url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
|---|---|---|---|
http://www.alecjacobson.com/weblog/?p=4690 | Like clockwork, my matlab updates have gotten out of sync with Xcode updates. It seems like fixing this SDK error always requires a different hack each year. This year I got the error:
No supported compiler or SDK was found. For options, visit
http://www.mathworks.com/support/compilers/current_release/.
To fix it, I replaced all occurrences of 10.9 with 10.11 in /Applications/MATLAB_R2017a.app/bin/maci64/mexopts/clang{++,}_maci64.xml
ld: warning: object file was built for newer OSX version (10.11) than being linked (10.9)
For now, I’m assuming that I can ignore them. We’ll see how far that gets me.
Tags: , , ,
### One Response to “”
1. ajx says:
Yet another time I’ve been frustrated trying to figure out where matlab is finding it’s mex options. This time I realized the answer is right in front of me. If you pass the -v (verbose) flag to your mex compile, you should see a whole bunch of debug messages including something like:
Options file: /Users/ajx/Library/Application Support/MathWorks/MATLAB/R2017a/mex_C++_maci64.xml
Replacing all 10.9s with 10.11s fixed the remaining problem above. | 2017-08-17 07:40:07 | {"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": 0.5247085690498352, "perplexity": 3851.200212552471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102993.24/warc/CC-MAIN-20170817073135-20170817093135-00046.warc.gz"} |
https://compas.dev/compas/latest/api/generated/compas.geometry.sum_vectors.html | sum_vectors
compas.geometry.sum_vectors(vectors, axis=0)[source]
Calculate the sum of a series of vectors along the specified axis.
Parameters
• vectors (list) – A list of vectors.
• axis (int, optional) – If axis == 0, the sum is taken per column. If axis == 1, the sum is taken per row.
Returns
list – The length of the list is len(vectors[0]), if axis == 0. The length is len(vectors), otherwise.
Examples
>>> vectors = [[1.0, 2.0, 3.0], [1.0, 2.0, 3.0], [1.0, 2.0, 3.0]]
>>> sum_vectors(vectors)
[3.0, 6.0, 9.0]
>>> sum_vectors(vectors, axis=1)
[6.0, 6.0, 6.0] | 2021-12-08 05:03:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6070293188095093, "perplexity": 4658.537426878553}, "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/1637964363437.15/warc/CC-MAIN-20211208022710-20211208052710-00466.warc.gz"} |
http://develop.spacemacs.org/layers/+lang/julia/README.html | # julia layer
## 1 Description
This layer adds support for Julia to Spacemacs.
### 1.1 Features:
• Syntax highlighting
• Repl support
• Linting
• Completion
• Jump-to-definition
• Documentation on hover
## 2 Install
To use this configuration layer, add it to your ~/.spacemacs. You will need to add julia to the existing dotspacemacs-configuration-layers list in this file.
### 2.1 LSP
Warning: enabling the LSP functionality within this package can cause emacs to hang for significant lengths of time when opening a Julia file. See tip below on using PackageCompiler.jl to mitigate this issue.
This layer can be used with LanguageServer.jl and emacs lsp-mode to provide richer, IDE-like capabilities. To use this layer with lsp, you must do the following:
1. Add lsp to dotspacemacs-configuration-layers.
2. Install LanguageServer.jl by invoking Pkg.add("LanguageServer") in the Julia REPL.
3. Enable layer integration with lsp as described in section below.
LanguageServer.jl tends to have a very long startup time. In the worst case, lsp-mode might give up on the language server before its started, but regardless usage of lsp-mode with Julia can cause long delays when first opening files. To mitigate this issue, you can try compiling LanguageServer.jl ahead of time using PackageCompiler.jl. This drastically reduces startup time if successful:
julia> Pkg.add("PackageCompiler")
julia> using PackageCompiler
julia> compile_package("LanguageServer")
## 3 Options
While julia-mode is perfectly usable without configuration or other packages, you can choose to replace the default functionality of using julia-repl as the layer’s interactive REPL with the ess REPL and other functionality that comes with it. To do so, install this layer with:
(setq-default
dotspacemacs-configuration-layers
'(
(julia :variables julia-mode-enable-ess t)))
This layer also includes experimental support for lsp-mode. To enable language server functionality with this layer, follow the instructions for installation, and then install this layer with:
(setq-default
dotspacemacs-configuration-layers
'(
(julia :variables julia-mode-enable-lsp t)))
Warning: LanguageServer.jl has not yet been released for Julia 1.0. If you are using this layer with Julia 1.0, julia-mode-enable-lsp should be set to nil.
## 4 Key bindings
Key binding Description
SPC m h h Calls @doc macro on symbol under cursor.
SPC m e m Expands macro under cursor.
SPC m w Calls workspace() in Repl to give clean namespace.
SPC e l Expands latex macro (e.g. \delta).
SPC m s b Sends buffer to REPL.
SPC m s l Sends line to REPL.
SPC m s r Sends region to REPL.
SPC m r Brings up Julia Repl (starts new one or focuses existing).
Created: 2019-01-22 Tue 20:43
Validate | 2019-04-23 06:24:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25278961658477783, "perplexity": 12467.1389404615}, "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-2019-18/segments/1555578593360.66/warc/CC-MAIN-20190423054942-20190423080942-00355.warc.gz"} |
http://www.mapleprimes.com/questions/200606-How-To-Write-Procedures-In-MAPLE-To?ref=Feed:MaplePrimes:New%20Questions | # Question:How to write procedures in MAPLE to deal with pseudo differential operators?
December 23 2013
1
I am trying to write procedures in MAPLE to deal with pseudo differential operators such as P=1+f_1D^{-1}+f_2D^{-2}+f_3D^{-3}+..., where D=d/dx, f_i are functions of x. D^n for n<=0 work like usual differetial operators for n>=0. For example D^nD^m=D^{m+n}, D^n f=\sum_{i>=0}\binom{n}{i}(D^i f)D^{n-i}, where \binom{n}{i} are binomial coefficients given by n(n-1)...(n-i+1)/i! for n<= 0. I am wondering which package in MAPLE I could begin with, and how can I define D^{-1} and generally D^{-n}.
Thank you very much!
| 2014-04-24 05:53:40 | {"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": 0.887948215007782, "perplexity": 3957.5429261793593}, "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-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00419-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1442277/show-that-ci-r-has-a-countable-dense-subset-and-therefore-a-countable-basi | Show that $C(I, R)$ has a countable dense subset, and therefore a countable basis
Give $R^I$ the uniform metric, where $I = [0, 1]$. Let $C(I, R)$ be the subspace consisting of continuous functions. Show that $C(I, R)$ has a countable dense subset, and therefore a countable basis.
I know to show that having a countable dense subset implies having a countable basis.
I'm doing this exercise in Munkres book and got no clue about the solution. Hope someone can help me solve this.
Hint Given in the Book is...Consider those continuous functions whose graphs consist of finitely many line segments with rational end points.
• Do you know the Weierstraß approximation theorem (or the more general Stone-Weierstraß theorem)? – Daniel Fischer Sep 19 '15 at 14:34
• In my book, there is a hint: "Consider those continuous functions whose graphs consist of finitely many line segments with rational end points." – Eclipse Sun Sep 19 '15 at 14:36
• Yah the statement of the theorem I know...but which version is reqd here? – user8795 Sep 19 '15 at 14:37
• "Show that $C(I,R)$ has a countable dense subset, and therefore a countable basis.": Do you mean that there is a countable basis for the topology on $C(I,R)$, or a basis for $C(I,R)$ in the vector-space sense? It's true that $C(I,R)$ has a countable basis in the second sense, but it's not as simple as "has a countable dense subset, therefore"; there are spaces with countable dense subsets but no countable basis. If otoh you meant basis for the topology then never mind (although you probably should have explained that's what you meant). – David C. Ullrich Sep 19 '15 at 14:45
1) Prove that linear combination of a finite number of $\chi_{[a,b)}$ with $a<b$ in $I$ are dense. Here I would use uniform continuity.
2) Prove that you can take the intervals of the above linear combination in $\mathbb{Q}$ instead that in $\mathbb{R}$.
3) Prove that you can take the coefficients of the combination in $\mathbb{Q}$ instead that in $\mathbb{R}$. | 2020-02-23 10:56:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9130164384841919, "perplexity": 119.01932210328853}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145767.72/warc/CC-MAIN-20200223093317-20200223123317-00477.warc.gz"} |
https://stm.sciencemag.org/content/4/138/138ra78?ijkey=0fbaa5f0838a6e22d562a253a73bdfbc888d409b&keytype2=tf_ipsecsha | Research ArticleGenomics
# Exome Sequencing Can Improve Diagnosis and Alter Patient Management
See allHide authors and affiliations
Science Translational Medicine 13 Jun 2012:
Vol. 4, Issue 138, pp. 138ra78
DOI: 10.1126/scitranslmed.3003544
## Abstract
The translation of “next-generation” sequencing directly to the clinic is still being assessed but has the potential for genetic diseases to reduce costs, advance accuracy, and point to unsuspected yet treatable conditions. To study its capability in the clinic, we performed whole-exome sequencing in 118 probands with a diagnosis of a pediatric-onset neurodevelopmental disease in which most known causes had been excluded. Twenty-two genes not previously identified as disease-causing were identified in this study (19% of cohort), further establishing exome sequencing as a useful tool for gene discovery. New genes identified included EXOC8 in Joubert syndrome and GFM2 in a patient with microcephaly, simplified gyral pattern, and insulin-dependent diabetes. Exome sequencing uncovered 10 probands (8% of cohort) with mutations in genes known to cause a disease different from the initial diagnosis. Upon further medical evaluation, these mutations were found to account for each proband’s disease, leading to a change in diagnosis, some of which led to changes in patient management. Our data provide proof of principle that genomic strategies are useful in clarifying diagnosis in a proportion of patients with neurodevelopmental disorders.
## Introduction
Next-generation sequencing (NGS), in which the whole genome, or a portion thereof, is sequenced, has proven extraordinarily useful for identifying new causes of genetic disease, especially for Mendelian disorders. However, the application of NGS directly in the clinic is not straightforward because of the difficulties in determining which of the thousands of variants of unknown significance (1, 2) are relevant to the individual patient’s presenting signs and symptoms. Still, there is great anticipation that NGS, especially whole-exome sequencing, in which the 1% of the genome that codes for proteins is sequenced (3), will improve diagnostic approaches in genetic disease.
Neurodevelopmental disorders affecting 4 to 6% of the general population, most notably children, include intellectual disability, epilepsy, autism, structural brain diseases, and neuromuscular disorders. The U.S. Centers for Disease Control estimates that the annual cost of neurodevelopmental disorders accounts for 5 to 10% of total health care expenditure in the United States owing to the lifelong care required for these patients (4). The inaccessibility of neural tissue makes it difficult to arrive at a specific diagnosis for patients with neurodevelopmental disorders, so clinicians are left with categorical diagnoses or long differential diagnoses lists. Low-yield and expensive radiographic, electrophysiological, biochemical, and biopsy evaluations are the only prospect of narrowing these lists, often costing in excess of $10,000 per patient (5). Neurodevelopmental disorders exhibit both clinical and locus heterogeneity; therefore, genetic investigations are often limited to candidate sequencing of a single gene, or a small panel of genes, that requires the clinician to have a clear sense of the likely genetic cause before testing. Although chromosomal and copy number variations account for 10 to 20% of these cases (6, 7), the remaining cases have relatively little chance of achieving a genetic diagnosis. The failure to make a specific diagnosis for neurodevelopmental disorders is a major clinical problem because it limits prognostic information, anticipatory counseling, prevention strategies, quality of life, and initiation of potentially beneficial therapies (8). For these reasons, and the finding that many neurodevelopmental disorders have a genetic basis, the neurodevelopmental disorders clinic represents a fruitful area to explore the use of whole-exome sequencing. For this project, we identified 118 probands and their families from regions of the world with high rates of consanguinity, which enhances the power to identify recessive genetic mutations using homozygosity mapping (9). About one-sixth of the world’s population resides in these areas, making this an important population to study. In such populations displaying recessive disease, heterozygous alleles can usually be excluded as causative (table S1), greatly reducing the number of variants to be considered, and overcoming one of the potential drawbacks of whole-exome sequencing (5). It is estimated that 80% of the variants causing Mendelian disease are located within the exome, making whole-exome sequencing an attractive method to interrogate variants of high effect (10, 11). Furthermore, about 15% of suspected Mendelian disease has a recessive mode of inheritance, and genomic carrier burden for such disease is estimated at 2.8 mutations per genome in outbred populations, making this class of diseases an important part of the neurodevelopmental disorder spectrum (12). Here, we present data on the application of whole-exome sequencing to a large clinical cohort. Our data show that not only is whole-exome sequencing a useful tool for identifying disease-causing genes, but it is also able to correct or modify the diagnosis in ~10% of the families studied (n = 118), thereby providing proof of principle that whole-exome sequencing can be a useful tool for diagnosis in the clinic. ## Results ### Patient recruitment and diagnostic sequencing We analyzed a total of 188 families by collecting pedigrees, phenotype information, and blood samples on each genetically informative subject. Initial medical diagnoses were generated by the collective medical team (that is, treating physician, geneticist, and medical specialists) at case conferences consistent with current medical practice (Table 1) and were termed “initial diagnosis” for the purposes of this study. In some instances, the presenting features were too nonspecific to suggest a unique diagnosis, and in such cases, a categorical diagnosis was assigned. All families contained two or more affected individuals born to consanguineous parents. We used a standard protocol to exclude known disease-causing genes either by direct sequencing of all coding exons and splice sites or by excluding known loci with linkage exclusion mapping. Of the 188 probands (Fig. 1), 40 had mutations in one of the genes associated with the initial diagnosis and the mutation segregated with the phenotype in the family according to a recessive model. Such mutations were reported to the referring physician as part of this research protocol, and the families were not further studied. For the remaining families, mutations in known genes were not identified, and these families moved on to the next phase of analysis. In hindsight, whole-exome sequencing analysis of the 40 probands with mutations in known genes might have been more efficient and cost-effective than single gene sequencing methods because most subjects required evaluation at three or more genetic loci (Table 1). Table 1 Summary of 10 probands and their families in which whole-exome sequencing corrected diagnosis. View this table: ### Linkage analysis The remaining 148 probands and their families were subjected to genome-wide parametric linkage analysis using a panel of highly informative single-nucleotide polymorphism (SNP) markers. In 30 families, a single linkage peak was identified, and such families were not considered further because we viewed strategies other than whole-exome sequencing to be a more direct method of mutant gene identification. For these families, in no instance did the identified peak overlap with a genetic locus known to cause the initial diagnosis, suggesting that many of these peaks should reveal previously unidentified causes of disease. In the remaining 118 families, we uncovered between two and eight peaks consistent with linkage, although about 30% of these families were not analyzed with linkage because they came to the study relatively late. These families were instead analyzed using homozygosity mapping from exome data (9). ### Exome sequencing and variant discovery From the 118 families without single linkage peaks, one proband per family was evaluated using whole-exome sequencing, producing an average coverage at >10× read depth for 96% of the exome, which is within the expected coverage and depth for whole-exome sequencing studies (13) and is sufficient to assess most recessive disease variants. On average, a total of 26,393 ± 4971 (SD) variants were identified per proband for evaluation. Tabulation of the <10% of the genome that failed adequate recovery from whole-exome sequencing (<10× depth) was generated in case a causative variant could not be identified among those recorded. Variants were then filtered and prioritized according to the presumed recessive disease model to identify variants of high effect size (Fig. 2). On the basis of the HapMap project, the average haplotype block size from an offspring resulting from a first-cousin marriage is >10 centimorgan (cM) (14), so we focused on such blocks of homozygosity identified from either parametric [LOD (logarithm of the odds ratio for linkage) scores] or nonparametric (homozygosity mapping) linkage. The remaining variants were then prioritized according to type of mutation (deletion/insertion > nonsense > missense), amino acid conservation, predicted damage to the protein, and relevance of the candidate gene to the given disease. The final variant list contained a mean of 9 (range, 4 to 21) new, coding, homozygous variants in linkage or homozygous intervals per proband (tables S1 and S2). Variants on the final filtered list were validated by Sanger sequencing, verified as homozygous in affecteds, and tested for segregation in the family to be consistent with the pedigree structure. ### Disease gene identification In 22 of the 118 probands who were analyzed by whole-exome sequencing, we identified a single variant in a gene not previously implicated in disease, which fell within a region of homozygosity, and suggested a previously unidentified disease gene as the cause of the disorder. Two of these variants in which we have validated segregation are listed in Table 2. Specifically, we identified a mutation in GFM2 in a family with microcephaly, simplified gyral pattern, and insulin-dependent diabetes and a mutation in EXOC8 in a family with Joubert syndrome [Mendelian Inheritance in Man (MIM) number 213300]. Table 2 Whole-exome sequencing is a useful technique for identifying new disease-causing genes. Summary of two families analyzed in which whole-exome sequencing identified a causative gene not previously associated with disease. View this table: GFM2 (also called EFG2) encodes the mitochondrial elongation factor G2 and is part of the mitochondrial translation complex essential for maintaining energy metabolism. The identified c.T2032A variant in family 650 changes p.D576E, but this variant also occurs in a conserved predicted splice site at the acceptor for exon 17 and is predicted to destroy the splice acceptor function based on NetGene2 and BDGP prediction algorithms (15, 16). The presentation is overlapping with Wolcott-Rallison syndrome (MIM 226980) (17), characterized by early-onset insulin-dependent diabetes and occasional microcephaly. Mutations in EIF2AK3 and IER3IP1, encoding a translational initiation factor kinase and an endoplasmic reticulum stress response factor, respectively, have been linked to Wolcott-Rallison syndrome (18, 19). The p.D576E variant is the single variant found in a homozygous interval that segregates in the family, is not present in 200 ethnically matched controls, is predicted to damage the protein, and occurs in an evolutionarily conserved residue (Fig. 3). This mutation in a mitochondrial elongation factor is consistent with the model of Wolcott-Rallison syndrome as a defect in energy and cellular stress homeostasis, leading to altered neurogenesis and apoptosis. These findings suggest that GFM2 is a rational candidate for the disease and further support the use of whole-exome sequencing in identifying previously unidentified disease-causing genes for Mendelian disorders. EXOC8 encodes the exocyst 84-kD subunit, one of the critical members of the eight-subunit complex required for targeting secretory vesicles to the plasma membrane during exocytosis (20). The p.E265G variant found in family 982 occurs in the B6 loop of the highly conserved pleckstrin homology (PH) domain, which is involved in binding phosphatidylinositol lipids for vesicular transport. This is the single, segregating variant in the family and is not present in 200 ethnically matched controls. It is predicted to be damaging according to POLYPHEN-2 (15, 16) and occurs in a fully conserved residue. Joubert syndrome is one of the “ciliopathy” diseases, and EXOC8 is part of the ciliary proteome (21). Further, the exocyst complex has been implicated in ciliary function (21). For these reasons, EXOC8 is a rational candidate for this disorder (Fig. 4). In the remaining 86 probands, we found 2 to 10 variants of unknown significance per proband, some of which are good disease-causing candidates. Studies are ongoing in the lab to improve variant annotation and search for probands with similar phenotypes displaying variants in the same gene in an effort to demonstrate causality, similar to published work (22). ### Corrected patient diagnoses In 10 of the 118 probands (Table 3), it was apparent that one of the variants occurred within a gene already listed in Online MIM (OMIM) to cause a neurodevelopmental disease phenotype that at least partially overlapped with the phenotype of the proband, suggesting that it might represent the causative mutation (figs. S1 to S10). In each of these 10 patients, however, the genetic diagnosis suggested from whole-exome sequencing differed from the initial diagnosis, leading us to question the veracity of the initial diagnosis. It was initially surprising to identify mutations in known disease genes, because for each initial diagnosis, we had excluded the genes most frequently mutated. For instance, in a family diagnosed with microcephaly, we excluded the genes for primary microcephaly (MCPH1, CDK5RAP2, MCPH4, ASPM, CENPJ, and STIL); in a family displaying ataxia with vitamin E deficiency, we excluded the causative gene (TTPA); and in a family with intellectual disability, we excluded the most commonly mutated gene for the recessive form of the disease (VSP13B) (Table 1). Table 3 Initial diagnosis compared to genetic diagnosis after whole-exome sequencing in 10 probands. Summary of 10 families analyzed in which whole-exome sequencing corrected diagnosis. In each family, an identified mutation in a known disease-causing gene led to a modification of the diagnosis. Only G726E (family 1436) is a previously reported disease mutation. For all mutations leading to a premature stop codon (families 928, 992, 890, 1409, and 951), other stop codons have been reported with the respective disease. For the missense mutations not previously reported (families 1004, 995, and 1002), and the splice mutation (family 702), each was located in an amino acid/base pair that is fully conserved across evolution (Supplementary Materials), located in a protein domain essential for protein function or splicing, is predicted to be damaging, and is not found in 200 ethnically matched controls. All mutations segregated normally with the phenotype in these families. These data, in addition to further scrutiny of the patient’s clinical profile, provide evidence that these mutations are the cause of the disorders seen in each family. View this table: To understand this paradox, we returned to the patient charts to review the presentation and clinical course. In each case, we found that the genetic variant was sufficient to explain the full clinical presentation, suggesting that whole-exome sequencing was able to either modify or correct an initial diagnosis for each of these 10 cases. ### Clinical presentations Family 890: Mutation in VLDLR. This family from Trabzon, Turkey, presented two affecteds at birth with microcephaly, nystagmus, congenital ataxia, mild spasticity, and arachnodactyly. Brain magnetic resonance imaging (MRI) analysis demonstrated severe hypoplasia of the midbrain, consistent with a diagnosis of pontocerebellar hypoplasia (MIM 607596), published as such in 2002 (23). The family was negative for mutations in the three known genes for pontocerebellar hypoplasia—TSEN2, TSEN34, and TSEN54—encoding transfer RNA (tRNA) splicing endonucleases (24), and linkage analysis demonstrated three potential linkage peaks not associated with any known pontocerebellar hypoplasia genes. Whole-exome sequencing identified a homozygous p.G1246fsX1305 alteration, which segregated in the family, leading to a protein frameshift in the VLDLR gene, encoding the very low-density lipoprotein receptor (fig. S1). Reevaluation of the brain MRI was completely consistent with VLDLR-associated congenital cerebellar ataxia with intellectual disability syndrome (MIM 224050), demonstrating the classical very small, smooth cerebellum (25). The team concluded that the initial diagnosis was incorrect because the clinical phenotype in this family was different from the spectrum previously described for VLDLR-associated disease. Family 951: Mutation in MAN2B1. This family from Islamabad, Pakistan, presented four affected children with intellectual disability. After a normal pregnancy, labor, and delivery except for low birth weight, there was intellectual disability noted by 2 years of age, as well as mild dysmorphic features including prominent forehead, wide-set eyes, and defects in hearing and speech. Routine metabolic screening and mass spectrometry were noncontributory. The affecteds received an initial diagnosis of recessive intellectual disability and were negative for alterations in the VPSB13B gene, tested because of concordant obesity (26). SNP-based linkage analysis pointed to two potential linkage peaks, neither containing genes for autosomal recessive intellectual disability. Whole-exome sequencing demonstrated a homozygous p.W695* truncating mutation in the MAN2B1 gene that segregated fully in the family (fig. S2). The MAN2B1 gene is mutated in α-mannosidosis (MIM 248500) (27), a metabolic lysosomal storage condition caused by an inability to cleave α-linked mannose residues from the nonreducing end of N-linked glycoproteins. Reevaluation of the phenotype in light of this finding confirmed the typical facial appearance, enlarged liver, and vacuolated lymphocytes typical of type I α-mannosidosis (28). The anticipatory guidance and direction of therapy has been changed to reflect this genetic diagnosis (29). The team concluded that the initial diagnosis did not take into account this disease because of the nonspecific presenting features. Family 1002: Mutation in SPG11. Family 1002 from Marrakech, Morocco, presented three affected members with progressively unsteady gait from the age of 5 years, interpreted as ataxia. There was areflexia, positive Babinski sign, and loss of proprioception with intact cognition, and a normal brain computed tomography (CT) scan, leading to the initial diagnosis of a progressive ataxia or spasticity. Initial workup included reduced serum levels of ApoA1, high-density lipoprotein (HDL), and vitamin E, consistent with a diagnosis of ataxia with vitamin E deficiency (MIM 277460). The reduced serum levels were within the range of other patients we have evaluated with this condition, although they lacked the common Moroccan 744delA mutation in the TTPA gene (30). However, patients showed nominal improvement in function upon administration of daily exogenous vitamin E, supporting the diagnosis. Full sequence of the TTPA gene was negative for variation, and SNP-based linkage analysis suggested two potential peaks, neither of which contained the TTPA gene or known modulators of vitamin E metabolism. Whole-exome sequencing analysis identified one splice and two missense variants, two of which were fully conserved across species and one predicted to be damaging. Only a homozygous c.T5088G variant leading to a p.A1696G amino acid transversion in the SPG11 gene segregated according to the predicted mode of inheritance in the seven children in the family, providing compelling evidence that this mutation may cause this neurodevelopmental disorder (fig. S3). The SPG11 gene is a recently reported cause of hereditary spastic paraplegia with thin corpus callosum (MIM 604360) (31). The p.A1696G changes a nonpolar neutral amino acid to a polar negative amino acid and is predicted to be damaging to protein function according to POLYPHEN-2 software (15, 16). The p.A1696 residue is perfectly conserved across evolution and occurs within the leucine-rich repeat 3 domain, supporting its pathogenicity. This variant was not detected in chromosomes from 200 control Moroccan individuals. Subsequent reevaluation of the family led to reinterpretation of the ataxia as spasticity, and brain MRI analysis in two affecteds demonstrated the characteristic thin corpus callosum, consistent with a diagnosis of SPG11-associated disease. Vitamin E therapy has subsequently been halted without clinical consequence. In this situation, the team concluded that the original initial diagnosis was incorrect due to an initial misinterpretation of the clinical signs and false-positive chemistry studies. Family 1004: Mutation in GJC2. This family from Cairo, Egypt, presented two affecteds with microcephaly and intellectual disability. The initial diagnosis of microcephaly was assigned on the basis of a head circumference of 48 cm at age 8 years (−2.5 SD) in an older male sibling and 45 cm at age 3 years (−2.5 SD) in a younger female sibling. Brain MRI showed thin corpus callosum, mild ventriculomegaly, and cerebellar hypoplasia. The family tested negative for mutations in the known primary microcephaly genes MCPH1, CDK5RAP2, MCPH4, ASPM, CENPJ, and STIL. As the children aged, they displayed signs of nystagmus, hyperreflexia, and spasticity, atypical for primary microcephaly, and the three linkage peaks identified from SNP-based analysis did not suggest any other microcephaly loci. Whole-exome sequencing demonstrated a homozygous c.C94T alteration in the GJC2 gene that segregated fully in the family and is known to cause hypomyelinating leukodystrophy II (MIM 608804) (32). This mutation leads to a p.R35C amino acid transversion in the connexin domain. The p.R35 residue is perfectly conserved across evolution, is predicted to be damaging, and was not found in chromosomes from 200 Egyptian control individuals (fig. S4). Subsequent reevaluation of the family focusing on this variant led us to conclude that the spasticity and nystagmus were progressively worsening, along with the presence of mild peripheral axonal neuropathy. MRI reinterpretation showed a hypomyelinating leukodystrophy consistent with GJC2-associated disease. The team concluded that the initial diagnosis was too broadly categorized due to nonspecific presenting features, which precluded a more accurate diagnosis. ## Discussion The main finding of this work is that whole-exome sequencing is beneficial over individual candidate gene sequencing in identifying mutations in genes not previously suspected in a given patient. This finding provides proof of principle that whole-exome sequencing has the potential to change clinical practice for genetic disease. Specifically, this work demonstrates the use of whole-exome sequencing in the clinic when applied to a group of patients with likely genetic disease for which the cause remained elusive before study. In our study, we found that in 10 cases of 118 probands undergoing whole-exome sequencing, there was a revision of the diagnosis and, in some cases, a change in management. Furthermore, in each of these 10 cases, genetic counseling, prenatal diagnostic options, and carrier testing were altered after diagnosis. We also identified likely causative mutations in two other families with neurodevelopmental disorders, which have the potential to lead to new therapies. Although the ability of NGS to provide an accurate genetic diagnosis has been established for single cases like 3,4-dihydroxyphenylalanine (dopa)–responsive dystonia and Charcot-Marie-Tooth neuropathy (33, 34), this report addresses the benefits of NGS in a large clinical cohort. In our cohort, we first excluded known genetic causes of disease by sequencing likely mutated genes on the basis of the initial diagnosis. This enriched for patients with new genetic causes of disease and with an incorrect or partially correct diagnosis. From this group, we used whole-exome sequencing to further stratify patients into those with a likely new disease gene (22 of 118, or 19%), those with no obvious single disease gene candidate but rather numerous candidates (86 of 118, or 73%), and those with a mutation in a disease gene known to cause a disorder different from the initial diagnosis (10 of 118, or 8%). These findings should be compared to other diagnostic modalities, such as copy number variant (CNV) or de novo mutation identification in neurodevelopmental disorders, where success rates fall between 10 and 60% in selected populations (3537). Although it is difficult to compare success rates due to differences in cohort structures, whole-exome sequencing in probands with recessive disease is likely to emerge as an attractive alternative approach to candidate gene sequencing. Our data show that in a substantial portion of patients in the neurodevelopmental disorders clinic, the initial diagnosis might not be accurate or might be too broadly classified. There are several potential reasons why an initial diagnosis might be incorrect or partially correct in the neurodevelopmental disorders spectrum. In our study, we attributed these differences to the following, and it is our experience that these limitations exist in the clinical setting in general: (i) Patient phenotypes differed partially or substantively from the spectrum previously described for a given gene, (ii) medical information or history was incomplete, and (iii) nonspecific clinical features were found in the patients. The field of genetic medicine has literally tens of thousands of unique syndromes. Even an experienced professional might not entertain all possible diagnoses for a given presentation due to the vast number of syndromes to consider. Medical diagnostic software that helps to maintain a broad and systematic differential could help with this issue (38) and would make a powerful partner to help prioritize and filter data. In each case presented here, the medical team returned to the clinical information to determine why the initial diagnosis differed from the genetic diagnosis, and it was found that most differences were due to limitations in the clinical practice of medicine. Whole-exome sequencing was able to overcome many of these limitations. This study demonstrates the clinical use of whole-exome sequencing and points out potential benefits in correcting patient diagnosis. The current cost for whole-exome sequencing is ~$2000 to \$4000 per patient (39) and is expected to drop substantially in the coming years. With similar costs for candidate gene sequencing, whole-exome sequencing should be considered an attractive alternative in families with a presumed genetic cause of disease. Whereas whole-genome sequencing is another technology that is sure to change the face of medicine in the future, whole-exome sequencing has captured the attention of the clinical genetics community because most genetic variants of large effect reside in the exome, because intronic mutations are difficult to interpret and to model, and because genome sequencing is still more expensive than whole-exome sequencing (10, 34). The data presented here suggest that whole-exome sequencing should be considered in a diagnostic context in the appropriate clinical settings.
Whereas whole-exome sequencing was used with some success in this study, it does suffer from limitations—even in the field of Mendelian genetics. Whole-exome sequencing in clinical applications lacks some sensitivity due to its inability to interrogate intronic sequence, the absence of recovery of some exons, and sequencing errors. Even more important is the difficulty in separating causative variants from the vast number of variants of unknown significance identified per patient (10, 13, 15). Furthermore, the recent finding that more than 25% of putative disease-causing variants in available databases are erroneous (12) makes interpretation all that much more difficult. These limitations were partially overcome by restricting analysis to consanguineous families with recessively inherited disease, and allowing exclusion of most variants of unknown significance using criteria specific to this model of disease. In addition, these families allowed for testing of segregation of each variant, thereby providing an additional level of certainty about the causation of each mutation. As human mutation databases become sufficiently populated and carefully curated, the ability to interpret whole-exome sequencing data will greatly improve. The introduction of whole-exome sequencing into routine clinical practice will require careful assessments of these issues. As for the future, the limitations of whole-exome sequencing seem tractable and there are solutions that should allow whole-exome sequencing to ultimately be used routinely in a clinical setting.
## Materials and Methods
### Study participants
The probands for this study were ascertained from the Middle East, North Africa, and Central Asia and were selected based on the criteria of (i) healthy parents with documented consanguinity, (ii) more than one affected child to enrich for recessive disease, and (iii) an initial diagnosis of a neurodevelopmental disorder of unknown genetic etiology. We excluded patients with clear single gene defects in which clinical features are absolutely characteristic, such as for Tay-Sachs disease, Niemann-Pick type C, and spinal muscular atrophy. This study was approved by the Institutional Review Board at the University of California, San Diego, and collaborating institutions; all study participants signed informed consent documents; and the study was performed in accordance with Health Insurance Portability and Accountability Act privacy rules.
### Phenotypic assessments
One or more of the authors who are board-certified in pediatrics, neurology, and/or genetics evaluated each patient. Standard evaluation included full general and neurological examination, height, weight, head circumference measurements, intelligence quotient (IQ), brain MRI or CT, and electroencephalogram (EEG) when clinically indicated. Diagnostic criteria were based on those standards in the field, and initial diagnoses were determined by consensus at clinical team meetings where differential diagnoses, genetic counseling, and care plans were discussed.
Linkage analysis was performed by genotyping all available and consenting members of the family in the affected and parental generations of the pedigree. DNA was extracted from peripheral blood leukocytes with salt extraction, genotyped with the Illumina Linkage IVb mapping panel (40), and analyzed with easyLINKAGE-Plus (41) software to generate multipoint LOD scores.
### Exome sequencing and variant analysis
For each sample, whole-exome sequencing was performed with the Agilent SureSelect Human All Exome 50 Mb Kit, which captures up to 50 Mb of the human exome and includes all exons annotated in the consensus CDS database, as well as 10 bases flanking each targeted exon and small noncoding RNAs. This kit provides >95% coverage at 1×, and >88% at 10× coverage, and paired-end sequencing of 75–base pair sequencing length was done with Illumina GAIIx or HiSeq2000 instruments (42). Depth of sequencing was 30 ± 16 (SD) per exome.
Whole-exome sequencing generated large data sets that required extensive analysis to identify the important genetic variants. The process of isolating potential disease-causing variants involved several major steps: (i) align and ensure quality of DNA sequences for each exome; (ii) identify polymorphisms in the patient’s DNA sequence compared to a reference sequence with tested SNP and insertions/deletions (INDEL) calling software [that is, Genome Analysis Tool Kit (GATK) and in-house generated tools]; (iii) verify the quality, repeatability, and comparability to published results of SNP and INDEL data; (iv) filter out variants that are outside runs of homozygosity, outside of coding/splice regions, and are found in homozygous form in the healthy population; (v) prioritize potential disease-causing variants by type of mutation, conservation of amino acid residue(s) across species, predicted damage to the protein, and relevance to the neurodevelopmental disorder; (vi) validate potential disease-causing variants by assuring Mendelian segregation in the family, screening healthy, ethnically matched controls, and identifying mutations in the same gene in other families with the disorder.
Genetic variants were delineated with the GATK software for both SNPs and INDELs (43). Briefly, Illumina “qseq.txt” files from each exome were converted to FASTQ format with Illumina Pipeline Software and then aligned to a reference sequence with Burrows-Wheeler Aligner (BWA) software (SourceForge). Duplicate sequencing reads, which can be produced by polymerase chain reaction amplification, were then removed, and sequence quality scores were recalibrated to correct for sequencing errors and artifacts. Alignments that contain INDELs can lead to mismatches that resemble SNPs; therefore, alignment of sequences with Maq was also necessary to identify and isolate INDEL-containing reads. Next, a Bayesian-based SNP caller and INDEL Genotyper (GATK) were used to filter potential disease-causing variants from the aligned sequences. Each aligned exome was then assessed for sufficient quality with the following parameters: number of SNPs called in each lane (average ~26,000), accuracy, depth of coverage, and error rate per read position. Only exomes of high quality were analyzed further.
