future of \cmd{forscript} and describes the next steps that should probably be taken to make it even more useful. \section{\cmd{script}}\label{script} \pack{util-linux} is the name of a collection of command-line utilities for Linux systems. It includes essential software like \cmd{dmesg}, \cmd{fdisk}, \cmd{mkswap}, \cmd{mount} and \cmd{shutdown} as well as the \cmd{script} and \cmd{scriptreplay} utilities. The original \pack{util-linux} package~\cite{util} was abandoned in 2006. Today, it has been replaced by its successor \pack{util-linux-ng}~\cite{utilng}, a \term{fork} based on the last available \pack{util-linux} version. \pack{util-linux-ng} is under active development. The analysis of the original \cmd{script} utility in this thesis is based on the most recent \pack{util-linux-ng} release as of the time of writing, version 2.17. \subsection{Purpose} The purpose of \cmd{script} is to record everything printed to the user’s terminal into a file. According to its manual, “[i]t is useful for students who need a hardcopy record of an interactive session as proof of an assignment”. It can also record timing data, specifying the chronological progress of the terminal session, into a second file. Using both of these files, the accompanying utility \cmd{scriptreplay} can display the recorded data in a video-like way. \subsection{Mode of Operation} In order to record the terminal session, \cmd{script} creates a new \term{pseudo terminal} (PTY), which is a virtual, software-based representation of a terminal line, and attach itself to the “master” side of it, thereby being able to send and receive data to and from an application connected to the “slave” side of the PTY. It launches a subprocess (also known as \term{child}), which launches the actual client application as its own subchild and then records the client application’s output stream. The parent process forwards the user’s input to the client application. Recording terminates as soon as the child process exits. \subsection{Invocation} \cmd{script} takes one optional argument, the file name of the output file (also called \term{typescript} file) to generate. If the argument is omitted, the file will be named \str{typescript}, except when the file already exists and is a symbolic or hard link: \cmd{script} then refuses to overwrite the file, apparently for safety reasons. This check can be avoided by explicitly providing the file name on the command line. There are several command-line switches that modify \cmd{script}’s behavior. The \param{-a} switch will pass the \str{a} flag instead of \str{w} to [[fopen()]]’s [[mode]] parameter. If a typescript file does already exist, it will then not be overwritten; instead, the new content will be appended to the existing file. By default, \cmd{script} will launch the shell specified by the environment variable \str{\$SHELL}. If \str{\$SHELL} it is not set, a default shell selected at compile time (usually \str{/bin/sh}). The shell will be called with \param{-i} as its first parameter, making it an interactive shell. However, if \cmd{script} is called with the \param{-c} option, followed by a command, it will launch the shell with \param{-c} and the command instead of \param{-i}. The shell will then be non-interactive and only run the specified command, then exit. For example, called with the parameters \param{-c 'last -5'}, \cmd{script} will launch \str{/bin/sh -c 'last -5'} (or whatever shell is defined in \str{\$SHELL}). Note that all POSIX-compatible shells have to support the \param{-i} and \param{-c} parameters. If the \param{-f} switch is used, \cmd{script} will call [[fflush()]] on the typescript file after new data has been written to it, resulting in instant updates to the typescript file, at the expense of performance. This is for example useful for letting another user watch the actions recorded by \cmd{script} in real time. If the \param{-q} switch is not specified, \cmd{script} will display a message when it starts or quits and also record its startup and termination it the typescript file. With \param{-q}, all of these messages will not appear, with one exception: Since \cmd{scriptreplay} will unconditionally discard the first line in a typescript file, writing the startup message (\str{"Script started on …"}) cannot be disabled. The \param{-t} switch will make \cmd{script} output timing information to \term{stderr}. Its format is described in section~\ref{scripttiming}. If \cmd{script} is called with \param{-V} or \param{--version} as only parameter, it will print its version and exit. Any other parameter will make \cmd{script} display an error message and exit. \subsection{File Formats} \subsubsection{Typescript} The current implementation of \cmd{script} uses a very simple typescript file format: Everything the client application sends to the terminal, i.e. everything printed on screen, will be written to the file, byte by byte, including control characters that are used for various tasks like setting colors, positioning the cursor etc. Additionally, a header \str{"Script started on XXX\bs{}n"} is written, where \str{XXX} is the human-readable date and time when \cmd{script} was invoked. If \cmd{script} was invoked without the \param{-q} flag, an additional footer \str{"Script done on YYY\bs{}n"}, where \str{YYY} is the human-readable date and time when \cmd{script} terminated, is written. \subsubsection{Timing} \label{scripttiming} Since this typescript format completely lacks timing information, the \param{-t} flag will output timing data to stderr. The user has to capture this output to a file by calling \cmd{script} like this: \str{script -t 2>timingfile}. The timing file consists of tuples of delay and byte count (space-separated), one per line: \begin{verbatim} 0.725168 56 0.006549 126 0.040017 1 4.727988 1 0.047972 1 \end{verbatim} Each line can be read like \emph{“\emph{x} seconds after the previous output, \emph{n} more bytes were sent to the terminal”}. If there was no previous output (because it is the first line of timing information), the delay specifies the time between \cmd{script} invocation and the first chunk of output. \subsection{Disadvantages}\label{scriptissues} The two file formats produced by \cmd{script}, typescript and timing, show several shortcomings with regard to forensic usage: \begin{itemize} \item Input coming from the user’s keyboard is not logged at all. A common example is the user entering a command in the shell but then pressing \ctrl{C} instead of return. The shell will move to the next line and display the prompt again; there is no visible distinction whether the command was run or not.\footnote{With more recent versions of Linux and Bash, terminals which have the ECHOCTL bit set (for example via stty) will show \ctrl{C} at the end of an interrupted line, which fixes this problem to some degree. Similar issues, like finding out whether the user entered or tab-completed some text, still persist.} \item Metadata about the environment \cmd{script} runs in is not logged. This leads to a high level of uncertainty when interpreting the resulting typescript, because even important information like the character set and encoding or the terminal size and type is missing. \item Typescript and timing are separate files, but one logical entity. They should reside in one file to protect the user from confusion and mistakes. \item Appending to a typescript file is possible, but ambigious, since the beginning of a new part is determined only by the string \str{"Script started on~…"}. Also, appending to a typescript and recording timing information are incompatible, because \cmd{scriptreplay} will only ignore the first header line in a typescript file. Subsequent ones will disturb the timing’s byte counter. \end{itemize} \subsection*{Summary} This section has presented the background, purpose and operation of \cmd{script}. We have learned that because of several lacking features, using it in computer forensics is problematic. The next section will introduce a software without these disadvantages. \section{Design of \cmd{forscript}}\label{design} In this section, the new file format as used by \cmd{forscript} will be specified. You will learn about how input, output and metadata are combined into a single output file. After describing the format’s characteristics, the invocation syntax, which is designed to be compatible to \cmd{script}, will be presented. \subsection{File Format} A \cmd{forscript} data file (called a \emph{transcript file}) consists of the mostly unaltered output stream of the client application, but includes blocks of additional data (called \emph{control chunks}) at arbitrary positions. A control chunk is started by a \emph{shift out} byte (\str{0x0e}) and terminated by a \emph{shift in} byte (\str{0x0f}). Each control chunk is either an input chunk or a metadata chunk. \subsubsection{Input Chunks} Input chunks contain the data that is sent to the client application’s input stream, which is usually identical to the user’s keyboard input. They are of arbitrary length and terminate at the \emph{shift in} byte. If a literal \emph{shift out} or \emph{shift in} byte needs to appear in an input chunk’s data, it is escaped by prepending a \emph{data link escape} byte (\str{0x10}). If a literal \emph{data link escape} byte needs to appear in an input chunk’s data, it has to be doubled (i.e., \str{0x10 0x10}). For example, if the user sends the byte sequence \str{0x4e 0x0f 0x00 0x61 0x74 0x10}, the complete input chunk written to the transcript file is \str{0x0e 0x4e 0x10 0x0f 0x00 0x61 0x74 0x10 0x10 0x0f}. \subsubsection{Metadata Chunks} Metadata chunks, also called meta chunks, contain additional information about the file or the application’s status, for example environment variables, terminal settings or time stamps. They contain an additional \emph{shift out} byte at the beginning, followed by a byte that determines the type of metadata that follows. The available types are described below. Meta chunks are of arbitrary length and terminate at the \emph{shift in} byte. The same escaping of \emph{shift out}, \emph{shift in} and \emph{data link escape} that is used for input chunks is also used for meta chunks. For example, the “terminal size” meta type is introduced by its type byte \str{0x11}, followed by width and heigth of the terminal, represented as two unsigned big-endian 16-bit integers. The information “terminal size is 80×16 characters” would be written to the transcript file as \str{0x0e 0x0e 0x11 0x00 0x50 0x00 0x10 0x10 0x0f}. Note that the least significant byte of the number 16 has to be written as \str{0x10 0x10} to prevent the special meaning of \str{0x10} to escape the following \str{0x0f}. \subsubsection{Properties of the File Format} This basic file format design has several advantages: \begin{itemize} \item New meta chunk types can be introduced while still allowing older tools to read the file, because the escaping rules are simple and the parsing application need not know a fixed length of each type. \item Since switching between input and output data occurs very often in a usual terminal session, the format is designed to require very little storage overhead for these operations. \item The format is very compact and easy to implement. Using a format like XML would decrease performance and require sophisticated libraries on the machine \cmd{forscript} is run on. However, for forensic usage it is best to be able to use a small statically linked executable. \item Converting a \cmd{forscript} file to a \cmd{script} file is basically as easy as removing everything between \emph{shift out} and \emph{shift in} bytes (while respecting escaping rules, of course). \end{itemize} \subsection{Metadata Chunk Types} The next sections will describe the available metadata chunk types. Integers are unsigned and big endian, except where noted otherwise. In the resulting file, numbers are represented in binary form, not as ASCII digits. For better understanding, the code \cmd{forscript} uses to write each meta chunk appears after the chunk’s explanation. The three functions [[chunkwh()]], [[chunkwf()]] and [[chunkwd()]] that are used for actually writing the data to disk will be explained in section~\ref{def:chunkwriters}. To be able to understand the code, it is sufficient to know that [[chunkwh()]] takes one parameter (the chunk’s type) and writes the header bytes. [[chunkwf()]] writes the footer byte and takes no parameters, while [[chunkwd()]] writes the payload data, escaping it on the fly, and requires a pointer and byte count. There is an additional convenience function [[chunkwm()]] that takes all three parameters and will write a complete metadata chunk. All chunk functions return a negative value if an error occured, for example if an environment setting could not be retrieved or if writing to the transcript file failed. Since only a partial metadata chunk may have been written to the transcript, the file is no longer in a consistent state. Therefore, \cmd{forscript} should terminate whenever a chunk function returns a negative value. A transcript file needs to begin with a \emph{file version} meta chunk, followed directly by the first \emph{start of session} chunk. \subsubsection*{\str{0x01} File Version (1 byte)} The transcript file must start with a meta chunk of this type; there may be no other data before it. Denotes the version of the \cmd{forscript} file format that is being used for this file. In order to guarantee a length of exactly one byte, the version numbers 0, 14, 15 and 16 are not allowed, therefore no escaping takes place. This document describes version 1 of the format, therefore currently the only valid value is \str{0x01}. <>= int chunk01() { unsigned char ver = 0x01; return chunkwm(0x01, &ver, sizeof(ver)); } @ \subsubsection*{\str{0x02} Begin of Session (10 bytes)} Denotes the start of a new \cmd{forscript} session. The first four data bytes represent the start time as the number of seconds since the Unix Epoch. The next four bytes contain a signed representation of the nanosecond offset to the number of seconds. If these four bytes are set to \str{0xffffffff}, there was an error retrieving the nanoseconds. The last two bytes specify the machine’s (signed) time zone offset to UTC in minutes. If these two bytes are set to \str{0xffff}, the machine’s timezone is unknown. <>= int chunk02() { struct