Exomes were then filtered to highlight false-positive variants. Likely false positives were flagged with the following criteria: low SNP confidence, frequency of the reference base in the population is overwhelmingly high, low depth of coverage for the SNP, and presence of the SNP in homopolymer runs longer than three bases. Flagged variants were included in the subsequent “variant filtering and prioritization pipeline” but were pursued with caution in the “validation pipeline.” Both pipelines are described below.
Exome call sets underwent the following evaluations: compared number of SNPs to the quantities usually found per cleaned exome data set, determined overlap with the dbSNP database, determined the transition/transversion ratio as a measure of the false-positive rate found in the data set, compared the data to other exomes sequenced in the lab, and compared random variants found in the exome to the Human Genome Browser. These data were used as quality measures to determine the overall integrity of the data set compared to published studies. The end result of this pipeline was a list of quality potential disease-causing variants that were further filtered and prioritized.
The “sequencing analysis pipeline” identified numerous potential disease-causing variants, and on average, an exome from one individual from a first-cousin marriage contains about 26,000 SNPs and 1000 INDELs (from our data sets). Most of the variants identified were known and/or nondeleterious polymorphisms; however, a small number of them represented frameshift, missense, and nonsense mutations in a potential disease-causing gene. Therefore, variants were further filtered with our “potential disease-causing variants filtering and prioritization pipeline.” First, SNPs and INDELs present in stretches of homozygosity, as determined by either linkage analysis or publicly available homozygosity mapping software (Homozygosity Mapper), were isolated (44). Homozygosity mapping in consanguineous families with recessive disease is a proven method for identifying disease-causing mutations given that the DNA sequence flanking the mutation will be preferentially homozygous by descent in children from consanguineous marriages. Next, variants present in coding regions that lead to nonsynonymous amino acid changes, or found in splice sites, were isolated. Only those not found in the homozygous state in dbSNP genotype studies were pursued. The remaining variants were cross compared to the Genome Variation Server (http://gvs.gs.washington.edu/GVS/) and SIFT (http://sift.jcvi.org/) databases to determine single-nucleotide evolutionarily constraint and conservation scores (GERP, PhastCons), protein damage prediction determinations (Polyphen), and relationship of variants to OMIM classifications (45). These measures were used to prioritize variants by high conservation across species, predicted damage to the protein (or causing nonsense-mediated decay), and are not associated with any known, nonrelated diseases. Finally, variants were reviewed for their expression profiles in the brain and their relevance to the neurodevelopmental disorders of interest (Fig. 2). The end result of this pipeline was a list of key potential disease-causing variants that underwent follow-up validation in patients and family members.
Once key potential disease-causing variants were identified, it was then necessary to validate each candidate to determine whether they were the disease-causing mutation in our “potential disease-causing variants validation pipeline.” Specifically, we used direct sequencing to confirm that the polymorphism segregates within the family in a manner that is consistent with recessive inheritance, and we verified that the mutation was not present in a large cohort of healthy, ethnically matched control individuals. Thus, each variant was consistent with variant interpretation category 2 (unreported) and is of the type that is expected to cause the disorder according to the American College of Medical Genetics and Genomics guidelines (1, 2).
## Supplementary Materials
www.sciencetranslationalmedicine.org/cgi/content/full/4/138/138ra78/DC1
Fig. S1. Genetic data for family 890.
Fig. S2. Genetic data for family 951.
Fig. S3. Genetic data for family 1002.
Fig. S4. Genetic data for family 1004.
Fig. S5. Genetic data for family 702.
Fig. S6. Genetic data for family 928.
Fig. S7. Genetic data for family 992.
Fig. S8. Genetic data for family 995.
Fig. S9. Genetic data for family 1409.
Fig. S10. Genetic data for family 1436.
Table S1. Estimated variants to be considered as causative from whole-exome sequencing are greatly reduced in recessive disease with documented consanguinity.
Table S2. Number of variants identified in each family at each step of the variant filtering and prioritization pipeline.
## References and Notes
1. Acknowledgments: We thank the families for their participation. Funding: Supported by Howard Hughes Medical Institute, NIH [National Institute of Neurological Disorders and Stroke grants R01NS041537, R01NS048453, and R01NS052455; National Human Genome Research Institute grants P01HD070494 (to J.G.G.) and U54HG003067 (to S.B.G. and C.R.); National Institute on Alcohol Abuse and Alcoholism/Center for Inherited Disease Research grant N01-HG-65403] for SNP genotyping, and NSF (grants III-081905 and CCF-1115206 to V.B.). Author contributions: T.J.D.-S., J.L.S., S.B., and J.G.G. recruited patients and designed and analyzed the experiments. N.U., J.S., A.E.S., J.O., V.B., A.G.F., G.N., and N.A. generated the bioinformatic pipeline for data analysis. M.S.Z., G.H.A.-S., L.A.M., L.S., S.A.-H., N.M., T.B.-O., N.A.A.-S., F.M.S., F.C., and M.A. identified and recruited families for study and ascertained clinical information. K.J.H. and A.C. assembled clinical data in tabular format. K.V.G., C.S., C.R., and S.B.G. generated sequencing results and provided preliminary analysis. T.J.D.-S. and J.G.G. wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: Data have been deposited into dbGap (phs000288).
View Abstract | 2020-01-21 21:05:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26399415731430054, "perplexity": 8668.553320559366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250605075.24/warc/CC-MAIN-20200121192553-20200121221553-00236.warc.gz"} |
http://annals.math.princeton.edu/articles/11352 | # The André-Oort conjecture for $\mathcal{A}_g$
### Abstract
We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ – the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged’ version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.
## Authors
Jacob Tsimerman
Department of Mathematics, University of Toronto,Toronto, Ontario, Canada M5S 2E4 | 2017-10-17 01:44:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6880037188529968, "perplexity": 591.390470070325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187820556.7/warc/CC-MAIN-20171017013608-20171017033608-00587.warc.gz"} |
https://blog.andersonbanihirwe.dev/posts/2020/advent-of-code-day-1/ | # Advent of Code - Day 1: Report Repair¶
## Part One¶
After saving Christmas five years in a row, you’ve decided to take a vacation at a nice resort on a tropical island. Surely, Christmas will go on without you.
The tropical island has its own currency and is entirely cash-only. The gold coins used there have a little picture of a starfish; the locals just call them stars. None of the currency exchanges seem to have heard of them, but somehow, you’ll need to find fifty of these coins by the time you arrive so you can pay the deposit on your room.
To save your vacation, you need to get all fifty stars by December 25th.
Collect stars by solving puzzles. Two puzzles will be made available on each day in the Advent calendar; the second puzzle is unlocked when you complete the first. Each puzzle grants one star. Good luck!
Before you leave, the Elves in accounting just need you to fix your expense report (your puzzle input); apparently, something isn’t quite adding up.
Specifically, they need you to find the two entries that sum to 2020 and then multiply those two numbers together.
For example, suppose your expense report contained the following:
• 1721
• 979
• 366
• 299
• 675
• 1456
In this list, the two entries that sum to 2020 are 1721 and 299. Multiplying them together produces 1721 * 299 = 514579, so the correct answer is 514579.
Of course, your expense report is much larger. Find the two entries that sum to 2020; what do you get if you multiply them together?
Note
The input data can be found here. :::
with open("../../data/advent-of-code/2020/day-1-input") as fid:
data = sorted([int(x.strip()) for x in data if x != "\n"])
data[:4], len(data)
([61, 156, 166, 279], 200)
### Brute force solution¶
My naive approach to this problem is to use two nested loops.
def pairs_addup(data, target):
"""Find indexes of pairs from data that add to target
Parameters
----------
data : list
Input data
target : int
Returns
-------
n1 : int
n2 : int
"""
n1, n2 = None, None
size = len(data)
for x in range(size):
for y in range(x + 1, size):
if data[x] + data[y] == target:
n1, n2 = data[x], data[y]
break
return n1, n2
a = %timeit -qo pairs_addup(data, 2020)
a
<TimeitResult : 2.07 ms ± 121 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)>
n1, n2 = pairs_addup(data, 2020)
if n1 is not None:
print(f"Two numbers: {n1} and {n2}\nTheir product: {n1 * n2}")
else:
print("Found Nothing....")
Two numbers: 156 and 1864
Their product: 290784
• This yields a solution with $$O(N^2)$$ time complexity in the worst case scenario
Can we do better?
### Improved solution using set logic¶
Rather than using two nested for loops, a much better solution consists of using set logic to find the two numbers that add up to the target:
def pairs_addup(data, target):
rest = set()
n1, n2 = None, None
for num in data:
remain = target - num
if remain in rest:
n1, n2 = num, remain
break
else:
return n1, n2
b = %timeit -qo pairs_addup(data, 2020)
b
<TimeitResult : 21.9 µs ± 316 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)>
pairs_addup(data, 2020)
n1, n2, n1 * n2
(156, 1864, 290784)
Compared to the brute force approach, this is a much faster approach with a considerable speedup:
print(f"{(a.worst / b.worst):.0f}x Speedup")
99x Speedup
• This yields a solution with $$O(N * log(N))$$ time complexity in the worst case scenario. This includes the time complexity for sorting the list (as the list was sorted during the data loading step).
## Part Two¶
The Elves in accounting are thankful for your help; one of them even offers you a starfish coin they had left over from a past vacation. They offer you a second one if you can find three numbers in your expense report that meet the same criteria.
Using the above example again, the three entries that sum to 2020 are 979, 366, and 675. Multiplying them together produces the answer, 241861950.
In your expense report, what is the product of the three entries that sum to 2020?
### Brute force solution¶
A naive/brute force solution consists of using three nested for loops:
def triplets_addup(data, target):
"""Find indexes of triplets from data that add to target
Parameters
----------
data : list
Input data
target : int
Returns
-------
n1 : int
n2 : int
n3 : int
"""
n1, n2, n3 = None, None, None
size = len(data)
for x in range(size):
for y in range(x + 1, size):
for z in range(y + 1, size):
if data[x] + data[y] + data[z] == target:
n1, n2, n3 = data[x], data[y], data[z]
break
return n1, n2, n3
n1, n2, n3 = triplets_addup(data, 2020)
if n1 is not None:
print(
f"Three numbers: {n1} and {n2} and {n3}\nTheir product: {n1 * n2 * n3}"
)
else:
print("Found Nothing....")
Three numbers: 279 and 521 and 1220
Their product: 177337980
• This yields a solution with 𝑂(𝑁^3) time complexity in the worst case scenario
c = %timeit -qo triplets_addup(data, 2020)
c
<TimeitResult : 192 ms ± 10.1 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)>
### Improved solution¶
Rather than using the nested three loops, we can save some work by:
• Looping over our input data,
• Saving the sums of all pairs of numbers in a dictionary with keys being the sum of each pair, and the values consisting of a tuple of numbers in each pair.
• Looping over our input data again
• Checking if target - num is in the dictionary
def triplets_addup(data, target):
two_nums_sum = {}
size = len(data)
for x in range(size):
for y in range(x + 1, size):
two_nums_sum[data[x] + data[y]] = (data[x], data[y])
n1, n2, n3 = None, None, None
for num in data:
remain = target - num
if remain in two_nums_sum:
n1, n2, n3 = num, *two_nums_sum[remain]
break
return n1, n2, n3
n1, n2, n3 = triplets_addup(data, 2020)
n1, n2, n3, n1 * n2 * n3
(279, 521, 1220, 177337980)
d = %timeit -qo triplets_addup(data, 2020)
d
<TimeitResult : 3.91 ms ± 356 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)>
This solution provides a huge improvement over our brute force solution:
print(f"{(c.worst / d.worst):.0f}x Speedup")
47x Speedup
• This yields a solution with 𝑂(𝑁^2) time complexity in the worst case scenario | 2021-01-23 23:14:00 | {"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": 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": 0.2677742540836334, "perplexity": 4267.210774440063}, "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/1610703538741.56/warc/CC-MAIN-20210123222657-20210124012657-00324.warc.gz"} |
https://www.or-exchange.org/questions/3525/estimating-population-proportions-with-simulation-models | # Estimating population proportions with simulation models
1 I'm using a simulation that includes reliability models of different types of entities. I want to estimate the proportion of failures among those entities under certain conditions. The textbook formulas for developing a CI on a population proportion assume that each observation is a binomial random variable and that if the sample is large enough the binomial approximation to the normal suffices to build a CI with a z score and standard deviation based on the binomial distribution. My question is: since i have a simulation model and can perform N independent replications each of n observations, and then have N estimates for the population proportion, would the standard CI formula using the student's t distribution, mean over all N, and empirical estimate of s be more appropriate? or should i stick with the text book method? Thanks, Jon asked 19 Aug '11, 22:28 Jon Davis 156●4 accept rate: 16%
Be the first one to answer this question!
toggle preview community wiki
By Email:
Markdown Basics
• *italic* or _italic_
• **bold** or __bold__
• image?
• numbered list: 1. Foo 2. Bar
• to add a line break simply add two spaces to where you would like the new line to be.
• basic HTML tags are also supported
Tags: | 2019-12-09 07:19:25 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8744770288467407, "perplexity": 1172.833682522227}, "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/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00074.warc.gz"} |
https://www.tutorialspoint.com/Drawing-an-image-from-a-data-URL-to-a-HTML5-canvas | # Drawing an image from a data URL to a HTML5 canvas
HTMLWeb DevelopmentFront End Technology
If you have a data url, you can create an image to a canvas. This can be done as shown in the following code snippet −
var myImg = new Image;
myImg.src = strDataURI;
The drawImage() method draws an image, canvas, or video onto the canvas. The drawImage() method can also draw parts of an image, and/or increase/reduce the image size.
The code given below is also appropriate with the sequence - create the image, set the onload to use the new image, and then set the src −
// load image from data url
Var Obj = new Image();
context.drawImage(myImg, 0, 0);
};
Obj.src = dataURL;
Published on 20-Mar-2018 11:27:00 | 2021-07-30 19:54:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3405165374279022, "perplexity": 4144.569314685517}, "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-2021-31/segments/1627046153980.55/warc/CC-MAIN-20210730185206-20210730215206-00478.warc.gz"} |
https://www.aanda.org/articles/aa/full_html/2012/08/aa19662-12/aa19662-12.html | Subscriber Authentication Point
Menu
Free Access
Issue A&A Volume 544, August 2012 A131 8 Planets and planetary systems https://doi.org/10.1051/0004-6361/201219662 13 August 2012
© ESO, 2012
## 1. Introduction
Direct imaging constitutes an attractive technique for exoplanet detection as it provides straightforward means to characterize planets and their host system (Absil & Mawet 2010) through, e.g., orbital motion (Soummer et al. 2011; Chauvin et al. 2012), spectro-photometry of planetary atmospheres (Janson et al. 2010; Galicher et al. 2011; Bonnefoy et al. 2011), or planet-disk interactions (Lagrange et al. 2012). Direct imaging has also the potential of understanding and bridging the gap between the population of extremely close planets discovered by radial velocity or transit techniques and the free floating planets discovered by microlensing observations (Sumi et al. 2011; Quanz et al. 2012). Indeed, many exoplanet candidates directly imaged so far have projected distances up to several hundreds of AU. On the other hand, some free floating low-mass objects have been found to be kinematically associated at projected distances of thousands of AU (Caballero et al. 2006). This raises the questions of their formation and the very definition of planets, on which direct imaging is key to shed more light.
However, imaging extra-solar planets around other stars constitutes a multiple challenge, and the practical hurdles are numerous. First of all, the angular separation between planets and stars is very small (e.g. < 500 mas for a 5-AU distance at 10 pc), usually requiring diffraction limited capabilities on 8-meter class telescopes1. Second, the contrast between a planet and its host star ranges from ≃ 10-3 for hot giant planets in the infrared to ≃ 10-10 for Earth-like planets in the visible. The contrast issue requires exquisite image (hence wavefront) quality to feed coronagraphic devices, most of the time very specialized observing strategies (e.g., angular differential imaging or ADI, Marois et al. 2006), and corresponding data reduction techniques such as the locally optimized combination of images (LOCI, Lafrenière et al. 2007).
Once a faint point source has been detected around a star, pointing to the potential discovery of a companion candidate, precise differential astrometric monitoring of the latter needs to be carried over a sufficiently long time so that the stellar proper motion overcomes the astrometric precision of the detected object by a sufficient margin (if the object is bound, it moves with its host star). Neuhäuser & Schmidt (2012) also argues that a spectrum, when possible (proximity to the host star often prevents to take clean uncontaminated spectra), can determine the spectral type and temperature of the companion, and thus indicates a planetary mass or sub-stellar body, but still possibly a cool background object. Both tests might sometimes be necessary, especially when targeting young associations where objects can potentially share common proper motion, likely to be small and rather uncertain (at the distance of star forming regions), making this astrometric process more difficult and the required time baseline longer. The T Tauri star ScoPMS 214 is a typical example, where a candidate companion was shown to share common proper motion, but was spectroscopically identified as a foreground M dwarf (Metchev & Hillenbrand 2009). In young associations, the probability for small and/or shared proper motion is thus significant. A third possibility for confirming the bound character of the companion is the detection of the orbital motion, but that implies that the candidate is on a reasonably tight orbit (period <1000 years), in order to be sampled with sufficient accuracy over a time baseline of a few years.
Most of the objects imaged so far are orbiting young stars (see exoplanet.eu for a thorough and up-to-date list, and Neuhäuser & Schmidt (2012) for a recent detailed review). Youth is the current bias of high contrast imaging, as short period, inclination or distances (orbital and/or parallactic) are the biases of radial velocity, transit and micro-lensing techniques, respectively. Indeed, the thermal radiation of young exoplanets peaks in the near-infrared, making them more easily detectable by several orders of magnitude than if we were to observe them in reflected light in the visible. Since the detected emission comes from the intrinsic thermal radiation of the planet, its physical properties (temperature, mass and radius) can only be inferred based on cooling track models, which critically depend on age and formation mechanisms/history (Allard et al. 2003; Marley et al. 2007; Fortney et al. 2008; Spiegel & Burrows 2012). Due to this very high sensitivity to initial conditions, deriving the nature of low-mass objects and young planets is indeed more problematic than for older ones, especially for long-period companions where the dynamical mass is more difficult to infer. For instance, Stassun et al. (2006) found a young eclipsing brown-dwarf binary in which the cooler object is the more massive one, which is very surprising, because most theoretical models predict that a brown dwarf of a given mass will at all times be warmer than a lower-mass brown dwarf of the same age. Stassun et al. (2008) also found a binary where both stars have the same mass within 2% but their surface temperatures differ by 300 K. Therefore, dynamical mass, hence astrometric precision and proper motion knowledge, age, together with distance, become key parameters that will determine the final precision and confidence on the companion physical characteristics.
In the present paper, we discuss these issues in details, and illustrate them with our search for planetary candidates around the young T Tauri star IM Lup. The paper is organized as follows: after presenting the IM Lup system in Sect. 2, we describe our VLT/NACO 2008 observations and discovery of a putative candidate companion in Sect. 3. Then, in Sect. 4, we present our re-analysis of the pre-discovery data taken with HST/NICMOS in 2005, followed by epoch 3 and 4 images taken with VLT/NACO in 2010 and 2011 in Sect. 5. In Sect. 6, we discuss our discovery in details and underline the difficulties of planet searches around this type of objects, in terms of astrometry and related proper motion analysis. We also establish benchmark detection limits around IM Lup after reestablishing its age (Sect. 7), before concluding in Sect. 8.
Table 1
Fundamental properties of IM Lup and associated optically thick circumstellar disk.
## 2. IM Lup: a young T Tauri star with a massive circumstellar disk
IM Lup (Table 1) is a young M0 (T = 3900 K) T Tauri star (TTS) with an equivalent width of the Hα emission known to vary from 7.5 to 21.5 Å, confirming its status as a borderline weak-line/classical TTS. Part of the Lupus association (140−190 pc, see Sect. 7), it is one of four young stellar objects in the small 13CO(1−0) Lupus 2 core near the extreme T Tauri star RU Lup (Tachihara et al. 1996). Our age estimate described in Sect. 7 yields 0.5−1.75 Myr.
Table 2
Observing log for IM Lup and the reference stars used in this work.
Despite the low accretion-related activity of IM Lup (Reipurth et al. 1996; Wichmann et al. 1999), long wavelength observations from the millimeter (Nuernberger et al. 1997; van Kempen et al. 2007; Lommen et al. 2007) to the infrared (Padgett et al. 2006) reveal ample evidence for gas-rich circumstellar material in the system. IM Lup’s protoplanetary disk scattered light was imaged in 1999 in the visible with HST/WFPC2 (PI: Stapelfeldt, Prog. ID 7387). It was followed in the near-infrared by HST/NICMOS images obtained in 2005 (PI: G. Schneider, Prog. ID 10177). An extensive modeling study of the IM Lup disk was performed by Pinte et al. (2008), using multi-wavelength spectro-photometry and images in a global fit with a 3D radiative transfer model, and led to quantitative evidence for dust processing and evolution in the disk.
A more recent paper by Panić et al. (2009) presents Submillimeter Array (SMA) observations showing a break in the gas and dust surface density of the IM Lup disk, seen to extend much further than the 400 AU outer edge determined by Pinte et al. (2008). One of the proposed explanations for the break is a companion body near the break at 400 AU. Indeed, a companion of 1 MJup could open a gap in the disk and affect its spreading. Panić et al. (2009) however argue that no candidate companion at this separation is visible in the HST image of Pinte et al. (2008).
## 3. Discovery of a candidate companion with VLT/NACO in 2008
As part of a coronagraphic study of young stars (prog. ID 380.C-0910(A), PI: Mawet), we observed IM Lup in March 2008 with NAOS-CONICA, the adaptive optics (Nasmyth Adaptive Optics System) and near-infrared spectrograph and imager of the Very Large Telescope (VLT).
### 3.1. Observing strategy for the 2008 discovery data set
For our discovery image in 2008, we used the four-quadrant phase-mask (FQPM) coronagraph (Rouan et al. 2007) in the Ks band. The FQPM is a phase-mask coronagraph applying a π phase shift between adjacent quadrants. The starlight, when centered on the FQPM cross-hair, undergoes a destructive interference upon propagation to the downstream pupil plane.
All of our frames for 2008 were taken with the Ks filter and the S13 camera (13.27 mas/pixel). This fine sampling (4 pixels per resolution element λ/d, where λ is the observing wavelength and d the telescope diameter) was necessary to center the target star on the FQPM cross-hair precisely. The main calibrator stars were carefully selected to present roughly the same V − K color as the target (Table 2). Matching the V magnitudes is important to ensure similar AO corrections between the target and reference stars, as the visible wavefront sensor of NAOS is mostly sensitive at V and R. Ks magnitudes have also to be matched to ensure SNR matching for the quasi-static speckles. Also, to avoid flexure-induced semi-static speckle variations as much as possible and to ensure a consistent telescope orientation with respect to the instrument between the target and the reference, the calibration stars were chosen and observed at the same parallactic angle as the target star. This condition was met on a best effort basis since the availability of a suitable reference fulfilling the set of constraints is never guaranteed, which is one of the drawbacks of the reference star differential imaging strategy (RDI).
The observing conditions for IM Lup and the reference stars were very good with a visible seeing between and . The total integration time was about 1350 s and 1200 s for each target (see Table 2). Respecting the consideration discussed here above to calibrate time-dependent PSF variations (speckle), we acquired coronagraphic images of the reference star 90 min after the science target at roughly the same parallactic angle. To reduce drift and pupil rotation, the centering was checked and corrected every 80 s.
Fig. 1 The upper strip presents the simple reference star subtracted Ks-band image (left) and the damped-LOCI processed image (middle) obtained with NACO in 2008, which show a very faint off-axis point source (mKs ≃ 19.1) to the northeast of IM Lup. The upper right image presents the 2005 F160W NICMOS observations, using an iterative roll-subtraction technique. The lower strip presents the NACO Lp-band image of 2010 (left), the 2011 NACO H-band image (middle), and the 2011 NACO Ks-band image (right). The off-axis point source is detected in all images (identified with an arrow). Open with DEXTER
### 3.2. Data analysis for the 2008 discovery data set
The data reduction proceeds as follows. NACO coronagraphic acquisition template moves the telescope alternatively between a fixed object position and a jittered set of sky positions which are median combined and subtracted to the object, removing the background and dark contributions at the same time. The normalization of the resulting image with the flat provides the first stage of data processing. However, this treatment does not completely remove the electronic noise of the readout process nor the two electronic ghosts which are characteristic of CONICA.
The second stage of the data reduction process consists in co-adding of the images with a sub-pixel centering procedure. For that, we applied a hybrid method which correlates the centroid of the unsaturated coronagraphic pattern with a pre-computed look-up table based on a diffraction model of the FQPM. Using this sophisticated method, we routinely achieve a centering precision of σ = 0.1 pixel or 1.4 mas rms. Despite the coronagraph starlight 10-fold attenuation, scattered starlight still dominates the extended source flux. Since the primary objective of our original program was disk imaging, and given the relative novelty of the ADI technique at the time of the first observations, and the fact that ADI is not an optimal strategy for nearly face-on disk (Milli et al. 2012), we chose to use classical RDI.
The final VLT/NACO 2008 image (see Fig. 1, top middle) was obtained using an enhanced version of the LOCI algorithm (Lafrenière et al. 2007). In short, LOCI finds the optimal linear combination of reference frames (here from the two reference stars CD-37 8989 and CD-35 9033) to minimize the noise in a given zone of the target image. The process is repeated until the area of interest in the target image is completely reduced. LOCI in its original form was conceived to find point sources, and has a known tendency to attenuate signal from extended sources such as circumstellar disks. However, this defect of the generic LOCI algorithm can be brought under control by a fine tuning of the geometrical parameters such as the size of the optimization zone, the number of reference frames used in the correlation matrix, as well as the introduction of a damping parameter (Lagrange multiplier) to balance flux conservation with noise attenuation as in Pueyo et al. (2012).
Table 3
Astrometry of the point source to the NE, along with its relative and absolute photometry.
### 3.3. Candidate companion and new image of the disk
In our 2008 Ks-band dataset (Table 2), we detected a very faint companion at a signal-to-noise ratio (SNR) of ≃ 10 (Fig. 1, top left). Applying the damped-LOCI of Pueyo et al. (2012), both the companion and the disk SW arc are detected simultaneously with minimum contamination from starlight scattering (Fig. 1, top middle). For the first time, we note that damped-LOCI, originally invented to detect point-source in multi-spectral data, can be successfully applied to the detection of circumstellar disks, improving upon the original LOCI of Lafrenière et al. (2007).
We performed relative astrometry on the final reduced image, using centroiding and a specific pre-computed look-up table for the star position behind the FQPM (see Sect. 3.2), and gaussian fitting for the candidate companion (with a subpixel precision). The companion is located to the northeast of IM Lup, at a radius of ≃ , and a position angle (PA) of ≃ 58° (see Table 3).
We then performed aperture photometry using the function APER of IDL, and found a relative Ks magnitude of . Naively assuming that the point source is physically associated to IM Lup (Sect. 6), this corresponds to an absolute Ks magnitude of , where the uncertainty is mostly due to the poor knowledge of the star distance (140–190 pc, see Sect. 7).
Note that the disk is detected as an arc to the SW, extending up to ≃ along the major axis, and ≃ along the minor axis, consistent with previous HST observations. We also measure a position angle of 140° ± 10° for the major axis, which is consistent with the value reported in Pinte et al. (2008).
## 4. Pre-imaging of the candidate companion with HST/NICMOS in 2005
Following the NACO discovery, we retrieved the archival HST/NICMOS data of IM Lup obtained in 2005 in the F160W filter (PI: G. Schneider, Prog. ID 10177). The reason for the non-detection of the companion by successive groups and notably by Pinte et al. (2008) is rather simple: they primarily aimed at characterizing the extended structure, and applied corresponding soft data reduction techniques (single reference star subtraction).
Taking advantage of the availability of two images taken at 2 different roll angles separated by 30 degrees, we used the iterative roll-subtraction algorithm introduced in Heap et al. (2000), and described in Krist et al. (2010). A simple roll-subtracted image contains a positive image of a companion and a negative one rotated by an angle equal to the difference between the telescope orientations. Ideally, these two would be combined to form a single, positive image. Instead of directly subtracting images from the two orientations and then trying to combine the results, an iterative technique can be used that solves for those portions of the two unsubtracted images that appear static on the detector (i.e., the PSF) and those that appear to rotate as the telescope rolls (i.e., the sky, including any companion or disk).
This method, which has been successfully used for β Pictoris (Heap et al. 2000) and HD 207129 (Krist et al. 2010), clearly ( ≃ 3σ) reveals the companion in the NICMOS data as well (Fig. 1, top right). The position of the star behind the opaque Lyot coronagraph of NICMOS was determined by two different methods. The first position was obtained by iteratively shifting a Tiny Tim model (Krist 1993) PSF by subpixel amounts until the residuals were visibly minimized. The second position estimate was obtained by measuring the intersection of the diffraction spikes of the secondary mirror support structures. Both methods yield similar result within half a pixel, which we considered as a systematic error folded in the error bar calculation (Table 3).
## 5. Epoch 3 (2010) and 4 (2011) with VLT/NACO
Given the insufficient time baseline and astrometric precision of the NICMOS data point, we waited for a few years and re-observed IM Lup in 2010 (Lp band) and in 2011 (H and Ks band), using NACO again (Table 2). The Lp-band data only marginally (SNR ≃ 2) shows the companion (Fig. 1, bottom left), with an estimated Lp magnitude of . On the other hand, it is easily detected in H and Ks bands in the 2011 data (Fig. 1, bottom middle and right). The strategy we chose for the most recent data set was to perform simple saturated-unsaturated imaging in order to enable precise astrometric and photometric analysis.
Results of the astrometry of the candidate companion relative to the host star are presented in Table 3. Note that the relative astrometry is somewhat different between both filters. Slight differences are expected, due for instance to differential aberrations between filters, and the difference in data quality (the Strehl ratio is naturally lower in the H band). However, it appears that the H-band astrometry is affected by the presence of the diffraction pattern of the telescope spiders (secondary mirror support structure), close to the point source. For these reasons, we only retained the Ks band astrometry in our final proper motion analysis of Sect. 6.1, which also corresponds to the filter of the plate scale calibration described below.
## 6. Discussion
In this section, we elaborate on the difficulty of exoplanet candidate confirmation and characterization for young distant stars.
### 6.1. Astrometry and proper motion
To test the hypothesis of an object linked to IM Lup we can use astrometry. If the companion is linked to IM Lup, no significant variation of their angular separation should be observed along time due to proper motion, since they will be co-moving. The only cause of variation would result from the orbital motion of the companion around the center of mass of the system. The apparent angular separation of ≃ corresponds to a deprojected physical separation of about 350–480 AU at 140–190 pc. We can expect a planet orbiting at such a distance from its central star to have extremely long periods (several thousand years at the minimum) so that its orbital motion is not detectable in our astrometric observations spread over 7 years.
#### 6.1.1. Prior astrometric calibration of CONICA
To calibrate the NACO plate scale and detector orientation in a consistent and precise way between the 2008 and 2011 epochs, we used the star clusters Theta Orionis/“Trapezium” (2008) and Omega Centauri (2011). The reference positions of the stars in the each cluster were derived in a different program (Montagnier et al., in prep.). For each cluster, many images were taken at various positions and orientations to establish a distortion solution of the NACO plate-scale (see the method described by Anderson & King 2003). The linear terms of the distortion (detector axes orientations and pixel dimensions) were then derived by observing appulses of transneptunian objects. Finally, the star positions were derived with an accuracy better than 1 mas (about 200 object for the Omega Centauri cluster field, and about 50 for the Theta Orionis cluster). On the calibration images of the 2 epochs needed in our astrometric analysis, the position of the centroid (sx,sy) of each non-saturated star (about 20 stars in the Theta Orionis’s 2008 epoch, and about 40 for the Omega Centauri’s 2011 epoch) on the reduced image of the field is measured in pixels; these values are then compared to the position on the sky (ρx,ρy) in arcseconds with the following equations: where \begin{lxirformule}$p_x$\end{lxirformule} (py) is the plate scale along the x- (y-) axis, θx (θy) are the orientations of the detector on sky along the x- (y-) axis, and x0 and y0 are offsets giving the correspondence between the absolute positions (it is only used to solve the equation). A Levenberg-Marquardt minimization is then applied to find the plate scale solution. Using this calibration method, the final precision on the plate scale is ≃ 50 μas, and ± 0.05° on the detector orientation. Note that this accuracy is only one item in the error budget of the final astrometric precision, which depends on many other terms all summed quadratically, such as the precision of the star position determination, fit of the off-axis location (itself dependent on the SNR), etc.
#### 6.1.2. Proper motion of IM Lup
We have considered the proper motions of IM Lup available in the literature (see Table 4) to test our hypothesis of co-moving objects. We notice that the measured proper motions vary in a large range from one author to the other. The putative binary nature of IM Lup wrongly reported by Hipparcos (Wichmann et al. 1998; Lasker et al. 1996; Kharchenko & Roeser 2009) indicates a potential disturbance in the measurements that led to a poor astrometry. Indeed, our observations do not reveal the second component down to a magnitude MKs ≃ 19, indicating that the Hipparcos detection, and other reports of the binarity might have been potentially contaminated by the presence of the optically thick circumstellar disk of IM Lup, as already suggested in Pinte et al. (2008).
Other authors have measured the proper motion of this unresolved object with unequal precisions, leading to a consensual value in declination (~22.4 mas/yr) while the proper motion in right ascension varies largely between authors (from –3 to –15.5 mas/yr for values with reasonable precisions). The origin of such discrepancies is difficult to pinpoint since proper motion quality is not only related to the time base but also to the number of different epochs of observation and evidently to the quality of each epoch measurement. In the case of pre-main sequence stars the situation is even more difficult since depending on the target, the object may be embedded into a dust and/or gas cloud (which is the case for IM Lup) perturbing the photo-center measurement. Moreover the morphology of the cloud may vary with time and lead to variable photo-centers at different epochs. In the data presented in Table 4, we chose to adopt the third one (PMS), but the four last values may be considered (PMS, SPM4, PPMXL and UCAC3) for the astrometric test of co-moving objects.
Table 4
Proper motions for IM Lup found in major astrometric catalogues.
#### 6.1.3. Bound or not bound?
With these proper motions of IM Lup, we would expect a background source to have moved by 61 ± 9 mas with respect to IM Lup between the two observations (NICMOS-2005 and NACO-2008). Such a motion is not detected within our error bars, meaning to first order that the companion is likely co-moving with IM Lup (Fig. 2, left). However, the average SNR on the NICMOS detection, the very slow proper motion and the large astrometric uncertainty mentioned above, do not allow us to firmly and definitely conclude on its bound character. Note that a galactic starcount model for the direction toward IM Lup (Girardi et al. 2005) yields a surface density of stars with 19 < Ks < 19.5 of ≃ 2.2 × 105 per square degree. This makes the chance of a random background source being located within of IM Lup ≃ 17%.
Summer 2011 was the first opportunity to firmly get closure on the bound aspect of this discovered candidate. We then used the epoch 4 NACO observation to redo the common proper motion astrometric analysis. This time, since the analysis is based on a single, well calibrated instrument, our astrometric precision can be trusted down to a conservative ≃ 10 mas per coordinate. With a time baseline of 1210 days, the background object should have moved by 68 ± 10 mas with respect to IM Lup, which is about the observed variation of separation (74 ± 20 mas) in the same direction. We conclude that the candidate companion is likely to be a background object, and is therefore not associated with IM Lup (Fig. 2, right).
Fig. 2 Left: astrometric analysis of the candidate companion proper motion between 2005 and 2008, using NICMOS and NACO. Right: astrometric analysis of the candidate companion proper motion between 2008 and 2011 (and the marginal Lp-band detection of 2010), using NACO only. The continuous line represents the combined parallactic motion and the proper motion of the background object in the reference frame of the primary target. The filled dots represent the positions of the candidate companion with error bars at the different epochs while the empty dots (“IF BKGD”) represent the positions of a putative background object at the same epochs. Open with DEXTER
### 6.2. Probable nature of the point source
Based on the combined H, Ks, Lp photometry, we analyzed the SED of the likely background object to verify that it is consistent with a blackbody. For that, we first checked that the extinction in the direction of IM Lup (≃25° from the galactic center bulge) is very small and can indeed be neglected in the near-infrared: AH ≃ 0.3, AK ≃ 0.2, and ALp ≃ 0.1 (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). The SED would be compatible with many possible stellar objects. For instance, the fit to a 3000 K blackbody is satisfying, with the Lp-band point falling only a little more than one sigma above the model. Any blackbody warmer than about 3000 K would actually fit the SED in almost the same way. The reduced χ2 of the fit is ≃2 when both the size and temperature of the blackbody are simultaneously fitted. If the background object is actually an M5V star, it would be located far away in our galaxy ( ≃ 4 kpc).
Finally, confirming the stellar nature of the point source, we verified that the H − Ks color derived from our data is not consistent with usual evolutionary models for planetary-mass objects (Baraffe et al. 2003; Fortney et al. 2008), which are generally much redder. Note that the Ks − Lp color of ≃ 1.5–2 we measured is marginally consistent with both hot start and core accretion models, but the error bars are such that the Lp photometric point is not significant, hence constraining.
## 7. Age, distance, and detection limits
Preliminary age estimates for IM Lup range from 0.1 Myr to 10 Myr (Hughes et al. 1994). To reduce the uncertainty associated with this large range, we re-estimated the age of IM Lup as follows. We placed the object on a Hertzsprung-Russell diagram (HRD). The effective temperature was given by the spectral type (conversion from Luhman et al. 2003, for M dwarfs). I and J magnitudes, which are not too much affected by accretion nor disk emission, were converted to bolometric magnitudes based on bolometric corrections and intrinsic colors of Kenyon & Hartmann (1995).
The bolometric luminosity was then deduced using an estimated distance most probably comprised between 140 pc (Hughes et al. 1994) and 190 pc (Wichmann et al. 1998)2, and corrected for extinction using an Av = 0.5 (Pinte et al. 2008) with the law presented in Draine (2003). We then used evolutionary models from Baraffe et al. (1998) and Siess et al. (2000) to draw isochrones and evolutionary tracks in the HRDs and to interpolate for the observed object. The age estimation was performed independently for I and J photometry, and then folded into error bars. We arrived at the following estimates: for a distance of 140 pc, we get an age range of 0.8–1.75 Myr, while for a distance of 190 pc, we get 0.5–1 Myr.
Our 2008 FQPM data set taken in the Ks band features the best contrast ever achieved around IM Lup. This data set is therefore suitable to derive detection limits for IM Lup (Fig. 3). We proceeded as follows. For increasing angular separations, we derived the standard deviation in annuli 1 resolution element wide. This profile was then multiplied by 5 to derive the 5σ detection limit associated with the corresponding data set.
In Fig. 3, we also overplot the level of contrast for two planet masses: 1, 2 MJup, assuming a median age of ≃ 1 Myr and considering the COND03 model from Baraffe et al. (2003). Thanks to the young age of IM Lup, our (model-dependent) detectability limits are excellent, down to less than one Jupiter mass beyond one arcsecond, and less than two Jupiter masses beyond . Note that the “core accretion” model of Fortney et al. (2008) yields much higher masses, reflecting the large uncertainties still plaguing evolutionary models for early ages (Marley et al. 2007).
Fig. 3 5σ detection limit around IM Lup, derived from the Ks-band 2008 data, assuming an age of ≃ 1 Myr and the COND03 model from Baraffe et al. (2003). The dashed curves show the limits for distances of 140 and 190 pc, respectively. The dashed lines shows the level of contrast for two planet masses: 1, 2 MJup. The black circle and associated error bar shows the point-source Ks band absolute magnitude if associated. Open with DEXTER
## 8. Conclusion
This paper presented a planet search we conducted with VLT/NACO around the young T Tauri star IM Lup between 2008 and 2011, using a pre-discovery image obtained with HST/NICMOS in 2005. IM Lup is the perfect prototype system for planet search since it has a massive optically thick circumstellar disk, likely at the stability limit. It also features a break in the gas and dust density at about 400 AU, which could indicate the presence of a Jupiter-mass body at the location of the discontinuity. A candidate companion was detected by NACO in 2008, and also seen in the 2005 HST/NICMOS data.
The candidate companion is located to the northeast of IM Lup, at a radius of ≃ , and a position angle (PA) of ≃ 58°. Tentatively and naively assuming association, this corresponds to a de-projected physical separation of about 350–480 AU at 140–190 pc. With our redetermined age of about 1 Myr, the mass of the putative off-axis companion using the usual “hot start” evolutionary models (Baraffe et al. 2003; Fortney et al. 2008) would be between 1–2 MJup.
However, and unfortunately, the candidate was later on proven to be a background object based on the NACO 2011 observations, a common proper motion analysis and a careful calibration of the NACO plate scale and detector orientation. This cautionary tale taught us the difficulty of planet search around young, distant and obscured stars, where proper motion might not be very well constrained, and where the age and distance determinations are tricky.
1
Note the exception presented in Serabyn et al. (2010), who showed a snapshot of 3 out of the 4 planets of HR 8799 taken with an adaptively-corrected 1.5-m telescope and a next-generation vector vortex phase mask coronagraph (Mawet et al. 2010).
2
Recently confirmed by Galli et al. (in prep.), who measured a kinematic distance of 179 pc for IM Lup.
## Acknowledgments
This work was carried out at the European Southern Observatory (ESO) site of Vitacura (Santiago, Chile), and the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech), under contract with the National Aeronautics and Space Administration (NASA). O.A. and J.S. acknowledge support from the Communauté française de Belgique – Actions de recherche concertées – Académie universitaire Wallonie-Europe. This research has made use of the NASA/IPAC/NExScI Star and Exoplanet Database, which is operated by the JPL, Caltech, under contract with NASA, and NASA’s Astrophysics Data System and of the SIMBAD database, operated at CDS (Strasbourg, France).
## References
1. Absil, O., & Mawet, D. 2010, A&ARv, 18, 317 [NASA ADS] [CrossRef] [Google Scholar]
2. Allard, N. F., Allard, F., Hauschildt, P. H., Kielkopf, J. F., & Machin, L. 2003, A&A, 411, L473 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
3. Anderson, J., & King, I. R. 2003, PASP, 115, 113 [NASA ADS] [CrossRef] [Google Scholar]
4. Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A, 337, 403 [NASA ADS] [Google Scholar]
5. Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, A&A, 402, 701 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
6. Bonnefoy, M., Lagrange, A.-M., Boccaletti, A., et al. 2011, A&A, 528, L15 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
7. Caballero, J. A., Martín, E. L., Dobbie, P. D., Barrado, Y., & Navascués, D. 2006, A&A, 460, 635 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
8. Chauvin, G., Lagrange, A.-M., Beust, H., et al. 2012, A&A, 542, A41 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
9. Draine, B. T. 2003, ARA&A, 41, 241 [NASA ADS] [CrossRef] [Google Scholar]
10. Ducourant, C., Teixeira, R., Périé, J. P., et al. 2005, A&A, 438, 769 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
11. Fortney, J. J., Marley, M. S., Saumon, D., & Lodders, K. 2008, ApJ, 683, 1104 [NASA ADS] [CrossRef] [Google Scholar]
12. Galicher, R., Marois, C., Macintosh, B., Barman, T., & Konopacky, Q. 2011, ApJ, 739, L41 [NASA ADS] [CrossRef] [Google Scholar]
13. Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & da Costa, L. 2005, A&A, 436, 895 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
14. Heap, S. R., Lindler, D. J., Lanz, T. M., et al. 2000, ApJ, 539, 435 [NASA ADS] [CrossRef] [Google Scholar]
15. Hughes, J., Hartigan, P., Krautter, J., & Kelemen, J. 1994, AJ, 108, 1071 [NASA ADS] [CrossRef] [Google Scholar]
16. Janson, M., Bergfors, C., Goto, M., Brandner, W., & Lafrenière, D. 2010, ApJ, 710, L35 [NASA ADS] [CrossRef] [Google Scholar]
17. Kenyon, S. J., & Hartmann, L. 1995, ApJS, 101, 117 [NASA ADS] [CrossRef] [Google Scholar]
18. Kharchenko, N. V., & Roeser, S. 2009, VizieR Online Data Catalog, I/280B [Google Scholar]
19. Krist, J. 1993, in Astronomical Data Analysis Software and Systems II, eds. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, ASP Conf. Ser., 52, 536 [Google Scholar]
20. Krist, J. E., Stapelfeldt, K. R., Bryden, G., et al. 2010, AJ, 140, 1051 [NASA ADS] [CrossRef] [Google Scholar]
21. Lafrenière, D., Marois, C., Doyon, R., Nadeau, D., & Artigau, É. 2007, ApJ, 660, 770 [NASA ADS] [CrossRef] [Google Scholar]
22. Lagrange, A.-M., Boccaletti, A., Milli, J., et al. 2012, A&A, 542, A40 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
23. Lasker, B. M., Sturch, C. R., Lopez, C., et al. 1996, VizieR Online Data Catalog, II/143A [Google Scholar]
24. Lommen, D., Wright, C. M., Maddison, S. T., et al. 2007, A&A, 462, 211 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
25. Luhman, K. L., Briceño, C., Stauffer, J. R., et al. 2003, ApJ, 590, 348 [NASA ADS] [CrossRef] [Google Scholar]
26. Marley, M. S., Fortney, J. J., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2007, ApJ, 655, 541 [NASA ADS] [CrossRef] [Google Scholar]
27. Marois, C., Lafrenière, D., Doyon, R., Macintosh, B., & Nadeau, D. 2006, ApJ, 641, 556 [NASA ADS] [CrossRef] [Google Scholar]
28. Mawet, D., Serabyn, E., Liewer, K., et al. 2010, ApJ, 709, 53 [NASA ADS] [CrossRef] [Google Scholar]
29. Metchev, S. A., & Hillenbrand, L. A. 2009, ApJS, 181, 62 [NASA ADS] [CrossRef] [Google Scholar]
30. Milli, J., Mouillet, D., Lagrange, A.-M., et al. 2012, A&A, in press, DOI: 10.1051/0004-6361/201219687 [Google Scholar]
31. Neuhäuser, R., & Schmidt, T. O. B. 2012 [arXiv:1201.3537] [Google Scholar]
32. Nuernberger, D., Chini, R., & Zinnecker, H. 1997, A&A, 324, 1036 [NASA ADS] [Google Scholar]
33. Padgett, D. L., Cieza, L., Stapelfeldt, K. R., et al. 2006, ApJ, 645, 1283 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
34. Panić, O., Hogerheijde, M. R., Wilner, D., & Qi, C. 2009, A&A, 501, 269 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
35. Pinte, C., Padgett, D. L., Ménard, F., et al. 2008, A&A, 489, 633 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
36. Pueyo, L., Crepp, J. R., Vasisht, G., et al. 2012, ApJS, 199, 6 [NASA ADS] [CrossRef] [Google Scholar]
37. Quanz, S. P., Lafrenière, D., Meyer, M. R., Reggiani, M. M., & Buenzli, E. 2012, A&A, 541, A133 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
38. Reipurth, B., Pedrosa, A., & Lago, M. T. V. T. 1996, A&AS, 120, 229 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
39. Roeser, S., Demleitner, M., & Schilbach, E. 2010, AJ, 139, 2440 [NASA ADS] [CrossRef] [Google Scholar]
40. Rouan, D., Baudrand, J., Boccaletti, A., et al. 2007, C. R. Phys., 8, 298 [NASA ADS] [CrossRef] [Google Scholar]
41. Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103 [NASA ADS] [CrossRef] [Google Scholar]
42. Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 [NASA ADS] [CrossRef] [Google Scholar]
43. Serabyn, E., Mawet, D., & Burruss, R. 2010, Nature, 464, 1018 [NASA ADS] [CrossRef] [Google Scholar]
44. Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593 [NASA ADS] [Google Scholar]
45. Soummer, R., Brendan Hagan, J., Pueyo, L., et al. 2011, ApJ, 741, 55 [NASA ADS] [CrossRef] [Google Scholar]
46. Spiegel, D. S., & Burrows, A. 2012, ApJ, 745, 174 [NASA ADS] [CrossRef] [Google Scholar]
47. Stassun, K. G., Mathieu, R. D., & Valenti, J. A. 2006, Nature, 440, 311 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
48. Stassun, K. G., Mathieu, R. D., Cargile, P. A., et al. 2008, Nature, 453, 1079 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
49. Sumi, T., Kamiya, K., Bennett, D. P., et al. 2011, Nature, 473, 349 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
50. Tachihara, K., Dobashi, K., Mizuno, A., Ogawa, H., & Fukui, Y. 1996, PASJ, 48, 489 [NASA ADS] [CrossRef] [Google Scholar]
51. van Kempen, T. A., van Dishoeck, E. F., Brinch, C., & Hogerheijde, M. R. 2007, A&A, 461, 983 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
52. van Leeuwen, F. 2007, A&A, 474, 653 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
53. Wichmann, R., Bastian, U., Krautter, J., Jankovics, I., & Rucinski, S. M. 1998, MNRAS, 301, L39 [NASA ADS] [CrossRef] [Google Scholar]
54. Wichmann, R., Covino, E., Alcalá, J. M., et al. 1999, MNRAS, 307, 909 [NASA ADS] [CrossRef] [Google Scholar]
55. Zacharias, N., Finch, C., Girard, T., et al. 2010, AJ, 139, 2184 [NASA ADS] [CrossRef] [Google Scholar]
## All Tables
Table 1
Fundamental properties of IM Lup and associated optically thick circumstellar disk.
Table 2
Observing log for IM Lup and the reference stars used in this work.
Table 3
Astrometry of the point source to the NE, along with its relative and absolute photometry.
Table 4
Proper motions for IM Lup found in major astrometric catalogues.
## All Figures
Fig. 1 The upper strip presents the simple reference star subtracted Ks-band image (left) and the damped-LOCI processed image (middle) obtained with NACO in 2008, which show a very faint off-axis point source (mKs ≃ 19.1) to the northeast of IM Lup. The upper right image presents the 2005 F160W NICMOS observations, using an iterative roll-subtraction technique. The lower strip presents the NACO Lp-band image of 2010 (left), the 2011 NACO H-band image (middle), and the 2011 NACO Ks-band image (right). The off-axis point source is detected in all images (identified with an arrow). Open with DEXTER In the text
Fig. 2 Left: astrometric analysis of the candidate companion proper motion between 2005 and 2008, using NICMOS and NACO. Right: astrometric analysis of the candidate companion proper motion between 2008 and 2011 (and the marginal Lp-band detection of 2010), using NACO only. The continuous line represents the combined parallactic motion and the proper motion of the background object in the reference frame of the primary target. The filled dots represent the positions of the candidate companion with error bars at the different epochs while the empty dots (“IF BKGD”) represent the positions of a putative background object at the same epochs. Open with DEXTER In the text
Fig. 3 5σ detection limit around IM Lup, derived from the Ks-band 2008 data, assuming an age of ≃ 1 Myr and the COND03 model from Baraffe et al. (2003). The dashed curves show the limits for distances of 140 and 190 pc, respectively. The dashed lines shows the level of contrast for two planet masses: 1, 2 MJup. The black circle and associated error bar shows the point-source Ks band absolute magnitude if associated. Open with DEXTER In the text
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while. | 2021-04-23 15:20:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5964590311050415, "perplexity": 4028.3869450913635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594808.94/warc/CC-MAIN-20210423131042-20210423161042-00135.warc.gz"} |
https://codegolf.stackexchange.com/tags/perl/hot | # Tag Info
30
eval eval '"'. ('['^"\+").( ('[')^ ')').(''|')'). (''|'.'). ('['^'/').('{'^'[' ).'\\'.'"' .(''^ '*').('['^'.').('['^ '(').('['^'/').('{'^ '[').(''|'!').(('')| '.').(''|'/').("\["^ '/').(''|'(').(''|'%' ).('['^')').(('{')^ ...
21
-n and unmatched curly brackets It is well known that the command line switch -n can be used to execute the script once for every line. perl --help says: -n assume "while (<>) { ... }" loop around program What it doesn't say explicitly is that Perl doesn't just assume a loop around the program; it literally wraps while (<>) { ...
15
Don't use qw. This is waste of two characters that could be used in better way. For example, don't write the following. @i=qw(unique value); Instead use barewords. @i=(unique,value); Or if you cannot use barewords, use glob syntax. @i=<unique value>; glob syntax can also be used for interesting effects. @i=<item{1,2,3}>;
11
Use non-word characters as variable names Using $% instead of$a can allow you to place the variable name right next to an if, for or while construct as in: @r=(1,2,3,4,5);$%=4; print$_*$%for@r Many can be used, but check the docs and @BreadBox's answer for which ones have magic effects! Use map when you can't use statement modifiers If you can't use ... 9 redo adds loop behavior to a block without for or while. {redo} is an infinite loop. 5 Perl 53 bytes You can save two more strokes: <>;print${\map$a-=(-1)**$_/(2*$_-1),1..$_},$a=$/for<>
5
You can run multiple different statements within nested ternary logic. Suppose you have a big if-elsif statement. This could be any logic and any number of statements. if( $_ < 1 ) {$a=1; $b=2;$c=3; say $a.$b.$c; } elsif($_ < 2 ) { $a=3;$b=2; $c=1; say$a.$b.$c; } elsif( $_ < 3) {$a=2; $b=2;$c=2; ...
5
Try to use the value of an assignment expression, like so: # 14 characters $n=pop;$o=$n&1 # 13 characters, saves 1$o=($n=pop)&1 This works because$n is 2 characters in Perl. You may change $n to () at no cost, and save 1 semicolon by moving the assignment into the parentheses. 4 Less than 70 character on 3 lines:$_=unpack("H21","9Ø HvÂláµöÂ");s/(.)/" "x(hex$1<3).substr "AHPJocehtunarslk",hex$1,1/eg;say Care: It's ISO-8859-1 encoded. perl -E '$_=unpack("H21","9Ø HvÂláµöÂ");s/(.)/" "x(hex$1<3).substr "AHPJocehtunarslk",hex$1,1/eg;say' Just Another Perl Hacker More than 70 chars at all, but less than 100! wc -clL <<&... 4 Use select(undef,undef,undef,$timeout) instead of Time::HiRes (Taken from https://stackoverflow.com/a/896928/4739548) Many challenges require you to sleep with greater precision than integers. select()'s timeout argument can do just that. select($u,$u,$u,0.1) is much more efficient than: import Time::HiRes qw(sleep);sleep(0.1) The former is only 20 ... 3 Perl 55 bytes <>;map($s-=(-1)**$_/(2*$_-1),1..$_),$s=!print$s,$/for<> Without using any Perl non-standard features (such as say). 3 bytes are wasted removing the first value from the input. Sample usage: in.dat 2 10 20 $perl leibniz.pl < in.dat 0.760459904732351 0.77290595166696 Alternative, also 55 bytes map{$s-=(-1)**$_/($_*2-1)for ...
2
Scala, 231 214 199 193 bytes Immutability always wins! (turns out the long package name of collection.mutable.Map wasn't worth the hassle...) class A{var a=Map[String,Any]() def g(s:String*):Any={val y=s(0);if(s.size>1)g(y).asInstanceOf[A].g(s.drop(1):_*)else if(a.contains(y))a(y)else{a+=y->new A;a(y)}} def s(s:String,v:Any)=a+=s->v} Only thing ...
1
C#, 215 bytes class A:System.Collections.Generic.Dictionary<string,object>{public void B(string a,object b){if(ContainsKey(a))this[a]=b;else Add(a,b);}public dynamic C(string a){if(!ContainsKey(a))Add(a,new A());return this[a];}} The class is named A. Use hash.B("key", "val"); to set a value, and hash.C("key"); to get a value. Nested keys can be ...
1
Shorten your print statements Unless the challenge specifies otherwise, you don't need trailing newlines. Our 'challenge' says 'output a random number from 0 to 9 to STDOUT'. We can take this code (28 bytes): $s=int(rand(10));print"$s\n" And shorten it to this (25 bytes): $s=int(rand(10));print$s by simply printing the variable. This last one only ...
1
use strict;*1=*CORE'die,*!=*=,@=='hacker',s??'&1(@!,$/)'?ee;s;;%ENV=~m ,..$,,$&+10;e,@!=(chr.'ust',~~reverse('rehtona'),'Perl',$@);&1("@{=}") It works with strictures enabled. Although it does not work on all machines/perls, because of CORE'die and ~~%ENV. $perl use strict;*1=*CORE'die,*!=*=,@=='hacker',s??'&1(@!,$/)'?ee;s;;%ENV=~m ,..$,,$&...
Only top voted, non community-wiki answers of a minimum length are eligible | 2020-09-18 20:48:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2098577618598938, "perplexity": 11846.664998769662}, "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/1600400188841.7/warc/CC-MAIN-20200918190514-20200918220514-00707.warc.gz"} |
http://bootmath.com/how-many-associative-binary-operations-there-are-on-a-finite-set.html | # How many associative binary operations there are on a finite set?
I am studying Scott’s book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But he doesn’t talk about how many associative binary operations there are on a finite set.
Is there an answer to that question? I mean, how many associative binary operations there are on a finite set?
#### Solutions Collecting From Web of "How many associative binary operations there are on a finite set?"
Unlike the example you give (of commutative binary operations), there is no closed formula for the number of associative binary operations on a finite set.
A semigroup is a set with an associative binary operation. In what follows I will write “semigroup” rather than “associative binary operation”.
It is shown in
Kleitman, Daniel J.; Rothschild, Bruce R.; Spencer, Joel H. The number of semigroups of order n.
Proc. Amer. Math. Soc. 55 (1976), no. 1, 227–232.
that almost all semigroups of order $n$ are $3$-nilpotent, and in Theorem 2.1(i) of
Distler, Andreas; Mitchell, J. D.
The number of nilpotent semigroups of degree 3.
Electron. J. Combin. 19 (2012), no. 2, Paper 51, 19 pp.
that the number of $3$-nilpotent semigroups of order $n$ is:
\sigma(n)=\sum_{m=2}^{a(n)}{n \choose m}m\sum_{i=0}^{m-1}(-1)^i{m-1 \choose
i}(m-i)^{\left((n-m)^2\right)}
where $a(n)=\left\lfloor n+1/2-\sqrt{n-3/4}\,\right\rfloor$.
So, $\sigma(n)$ is approximately the number of distinct associative binary operations on a set of size $n$.
The value $\sigma(n)$ appears to converge very quickly to the number $\tau(n)$ of semigroups with $n$ elements:
\begin{array}{l|llllllll}
n&1&2&3&4&5&6&7&8\\\hline
\tau(n)& 1& 8& 113& 3492& 183732& 17061118& 7743056064& 148195347518186\\
\sigma(n)&0& 0& 6& 180& 11720& 3089250& 5944080072& 147348275209800
\end{array}
So, by $n=8$, the ratio $\sigma(n)/\tau(n)>0.994$. | 2018-06-24 22:44:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8551821112632751, "perplexity": 431.4631038824575}, "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/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00558.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Sandbox&diff=prev&oldid=150674 | # Difference between revisions of "AoPS Wiki:Sandbox"
Welcome to the sandbox, a location to test your newfound wiki-editing abilities.
Please note that all contributions here may be deleted periodically and without warning.
$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}$ | 2021-05-13 11:51:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5014312267303467, "perplexity": 12676.507672415548}, "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-2021-21/segments/1620243989916.34/warc/CC-MAIN-20210513111525-20210513141525-00394.warc.gz"} |
https://www.nature.com/articles/s41598-017-15817-4?error=cookies_not_supported&code=acecc35c-400f-4b03-89e7-8a53a79417a6 | Article | Open | Published:
# Planar array with bidirectional elements for tunnel environments
## Abstract
The growing demands for wireless communications in tunnel environments are driven by requests for maintaining uninterrupted internet access for users in tunnels as well as wireless connections for wireless sensors, security, and control networks. Nevertheless, wireless networks in tunnel environments are associated with technical challenges related to elongated wireless coverage in two opposite near-end-fire directions. Here, we introduce a low-profile bidirectional antenna that can be mounted on a large ground plane and that has a dual-magnetic-current mode exhibiting radiation patterns with 3-dB beamwidth coverage at near-end-fire angles. Furthermore, we realized a planar array with eight such bidirectional antennas that were configured as a sparse array in order to reduce the commonplace issues of mutual coupling and grating lobes. The radiation beams of the proposed antenna array can scan in the end-fire area (60° < φ < 120°, 45° < θ < 80°), with scanning gains of 11~15 dB in the near-end-fire directions. In addition, we demonstrate that the proposed array can adaptively generate a directional single beam or multiple beams according to the positions of users, which is suitable for intelligent communication systems with low power consumptions and high communication bandwidths in tunnel environments.
## Introduction
Although the above-mentioned antennas have superior bidirectional radiation performance in free space, they are less appropriate for on-wall or on-vehicle applications because they lack a large back ground plane to eliminate the influence of back structures and materials on radiation. As a result, a linear reconfigurable array with two isolated ports and four L-shaped monopoles vertical to a metallic ground was proposed; the array had two opposite end-fire radiation patterns that could be switched by switching the feeding signals at two ports25. However, the monopole elements have a high profile and cannot be easily fabricated using standard techniques compared to printed patch antennas. Printed leaky-wave antennas have low profiles, can be effortlessly-fabricated to generate near-end-fire radiation beams and can be backed by larger ground planes. Consequently, coplanar waveguide26 and substrate integrated waveguide27 approaches were proposed to design end-fire leaky-wave antennas. By using tapered substrate-integrated waveguides, leaky-wave antennas can also be designed for conformal applications28. A fixed-frequency beam-steering leaky-wave antenna was proposed that can scan its beams in near-end-fire directions using binary switches29. The combination of two groups of end-fire leaky-wave radiators can generate bidirectional radiation patterns. As a special kind of leaky-wave antenna, surface-wave antennas have recently been widely investigated to produce end-fire bidirectional radiation patterns30,31,32 using large ground planes33 or high impedance surfaces34,35. In addition, microstrip magnetic dipoles with a large ground plane are another method for generating near-end-fire radiation patterns with vertical polarizations36,37,38,39.
The above antennas and arrays that can be mounted on a large ground plane can produce near-end-fire radiation beams or bidirectional radiation beams, which are highly needed for tunnel environments. Nevertheless, it is much more energy-efficient to automatically direct radiation beams to users in modern intelligent communication systems. Therefore, phased arrays capable of beam steering are highly needed in communication systems40,41,42. When we focus on tunnel communication environments, a new challenge arises. Generally, phased arrays can scan their beams in broadside directions, and scanning beams in near-end-fire directions remains a technological challenge43,44,45. To solve that problem, some methods have been proposed to increase element beamwidth, including the high impedance surface method46, image-theory method47, magnetic current method48, and reconfigurable element method49,50 for open space applications. In addition to approaches for widening element beamwidths, metallic wall51,52,53,54,55,56 and wide-angle impedance matching (WAIM) approaches57,58,59 were proposed to match the active impedances of elements at different scanning angles. However, those widening element beamwidth approaches are not adapted to tunnel environments because scanning range in broadside directions is redundant in tunnel environments and the arrays can only scan beams in a certain plane rather than over bidirectional areas.
In this paper, a low-profile, easily fabricated dual-mode bidirectional antenna and a planar phased array with bidirectional elements are proposed for tunnel environments. The design methodology is first analysed considering three important key factors that are introduced to guide the design of an array for use in a tunnel: designing bidirectional elements, dealing with the strong coupling, and avoiding grating lobes. Furthermore, a dual-magnetic-current antenna with two bidirectional modes is proposed based on the operational principle of phase-inversed binary arrays. By using a patch with opposite short edges and opposite open edges, a dual-magnetic-current antenna is realized. The 3-dB beam-width can cover bidirectional near-end-fire directions from 40° to 85° and from −40° to −85°. A planar array with eight bidirectional elements is then presented. Decoupling cavities and a sparse array configuration are used to achieve weak mutual coupling and avoid grating lobes in the array. Finally, the time reversal synthesis method is applied to adaptively synthesize single beam and multiple beams according to the users, which is suitable for intelligent communication systems used in tunnel environments.
## Design Procedures and Results
### Design procedures
The arrays used for tunnel environments in modern intelligent communications systems should have three key performance traits: (1) large back ground planes for on-wall or on-vehicle applications, (2) the ability to generate radiation that can cover near-end-fire directions, (3) ideally, the radiation beams that are automatically directed to users automatically. The schematic diagram of the planar phased array for use in tunnel environments is shown in Fig. 1.
To design an array for use in tunnels, there are three challenges: (1) designing low-profile bidirectional elements with a large back ground plane, (2) dealing with the strong coupling between bidirectional elements, especially in the along-tunnel direction and (3) avoiding grating lobes and large sidelobes. To homogenize the radiation distribution in a tunnel, the radiation pattern of each antenna array should cover its near-end-fire directions, and the directions along the broadside of one antenna array are covered by another antenna array, which means that the broadside radiation patterns of the array elements are redundant and that the bidirectional element is suitable for tunnel applications. To design the bidirectional antennas with large ground planes, a dual-magnetic-current structure based on an inversed-phase binary array is proposed in this paper. The radiation pattern of the bidirectional element is mainly focused in the two near-end-fire directions along the tunnel; therefore, the coupling between the elements arranged along the tunnel direction may be very strong. In this paper, two techniques are used to decrease coupling between elements. First, a grounded decoupling cavity is placed around the elements, and second, the distance between the elements along the tunnel direction is approximately one wavelength. However, according to the product principle of array patterns, grating lobes may appear when the distance between array elements exceeds a half wavelength. Therefore, decreasing coupling is contradictory to avoiding grating lobes. To solve these problems, we propose that the elements along the vertical direction to the tunnel are arranged as stair steps, such that the element distance in the tunnel direction decreases and grating lobes can be avoided.
### Dual-mode dual-magnetic-current bidirectional antenna
The geometry of the bidirectional antenna is shown in Fig. 2, and its optimum dimensions are indicated in the figure. A patch with two slots is printed on a substrate with a thickness of 4 mm and a relative permittivity of 2.65. Two edges of the patch are shorted and the other two edges are partially shorted at the ends. The open area of two opposite edges can support two inversed equivalent magnetic currents, which compose an inversed-phase binary array and produce a bidirectional radiation pattern. In this antenna, there are two close resonant modes, which are denoted TM12 and TM14. The two slots on the patch are used to excite the TM14 mode, which has a resonance frequency close to that of TM12. The feed position is at the centre of the patch in the y direction but not at the centre in the x direction. The operational principle will be analysed step by step below.
To clearly display the operational principle of the proposed bidirectional antenna, the evolutionary process of magnetic-current antennas is shown in Fig. 3. Antenna 1 is a fundamental magnetic-current antenna and consists of an open edge and three short edges. The open edge can be equivalent to a magnetic current and generate a dipole-like radiation pattern. The fields on the open edge of the patch can be equivalent to a magnetic current by analogy to the radiation principle of slot antennas:
$${J}_{m}=-\,{n}_{y}\times {E}_{t}$$
(1)
where, $${n}_{y}$$ is a unit vector in the z direction and $${E}_{t}$$ is the electric field in the slots. According to the operational principle of inversed-phase binary arrays, two single-magnetic-current antennas are arranged back-to-back and are labeled Antenna 2 in the figure. When the two single-magnetic-current antennas of Antenna 2 are fed with equal magnitude and phase, the equivalent magnetic currents on the two open edges will have inversed phase because of the opposite directions of the open edges. As we know, inversed-phase binary array can produce a bidirectional radiation pattern, and Antenna 2 therefore emits a bidirectional radiation pattern. The structure of Antenna 3, which has two open edges and two short edges, is a simplification of the structure of Antenna 2. Antenna 3 has only one off-centre feed, the position of which controls impedance matching. Antenna 4 is an example of the practical antenna shown in Fig. 2. The sheets on the short edges of Antenna 3 are replaced with grounded holes, and the two slots that are etched on the top patch excite two binary magnetic-current modes and widen the operational frequency band.
The reflection coefficients and electric field distributions of the bidirectional antenna in Fig. 2 with/without slots on the patch are shown in Fig. 4. When there are two slots on the top patch, which is the structure shown in Fig. 2, we can see two resonant valleys at 5.61 GHz (Position B) and 5.78 GHz (Position C), and when there are no slots on the patch, only one resonant valley at 5.48 GHz (Position A) can be seen. At Position A, the electric field distributions are from the ground to the patch in the central area and from the patch to the ground at the open edges, which indicates a TM12 mode. At Position B, the electric field distributions are similar to those at Position A. At Position C, the electric field distributions in the central area and at the open edges are from the ground to the patch, and the electric fields between the open edges and slots are from the patch to the ground, which indicates a TM14 mode. Therefore, the modes at Positions A, B, and C are the TM12 mode, the TM12 mode, and the TM14 mode, respectively. The slots on the patch result in a higher-order mode close to the resonant frequency of TM12 mode, and the two resonant modes induce a relatively wide operational frequency band.
Aided by the slots on the top patch, the two modes resonate at two frequencies that are close to each other. To increase the operational frequency bandwidth, the feed position and grounded holes at the ends of the open edges are also important for impedance matching. From Fig. 2, we can see that there are 3 grounded holes (noted as side additional grounded holes) at each end of the two open edges, in addition to the grounded holes at the two short edges on the patch. The relation between the reflection coefficient and the number of additional side grounded holes, n, is shown in Fig. 5(a). When n increases from 1 to 3, the resonant depth of the TM14 mode gradually increases, and when n further increases from 3 to 5, the resonant depths of the two modes gradually decrease. The relation between the reflection coefficient and feed position is shown in Fig. 5(b). With increasing distance D, the two resonant points in the map gradually become closer. For D ~ 2 mm, the antenna is matched to the TM12 mode, for D ~ 3.3 mm, the antenna is matched to the TM14 mode, and for intermediate values of D, the antenna has a compromised impedance matching between the two resonant modes. When D = 2.9 mm, a relatively wide operational frequency bandwidth can be achieved, as the vertical line shows.
The simulated normalized patterns of the proposed dual-mode bidirectional antenna with infinite ground plane are shown in Fig. 6. For this antenna, the yz and xy planes are the E and H planes, respectively. In the yz plane, the vertically polarized components are shaped like the letter “m” at both 5.6 GHz and 5.8 GHz. The horizontally polarized component at 5.6 GHz is less than −30 dB, and that at 5.8 GHz is less than −25 dB. In the xy plane, the vertically polarized components are shaped like the number “8” at both 5.6 GHz and 5.8 GHz, and the horizontally polarized components are very small around the two end-fire directions. Figure 6 indicates that the radiation is bidirectional over a relatively wide frequency range, and the horizontally polarized component in the near-end-fire directions is much smaller than the vertically polarized component.