timespec now; extern long timezone; int ret; unsigned char data[10]; uint32_t secs; int32_t nanos = ~0; int16_t tzone = ~0; if ((ret = clock_gettime(CLOCK_REALTIME, &now)) < 0) return ret; secs = htonl(now.tv_sec); if (now.tv_nsec < 1000000000L && now.tv_nsec > -1000000000L) nanos = htonl(now.tv_nsec); tzset(); tzone = htons((uint16_t)(timezone / -60)); memcpy(&data[0], &secs, sizeof(secs)); memcpy(&data[4], &nanos, sizeof(nanos)); memcpy(&data[8], &tzone, sizeof(tzone)); return chunkwm(0x02, data, sizeof(data)); } @ This chunk requires the headers \str{time.h} for [[clock_gettime()]], \str{inet.h} for [[htonl()]] and \str{string.h} for [[memcpy()]]: <>= #include #include #include @ \subsubsection*{\str{0x03} End of Session (1 byte)} Denotes the end of a \cmd{forscript} session. The data byte contains the return value of the child process. The usual exit code convention applies: If the child exited normally, use its return value. If the child was terminated as a result of a signal (like \str{SIGSEGV}), use the number of the signal plus$128$. The parameter [[status]] should contain the raw status value returned by [[wait()]], not only the child’s return value. If the exit code of the child could not be determined, \str{0xff} is used instead. <>= int chunk03(int status) { unsigned char data = ~0; if (WIFEXITED(status)) data = WEXITSTATUS(status); else if (WIFSIGNALED(status)) data = 128 + WTERMSIG(status); return chunkwm(0x03, &data, sizeof(data)); } @ \subsubsection*{\str{0x11} Terminal Size (two 2-byte values)} Is written at session start and when the size of the terminal window changes. The first data word contains the number of colums, the second one the number of rows. Since the terminal size has to be passed to the running client application, the chunk itself does not request the values, but receives them as a parameter. <>= int chunk11(struct winsize *size) { uint32_t be; be = htonl((size->ws_col << 16) | size->ws_row); return chunkwm(0x11, (unsigned char *)&be, sizeof(be)); } @ \subsubsection*{\str{0x12} Environment Variables (arbitrary number of C strings)} Is written at session start. Contains the environment variables and their values as \str{NAME=value} pairs, each pair is terminated by a null byte (\str{0x00}). Since variable names may not contain the \str{=} character and neither variables names nor the values may include a null byte, the list needs no special escaping. <>= int chunk12() { extern char **environ; int i = 0; int ret; while (environ[i] != NULL) { if (i == 0) { if ((ret = chunkwh(0x12)) < 0) return ret; } if ((ret = chunkwd((unsigned char *)environ[i], strlen(environ[i]) + 1)) < 0) return ret; i++; } if (i != 0) { if ((ret = chunkwf()) < 0) return ret; } return 1; } @ \subsubsection*{\str{0x13} Locale Settings (seven C strings)} Is written at session start. Contains the string values of several locale settings, namely \str{LC\_ALL}, \str{LC\_COLLATE}, \str{LC\_CTYPE}, \str{LC\_MESSAGES}, \str{LC\_MONETARY}, \str{LC\_NUMERIC} and \str{LC\_TIME}, in that order, each terminated by a null byte. <>= int chunk13() { int cat[7] = { LC_ALL, LC_COLLATE, LC_CTYPE, LC_MESSAGES, LC_MONETARY, LC_NUMERIC, LC_TIME }; char *loc; int ret; if ((ret = chunkwh(0x13)) < 0) return ret; for (int i = 0; i < 7; i++) { if ((loc = setlocale(cat[i], "")) == NULL) return -1; if ((ret = chunkwd((unsigned char *)loc, strlen(loc) + 1)) < 0) return ret; } if ((ret = chunkwf()) < 0) return ret; return 0; } @ [[setlocale()]] requires \str{locale.h}: <>= #include @ \subsubsection*{\str{0x16} Delay (two 4-byte values)} Contains the number of seconds and nanoseconds that have passed since the last delay chunk (or, if this is the first one, since the session started). A replaying application should wait for the time specified in this chunk before advancing further in the transcript file. Since the seconds and nanoseconds are represented as integers, converting to a floating-point number would mean a loss of precision. Therefore both integers are subtracted independently. If the nanoseconds part of [[now]] is less than that of [[ts]], the seconds part has to be decreased by one for the result to be correct. <>= int chunk16(struct timespec *ts) { unsigned char buf[2 * sizeof(uint32_t)]; uint32_t secs, nanos; struct timespec now; if (clock_gettime(CLOCK_MONOTONIC, &now) < 0) return -1; secs = now.tv_sec - ts->tv_sec; if (now.tv_nsec > ts->tv_nsec) { nanos = now.tv_nsec - ts->tv_nsec; } else { nanos = 1000000000L - (ts->tv_nsec - now.tv_nsec); secs--; } *ts = now; secs = htonl(secs); nanos = htonl(nanos); memcpy(&buf[0], &secs, sizeof(secs)); memcpy(&buf[sizeof(secs)], &nanos, sizeof(nanos)); return chunkwm(0x16, buf, sizeof(buf)); } @ \subsection{Magic Number} Since a \cmd{forscript} file has to start with a file version chunk followed by a begin of session chunk, there is a distinctive eight-byte signature at the beginning of each file: \begin{verbatim} 0x0e 0x0e 0x01 0x?? 0x0f 0x0e 0x0e 0x02 \end{verbatim} The first two bytes start a metadata chunk, the third one identifies it as a file version chunk. The fourth byte contains the version number, which is currently \str{0x01} but may change in the future. Byte 5 closes the version chunk, 6 to 8 start a begin of session chunk. \subsection{Invocation} \cmd{forscript}’s invocation syntax has been designed to be compatible to \cmd{script}, most parameters result in the same behavior. The following list contains additional notes and describes the differences to \cmd{script}: \begin{itemize} \item \param{-a} (append): If the target transcript file already exists and is non-empty, it has to start with a valid and supported \emph{file version} header. \item \param{-c} (command) and \param{-f} (flush): Identical to \cmd{script}. \item \param{-q} (quiet): In contrast to \cmd{script}, no startup message will be written to the transcript file. \item \param{-t} (timing): This parameter will be accepted, but ignored. \cmd{forscript} always records timing information. \item \param{-V} and \param{--version}: Identical to \cmd{script}, both will make \cmd{forscript} output its version information and terminate. The parameter has to be the only one specified on the command line, else an error message will be printed. \end{itemize} If unsupported parameters are passed, \cmd{forscript} will print a short usage summary to \emph{stderr} and exit. While running, the client application’s output will be printed to \emph{stdout}. Error messages will be printed to \emph{stderr}. \subsection*{Summary} Now you know how \cmd{forscript} stores the recorded terminal session and how it will be called by the user. You have seen the code that writes the various metadata chunks. After this soft introduction to \cmd{forscript}’s implementation, the next section contains the rest of the code and will talk in detail about how the software works. \pagebreak \section{Implementation of \cmd{forscript}}\label{implementation} This section will describe the code of \cmd{forscript} in detail. You will learn how the software hooks into the input and output stream of the client application and how it reacts to things like window size changes or the child terminating. Other interesting topics include how to launch a subprocess and change its controlling terminal as well as how to read from multiple data streams at one without having to run separate processes. \subsection{Constants} For improved readability, we define the special characters introduced in the previous section as constants: <>= const unsigned char SO = 0x0e; const unsigned char SI = 0x0f; const unsigned char DLE = 0x10; @ It is by design that the three special characters have consecutive byte numbers. This allows us to define a minimum and maximum byte value that requires special escape handling: <>= const unsigned char ESCMIN = 0x0e; const unsigned char ESCMAX = 0x10; @ \subsection{Writing Metadata Chunks to Disk} \label{def:chunkwriters} The function \emph{chunkwd()} takes a pointer and a byte count as arguments and writes chunk data to the transcript file, applying required escapes on the fly. To improve performance, it does not write byte-by-byte, but instead scans the input data until it finds a special character. When it does, it writes everything up to, but not including, the special character to the file and then adds a DLE character. The search then goes on. If another special character is found, everything from the last special character (inclusive) to the current one (exclusive) plus a DLE is written. Eventually the whole input data will have been scanned and the function terminates after writing everything from the last special character (inclusive) or the beginning of the data (if there were no special characters) to the end of the input data. This is the code: <>= int chunkwd(unsigned char *data, int count) { int escaped = 0; int pos = 0; int start = 0; while (pos < count) { if (data[pos] <= ESCMAX && data[pos] >= ESCMIN) { if (pos > start) { if (!swrite(&data[start], sizeof(char), pos - start, OUTF)) return -1; } if (!swrite(&DLE, sizeof(DLE), 1, OUTF)) return -2; start = pos; escaped++; } pos++; } if (!swrite(&data[start], sizeof(char), pos - start, OUTF)) return -3; return escaped; } @ \emph{OUTF} is the already opened transcript file and a global variable: <>= FILE *OUTF; @ The \emph{swrite()} function (“safe write”) that is being used here will return zero if the number of items written is not equal to the number of items that \emph{should} have been written: <>= int swrite(const void *ptr, size_t size, size_t nmemb, FILE *stream) { return (fwrite(ptr, size, nmemb, stream) == nmemb); } @ To be able to use [[fwrite()]], \str{stdio.h} has to be included: <>= #include @ There are functions to write chunk headers and footers: <>= int chunkwh(unsigned char id) { int ret; for (int i = 0; i < 2; i++) { ret = swrite(&SO, sizeof(SO), 1, OUTF); if (!ret) return -1; } return (swrite(&id, sizeof(unsigned char), 1, OUTF)) ? 1 : -1; } int chunkwf() { return (swrite(&SI, sizeof(SI), 1, OUTF)) ? 1 : -1; } @ There is also a convenience function that writes a meta chunk’s header and footer as well as the actual data: <>= int chunkwm(unsigned char id, unsigned char *data, int count) { int ret; if (!chunkwh(id)) return -11; if ((ret = chunkwd(data, count)) < 0) return ret; if (!chunkwf()) return -12; return 1; } @ \subsection{Error Handling} If the program has to terminate abnormally, the function [[die()]] will be called. After resetting the terminal attributes and telling a possible child process to exit, it will output an error message and exit the software. <>= void die(char *message, int chunk) { if (TTSET) tcsetattr(STDERR_FILENO, TCSADRAIN, &TT); if (CHILD > 0) kill(CHILD, SIGTERM); fprintf(stderr, "%s: ", MYNAME); if (chunk != 0) { fprintf(stderr, "metadata chunk %02x failed", chunk); if (message != NULL) fprintf(stderr, ": "); } else { if (message == NULL) fprintf(stderr, "unknown error"); } if (message != NULL) fprintf(stderr, "%s", message); fprintf(stderr, "; exiting.\n"); exit(EXIT_FAILURE); } @ [[exit()]] requires \str{stdlib.h}: <>= #include @ The global variable [[MYNAME]] contains a pointer to the name the binary was called as and is set in [[main()]]. <>= char *MYNAME; @ \subsection{Startup and Shutdown Messages} The [[statusmsg()]] function writes a string to both the terminal and the transcript: <>= void statusmsg(const char *msg) { char date[BUFSIZ]; time_t t = time(NULL); struct tm *lt = localtime(&t); if (lt == NULL) die("localtime failed", 0); if (strftime(date, sizeof(date), "%c", lt) < 1) die("strftime failed", 0); if (printf(msg, date, OUTN) < 0) { perror("status stdout"); die("statusmsg stdout failed", 0); } if (fprintf(OUTF, msg, date, OUTN) < 0) { perror("status transcript"); die("statusmsg transcript failed", 0); } } @ \subsection{Initialization} \subsubsection{Determining the Binary’s Name} To be able to output its own name (e.g. in error messages), \cmd{forscript} determines the name of the binary that has been called by the user. This value is stored in [[argv[0]]]. The global variable [[MYNAME]] will be used to reference that value from every function that needs it. <>= MYNAME = argv[0]; @ If \cmd{forscript} was called using a path name (e.g. \str{/usr/bin/forscript}), everything up to the final slash needs to be cut off. This is done by moving the pointer to the character immediately following the final slash. <>= { char *name; if ((name = basename(MYNAME)) != NULL) MYNAME = name; } @ [[basename()]] requires \str{libgen.h}: <>= #include @ \subsubsection{Command Line Arguments} Since \cmd{forscript}’s invocation tries to mimic \cmd{script}’s as far as possible, command line argument handling is designed to closely resemble \cmd{script}’s behavior. Therefore, like in \cmd{script}, the command line switches \str{--version} and \str{-V} are treated separately. If there is exactly one command line argument and it is one of these, \cmd{forscript} will print its version and terminate. <>= if ((argc == 2) && (!strcmp(argv[1], "-V") || !strcmp(argv[1], "--version"))) { printf("%s %s\n", MYNAME, MYVERSION); return 0; } @ [[MYVERSION]] is defined as a global constant: <>= const char *MYVERSION = "1.0.0-git"; @ The other options are parsed using the normal [[getopt()]] method, which requires \str{unistd.h}: <>= #include @ [[getopt()]] returns the next option character each time it is called, and$-1$if there are none left. The option characters are handled in a [[switch]] statement. As in \cmd{script}, flags that turn on some behavior cause a respective global [[int]] variable to be increased by one. These flags are: <>= int aflg = 0, fflg = 0, qflg = 0; @ The value of the \str{-c} parameter is stored in a global string pointer: <>= char *cflg = NULL; @ The \str{-t} flag is accepted for compatibility reasons, but has no effect in \cmd{forscript} because timing information is always written. After the loop terminates, [[optind]] arguments have been parsed. [[argc]] and [[argv]] are then modified accordingly to only handle non-option arguments (in \cmd{forscript} this is only the file name). The parsing loop therefore looks like this: <>= { int c; extern char *optarg; extern int optind; while ((c = getopt(argc, argv, "ac:fqt")) != -1) switch ((char)c) { case 'a': aflg++; break; case 'c': cflg = optarg; break; case 'f': fflg++; break; case 'q': qflg++; break; case 't': break; case '?': default: fprintf(stderr, "usage: %s [-afqt] [-c command] [file]\n", MYNAME); exit(EXIT_FAILURE); break; } argc -= optind; argv += optind; } @ After the options have been parsed, the output file name will be determined and stored in the global string [[OUTN]]: <>= char *OUTN = "transcript"; @ If there was no file name supplied on the command line, the default name is \str{transcript}. This differs from \cmd{script}’s default name \str{typescript} intentionally, because the file format is different and can, for example, not be displayed directly using \cmd{cat}. If there are any scripts or constructs that assume the default output file name to be \str{typescript}, the chance that replacing \cmd{script} with \cmd{forscript} will break their functionality anyway is quite high. \subsubsection{Opening the Output File} As in \cmd{script}, there is a safety warning if no file name was supplied and \str{transcript} exists and is a (hard or soft) link. <>= if (argc > 0) { OUTN = argv[0]; } else { struct stat s; if (lstat(OUTN, &s) == 0 && (S_ISLNK(s.st_mode) || s.st_nlink > 1)) { fprintf(stderr, "Warning: %s' is a link.\n" "Use `%s [options] %s' if you really " "want to use it.\n" "%s not started.\n", OUTN, MYNAME, OUTN, MYNAME); exit(EXIT_FAILURE); } } @ [[lstat()]] needs \str{types.h} and \str{stat.h} as well as \str{\_XOPEN\_SOURCE}: <>= #include #include @ <>= #define _XOPEN_SOURCE 500 @ The file will now be opened, either for writing or for appending, depending on [[aflg]]. Note that if appending, the file will be opened for reading as well. This is because \cmd{forscript} checks the file version header before appending to a file. <>= if ((OUTF = fopen(OUTN, (aflg ? "a+" : "w"))) == NULL) { perror(OUTN); die("the output file could not be opened", 0); } @ If the file has been opened for appending, check whether it starts with a compatible file format. Currently, the only format allowed is \str{0x01}. If the file is empty, appending is possible, but the \emph{file version} chunk has to be written. This is done by setting [[aflg]] to$0$, which will cause [[doio()]] to write the chunk. <>= if (aflg) { char buf[5]; size_t count; count = fread(&buf, sizeof(char), 5, OUTF); if (count == 0) aflg = 0; else if (count != 5 || strncmp(buf, "\x0e\x0e\x01\x01\x0f", 5) != 0) die("output file is not in forscript format v1, " "cannot append", 0); } @ \subsection{Preparing a New Pseudo Terminal} While \cmd{script} uses manual PTY allocation (by trying out device names) or BSD’s [[openpty()]] where available, \cmd{forscript} has been designed to use the Unix~98 PTY multiplexer (\str{/dev/ptmx}) standardized in POSIX.1-2001 to create a new PTY. This method requires \str{fcntl.h} and a sufficiently high feature test macro value for POSIX code. <>= #include @ <>= #define _POSIX_C_SOURCE 200112L @ The PTY’s master and slave file descriptors will be stored in these global variables: <>= int PTM = 0, PTS = 0; @ Additionally, the settings of the terminal \cmd{forscript} runs in will be saved in the global variable [[TT]]. This variable is used to duplicate the terminal’s settings to the newly created PTY as well as to restore the terminal settings as soon as \cmd{forscript} terminates. There is also a variable [[TTSET]] which stores whether the settings have been written to [[TT]]. This is important when restoring the terminal settings after a failure: If the settings have not yet been written to [[TT]], applying them will lead to undefined behavior. <>= struct termios TT; int TTSET = 0; @ <>= if (tcgetattr(STDIN_FILENO, &TT) < 0) { perror("tcgetattr"); die("tcgetattr failed", 0); } TTSET = 1; @ The \str{termios} structure is defined in \str{termios.h}. <>= #include @ A new PTY master is requested like this: <>= if ((PTM = posix_openpt(O_RDWR)) < 0) { perror("openpt"); die("openpt failed", 0); } @ Then, access to the slave is granted. <>= if (grantpt(PTM) < 0) { perror("grantpt"); die("grantpt failed", 0); } if (unlockpt(PTM) < 0) { perror("unlockpt"); die("unlockpt failed", 0); } @ The slave’s device file name is requested using [[ptsname()]]. Since the name is not needed during further execution, the slave will be opened and its file descriptor stored. <>= { char *pts = NULL; if ((pts = ptsname(PTM)) != NULL) { if ((PTS = open(pts, O_RDWR)) < 0) { perror(pts); die("pts open failed", 0); } } else { perror("ptsname"); die("ptsname failed", 0); } } @ The “parent” terminal will be configured into a “raw” mode of operation. \cmd{script} does this by calling [[cfmakeraw()]], which is a nonstandard BSD function. For portability reasons \cmd{forscript} sets the corresponding bits manually, thereby emulating [[cfmakeraw()]]. The list of settings is taken from the \emph{termios(3)} Linux man page~\cite{linuxman} and should be equivalent. Afterwards, the settings of the terminal \cmd{forscript} was started in will be copied to the new terminal. This means that in the eyes of the user the terminal’s behavior will not change, but \cmd{forscript} can now document the terminal’s data stream with maximum accuracy. <>= { struct termios rtt = TT; rtt.c_iflag &= ~(IGNBRK | BRKINT | PARMRK | ISTRIP | INLCR | IGNCR | ICRNL | IXON); rtt.c_oflag &= ~OPOST; rtt.c_lflag &= ~(ECHO | ECHONL | ICANON | ISIG | IEXTEN); rtt.c_cflag &= ~(CSIZE | PARENB); rtt.c_cflag |= CS8; if (tcsetattr(STDIN_FILENO, TCSANOW, &rtt) < 0) { perror("tcsetattr stdin"); die("tcsetattr stdin failed", 0); } if (tcsetattr(PTS, TCSANOW, &TT) < 0) { perror("tcsetattr pts"); die("tcsetattr pts failed", 0); } } @ \subsubsection{Managing Window Size} If the size of a terminal window changes, the controlling process receives a \str{SIGWINCH} signal and should act accordingly. \cmd{forscript} handles this signal in the [[resized()]] function by writing the new size to the transcript and forwarding it to the client terminal. <>= void resized(int signal) { UNUSED(signal); winsize(3); } @ The actual reading and writing of the window size is done by [[winsize()]], which takes a [[mode]] parameter. If the mode is$1$, the client application’s terminal size will be set. If the mode is$2$, the terminal size will be written to the transcript. If the mode is$3$, both operations will be done, which is the usual case. <>= void winsize(unsigned int mode) { struct winsize size; ioctl(STDIN_FILENO, TIOCGWINSZ, &size); if (mode & 2) if (chunk11(&size) < 0) die("writing window size", 0x11); if ((mode & 1) && PTM) ioctl(PTM, TIOCSWINSZ, &size); } @ Retrieving the window size requires \str{ioctl.h} for [[ioctl()]]: <>= #include @ The client PTY’s window size will be initialized now. This needs to take place before the client application is launched, because it probably requires an already configured terminal size when starting up. Writing the size to the transcript however would put the window size meta chunk before the start of session chunk, therefore [[winsize()]]’s mode$1$is used. <>= winsize(1); @ \subsection{Launching Subprocesses} The original \cmd{script} uses one process to listen for input, one to listen for output and one to initialize and [[execl()]] the command to be recorded. \cmd{forscript} in contrast uses only the [[select()]] function to be notified of pending input and output and therefore only needs two processes: Itself and the subcommand. \subsubsection*{Registering Signal Handlers} To be notified of an exiting subprocess, a handler for the \str{SIGCHLD} signal needs to be defined. This signal is usually sent by the operating system if any child process’s run status changes, i.e. it is stopped (\str{SIGSTOP}), continued (\str{SIGCONT}) or it exits. \cmd{script} terminates if the child is stopped, but \cmd{forscript} does not, because it uses the \str{SA\_NOCLDSTOP} flag to specify that it wishes not to be notified about the child stopping or resuming. The function [[finish()]] handles the child’s termination. The second signal handler, [[resized()]], handles window size changes. <>= { struct sigaction sa; sigemptyset(&sa.sa_mask); sa.sa_flags = SA_NOCLDSTOP; sa.sa_handler = finish; sigaction(SIGCHLD, &sa, NULL); sa.sa_handler = resized; sigaction(SIGWINCH, &sa, NULL); } @ These functions and constants require \str{signal.h}. <>= #include @ \subsubsection*{Forking} When a progam calls the [[fork()]] function, the operating system basically clones the program into a new process that is a subprocess of the caller. Both processes continue to run at the next command after the [[fork()]] call, but the value [[fork()]] returned will be different: The child will see a return value of [[0]], while the parent will retrieve the process ID of the child. A negative value will be returned if the fork did not succeed. <>= if ((CHILD = fork()) < 0) { perror("fork"); die("fork failed", 0); } @ [[CHILD]] is used in several places when dealing with the subprocess, therefore it is a global variable. <>= int CHILD = 0; @ After forking, the child launches (or, to be exact, becomes) the process that should be logged, while the parent does the actual input/output logging. <>= if (CHILD == 0) doshell(); else doio(); @ \subsection{Running the Target Application} The [[doshell()]] function is run in the child process, whose only task it is to set up all required PTY redirections and then execute the client command. Therefore, open file descriptors from the parent process which are no longer needed are closed early. <>= void doshell() { close(PTM); fclose(OUTF); @ \subsubsection*{Changing the Terminal} Next, the child process changes its controlling terminal to be the PTY slave. In order to do that, it has to be placed in a separate session. <>= setsid(); if (ioctl(PTS, TIOCSCTTY, 0) < 0) { perror("controlling terminal"); die("controlling terminal failed", 0); } @ Standard input, output and error are bound to the PTY slave, which can then be closed. <>= if ((dup2(PTS, STDIN_FILENO) < 0) || (dup2(PTS, STDOUT_FILENO) < 0) || (dup2(PTS, STDERR_FILENO) < 0)) { perror("dup2"); die("dup2 failed", 0); } close(PTS); @ \subsubsection*{Determining the Shell} If the environment variable \str{\$SHELL} is set, its value is used. Otherwise the default is \str{/bin/sh}, which should exist on all Unix systems. <>= char *shell; if ((shell = getenv("SHELL")) == NULL) shell = "/bin/sh"; @ Next, the name of the shell, without any path components, is determined to be used as argument zero when executing the client command. <>= char *shname; if ((shname = basename(shell)) == NULL) shname = shell; @ \subsubsection*{Executing the Shell} Finally, the [[execl()]] function is used to replace the currently running \cmd{forscript} process with the shell that has just been selected. If a target command has been specified using the \str{-c} option, it will be passed to the shell. Else, an interactive shell is launched using the \str{-i} option. <>= if (cflg != NULL) execl(shell, shname, "-c", cflg, NULL); else execl(shell, shname, "-i", NULL); @ The \cmd{forscript} child process should now have been replaced with the shell. If execution reaches code after [[execl()]], an error occured and the child process will terminate with an error message. <>= perror(shell); die("execing the shell failed", 0); } @ \subsection{Handling Input and Output} While \cmd{script} forks twice and utilizes separate processes to handle input and output to and from the client application, \cmd{forscript} uses a single process for both tasks, taking advantage of the [[select()]] function (defined in \str{select.h}) that allows it to monitor several open file descriptors at once. <>= #include @ Input and output data will never be read simultaneously. Therefore, a single data buffer is sufficient. Its size is [[BUFSIZ]]~bytes, which is a constant defined in \str{stdio.h} and contains a recommended buffer size, for example 8192~bytes. The number of bytes that have been read into the buffer by [[read()]] will be stored in [[count]]. <>= void doio() { char iobuf[BUFSIZ]; int count; @ The [[select()]] function is supplied with a set of file descriptors to watch, stored in the variable [[fds]]. It returns in [[sr]] the number of file descriptors that are ready, or $-1$ if an error occured (for example, a signal like \str{SIGWINCH} was received). Additionally, it requires the number of the highest-numbered file descriptor plus one as its first parameter. On all Unix systems, stdin should be file descriptor~$0$, but for maximum portability, \cmd{forscript} compares both descriptors and stores the value to pass to [[select()]] in the variable [[highest]]. <>= fd_set fds; int sr; int highest = ((STDIN_FILENO > PTM) ? STDIN_FILENO : PTM) + 1; @ The variable [[drain]] determines whether the child has already terminated, but the buffers still have to be drained. <>= int drain = 0; @ Several metadata chunks need to be written. If the \str{-a} flag is not set, a \emph{file version} chunk is written. Then \emph{begin of session}, \emph{environment variables} and \emph{locale settings}. Finally [[winsize()]]’s mode $2$ is used to only write the window size to the transcript without sending a second \str{SIGWINCH} to the client. <>= if (!aflg) if (chunk01() < 0) die(NULL, 0x01); if (chunk02() < 0) die(NULL, 0x02); if (chunk12() < 0) die(NULL, 0x12); if (chunk13() < 0) die(NULL, 0x13); winsize(2); @ To be able to calculate the delay between I/O~chunks, the monotonic clock available via [[clock_gettime()]] is used. The following code will initialize the timer: <>= struct timespec ts; if (clock_gettime(CLOCK_MONOTONIC, &ts) < 0) { perror("CLOCK_MONOTONIC"); die("retrieving monotonic time failed", 0); } @ If the \str{-q} flag has not been supplied, \cmd{forscript} will display a startup message similar to \cmd{script}’s and write the same message to the transcript file. Note that this behavior differs from \cmd{script}’s: When called with \str{-q}, \cmd{script} would not output the startup message to the terminal, but record it to the typescript file nevertheless. This is required because \cmd{scriptreplay} assumes that the first line in the typescript is this startup message and will unconditionally suppress its output. \cmd{forscript}, however, has no such limitation and will not write the startup line to the transcript if the \str{-q} flag is set. <>= if (!qflg) statusmsg(STARTMSG); @ <>= const char *STARTMSG = "forscript started on %s, " "file is %s\r\n"; @ The main loop, which handles input and output, will run until the child process exits. <>= while ((CHILD > 