From the above analysis, we can see that the significant challenge in designing low-profile bidirectional elements for use in tunnel arrays can be overcome by using the proposed dual-magnetic-current method. In the next section, a planar array with bidirectional elements will be proposed for use in tunnel.
### Array with bidirectional elements for use in tunnels
The eight elements shown in Fig. 7(a) were arranged as a planar array, as shown in Fig. 8. From the last section, we know that two main beams of the bidirectional elements are located in the yz plane and are directed close to ±y directions. Therefore, the coupling between elements arranged along the y axis may be very strong, even with a decoupling cavity. To decrease the coupling, the element distance is set to 52 mm, which is approximately one wavelength at 5.8 GHz. However, on the basis of the product principle of array patterns, grating lobes may appear when the distance between array elements exceeds a half wavelength. To solve this problem, the elements along the x direction are arranged as stair-steps. Therefore, the element distance in the y direction is smaller than half wavelength. This sparse distribution allows the grating lobes to be avoided and makes the coupling between the elements in x the direction to decrease. The details of the array arrangement will be analysed below. The measured active S parameters of the array are shown in Fig. 8(c). All the reflection coefficients are smaller than −10 dB from 5.75–5.86 GHz, and the couplings between elements are smaller than −15 dB. The weak coupling between the elements indicates that the decoupling cavity and sparse arrangement are effective, which can also be verified by the magnetic field distribution plots shown in Fig. 8(d).
To clearly show the effect of the decoupling cavity and element arrangement on array coupling, two elements arranged side by side and two elements arranged end to end were analysed, as shown in Figs 9 and 10, respectively. From Fig. 9, we can observe that the mutual coupling decreases with distance Dy in both two cases, with and without the decoupling cavities, and S21 in the case with the decoupling cavity is approximately 5 dB weaker than that in the case without the decoupling cavity, which means that both the decoupling cavity and the large element distance can weaken mutual coupling. From Fig. 10, we can see that S21 in the case with the decoupling cavity is approximately 3 dB weaker than that in the case without the decoupling cavity, which is similar to Fig. 9. However, the trend in the variation in mutual coupling with Dx differs from that with Dy. In the operational frequency band, the mutual coupling decreases when the distance Dx changes from 0 to 10 mm and then increases in both two cases (with and without decoupling cavities). When Dx is near 0, the distance between the two elements is very small, and the coupling is therefore strong; as Dx increases, the distance increases, and the coupling decreases. However, when the right element moves further in the –y direction, the direction of the radiation emitted by the proposed bidirectional element becomes significant. We know that the radiation of the element is near the ± y direction. In the near field, the electric fields are strong around the open edges and weak around the short edges, as shown in Fig. 4. When Dx increases above 10 mm, the two elements approach the direction of strong radiation and enter the strong field area, where the mutual coupling increases. Therefore, our proposed stair-step distribution for the antenna array is able to not only significantly avoid scanning grating lobes but also efficiently decreases mutual coupling.
The active far-field radiation patterns of each element in the proposed array are shown in Fig. 11. All the elements have bidirectional radiation patterns in both the simulated and measured cases shown in the figures, and the vertically polarized component is 15 dB stronger than the horizontally polarized component. For each element, the two near-end-fire beams are asymmetrical in both the simulated and measured results because of the effect of the other elements in the array.
Subsequently, the time reversal synthesis method was used to adaptively determine the feeding amplitude and phase corresponding to the desired radiation pattern. Due to its ability to automatically focus, the time reversal technique has been widely used in super-resolution imaging60,61,62, high-capacity communications63,64,65,66, and phased array synthesis47,67. The time reversal synthesis method consists of the following characteristics47,67: (1) The excitation amplitude and phase of each element can be adaptively determined regardless of array arrangement and mutual coupling. (2) The excitation signals of all the elements can be determined by a time reversal operation both in simulations and in practical communication systems; therefore, time reversal is a solution for intelligent communication systems. It should be noted that the radiation efficiency of the antenna array can be reduced by the mutual coupling, although the main radiation beam of the array can be adaptively pointed to the desired directions using the time-reversal approach in situations of strong mutual coupling. Therefore, the strategies for reducing mutual coupling discussed in the previous paragraphs remain necessary. The basic process of the time reversal synthesis method is described as follows: In Step 1, the phased array is illuminated by detecting signals with temporal waveforms that cover the operating frequency bandwidth of the array from the desired steering directions. In Step 2, each element of the array receives the signal at the same time, and the time-reversal operation is applied. In Step 3, each element of the array transmits the time-reversed signal. In the frequency domain, the time reversal operation is equivalent to phase conjugation. Thus, the process can be finished in the frequency domain. The amplitude $${{\rm{A}}}_{n}$$ and phase $${{\rm{\phi }}}_{n}$$ at operational frequency $${f}_{0}$$ of the received signal of Element n can be obtained by Fourier transform and the amplitude and phase of the corresponding exciting signal in Step 3 can be determined as $${{\rm{A}}}_{n}$$ and $$-{{\rm{\phi }}}_{n}$$. It is difficult to control amplitude excitations in practical applications. It has been proven that the amplitude $${{\rm{A}}}_{n}$$ of each element can be set uniformly to 1 in the time reversal synthesis process, with a weak effect on the radiation pattern of the array47. Therefore, only the phase is controlled using the time reversal synthesis method in this paper. In other words, in Step 3, each element of the array transmits a signal with an amplitude of 1 and a phase of $$-{{\rm{\phi }}}_{n}$$. In Step 4, the signal is focused in the directions of the detecting signals, so that the desired main beams will be obtained. The process of the time reversal synthesis method is shown in supplementary Movie 1.
Using the time reversal synthesis method, radiation patterns of the proposed array were synthesised. The simulated scanning patterns of the proposed array are shown in Figs 12 and 13. Given the structural symmetry of the proposed array, the scanning patterns around only one end-fire direction are shown in the figures; the scanning performances around the other end-fire direction are similar. From Fig. 12, we can see that when the main beam is directed to $${\rm{\theta }}=45^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=35^\circ$$ to $$\,{\rm{\theta }}=60^\circ$$. When the main beam is directed to $${\rm{\theta }}=60^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=45^\circ$$ to $$\,{\rm{\theta }}=80^\circ$$. When the main beam is directed to $${\rm{\theta }}=70^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=55^\circ$$ to $$\,{\rm{\theta }}=90^\circ$$. From Fig. 13, we observe that the main beam of the array can be scanned from $${\rm{\theta }}=60^\circ$$ to $${\rm{\phi }}=120^\circ$$ and that the radiation beams of the proposed array can be scanned in the end-fire area. In addition, the horizontally polarized components in all the scanning cases are more than 10 dB smaller than the vertically polarized components.
The measured scanning radiation patterns are shown in Fig. 14 around only one end-fire direction of the proposed array as a result of the symmetry of the structure. When the main beam is directed to $${\rm{\theta }}=45^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB radiation beam in the yz plane approximately covers $${\rm{\theta }}=36^\circ$$ to $$\,{\rm{\theta }}=65^\circ$$. When the main beam is directed to $${\rm{\theta }}=60^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=44^\circ$$ to $$\,{\rm{\theta }}=84^\circ$$. When the main beam is directed to $${\rm{\theta }}=70^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=50^\circ$$ to $$\,{\rm{\theta }}=90^\circ$$. When the radiation beam is scanned in the yz plane, the horizontally polarized components in all the scanning cases are approximately 15 dB smaller than the vertically polarized components. When the beam is scanned in the xy plane, the main beam of the array can be scanned from $${\rm{\phi }}=60^\circ$$ to $${\rm{\phi }}=120^\circ$$ and the horizontally polarized components in all the scanning cases are more than 6 dB smaller than the vertically polarized components. The measured scanning gains of the proposed array are shown in Fig. 15. From Fig. 15, we can see that the scanning gain decreases when the main beam direction is away from the end-fire direction. The scanning gains in the near-end-fire directions (60° < φ < 120°, 45° < θ < 80°) are within 11~15 dB, and the gain fluctuations are small, which indicates good scanning performance.
The above results show the scanning performances in the near-end-fire directions of the proposed array, which indicate that the proposed array can scan its main beam in hotspot areas of tunnel environments. In a time reversal communication system, after a user transmits a detection signal in the tunnel, the proposed array can automatically focus its main beam to the user using a tracking mechanism. In this situation, one main radiation beam can increase the transmission efficiency and avoid signal mutual interference. In other situations, for example, when users are located in two opposite directions and transmit detection signals simultaneously, the proposed array is expected to have multi-beam performance. To demonstrate the multi-beam performance, the proposed array is simultaneously illuminated by two plane waves from ±y, i.e., $${\rm{\theta }}=90^\circ ,{\rm{\phi }}=90^\circ$$ and $$\,{\rm{\theta }}=90^\circ ,{\rm{\phi }}=270^\circ$$. Based on the time reversal synthesis method introduced above, a bidirectional end-fire radiation pattern can be realized, as shown in Fig. 16. The 3-dB beamwidth of the vertically polarized component can cover the bidirectional near-end-fire directions, including $${\rm{\theta }}=90^\circ ,{\rm{\phi }}=90^\circ$$ and $$\,{\rm{\theta }}=90^\circ ,{\rm{\phi }}=270^\circ$$. The horizontally polarized component is 10 dB smaller than the vertically polarized component, and an excellent bidirectional radiation pattern is demonstrated. From the above analysis, we can see that the proposed array can be used in both single-user and multi-user situations in tunnel environments.
## Discussion
In this study of a planar phased array for use in tunnel environments, we implemented three key techniques for the proposed array: the bidirectional element design method, the bidirectional array arrangement method, and the adaptive array synthesis method. To design the bidirectional element, the primary concept was to manipulate dual-magnetic currents based on a phase-inversed binary array. Moreover, the dual-magnetic-current antenna was realized using a patch with opposite short edges and opposite open edges. To increase the operational frequency bandwidth, two slots were etched on the patch of the antenna to excite two bidirectional modes. In addition, the two contradictory issues of reducing strong coupling and avoiding grating lobes were solved using the decoupling cavity and the sparse array configuration in the proposed tunnel-used array arrangement process. Finally, the time reversal synthesis method was applied to synthesize array patterns. The time reversal method can be used not only to synthesize desired radiation patterns in simulations but also to manipulate the focus of radiation beams in practical intelligent communication systems. With the time reversal method, the proposed array can adaptively generate a single beam or multiple beams according to the positions of users in tunnel environments. In conclusion, excellent beam-scanning performances can be realized using the proposed array and methods. The proposed methods can be used to improve bandwidth and reduce energy consumption in demanding wireless communication applications in tunnel environments.
## Methods
### Simulation method
The proposed bidirectional antenna and planar phased array were simulated and optimized using CST Microwave Studio. The selected operating frequency band of the proposed antenna and array was centred around 5.8 GHz, which is a part of the spectrum open for academic research in China. The time reversal synthesis method was implemented in CST Microwave Studio using the following steps: (1) The proposed array is illuminated by a plane wave of 0–10 GHz from the desired steering directions. (2) Each element of the array receives the signal at the same time and the received signals are exported. (3) The amplitude $${{\rm{A}}}_{n}$$ and phase $${{\rm{\phi }}}_{n}$$ at 5.8 GHz of the received signal of Element n are calculated by Fourier transform in Matlab. (4) Array Element n transmits a signal with an amplitude of 1 and a phase of $$-{{\boldsymbol{\phi }}}_{{\boldsymbol{n}}}$$, and the desired far-field radiation pattern is obtained. When multiple beams are required, the illumination process should be performed several times because only one plane wave can be set during a calculation cycle in CST Microwave Studio. For example, two illumination calculations were performed in CST Microwave Studio to obtain Fig. 16. First, the proposed array was illuminated by a plane wave from the y direction, and the received signal at each port was exported. Subsequently, the proposed array was illuminated by a plane wave from the -y direction, and the received signal at each port was exported. Finally, the signals received in the two calculation steps at each port were added, and the Fourier transform of the added signals was calculated in Matlab to obtain the exciting phase for each port.
### Measurement method
The scattering matrices S mn (m, n = 1, 2, 3, …, 8) for the proposed planar phased array were measured using an E8361A Vector Network Analyzer. When the S matrices of the two ports were measured, the other ports were connected to matched loads. The far-field radiation patterns of the proposed antenna and array were measured in a microwave anechoic chamber equipped with a SATIMO measurement system. When the active patterns of one element were measured, the other ports were connected to matched loads. The scanning performances were measured using the feed system shown in Fig. 14(e). Eight ports of the measured array were connected to an eight-channel phase shifter, which was fed by a source through a one-to-eight power divider. The phase shifter was controlled by computer software, and each phase channel could be controlled independently. In the measurement, the feed amplitude and phase of Element n in each case were set to 1 and $$-{{\boldsymbol{\phi }}}_{{\boldsymbol{n}}}$$, respectively, based on the results simulated in CST Microwave Studio and the Matlab calculations (see the section on the simulation method).
### Data availability
The data from this paper can be obtained from the University of Southampton ePrints research repository: http://doi.org/10.5258/SOTON/D0292.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Change history
• ### 13 June 2018
A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.
## References
1. 1.
Guan, K., Zhong, Z., Alonso, J. I. & Briso-Rodríguez, C. Measurement of distributed antenna systems at 2.4 GHz in a realistic subway tunnel environment. IEEE Trans. Veh. Technol. 61, 834–837 (2012).
2. 2.
Forooshani, A. E., Lotfi-Neyestanak, A. A. & Michelson, D. G. Optimization of antenna placement in distributed MIMO systems for underground mines. IEEE Trans. Wireless Commun. 13, 4685–4692 (2014).
3. 3.
Fuschini, F. & Falciasecca, G. A mixed rays-modes approach to the propagation in real road and railway tunnels. IEEE Trans. Antennas Propag. 60, 1095–1105 (2012).
4. 4.
Zhang, J. et al. Dual-band bidirectional high gain antenna for WLAN 2.4/5.8 GHz applications. Elect. Lett. 45, 6–7 (2009).
5. 5.
Li, X., Yang, L., Gong, S. X. & Yang, Y. J. Bidirectional high gain antenna for WLAN applications. Prog. Electromag. Res. 6, 99–106 (2009).
6. 6.
Cho, K. & Hori, T. Bidirectional rod antenna composed of narrow patches. ISAP 1, 174–177 (1994).
7. 7.
Iwasaki, H. Microstrip antenna with back-to-back configuration relative to a slot on a ground plane. Elect. Lett. 34, 1373–1374 (1998).
8. 8.
Deng, C., Li, Y., Zhang, Z., Wang, J. & Feng, Z. A bidirectional left-hand circularly polarized antenna using dual rotated patches. Microwave Opt. Technol. Lett. 55, 2044–2047 (2013).
9. 9.
Liu, L., Zhang, Z., Tian, Z. & Feng, Z. A bidirectional endfire array with compact antenna elements for coal mine/tunnel communication. IEEE Antennas Wireless Propag. Lett. 11, 342–345 (2012).
10. 10.
Liu, L., Liu, W., Li, Y., Zhang, Z. & Feng, Z. A planar reconfigurable antenna with bidirectional end‐fire and broadside radiation patterns. Microwave Opt. Technol. Lett. 56, 1942–1946 (2014).
11. 11.
Liu, W., Li, Y., Zhang, Z. & Feng, Z. A bidirectional array of the same left-handed circular polarization using a special substrate. IEEE Antennas Wireless Propag. Lett. 12, 1543–1546 (2013).
12. 12.
Liu, W., Zhang, Z., Tian, Z. & Feng, Z. A bidirectional high-gain cascaded ring antenna for communication in coal mine. IEEE Antennas Wireless Propag. Lett. 12, 761–764 (2013).
13. 13.
Kosulvit, S., Phongcharoenpanich, C., Krairiksh, M. & Wakabayashi, T. Radiation characteristics of a bidirectional antenna using a linear probe in a rectangular ring. ICMMT 337–340 (1998).
14. 14.
Kosulvit, S., Phongcharoenpanich, C. & Krairiksh, M. A bidirectional antenna using a probe excited rectangular ring. Science Asia 27, 285–291 (2001).
15. 15.
Phongcharoenpanich, C., Lamultree, S., Kosulvit, S. & Krairiksh, M. Theory and experiment of an antenna using a probe excited rectangular ring. ISAP 3, 737–740 (2003).
16. 16.
Lamultree, S., Phongcharoenpanich, C., Kosulvit, S. & Krairiksh, M. Investigations of a bidirectional antenna using a probe excited rectangular ring. APMC 5, 1–4 (2005).
17. 17.
Vongsack, S., Phongcharoenpanich, C., Kosulvit, S. & Wakabayashi, T. A bent probe excited circular ring antenna above cylindrical reflector radiating circular polarization. APMC 1–4(2007).
18. 18.
Liu, W., Zhang, Z. & Feng, Z. Bidirectional rectangular ring antenna for coal mine/tunnel communication. Microwave Opt. Technol. Lett. 55, 1412–1416 (2013).
19. 19.
Chawanonphithak, K., Phongcharoenpanich, C., Kosulvit, S. & Krairiksh, M. An ultra-wideband bidirectional antenna with modified circular disc monopole excited elliptical ring. RWS 261–264 (2007).
20. 20.
Krairiksh, M., Keowsawat, P., Phongcharoenpanich, C. & Kosulvit, S. Two-probe excited circular ring antenna for MIMO application. Prog. Electromag. Res. 97, 417–431 (2009).
21. 21.
Zhao, Y., Wei, K., Zhang, Z. & Feng, Z. A waveguide antenna with bidirectional circular polarizations of the same sense. IEEE Antennas Wireless Propag. Lett. 12, 559–562 (2013).
22. 22.
Sammeta, R. & Filipovic, D. S. Reduced size planar dual-polarized log-periodic antenna for bidirectional high power transmit and receive applications. IEEE Trans. Antennas Propag. 62, 5453–5461 (2014).
23. 23.
Chieh, J. C. S. & Pham, A. V. A bidirectional microstrip X-band antenna array on liquid crystal polymer for beamforming applications. IEEE Trans. Antennas Propag. 61, 3364–3368 (2013).
24. 24.
Zhao, Y., Zhang, Z. & Feng, Z. Design of a CPW-fed C-Shaped slot array antenna for coal mine/tunnel applications. Microwave Opt. Technol. Lett. 55, 1784–1789 (2013).
25. 25.
Liu, F., Zhang, Z., Chen, W., Feng, Z. & Iskander, M. F. An endfire beam-switchable antenna array used in vehicular environment. IEEE Antennas Wireless Propag. Lett. 9, 195–198 (2010).
26. 26.
Grbic, A. & Eleftheriades, G. V. Leaky CPW-based slot antenna arrays for millimeter-wave applications. IEEE Trans. Antennas Propag. 50, 1494–1504 (2002).
27. 27.
Liu, J., Jackson, D. R., Li, Y., Zhang, C. & Long, Y. Investigations of SIW leaky-wave antenna for endfire-radiation with narrow beam and sidelobe suppression. IEEE Trans. Antennas Propag. 62, 4489–4497 (2014).
28. 28.
Martinez-Ros, A. J., Gómez-Tornero, J. L. & Goussetis, G. Conformal tapered substrate integrated waveguide leaky-wave antenna. IEEE Trans. Antennas Propag. 62, 5983–5991 (2014).
29. 29.
Karmokar, D. K., Esselle, K. P. & Hay, S. G. Fixed-frequency beam steering of microstrip leaky-wave antennas using binary switches. IEEE Trans. Antennas Propag. 64, 2146–2154 (2016).
30. 30.
Chen, Z. & Shen, Z. Wideband flush-mounted surface wave antenna of very low profile. IEEE Trans. Antennas Propag. 63, 2430–2438 (2015).
31. 31.
Hu, Z., Shen, Z. & Wu, W. Reconfigurable leaky-wave antenna based on periodic water grating. IEEE Antennas Wireless Propag. Lett. 13, 134–137 (2014).
32. 32.
Yang, R., Lei, Z., Chen, L., Wang, Z. & Hao, Y. Surface wave transformation lens antennas. IEEE Trans. Antennas Propag. 62, 973–977 (2014).
33. 33.
Zhang, S. & Pedersen, G. F. Compact wideband and low-profile antenna mountable on large metallic surfaces. IEEE Trans. Antennas Propag. 65, 6–16 (2017).
34. 34.
Wang, X. C., Zhao, W. S., Hu, J. & Yin, W. Y. Reconfigurable terahertz leaky-wave antenna using graphene-based high-impedance surface. IEEE Trans. Nanotechnol. 14, 62–69 (2015).
35. 35.
Li, M., Xiao, S. Q., Xiong, J. & Wang, B.-Z. Horizontal dipole located close to ground plane with bidirectional endfire radiation. IEEE Antennas Wireless Propag. Lett. 13, 1144–1147 (2014).
36. 36.
Wang, R., Wang, B.-Z., Hu, C., Gong, C. & Ding, X. Low-profile on-board antenna with a broad beam based on three-current model. Prog. Electromag. Res 156, 13–24 (2016).
37. 37.
Liu, J. & Xue, Q. Microstrip magnetic dipole Yagi array antenna with endfire radiation and vertical polarization. IEEE Trans. Antennas Propag 61, 1140–1147 (2013).
38. 38.
Liu, J. & Xue, Q. Broadband long rectangular patch antenna with high gain and vertical polarization. IEEE Trans. Antennas Propag 61, 539–546 (2013).
39. 39.
Liang, Z., Li, Y., Liu, J., Zheng, S. Y. & Long, Y. Microstrip magnetic monopole endfire array antenna with vertical polarization. IEEE Trans. Antennas Propag 64, 4208–4217 (2016).
40. 40.
Waterhouse, R. & Novack, D. Realizing 5 G: microwave photonics for 5 G mobile wireless systems. IEEE Microwave Mag 16, 84–92 (2015).
41. 41.
Sun, S., Rappaport, T. S., Heath, R. W., Nix, A. & Rangan, S. MIMO for millimeter-wave wireless communications: beamforming, spatial multiplexing, or both. IEEE Commun. Mag. 52, 110–121 (2014).
42. 42.
Yaacobi, A. et al. Integrated phased array for wide-angle beam steering. Opt. Let. 39, 4575–4578 (2014).
43. 43.
Toshev, A. G. Multipanel concept for wide-angle scanning of phased array antennas. IEEE Trans. Antennas Propag. 56, 3330–3333 (2008).
44. 44.
Bai, Y. Y., Xiao, S. Q., Tang, M. C., Ding, Z. F. & Wang, B.-Z. Wide-angle scanning phased array with pattern reconfigurable elements. IEEE Trans. Antennas Propag. 59, 4071–4076 (2011).
45. 45.
Ding, X., Wang, B.-Z. & He, G.-Q. Research on a millimeter-wave phased array with wide-angle scanning performance. IEEE Trans. Antennas Propag. 61, 5319–5324 (2013).
46. 46.
Li, M., Xiao, S. Q. & Wang, B.-Z. Investigation of using high impedance surfaces for wide-angle scanning arrays. IEEE Trans. Antennas Propag. 63, 2895–2901 (2015).
47. 47.
Wang, R., Wang, B.-Z., Ding, X. & Yang, X. S. Planar phased array with wide-angle scanning performance based on image theory. IEEE Trans. Antennas Propag. 63, 3908–3917 (2015).
48. 48.
Wen, Y. Q., Wang, B.-Z. & Ding, X. A wide-angle scanning and low side lobe level microstrip phased array based on genetic algorithm optimization. IEEE Trans. Antennas Propag. 64, 805–810 (2016).
49. 49.
Ding, X. et al. A wide-angle scanning planar phased array with pattern reconfigurable magnetic current element. IEEE Trans. Antennas Propag. 65, 1434–1439 (2017).
50. 50.
Xiao, S., Zheng, C., Li, M., Xiong, J. & Wang, B.-Z. Varactor-loaded pattern reconfigurable array for wide-angle scanning with low gain fluctuation. IEEE Trans. Antennas Propag. 63, 2364–2369 (2015).
51. 51.
Edelberg, S. & Oliner, A. Mutual coupling effects in large antenna arrays II: Compensation effects. IRE. Trans. Antennas Propag. 8, 360–367 (1960).
52. 52.
Awide, M. H., Kamel, A. H. & Fathy, A. E. Analysis and design of wide-scan angle wide-band phased arrays of substrate-integrated cavity-backed patches. IEEE Trans. Antennas Propag. 61, 3034–3041 (2013).
53. 53.
Valavan, S. E., Tran, D., Yarovoy, A. G. & Roederer, A. G. Planar dual-band wide-scan phased array in X-band. IEEE Trans. Antennas Propag. 62, 5370–5375 (2014).
54. 54.
Valavan, S. E., Tran, D., Yarovoy, A. G. & Roederer, A. G. Dual-band wide-angle scanning planar phased array in X/Ku-bands. IEEE Trans. Antennas Propag. 62, 2514–2521 (2014).
55. 55.
Chaloun, T., Ziegler, V. & Menzel, W. Design of a dual-polarized stacked patch antenna for wide-angle scanning reflectarrays. IEEE Trans. Antennas Propag. 64, 3380–3390 (2016).
56. 56.
Arand, B. A., Bazrkar, A. & Zahedi, A. Design of a phased array in triangular grid with an efficient matching network and reduced mutual coupling for wide-angle scanning. IEEE Trans. Antennas Propag. 65, 2983–2991 (2017).
57. 57.
Magill, E. & Wheeler, H. A. Wide-angle impedance matching of a planar array antenna by a dielectric sheet. IEEE Trans. Antennas Propag. 14, 49–53 (1966).
58. 58.
Sajuyigbe, S. et al. Wide angle impedance matching metamaterials for waveguide-fed phased-array antennas. IET Microw. Antennas Propag. 4, 1063–1072 (2010).
59. 59.
Oliveri, G., Viani, F., Anselmi, N. & Massa, A. Synthesis of multi-layer WAIM coatings for planar phased arrays within the system-by-design framework. IEEE Trans. Antennas Propag. 63, 2482–2496 (2015).
60. 60.
Rosny, J. & Fink, M. Focusing properties of near-filed time reversal. Phys. Rev. A 76, 065801 (2007).
61. 61.
Gao, Q., Wang, B.-Z. & Wang, X. H. Far-field super-resolution imaging with compact and multi-frequency planar resonant lens based on time reversal. IEEE Trans. Antennas Propag. 63, 5586–5592 (2015).
62. 62.
Gong, Z. S. et al. Far-field super-resolution imaging of scatterers with a time-reversal system aided by a grating plate. IEEE Photon. J. 9, 6900108 (2017).
63. 63.
Lerosey, G., De Rosny, J., Tourin, A. & Fink, M. Focusing beyond the diffraction limit with far-field time reversal. Science 315, 1120–1122 (2007).
64. 64.
Naqvi, H. et al. Experimental validation of time reversal ultra wide-band communication system for high data rates. IET Microw. Antennas Propag. 4, 643–650 (2010).
65. 65.
Ding, S., Wang, B.-Z., Ge, G. & Zhao, D. Sub-wavelength array with embedded chirped delay lines based on time reversal technique. IEEE Trans. Antennas Propag. 61, 2868–2873 (2013).
66. 66.
Wang, R., Wang, B.-Z., Gong, Z. S. & Ding, X. Compact multiport antenna with radiator-sharing approach and its performance evaluation of time reversal in an intra-car environment. IEEE Trans. Antennas Propag. 63, 4213–4219 (2015).
67. 67.
Zhao, D., Jin, Y., Wang, B.-Z. & Zang, R. Time reversal based broadband synthesis method for arbitrarily structured beam-steering arrays. IEEE Trans. Antennas Propag. 60, 164–173 (2012).
## Acknowledgements
This work was supported by funds from National Natural Science Foundation of China (NSFC Grant No. 61331007, 61361166008, and 61401065), and University of Southampton Zepler Institute Research Collaboration Stimulus Fund (Grant No. 516719104), enabled by the 2016/17 UK Higher Education Innovation Fund.
## Author information
### Affiliations
1. #### Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China
• Ren Wang
• , Bing-Zhong Wang
• & Xiao Ding
• Jun-Yu Ou
### Contributions
B.W. initiated and supervised the project, R.W. performed the simulations and experiments, R.W. and X.D. analyzed the results and developed the theory, R.W. and J.Y.O. wrote the main manuscript text. All authors reviewed the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Bing-Zhong Wang. | 2018-10-18 08:00:13 | {"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": 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": 0.6542603969573975, "perplexity": 2866.2499367766195}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511744.53/warc/CC-MAIN-20181018063902-20181018085402-00122.warc.gz"} |
https://elementald6.com/doku.php?id=wiki:cead&rev=1517464315&do=diff | # Differences
This shows you the differences between two versions of the page.
Line 13: Line 13:
**Coordinates**: 36° N, 30° W\\ **Coordinates**: 36° N, 30° W\\
\\ \\
Line 23: Line 23:
\\ \\ | 2020-09-18 21:00:56 | {"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": 0.8093186616897583, "perplexity": 14944.317898060106}, "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-2020-40/segments/1600400188841.7/warc/CC-MAIN-20200918190514-20200918220514-00038.warc.gz"} |
https://bout-dev.readthedocs.io/en/stable/user_docs/variable_init.html | Variable initialisation¶
Variables in BOUT++ are not initialised automatically, but must be explicitly given a value. For example the following code declares a Field3D variable then attempts to access a particular element:
Field3D f; // Declare a variable
f(0,0,0) = 1.0; // Error!
This results in an error because the data array to store values in f has not been allocated. Allocating data can be done in several ways:
1. Initialise with a value:
Field3D f = 0.0; // Allocates memory, fills with zeros
f(0,0,0) = 1.0; // ok
This cannot be done at a global scope, since it requires the mesh to already exist and have a defined size.
2. Set to a scalar value:
Field3D f;
f = 0.0; // Allocates memory, fills with zeros
f(0,0,0) = 1.0; // ok
Note that setting a field equal to another field has the effect of making both fields share the same underlying data. This behaviour is similar to how NumPy arrays behave in Python.
Field3D g = 0.0; // Allocates memory, fills with zeros
Field3D f = g; // f now shares memory with g
f(0,0,0) = 1.0; // g also modified
To ensure that a field has a unique underlying memory array call the Field3D::allocate() method before writing to individual indices.
3. Use Field3D::allocate() to allocate memory:
Field3D f;
f.allocate(); // Allocates memory, values undefined
f(0,0,0) = 1.0; // ok
In a BOUT++ simulation some variables are typically evolved in time. The initialisation of these variables is handled by the time integration solver.
Initialisation of time evolved variables¶
Each variable being evolved has its own section, with the same name as the output data. For example, the high-$$\beta$$ model has variables “P”, “jpar”, and “U”, and so has sections [P], [jpar], [U] (not case sensitive).
Expressions¶
The recommended way to initialise a variable is to use the function option for each variable:
[p]
function = 1 + gauss(x-0.5)*gauss(y)*sin(z)
This evaluates an analytic expression to initialise the $$P$$ variable. Expressions can include the usual operators (+,-,*,/), including ^ for exponents. The following values are also already defined:
Name Description
x $$x$$ position between $$0$$ and $$1$$
y $$y$$ position between $$0$$ and $$2\pi$$ (excluding the last point)
z $$z$$ position between $$0$$ and $$2\pi$$ (excluding the last point)
pi π $$3.1415\ldots$$
Table: Initialisation expression values
By default, $$x$$ is defined as i / (nx - 2*MXG), where MXG is the width of the boundary region, by default 2. Hence $$x$$ actually goes from 0 on the leftmost point to (nx-1)/(nx-4) on the rightmost point. This is not a particularly good definition, but for most cases its sufficient to create some initial profiles. For some problems like island reconnection simulations, it’s useful to define $$x$$ in a particular way which is more symmetric than the default. To do this, set in BOUT.inp
[mesh]
symmetricGlobalX = true
This will change the definition of $$x$$ to i / (nx - 1), so $$x$$ is then between $$0$$ and $$1$$ everywhere.
By default the expressions are evaluated in a field-aligned coordinate system, i.e. if you are using the [mesh] option paralleltransform = shifted, the input f will have f = fromFieldAligned(f) applied before being returned. To switch off this behaviour and evaluate the input expressions in coordinates with orthogonal x-z (i.e. toroidal $$\{\psi,\theta,\phi\}$$ coordinates when using paralleltransform = shifted), set in BOUT.inp
[input]
transform_from_field_aligned = false
The functions in Table 1 are also available in expressions.
Table 1 Initialisation expression functions
Name Description
abs(x) Absolute value $$|x|$$
asin(x), acos(x), atan(x), atan(y,x) Inverse trigonometric functions
ballooning(x) Ballooning transform ((1), Fig. 3)
ballooning(x,n) Ballooning transform, using $$n$$ terms (default 3)
cos(x) Cosine
cosh(x) Hyperbolic cosine
exp(x) Exponential
tanh(x) Hyperbolic tangent
gauss(x) Gaussian $$\exp(-x^2/2) / \sqrt{2\pi}$$
gauss(x, w) Gaussian $$\exp[-x^2/(2w^2)] / (w\sqrt{2\pi})$$
H(x) Heaviside function: $$1$$ if $$x > 0$$ otherwise $$0$$
log(x) Natural logarithm
max(x,y,...) Maximum (variable arguments)
min(x,y,...) Minimum (variable arguments)
mixmode(x) A mixture of Fourier modes
mixmode(x, seed) seed determines random phase (default 0.5)
power(x,y) Exponent $$x^y$$
sin(x) Sine
sinh(x) Hyperbolic sine
sqrt(x) $$\sqrt{x}$$
tan(x) Tangent
erf(x) The error function
TanhHat(x, width, centre, steepness) The hat function $$\frac{1}{2}(\tanh[s (x-[c-\frac{w}{2}])]$$ $$- \tanh[s (x-[c+\frac{w}{2}])] )$$
fmod(x) The modulo operator, returns floating point remainder
For field-aligned tokamak simulations, the Y direction is along the field and in the core this will have a discontinuity at the twist-shift location where field-lines are matched onto each other. To handle this, the ballooning function applies a truncated Ballooning transformation to construct a smooth initial perturbation:
(1)$U_0^{balloon} = \sum_{i=-N}^N F(x)G(y + 2\pi i)H(z + q2\pi i)$
There is an example code test-ballooning which compares methods of setting initial conditions with the ballooning transform.
The mixmode(x) function is a mixture of Fourier modes of the form:
$\mathrm{mixmode}(x) = \sum_{i=1}^{14} \frac{1}{(1 + |i-4|)^2}\cos[ix + \phi(i, \mathrm{seed})]$
where $$\phi$$ is a random phase between $$-\pi$$ and $$+\pi$$, which depends on the seed. The factor in front of each term is chosen so that the 4th harmonic ($$i=4$$) has the highest amplitude. This is useful mainly for initialising turbulence simulations, where a mixture of mode numbers is desired.
Initalising variables with the FieldFactory class¶
This class provides a way to generate a field with a specified form. For example to create a variable var from options we could write
FieldFactory f(mesh);
Field2D var = f.create2D("var");
This will look for an option called “var”, and use that expression to initialise the variable var. This could then be set in the BOUT.inp file or on the command line.
var = gauss(x-0.5,0.2)*gauss(y)*sin(3*z)
To do this, FieldFactory implements a recursive descent parser to turn a string containing something like "gauss(x-0.5,0.2)*gauss(y)*sin(3*z)" into values in a Field3D or Field2D object. Examples are given in the test-fieldfactory example:
FieldFactory f(mesh);
Field2D b = f.create2D("1 - x");
Field3D d = f.create3D("gauss(x-0.5,0.2)*gauss(y)*sin(z)");
This is done by creating a tree of FieldGenerator objects which then generate the field values:
class FieldGenerator {
public:
virtual ~FieldGenerator() { }
virtual FieldGenerator* clone(const list<FieldGenerator*> args) {return NULL;}
virtual BoutReal generate(int x, int y, int z) = 0;
};
All classes inheriting from FieldGenerator must implement a FieldGenerator::generate() function, which returns the value at the given (x,y,z) position. Classes should also implement a FieldGenerator::clone() function, which takes a list of arguments and creates a new instance of its class. This takes as input a list of other FieldGenerator objects, allowing a variable number of arguments.