0) || drain) { @ Since [[select()]] manipulates the value of [[fds]], it has to be initialized again in each iteration. First its value is cleared, then the file descriptors for standard input and the PTY’s master are added to the set, then [[select()]] is called to wait until one of the file descriptors has data to read available. When in drain mode, [[select()]] may not be called to avoid blocking. <>= if (!drain) { FD_ZERO(&fds); FD_SET(STDIN_FILENO, &fds); FD_SET(PTM, &fds); sr = select(highest, &fds, NULL, NULL, NULL); @ If the child process has terminated, there may still be data left in the buffers, therefore the terminal’s file descriptor is set to non-blocking mode. Reading will then continue until no more data can be retrieved. If drain mode is already active, this code will not be executed. <>= if (CHILD < 0) { int flags = fcntl(PTM, F_GETFL); if (fcntl(PTM, F_SETFL, (flags | O_NONBLOCK)) == 0) { drain = 1; continue; } } @ If select returns $0$ or less, none of the file descriptors are ready for reading. This can for example happen if a signal was received and should be ignored. If the signal was \str{SIGCHLD}, notifying the parent thread of the child’s termination, the signal handler will have set [[CHILD]] to $-1$ and the loop will finish after the buffers have been drained. If drain mode is already active, [[select()]] will not have been run, therefore this test is not needed then. <>= if (sr <= 0) continue; @ Execution does not reach this point if none of the file descriptors had data available. Thus it can be assumed that data will be written to the transcript file. Therefore [[chunk16()]] is called to calculate and write a delay meta chunk. After it has calculated the time delta, it will automatically update [[ts]] to contain the current time. <>= if (chunk16(&ts) < 0) die(NULL, 0x16); @ If user input is available, it will be read into the buffer. The data will then be written to the transcript file, having [[SO]] prepended and [[SI]] appended. Then it will be sent to the client application. When in drain mode, user input is irrelevant since the child has already terminated. <>= if (FD_ISSET(STDIN_FILENO, &fds)) { count = read(STDIN_FILENO, iobuf, BUFSIZ); if (count > 0) { fwrite(&SO, sizeof(SO), 1, OUTF); chunkwd((unsigned char *)iobuf, count); fwrite(&SI, sizeof(SI), 1, OUTF); write(PTM, iobuf, count); } } @ Regardless of whether in drain mode or not, if output from the client application is available, it will be read into the buffer and written to the transcript file and standard output. If there was no data to read, the buffer has been drained, drain mode ends and the main loop will terminate. <>= } // if (!drain) if (FD_ISSET(PTM, &fds)) { count = read(PTM, iobuf, BUFSIZ); if (count > 0) { fwrite(iobuf, sizeof(char), count, OUTF); write(STDOUT_FILENO, iobuf, count); } else drain = 0; } @ If the \str{-f} flag has been specified on the command line, the file should be flushed now that data has been written. <>= if (fflg) fflush(OUTF); @ If the main loop exits, the child has terminated. [[done()]] is called to flush data and tidy up the environment. <>= } done(); } @ \subsection{Finishing Execution} Since a signal handler can handle more than one signal, its number is passed as an argument. However, [[finish()]] only handles \str{SIGCHLD}, therefore it will ignore its argument. Its only task is setting [[CHILD]] to $-1$, which will cause the main loop to exit as soon as possible. <>= void finish(int signal) { UNUSED(signal); CHILD = -1; } @ [[UNUSED]] is a macro that causes the compiler to stop warning about an unused parameter: <>= #define UNUSED(var) while (0) { (void)(var); } @ The function [[done()]] is called as soon as the main loop terminates. It cleans up the environment, resets the terminal and finishes execution. First, it has to fetch the exit status of the child process using [[wait()]]. <>= void done() { int status; wait(&status); @ To be able to use [[wait()]], \str{wait.h} must be included. <>= #include @ If the \str{-q} flag has not been supplied, \cmd{forscript} will write a shutdown message to both the terminal and the transcript file. <>= if (!qflg) statusmsg(STOPMSG); @ <>= const char *STOPMSG = "forscript done on %s, " "file is %s\r\n"; @ Finally, it will write an \emph{end of session} chunk, close open file descriptors, reset the terminal and exit. <>= if (chunk03(status) < 0) die(NULL, 0x03); fclose(OUTF); close(PTM); close(PTS); if (tcsetattr(STDIN_FILENO, TCSADRAIN, &TT) < 0) { perror("tcsetattr on exit"); die("tcsetattr on exit failed", 0); } exit(EXIT_SUCCESS); } @ \subsection{Putting It All Together} The code contained in the last sections is assembled into a single C source file, starting with feature test macros, ordinary macros and include statements. <>= <> <> <> @ Afterwards, constants and global variables are defined. <>= <> <> @ The functions used in the code are put in an order that makes sure every function is defined before it is called. Since [[die()]] is required at many places, it is put first. Next, all the chunk writing functions appear (the helper functions first). <>= <> <> <> <> <> <> @ The code continues with the startup and shutdown functions. <>= <> <> @ Next, the signal handlers. <>= <> <> <> @ The two functions that represent the parent and child processes are defined next. <>= <> <> @ Finally, the [[main()]] function decides the order in which the steps described in this chapter are executed. Since neither the parent nor the child process should ever reach the end of [[main()]], it returns [[EXIT_FAILURE]]. <>= int main(int argc, char *argv[]) { <> <> <> <> <> <> return EXIT_FAILURE; } @ \section{Evaluation}\label{evaluation} In order to show you what the code you have just seen actually does, this section contains instructions on how to compile it, and it features an example transcript file analyzed in detail. \subsection{Compiling \cmd{forscript}} \cmd{forscript} is written conforming to the C99 and POSIX-1.2001 standards, with portability in mind. It has been developed on a machine running Linux~\cite{kernel} 2.6.32, using glibc~\cite{glibc} 2.10 and GCC~\cite{gcc} 4.4.3. The following command line is an example of how to compile \cmd{forscript}: \begin{verbatim} gcc -std=c99 -Wl,-lrt -g -o forscript -Wall \ -Wextra -pedantic -fstack-protector-all -pipe forscript.c \end{verbatim} To generate \str{forscript.c} out of the \pack{noweb} source code, the following command line can be used: \begin{verbatim} notangle -Rforscript.c thesis.nw > forscript.c \end{verbatim} On the author’s machine, \cmd{forscript} can be compiled without any compiler warnings. It has also been successfully compiled on NetBSD. Since Apple Mac OS~X in its current version 10.6.2 lacks support for the real-time extension of POSIX, the [[clock_gettime()]] function required by \cmd{forscript} is not natively available. Therefore the code described in this thesis can in its current state not be compiled on OS~X. However, it should be possible to create a function emulating [[clock_gettime()]] and then port \cmd{forscript} to OS~X. \subsection{Example Transcript File} To demonstrate \cmd{forscript}’s output, the following pages contain a commented \term{hex dump} of a transcript file created on the author’s machine. The dump has been created using \str{hexdump -C transcript}. Since metadata chunks do not necessarily start or end at a 16-byte border, the dump has been cut into distinct pieces, bytes not belonging to the current logical unit being replaced by whitespace. The hex dump consists of several three-colum lines. The first two columns contain 16~bytes of data represented in hexadecimal form, eight bytes each. The third column represents these 16~bytes interpreted as ASCII characters, nonprintable characters are replaced with a single dot. The transcript starts with a \emph{file version} chunk, specifying that version 1 is used: \begin{verbatim} 0e 0e 01 01 0f |..... | \end{verbatim} Then a \emph{start of session} chunk follows. \begin{verbatim} 0e 0e 02 4b 82 d0 f3 04 4d 8b e3 | ...K....M..| 00 3c 0f |.<. | \end{verbatim} Its first eight bytes, (\str{4b} to \str{e3}) tell you that the time is $1266864371.072190947$ seconds after the epoch, which is February 22, 2010, 18:46:11 UTC. The next two bytes, \str{00 3c} represent a timezone of $60$ which translates to UTC+01:00. After this chunk, the environment variables are listed. These are \str{name=value} pairs, separated by null bytes. This information is important to interpret the actual terminal data: For example, different control codes are used depending on the \str{TERM} variable’s setting. \begin{verbatim} 0e 0e 12 53 53 48 5f 41 47 45 4e 54 5f | ...SSH_AGENT_| 50 49 44 3d 31 36 33 30 00 47 50 47 5f 41 47 45 |PID=1630.GPG_AGE| 4e 54 5f 49 4e 46 4f 3d 2f 74 6d 70 2f 67 70 67 |NT_INFO=/tmp/gpg| 2d 4b 50 62 79 65 43 2f 53 2e 67 70 67 2d 61 67 |-KPbyeC/S.gpg-ag| 65 6e 74 3a 31 36 33 31 3a 31 00 54 45 52 4d 3d |ent:1631:1.TERM=| 72 78 76 74 00 53 48 45 4c 4c 3d 2f 62 69 6e 2f |rxvt.SHELL=/bin/| 62 61 73 68 00 57 49 4e 44 4f 57 49 44 3d 32 37 |bash.WINDOWID=27| 32 36 32 39 38 34 00 55 53 45 52 3d 73 63 79 00 |262984.USER=scy.| 53 53 48 5f 41 55 54 48 5f 53 4f 43 4b 3d 2f 74 |SSH_AUTH_SOCK=/t| 6d 70 2f 73 73 68 2d 64 63 74 77 4b 42 31 36 30 |mp/ssh-dctwKB160| 37 2f 61 67 65 6e 74 2e 31 36 30 37 00 50 41 54 |7/agent.1607.PAT| 48 3d 2f 68 6f 6d 65 2f 73 63 79 2f 62 69 6e 3a |H=/home/scy/bin:| 2f 75 73 72 2f 6c 6f 63 61 6c 2f 62 69 6e 3a 2f |/usr/local/bin:/| 75 73 72 2f 62 69 6e 3a 2f 62 69 6e 3a 2f 75 73 |usr/bin:/bin:/us| 72 2f 67 61 6d 65 73 00 50 57 44 3d 2f 68 6f 6d |r/games.PWD=/hom| 65 2f 73 63 79 00 4c 41 4e 47 3d 65 6e 5f 55 53 |e/scy.LANG=en_US| 2e 55 54 46 2d 38 00 43 4f 4c 4f 52 46 47 42 47 |.UTF-8.COLORFGBG| 3d 37 3b 64 65 66 61 75 6c 74 3b 30 00 48 4f 4d |=7;default;0.HOM| 45 3d 2f 68 6f 6d 65 2f 73 63 79 00 53 48 4c 56 |E=/home/scy.SHLV| 4c 3d 32 00 4c 4f 47 4e 41 4d 45 3d 73 63 79 00 |L=2.LOGNAME=scy.| 57 49 4e 44 4f 57 50 41 54 48 3d 37 00 44 49 53 |WINDOWPATH=7.DIS| 50 4c 41 59 3d 3a 30 2e 30 00 43 4f 4c 4f 52 54 |PLAY=:0.0.COLORT| 45 52 4d 3d 72 78 76 74 2d 78 70 6d 00 5f 3d 75 |ERM=rxvt-xpm._=u| 6e 69 2f 62 61 63 68 65 6c 6f 72 2f 66 6f 72 73 |ni/bachelor/fors| 63 72 69 70 74 00 0f |cript.. | \end{verbatim} The next chunk contains the locale settings the C library uses for messages, number and currency formatting and other things. Although the user may choose different locales for either category, they are usually all the same. This example makes no difference: The system is configured for US English and a character encoding of UTF-8. \begin{verbatim} 0e 0e 13 65 6e 5f 55 53 2e | ...en_US.| 55 54 46 2d 38 00 65 6e 5f 55 53 2e 55 54 46 2d |UTF-8.en_US.UTF-| 38 00 65 6e 5f 55 53 2e 55 54 46 2d 38 00 65 6e |8.en_US.UTF-8.en| 5f 55 53 2e 55 54 46 2d 38 00 65 6e 5f 55 53 2e |_US.UTF-8.en_US.| 55 54 46 2d 38 00 65 6e 5f 55 53 2e 55 54 46 2d |UTF-8.en_US.UTF-| 38 00 65 6e 5f 55 53 2e 55 54 46 2d 38 00 0f |8.en_US.UTF-8.. | \end{verbatim} The terminal \cmd{forscript} is running in is 168 characters wide (\str{00 a8}) and 55 characters high (\str{00 37}), as the \emph{terminal size} chunk shows: \begin{verbatim} 0e | .| 0e 11 00 a8 00 37 0f |.....7. | \end{verbatim} After all these metadata chunks, this is where actual terminal output starts. Since the \param{-q} flag was not used, \cmd{forscript} writes a startup message both to the terminal and the transcript, containing date and time and the file name. The final two bytes \str{0d 0a} represent the control codes \emph{carriage return} and \emph{line feed}. Note that in contrast to the Unix convention of using just \emph{line feed} (\str{\bs{}n}) to designate “new line” in text files, a terminal (or at least the terminal the author’s machine is using) requires both bytes to be present. \begin{verbatim} 66 6f 72 73 63 72 69 70 74 | forscript| 20 73 74 61 72 74 65 64 20 6f 6e 20 4d 6f 6e 20 | started on Mon | 32 32 20 46 65 62 20 32 30 31 30 20 30 37 3a 34 |22 Feb 2010 07:4| 36 3a 31 31 20 50 4d 20 43 45 54 2c 20 66 69 6c |6:11 PM CET, fil| 65 20 69 73 20 74 72 61 6e 73 63 72 69 70 74 0d |e is transcript.| 0a |. | \end{verbatim} Now the shell is started. It requires some time to read its configuration files and initialize the environment, therefore \cmd{forscript} has to wait for it and starts measuring the time until the next piece of data arrives. After the shell has initialized, it prints out its \term{prompt}. On this machine, the prompt (\str{scy@bijaz \textasciitilde{} master ? 0.11 19:46 \$}) is a rather complicated, colored one and therefore contains lots of ISO~6429 control codes (also known as “ANSI escape codes”) to define the visual appearance. However, before the prompt is written to the data file, \cmd{forscript} writes a \emph{delay} meta chunk: It took$0.065087679$seconds before the prompt was printed. \begin{verbatim} 0e 0e 16 00 00 00 00 03 e1 28 bf 0f 1b 5d 30 | .........(...]0| 3b 73 63 79 40 62 69 6a 61 7a 3a 7e 07 1b 5b 31 |;scy@bijaz:~..[1| 3b 33 32 6d 73 63 79 1b 5b 30 3b 33 32 6d 40 1b |;32mscy.[0;32m@.| 5b 31 3b 33 32 6d 62 69 6a 61 7a 1b 5b 31 3b 33 |[1;32mbijaz.[1;3| 34 6d 20 7e 20 1b 5b 30 3b 33 36 6d 6d 61 73 74 |4m ~ .[0;36mmast| 65 72 20 3f 20 1b 5b 31 3b 33 30 6d 30 2e 31 31 |er ? .[1;30m0.11| 20 1b 5b 30 3b 33 37 6d 31 39 3a 34 36 20 1b 5b | .[0;37m19:46 .[| 30 3b 33 33 6d 1b 5b 31 3b 33 32 6d 24 1b 5b 30 |0;33m.[1;32m$.[0| 6d 20 |m | \end{verbatim} Next, $1.291995750$ seconds after the prompt has been printed, the user types the letter \str{e} on the keyboard. The letter is enclosed by \str{0e} and \str{0f} in order to mark it as input data. \begin{verbatim} 0e 0e 16 00 00 00 01 11 67 80 66 0f 0e 65 |m ........g.f..e| 0f |. | \end{verbatim} After the letter has been typed, the kernel will usually \emph{echo} the character, that is, put it into the terminal’s output stream to make it appear on screen. It will take a small amount of time (in this case $0.0079911$ seconds) until \cmd{forscript} receives the character and write it to the transcript file, this time declaring it as output. \begin{verbatim} 0e 0e 16 00 00 00 00 00 79 ef 3c 0f 65 | ........y.<.e | \end{verbatim} The user now continues to type the characters \str{echo -l}, which will be echoed as well. \begin{verbatim} 0e 0e | ..| 16 00 00 00 00 05 b9 48 10 10 0f 0e 63 0f 0e 0e |.......H....c...| 16 00 00 00 00 00 79 a5 09 0f 63 0e 0e 16 00 00 |......y...c.....