The simplest generator is a fixed numerical value, which is represented by a FieldValue object:
class FieldValue : public FieldGenerator {
public:
FieldValue(BoutReal val) : value(val) {}
BoutReal generate(int x, int y, int z) { return value; }
private:
BoutReal value;
};
To add a new function to the FieldFactory, a new FieldGenerator class must be defined. Here we will use the example of the sinh function, implemented using a class FieldSinh. This takes a single argument as input, but FieldPI takes no arguments, and FieldGaussian takes either one or two. Study these after reading this to see how these are handled.
First, edit src/field/fieldgenerators.hxx and add a class definition:
class FieldSinh : public FieldGenerator {
public:
FieldSinh(FieldGenerator* g) : gen(g) {}
~FieldSinh() {if(gen) delete gen;}
FieldGenerator* clone(const list<FieldGenerator*> args);
BoutReal generate(int x, int y, int z);
private:
FieldGenerator *gen;
};
The gen member is used to store the input argument, and to make sure it’s deleted properly we add some code to the destructor. The constructor takes a single input, the FieldGenerator argument to the sinh function, which is stored in the member gen .
Next edit src/field/fieldgenerators.cxx and add the implementation of the clone and generate functions:
FieldGenerator* FieldSinh::clone(const list<FieldGenerator*> args) {
if(args.size() != 1) {
throw ParseException("Incorrect number of arguments to sinh function. Expecting 1, got %d", args.size());
}
return new FieldSinh(args.front());
}
BoutReal FieldSinh::generate(double x, double y, double z, double t) {
return sinh(gen->generate(x,y,z,t));
}
The clone function first checks the number of arguments using args.size() . This is used in FieldGaussian to handle different numbers of input, but in this case we throw a ParseException if the number of inputs isn’t one. clone then creates a new FieldSinh object, passing the first argument ( args.front() ) to the constructor (which then gets stored in the gen member variable).
The generate function for sinh just gets the value of the input by calling gen->generate(x,y,z), calculates sinh of it and returns the result.
The clone function means that the parsing code can make copies of any FieldGenerator class if it’s given a single instance to start with. The final step is therefore to give the FieldFactory class an instance of this new generator. Edit the FieldFactory constructor FieldFactory::FieldFactory() in src/field/field_factory.cxx and add the line:
addGenerator("sinh", new FieldSinh(NULL));
That’s it! This line associates the string "sinh" with a FieldGenerator . Even though FieldFactory doesn’t know what type of FieldGenerator it is, it can make more copies by calling the clone member function. This is a useful technique for polymorphic objects in C++ called the “Virtual Constructor” idiom.
Parser internals¶
When a FieldGenerator is added using the addGenerator function, it is entered into a std::map which maps strings to FieldGenerator objects (include/field_factory.hxx):
map<string, FieldGenerator*> gen;
Parsing a string into a tree of FieldGenerator objects is done by first splitting the string up into separate tokens like operators like ’*’, brackets ’(’, names like ’sinh’ and so on, then recognising patterns in the stream of tokens. Recognising tokens is done in src/field/field_factory.cxx:
char FieldFactory::nextToken() {
...
This returns the next token, and setting the variable char curtok to the same value. This can be one of:
• -1 if the next token is a number. The variable BoutReal curval is set to the value of the token
• -2 for a string (e.g. “sinh”, “x” or “pi”). This includes anything which starts with a letter, and contains only letters, numbers, and underscores. The string is stored in the variable string curident .
• 0 to mean end of input
• The character if none of the above. Since letters and numbers are taken care of (see above), this includes brackets and operators like ’+’ and ’-’.
The parsing stage turns these tokens into a tree of FieldGenerator objects, starting with the parse() function:
FieldGenerator* FieldFactory::parse(const string &input) {
...
which puts the input string into a stream so that nextToken() can use it, then calls the parseExpression() function to do the actual parsing:
FieldGenerator* FieldFactory::parseExpression() {
...
This breaks down expressions in stages, starting with writing every expression as:
expression := primary [ op primary ]
i.e. a primary expression, and optionally an operator and another primary expression. Primary expressions are handled by the parsePrimary() function, so first parsePrimary() is called, and then parseBinOpRHS which checks if there is an operator, and if so calls parsePrimary() to parse it. This code also takes care of operator precedence by keeping track of the precedence of the current operator. Primary expressions are then further broken down and can consist of either a number, a name (identifier), a minus sign and a primary expression, or brackets around an expression:
primary := number
:= identifier
:= '-' primary
:= '(' expression ')'
:= '[' expression ']'
The minus sign case is needed to handle the unary minus e.g. "-x" . Identifiers are handled in parseIdentifierExpr() which handles either variable names, or functions
identifier := name
:= name '(' expression [ ',' expression [ ',' ... ] ] ')'
i.e. a name, optionally followed by brackets containing one or more expressions separated by commas. names without brackets are treated the same as those with empty brackets, so "x" is the same as "x()". A list of inputs (list<FieldGenerator*> args; ) is created, the gen map is searched to find the FieldGenerator object corresponding to the name, and the list of inputs is passed to the object’s clone function. | 2020-10-21 05:29:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.33227404952049255, "perplexity": 3413.1858473609414}, "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/1603107875980.5/warc/CC-MAIN-20201021035155-20201021065155-00107.warc.gz"} |
https://zchee.github.io/golang-wiki/Iota/ | # Summary
Go's iota identifier is used in const declarations to simplify definitions of incrementing numbers. Because it can be used in expressions, it provides a generality beyond that of simple enumerations.
The value of iota is reset to 0 whenever the reserved word const appears in the source (i.e. each const block) and incremented by one after each ConstSpec e.g. each Line. This can be combined with the constant shorthand (leaving out everything after the constant name) to very concisely define related constants.
Constant declarations: https://go.dev/ref/spec#Constant_declarations
# Examples
The official spec has two great examples:
https://go.dev/ref/spec#Iota
Here's one from Effective Go:
type ByteSize float64
const (
_ = iota // ignore first value by assigning to blank identifier
KB ByteSize = 1 << (10 * iota)
MB
GB
TB
PB
EB
ZB
YB
)
Weekday enum example - How iota is calculated - From Learn Go Programming Blog:
# Articles
Last update: March 1, 2022 | 2022-10-01 11:30:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.43234628438949585, "perplexity": 6465.582869084791}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335609.53/warc/CC-MAIN-20221001101652-20221001131652-00087.warc.gz"} |
https://www.semanticscholar.org/paper/Observation-of-contemporaneous-optical-radiation-a-Akerlof-Balsano/891ca6fb27d83427082960a67cf6e04cd3d31e29 | Observation of contemporaneous optical radiation from a γ-ray burst
@article{Akerlof1999ObservationOC,
title={Observation of contemporaneous optical radiation from a $\gamma$-ray burst},
author={Carl W. Akerlof and Richard Balsano and Scott Douglas Barthelmy and Jeffrey J. Bloch and Paul S. Butterworth and Donald E. Casperson and Thomas L. Cline and Sandra Fletcher and Filippo Frontera and Galen R. Gisler and John Heise and JACK G. Hills and Robert Kehoe and B. L. Lee and Stuart L. Marshall and Timothy A. Mckay and R. S. Miller and Luigi Piro and William C. Priedhorsky and John J. Szymanski and James Wren},
journal={Nature},
year={1999},
volume={398},
pages={400-402}
}
• Published 17 March 1999
• Physics
• Nature
The origin of γ-ray bursts (GRBs) has been enigmatic since their discovery. The situation improved dramatically in 1997, when the rapid availability of precise coordinates, for the bursts allowed the detection of faint optical and radio afterglows — optical spectra thus obtained have demonstrated conclusively that the bursts occur at cosmological distances. But, despite efforts by several groups, optical detection has not hitherto been achieved during the brief duration of a burst. Here we…
366 Citations
Prompt Optical Emission from Gamma-ray Bursts
• Physics
• 1999
The Robotic Optical Transient Search Experiment (ROTSE) seeks to measure contemporaneous and early afterglow optical emission from gamma-ray bursts (GRBs). The ROTSE-I telescope array has been fully
Early optical emission from the γ-ray burst of 4 October 2002
• Physics
Nature
• 2003
Observations of the long-lived emission—or ‘afterglow’—of long-duration γ-ray bursts place them at cosmological distances, but the origin of these energetic explosions remains a mystery. Observations
Discovery of Early Optical Emission from GRB 021211
• Physics
• 2003
We report our discovery and early-time optical, near-infrared, and radio wavelength follow-up observations of the afterglow of the gamma-ray burst GRB 021211. Our optical observations, beginning 21
Gamma-ray Bursts
Since their discovery in 1967, Gamma-ray bursts (GRBs) have been puzzling to astrophysicists. With the advent of a new generation of X-ray satellites in the late 90’s, it was possible to carry out
Multiwavelength Observations of GRB 050820A: An Exceptionally Energetic Event Followed from Start to Finish
• Physics
• 2006
We present observations of the unusually bright and long γ-ray burst GRB 050820A, one of the best sampled broadband data sets in the Swift era. The γ-ray light curve is marked by a soft precursor
Very Early Optical Afterglows of Gamma-Ray Bursts: Evidence for Relative Paucity of Detection
• Physics
• 2005
Comparisons of the X-ray afterglow flux to the optical after glow flux and prompt γ-ray fluence is used to quantify the seemingly deficient optical, and in some cases X-rays, light at these early epochs.
A Search for Early Optical Emission from Short- and Long-Duration Gamma-Ray Bursts
• Physics
• 2001
Gamma-ray bursts of short duration may harbor vital clues to the range of phenomena producing bursts. However, recent progress from the observation of optical counterparts has not benefited the study
Prompt Optical Observations of GRB 050319 with the Swift UVOT
• Physics
• 2005
The UVOT telescope on the Swift observatory has detected optical afterglow emission from GRB 050319. The flux declined with a power-law slope of α = -0.57 between the start of observations some 230 s
GRB 061121: Broadband Spectral Evolution through the Prompt and Afterglow Phases of a Bright Burst
• Physics
• 2007
Swift triggered on a precursor to the main burst of GRB 061121 (z = 1.314), allowing observations to be made from the optical to gamma-ray bands. Many other telescopes, including Konus-Wind,
GRB 081008: FROM BURST TO AFTERGLOW AND THE TRANSITION PHASE IN BETWEEN
• Physics
• 2010
A multi-wavelength study of GRB 081008 by Swift, ROTSE-III, and Gamma-Ray Burst Optical/NearInfrared Detector suggests the early optical variability may be from continuous energy injection into the forward shock front by later shells producing the second epoch of burst activities.
References
SHOWING 1-10 OF 21 REFERENCES
Discovery of an X-ray afterglow associated with the γ-ray burst of 28 February 1997
• Physics
Nature
• 1997
Establishing the nature of γ-ray bursts is one of the greatest challenges in high-energy astrophysics. The distribution of these bursts is isotropic across the sky, but inhomogeneous in space, with a
The first X – ray localization of a γ – ray burst by BeppoSAX and its fast spectral evolution
• Physics
• 1997
In this paper we present the observations performed by the BeppoSAX Gamma–Ray Burst Monitor (GRBM) and Wide Field Cameras (WFC) of GB960720. We derive a precise localization (3 arcmin radius) and
GRB 990123: Evidence that the Gamma Rays Come from a Central Engine
• Physics
• 1999
GRB 990123 was a long, complex gamma-ray burst with an optical transient that started early within the gamma-ray phase. The peak and power-law decay of the early optical emission strongly indicate
New Constraints on Simultaneous Optical Emission From Gamma-Ray Bursts Measured by the Livermore Optical Transient Imaging System Experiment
• Physics
• 1997
LOTIS is a gamma-ray burst optical counterpart search experiment located near Lawrence Livermore National Laboratory in California. Since operations began in 1996 October, LOTIS has responded to five
Searches for optical counterparts of BATSE gamma-ray bursts with the Explosive Transient Camera
• Physics
• 1996
The Explosive Transient Camera (ETC) is a wide-field CCD camera system capable of detecting short (1–10 s) celestial optical flashes as faint as m∼10 over a field-of-view of 0.75 steradians between
Results from Gamma-Ray Optical Counterpart Search Experiment: A Real Time Search for Gamma-Ray Burst Optical Counterparts
• Physics
• 1997
The Gamma-Ray Optical Counterpart Search Experiment (GROCSE) has searched for contemporaneous optical counterparts to gamma-ray bursts (GRBs) using an automated rapidly slewing wide field of view
THE FIRST X-RAY LOCALIZATION OF A GAMMA -RAY BURST BY BEPPOSAX AND ITS FAST SPECTRAL EVOLUTION
• Physics
• 1997
In this paper we present the observations performed by the BeppoSAX Gamma-Ray Burst Monitor (GRBM) and Wide Field Cameras (WFC) of GB960720. We derive a precise
Optical and Long-Wavelength Afterglow from Gamma-Ray Bursts
• Physics
• 1997
We discuss the evolution of cosmological gamma-ray burst remnants, consisting of the cooling and expanding fireball ejecta together with any swept-up external matter, after the gamma-ray event. We
THE EARLY AFTERGLOW
• Physics
• 1999
We calculate the expected spectrum and light curves of the early afterglow. For short GRBs the peak of the afterglow will be delayed, typically, by few dozens of seconds after the burst. The X-ray
Observations of Gamma-Ray Bursts of Cosmic Origin
• Physics
• 1973
Sixteen short bursts of photons in the energy range 0.2–1.5 MeV have been observed between 1969 July and 1972 July using widely separated spacecraft. Burst durations ranged from less than 0.1 s to | 2022-10-01 21:28:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4843617081642151, "perplexity": 11010.46535143159}, "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-2022-40/segments/1664030336921.76/warc/CC-MAIN-20221001195125-20221001225125-00674.warc.gz"} |
http://www.citeulike.org/user/victor_rodriguez_toro/article/12033938 | CiteULike is a free online bibliography manager. Register and you can start organising your references online.
Tags
# Scattering of Graphene plasmons by defects in the graphene sheet
(22 Jan 2013) Key: citeulike:12033938
## Likes (beta)
This copy of the article hasn't been liked by anyone yet.
### Abstract
A theoretical study is presented on the scattering of graphene surface plasmons by defects in the graphene sheet they propagate in. These defects can be either natural (as domain boundaries, ripples and cracks, among others) or induced by an external gate. The scattering is shown to be governed by an integral equation, derived from a plane wave expansion of the fields, which in general must be solved numerically but it provides useful analytical results for small defects. Two main cases are considered: smooth variations of the graphene conductivity (characterized by a Gaussian conductivity profile) and sharp variations (represented by islands with different conductivity). In general, reflection largely dominates over radiation out of the graphene sheet. However, in the case of sharply defined conductivity islands there are some values of island size and frequency where the reflectance vanishes and, correspondingly, the radiation out of plane is the main scattering process. For smooth defects, the reflectance spectra present a single maximum at the condition $k_p a ≈ \sqrt2$, where $k_p$ is the GSP wavevector and $a$ the spatial width of the defect. In contrast, the reflectance spectra of sharp defects present periodic oscillations with period $k_p' a$, where $k_p'$ is the GSP wavelength inside the defect. Finally, the case of cracks (gaps in the graphene conductivity) is considered, showing that the reflectance is practically unity for gap widths larger than one tenth of the GSP wavelength.
### Citations (CiTO)
No CiTO relationships defined | 2013-05-23 18:38:28 | {"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": 0.6812351942062378, "perplexity": 1416.3465837282004}, "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-2013-20/segments/1368703682988/warc/CC-MAIN-20130516112802-00093-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/85684/the-question-is-about-elasticity-closed | ## The question is about Elasticity [closed]
Hi all please help me to find the best way to solve this question I have a solution but I don't know how can I calculate it Thanks
Management of first league football team want to calculate number of tickets sold in the next session and think about changing price of the tickets.
a). Elasticity of demand of tickets related to local population is about 0,7. Explain what does it mean. What will be the planning amount of sold tickets when the population in the town will change from 60000 to 61000? (1,16%)
b). Average fan pays nowadays 5 $per ticket. Price elasticity of demand is (–) 0,6. Management wants to increase the price to 5,5$. Calculate the percentage change of sold tickets. Do you expect increase or decrease of total revenue from sold tickets? (-6% ; 5q to 5,17 q)
c). Average fan during football match spends 4$on beer. When the ticket price equals 5$ total revenue per fan is 9$. Calculate if the increase of ticket price to 5,5$ will cause increase or decrease or total revenue for the club. (9q to 8,93q)
-
This is not a question about PDEs. In any case, it is not suited to MathOverflow, which is for maths research questions, and it would belong better on math.stackexchange.com – Yemon Choi Jan 14 2012 at 19:56
This is about homework, so does not belong anywhere. – Igor Rivin Jan 14 2012 at 20:48 | 2013-05-20 00:42:20 | {"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": 0.30790749192237854, "perplexity": 1043.385990405256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698196686/warc/CC-MAIN-20130516095636-00039-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://tailhq.github.io/DynaML/core/core_dtflearn/ | # Building Blocks
Summary
The dtflearn object makes it easy to create and train neural networks of varying complexity.
## Activation Functions¶
Apart from the activation functions defined in tensorflow for scala, DynaML defines some additional activations.
• Hyperbolic Tangent
val act = dtflearn.Tanh("SomeIdentifier")
• Cumulative Gaussian
val act = dtflearn.Phi("OtherIdentifier")
• Generalized Logistic
val act = dtflearn.GeneralizedLogistic("AnotherId")
## Layers¶
DynaML aims to supplement and extend the collection of layers available in org.platanios.tensorflow.api.layers, all the layers defined in DynaML's tensorflow package extend the Layer[T, R] class in org.platanios.tensorflow.api.layers.
Radial Basis Function (RBF) networks are an important class of basis functions, each of which are expressed as decaying with distance from a defined central node.
\begin{align} f(x) & = \sum_{i}{w_{i} \varphi(||x - c_{i}||/\sigma)} \\ \varphi(u) & = exp(-u^2/2) \end{align}
The RBF layer implementation in DynaML treats the node center positions $c_i$ and length scales $\sigma_i$ as parameters to be learned via gradient based back-propagation.
import io.github.mandar2812.dynaml.tensorflow._
val rbf = dtflearn.rbf_layer(name = "rbf1", num_units = 10)
### Continuous Time RNN¶
Continuous time recurrent neural networks (CTRNN) are an important class of recurrent neural networks. They enable the modelling of non-linear and potentially complex dynamical systems of multiple variables, with feedback.
Each state variable is modeled by a single neuron $y_i$, the evolution of the system $y = (y_1, \cdots, y_n)^T$ is governed by a set of coupled ordinary differential equations. These equations can be expressed in vector form as follows.
\begin{align} dy/dt & = - \Lambda . y + W . \sigma(G.y + b) \\ \end{align}
The parameters of the system above are.
• Time Constant/Decay Rate
$$\Lambda = \begin{pmatrix} \lambda_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_n \end{pmatrix}$$
• Gain
$$G = \begin{pmatrix} g_{11} & \cdots & g_{1n} \\ \vdots & \ddots & \vdots \\ g_{n1} & \cdots & g_{nn} \end{pmatrix}$$
• Bias
$$b = \begin{pmatrix} b_{1}\\ \vdots\\ b_{n} \end{pmatrix}$$
• Weights
$$W = \begin{pmatrix} w_{11} & \cdots & w_{1n} \\ \vdots & \ddots & \vdots \\ w_{n1} & \cdots & w_{nn} \end{pmatrix}$$
In order to use the CTRNN model in a modelling sequences of finite length, we need to solve its governing equations numerically. This gives us the trajectory of the state upto $T$ steps $y^{0}, \cdots, y^{T}$.
y^{k+1} = y^{k} + \Delta t (- \Lambda . y^{k} + W . \sigma(G.y^{k} + b))
DynaML's implementation of the CTRNN can be used to learn the trajectory of dynamical systems upto a predefined time horizon. The parameters $\Lambda, G, b, W$ are learned using gradient based loss minimization.
The CTRNN implementations are also instances of Layer[Output, Output], which take as input tensors of shape $n$ and produce tensors of shape $(n, T)$, there are two variants that users can choose from.
#### Fixed Time Step Integration¶
When the integration time step $\Delta t$ is user defined and fixed.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val ctrnn_layer = dtflearn.ctrnn(
name = "CTRNN_1", units = 10,
horizon = 5, timestep = 0.1)
#### Dynamic Time Step Integration¶
When the integration time step $\Delta t$ is a parameter that can be learned during the training process.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val dctrnn_layer = dtflearn.dctrnn(
name = "DCTRNN_1", units = 10,
horizon = 5)
### Stack & Concatenate¶
Often one would need to combine inputs of previous layers in some manner, the following layers enable these operations.
#### Stack Inputs¶
This is a computational layer which performs the function of dtf.stack().
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val stk_layer = dtflearn.stack_outputs("StackTensors", axis = 1)
#### Concatenate Inputs¶
This is a computational layer which performs the function of dtf.concatenate().
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val concat_layer = dtflearn.stack_outputs("ConcatenateTensors", axis = 1)
#### Collect Layers¶
A sequence of layers can be collected into a single layer which accepts a sequence of symbolic tensors.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val layers = Seq(
tf.learn.Linear("l1", 10),
dtflearn.identity("Identity"),
dtflearn.ctrnn(
name = "CTRNN_1", units = 10,
horizon = 5, timestep = 0.1
)
)
val combined_layer = dtflearn.stack_layers("Collect", layers)
#### Input Pairs¶
To handle inputs consisting of pairs of elements, one can provide a separate layer for processing each of the elements.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val sl = dtflearn.tuple2_layer(
"tuple2layer",
dtflearn.rbf_layer("rbf1", 10),
tf.learn.Linear("lin1", 10))
Combining the elements of Tuple2 can be done as follows.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
//Stack elements of the tuple into one tensor
val layer1 = dtflearn.stack_tuple2("tuple2layer", axis = 1)
//Concatenate elements of the tuple into one tensor
val layer2 = dtflearn.concat_tuple2("tuple2layer", axis = 1)
## Stoppage Criteria¶
In order to train tensorflow models using iterative gradient based models, the user must define some stoppage criteria for the training process. This can be done via the method tf.learn.StopCriteria(). The following preset stop criteria call tf.learn.StopCriteria() under the hood.
### Iterations Based¶
val stopc1 = dtflearn.max_iter_stop(10000)
### Change in Loss¶
#### Absolute Value of Loss¶
val stopc2 = dtflearn.abs_loss_change_stop(0.1)
#### Relative Value of Loss¶
val stopc2 = dtflearn.rel_loss_change_stop(0.1)
## Network Building Blocks¶
To make it convenient to build deeper stacks of neural networks, DynaML includes some common layer design patterns as ready made easy to use methods.
### Convolutional Neural Nets¶
Convolutional neural networks (CNN) are a crucial building block of deep neural architectures for visual pattern recognition. It turns out that CNN layers must be combined with other computational units such as rectified linear (ReLU) activations, dropout and max pool layers.
Currently two abstractions are offered for building large CNN based network stacks
#### Convolutional Unit¶
A single CNN unit is expressed as a convolutional layer followed by a ReLU activation and proceeded by a dropout layer.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
//Learn 16 filters of shape (2, 2, 4), suitable for 4 channel jpeg images.
//Slide the filters over the image in steps of 1 pixel in each direction.
val cnn_unit = dtflearn.conv2d_unit(
shape = Shape(2, 2, 4, 16), stride = (1, 1),
relu_param = 0.05f, dropout = true,
keep_prob = 0.55f)(i = 1)
#### Convolutional Pyramid¶
A CNN pyramid builds a stack of CNN units each with a stride multiplied by a factor of 2 and depth divided by a factor of 2 with respect to the previous unit.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
//Start with a CNN unit of shape (2, 2, 3, 16) stride (1, 1)
//End with a CNN unit of shape (2, 2, 8, 4) and stride of (8, 8)
val cnn_stack = dtflearn.conv2d_pyramid(
size = 2, num_channels_input = 3)(
start_num_bits = 4, end_num_bits = 2)(
relu_param = 0.1f, dropout = true,
keep_prob = 0.6F)
### Feed-forward Neural Nets¶
Feed-forward networks are the oldest and most frequently used components of neural network architectures, they are often stacked into a number of layers. With dtflearn.feedforward_stack(), you can define feed-forward stacks of arbitrary width and depth.
import io.github.mandar2812.dynaml.tensorflow._
import org.platanios.tensorflow.api._
val net_layer_sizes = Seq(10, 20, 13, 15)
val architecture = dtflearn.feedforward_stack(
(i: Int) => dtflearn.Phi("Act_"+i), FLOAT64)(
net_layer_sizes)
## Building Tensorflow Models¶
After defining the key ingredients needed to build a tensorflow model, dtflearn.build_tf_model() builds a new computational graph and creates a tensorflow model and estimator which is trained on the provided data. In the following example, we bring together all the elements of model training: data, architecture, loss etc.
import ammonite.ops._
import io.github.mandar2812.dynaml.tensorflow.dtflearn
import org.platanios.tensorflow.api._
import java.nio.file.Paths
val tempdir = home/"tmp"
Paths.get(tempdir.toString()),
val trainImages = tf.data.TensorSlicesDataset(dataSet.trainImages)
val trainLabels = tf.data.TensorSlicesDataset(dataSet.trainLabels)
val trainData =
trainImages.zip(trainLabels)
.repeat()
.shuffle(10000)
.batch(128)
.prefetch(10)
println("Building the classification model.")
val input = tf.learn.Input(
UINT8,
Shape(
-1,
dataSet.trainImages.shape(1),
dataSet.trainImages.shape(2),
dataSet.trainImages.shape(3))
)
val trainInput = tf.learn.Input(UINT8, Shape(-1))
val architecture = tf.learn.Cast("Input/Cast", FLOAT32) >>
dtflearn.conv2d_pyramid(2, 3)(4, 2)(0.1f, true, 0.6F) >>
tf.learn.MaxPool(
"Layer_3/MaxPool",
Seq(1, 2, 2, 1),
tf.learn.Flatten("Layer_3/Flatten") >>
dtflearn.feedforward(256)(id = 4) >>
tf.learn.ReLU("Layer_4/ReLU", 0.1f) >>
dtflearn.feedforward(10)(id = 5)
val trainingInputLayer = tf.learn.Cast("TrainInput/Cast", INT64)
val loss = tf.learn.SparseSoftmaxCrossEntropy("Loss/CrossEntropy") >>
tf.learn.Mean("Loss/Mean") >>
tf.learn.ScalarSummary("Loss/Summary", "Loss")
println("Training the linear regression model.")
val summariesDir = java.nio.file.Paths.get(
(tempdir/"cifar_summaries").toString()
)
val (model, estimator) = dtflearn.build_tf_model(
architecture, input, trainInput, trainingInputLayer,
loss, optimizer, summariesDir,
dtflearn.max_iter_stop(1000),
100, 100, 100)(trainData) | 2022-08-14 15:45:52 | {"extraction_info": {"found_math": true, "script_math_tex": 11, "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": 4, "equation": 8, "x-ck12": 0, "texerror": 0, "math_score": 0.523727297782898, "perplexity": 6229.435697128444}, "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/1659882572043.2/warc/CC-MAIN-20220814143522-20220814173522-00203.warc.gz"} |
https://docs.sciml.ai/stable/analysis/global_sensitivity/ | # Global Sensitivity Analysis
Global Sensitivity Analysis (GSA) methods are used to quantify the uncertainty in output of a model w.r.t. the parameters, their individual contributions, or the contribution of their interactions. The GSA interface allows for utilizing batched functions for parallel computation of GSA quantities.
## Installation
This functionality does not come standard with DifferentialEquations.jl. To use this functionality, you must install DiffEqSensitivity.jl:
]add DiffEqSensitivity
using DiffEqSensitivity
## General Interface
The general interface for calling a global sensitivity analysis is either:
effects = gsa(f, method, param_range; N, batch=false)
where:
• y=f(x) is a function that takes in a single vector and spits out a single vector or scalar. If batch=true, then f takes in a matrix where each row is a set of parameters, and returns a matrix where each row is a the output for the corresponding row of parameters.
• method is one of the GSA methods below.
• param_range is a vector of tuples for the upper and lower bound for the given parameter i.
• N is a required keyword argument for the number of samples to take in the trajectories/design.
Note that for some methods there is a second interface where one can directly pass the design matrices:
effects = gsa(f, method, A, B; batch=false)
where A and B are design matrices with each row being a set of parameters. Note that generate_design_matrices from QuasiMonteCarlo.jl can be used to generate the design matrices.
### Morris Method
Morris has the following keyword arguments:
• p_steps - Vector of $\Delta$ for the step sizes in each direction. Required.
• relative_scale - The elementary effects are calculated with the assumption that the parameters lie in the range [0,1] but as this is not always the case scaling is used to get more informative, scaled effects. Defaults to false.
• total_num_trajectory, num_trajectory - The total number of design matrices that are generated out of which num_trajectory matrices with the highest spread are used in calculation.
• lendesignmat - The size of a design matrix.
#### Morris Method Details
The Morris method also known as Morris’s OAT method where OAT stands for One At a Time can be described in the following steps:
We calculate local sensitivity measures known as “elementary effects”, which are calculated by measuring the perturbation in the output of the model on changing one parameter.
$EE_i = \frac{f(x_1,x_2,..x_i+ \Delta,..x_k) - y}{\Delta}$
These are evaluated at various points in the input chosen such that a wide “spread” of the parameter space is explored and considered in the analysis, to provide an approximate global importance measure. The mean and variance of these elementary effects is computed. A high value of the mean implies that a parameter is important, a high variance implies that its effects are non-linear or the result of interactions with other inputs. This method does not evaluate separately the contribution from the interaction and the contribution of the parameters individually and gives the effects for each parameter which takes into consideration all the interactions and its individual contribution.
### Sobol Method
The Sobol object has as its fields the order of the indices to be estimated.
• order - the order of the indices to calculate. Defaults to [0,1], which means the Total and First order indices. Passing 2 enables calculation of the Second order indices as well.
• Ei_estimator - Can take :Homma1996, :Sobol2007 and :Jansen1999 for which Monte Carlo estimator is used for the Ei term. Defaults to :Jansen1999.
#### Sobol Method Details
Sobol is a variance-based method and it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. This helps to get not just the individual parameter's sensitivities but also gives a way to quantify the affect and sensitivity from the interaction between the parameters.
$Y = f_0+ \sum_{i=1}^d f_i(X_i)+ \sum_{i < j}^d f_{ij}(X_i,X_j) ... + f_{1,2...d}(X_1,X_2,..X_d)$
$Var(Y) = \sum_{i=1}^d V_i + \sum_{i < j}^d V_{ij} + ... + V_{1,2...,d}$
The Sobol Indices are "order"ed, the first order indices given by $S_i = \frac{V_i}{Var(Y)}$ the contribution to the output variance of the main effect of $X_i$, therefore it measures the effect of varying $X_i$ alone, but averaged over variations in other input parameters. It is standardised by the total variance to provide a fractional contribution. Higher-order interaction indices $S_{i,j}, S_{i,j,k}$ and so on can be formed by dividing other terms in the variance decomposition by $Var(Y)$.
### eFAST Method
eFAST has num_harmonics as the only argument, it is the number of harmonics to sum in the Fourier series decomposition and defaults to 4.
#### eFAST Method Details
eFAST offers a robust, especially at low sample size, and computationally efficient procedure to get the first and total order indices as discussed in Sobol. It utilizes monodimensional Fourier decomposition along a curve exploring the parameter space. The curve is defined by a set of parametric equations,
$x_{i}(s) = G_{i}(sin ω_{i}s), ∀ i=1,2 ,..., n,$
where s is a scalar variable varying over the range $-∞ < s < +∞$, $G_{i}$ are transformation functions and ${ω_{i}}, ∀ i=1,2,...,n$ is a set of different (angular) frequencies, to be properly selected, associated with each factor. For more details on the transformation used and other implementation details you can go through A. Saltelli et al..
### Regression Method
RegressionGSA has the following keyword arguments:
• rank: flag which determines whether to calculate the rank coefficients. Defaults to false.
It returns a RegressionGSAResult, which contains the pearson, standard_regression, and partial_correlation coefficients, described below. If rank is true, then it also contains the ranked versions of these coefficients. Note that the ranked version of the pearson coefficient is also known as the Spearman coefficient, which is returned here as the pearson_rank coefficient.
For multi-variable models, the coefficient for the $X_i$ input variable relating to the $Y_j$ output variable is given as the [i, j] entry in the corresponding returned matrix.
#### Regression Details
It is possible to fit a linear model explaining the behavior of Y given the values of X, provided that the sample size n is sufficiently large (at least n > d).
The measures provided for this analysis by us in DiffEqSensitivity.jl are
a) Pearson Correlation Coefficient:
$r = \frac{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2(y_i - \overline{y})^2}}$
b) Standard Regression Coefficient (SRC):
$SRC_j = \beta_{j} \sqrt{\frac{Var(X_j)}{Var(Y)}}$
where $\beta_j$ is the linear regression coefficient associated to $X_j$. This is also known as a sigma-normalized derivative.
c) Partial Correlation Coefficient (PCC):
$PCC_j = \rho(X_j - \hat{X_{-j}},Y_j - \hat{Y_{-j}})$
where $\hat{X_{-j}}$ is the prediction of the linear model, expressing $X_{j}$ with respect to the other inputs and $\hat{Y_{-j}}$ is the prediction of the linear model where $X_j$ is absent. PCC measures the sensitivity of $Y$ to $X_j$ when the effects of the other inputs have been canceled.
If rank is set to true, then the rank coefficients are also calculated.