| 00 00 0a 7d bf 1e 0f 0e 68 0f 0e 0e 16 00 00 00 |...}....h.......| 00 00 79 db 51 0f 68 0e 0e 16 00 00 00 00 0b 71 |..y.Q.h........q| c4 94 0f 0e 6f 0f 0e 0e 16 00 00 00 00 00 79 fc |....o.........y.| 54 0f 6f 0e 0e 16 00 00 00 02 09 89 aa a1 0f 0e |T.o.............| 20 0f 0e 0e 16 00 00 00 00 00 79 f2 83 0f 20 0e | .........y... .| 0e 16 00 00 00 01 2f 35 2a bc 0f 0e 2d 0f 0e 0e |....../5*...-...| 16 00 00 00 00 00 79 bb 20 0f 2d 0e 0e 16 00 00 |......y. .-.....| 00 00 14 fb 28 4d 0f 0e 6c 0f 0e 0e 16 00 00 00 |....(M..l.......| 00 00 7a 01 3d 0f 6c 0e 0e 16 00 00 00 00 2b 64 |..z.=.l.......+d| b7 45 0f |.E. | \end{verbatim} Since typing the \str{l} was a mistake, the user presses the “backspace” key (ASCII value $127$) to remove the last character. \begin{verbatim} 0e 7f 0f | ... | \end{verbatim} After the usual delay, the shell will send two things to the terminal: First, an ASCII backspace character (\str{08}) to position the cursor on the \str{l}, then the ANSI code \emph{CSI~K}, represented by the bytes \str{1b 5b 4b}, which will cause the terminal to make all characters at or right of the cursor’s position disappear. \begin{verbatim} 0e 0e 16 00 00 00 00 00 79 c2 | ........y.| 7e 0f 08 1b 5b 4b |~...[K | \end{verbatim} The user now enters the letter \str{n} and hits the return key (represented as ASCII byte \str{0d}) in order to execute the command \str{echo -n}. After executing the command (which produces no output), the shell displays the prompt again. \begin{verbatim} 0e 0e 16 00 00 00 00 37 50 74 | .......7Pt| a3 0f 0e 6e 0f 0e 0e 16 00 00 00 00 00 79 c4 67 |...n.........y.g| 0f 6e 0e 0e 16 00 00 00 00 2e bb 20 01 0f 0e 0d |.n......... ....| 0f 0e 0e 16 00 00 00 00 00 79 f9 df 0f 0d 0a 0e |.........y......| 0e 16 00 00 00 00 02 25 be d3 0f 1b 5d 30 3b 73 |.......%....]0;s| 63 79 40 62 69 6a 61 7a 3a 7e 07 1b 5b 31 3b 33 |cy@bijaz:~..[1;3| 32 6d 73 63 79 1b 5b 30 3b 33 32 6d 40 1b 5b 31 |2mscy.[0;32m@.[1| 3b 33 32 6d 62 69 6a 61 7a 1b 5b 31 3b 33 34 6d |;32mbijaz.[1;34m| 20 7e 20 1b 5b 30 3b 33 36 6d 6d 61 73 74 65 72 | ~ .[0;36mmaster| 20 3f 20 1b 5b 31 3b 33 30 6d 30 2e 31 30 20 1b | ? .[1;30m0.10 .| 5b 30 3b 33 37 6d 31 39 3a 34 36 20 1b 5b 30 3b |[0;37m19:46 .[0;| 33 33 6d 1b 5b 31 3b 33 32 6d 24 1b 5b 30 6d 20 |33m.[1;32m$.[0m | \end{verbatim} Note that without recording the user’s input, it would be impossible to determine whether the user pressed return to actually run the command or whether entering the command was cancelled, for example by pressing \ctrl{C}.$1.587984366$seconds later, the user decides to end the current session by pressing \ctrl{D}, which is equivalent to the byte value \str{04}. \begin{verbatim} 0e 0e 16 00 00 00 01 23 0b ed ee 0f 0e 04 0f |.......#....... | \end{verbatim} The shell reacts by printing \str{exit} and terminating. Then, \cmd{forscript} prints its shutdown message. \begin{verbatim} 65 | e| 78 69 74 0d 0a 66 6f 72 73 63 72 69 70 74 20 64 |xit..forscript d| 6f 6e 65 20 6f 6e 20 4d 6f 6e 20 32 32 20 46 65 |one on Mon 22 Fe| 62 20 32 30 31 30 20 30 37 3a 34 36 3a 32 31 20 |b 2010 07:46:21 | 50 4d 20 43 45 54 2c 20 66 69 6c 65 20 69 73 20 |PM CET, file is | 74 72 61 6e 73 63 72 69 70 74 0d 0a |transcript.. | \end{verbatim} Finally, the exit status ($0\$) of the shell is recorded in an \emph{end of session} metadata chunk and the transcript file ends. \begin{verbatim} 0e 0e 03 00 | ....| 0f |.| \end{verbatim} \section{Summary}\label{summary} In this thesis it has been presented why \cmd{script}, although often used for forensic investigations, lacks features that are crucial for reliable documentation. A new software, \cmd{forscript}, has been designed and implemented, the weaknesses of \cmd{script} have been eliminated. \subsection{Future Tasks}\label{future} The primary reason to develop \cmd{forscript} was the need to create a software that enables a forensic investigator to convert an interactive command-line session into a version suitable for inclusion in a printed report. While thinking about possible approaches, it became apparent that the output generated by \cmd{script} does not suffice to provide such a software with the information it needs to unambigously reconstruct what the user did. A tool that records the required information had to be developed first. This task has been solved in this bachelor thesis. Next, a tool that is able to parse the output \cmd{forscript} generates is to be written. \cmd{forscript} will be released by the author as free software, available at~\cite{forscript}. Corrections and improvements are encouraged: \cmd{forscript} is far from being perfect and it is quite possible that during the development of additional tools, bugs and shortcomings will need to be fixed. Additionally, we will approach the maintainers of \cmd{script} and the forensic community as they can probably benefit from \cmd{forscript}’s existence. \pagebreak \begin{thebibliography}{99} \bibitem{casey} Casey, Eoghan: \emph{Digital Evidence and Computer Crime} (2nd edition, 2004), Academic Press, ISBN~978-0121631048. \bibitem{literate} Knuth, Donald~E.: \emph{Literate Programming} (1992), Center for the Study of Language and Information, ISBN~978-0937073803. \bibitem{noweb} \emph{Noweb~— A simple, Extensible Tool for Literate Programming}, created and maintained by Norman Ramsey, current release 2.11b. \\\url{http://www.cs.tufts.edu/~nr/noweb/} \bibitem{util} \emph{The \pack{util-linux} project}, no longer maintained, last release 2.13-pre7 in 2006. \\\url{http://www.kernel.org/pub/linux/utils/util-linux/} \bibitem{utilng} \emph{The \pack{util-linux-ng} project}, maintained by Karel Zak, current release 2.17. \\\url{http://userweb.kernel.org/~kzak/util-linux-ng/} \bibitem{linuxman} \emph{The Linux \emph{man-pages} project}, maintained by Michael Kerrisk, release~3.23. \\\url{http://www.kernel.org/doc/man-pages/} \bibitem{kernel} \emph{The Linux kernel}, maintained by Linus Torvalds, release~2.6.31. \\\url{http://www.kernel.org/} \bibitem{glibc} \emph{GNU C Library}, maintained by Ulrich Drepper, release~2.10. \\\url{http://www.gnu.org/software/libc/} \bibitem{gcc} \emph{GNU Compiler Collection}, maintained by its steering committee, release~4.4.3. \\\url{http://gcc.gnu.org/} \bibitem{forscript} \emph{forscript}, created and maintained by Tim Weber, release~1.0.0. \\\url{http://scytale.name/proj/forscript/} \end{thebibliography} % This is where it all ends. \end{document} | 2015-10-09 21:57:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.3740478754043579, "perplexity": 4745.38385917146}, "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-2015-40/segments/1443737935954.77/warc/CC-MAIN-20151001221855-00058-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://settheory.mathtalks.org/natasha-dobrinen-mini-course-on-infinitary-ramsey-theory/ | Natasha Dobrinen: Mini-course on Infinitary Ramsey theory
Time and Place: Tuesday, January 8 and Wednesday, January 9 at 10:30am in the KGRC lecture room (both parts) at the KGRC.
Part I. Topological Ramsey spaces and applications to ultrafilters
Part II. Ramsey theory on trees and applications to big Ramsey degrees
The Infinite Ramsey Theorem states that given $n,r\ge 1$ and a coloring of
all $n$-sized subsets of $\mathbb{N}$ into $r$ colors, there is an
infinite subset of $\mathbb{N}$ in which all $n$-sized subsets have the
same color. There are several natural ways of extending Ramsey’s Theorem.
One extension is to color infinite sets rather than finite sets. In this
case, the Axiom of Choice precludes a full-fledged generalization, but
upon restricting to definable colorings, much can still be said. Another
way to extend Ramsey’s Theorem is to color finite sub-objects of an
infinite structure, requiring an infinite substructure isomorphic to the
original one. While it is not possible in general to obtain substructures
on which the coloring is monochromatic, sometimes one can find bounds on
the number of colors, and this can have implications in topological
dynamics.
In Part I, we will trace the development of Ramsey theory on the Baire
space, from the Nash-Williams Theorem for colorings of clopen sets to the
Galvin-Prikry Theorem for Borel colorings, culminating in Ellentuck’s
Theorem correlating the Ramsey property with the property of Baire in a
topology refining the metric topology on the Baire space. This refinement
is called the Ellentuck topology and is closely connected with Mathias
forcing. Several classical spaces with similar properties will be
presented, including the Carlson-Simpson space and the Milliken space of
block sequences. From these we shall derive the key properties of
topological Ramsey spaces, first abstracted by Carlson and Simpson and
more recently given a refined presentation by Todorcevic in his book {\em
Introduction to Ramsey spaces}. As the Mathias forcing is closely
connected with Ramsey ultrafilters, via forcing mod finite initial
segments, so too any Ramsey space has a $\sigma$-closed version which
forces an ultrafilter with partition properties. Part I will show how
Ramsey spaces can be used to find general schemata into which disparate
results on ultrafilters can be seen as special cases, as well as obtain
fine-tuned results for structures involving ultrafilters.
Part II will focus on Ramsey theory on trees and their applications to
Ramsey theory of homogeneous structures. An infinite structure is {\em
homogeneous} if each isomorphism between two finite substructures can be
extended to an automorphism of the infinite structure. The rationals as a
linearly ordered structure and the Rado graph are prime examples of
homogeneous structures. Given a coloring of singletons in the rationals,
one can find a subset isomorphic to the rationals in which all singletons
have the same color. However, when one colors pairs of rationals, there
is a coloring due to Sierpinski for which any subset isomorphic to the
rationals has more than one color on its pairsets. This is the origin of
the theory of {\em big Ramsey degrees}, a term coined by Kechris, Pestov
and Todorcevic, which investigates bounds on colorings of finite
structures inside infinite structures. Somewhat surprisingly, a theorem
of Halpern and L\”{a}uchli involves colorings of products of trees,
discovered en route to a proof that the Boolean Prime Ideal Theorem is
strictly weaker than the Axiom of Choice, is the heart of most results on
big Ramsey degrees. We will survey big Ramsey degree results on countable
and uncountable structures and related Ramsey theorems on trees, including
various results of Dobrinen, Devlin, D\v{z}amonja, Hathaway, Larson,
Laver, Mitchell, Shelah, and Zhang.
Attachments
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2019-01-21 00:57:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7698807716369629, "perplexity": 2446.9653774903327}, "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-2019-04/segments/1547583745010.63/warc/CC-MAIN-20190121005305-20190121031305-00252.warc.gz"} |
https://answers.ros.org/answers/117456/revisions/ | If I understand your question, I think you should clone this fork into your workspace source folder(your_ws/src/) and source the setup file found in your_ws/devel/setup.bash. This will overlay the fork package over the original package. | 2021-12-05 08:50:07 | {"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": 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": 0.5896143913269043, "perplexity": 2724.5041614321954}, "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/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00610.warc.gz"} |
https://hal.inria.fr/hal-00673801v3 | New interface
# Bayesian Pursuit Algorithms
1 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
CEMAGREF - Centre national du machinisme agricole, du génie rural, des eaux et forêts, Inria Rennes – Bretagne Atlantique
Abstract : This paper addresses the sparse representation (SR) problem within a general Bayesian framework. We show that the Lagrangian formulation of the standard SR problem, i.e. $\x^\star=\argmin_\x \lbrace \| \y-\D\x\|_2^2+\lambda\| \x\|_0 \rbrace$, can be regarded as a limit case of a general maximum a posteriori (MAP) problem involving Bernoulli-Gaussian variables. We then propose different tractable implementations of this MAP problem that we refer to as ''Bayesian pursuit algorithms". The Bayesian algorithms are shown to have strong connections with several well-known pursuit algorithms of the literature (e.g., MP, OMP, StOMP, CoSaMP, SP) and generalize them in several respects. In particular, i) they naturally allow for atom deselection; ii) they can include any prior information about the probability of occurrence of each atom within the selection process; iii) they can encompass the estimation of unkown model parameters into their recursions.
Document type :
Preprints, Working Papers, ...
Domain :
Complete list of metadata
Cited literature [44 references]
https://hal.inria.fr/hal-00673801
Contributor : Cédric Herzet Connect in order to contact the contributor
Submitted on : Monday, August 6, 2012 - 11:10:22 AM
Last modification on : Wednesday, October 26, 2022 - 4:07:44 PM
Long-term archiving on: : Friday, March 31, 2017 - 11:43:49 AM
### File
technical_report.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : hal-00673801, version 3
### Citation
Cedric Herzet, Angélique Drémeau. Bayesian Pursuit Algorithms. 2012. ⟨hal-00673801v3⟩
Record views | 2022-12-03 20:24:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5476200580596924, "perplexity": 6277.492521766699}, "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-2022-49/segments/1669446710936.10/warc/CC-MAIN-20221203175958-20221203205958-00269.warc.gz"} |
https://math.stackexchange.com/questions/830679/vector-bundle-as-a-smooth-manifold | # Vector Bundle as a smooth manifold
Definition: It is said that a section $F:M\to E$ of a vector bundle $E$ is smooth if it is smooth as a map between manifolds.
Possible Issue: A vector bundle is defined to be, a priori, a smooth manifold, which means that it has some implicit smooth structure $\mathcal{A}$. However, it then has additional structure, i.e. local trivializations
$$\Phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb{R}^k.$$
Since the $\Phi_i$ are defined as diffeomorphisms, $(\Phi_i,\pi^{-1}(U_i))$ is a smooth atlas, which defines another smooth structure $\mathcal{B}$ on $E$.