## GSA examples
### Lotka-Volterra Global Sensitivities
Let's run GSA on the Lotka-Volterra model to and study the sensitivity of the maximum of predator population and the average prey population.
using DiffEqSensitivity, Statistics, OrdinaryDiffEq #load packages
First let's define our model:
function f(du,u,p,t)
du[1] = p[1]*u[1] - p[2]*u[1]*u[2] #prey
du[2] = -p[3]*u[2] + p[4]*u[1]*u[2] #predator
end
u0 = [1.0;1.0]
tspan = (0.0,10.0)
p = [1.5,1.0,3.0,1.0]
prob = ODEProblem(f,u0,tspan,p)
t = collect(range(0, stop=10, length=200))
Now let's create a function that takes in a parameter set and calculates the maximum of the predator population and the average of the prey population for those parameter values. To do this, we will make use of the remake function which creates a new ODEProblem, and use the p keyword argument to set the new parameters:
f1 = function (p)
prob1 = remake(prob;p=p)
sol = solve(prob1,Tsit5();saveat=t)
[mean(sol[1,:]), maximum(sol[2,:])]
end
Now let's perform a Morris global sensitivity analysis on this model. We specify that the parameter range is [1,5] for each of the parameters, and thus call:
m = gsa(f1,Morris(total_num_trajectory=1000,num_trajectory=150),[[1,5],[1,5],[1,5],[1,5]])
Let's get the means and variances from the MorrisResult struct.
m.means
2×2 Array{Float64,2}:
0.474053 0.114922
1.38542 5.26094
m.variances
2×2 Array{Float64,2}:
0.208271 0.0317397
3.07475 118.103
Let's plot the result
scatter(m.means[1,:], m.variances[1,:],series_annotations=[:a,:b,:c,:d],color=:gray)
scatter(m.means[2,:], m.variances[2,:],series_annotations=[:a,:b,:c,:d],color=:gray)
For the Sobol method we can similarly do:
m = gsa(f1,Sobol(),[[1,5],[1,5],[1,5],[1,5]],N=1000)
### Design Matrices
For the Sobol Method, we can have more control over the sampled points by generating design matrices. Doing it in this manner lets us directly specify a quasi-Monte Carlo sampling method for the parameter space. Here we use QuasiMonteCarlo.jl to generate the design matrices as follows:
N = 10000
lb = [1.0, 1.0, 1.0, 1.0]
ub = [5.0, 5.0, 5.0, 5.0]
sampler = SobolSample()
A,B = QuasiMonteCarlo.generate_design_matrices(N,lb,ub,sampler)
and now we tell it to calculate the Sobol indices on these designs:
sobol_result = gsa(f1,Sobol(),A,B)
We plot the first order and total order Sobol Indices for the parameters (a and b).
p1 = bar(["a","b","c","d"],sobol_result.ST[1,:],title="Total Order Indices prey",legend=false)
p2 = bar(["a","b","c","d"],sobol_result.S1[1,:],title="First Order Indices prey",legend=false)
p1_ = bar(["a","b","c","d"],sobol_result.ST[2,:],title="Total Order Indices predator",legend=false)
p2_ = bar(["a","b","c","d"],sobol_result.S1[2,:],title="First Order Indices predator",legend=false)
plot(p1,p2,p1_,p2_)
### Parallelized GSA Example
In all of the previous examples, f(p) was calculated serially. However, we can parallelize our computations by using the batch interface. In the batch interface, each column p[:,i] is a set of parameters, and we output a column for each set of parameters. Here we showcase using the Ensemble Interface to use EnsembleGPUArray to perform automatic multithreaded-parallelization of the ODE solves.
f1 = function (p)
prob_func(prob,i,repeat) = remake(prob;p=p[:,i])
ensemble_prob = EnsembleProblem(prob,prob_func=prob_func)
# Now sol[i] is the solution for the ith set of parameters
out = zeros(2,size(p,2))
for i in 1:size(p,2)
out[1,i] = mean(sol[i][1,:])
out[2,i] = maximum(sol[i][2,:])
end
out
end
And now to do the parallelized calls we simply add the batch=true keyword argument:
sobol_result = gsa(f1,Sobol(),A,B,batch=true)`
This user-side parallelism thus allows you to take control, and thus for example you can use DiffEqGPU.jl for automated GPU-parallelism of the ODE-based global sensitivity analysis! | 2020-07-06 03:13:15 | {"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7170119881629944, "perplexity": 1166.2924674810042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890092.28/warc/CC-MAIN-20200706011013-20200706041013-00159.warc.gz"} |
https://www.physicsoverflow.org/34704/dysons-argumet-about-divergent-nature-perturbative-expansion | # Doubt in Dyson's argumet about the divergent nature of the perturbative expansion in QED
+ 2 like - 0 dislike
268 views
I am trying to understand Dyson's argument about the divergent nature of the perturbative expansion in QED. Quoting his own words [Source: hep-ph:0508017 (PDF)]
... let
$$F(e^2)=a_0+a_1e^2+a_2e^4+\ldots$$
be a physical quantity which is calculated as a formal power series in $e^2$ by integrating the equations of motion of the theory over a finite or infinite time.Suppose, if possible, that the series... converges for some positive
value of $e^2$; this implies that $F(e^2)$ is an analytic function
of $e$ at $e=0$. Then for sufficiently small value of $e$, $F(−e^2)$ will also be a well-behaved analytic function with a convergent power series expansion.
My question is, why does the convergence of the series for some positive value of $e^2$ imply that it must be analytic at $e=0$?
edited Dec 25, 2015
Your question in the text is purely mathematical. It concerns all functions.
As to real doubts about Dyson's argument, I have already written many times that the used in practice series are not in powers of $e^2$ or $\alpha$. For example, the meaningful inclusive cross section is a selective sum over all orders of $\alpha$: $F(\alpha)=G(\alpha) + b_0+b_1\cdot\alpha+...=G(1/137) + b_0+b_1\cdot\alpha+...$. First, G(1/137) is finite and the remaining series is IR finite too. Next, all non-analyticity may be contained in $G(\alpha)$ and the remaining series may represent an analytical function. This question has never been studied, so the Dyson's argument is just wrong as dealing with a wrong object - an IR-divergent series.
@VladimirKalitvianski: The Dyson series is being used in may places for finding the magnetic moment of the electron at first nontrivial order. Higher order computations of magnetic moments, lamb shifts, etc. in QED are done in NRQED, which is a joint expansion in $\alpha$ and $Z\alpha$; see. e.g., this article by Kinoshita.
Your method, on the other hand, has never been used in QED, only in your toy models.
@ArnoldNeumaier : The question is not about "my method", but about QED series. And "my method" is about possibilities of representation of a function in different ways. Factually, it is already written generally as $F(x)=G(x)$+another series in my comment above, without necessity to look into details of "my method".
@VladimirKalitvianski: The question is explicitly about the Dyson series in QED, not about resummation techniques that rearrange terms. So strictly speaking your comment was off-topic.
@ArnoldNeumaier : "Resummation" is a necessity and banality in QED; it is given in all QED textbooks, but if you think that I am off-topic, you may delete my comments entirely.
Analytic at $x=0$ means convergent for $x$ in the interior of some disk around zero; its maximal radius $r$ is the radius of convergence of the series. The root test for the convergence of an infinite series implies that if a power series in $x$ converges for some nonzero $x$ then the radius of convergence is at least $|x|$. In particular, if the Dyson series converges for some $e^2=x>0$, it also converges for $e^2=-x/2<0$.
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register. | 2018-08-16 02:10:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7908790111541748, "perplexity": 642.5218556967094}, "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/1534221210408.15/warc/CC-MAIN-20180816015316-20180816035316-00265.warc.gz"} |
https://www.physicsforums.com/threads/basic-trigonometric-limit.271382/ | # Basic Trigonometric Limit
1. Nov 12, 2008
### optics.tech
Hi everyone,
I would like to know, since the lim theta --> 0 (sine theta / theta) = 1, what is the limit theta --> 0 of cosine and tangent?
Thank you very much in advance
2. Nov 12, 2008
### Staff: Mentor
What do you think they are? The cos(theta)/theta case seems pretty easy, right? And how is the sin(theta)/theta limit calculated? Can you use the same technique in the tan(theta)/theta case?
3. Nov 12, 2008
### HallsofIvy
Staff Emeritus
What do you mean by "limit of cosine and tangent"? If you mean those functions separately, they are continuous so the limit of cosine(theta) as theta goes to 0 is cos(0)= 1. The limit of tan(theta) as theta goes to 0 is tan(0)= 0.
If you mean, as berkeman says, cos(theta)/theta, and tan(theta)/theta, all you need to know is that cos(theta)/theta has numerator close to 1 and denominator close to 0 for theta close to 0. For tan(theta)/theta, think tan(theta)= sin(theta)/cos(theta) so tan(theta)/theta= (sin(theta)/theta)/cos(theta). | 2017-01-16 13:23:09 | {"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": 0.8357263207435608, "perplexity": 3458.7872727365834}, "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/1484560279176.20/warc/CC-MAIN-20170116095119-00311-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/149542/is-this-the-nmr-spectrum-of-3-methyl-1-butanol | # Is this the NMR spectrum of 3-methyl-1-butanol?
It is a $$\ce{^1H}$$-NMR spectrum and I think the signal near 3,61 ppm is the CH2 group next to the hydroxy group. But which signal is the hydroxy group and which protons are in the multiplet?
Welcome to ChemistrySE.
In short: is the spectrum consistent with 3-methyl-1-butanol? No. The triplet splitting and integral of the signal at 0.86ppm is inconsistent with an isopropyl sidechain. This suggests a methyl group attached to a methylene, $$\ce{-CH2-CH3}$$.
The signal at $$\pu{3.6 ppm}$$ has a chemical shift symptomatic of an alcohol, as you suggest. It has diagnostic splitting of $$\pu{4.2 Hz}$$, also consistent with coupling to the alcohol peak. So, this peak is a doublet of triplets..... which makes it look a bit like a quartet. To see this coupling, the sample needs to very dry to eliminate exchange decoupling. So where is the alcohol peak? In the absence of any exchange, and in chloroform (most likely solvent used given the spectrum shown), alcohol peaks usually come in the region of about $$1.2$$-$$\pu{1.5 ppm}$$. So possibly under the multiplet. Which would then leave you with 6 protons to account for......or $$3 \times \ce{CH2}$$ groups. The peak at $$\pu{1.5 ppm}$$ has coupling that suggests it is attached to the $$\ce{-CH2OH}$$, giving a $$\ce{-CH2CH2OH}$$ group. There are 14 protons in all. You've got a $$\ce{-CH2-CH3}$$, a $$\ce{-CH2CH2OH}$$ and the rest are probably $$\ce{-CH2\!-}$$. How might you put them together?
In the real world, identification of complete unknowns are rarely done using $$\mathrm{^1H \ NMR}$$ alone. Mass Spec, $$\mathrm{^{13}C}$$, and a range of $$\mathrm{2D}$$ options are very useful.
And to answer a follow-up question below, the splitting at 1.54 is from the protons on the beta carbon, and is a triplet of triplets. When coupling constants are close (6.5 and 6.8) this will appear as a pseudo-pentet, as shown below:
• Do you also know which solvent is used? – Chemistrygirl Apr 15 at 11:39
• Not conclusively. CDCl3 is the most common solvent used for small organics - its residual peak comes at 7.26ppm, and so not visible in this spectrum. Other common possible solvents (dmso, methanol, acetone) are not visible in this spectrum. – long Apr 18 at 7:07
• Thanks for your answer! One last thing I want to know.... Can you explain the splitting at 1,54Hz? – Chemistrygirl Apr 29 at 8:21
• This is due to the protons on the beta carbon (the middle ones in this group) -CH2-CH2-CH2OH. They are coupled to the terminal CH2 and also the gamma CH2, so will be a triplet of triplets. When the coupling is very close, (6.8 and 6.5), this pattern will look like a pseudo-pentet. I'll put the splitting pattern in the answer above. Feel free to accept the answer to close it off if it addresses what you were after. – long Apr 29 at 22:13 | 2021-07-23 23:26:39 | {"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": 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": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49621716141700745, "perplexity": 1295.8064769827165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150067.51/warc/CC-MAIN-20210723210216-20210724000216-00588.warc.gz"} |
https://www.preprints.org/manuscript/201809.0608/v1 | Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed
# Hyers-Ulam Stability of Lagrange's Mean Value Points in Two Variables
Version 1 : Received: 30 September 2018 / Approved: 30 September 2018 / Online: 30 September 2018 (10:42:59 CEST)
A peer-reviewed article of this Preprint also exists.
Jung, S.-M.; Kim, J.-H. Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables. Mathematics 2018, 6, 216. Jung, S.-M.; Kim, J.-H. Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables. Mathematics 2018, 6, 216.
Journal reference: Mathematics 2018, 6, 216
DOI: 10.3390/math6110216
## Abstract
Using a theorem of Ulam and Hyers, we will prove the Hyers-Ulam stability of two-dimensional Lagrange's mean value points $(\eta, \xi)$ which satisfy the equation, $f(u, v) - f(p, q) = (u-p) f_x(\eta, \xi) + (v-q) f_y(\eta, \xi)$, where $(p, q)$ and $(u, v)$ are distinct points in the plane. Moreover, we introduce an efficient algorithm for applying our main result in practical use.
## Keywords
Hyers-Ulam stability; mean value theorem; Lagrange's mean value point; two-dimensional Lagrange's mean value point
## Subject
MATHEMATICS & COMPUTER SCIENCE, Analysis
Views 0 | 2021-12-07 12:09:25 | {"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": 0.6459468007087708, "perplexity": 2603.5960624562995}, "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/1637964363376.49/warc/CC-MAIN-20211207105847-20211207135847-00174.warc.gz"} |
http://en.wikipedia.org/wiki/Triangular_distribution | # Triangular distribution
Parameters Probability density function Cumulative distribution function $a:~a\in (-\infty ,\infty )$ $b:~a $c:~a\leq c\leq b\,$ $a\leq x\leq b\!$ ${\begin{cases}0&{\mathrm {for\ }}x ${\begin{cases}0&{\mathrm {for\ }}x ${\frac {a+b+c}{3}}$ ${\begin{cases}a+{\frac {{\sqrt {(b-a)(c-a)}}}{{\sqrt {2}}}}&{\mathrm {for\ }}c\geq {\frac {a+b}{2}},\\[6pt]b-{\frac {{\sqrt {(b-a)(b-c)}}}{{\sqrt {2}}}}&{\mathrm {for\ }}c\leq {\frac {a+b}{2}}.\end{cases}}$ $c\,$ ${\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}$ ${\frac {{\sqrt 2}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{{\frac {3}{2}}}}}$ $-{\frac {3}{5}}$ ${\frac {1}{2}}+\ln \left({\frac {b-a}{2}}\right)$ $2{\frac {(b\!-\!c)e^{{at}}\!-\!(b\!-\!a)e^{{ct}}\!+\!(c\!-\!a)e^{{bt}}}{(b-a)(c-a)(b-c)t^{2}}}$ $-2{\frac {(b\!-\!c)e^{{iat}}\!-\!(b\!-\!a)e^{{ict}}\!+\!(c\!-\!a)e^{{ibt}}}{(b-a)(c-a)(b-c)t^{2}}}$
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. The probability density function is given by $Probability density function for the triangular distribution.$
whose cases avoid division by zero if c = a or c = b.
## Special cases
### Two points known
The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the equations above become:
$\left.{\begin{matrix}f(x)&=&2x\\[8pt]F(x)&=&x^{2}\end{matrix}}\right\}{\text{ for }}0\leq x\leq 1$
{\begin{aligned}E(X)&={\frac {2}{3}}\\[8pt]{\mathrm {Var}}(X)&={\frac {1}{18}}\end{aligned}}
### Distribution of mean of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0.5 is distribution of X = (X1 + X2)/2, where X1, X2 are two independent random variables with standard uniform distribution.
$f(x)={\begin{cases}4x&{\text{for }}0\leq x<{\frac {1}{2}}\\4-4x&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}$
$F(x)={\begin{cases}2x^{2}&{\text{for }}0\leq x<{\frac {1}{2}}\\1-2(1-x)^{2}&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}$
{\begin{aligned}E(X)&={\frac {1}{2}}\\[6pt]\operatorname {Var}(X)&={\frac {1}{24}}\end{aligned}}
### Distribution of the absolute difference of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0 is distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard uniform distribution.
{\begin{aligned}f(x)&=2-2x{\text{ for }}0\leq x<1\\[6pt]F(x)&=2x-x^{2}{\text{ for }}0\leq x<1\\[6pt]E(X)&={\frac {1}{3}}\\[6pt]\operatorname {Var}(X)&={\frac {1}{18}}\end{aligned}}
## Generating Triangular-distributed random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
${\begin{matrix}{\begin{cases}X=a+{\sqrt {U(b-a)(c-a)}}&{\text{ for }}0[1]
Where F(c) = (c-a)/(b-a)
has a Triangular distribution with parameters a, b and c. This can be obtained from the cumulative distribution function.
## Use of the distribution
The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on a knowledge of the minimum and maximum and an "inspired guess" [2] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution. | 2014-03-08 05:03:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 23, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8947639465332031, "perplexity": 10265.807388125504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999653202/warc/CC-MAIN-20140305060733-00055-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.sasbury.com/notes/omnibus/finite_automata.html | ## Finite Automata
Chapter 2 of the The New Turing Omnibus is about Finite Automata. I wanted to do some of the exercises on the blog so I looked around for a good way to draw finite state diagrams. Looks like graphviz is the standard. I thought it would be easier if I could just write my graphviz code in my markdown, so I looked around some more and found this discussion on stack overflow. There is a pointer to a version of graphviz compiled with llvm and emscripten on github. So I updated my pages to automatically process pre.graphviz tags as input for graphviz. Below are the results for some of the exercises at the end of chapter 2.
The core code is:
svg = Viz(graphvizCode, 'svg');
\$('#svg-'+index).html(svg);
Each exercise is to describe the finite automata that accept the following languages.
I haven't drawn a lot of these, so comments are welcome.
### (ab)*(a+b)*
The first exercise is to find the automata that processes (ab)\*(a+b)\*. I read this as:
3. go back to 1, or continue
5. go back to 4 or finish
As a finite state machine, i see this as:
but this is the one I am least sure about. I am not sure if I am using the states 1 and 2 correctly.
### 0 + 1
The second language is 0 + 1 which I see as match 0 or 1 and then you are done.
### 0* + 1
The final exercise is the language 0* + 1. This means match "1 or more" zeros or a 1. After matching, the language is finished.
I really like thinking about these finite automata because, as Dewdney says, finite automata are basically equivalent to regular expressions. I have started to use regular expressions a lot more over the last year and find them to be a really amazing tool once you start to internalize them. | 2020-10-20 22:17:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2898094952106476, "perplexity": 610.1240114571069}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874340.10/warc/CC-MAIN-20201020221156-20201021011156-00020.warc.gz"} |
https://jp.maplesoft.com/support/help/maple/view.aspx?path=DifferentialGeometry%2FHook | Hook - Maple Help
DifferentialGeometry
Hook
form the interior product of a vector or a list of vectors with a differential form
Calling Sequence Hook(X, omega) Hook(Y, T, indexlist)
Parameters
X - a vector or a list of vectors omega - a differential p-form Y - a list of vectors or differential 1-forms T - a tensor indexlist - (optional) a list of positive integers labeling various arguments of the tensor T
Description
• If X is a vector and omega a differential p-form, then the interior product or hook of X and omega is the differential (p - 1)-form theta = Hook(X, omega) defined by theta(Y1, Y2, ..., Yq) = omega(X, Y1, Y2, ..., Yq), where q = p - 1 and Y1, Y2, ..., Yq are arbitrary vectors.
• More generally, given a list of vectors [X1, X2, ... , Xr] and a differential p-form omega then theta = Hook([X1, X2, ..., Xr], omega) is the differential form of degree p - r defined by theta(Y1, Y2, ..., Yq) = omega(X1, X2, ..., Xr, Y1, Y2, ..., Yq), where q = p - r and Y1, Y2, ...,Yq are arbitrary vectors.
• If Y = [X1, X2, alpha1, ..., Xr] is a list of vectors or differential 1-forms and T is a tensor of total rank s = r, then the second calling sequence evaluates the scalar T(X1, X2, alpha1, ... , Xr). If s > r and indexlist = [i_1, i_2, ..., i_r], then Hook(Y, T, indexlist) calculates the rank s - r tensor obtained by evaluating the i_k-th argument of the tensor T on the k-th element of the list Y, for k = 1, 2, ... r.
• This command is part of the DifferentialGeometry package, and so can be used in the form Hook(...) only after executing the command with(DifferentialGeometry). It can always be used in the long form DifferentialGeometry:-Hook.
Examples
> $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$
> $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
Example 1.
Define vectors X1, X2, X3.
> $\mathrm{X1}≔\mathrm{evalDG}\left(a\mathrm{D_x}+b\mathrm{D_y}+c\mathrm{D_z}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]\right]\right)$ (1)
> $\mathrm{X2}≔\mathrm{evalDG}\left(2\mathrm{D_y}-\mathrm{D_z}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{2}\right]{,}\left[\left[{3}\right]{,}{-}{1}\right]\right]\right]\right)$ (2)
> $\mathrm{X3}≔\mathrm{evalDG}\left(\mathrm{D_x}+\mathrm{D_z}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)$ (3)
Define a 1-form alpha, a 2-form beta and a 3-form omega.
> $\mathrm{\alpha }≔\mathrm{evalDG}\left(3\mathrm{dx}+4\mathrm{dy}-7\mathrm{dz}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{3}\right]{,}\left[\left[{2}\right]{,}{4}\right]{,}\left[\left[{3}\right]{,}{-}{7}\right]\right]\right]\right)$ (4)
> $\mathrm{\beta }≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-3\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (5)
> $\mathrm{\omega }≔\mathrm{evalDG}\left(\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (6)
Compute various interior products. Note that Hook(Y, Hook(X, omega) = Hook([X, Y], omega).
> $f≔\mathrm{Hook}\left(\mathrm{X1},\mathrm{\alpha }\right)$
${f}{≔}{-}{7}{}{c}{+}{4}{}{b}{+}{3}{}{a}$ (7)
> $\mathrm{\theta }≔\mathrm{Hook}\left(\mathrm{X1},\mathrm{\beta }\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{3}{}{c}{-}{b}\right]{,}\left[\left[{2}\right]{,}{a}\right]{,}\left[\left[{3}\right]{,}{-}{3}{}{a}\right]\right]\right]\right)$ (8)
> $\mathrm{Hook}\left(\mathrm{X2},\mathrm{\theta }\right)$
${5}{}{a}$ (9)
> $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2}\right],\mathrm{\beta }\right)$
${5}{}{a}$ (10)
> $\mathrm{Hook}\left(\left[\mathrm{X2},\mathrm{X1}\right],\mathrm{\beta }\right)$
${-}{5}{}{a}$ (11)
> $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2},\mathrm{X3}\right],\mathrm{\omega }\right)$
${2}{}{a}{-}{b}{-}{2}{}{c}$ (12)
Example 2.
Evaluate a type (1,3) tensor T on various vectors and differential forms.
> $T≔\mathrm{evalDG}\left(2\left(\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}-3\left(\left(\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{2}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (13)
> $\mathrm{Hook}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z},\mathrm{dy}\right],T\right)$
${2}$ (14)
> $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X3}\right],T\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{-}{3}{}{b}\right]\right]\right]\right)$ (15)
> $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2},\mathrm{X3},\mathrm{\alpha }\right],T\right)$
${16}{}{a}$ (16)
> $\mathrm{Hook}\left(\left[\mathrm{D_y},\mathrm{dz}\right],T,\left[1,4\right]\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (17)
> $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2}\right],T,\left[1,3\right]\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{3}{}{b}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{2}{}{a}\right]\right]\right]\right)$ (18)
Example 3.
The interior product can be calculated for abstract forms.
> $\mathrm{DGsetup}\left(\left[\left[\mathrm{ω1},\mathrm{ω2},\mathrm{ω3}\right],\mathrm{β1}=\mathrm{dgform}\left(2\right)\right],\left[d\left(\mathrm{ω1}\right)=\mathrm{ω2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ω3},d\left(\mathrm{ω2}\right)=\mathrm{β1}\right],\mathrm{M3}\right)$
${\mathrm{frame name: M3}}$ (19)
> $\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{ω1}\right);$$\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)$
${1}$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M3}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)$ (20)
Structure equations for interior products can be specified.
> $\mathrm{DGsetup}\left(\mathrm{M3},\left[\right],\left[\mathrm{hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)=\mathrm{ω3}\right]\right)$
${\mathrm{updated frame: M3}}$ (21)
> $\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)$
${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M3}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)$ (22)
M3 > | 2023-03-24 18:50:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 49, "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": 0.858972430229187, "perplexity": 3753.078831076451}, "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-2023-14/segments/1679296945288.47/warc/CC-MAIN-20230324180032-20230324210032-00166.warc.gz"} |
https://dsp.stackexchange.com/questions/37809/how-will-the-signal-sum-n-infty-infty-delta-t-nu-1t-n-look-like | # How will the signal $\sum_{n=-\infty}^\infty \Delta (t-n)u_{-1}(t-n)$ look like?
How will the signal $\sum_{n=-\infty}^\infty \Delta (t-n)u_{-1}(t-n)$ look like?
1. I understand that $\sum_{n=-\infty}^\infty \Delta (t-n)$ is a periodic signal with period $1$, where the triangular pulse $\Delta(t)$ is shifted by $1$ units for each value of $n$.
where, $$\Delta(t) = \begin{cases} 1-2\vert t\rvert,& \lvert t\rvert<\frac{1}{2} \\ 0,&\lvert t\rvert>\frac{1}{2} \end{cases}$$
2. From this I infer that $\sum_{n=-\infty}^\infty \Delta (t-n)u(t-n)$ would probably be a signal in which the RIGHT HALF of $\Delta (t)$ will be shifted by $1$ unit for every value of $n$. But here I have a doubt that $u(t-n)$ will be added with the previous values of $u(t-n)$ i.e $u(t-(n-1))$ and will have a staircase effect. Thus the signal will no longer be periodic.(Which is a cause of concern as we have been asked to find the Fourier series of the signal, which means the signal should be periodic)
3. I also don't understand what the subscript $-1$ in $u_{-1}(t-n)$ means.
Please help me visualize this signal(It would be highly helpful if you would draw the graph as well to help me visualize the signal)
• As long as we don't know what $u_{-1}(t)$ means I'm afraid nobody can help you. If that exercise is from a book, they should define it somewhere. My guess would be that it's a time-inverted step (but I don't know why one would need such a notation), i.e., $u_{-1}(t)=u(-t)$, but that's just a wild guess. – Matt L. Feb 22 '17 at 7:20
• @MattL. I too was thinking of that possibility, but when I saw the Fourier coefficients being calculated as $x_n=\int_{0}^1(-t+1)e^{-j2\pi nt}$ in the solution manual(this one is authorised), I ruled out that possibility.(Their definition of $\Delta (t)$ is slightly different from what I have defined above) Sir, what if the subscript -1 had not been present, what would the case have been? Would my interpretation have been correct in that case? – Soumee Feb 22 '17 at 7:35
Initially, don't worry about the sum. Just try to figure out how the signal $$\Delta(t)u_{-1}(t)$$ looks like. As soon as you know that, you just need to shift it by integer values of $$t$$, and add them all up. If the shifted versions do not overlap, you simply get copies of the basic signal shifted to integer values of $$t$$. From the solution in the solution manual (as given in a comment), the basic signal is the right half of a triangle, and the period equals $$1$$. The symbol $$u_{-1}(t)$$ refers to the unit step function, which is more frequently referred to as $$u(t)$$ (see also this question and its answer(s)).
Assuming that you're referring to the conventions used for the unit doublets (see Signals and Systems,2e,Ch.2,Sec 2.5.3, A.Opp) then $u_{-1}(t) = u(t)$, where the right hand side is the conventional unit step (heaviside) function.
Therefore given the definition of that triangular pulse as: $$\Delta(t) = \begin{cases} 1-2|t|, \, |t|<\frac{1}{2} \\ 0,\,|t|>\frac{1}{2} \end{cases}$$
we can see that the required signal is a periodic extension, with period 1, of the following base period : $$\Delta_{B}(t) = \begin{cases} 1-2t, ~~~~ 0<t<\frac{1}{2} \\ 0 ~~~~~,~~~~~~\frac{1}{2} < t <1 \end{cases}$$
The resulting sum can be seen to be the following:
Note that the pulses extend from $-\infty$ to $\infty$ ...
• Strange notation ... so $u_0(t)=\delta(t)$?? I prefer $\delta'(t)$ or $\delta^{(n)}(t)$ for the first or n'th derivative of $\delta(t)$. By the way, according to the formula for the Fourier coefficients given in a comment under the question, the half-triangle hits zero at $t=1$, not at $t=1/2$, doesn't it? – Matt L. Feb 22 '17 at 10:32
• Yes true. Just a notation. And $u_1(t)$ is $\delta'(t)$ and it has the following properties; $u_m(t) = u_1(t) \star u_1(t) \star ... \star u_1(t)$ m-times, for $m$ pos integer and similarly $u_m(t) = u_{-1}(t) \star u_{-1}(t) \star ... \star u_{-1}(t)$ for $m$ neg int. $u_k(t) \star u_m(t) = u_{k+m} (t)$. He seems to make a typo either in the time domain definition or in the Fourier series integrand. I have choosen the time domain definition to be correct. If the latter is correct then as you pointed, the triangle will hit zero at $t=1$ and the periodic wave will be kind of a sawtooth. – Fat32 Feb 22 '17 at 12:01
• At this point I don't understand this discussion(those in the comments,though I have understood the answer by Mr.Fat32),but would like to mention that the Fourier coefficients has been calculated considering the following function of $\Delta (t)$ $$\Delta(t) = \begin{cases} 1-\vert t\rvert,& \lvert t\rvert<=1 \\ 0,&\lvert t\rvert>1 \end{cases}$$ which results in $x_n=\int_{0}^1(-t+1)e^{-j2\pi nt}$ – Soumee Feb 22 '17 at 12:49
• Sir, would you mind having a look at this question: dsp.stackexchange.com/q/66740/26549 – Soumee Apr 21 '20 at 18:13 | 2021-05-15 11:32:35 | {"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": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.855461597442627, "perplexity": 246.24498267787263}, "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/1620243991801.49/warc/CC-MAIN-20210515100825-20210515130825-00120.warc.gz"} |
http://math.stackexchange.com/tags/propositional-calculus/new | # Tag Info
1
I will show that 1 is true and therefore 2 is false. Note that since $\Sigma=\{p_2\to p_1, p_3\to p_2,\, \dots\,\}$, we have that $\Sigma\equiv\{\neg p_1\to \neg p_2, \neg p_2\to \neg p_3,\, \dots\,\}$. In particular, we have $\Sigma \equiv \bigcup_{1 \leq n<\omega}\{\neg p_n \to \neg p_{n+1}\}$. (1) Claim: for all $n<\omega$ such that $\Sigma \cup ... 0 Both of them are false, this is due to exact definition: 1.for a formula$\psi$we say$\vDash\psi$if for all valuations$w$,$w(\psi)=1$not for just a valuation$v$. 2.We say a set is consistent if we could find a valuation$v$such that for all formula$\psi$in it$v(\psi)=1$. But as you said formula$\varphi_1$could be a contradiction. 4 This is a logic "confused" problem ... We approach it "by steps". Question 1 We assume that the meaning of the first part of the question is : for a valuation$v$there is some$n$for which :$v(\varphi_n)=1$. Is so, due to the fact that :$v(\varphi_1 \lor \ldots \lor \varphi_n) = max_i \{ v(\varphi_i) \}$, we have that for the valuation$v$: ... 0 The convention in play is that in the main part of the table (after recording the truth-values of atoms -- if we bother to that again, since that's just copying across the values assigned at the beginning of the line) we write the truth-value under the main connective of the (sub)formula we are evaluating at that step. So it is indeed wffs that are being ... 2 Presumably one is only interested in$1$s in the output. There are$8$where$p_4$is true. Or else we want the antecedent of$p_4$false, forcing$p_3$false, and$p_1\longrightarrow p_2$true. The count of cases where$p_1\longrightarrow p_2is true, and the rest false is easy. One uses a similar analysis in producing a "quick" disjunctive normal ... 1 Both are correct given that you have "0" as allowable in expressions since (p→0) is logically equivalent to ¬p. ¬φ is logically equivalent to (p→(q→r)). ¬(p∧q→r) is logically equivalent to ¬(¬(p→¬q)→r) which is logically equivalent ¬(¬(p→(q→0))→r) which is logically equivalent to ¬(((p→(q→0))→0)→r) which is logically equivalent to ... 0 In general $$A \rightarrow B = \lnot A \lor B$$ Also, notice that $$\lnot(\lnot A \lor B) = A \land \lnot B ~\text{(De Morgan theorem)}$$ In your case, you have: $$\varphi = \lnot((p \land q) \rightarrow r) = p \land q \land \lnot r$$ and $$\lnot\varphi = (p \land q) \rightarrow r = \lnot p \lor \lnot q \lor r ~\text{(De Morgan theorem)}$$ In the ... 4 \begin{align}\lnot \varphi & \equiv \lnot \Big(\lnot((p\land q)\rightarrow r)\Big) \\ & \equiv (p \land q)\rightarrow r\\ & \equiv p\rightarrow(q\rightarrow r) \end{align} 0 Hint: $$x\lor y \equiv \lnot x\to y \\ x\land y \equiv \lnot(\lnot x \lor \lnot y)$$ 0 I'd say this proof is more easily understood if you generalize it by replacing\bot$with an arbitrary proposition$C$: $$\left((A\Rightarrow C)\land (B\Rightarrow C)\right)\Rightarrow \left((A\lor B)\Rightarrow C\right)$$ But this is just the elimination rule for$\lorin disguise in natural deduction: $$\underline{\Gamma\vdash A\lor B \quad ... 0 You can prove it in Sequent Calculus as described in the pdf mentioned in the comments (http://zll22.user.srcf.net/talks/2011-12-01-CategoricalLogic.pdf ) 1 A |- A Identity 2 B |- B Identity 3 C |- C ... 1 (p\to q) \wedge (q\to b) implies p\to b, but not the other way around. Hence those two justifications are false as written. Fortunately, you only need them in one direction, the one that is true. 1 I'll give you a hint. Recall that for f : A\rightarrow B$$f^{-1}(C) = \{x\in A : f(x)\in C\}$$Therefore$$\begin{align}x\in f^{-1}(B\setminus Y)&\Leftrightarrow x\in A\ \text{and}\ f(x)\in B\setminus Y \\ &\Leftrightarrow x\in A\ \text{and}\ f(x)\in B\ \text{and}\ f(x)\not\in Y \end{align}From here, there are just two small steps left: can ... 0 Hint 1: B\setminus Y = B \cap Y^c. Hint 2: \begin{align*} f^{-1}(B\setminus Y) &= f^{-1}(B\cap Y^c)\\ &= \{a \in A : f(a) \in B \cap Y^c)\} \\ &= \{a \in A : f(a) \in B \text{ and } f(a) \notin Y\} \end{align*} Hint 3: To show that the two sets are equal, you must take a \in f^{-1}(B\setminus Y) and show it is in f^{-1}(B)\setminus ... 1 Whenever you want to show that two sets are equal, try element chasing. That means, assume your element is in the set on the left, f^{-1}(B \setminus Y), and then show that it must be in the set on the right, f^{-1}(B) \setminus f^{-1}(Y). That's half of the proof. Chase an element in the other direction, and you've done both halves. First assume x\in ... 2 \Rightarrow:\;\;(1) Distribute. \;\;(2) A\lor A \equiv A\land A \equiv A. (Simplification.) \;\;(3)\land-Elimination.A\lor (A\land B) \overset{(1)}{\iff} (A \lor A)\land (A\lor B) \overset{(2)}\iff A \land (A\lor B) \overset{(3)}\implies A$$\Leftarrow:\;\; We use disjunction-Introduction.$$A \implies A\lor (\text{anything}.)\;\;\text{So,}\; ... 1 For some general remarks on how to prove that sets of connectives are/are not complete, see How to prove that a set of logical connectives is functionally complete(incomplete)? In this particular case, it is fairly easy to figure out how to build\vee$and$\neg$out of$\{\$,\top,\bot\}$, once you see that $\$(p,q,r)$equals$q$if$p$is true and$r$... 4 I think you are looking for an example of statements here. That is$\phi, \psi, \chi$s.t. these properties hold. Let$L=\{\cdot\}$. Let$T=\text{group axioms}\cup \text{there are only twelve distinct elements}$. Now Let$\phi$say that "I'm is abelian and I'm not is not cyclic" (i.e.$\forall a,b, a\cdot b = b\cdot a$and a fairly long sentence that says ... 1 If ¬ (ψ → ϕ), then ψ, and ¬ϕ. If ¬ (χ → ϕ), then χ, and ¬ϕ. If ψ, and χ, then (ψ ∧ χ). If (ψ ∧ χ), and ¬ϕ, then ¬[(ψ ∧ χ)→ϕ]. If [ϕ ⟷ (ψ ∧ χ)], then [(ψ ∧ χ)→ϕ]. Thus, ϕ ⟷ (ψ ∧ χ), ¬ (ψ → ϕ), and ¬ (χ → ϕ) do not simultaneously hold true. 1 We have that :$\lnot (ψ \rightarrow ϕ)$is equivalent to :$\lnot (\lnot ψ \lor ϕ)$i.e. to :$ψ \land \lnot ϕ$. And$\lnot (χ \rightarrow ϕ)$is equivalent to :$χ \land \lnot ϕ$. Thus, there is no way to satisfy both formuale together with :$ϕ \leftrightarrow (ψ \land χ)$. Comment The "problem" in your argument is the ... 2 As mentioned in the comment, the translation here is indeed crucial. The use of 'but' does not seem to fit in there (but I have to say that I'm also not a native English speaker). In any case, spoken languages can hardly cover the subtleties of logic (except for Lojban...). There are several interpretations or conventions for natural languages that are not ... 1 After chugging away on this for a while, I do think that the difficulty comes in keeping any “in-scope assumptions” available in each of the cases. The proof I outlined in the question does this by making the codomain of the coproduct arrow an exponential. That is, in trying to show$C$from the assumptions$A \lor B$,$A \lif C$, and$B \lif C$, I ended ... 3$\;\lnot (r \land t)\lor u,\;$(premise)$\;r\land t,\;$(premise)$\lnot \lnot (r \land t)$from$(2),\;$(double negation)$\therefore \;u\;$(disjunctive syllogism) 1 It is all done with sentences and connectives: A sentence is usually denoted by Capital letters like$A,B$, etc., the connectives are or:=$ \lor$; and:=$\land$, and if-then:=$ \rightarrow$, and negation connective is$\lnot$so that, e.g., If I go, then I will see a movie is transcribed as : say$A$is " If I go" , and$B$is , "I will see a movie" is ... 1 Using OTTER [1], I've found a 7 step, level 4 proof. axiom 4 CxCyx level 0 axiom 5 CCxyCCyzCxz 0 axiom 6 CCCxyxx 0 D4.4 8 CxCyCzy 1 S5.5 9 CCxyCCCxzuCCyzu 1 D5.8 31 CCCxCyxzCuz 2 D9.6 ... 0 Combining the subformula strategy, weighting of the axioms, the assumptions, and the goal, and a (partial) level saturation search, I've found a 20 step (excluding the axioms and assumptions), level 6 proof using OTTER [1] (OTTER treats "p" and "q" here as nullary functions or in other words, constants). axiom 4 CxCyx. level 0 ... 0 Yes, all of those systems have single axioms. Rezus's paper On a Theorem of Tarski makes this clear. 1 Combining the subformula strategy, weighting of the axioms and the goal, and a (partial) level saturation search, I've found a 15 step (excluding the axioms), level 5 proof using OTTER [1]. Actually, instead of proving CCCp0Cq0CCCp0qp. OTTER proved the more general CCCpqCr0CCCpqrp. (we can substitute q/0, r/q in CCCpqCr0CCCpqrp to get CCCp0Cq0CCCp0qp, ... 3 Let's assume the following axiom: Negatives Reverse Inequalities: If$x < y$and$z < 0$, then$xz > yz$. Now suppose that$a < b < 0$. Then since$a < b$and$a < 0$, it follows from the above axiom that$a^2 > ab$. Likewise, since$a < b$and$b < 0$, it follows from the above axiom that$ab > b^2. Combining ... 3 We have \begin{align} \neg(a < b < 0) &\iff \neg(\,(a < b) \,\&\, (b < 0)\,) \\&\iff (\,\neg(a < b)\,) \vee (\,\neg(b < 0)\,) \\&\iff (a \geq b) \vee (b \geq 0) \,\,. \end{align} 1 You are right. Your second example is [page 28] : Example 2.7.1 :We prove the sequent(¬(¬(φ ∨ (¬φ))))$. See the final comment : [...] At first sight it looks as if the two ¬ signs at the beginning of the conclusion have made extra work for us. This is not so. The sequent$(φ ∨ (¬φ))$is certainly valid, but it is just as hard to prove; in fact ... 2 When it comes to showing that logical statements are independent of certain variables I like to cycle through them and show it this way. Notice that you've shown that your statement is independent of the truth or falsity of$R$so I'll take that approach here. Let's suppose$R = T, then \neg Q\vee(\neg Q\wedge R) = \neg Q\vee(\neg Q\wedge T) = \neg ... 6 \begin{align*} \neg Q \vee (\neg Q\wedge R) &= (\neg Q\wedge 1) \vee (\neg Q\wedge R) \\ &= \neg Q \wedge (1\vee R) \\ &= \neg Q \wedge 1 \\ &= \neg Q \end{align*} 1 If you don't mind using order theory, you can show it without any reference to true or false. Let \leq be an order with A \vee B \geq A and A \wedge B \leq A. Then, \neg Q \vee (\neg Q \wedge R) \geq \neg Q and \neg Q \wedge (\neg Q \wedge R) \leq \neg Q. If the two expressions are equal, you can see that they must equal \neg Q. 1 Assume that Q is true. Then it follows that \lnotQ is false, and (\lnotQ\landR) is false also. So, ¬Q∨(¬Q∧R) is false. Assume that Q is false. Then \lnotQ is true, and thus so is ¬Q∨(¬Q∧R). Since Q is either true or false, and since in either case the equality here ¬Q∨(¬Q∧R)=¬Q, it follows that ¬Q∨(¬Q∧R)=¬Q in all cases. 1 Q\implies\lnot Q \wedge R\implies\lnot Q Contradiction! 1 One might argue that \lnotI corresponds to the formula ((\phi\rightarrow0)$\rightarrow$$\lnot$$\phi$). This can get thought of as one-half of the definition of $\lnot$. On the other hand, RAA similarly corresponds to the formula (($\lnot$$\phi$$\rightarrow$0)$\rightarrow$$\phi). This doesn't correspond to any definition quite so easily. 2 Well, Chiswell and Hodges in the exercises immediately following the quoted remarks in their book explicitly give some examples of sequents whose proofs in their system depend essentially on RAA, including$$\vdash \neg(\phi \to \psi) \to \phi\vdash \phi \to (\neg\phi \to \psi)$$So: If a conditional is false, its antecedent has to be true(?). A ... 1 Yes. You have to follow a "typical" proof of "equivalence" between Natural Deduction and Hilbert-style. See : Sara Negri & Jan von Plato, Structural Proof Theory (2001), page 41-on. An alternative approach is through soundness and completeness. Both proof systems are sound and complete regarding valid formulae; thus, a formula unprovable in ND, ... 1 Also in the Case 3, i.e.$\mathbf A \equiv \mathbf B \land \mathbf C$, we have to apply the Induction Hypotheses. We have that, by IH both$\mathbf B$and$\mathbf C$satisfy the theorem, i.e. :$\mathbf B \equiv (B_1 \lor \ldots \lor B_n)$and$\mathbf C \equiv (C_1 \lor \ldots \lor C_k)$where the RHS are DNF (disjunctions of conjuncts of ... 2 See Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), page 23 : take the special case in which$\Sigma$is the empty set$\emptyset$. Observe that it is vacuously true that any truth assignment satisfies every member of$\emptyset$. (How could this fail? Only if there was some unsatisfied member of$\emptyset$, which is absurd.) ... 0 You cannot prove$P \leftrightarrow Q$"alone", and you cannot prove$P \rightarrow Q$alone. There must be some other assumption missing ! This is the reason why (in this you are right !) you cannot understand the claim that : "not-Q contradicts a given P". Referring to your attachment, I assume that you are interested to the following proof : ... 3 The only way to prove that proposition$a\to b$is false is if$a$is true and$b$is false. Hence, you must have$p$true and$r$false, by the conditions of the task you were set. If$q$were true, then$r$would be true (by the second hypothesis), so we must have$q$false. Now verify that$\{p$true,$q,r$false$\}\$ satisfy both hypothesis and ...
Top 50 recent answers are included | 2014-07-12 03:34:09 | {"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": 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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9990695118904114, "perplexity": 1744.3317222530068}, "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-2014-23/segments/1404776431056.5/warc/CC-MAIN-20140707234031-00067-ip-10-180-212-248.ec2.internal.warc.gz"} |
http://www.r-bloggers.com/implementing-the-exact-binomial-test-in-julia/ | # Implementing the Exact Binomial Test in Julia
April 14, 2012
By
(This article was first published on John Myles White » Statistics, and kindly contributed to R-bloggers)
One major benefit of spending my time recently adding statistical functionality to Julia is that I’ve learned a lot about the inner guts of algorithmic null hypothesis significance testing.
Implementing Welch’s two-sample t-test last week was a trivial task because of the symmetry of the null hypothesis, but implementing the exact binomial test has proven to be more challenging because the asymmetry of a skewed null defined over a bounded set means that one has to think a bit more carefully about what is being computed.
To see why, let’s first recap the logic of the standard two-sided hypothesis test. In all NHST situations, you assign a p-value to the observed data by working under the null hypothesis and using this assumption to calculate the probability of observing data sets that are as extreme or more extreme than the observed data.
For the normal, this calculation is easy: you find the probability of seeing an equal or higher z-score than the observed data’s z-score and then you double this probability to account for the lower tail in which the hypothetical z-score is lower than the observed z-score.
But for the binomial, the right quantity to use as a definition of extremity is less obvious (at least to me). Suppose that you’ve seen x successes after n samples from a Bernoulli variable with probability p of success.
You might try defining extremity by saying that a hypothetical data set y is more extreme than x if abs(y - n) > abs(x - n): in short, you could use the count space to assess extremity.
This approach will not work. Consider the case in which x = 4, n = 10 and p = 0.2. Under this definition y = 0 would be as extreme as y = 4, but p(y = 0) > p(y = 4), so 0 should not be considered as extreme as 4. You need to use probability space and not count space to assess extremity.
This logic leads to the conclusion that the proper definition is one in which y is as extreme or more extreme than x if p(y, n, p) < p(x, n, p). This is the correct definition for the exact binomial test. Implementing it leads to this piece of code in Julia for computing p-values for the binomial test:
1 2 3 4 5 6 7 8 9 load("extras/Rmath.jl") function binom_p_value(x, n, p) sum(filter(d -> d <= dbinom(x, n, p), map(i -> dbinom(i, n, p), [0:n]))) end binom_p_value(2, 10, 0.8)
As far as I know, this procedure may be as efficient as possible, but it seems odd to me that we should need to assess the PDF at n + 1 numbers when, in principle, we should only need to assess the CDF of the binomial distribution at two points to find a p-value.
For that reason, you might hope to replace your loop over n + 1 numbers to one that, for extreme data sets, is more efficient by estimating lower and upper bounds on the count values with lower PDF values than the observed data. For the moment, I’m experimenting with doing this as follows:
1 2 3 4 5 6 7 8 9 10 11 12 load("extras/Rmath.jl") function binom_p_value(x, n, p) lower_bound = floor(n * p - abs(n * p - x)) upper_bound = ceil(n * p + abs(n * p - x)) sum(filter(d -> d <= dbinom(x, n, p), map(i -> dbinom(i, n, p), vcat([0:lower_bound], [upper_bound:n])))) end binom_p_value(2, 10, 0.8)
Unfortunately, I haven’t yet done the analytic work to demonstrate that these bounds are actually correct. (One of them must be, since one of them is a and the strict monotonicity of the distribution function about n * p guarantees that a must be either a lower or an upper bound for itself.) Of course, if these bounds are sufficiently conservative, they’ll function to save computation without any risk of giving corrupt answers — even if they’re not the tightest possible bounds.
Note that, in principle, it should be possible to go further: if you know exact bounds, then the summing and filtering operations are entirely superfluous and we can run:
1 2 3 4 5 6 7 8 9 10 load("extras/Rmath.jl") function binom_p_value(x, n, p) lower_bound = exact_lower_bound(x, n, p) upper_bound = exact_upper_bound(x, n, p) pbinom(lower_bound, n, p) + 1 - pbinom(upper_bound - 1, n, p) end binom_p_value(2, 10, 0.8)
I don’t know that these exact bounds can be computed exactly without a lot of work, but if they can be, they give a much more efficient implementation of the exact binomial test.
I wrote all of this up because (a) I’d appreciate knowing if the exact bounds can be computed efficiently and (b) I thought it was a very nice example of (1) thinking through the logic of hypothesis testing in detail (including considerations of what extremity really means) and (2) the constant problem that mathematically equivalent definitions suggest algorithms with very different computational costs.
R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more... | 2014-11-23 20:09:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5932708382606506, "perplexity": 725.9203158635502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400379916.51/warc/CC-MAIN-20141119123259-00061-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://indico.cern.ch/event/443176/contributions/2155257/ | # SUSY 2016
Jul 3 – 8, 2016
The University of Melbourne
Australia/Melbourne timezone
## Analysis of $B \to PP,PV$ decays in Factorization Assisted Topological Amplitude Approach
Jul 7, 2016, 3:00 PM
20m
Alice Hoy Room 309
### Alice Hoy Room 309
Talk Flavour Physics
### Speaker
Cai-Dian Lu (IHEP, Beijing)
### Description
We analyze charmless two-body non-leptonic B decays $B \to PP, PV$ under the framework of factorization assisted topological amplitude approach, where $P(V)$ denotes a light pseudoscalar (vector) meson. Compared with the conventional flavor diagram approach, we consider flavor $SU(3)$ breaking effect assisted by factorization hypothesis for topological diagram amplitudes of different decay modes, factorizing out the corresponding decay constants and form factors from non-factorization (factorization) topological amplitudes after parameterizing (factorized in factorization framework) it as associated magnitude $\chi$ and strong phase $\phi$ (effective Wilson coefficient). These non-perturbative parameters are universal that can be extracted from current abundant experimental data of $B \to PP,PV$ simultaneously. With these best fitted parameters, we predict branching fractions and $CP$ asymmetries of 97 decay modes, which are in good agreement with measured data or to be test in the LHCb and the Belle-II experiments in the future. The long-standing $\pi \pi$ and $\pi K$-$CP$ puzzles are resolved with favorable color-suppressed tree emission diagram $C$.
### Primary author
Cai-Dian Lu (IHEP, Beijing) | 2022-09-27 01:25:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7277777194976807, "perplexity": 9084.854920672196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00737.warc.gz"} |
http://pari.math.u-bordeaux.fr/archives/pari-dev-0210/msg00023.html | Karim BELABAS on Fri, 4 Oct 2002 15:33:15 +0200 (MEST)
Re: bug in pari-gp precision? (fwd)
Hi,
accuracy in PARI. Two quick comments:
1) the problems reported are not due to my addrr() patches: I've checked
version 1.39.15, they were already there.
2) no worries about polroots: computations and checks are done using exact
arithmetic.
Karim.
--
Karim Belabas Tel: (+33) (0)1 69 15 57 48
Dép. de Mathematiques, Bat. 425 Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud Email: Karim.Belabas@math.u-psud.fr
F-91405 Orsay (France) http://www.math.u-psud.fr/~belabas/
--
---------- Forwarded message ----------
From: Walter Neumann <neumann@cpw.math.columbia.edu>
Date: Fri, 4 Oct 2002 02:21:32 -0400 (EDT)
To: Karim BELABAS <Karim.Belabas@math.u-psud.fr>
Dear Karim
A new one:
? 1. + 10^-50 - 1.
%2 = -2.52435489 E-29
What is happening here?
I visited pari-dev's archive and see there has been no discussion of your
post on the precision issue. You ask there if a correction would have
application. It would be very important for us.
Let me first say how we've been using pari:
Oliver Goodman, Craig Hodgson, and I developed, over the last eight years,
a 3-manifold package called Snap, based on Jeff Week's Snappea, using the
pari libraries, to do exact provable computations for hyperbolic
3-manifolds, computing hyperbolic structures and arithmetic invariants of
them. We promise (in a paper in Experimental Mathematics vol 9 (2000))
that Snap gives proofs of the hyperbolic structures it finds. The "proof"
involves giving an exact ideal hyperbolic triangulation of the manifold in
question. Each ideal simplex is given by a complex parameter which is an
algebraic number. We thus describe the parameter by giving an an
approximate numerical value (at least 50 digits pari precision by default)
plus an exact description as an element of an appropriate number field.
As I understand pari's real arithmetic, although we can be supremely
confident of Snap's results, our claim of "proof" is wrong, and is hard to
correct with current pari.
The numerical value of the parameter is computed as a root of an integer
polynomial and we write:
'we quote from the manual for the pari libraries: The algorithm used is
a modification of A. Sch\"onhage's remarkable root-finding algorithm,
due to and implemented by X. Gourdon. Barring bugs, it is guaranteed to
converge and to give the roots to the desired accuracy.''
This guarantee seems to me suspect?
Moreover, for geometric reasons we need to verify that the parameters have
positive imaginary part, which we do from the numerical value.
We must also confirm that the equations that say the tetrahedra fit
together correctly are exactly satisfied. Numerical issues come up here
because, these equations say that certain products of the parameters equal
1 (easily verified exactly) but also that the sum of their logs is 2\pi*i
which we verify numerically (the precision need only be better than \pi).
But, without analyzing carefully a complicated numerical computation that
pari has done, one has no proof that pari's answer and claimed precision
are even close to the truth, as in pari's
? (1. + 10.^-20 - 1. - 10.^-20 + 10.^-27 - 10.^-27 )*10.^29 + 1.
%13 = -0.2621774483536188886
? \x
[&=080e1eec] REAL(lg=4,CLONE):05000004 (-,expo=-2):c07ffffe 863c1f5c
dae42f94
We really need both the number theoretic features of pari and the real
arithmetic (the same is true for the undergraduate student Max Lipyanskiy,
who first raised this issue with you, in connection with an accurate
Dirichlet domain program he has written). Since one of the points of our
programs is that they are supposed to provide "proven" results, we need
honest report of the valid precision of real arithmetic.
Thus, I would welcome your suggestion of adding a bit-count of precision
to the data-structure for a pari real, and having this bit-count treated
honestly. Since it would be too conservative for "practical use" one would
probably want to make it accessible to the program, but take a more
pragmatic approach, as in the current implementation, to what one
displays.
Regards, Walter | 2016-07-31 09:24:24 | {"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": 0.4749341309070587, "perplexity": 5117.157263885898}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257828314.45/warc/CC-MAIN-20160723071028-00170-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://testbook.com/question-answer/%E2%80%8Bfind-the-value-offrac-4-sqrt--611cd8c8f3e271ccfcff7877 | # Find the value of $${\frac{{\; 4 }}{{\sqrt {7} + \sqrt{5}}}}$$.+ $${\frac{{\; \sqrt7 + \sqrt5 }}{{4}}}$$
This question was previously asked in
DSSSB PGT Geography Male General Section - 11 July 2021 Shift 2
View all DSSSB PGT Papers >
1. $${\frac{{\; 9√7 + 7√5 }}{{2}}}$$
2. $${\frac{{\; 9√7 - 7√5 }}{{2}}}$$
3. $${\frac{{\; 9√7 + 7√5 }}{{4}}}$$
4. $${\frac{{\; 9√7 - 7√5 }}{{4}}}$$
Option 4 : $${\frac{{\; 9√7 - 7√5 }}{{4}}}$$
## Detailed Solution
Formula used:
(a + b)2 = a2 + b2 + 2ab
a2 - b2 = (a + b)(a - b)
Concept used:
Rationalizing the Denominator
Calculation:
$${\frac{{\; 4 }}{{√ {7} + √{5}}}}$$ + $${\frac{{\; √7 + √5 }}{{4}}}$$
⇒ $${\frac{{\; 4 }}{{√ {7} + √{5}}}}$$× $${\frac{{\; √7 - √5 }}{{\sqrt7-\sqrt5}}}$$$${\frac{{\; √7 + √5 }}{{4}}}$$
⇒ $${\frac{{\; 4 \times ( √7-√5)}}{({√ {7}^2 -√{5}^2) }}}$$ + $${\frac{{\; √7 + √5 }}{{4}}}$$
⇒ $${\frac{{\; (8√7 -8√5 + √7+√5)}}{{4}}}$$
⇒ $${\frac{{\; 9√7 - 7√5 }}{{4}}}$$
∴ The required value is $${\frac{{\; 9√7 - 7√5 }}{{4}}}$$. | 2021-10-25 07:49:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.3962511420249939, "perplexity": 4153.33973981834}, "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/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00038.warc.gz"} |
http://helmut.knaust.info/paper/baire.html |
## Array convergence of functions of the first Baire-class
### Abstract
We show that every array $(x(i,j):\ 1\leq i < j < \infty)$ of elements in a pointwise compact subset of the Baire-1 functions on a Polish space, whose iterated pointwise limit $\lim_i \lim_j x(i,j)$ exists, is converging Ramsey-uniformly. An array $(x(i,j)_{i < j})$ in a Hausdorff space $\T$ is said to converge Ramsey-uniformly to some $x$ in $\T$, if every subsequence of the positive integers has a further subsequence $(m_i)$ such that every open neighborhood $U$ of $x$ in $\T$ contains all elements $x(m_i,m_j)$ with $i < j$ except for finitely many $i$.
### 1. Introduction
It is a well known consequence of Ramsey's Theorem that every array $(a_{ij})_{i < j}$ of real numbers with $\lim_i \lim_j a_{ij} =a$ for some $a\in \real$ has the following property: There is a subsequence $(m_i)$ so that for all $\epsilon>0$ there is an $n \in \nat$ such that $|a_{m_im_j}-a| < \epsilon$ for all $n < m_i < m_j$. This result generalizes easily to Hausdorff spaces which satisfy the first countability axiom. The purpose of our note is to show that a corresponding result holds for the space of functions of the first Baire-class $\bone$ on a Polish space $\Om$, given the topology of pointwise convergence. Let us say that an array $(x(i,j):1 \leq i < j < \infty )$ of elements in a Hausdorff space $\T$ converges Ramsey-uniformly to some $x \in \T$, if every subsequence of $\nat$ has a further subsequence $(m_i)$ such that for every open neighborhood $U$ of $x$ in $\T$ there is an $n \in \nat$ so that $x(m_i,m_j) \in U$ for all $n < m_i < m_j$. With this notation we can state our main result as follows:
Theorem 1. Let $\Om$ be a Polish space and let $K$ be a pointwise compact subset of $\bone$. If $x$ and $(x(i,j))_{i < j}$ are elements in $K$ with $\lim_i\lim_j x(i,j)=x$, then $(x(i,j))$ converges Ramsey-uniformly to $x$.
A topological space $\Om$ is Polish, if it is homeomorphic to a complete separable metric space. A real-valued function is of the first Baire-class on $\Om$, if it is the pointwise limit of a sequence of continuous functions on $\Om$. It is a fundamental result of Bourgain, Fremlin and Talagrand [BFT] that $\bone$ is an angelic space, if $\Om$ is Polish. A Hausdorff space $\T$ is angelic, if for every relatively compact subset $A$ of $\T$ each point in the closure of $A$ is the limit of a sequence in $A$ and if relatively countably compact sets in $\T$ are relatively compact. In angelic spaces the notions of (relative) compactness, (relative) countable compactness and (relative) sequential compactness coincide. Further basic results about angelic spaces can be found in [P]. Theorem 1 strengthens - in the case of functions of the first Baire-class on a Polish space - a result of Boehme and Rosenfeld [BR], which we phrase for our purposes as follows:
Lemma 2. Let $\T$ be an angelic space, and let $x$ and $(x(i,j))_{i < j}$ be elements in a compact subset of $\T$ with $\lim_i\lim_j x(i,j)=x$. Then there is a subsequence $(m_i)$ of $\nat$ with $\lim_k x(m_{2k-1},m_{2k})=x$.
Lemma 2 was also obtained independently - in the $\bone$-setting - by Rosenthal [R]. From Theorem 1 and a result by Odell and Rosenthal [OR] we obtain the following Banach space corollary:
Corollary 3. Let $X$ be a separable Banach space not containing $\ell_1$. If $x^{**}$ and $(x^{**}(i,j))_{i < j}$ are elements in a bounded subset of $X^{**}$ with $\omega^*$-$\lim_i\omega^*$-$\lim_j x^{**}(i,j)=x^{**}$, then $(x^{**}(i,j))$ converges Ramsey-uniformly to $x^{**}$ in the $\omega^*$-topology.
The proof of Theorem 1 utilizes Lemma 2 to extract "nice" converging subsequences out of the given array $(x(i,j))$. We use Ramsey theory to produce the subarray for which one obtains Ramsey-uniform convergence. If $M$ is an infinite subset of $\nat$, $\pinf{M}$ will denote the set of all infinite subsets of $M$. We give $\pinf{\nat}$ the topology, which is inherited by considering $\pinf{\nat}$ as a subspace of $\{0,1\}^{\nat}$ endowed with the product topology. A subset $\A \subset \pinf{\nat}$ is called a Ramsey set, if for all $L \in \pinf{\nat}$ there is an $M \in \pinf{L}$ such that either $\pinf{M} \subset \A$ or $\pinf{M} \cap \A =\emptyset$. It is known that analytic (and coanalytic) subsets of $\pinf{\nat}$ are Ramsey sets [E,S]. For a proof of this result, some history and more general results see [O].
I would like to thank D. Alspach, E. Odell and H. P. Rosenthal for useful discussions.
### 2. Proof
Proof of Theorem 1: Let $x$ and $x(i,j)$ with $1 \leq i < j < \infty$ be elements in $K$ such that $\lim_i\lim_j x(i,j)=x$. We let $\A=\left\{ M=(m_i) \in \pinf{\nat} :\ (x(m_{2k-1},m_{2k}))_{k=1}^{\infty} \mbox{ is pointwise convergent}\right\}$.
Lemma 4. $\A$ is coanalytic.
We postpone the proof of the lemma and proceed with the proof of the theorem. Since $\A$ is coanalytic, $\A$ is a Ramsey set. Let $L \in \pinf{\nat}$. We can thus find $M=(m_i)_{i=1}^{\infty} \in \pinf{L}$ so that $\pinf{M} \subset \A$ or $\pinf{M} \cap \A=\emptyset$. Lemma 2 shows that the first alternative holds. Moreover, Lemma 2 asserts that $\lim_k x(m_{2k-1}^{\prime},m_{2k}^{\prime})=x$ for some $M^{\prime}= (m_i^{\prime}) \in \pinf{M}$. Suppose now the conclusion of Theorem 1 fails for $M^{\prime}$. Then there is an open neighborhood $U$ of $x$ and a subsequence $M^{\prime\prime} \subset M^{\prime}$ with $x(m^{\prime\prime}_{2k-1},m^{\prime\prime}_{2k})\not\in U \mbox{ for all } k \in \nat .$ Since $M^{\prime\prime} \in \pinf{M}$, we have $M^{\prime\prime} \in \A$ and thus $\lim_k x(m^{\prime\prime}_{2k-1},m^{\prime\prime}_{2k}) =y$ for some $y \in \bone$. Note that $y \not=x$. We now construct a subsequence $N=(n_i)\in\pinf{M}$ inductively as follows: Let $n_1=m_1^{\prime}$ and $n_2=m_2^{\prime}$. Once $n_1,n_2,\ldots,n_{2k}$ have been chosen, we define $n_{2k+1}$ and $n_{2k+2}$ as follows: If $k$ is odd, we choose an $\ell \in \nat$ so that $m^{\prime\prime}_{2\ell -1}> n_{2k}$ and let $n_{2k+1}=m^{\prime\prime}_{2\ell -1},n_{2k+2}=m^{\prime\prime}_{2 \ell}$. If $k$ is even, we can find an $\ell \in \nat$ with $m_{2\ell -1}^{\prime}>n_{2k}$ and then let $n_{2k+1}=m_{2 \ell -1}^{\prime},n_{2k+2}=m_{2\ell}^{\prime}$. On the one hand the sequence $(x(n_{2k-1},n_{2k}))$ is pointwise convergent, on the other hand it contains two subsequences converging to $x$ and $y$ respectively. This yields a contradiction.
Proof of Lemma 4. The proof of Lemma 4 uses techniques similar to those employed in [St]. Let $Y$ be the set of all real-valued arrays $(a(i,j))_{i < j}$, endowed with the topology of pointwise convergence. We set $Z=\pinf{\nat} \times Y$ and denote by $\phi:\Om \longrightarrow Y$ the canonical map defined by $\phi(\omega)=(x(i,j)(\omega))_{i < j}$. Since $\phi$ is a Borel-measurable map and $\Om$ is Polish, $\phi(\Om)$ is analytic in $Y$ (see [Ku]). Consequently $Z_1:=\pinf{\nat} \times \phi(\Om)$ is analytic in $Z$. We define a set $Z_2 \subset \pinf{\nat} \times Y$ as follows: $Z_2=\left\{(M,(a(i,j))):\ (a(m_{2k-1},m_{2k}))_{k=1}^{\infty} \mbox{ is not Cauchy } \right\}$ Observing that the set \begin{eqnarray*} Z_2^{\ell,N}&:=&\left\{ (M,(a(i,j))):\mbox{ there are $k_1,k_2>N$ with } \right.\\ & & \left.|a(m_{2k_1-1},m_{2k_1})-a(m_{2k_2-1},m_{2k_2})|>2^{-\ell} \right\} \end{eqnarray*} is open, and that $Z_2= \bigcup_{\ell \in \nat}\bigcap_{N \in \nat} Z_2^{\ell,N},$ we obtain that $Z_2$ is a $G_{\delta\sigma}$-set in $Z$.
Consequently $Z_1 \cap Z_2$ is analytic in $Z$. We let $P:Z \longrightarrow \pinf{\nat}$ be the projection of $Z$ onto its first coordinate. One can see easily that the complement of $\A$ is equal to $P(Z_1 \cap Z_2)$. Thus $\pinf{\nat} \setminus \A$ is analytic in $\pinf{\nat}$ as the continuous image of an analytic set in $Z$ (see [Ku]).
Problem: Does Theorem 1 hold for arbitrary angelic spaces?
Lemma 2 reduces this problem to the apparently open question, whether the set $\A \subset \pinf{\nat}$, defined at the beginning of the proof, is still a Ramsey set for arbitrary angelic spaces.
### References:
[BFT] J. Bourgain, D. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), 845-886.
[BR] T. K. Boehme and M. Rosenfeld, An example of two compact Hausdorff Frèchet spaces whose product is not Frèchet, J. London Math. Soc. 8(1974), 339-344.
[E] E. E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163-165.
[Ku] K. Kuratowski, Topology, Academic Press, New York, 1966.
[O] E. Odell, Applications of Ramsey Theorems to Banach Space Theory, in: Notes in Banach spaces, ed. by H. E. Lacey, University of Texas Press, Austin, 1980, 379-404.
[OR] E. Odell and H. P. Rosenthal, A double dual characterization of separable Banach spaces containing $\ell^1$, Israel J. Math. 20 (1975), 375-384.
[P] J. D. Pryce, A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. 23 (1971), 532-546.
[R] H. P. Rosenthal, Some remarks concerning unconditional basic sequences, in: Longhorn Notes 1982-83, The University of Texas, Austin, 15-48.
[S] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64.
[St] J. Stern, A Ramsey theorem for trees, with an application to Banach spaces, Israel J. Math. 29 (1978), 179-188.
In: Proceedings of the American Mathematical Society, Volume 112, Issue 2 (June 1991), 529-532. | 2020-02-24 22:47:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.9483057260513306, "perplexity": 129.88200928103188}, "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/1581875145989.45/warc/CC-MAIN-20200224224431-20200225014431-00058.warc.gz"} |
https://www.physicsforums.com/threads/linear-equation.127509/ | # Linear Equation
1. Jul 29, 2006
### suspenc3
I am new to to this topic, hints?
$$y^1=x+5y$$
The only examples are in the form dy/dx+p[x(y)]=Q[x]
2. Jul 29, 2006
### StatusX
Try some simple forms for the solution. y=ax+b should work.
If you want to be less reliant on luck, you could try looking at the asymptotic behavior of the DE. It seems to resemble the equation for exponential growth, in which case the y term would soon dominate over the x term on the RHS, and the solution would approach true exponential growth more and more closely. This suggests trying a solution of the form y=f(x) e5x. Plugging this in and deriving a DE for f(x), you get something you can solve easily, but it turns out the exponential cancels, and you're left with something of the above form.
3. Jul 30, 2006
### HallsofIvy
Staff Emeritus
Do you mean the differential equation y'= x+ 5y?
You say "The only examples are in the form dy/dx+p[x(y)]=Q[x]". Surely you must mean dy/dx+ p(y(x))= Q(x). This is exactly of that form:
dy/dx- 5y= x. p(y)= 5y and Q(x)= x. StatusX's suggestion of trying y= f(x)e5x is excellent but you could do basically the same thing by multiplying the entire equation by e5x (an "integrating factor"):
$$e^{5x}\frac{dy}{dx}+ 5e^{5x}y= xe^{5x}$$
because
$$\frac{d(e^{5x}y)}{dx}= e^{5x}\frac{dy}{dx}+ 5e^{5x}y$$
by the product rule. The problem reduces to an integration by parts.
4. Jul 30, 2006
### StatusX
Right, sorry. I've been doing DEs lately where there's no obvious solution, and these are the methods I've used. I forgot that equations of this form have a general method of solution, namely integrating factors. Thanks Halls.
5. Aug 1, 2006
### suspenc3
soo..$$\int \frac{d}{dx}(e^5^xy) - \int 5e^5^x = 5e^5^xy$$? | 2017-01-23 00:48:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7271319031715393, "perplexity": 1400.215101295953}, "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/1484560281659.81/warc/CC-MAIN-20170116095121-00469-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://stats.stackexchange.com/questions?sort=newest | # All Questions
3 views
### finding Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$
let $X_1,X_2,\ldots,X_n$ are random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and $Y_1,Y_2,\ldots,Y_n$ are random sample of geometric ...
3 views
### Multi-armed bandit algorithms vs Uplift modeling
Multi-Armed Bandit: http://en.wikipedia.org/wiki/Multi-armed_bandit Uplift Modeling: http://en.wikipedia.org/wiki/Uplift_modelling How are these two approaches different? How are they similar? Is ...
12 views
### How to interpret output of poisson.test?
Chapter 11 of Introduction to Data Science is about Poisson distributions. One example sample has 58638 observations out of 100000 with a value less than or equal to 10. ...
22 views
### Forecasting irregular time series (with R)
There are several methods to make forecasts of equidistant time series (e.g. Holt-Winters, ARIMA, ...). However I am currently working on the following irregular spaced data set, which has a varying ...
17 views
### Kolmogorov-Smirnov with parameter estimates
Can someone explain in as clear a (formalish mathematical) way as possible why the KS test in original form with parameter estimates is unreliable?
86 views
### In left skewed data, what is the relationship between mean and median?
I found this question on the open MOOC from Stanford, however the answer is not present. I think the median $\leq$ mean. Is this the case?
12 views
### Need subspace partition algorithm, not necessarily a full classifier
The image above represents a hypothetical data set of interest. For some set of points in N-dimensional space (each attribute of the data set corresponds to one dimension), I want to identify ...
10 views
### Optimal weighting matrix instrumental variables estimator
The formula for the optimal weighting matrix when you perform regression with more instrumental variables than endogenous predictors is the following: $W_{opt} = (\frac{1}{N}Z'Z)^{-1}$ This tells ...
224 views
### Do I have to learn SAS if I want to go into industry?
I am currently a PhD student in Biostatistics (finishing my 2nd year). My original goal when entering graduate school was to go into academia. Recently I have been debating this choice and wondering ...
85 views
### Variance of Z for Z = X + Y, when X and Y correlated
So I'm trying to show that ${\rm Var}(Z) \le 2({\rm Var}(X)+{\rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble ...