Two questions: Does the rest of the definition of a vector bundle imply that $\mathcal{A}=\mathcal{B}$? If not, then when we say that vector fields are maps between smooth manifolds, then which manifold: $(E,\mathcal{A}),$ or $(E,\mathcal{B})$?
It is part of the (usual) definition of a smooth vector bundle (cf. e.g. http://en.wikipedia.org/wiki/Vector_bundle#Smooth_vector_bundles ) that the local trivializations $\Phi_i$ are required to be smooth (even Diffeomorphisms).
EDIT: Note that if all $\Phi_i$ are smooth w.r.t. $\mathcal{A}$, this implies that $\mathcal{A} \cup \{\Phi_i \mid i \in I\}$ is also an atlas, so that the smooth structures induced by $\mathcal{A}$ and $\{\Phi_i \mid i \in I\}$ are the same. | 2019-06-20 17:57:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.9950919151306152, "perplexity": 97.3781493512762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999263.6/warc/CC-MAIN-20190620165805-20190620191805-00124.warc.gz"} |
https://mathoverflow.net/questions/225948/do-differential-objects-form-triangulated-categories | # Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies" (so, we have a full functor $Diff(\mathcal{A})\to Diff'(\mathcal{A})$ for the category $Diff'(A)$ that I want to describe; there also exists a functor $Diff'(\mathcal{A})\to K(\mathcal{A})$). The shift functor for $Diff'(\mathcal{A})$ may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff'(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?
P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff'(\mathcal{A})$ is (Heller) triangulated.
P.S. The proof appears to be a rather simple application of differential graded arguments. However, I wonder whether this construction and its relation to periodic derived categories was considered somewhere. | 2019-10-18 18:10:52 | {"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": 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": 0.9463750123977661, "perplexity": 389.30109283762073}, "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/1570986684226.55/warc/CC-MAIN-20191018154409-20191018181909-00048.warc.gz"} |
http://www.math-only-math.com/linear-pair-of-angles.html | # Linear Pair of Angles
What is linear pair of angles?
Two angles form a linear pair if they have;
A common arm
A common vertex
Their interiors do not overlap
The sum of two angles is 180°.
Therefore, linear pair of angles are adjacent angles whose non-common arms are opposite rays.
Note:
All adjacent angles do not form a linear pair.
From the above figure we can observe; OX and OY are two opposite rays and ∠XOZ and ∠YOZ are the adjacent angles. Therefore, ∠XOZ and ∠YOZ form a linear pair.
If you measure ∠XOZ and ∠YOZ with the help of the protractor, you will find the sum of their measures equal to 180°.
Thus, the sum of the angles in a linear pair is 180°.
Worked-out problems on Linear Pair of angles:
In the given figure, ∠AOC and ∠ BOC form a linear pair if x - y = 60°, find the value of x and y.
Solution:
Given x - y = 60° ………… (i)
We know that, x + y = 180° ………… (ii)
2x = 240°
x = 240°/2
Therefore, x = 120°
Since, x - y = 60°
or, 120° - y = 60°
or, 120° - 120° - y = 60° - 120°
or, -y = -60°
Therefore, y = 60°
Lines and Angles
Fundamental Geometrical Concepts
Angles
Classification of Angles
Related Angles
Some Geometric Terms and Results
Complementary Angles
Supplementary Angles
Complementary and Supplementary Angles
Linear Pair of Angles
Vertically Opposite Angles
Parallel Lines
Transversal Line
Parallel and Transversal Lines | 2018-08-15 20:41:59 | {"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": 0.6692118048667908, "perplexity": 5142.412012241606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210304.2/warc/CC-MAIN-20180815200546-20180815220546-00350.warc.gz"} |
https://physics.stackexchange.com/questions/531774/what-is-the-general-prescription-for-constructing-the-quantum-mechanical-momentu | # What is the general prescription for constructing the quantum mechanical momentum operator conjugate to a given coordinate?
Obviously, if $$x$$ is a Cartesian coordinate, then the corresponding momentum operator is $$-i \hbar \partial_x$$. But what if $$x$$ is something more complicated, like some sort of curvilinear coordinate in 3D space?
There is this question elsewhere on this site. Quantum mechanical analogue of conjugate momentum But it's not clear to me that the accepted answer there is really answering the question I am asking. Nothing in the question or answer specifically addresses non-Cartesian coordinates, certainly not in any detail.
If an $$n$$-dimensional configuration manifold $$M$$ of some physical system is orientable and endowed with a positive volume form $$\Omega~=~\rho(x)\mathrm{d}x^1\wedge\ldots\wedge\mathrm{d}x^n, \qquad \rho(x)~>~0,$$ we can define a sesquilinear form $$\langle \phi | \psi\rangle~:=~\int_M \! \Omega ~\phi^{\ast} \psi.$$ The Schroedinger representation of the self-adjoint phase space operators is $$\hat{x}^j~=~x^j,\qquad \hat{p}_k~=~\frac{\hbar}{i\sqrt{\rho(x)}} \frac{\partial}{\partial x^k} \sqrt{\rho(x)}, \qquad [\hat{x}^j,\hat{p}_k]~=~i\hbar\delta^j_k \mathbb{1}.$$ See also this related Phys.SE post. | 2021-09-21 17:58:24 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.88946533203125, "perplexity": 149.8693822423729}, "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/1631780057225.57/warc/CC-MAIN-20210921161350-20210921191350-00367.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-4-an-introduction-to-functions-4-5-writing-a-function-rule-lesson-check-page-264/2 | ## Algebra 1
f = $\frac{h}{12}$
So you can find feet by knowing the height in inches. 12 inch equals to 1 feet so to find feet it's $\frac{inches}{12}$ | 2021-05-18 12:22:55 | {"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": 0.783769965171814, "perplexity": 1525.2741638916164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989819.92/warc/CC-MAIN-20210518094809-20210518124809-00213.warc.gz"} |
https://www.codingame.com/training/medium/ascii-cube | • 19
## Learning Opportunities
This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.
## Goal
Instructions
Your goal is to output a cube of dimensions w × h × d in ASCII Art, oriented so that the front face is in the direction of bottom left.
Therefore, the three visible faces are the front, the top and the right.
The visible edges are represented with the characters /, \ and _.
You'll have to draw the hidden edges too, with the characters , and ..
Notes
or are characters from braille alphabet. Their unicode are 0x280c and 0x2821, respectively.
• Because a console character is higher than large, a unit of 1 w (width) is represented with two characters instead of one (__ or ..).
• If a visible face has a width, a height or a depth of 1, you won't have to draw the hidden edges behind this face.
• If a solid line and a dotted line are layered on the same character, the priority always goes to the solid line, except for a hidden oblique edge ( or ), which must replace a visible horizontal edge (_) (see examples below).
Examples
Input:w = 4h = 2d = 4
Output: ________ /⠡ /\ / ⠡..../..\ / ⠌ / //___⠌___/ /\ ⠌ \ / \⠌______\/
Input:w = 6h = 2d = 2
Output: ____________ /⠡ /\/__⠡________/..\\ ⠌ \ / \⠌__________\/
Input:w = 1h = 1d = 1
Output: __/_/\\_\/
Input
Line 1: An integer w for the width.
Line 2: An integer h for the height.
Line 3: An integer d for the depth.
Output
An ASCII Art of a cube of dimensions w × h × d.
Constraints
w, h, d > 0
Example
Input
8
3
2
Output
________________
/⠡ /\
/__⠡____________/ \
\ ⠡...........\...\
\ ⠌ \ /
\⠌______________\/
A higher resolution is required to access the IDE
Join the CodinGame community on Discord to chat about puzzle contributions, challenges, streams, blog articles - all that good stuff!
Online Participants | 2021-09-23 11:27:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5127927660942078, "perplexity": 4723.288329114291}, "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/1631780057421.82/warc/CC-MAIN-20210923104706-20210923134706-00314.warc.gz"} |
https://www.nature.com/articles/s41598-017-15428-z?error=cookies_not_supported&code=66512ad2-72b2-4d9d-a1e9-c536c746cf00 | Article | Open | Published:
# Structure constrained by metadata in networks of chess players
## Abstract
Chess is an emblematic sport that stands out because of its age, popularity and complexity. It has served to study human behavior from the perspective of a wide number of disciplines, from cognitive skills such as memory and learning, to aspects like innovation and decision-making. Given that an extensive documentation of chess games played throughout history is available, it is possible to perform detailed and statistically significant studies about this sport. Here we use one of the most extensive chess databases in the world to construct two networks of chess players. One of the networks includes games that were played over-the-board and the other contains games played on the Internet. We study the main topological characteristics of the networks, such as degree distribution and correlations, transitivity and community structure. We complement the structural analysis by incorporating players’ level of play as node metadata. Although both networks are topologically different, we show that in both cases players gather in communities according to their expertise and that an emergent rich-club structure, composed by the top-rated players, is also present.
## Introduction
Chess is an iconic ancient game with a development intimately related to the history of mankind and, in modern times, to the evolution of computers1. It is known that high levels of expertise require hard training even for the most talented players. Due to the popularity and complexity of chess2,3 this game has been used to study several fields of human behavior. For instance, chess activity has been used to evaluate cognitive performance in professional and novice players4, decision-making processes5,6, and the relation between expertise and knowledge7,8. There are also interesting results regarding the collective behavior of a pool of players9,10,11,12. For example, by exploring chess databases Blasius and Tönjes9 observed that the pooled distribution of chess opening weights follows Zipf’s law with a universal exponent, and explained these findings in terms of an analytical treatment of a multiplicative process. Moreover, studying the growing dynamics of the game-tree, we have found that the emerging Zipf and Heaps laws can be explained in terms of nested Yule-Simon preferential growth processes10. Also, we have observed interesting memory effects in the history of the recorded games11,12.
Nowadays there are big active communities of chess players that comprise all levels of expertise playing in both, the traditional over-the-board way and in online portals. These very large world-wide communities of chess players produce extensive game records, providing a source of data useful for large-scale analyses. In these databases the expertise of each active player is statistically well characterized through the ‘Elo’ system introduced by the physicist Arpad Elo13. Elo is not just a rating system that allows to rank the players according to their level of play but is also a predictive system that permits to anticipate their future performance. Hence, it is useful, for instance, for pairing players in tournaments and to follow their evolution.
Like in almost every human activity, it is expected that chess players gather together to form networks structured in communities. The statistical properties of this kind of networks, including their community structure, have not been yet characterized to the best of our knowledge. When considering chess players as nodes of a network, the expertise takes the role of node information, or metadata and can give important information about the topological properties of the network. For instance, homophilic features are expected to be present in this kind of networks, as it is widely known14 that players tend to play against similar skilled opponents. Moreover, the relation between players’ expertise and network structure can provide useful insights about the mechanisms that work in the formation of communities in social systems.
The study of correlations between the topology of social networks and the metadata of its nodes is currently a prominent topic in complex networks15,16,17,18. Its importance arises because it can provide clues in the understanding of the structure and the dynamics of social communities. Moreover, the relation between network topology and metadata has been proved to be useful for detecting social stratification15, disassortative communities16 and data inconsistency such as missing-links17. It has also served in pointing out possible drawbacks of the currently employed community detection algorithms18. However, the data available to study this types of correlations is yet scarce and there are just a few studies of social systems that are analyzed from this point of view.
The aim of this paper is to describe the community of chess players in a world-wide scale in terms of the theory of complex networks and to combine network topology with metadata in order to study the similarities and differences that exist between online and over-the-board playing. The results are divided in four sections. First, we introduce two chess databases from which we performed our analysis and provide some relevant statistical information about them. Second, we describe how the networks of chess players were constructed from the data. Third, we present a characterization of the networks, giving their main structural properties and discussing the relation between structure and associated node metadata. Last, we perform a community detection analysis and show the correlations between community membership and players’ level of play.
## Results
### Statistics of the databases
Two databases were used to construct the networks. Both of them were provided by Opening Master, a chess database company that claims to have the world largest collection of chess games. The first database is a set of 7.7 million games played over-the-board, and we refer to it as ‘OTB’. The second one, which we call ‘Portal’, contains more than 15 million chess games played between humans in different websites or portals, such as chess.com, ICC-chessclub.com, Playchess.com, freechess.org and chesscube.com. Each database includes several information fields for each game, such as the name of the players, their Elo rating, the result, the date, the opening played and the sequence of moves. Since the Elo of each particular player is a varying quantity (it is updated after each game), we computed each player’s mean Elo value $$\langle {\rm{Elo}}\rangle$$ and its corresponding standard deviation $${\sigma }_{{\rm{Elo}}}$$. In Fig. 1(a) we show the mean Elo distributions for the complete set of players, computed for each database, and in Fig. 1(b) we plot the corresponding distributions of the standard deviations. Both networks exhibit centered $$\langle {\rm{Elo}}\rangle$$ distributions that can be well approached by Gaussian distributions. The mean Elo value for OTB is 1884 ± 276 while for Portal it is 1692 ± 228. Although we don’t show it here, the time evolution of the Elo is interesting on its own. Some players, for instance older top-rated players, maintain a rather constant value, except for fluctuations. Others, for example outstanding young professional players, show a more or less monotonic increasing evolution. Despite it could bring to interesting results, a thorough study of the Elo dynamics is beyond the scope of this work.
In the game of chess, players prefer to face opponents that have a similar level of play, i.e., similar Elo14. One of the reasons for this is motivational; players in general are not interested in playing a game if their chances of winning (or losing) are excessively high. The other reason is related to the way the Elo score is adjusted after a game. If a player with high Elo wins a game against a lower rated opponent, she/he will gain just a few points of Elo, but if she/he loses the game, she/he will lose a large amount of points. As a consequence, high Elo players -specially the top ranked ones- try to avoid playing against lower ranked opponents. This behavior can be visualized from the data by employing a two-dimensional density plot, corresponding to the values of Elo of White and Black players at the moment of a game, as depicted in Fig. 1(c,d). As it can be seen, most of the games where played between players whose Elo difference ΔE is not greater than 300. In fact, the Elo difference distribution is almost exponential, as can be seen in the insets of these figures. The mean value and standard deviation are $$\langle {\rm{\Delta }}E\rangle =\mathrm{168,}\,{\sigma }_{{\rm{\Delta }}E}=131$$ for OTB and $$\langle {\rm{\Delta }}E\rangle =\mathrm{145,}\,{\sigma }_{{\rm{\Delta }}E}=156$$ for Portal. As we move towards higher values of Elo, the dispersion narrows, meaning that top-rated players tend to form a closed community. A clear difference between both databases is that in OTB most of the games correspond to players with high Elo. On the contrary, in Portal most of the games are played between lower ranked players.
It is worth to point out that the previous statement does not imply that in OTB most of the players are top-rated–actually, it can be seen from the histogram of Fig. 1(a) that this is not the case. Instead, this result only implies that the few very active top-rated players contribute more to recorded events than the rest of the players. This also explains why the average Elo in OTB is lower than the standard scores of professional players and close to Portal’s mean value. In fact, in a recent published work12 in which we analyze a different over-the-board database we also found that a small fraction of players generate a significant fraction of the games recorded in the database.
### Network construction
With the information contained in the databases we built a weighted directed network for each database. As depicted in Fig. 2, each node in the network represents an individual player and contains two different metadata, namely the name of the player and her/his mean Elo. Links between nodes are established only if the players have played at least one game. Weights and directions of the connections are assigned according to the net flux of Elo between the players (each time a game is finished there is a net transfer of Elo from one player to the other, which depends on the result of the game and on the initial Elo of both players). For each pair of players we computed the net flux of Elo over all the games they have played against each other. If the net flux goes from player ‘A’ to player ‘B’ we added a link pointing from A to B with a weight equal to the flux value. We also computed a binarized version of the networks, i.e. the corresponding undirected and unweighted networks. Finally, we constructed different randomizations of the networks and employed them as null models (details on how the randomizations where performed can be seen in the Methods section).
### Network Characterization
We computed the degree distribution for the binarized version of both OTB and Portal networks. As shown in Fig. 3, both probability distributions exhibit a right-skewed tail, feature that is shared among several social networks19,20. Although similar, they exhibit some differences. While the OTB distribution is concave in the log-log plot, the degree distribution of the Portal network is well fitted by a power law with an exponent of ~1.5 across 3 orders of magnitude, followed by a gentle decay at the tail. As the original networks are directed and weighted, we computed the node strength21 distribution. The results (not shown) are similar to the ones obtained for the degree distribution.
In order to explore the presence of degree correlations we computed the Pearson correlation coefficient r as introduced by Newman22 (see Methods). We observe (Table 1) that OTB is assortative, whereas Portal exhibits a slightly disassortative behavior. As it has been previously mentioned in the literature22, many social networks with a long-tailed degree distribution are assortative like, for example, the science coauthorship and the film actor collaboration networks23. The noteworthy fact here is that Portal network strays from this behavior and behaves instead more like a technological or a biological network22.
We studied the transitivity within the networks24 by computing their clustering coefficients, following the definitions of Newman24 and Watts-Strogatz25 (see Methods). As can be seen in Table 1, both networks have similar values for these measures. The differences between OTB and Portal arise when comparing them with their random networks, as randomization substantially decreases the transitivity for OTB but has almost no effect on Portal. We also computed the clustering coefficient c(k) as a function of the degree26. In Fig. 3, we show that for both networks c(k) is a decreasing function, and that decreases more notoriously for Portal than for OTB. The behavior, which has been previously observed in cases such as world-trade27 and metabolic networks28, has been reported as a good indicator of an existing hierarchical structure28,29 in the network. In particular, the original model proposed by Ravasz, predicts a power law dependence $$c(k)\sim {k}^{-1}$$ for a hierarchical network30. The dotted lines in Fig. 3(c,d) indicate this relation
Since in disassortative networks the clustering of hubs is bounded from above31, disassortative networks are less clustered than assortative ones32. This is consistent with our results (OTB is assortative and more clustered than Portal, which is disassortative), supporting a more pronounced hierarchical structure in Portal than in OTB. After randomization, correlations for c(k) completely disappear in OTB, whichever randomization algorithm is employed. In Portal, the two randomization algorithms we employed substantially diminish dispersion and reduce correlations, but none of them can fully remove them.
Another aspect of the networks that we studied was the relation between nodes degree and the mean Elo of the corresponding players. According to Fig. 4(a), in OTB the number of opponents each player meets is positively correlated to the average rank of the player and negatively correlated to its variability. On the other hand, no such correlations are observed in Portal. We also analyzed the normalized rich-club coefficient33,34 p(k) defined in terms of the degree (see Methods) for both networks. Results are shown in Fig. 4(b), where it can be observed that the degree-based rich-club phenomenon is present in the OTB network whilst is absent in Portal. Moreover, higher degree nodes in Portal are less connected to each other as compared to random networks. As we discuss in the Methods section, the rich-club coefficient can be also defined in terms of a quantitative node metadata, such as the mean Elo of the players. Figure 4(c) shows that, under this definition, a rich-club structure emerges in both networks. This fact indicates that there exists a densely connected group, composed by the best players, in both chess communities. The individuals that belongs to this group, nevertheless, are not necessarily those who meet more opponents.
## Communities
### Communities based on modularity
In many complex networks the distribution of links tends to be inhomogeneous at different levels of organization. At a microscopic level, inhomogeneities manifest themselves in the form on long-tailed degree distributions. At a mesoscopic level, certain groups of nodes are found to be densely connected to each other and relatively separated from the rest of the network. Such groups of nodes are said to form communities–also called clusters or modules–, which can give useful information about the network, because it is probable that nodes that belong to the same community share common properties or play similar roles in the system17.
Given a network and a particular partition, i.e. a function that assigns each node to a given community, one can compute the so called modularity 35 Q. This property aims to measure how good the clustering is by evaluating the number of links that connect nodes lying within the same communities. If the fraction of inner links does not differ from what would be expected in a random network, the value of the modularity is zero. According to Clauset et al., a value over 0.3 indicates that the network has a significant community structure35. Another important quantity that measures whether the communities in a network are well defined or not is the mixing parameter36 μ, which is the fraction of inter-community links. More specifically, small (large) values of the mixing parameter indicate that the communities are well (loosely) defined.
In order to study the community structure of our networks we employed the Louvain37 algorithm, which can be used with weighted networks. The choice of the method was based on both its performance and its speed (for a comparison among several state-of-the-art algorithms, see38). The community structures obtained were characterized by means of their modularity, mixing parameter, number of communities and community size distribution. The first three measures are summarized in Table 1. We observed that communities are better defined in the OTB network as compared to Portal. This can be seen by noting that OTB has a larger value for Q and a lower value for μ than Portal. As expected, randomization considerable increases the mixing parameter and decreases the modularity in both networks. The number of communities is also reduced in the randomized networks.
We computed the mean Elo of the pool of players that belong to each community, and then ranked the communities according to that value. We observed that the range of Elo spanned by the communities is approximately contained in the interval (1400, 2600). As shown in Fig. 5(a,b), two regimes can be identified for each network in this ranking plot. Each of them is characterized by a linear relation between ranking and Elo, but with a different slope. For top-rated communities, the mean Elo steeply decreases as the ranking decreases, indicating a strong stratification of the communities. For lower ranked communities the slope shallows. As can be seen in the figure, this effect is more notorious in Portal than in OTB. After randomization, the effect completely disappears. As shown in the insets of the figures, the mean Elo is nearly the same for all communities and the standard deviation becomes larger than in the original networks.
Correlations between Elo and communities can be better visualized by computing the coefficient of variation CV of the Elo, that is, the quotient between the standard deviation of the Elo and its mean. In Fig. 5(c,d) we plot the CV for each community as a function of its mean Elo. It can be observed that the variability of the Elo is, on average, lower inside the communities than it is for the whole network. Moreover, by comparing the results to those obtained for the null models (see filled region in the figures), it can be seen that the difference is indeed significant. To put it in other words, Louvain algorithm is able to successively discriminate players according to their expertise.
In order to explicitly detect correlations between communities and the Elo of the players we analyzed the community structure using an algorithm recently introduced by Newman and Clauset16 (see Methods). The algorithm employs Bayesian inference to construct a generative model that links community structure of the network with node metadata. As opposed to the Louvain algorithm, the number of communities is in this case fixed to a predefined value K. We used, as before, the mean Elo as the metadata. The correlation between Elo and community membership was measured via the normalized mutual information NMI (refer to the Methods section).
When the number of communities is fixed to 2, the Elo distribution in each community is clearly different; one community includes mostly low Elo players and the other high Elo players, as can be seen in Fig. 6. This feature is observed in both OTB and Portal networks, being most notorious in the second one. Now, it could be argued that the communities found are just an artifact of the algorithm employed. In order to test this hypothesis we repeated the experiments using three different network randomizations or null models. First, we performed, as before, a double-edge swapping. With this approach, the algorithm becomes particularly sensitive to the random seed, finding spurious communities that may or may not be correlated with the Elo. Second, we shuffled node metadata, i.e. we randomly reassigned the Elo of the players. This case is depicted in Fig. 6(c,d), where it can be seen that the correlation between Elo and communities disappears. Third, we performed a double randomization; after swapping edges we randomized the Elo. This case gives results similar to those obtained in the first case. We repeated the analysis by varying the number of communities from 3 to 9 (not shown) and obtained results that are qualitatively similar to the ones obtained for $$K=2$$.
## Discussion
In addition to the traditional over-the-board (OTB), Internet portals have introduced a new way of playing chess, creating in the process two parallel communities of players. Although the game is the same, both communities evolved by their own, presenting important similarities and differences. When analyzing OTB games, it can be seen that recorded games are focused on professional players and include almost no amateur gaming. This is expected to happen because most of the amateur activity takes place in informal contexts where games are not recorded. As a consequence, players in OTB have large average Elos that slowly vary in time. On the Internet, this aspect is completely different, since every game is recorded–even if it has been carried on by players with low Elo. Thus, players on the Internet exhibit comparatively lower Elo values and greater variances.
When considering chess communities as complex networks, where nodes represent players and where links represent transfer of Elo between them, it can be seen that they have long-tailed degree distributions, meaning that there exist a huge variation between the number of games (and thus, the number of opponents) each player plays in her/his career. In the case of Internet games (Portal), the distribution is well fitted by a power law with exponent ~1.5, and a gently decay at the tail. The decay for large values of k can be understood by the fact that chess players cannot play with an arbitrarily large number of opponents throughout their life, but they are constrained by the duration of their career. In the case of OTB, the curve strays from a line in a log-log plot, presenting a concavity. The plateau for lower values of k can be understood in terms of the discussion of the previous paragraph. Most of the players who play very few games are amateurs, so their games are not always recorded and thus, they do not appear in the databases.
One of the main differences between the studied networks is related to their degree-degree correlations. Whilst OTB is an assortative network–feature that seems to be common to the majority of social networks22–, Portal exhibits a dissasortative structure. This difference can be understood by introducing players’ level of play (i.e. their Elo) to the networks as node metadata and considering the relationship between this measure and the number of opponents each player faces (the degrees of the corresponding nodes). In OTB, there is a positive correlation between these two variables, meaning that the most active players also turn out to be the best ones. Given that players face opponents with similar Elo, assortativity in terms of the degree becomes natural. On the other hand, Portal exhibits no correlation between Elo and degree, so it is equally likely for a player to face opponents with any level of activity. A complementary approach was performed by studying the rich-club phenomenon for each network. We observed that in both chess communities (OTB and Portal), elite players tend to form a densely connected group, which is reflected on a normalized rich-club coefficient that increases as a function of the Elo.
In terms of network transitivity, we showed that both networks present a decreasing value of the clustering coefficient as a function of the degree. Nevertheless, whilst in OTB this decrease is rather slow, it becomes more steep in Portal, being well approached by a relation $$c(k)\sim {k}^{-1}$$. As discussed by Ravasz28,29,30, this suggests a more pronounced hierarchical structure in Portal. This is consistent with the fact that correlations between clustering and degree cannot be fully removed by performing a network randomization, indicating that the observed transitivity is inherent to the topology of Portal. The presence of this hierarchical structure implies that hubs do not form a closed community. This is consistent with the result obtained from the rich-club analysis in terms of the degree, which shows that the normalized rich-club coefficient decays for higher degrees. The isolation of hubs is possible because they belong to the class of intermediate level players, which is the most popular one. In chess portals most of the games involve two players picked at random among the pool of players with similar Elo. Since there are few hubs, it is unlikely that in a random pairing they face each other forming closed communities.
The existence of communities in OTB and Portal networks was confronted against corresponding experiments performed with appropriate null models, which consisted in randomized versions of the networks preserving the degree distributions of the original ones. From this analysis we observed that the modularities of the null models are considerably lower than those of the corresponding original networks, and that the opposite relation occurs for the mixing parameter. The community detection analysis also confirmed the existence of a correlation between the community structure of the networks and the Elo of the players. As a general rule, the Elo of a community is less dispersed than the Elo of the complete network. Supporting the previous statement, the results obtained by employing the algorithm proposed in16 show that the Elo is significantly correlated to the community structure of the networks.
## Methods
### Network randomization
In order to compare our results with a null model, we performed two different randomization of the networks. In both cases we rearranged the links in a way that maintains the degree distribution invariant. One of the randomization employed was the Fabien-Viger39 algorithm, which is based on the known Molloy-Reed40,41 random graph generating model. It allows to construct a simple and connected random graph with an arbitrary degree distribution. Each realization of the algorithm gives a sample of the ensemble of such graphs, taken with uniform probability. The second randomization was based on double-edge swapping. The procedure is as follows. Suppose node u is connected to node v and node w is connected x. If u is not connected to w and neither v is connected to x, edges μv and wx are removed and replaced by edges μw and vx. This step is repeated as many times as needed so as to fully randomize the network. Based on Milo, et al. criterion42, we performed a number of steps equal to ten times the number of links of each network.
### Clustering coefficient and correlations
In social networks, it is common to see that when a node μ is connected to another node v, and v is connected to a third node w, it is likely that μ is also connected to w. In common parlance, this means that “the friend of my friend is also my friend”. This characteristic is called transitivity or clustering and can be quantified in different ways24,25. In this work, we employed the so called clustering coefficient and computed two definitions that are widespread in the literature. According to Watts and Strogatz25, the clustering $${C}_{WS}(i)$$ of node i can be defined as:
$${C}_{WS}(i)=\frac{{e}_{i}}{{k}_{i}({k}_{i}-\mathrm{1)/2}},$$
(1)
where k i is the degree of node i and e i is the number of edges between his neighbors. The average clustering coefficient of the complete network is then
$${C}_{WS}=\frac{1}{N}\sum _{i\mathrm{=1}}^{N}{C}_{WS}(i\mathrm{).}$$
(2)
Alternatively, the clustering coefficient of the complete network can be defined as:24
$$C=\frac{3\times (\mathrm{number}\,{\rm{of}}\,\mathrm{triangles})}{{\rm{number}}\,{\rm{of}}\,{\rm{connected}}\,{\rm{triples}}},$$
(3)
where connected triples means three nodes connected by at least two edges. Finally, as pointed out in26, it is sometimes worthy to discriminate the clusterization of the network according to the degree of the nodes. Thus a degree dependent clustering coefficient can be defined as
$$c(k)=\frac{1}{{N}_{k}}\sum _{i/{k}_{i}=k}^{{N}_{k}}{C}_{WS}(i),$$
(4)
where the sum ranges over all nodes in the network having degree k and N k is the number of such nodes.
Degree-degree correlations can be calculated through the Pearson coefficient22. This coefficient ranges from −1 and 1, and allows to quantify if nodes with similar degree are more prone to be connected to each other than nodes with very different degrees. In the first case, r > 0 and the network is said to be assortative. In the latter case, nodes are connected if its degrees are very different, hence r < 0 and the network is dissasortative.
### Rich-club phenomenon
The rich-club phenomenon is related to the tendency of nodes of high degree to connect to each other forming tightly interconnected communities. As proposed by Zhou and Mondragón33, this effect can be measured by introducing a coefficient defined as follows:
$$\varphi (k)=\frac{2{M}_{ > k}}{{N}_{ > k}({N}_{ > k}-\mathrm{1)}},$$
(5)
where $${M}_{ > k}$$ and $${N}_{ > k}$$ are the remaining links and nodes in the network after removing nodes with degree smaller or equal to a given value k. As it was mentioned by Colizza34, this coefficient is a monotonically increasing function even for uncorrelated networks, so it is necessary to compare it with the one obtained from an appropriated null model in order to detect a truly rich-club tendency. To correction of this spurious effect is obtained by defining $$\rho (k)=\varphi (k)/{\varphi }_{ran}(k)$$, where $${\varphi }_{ran}$$ is the rich-club coefficient of a fully randomized network. In the context of social systems a normalized rich-club coefficient $$\rho (k)$$ that increases with degree k corresponds to the existence of a kind of oligarchy in the organization form of the system.
The rich-club can be also defined in terms of a node property different from the degree. In this paper, we choose the mean Elo for each player and defined the rich-club phenomenon for the Elo property as
$$\varphi ({\rm{Elo}})=\frac{2{M}_{ > \mathrm{Elo}}}{{N}_{ > \mathrm{Elo}}({N}_{ > \mathrm{Elo}}-\mathrm{1)}},$$
(6)
where $${M}_{ > \mathrm{Elo}}$$ and $${N}_{ > \mathrm{Elo}}$$ are the remaining links and nodes in the network after removing nodes with mean Elo smaller or equal to a given value Elo. From this definition we derived the normalized coefficient $$\rho ({\rm{Elo}})$$ in the same way as done before.
Newman and Clauset16 introduced a community detection algorithm that makes use of Bayesian statistical inference to construct a generative network model possessing the specific features one hopes to find in the data, namely community structure and a correlation between that structure and node metadata. Then the model is fit to an observed network having metadata and the parameters of the fit give the information about the structure of the network.
The model is based on the stochastic block model43, but incorporates the dependence on node metadata via a set of prior probabilities. An important fact is that the number of communities to be find is a fixed parameter, in contrast with, for example, the Louvain algorithm. It is also worthy to stress that this algorithm does not suppose any a priori relation between structure and metadata, but if there exist any, the algorithm should be able to find it. Among the different alternatives of quantifying correlation between communities and metadata we employed here, as proposed by the authors, the normalized mutual information NMI between the membership of nodes to each metadata category and their membership to each community found by the algorithm. In order to incorporate the metadata to the algorithm, the mean Elo was divided into a set of categories, inspired in the nomenclature used by the Word Chess Federation. The corresponding assignation rule of range of Elo and category is summarized in Table 2.
### Data availability
The datasets that support the findings of this study are available from Opening Master (www.openingmaster.com) but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of Opening Master.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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## Acknowledgements
We thank Opening Master for providing the datasets. This work was partially supported by grants from CONICET (PIP 112 20150 10028), SeCyT–Universidad Nacional de Córdoba (Argentina).
## Author information
### Affiliations
1. #### Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba, 5000, Argentina
• Nahuel Almeira
• , Ana L. Schaigorodsky
• , Juan I. Perotti
• & Orlando V. Billoni
2. #### Instituto de Física Enrique Gaviola (IFEG-CONICET), Ciudad Universitaria, Córdoba, 5000, Argentina
• Nahuel Almeira
• , Ana L. Schaigorodsky
• , Juan I. Perotti
• & Orlando V. Billoni
### Contributions
N.A. performed the data analysis. All authors discussed the results. N.A. and O.B. wrote the manuscript, with input from all the authors.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Orlando V. Billoni. | 2018-10-21 19:27:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6802191734313965, "perplexity": 1165.3023314532984}, "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/1539583514314.87/warc/CC-MAIN-20181021181851-20181021203351-00546.warc.gz"} |
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Volume 9, Number 3, 2019, Pages 962-980 Multidimensional stability of planar waves for delayed reaction-diffusion equation with nonlocal diffusion Zhaohai Ma,Xin Wu,Rong Yuan,Yang Wang Keywords:Multidimensional stability, planar waves, nonlocal diffusion, weighted energy, Fourier transform. Abstract: In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$--dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\RR^n )$ in the form of $\ee^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$( $\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\RR^n)$ space and $H^k(\RR^n) (k \geq [\frac{n+1}{2}])$ space. PDF Download reader | 2020-09-21 19:35:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8414815664291382, "perplexity": 1364.1905808212111}, "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-2020-40/segments/1600400202007.15/warc/CC-MAIN-20200921175057-20200921205057-00504.warc.gz"} |
https://shapeofdata.wordpress.com/2016/01/20/continuous-bayes-theorem/ | ## Continuous Bayes’ Theorem
Bayes’ Rule is one of the fundamental Theorems of statistics, but up until recently, I have to admit, I was never very impressed with it. Bayes’ gives you a way of determining the probability that a given event will occur, or that a given condition is true, given your knowledge of another related event or condition. All the examples that I’ve read or heard about seemed somewhat contrived and unrelated to the sorts of data analysis I was interested in. But it turns out there’s also an interpretation of Bayes’ Theorem that’s not only much more geometric than the standard formulation, but also fits quite naturally into the types of things that I’ve been discussing on this blog. So in today’s post, I want to explain how I came to truly appreciate Bayes’ Theorem.
But rather than start with the statement of Bayes’ Theorem, I want to use an old math teacher trick (which I realize many students hate) of trying to derive it from scratch, without stating what we’re trying to derive. Rather, we’ll start by modifying a problem that I described in an earlier post on probability distributions.
Lets pretend we have a robot arm with two joints: The first is fixed to the center of a table and spins horizontally. There’s a bar from this first joint to the second joint, which also spins horizontally and is attached to a second bar. The two bars are the same length, as shown on the left in the Figure below. The game is to randomly pick a pair of angles for the two joints, then try to guess the x and y coordinates of the hand at the end of the arm. As I described in the earlier post, we can think of the (density function of the) probability distribution of all the possible (x, y) coordinates, and it would look something like what’s shown on the right of the Figure. Here, darker colors indicate larger values of the function.
In the earlier post, we determined that the best place to predict that the hand would be is near the center, since that’s where the probability density is highest. But now we’re going to modify the game a bit: What if each time after spinning the wheel, we are told something about the x-coordinate; either its exact value or a small range. Then how would that change our prediction? Note that if we’re given the exact x-value and asked to predict y, then this is essentially the regression problem except that the probability distribution doesn’t look anything like the one we used in the post on regression.
But lets start with the case where we’re given a range, for example if we knew that the x-value was between 1/4 and 1/3. Then the probability of getting any point with an x-value outside this range would be zero, so the probability density that we would use to guess the y-value would be equal to zero for all points with x-values outside this range. So it would look something like the Figure on the right.
But dropping those values to zero isn’t enough to get the new distribution; the problem is that when we add this restriction on x, the probability of any point with an x-value in the correct range will increase. The question is: By how much will they increase?
In order to answer this question, we need to look more closely at what the probability density function really means. The first thing to note is that the value of the probability density function at a point is not the probability of choosing that point. In fact, the probability of picking any one point is zero, since there are infinitely many possible x and y values.
In order to understand the meaning of the probability density function, we need to use integrals, but (as usual) we can avoid much of the technical details by describing things in terms of the geometry that underlies those integrals. In particular, we’re going to think of our probability density function as describing the elevations of a mountain whose base is the square in which our robot arm rotates. But it won’t be a normal looking mountain – because of the way the density function looks, it’ll have a high peak in the middle, surrounded by a deep moat, then a high circular ridge (shorter than the central peak) around the outside.
I’ve attempted to draw this on the left, but you’re probably better off using your imagination to picture it. Once we’ve transformed our density function into this mountain, we can replace the word “integral” with “volume” and we’ll be able to calculate some probabilities.
Now, as noted above, if we pick one specific point, the probability that the hand will end up there is zero. However, if we pick a particular region of the square, such as a rectangle defined by a range of x-values and a range of y-values, then there may be a non-zero probability that the hand will stop within A. (Though everything I will say below also holds true for more complex shapes A, as well as for shapes in higher-dimensional probability spaces.) In particular, the density function is defined specifically so that the probability will be equal to the volume of the part of the mountain above the shape A.
In other words, if we were to take a band saw to the mountain, following the outline of A, then the volume of the piece that we cut out would be equal to the probability of the robot’s hand stopping within that region. We’ll call this volume/probability P(A).
So, not only is the value of the probability density function at a point not the probability of getting that point (since it’s always zero), the value of the density function at a point doesn’t even need to be less than one. In particular, if there is a region A with a small area but a very high (though still less than 1) probability, the values of the density function would need to be very high in order to get the appropriate volume.
If we choose region A to be the entire square then P(A) = 1 because the arm is constrained to stay within that region. So the volume of the entire mountain is 1. If we choose a smaller region A, and an even smaller region B contained in A, then we’ll get a piece with smaller volume P(B) < P(A), and thus lower probability as we would expect.
But now lets return to the original question of how to modify our density function after we’ve narrowed down the set of possible outcomes to a smaller range of x-values. This function will define a different mountain that is low and flat everywhere outside of A, but has the same elevations as the original within A, as on the left in the Figure below. We want to modify this function to give us a new probability density defining a volume function which we’ll write P(*|A) where * can be any region of the square. Since we know the robot hand landed in A, the overall probability, i.e. the volume P(A|A) of the new mountain, should be 1. However, since all we did was flatten the parts of the mountain outside the region, its volume is initially quite a bit less than 1.
In order to get the correct volume, we’ll need to scale the function up, i.e. multiply each value of the function by a constant k. The resulting function will define a mountain more like the one shown on the right above. For any region B contained in A, we’ll have P(B|A) = kP(B). Since we want P(A|A) = 1 = P(A)/P(A), the only possible value for k is 1/P(A) and we get P(B|A) = P(B)/P(A).
This is the case when the region B is contained in A. But what if it isn’t? For example, if we want to predict a range of y-values once we know the robot hand is in a certain range of x-values, then A will be a vertical strip of the square, and B will be a horizontal strip of the square, with the two intersecting in a smaller rectangle. (But as I noted above, we could just as easily let A and B be arbitrary blobs in the square or even blobs in a higher-dimensional space, but lets not get too complicated…)
So, if we want to calculate P(B|A) in this case, we need to consider two parts of B separately: The density function above the part of B that is outside of A will all get flattened to zero, so P(B|A) is completely determined by the part of B inside of A, i.e. the intersection A ∩ B. In other words, we have P(B|A) = P(A ∩ B)/P(A). Note that there’s no difference between A and B in this formulation, so we also have P(A|B) = P(A ∩ B)/P(B). We can solve both equations for P(A ∩ B) to get P(B|A)P(B) = P(A ∩ B) = P(A|B)P(A). Finally, if we divide both sides by P(B), we get Bayes’ Theorem:
P(B|A) = P(A|B)P(A)/P(B)
Of course, for the problem we started out with, the original equation P(B|A) = P(A ∩ B)/P(B) may sometimes be more useful. But in the standard setting of Bayes’ Theorem, P(A ∩ B) is the probability that both events happen (or both statements are true) so it might be harder to calculate.
For extra credit, take a minute to think about how you might calculate the probabilities of different y-values if we knew the exact value of x rather than a range. I’ll give you two hints: First, note that the probability density function over the vertical line defined by a single x-value defines a single-variable function like you might find in Calculus I and II, and there is some area (rather than a volume) below this function. Second, note that you can take A to be a small rectangular strip around the line defined by the x-value, calculate its volume, then make the strip smaller and smaller and take a limit.
But this post is already long enough, and I expect that most of my readers either don’t want to read about limits, or would rather work it out themselves. (For me, it’s both.) So I’ll leave it there.
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### 2 Responses to Continuous Bayes’ Theorem
1. KlavierKatze says:
There’s some mistakes in deriving Bayes’ theorem in the general case. It should be \$P(B|A) = P(B \cap A|A) = P(B \cap A)/P(A)\$.
• Thanks for catching that! It should be correct now. | 2020-06-07 07:27:44 | {"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": 0.8708136081695557, "perplexity": 336.51424226580195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348523564.99/warc/CC-MAIN-20200607044626-20200607074626-00486.warc.gz"} |