13 views
### Using MANOVA for classification without separating training and test sets
In this study: Rosenblum, Sara, et al. "Handwriting as an objective tool for Parkinson’s disease diagnosis." Journal of neurology 260.9 (2013): 2357-2361 The researchers attempt to classify ...
2 views
### Statistic test on percentage correct classified by emotion recognition
For a potential emotion recognition bachelor-project I was wondering what statistical test I have to perform when I get my results to test whether it's significant. I will be testing which combination ...
17 views
### AIC BIC Mallows Cp Cross Validation Model Selection
If you have several linear models, say model1, model2 and model3, how would you cross-validate it to pick the best model? (In R) I'm wondering this because my AIC and BIC for each model are not ...
46 views
19 views
### Difference between iid data and non-iid data for a simple regression problem
I am trying to understand the difference between iid and non-iid data. Let's consider a given time series, and say it's reasonable to assume that at each time point the random variable $X_t$ depends ...
17 views
15 views
### R: One period our cross validation with time series
I have quarterly data with one causal variable (X) and one dependent variable (Y). 30 such observations. I have the X variable for a quarter, and I'm seeking to predict that quarter's Y. The ... | 2014-03-10 02:43:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8781753778457642, "perplexity": 1607.3178535076645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010557037/warc/CC-MAIN-20140305090917-00042-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.gap-system.org/Manuals/pkg/idrel-2.38/doc/chap2.html | Goto Chapter: Top 1 2 3 4 5 Bib Ind
### 2 Rewriting Systems
This chapter describes functions to construct rewriting systems for finitely presented groups which store rewriting information. The main example used is a presentation of the quaternion group q8 with generators a,b and relators [a^4, b^4, abab^-1, a^2b^2].
#### 2.1 Monoid Presentations of FpGroups
##### 2.1-1 FreeRelatorGroup
‣ FreeRelatorGroup( grp ) ( attribute )
‣ FreeRelatorHomomorphism( grp ) ( attribute )
The function FreeRelatorGroup returns a free group on the set of relators of the given fp-group G. If HasName(G) is true then a name is automatically assigned to the free group.
The function FreeRelatorHomomorphism returns the group homomorphism from the free group on the relators to the free group on the generators of G, mapping each generator to the corresponding word.
gap> F := FreeGroup( 2 );;
gap> a := F.1;; b:= F.2;;
gap> rels := [ a^4, b^4, a*b*a*b^-1, a^2*b^2];;
gap> q8 := F/rels;;
gap> SetName( q8, "q8" );
gap> frq8 := FreeRelatorGroup( q8 );
q8_R
gap> GeneratorsOfGroup( frq8 );
[ q8_R1, q8_R2, q8_R3, q8_R4]
gap> frhomq8 := FreeRelatorHomomorphism( q8 );
[ q8_R1, q8_R2, q8_R3, q8_R4] -> [ f1^4, f2^4, f1*f2*f1*f2^-1, f1^2*f2^2]
##### 2.1-2 MonoidPresentationFpGroup
‣ MonoidPresentationFpGroup( grp ) ( attribute )
‣ ArrangementOfMonoidGenerators( grp ) ( attribute )
‣ FreeGroupOfPresentation( mon ) ( attribute )
‣ GroupRelatorsOfPresentation( mon ) ( attribute )
‣ InverseRelatorsOfPresentation( mon ) ( attribute )
‣ HomomorphismOfPresentation( mon ) ( attribute )
A monoid presentation for a finitely presented group G has two monoid generators g^+,g^- for each group generator g. The relators of the monoid presentation comprise the group relators, and relators g^+g^- specifying the inverses. The function MonoidPresentationFpGroup returns the monoid presentation derived in this way from an fp-presentation.
The function FreeGroupOfPresentation returns the free group on the monoid generators.
The function GroupRelatorsOfPresentation returns those relators of the monoid which correspond to the relators of the group. All negative powers in the group relators are converted to positive powers of the g^-. The function InverseRelatorsOfPresentation returns relators which specify the inverse pairs of the monoid generators.
The function HomomorphismOfPresentation returns the homomorphism from the free group of the monoid presentation to the free group of the group presentation.
The attribute ArrangementOfMonoidGenerators will be discussed before the second example in the next section.
In the example below, the four monoid generators a^+, b^+, a^-, b^- are named q8_M1, q8_M2, q8_M3, q8_M4 respectively.
gap> mon := MonoidPresentationFpGroup( q8 );
monoid presentation for an fp-group with homomorphism
[ [ q8_M1, q8_M2, q8_M3, q8_M4 ], [ f1, f2, f1^-1, f2^-1 ] ]
gap> fgmon := FreeGroupOfPresentation( mon);
<free group on the generators [ q8_Ml, q8_M2, q8_M3, q8_M4]>
gap> genfgmon := GeneratorsOfGroup( fgmon);
[ q8_Ml, q8_M2, q8_M3, q8_M4]
gap> gprels := GroupRelatorsOfPresentation( mon );
[ q8_Ml^4, q8_M2^4, q8_Ml*q8_M2*q8_Ml*q8_M4, q8_Ml^2*q8_M2^2]
gap> invrels := InverseRelatorsOfPresentation( mon);
[ q8_Ml*q8_M3, q8_M2*q8_M4, q8_M3*q8_Ml, q8_M4*q8_M2]
gap> hompres := HomomorphismOfPresentation( mon );
[ q8_Ml, q8_M2, q8_M3, q8_M4] -> [ fl, f2, fl^-l, f2^-1 ]
#### 2.2 Rewriting systems for FpGroups
These functions duplicate the standard Knuth Bendix functions which are available in the GAP library. There are two reasons for this: (1) these functions were first written before the standard functions were available; (2) we require logged versions of the functions, and these are most conveniently extended versions of the non-logged code.
##### 2.2-1 RewritingSystemFpGroup
‣ RewritingSystemFpGroup( grp ) ( attribute )
This function attempts to return a complete rewrite system for the group G obtained from the monoid presentation mon, with a length-lexicographical ordering on the words in fgmon, by applying Knuth-Bendix completion. Such a rewrite system can be obtained for all finite groups. The rewrite rules are (partially) ordered, starting with the inverse relators, followed by the rules which reduce the word length the most.
In our q8 example there are 16 rewrite rules in the rewriting system rws:
gap> rws := RewritingSystemFpGroup( q8 );
[ [q8_Ml*q8_M3, <identity ...>], [ q8_M2*q8_M4, <identity ...> ],
[q8_M3*q8_Ml, <identity ...>], [ q8_M4*q8_M2, <identity ...> ],
[q8_M1^2*q8_M4, q8_M2], [q8_Ml^2*q8_M2, q8_M4], [ q8_Ml^3, q8_M3 ],
[ q8_M4^2, q8_Ml^2], [ q8_M4*q8_M3, q8_Ml*q8_M4],
[ q8_M4*q8_Ml, q8_Ml*q8_M2], [q8_M3*q8_M4, q8_Ml*q8_M2],
[ q8_M3^2, q8_Ml^2], [q8_M3*q8_M2, q8_Ml*q8_M4],
[ q8_M2*q8_M3, q8_Ml*q8_M2], [q8_M2^2, q8_Ml^2],
[ q8_M2*q8_Ml, q8_Ml*q8_M4] ]
The default ordering of the 2n monoid generators is [g_1^+,g_2^+,...,g_n^+,g_1^-,g_2^-,...,g_n^-]. In the case of the two-generator abelian group T = ⟨ a,b ~|~ [a,b] ⟩ the Knuth-Bendix process starts to generate infinite sets of relations such as {ab^ma^-1 -> b^m,~ m geqslant 1}. If, using the ArrangementOfMonoidGenerators function, we specify the alternative ordering [g_1^+,g_1^-,g_2^+,g_2^-,...], then a finite set of rules is obtained.
gap> F := FreeGroup(2);;
gap> T := F/[Comm(F.1,F.2)];
<fp group of size infinity on the generators [ f1, f2 ]>
gap> SetName( T, "T" );
gap> SetArrangementOfMonoidGenerators( T, [1,-1,2,-2] );
gap> monT := MonoidPresentationFpGroup(T);
monoid presentation for an fp-group with homomorphism
[ [ T_M1, T_M2, T_M3, T_M4 ], [ f1, f1^-1, f2, f2^-1 ] ]
gap> rwsT := RewritingSystemFpGroup( T );
[ [ T_M1*T_M2, <identity ...> ], [ T_M2*T_M1, <identity ...> ],
[ T_M3*T_M4, <identity ...> ], [ T_M4*T_M3, <identity ...> ],
[ T_M4*T_M2, T_M2*T_M4 ], [ T_M4*T_M1, T_M1*T_M4 ],
[ T_M3*T_M2, T_M2*T_M3 ], [ T_M3*T_M1, T_M1*T_M3 ] ]
The functions called by RewritingSystemFpGroup are as follows.
##### 2.2-2 OnePassReduceWord
‣ OnePassReduceWord( word, rules ) ( operation )
‣ ReduceWordKB( word, rules ) ( operation )
Assuming that word is an element of a free monoid and rules is a list of ordered pairs of such words, the function OnePassReduceWord searches the list of rules until it finds that the left-hand side of a rule is a subword of word, whereupon it replaces that subword with the right-hand side of the matching rule. The search is continued from the next rule in rules, but using the new word. When the end of rules is reached, one pass is considered to have been made and the reduced word is returned. If no matches are found then the original word is returned.
The function ReduceWordKB repeatedly applies the function OnePassReduceWord until the word remaining contains no left-hand side of a rule as a subword. If rules is a complete rewrite system, then the irreducible word that is returned is unique, otherwise the order of the rules in rules will determine which irreducible word is returned. In the example we see that b^9a^9 reduces to ba which is not a normal form as we have seen in 2.2-1, where b^+a^+ -> a^+b^- is the last rule.
gap> monrels := Concatenation( gprels, invrels );
[ q8_Ml^4, q8_M2^4, q8_Ml*q8_M2*q8_Ml*q8_M4, q8_Ml^2*q8_M2^2, q8_Ml*q8_M3,
q8_M2*q8_M4, q8_M3*q8_Ml, q8_M4*q8_M2]
gap> id := One( monrels[l] );;
gap> r0 := List( monrels, r -> [ r, id ] );
[ [ q8_Ml^4, <identity ...> ], [ q8_M2^4, <identity. ..> ],
[ q8_Ml*q8_M2*q8_Ml*q8_M4, <identity ...> ],
[ q8_Ml^2*q8_M2^2, <identity. ..>], [ q8_Ml*q8_M3, <identity ...> ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity. ..>],
[ q8_M4*q8_M2, <identity ...> ] ]
gap> ap := genfgmon[l];; bp := genfgmon[2];; ## p for plus
gap> am := genfgmon[3];; bm := genfgmon[4];; ## m for minus
gap> w0 := bp^9 * ap^9;
q8_M2^9*q8_M1^9
gap> w1 := OnePassReduceWord( w0, r0 );
q8_M2^5*q8_M1^5
gap> w2 := ReduceWordKB( w0, r0 );
q8_M2*q8_M1
##### 2.2-3 OnePassKB
‣ OnePassKB( rules ) ( operation )
‣ RewriteReduce( rules ) ( operation )
‣ KnuthBendix( rules ) ( operation )
‣ ShorterRule( rule1, rule2 ) ( operation )
The function OnePassKB implements the main loop of the Knuth-Bendix completion algorithm. Rules are compared with each other; all critical pairs are calculated; and the irreducible critical pairs are orientated with respect to the length-lexicographical ordering and added to the rewrite system.
The function RewriteReduce will remove unnecessary rules from a rewrite system. A rule is deemed to be unnecessary if it is implied by the other rules, i.e. if both sides can be reduced to the same thing by the remaining rules.
The function KnuthBendix implements the Knuth-Bendix algorithm, attempting to complete a rewrite system with respect to a length-lexicographic ordering. It calls first OnePassKB, which adds rules, and then (for efficiency) RewriteReduce which removes any unnecessary ones. This procedure is repeated until OnePassKB adds no more rules. It will not always terminate, but for many examples (all finite groups) it will be successful. The rewrite system returned is complete, that is: it will rewrite any given word in the free monoid to a unique irreducible; there is one irreducible for each element of the quotient monoid; and any two elements of the free monoid which are in the same class will rewrite to the same irreducible.
The function ShorterRule gives an ordering on rules. Rules (g_lg_2,id) that identify two generators (or one generator with the inverse of another) come first in the ordering. Otherwise one precedes another if it reduces the length of a word by a greater amount.
One pass of this procedure for our q8 example adds 13 relators to the original 8, and these 21 are then reduced to 9. A second pass and reduction gives the list of 16 rules seen above, forming a complete rewrite system for the group. Now b^9a^9 correctly reduces to ab^-1.
gap> r1 := OnePassKB( r0 );
[ [ q8_Ml^4, <identity ...> ], [ q8_M2^4, <identity ...> ],
[ q8_Ml*q8_M2*q8_Ml*q8_M4, <identity ...> ],
[ q8_Ml^2*q8_M2^2, <identity. ..> ], [ q8_Ml*q8_M3, <identity ...> ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_M2*q8_Ml*q8_M4, q8_Ml^3],
[ q8_Ml*q8_M2^2, q8_Ml^3 ], [ q8_M2^2, q8_Ml^2 ], [q8_Ml^3, q8_M3 ],
[ q8_M2^3, q8_M4 ], [ q8_Ml*q8_M2*q8_Ml, q8_M2],
[ q8_M2^3, q8_Ml^2*q8_M2], [ q8_M2^2, q8_Ml^2 ], [q8_Ml^2*q8_M2, q8_M4 ],
[ q8_Ml^3, q8_M3 ], [ q8_M2*q8_Ml*q8_M4, q8_M3 ], [q8_Ml*q8_M2^2, q8_M3 ],
[ q8_M2^3, q8_M4 ] ]
gap> Length( r1 );
21
gap> r1 := RewriteReduce( r1 );
[ [ q8_Ml*q8_M3, <identity ...> ], [ q8_M2^2, q8_Ml^2 ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_Ml^3, q8_M3 ],
[ q8_Ml^2*q8_M2, q8_M4 ], [ q8_Ml*q8_M2*q8_Ml, q8_M2 ],
[ q8_M2*q8_Ml*q8_M4, q8_M3 ] ]
gap> Length( r1 );
9
gap> r2 := KnuthBendix( r1 );
[ [ q8_Ml*q8_M3, <identity ...> ], [ q8_M2*q8_Ml, q8_Ml*q8_M4 ],
[ q8_M2^2, q8_Ml^2], [ q8_M2*q8_M3, q8_Ml*q8_M2 ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M3*q8_M2, q8_Ml*q8_M4 ], [ q8_M3^2, q8_Ml^2 ],
[ q8_M3*q8_M4, q8_Ml*q8_M2 ], [ q8_M4*q8_Ml, q8_Ml*q8_M2 ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_M4*q8_M3, q8_Ml*q8_M4 ],
[ q8_M4^2, q8_Ml^2], [ q8_Ml^3, q8_M3 ], [q8_Ml^2*q8_M2, q8_M4 ],
[ q8_Ml^2*q8_M4, q8_M2 ] ]
gap> Length( r2 );
16
gap> w2 := ReduceWordKB( w0, r2 );
q8_M1*q8_M4
#### 2.3 Enumerating elements
##### 2.3-1 ElementsOfMonoidPresentation
‣ ElementsOfMonoidPresentation( mon ) ( attribute )
The function ElementsOfMonoidPresentation returns a list of normal forms for the elements of the group given by the monoid presentation mon. The normal forms are the least elements in each equivalence class (with respect to length-lex order). When rules is a complete rewrite system for G the list returned is a set of normal forms for the group elements. For q8 this list is
[\; {\rm id},\; a^+,\; b^+,\; a^-,\; b^-,\; a^{+2},\; a^+b^+,\; a^+b^-\; ].
gap> elq8 := Elements( q8 );
[ <identity ...>, f1, f1^3, f2, f1^2*f2, f1^2, f1*f2, f1^3*f2 ]
gap> elmonq8 := ElementsOfMonoidPresentation( monq8 );
[ <identity. ..>, q8_Ml, q8_M2, q8_M3, q8_M4, q8_Ml^2, q8_Ml*q8_M2,
q8_Ml*q8_M4 ]
Goto Chapter: Top 1 2 3 4 5 Bib Ind
generated by GAPDoc2HTML | 2018-10-19 15:29:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4177045524120331, "perplexity": 12650.36937779229}, "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/1539583512411.13/warc/CC-MAIN-20181019145850-20181019171350-00017.warc.gz"} |
https://cs.stackexchange.com/questions/124596/n-numbers-n-2-pairs-minimizing-the-maximum-sum-of-a-pairing-proving-greedy-al | # N numbers, N/2 pairs. Minimizing the maximum sum of a pairing. Proving greedy algorithm
So say I have n numbers, where n is even. I want to pair the numbers such that the maximum sum of the pairs is minimized. For example -2, 3, 4, 5. The ideal pairing is (-2, 5), (3, 4), since its maximum sum is 3 + 4 = 7, and that is the minimal sum possible for a max sum in any pairing. The key to the algorithm is to sort the values from least to greatest. Then pair the least with the greatest, and so on, until you reach the center of the ordering.
Example: 3, -2, 4, 5
Algorithm sorts the values: -2 , 3, 4, 5
Then pairs first with last: (-2, 5)
Then pairs the next available first and last: (3, 4)
Terminates since no pairs left.
This is a greedy algorithm and I am trying to prove that it is always correct using a "greedy stays ahead" approach. My issue is that I am struggling to show that the algorithm's maximum sum is always $$\leq$$ optimal maximum sum. My intention was to suppose for contradiction that the optimal maximum sum is $$<$$ the algorithm's maximum sum. But I'm not sure how to find a contradiction. How would this proof go?
Can you see why $$\max((-2)+5, 3+4) \lt \max(-2+3, 4+5)$$?
The reason is simple. Because on the right hand side, the maximum number 5 is not paired with the minimum number.
Let the numbers are $$a_1\le a_2\le \cdots\le a_n$$. Let the numbers be paired in some way.
• If $$a_n$$ is paired with $$a_1$$, we are done at this round.
• Suppose $$a_n$$ is paired with $$a_j$$, $$j\not= 1$$. Then $$a_1$$ is paired with $$a_k$$ for some $$k\not= n$$. So we have two pairs, $$\{a_n, a_j\}$$ and $$\{a_1, a_k\}$$. The sums of these two pairs are $$a_n + a_j$$ and $$a_1 + a_k$$, the large one of which is $$a_n+a_j$$.
Let us switch $$a_j$$ and $$a_1$$ so that $$a_n$$ will pair with $$a_1$$, and $$a_j$$ will pair with $$a_k$$. The sums of the two new pairs are $$a_n + a_1$$ and $$a_j + a_k$$, both of which is at most $$a_n+a_j$$, i.e., the large one of them is at most $$a_n+a_j$$. So after the switch, the maximum sum of the pairs involving $$a_n, a_j, a_k, a_1$$ does not increase. Since other pairs stays the same, So after the switch, the maximum sum of all pairs does not increase.
Continuing this process, we will make sure the largest number remaining be paired with the smallest number remaining at each round. The maximum sum of the pairs will never increase at each round. After $$n/2$$ rounds, we will reach the pairing where $$a_k$$ is paired with $$a_{n+1-k}$$.
You can see the above approach is indeed the "greedy stays ahead" approach. | 2021-02-27 10:23:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 31, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8755931854248047, "perplexity": 191.8116470407475}, "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/1614178358798.23/warc/CC-MAIN-20210227084805-20210227114805-00443.warc.gz"} |
https://mathoverflow.net/questions/325798/monotonicity-of-schr%C3%B6dinger-eigenvalues | # Monotonicity of Schrödinger Eigenvalues
Let us consider the Schrödinger operator $$H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x)$$ on $$L^2[-\pi,\pi]$$ with Neumann boundary conditions $$f^\prime(\pm\pi)=0$$. Here, $$h\geq 0$$ is a parameter.
It is easy to see that (up to normalization) $$\psi_0^h(x)=e^{h\cos(x)}$$ is an eigenfunction for the eigenvalue $$0$$.
Running some numerics suggests that the mapping $$h\mapsto\lambda_n(H_h)$$ where $$\lambda_n$$ denotes the $$n^\text{th}$$ eigenvalue of $$H_h$$, $$n=0,1,2,\dots$$, is monotone non-decreasing. The question is how to prove this.
My approach was the following: By a famous result, $$\frac{d}{dh}\lambda_n(H_h)=\langle \psi_n^h,H_h^\prime\psi_n^h\rangle$$ where $$\psi_n^h$$ is the $$n^\text{th}$$ eigenfuntion of $$H_h$$ and $$H_h^\prime f(x)=(2h\sin^2(x)-\cos(x))f(x)$$. However, I am struggling to prove that $$\langle \psi_n^h,H_h^\prime\psi_n^h\rangle\geq 0$$.
Update
Maybe we should first focus on the second eigenvalue $$\lambda_1(H_h)$$. By general theory, $$\psi_1^h$$ is a differentiable odd function. Furthermore, $$\psi_1^h$$ has only one zero (at $$x=0$$). To obtain $$\frac{d}{dh}\lambda_n(H_h)\geq 0$$, it would thus be sufficient to prove $$\max_{x\in[0,x_0]}|\psi_1^h(x)|\leq\min_{x\in[x_0,\pi]}|\psi_1^h(x)|,$$ where $$x_0=2\arctan\left(\sqrt{\sqrt{16h^2 + 1} - 4 h}\right)$$ is the zero of $$2h\sin^2(x)-\cos(x)$$ in $$[0,\pi]$$, since $$\int_{0}^\pi 2h\sin^2(x)-\cos(x)\,dx=h\pi\geq 0.$$
Update 2
This is based on the comment by Carlo Beenakker. We transform the initial Neumann problem into a Dirichlet problem. A computation gives that the non-zero spectrum of $$H_h$$ coincides with the spectrum of \begin{align} K_hf(x)&=-f''(x)+h(h\sin^2(x)+\cos(x))f(x)\\&=-f''(x)+\left(h\cos(x)-\frac{h^2}{2}\cos(2x)+\frac{h^2}{2}\right)f(x) \end{align} acting on $$L^2[-\pi,\pi]$$ with Dirichlet boundary conditions $$f(\pm\pi)=0$$.
The potential in the OP is known as the "Razavy potential" or "double cosine potential". A recent study is Exact Solutions of the Razavy Cosine Type Potential (2018). The Schrödinger equation is $$-\psi''(x)+V(x)\psi(x)=E\psi(x),\;\;V(x)=\tfrac{1}{4}\xi^2\sin^2 x-(a+1)\xi\cos x,$$ on $$-\pi\leq x\leq\pi$$ with $$\psi(\pm\pi)=0$$. The differential equation in the OP is for $$\xi=2h$$ and $$a=-3/2$$. (I note that the cited paper assumes $$\xi,a>0$$, but that does not seem to be an essential condition on the solution.)
The substitution $$\psi(x)=\exp(\tfrac{1}{2}\xi\cos x)\phi(x)$$ and the change of variables $$z=\cos^2(x/2)$$ produces a confluent Heun differential equation with solution given by the Heun function: $$\psi(x)=\exp(\tfrac{1}{2}\xi\cos x)H\bigl(2\xi,-1/2,-1/2,-(2a+1)\xi,2(a+1)\xi+3/8-E;\cos^2(x/2)\bigr).$$ The energy $$E$$ should then be obtained from the boundary condition $$x=\pm\pi$$, so at the origin for the Heun function, but the cited paper does not succeed in obtaining a closed-form solution. | 2019-06-18 21:50:12 | {"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": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 42, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9850916862487793, "perplexity": 87.65416558354187}, "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-2019-26/segments/1560627998817.58/warc/CC-MAIN-20190618203528-20190618225528-00446.warc.gz"} |
https://www.rdocumentation.org/packages/rms/versions/5.1-3.1/topics/print.ols | # print.ols
0th
Percentile
##### Print ols
Formatted printing of an object of class ols using methods taken from print.lm and summary.lm. Prints R-squared, adjusted R-squared, parameter estimates, standard errors, and t-statistics (Z statistics if penalized estimation was used). For penalized estimation, prints the maximum penalized likelihood estimate of the residual standard deviation (Sigma) instead of the usual root mean squared error. Format of output is controlled by the user previously running options(prType="lang") where lang is "plain" (the default), "latex", or "html".
Keywords
print
##### Usage
# S3 method for ols
print(x, digits=4, long=FALSE, coefs=TRUE,
title="Linear Regression Model", …)
##### Arguments
x
fit object
digits
number of significant digits to print
long
set to TRUE to print the correlation matrix of parameter estimates
coefs
specify coefs=FALSE to suppress printing the table of model coefficients, standard errors, etc. Specify coefs=n to print only the first n regression coefficients in the model.
title
a character string title to be passed to prModFit
other parameters to pass to print or format
ols, lm,prModFit | 2021-01-24 13:30:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2354518622159958, "perplexity": 9463.165491454818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703548716.53/warc/CC-MAIN-20210124111006-20210124141006-00413.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2018228 | # American Institute of Mathematical Sciences
October 2018, 38(10): 5163-5188. doi: 10.3934/dcds.2018228
## Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms
LAGA - Université Paris 13, 99 Av. Jean-Baptiste Clément, 93430 Villetaneus, France
Received January 2018 Published July 2018
We consider $\mathit{Sp}\left( 2\mathit{d},\mathbb{R} \right)$ cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.
Citation: Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228
##### References:
show all references
##### References:
definition of $z_n$ and $x'$
definition $h:\mathcal{W}^c(p)\to \mathcal{W}^c(p)$ for class ${\bf{A}}$
definition $h:\mathcal{W}^c(p)\to \mathcal{W}^c(p)$ for class ${\bf{B}}$
perturbation of $H$
[1] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [2] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020434 [3] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 [4] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [5] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020444 [6] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020467 [7] Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $\Lambda$-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 [8] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020055 [9] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [10] Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 [11] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [12] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020168 [13] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [14] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 [15] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
2019 Impact Factor: 1.338 | 2020-12-04 11:56:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6151533722877502, "perplexity": 11634.652588582447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141735600.89/warc/CC-MAIN-20201204101314-20201204131314-00530.warc.gz"} |
https://golem.ph.utexas.edu/category/2008/10/the_nature_of_time.html | ## October 13, 2008
### The Nature of Time
#### Posted by John Baez
The Foundational Questions Institute is having an essay contest on The Nature of Time. The top prize is $10,000, the second prize is$5,000, and so on.
It’s a fascinating topic, but I can’t say I’m thrilled with most of the essays. In fact, that’s a polite way of expressing my feelings, in keeping with the civil atmosphere of this café. I’ll mention my favorite essay below, and keep quiet about the worst.
Maybe you could do better. In fact, maybe you should give it a try! Just make sure to submit your essay before December 1st, 2008.
I would write one myself, but I don’t have… time.
I haven’t read all the essays — so there could be some gems I haven’t seen. So far, my favorite is the one by my friend Carlo Rovelli:
• Carlo Rovelli, Forget time.
Abstract: Following a line of research that I have developed for several years, I argue that the best strategy for understanding quantum gravity is to build a picture of the physical world where the notion of time plays no role at all. I summarize here this point of view, explaining why I think that in a fundamental description of nature we must “forget time”, and how this can be done in the classical and in the quantum theory. The idea is to develop a formalism that treats dependent and independent variables on the same footing. In short, I propose to interpret mechanics as a theory of relations between variables, rather than the theory of the evolution of variables in time.
For those familiar with his work on quantum gravity and the ‘thermal time hypothesis’, there’s nothing drastically new about this essay. But, it makes a good case for the radical viewpoint that has motivated his work all along.
By the way: you can vote for your favorite essay! However, your vote will have no direct effect, unless you’re a member of the Foundational Questions Institute. It’s just like voting in the U.S. presidential election when you don’t live in a swing state.
Also by the way: I’m slightly suprised by the fact that right now, every essay except Rovelli’s has exactly 10 votes. How likely is that? Do you get 10 free votes just for playing?
Posted at October 13, 2008 10:16 PM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1816
### Re: The Nature of Time
When I checked, most essays had less than ten public votes, one had ten and there were a few with more than ten.
Posted by: Bruce Westbury on October 14, 2008 7:32 AM | Permalink | Reply to this
### Re: The Nature of Time
Hmm. Now they all have 10 except for three that have more: Bundy’s, Miller’s, and Rovelli’s.
Posted by: John Baez on October 14, 2008 6:49 PM | Permalink | Reply to this
### Re: The Nature of Time
0 restricted votes • $\lt 10$ public votes
The symbol before 10 means “fewer than”.
Posted by: Simon Willerton on October 14, 2008 7:18 PM | Permalink | Reply to this
### Re: The Nature of Time
Simon inquired:
I think it’s my brain — you know, that inert lump of matter a few inches behind my glasses. I didn’t expect them to be saying “$\lt 10$” votes, so I didn’t see the ‘less than’ sign. I must have thought it was meaningless cruft just like the $\bullet$.
I feel happier now, though I wonder why you need to get 10 votes or more before you can see how many you got. Maybe it’s to avoid people’s feelings being hurt?
Posted by: John Baez on October 14, 2008 8:49 PM | Permalink | Reply to this
### Re: The Nature of Time
It would be desireable to have a really good answer to
Why Lorentzian signature?
And: what is Wick rotation really?
Consider 2-dimensional QFT. There are two kinds of precise axiomatic definitions to capture it:
- one is local nets of operators: this requires Lorentzian signature.
- the other is vertex operator algebras: this requires Euclidean signature – but comes in two copies: left and right chiral parts.
What’s the relation? Something like this:
consider the real 4-dimensional space $\mathbb{C} \times \mathbb{C}$ with canonical (complex) coordinates $z_1$ and $z_2$ and equipped with a symplectic form
$\omega : (\mathbb{C} \times \mathbb{C}) \times (\mathbb{C} \times \mathbb{C}) \to \mathbb{R}$ given by $\omega((z_1,z_2),(z'_1,z'_2)) = Re(z_1)Im(z'_2) + Re(z_2)Im(z'_1) - Re(z'_1)Im(z_2) - Re(z'_2)Im(z_1) \,.$
There are the following two interesting polarizations of $\omega$ (i.e. 2-dimensional sub-vectorspaces of $\mathbb{C}\times \mathbb{C}$ on which $\omega$ vanishes):
one is spanned by $\{ \partial_{Im(z_1)}\,, \partial_{Im(z_2)} \}$ call this $P_1$.
The other is spanned by $\{ \partial_{Re(z_1)} - \partial_{Re(z_2)} \;,\;\; \partial_{Im(z_1)} + \partial_{Im(z_2)} \} \,.$
The first polarization imposes the constraint: “both $z_1$ and $z_2$ are real”.
In the sense that: if you know a function which is annihilated by $P_1$ on the subspace given by $Im(z_1) = Im(z_2) = 0$ then you know it everywhere.
The second polarization imposes the constraint “$z_2$ is the complex conjugate of $z_1$”.
In the sense that: if you know a function which is annihilated by $P_2$ on the subspace given by $\bar z_2 = z_1$ then you know it everywhere.
The first case is the Lorentzian case: $Re(z_1)$ and $Re(z_2)$ play the role of lightcone coordinates on $\mathbb{R}^2$.
The second case is the Euclidean case: $z_1$ and $z_2 = \bar z_1$ play the role of the two “chiral” coordinates.
Here “play the role” means: in 2d QFT, when switching between the Lorentzian and the Wick-rotated Euclidean setup, this is how the two variables in each case are related.
Posted by: Urs Schreiber on October 14, 2008 10:00 AM | Permalink | Reply to this
### Re: The Nature of Time
One nice thing about these questions is that they don’t even involve general relativity. General relativity brings up a lot of the deep questions about time. But, you’re pointing out that even 2d special relativity holds its mysteries.
Posted by: John Baez on October 14, 2008 6:53 PM | Permalink | Reply to this
### Re: The Nature of Time
It would be desireable to have a really good answer to
Why Lorentzian signature?
I think posets have something interesting to say regarding this question (as you know). A Lorenztian signature arises naturally from a causal structure. A causal structure can arise naturally via many means. My favorite is via directed graphs.
Posted by: Eric on October 14, 2008 7:16 PM | Permalink | Reply to this
### Re: The Nature of Time
It would be desireable to have a really good answer to
Why Lorentzian signature?
I think posets have something interesting to say regarding this question (as you know).
Yes, indeed, good point. There are various approaches by various people to pin down what precisely it is that makes Lorentzian signature so special, or maybe exceptional.
One is the observation that, roughly, a Lorentzian metric can be encoded in a poset structure on spacetime plus a volume density. Since a poset is a rather fundamental concept, this might indicate a deeper hidden meaning.
Interestingly, at least when restricting to certain dimensions, there are also characterizations of rather different flavor. For instance 3+1-dimensional Minkwoski space is closely related to the quaternions.
A causal structure can arise naturally via many means. My favorite is via directed graphs.
For others, let me say that in detail:
for any natural number $n$, consider the directed graph whose vertices are $\mathbb{Z}^n$ and whose edges consist of those going from the origin to the standard basis elements and all those obtained from these by translation along them.
To avoid irrelevant technical issues, truncate this to some finite subgraph.
Now consider the “graph algebra” of this graph: the associative algebra generated from the vertices and the edges, where the product from the left of an edge with its source vertex is the edge itself, same for the product from the right with the target vertex, and where all other products between vertices and edges vanish. The product of two edges which are not composable vanishes, while the product of two edges which do touch is a new generator of the algebra. Then divide out the relation that all sums of generators of length more than one with the same source and target vanish.
This algebra then is a model for differential forms on $\mathbb{Z}^n$. $k$-forms are generators of length $k$. It is noncommutative, where the non-commutativity is “proportional to the lattice spacing” in an obvious sense.
Due to the non-commutativity, the differential is an inner graded derivation: it is the graded commutator with the “graph operator”: the algebra element which is the sum of all edges.
(Notice that, while I can’t claim to see a direct relation at the technical level, at the moral level this is not unlike the phenomenon that type III von Neumann factors have a “canonical time evolution”.)
So we have a noncommutative DGA.
By taking the generators to be an orthonormal basis, there is naturally the structure of a Hilbert space on this algebra, which corresponds to the Hodge inner product $(\alpha,\beta) = \int_X \alpha \wedge \star \beta$, for $\star$ the Hodge star of standard flat $\mathbb{R}^n$.
On the algebra side one changes this metric by deforming the inner product by a hermitean operator $\hat g$: $(\alpha,\beta)' := (\alpha, \hat g \beta)$.
There is one god-given such operator in the game: the graph operator plus its ajoint times the graph operator minus its adjoint (with respect to the original metric). One checks that the metric this operator induces is the standard Minkowski metric on $\mathbb{Z}^n$ which regards all the edges of the graph as timelike and future-directed (or all as past directed, depending on convention).
Then what we end up with is a lattice version of differential form on Minkowski space with $\hat g$ being the Krein operator which changes the Minkowski Hodge inner product to the Euclidean one.
By the way, I think it is not inconceivable that there is a nice generalization of the picture of Lie-$\infty$-theory obtained by generalizing DGCAs to possibly non-commutative DGAs as above.
Posted by: Urs Schreiber on October 15, 2008 9:45 AM | Permalink | Reply to this
### Re: The Nature of Time
Concerning the question of Wick rotation and passage from Euclidean to Lorentzian QFT:
Dirk Schlingemann, From Euclidean field theory to quantum field theory
which, after a quick review of the Osterwalder-Schrader theorem which relates Euclidean Schwinger correlation functions with Minkwoskian Wightman correlation functions, gives a definition of Euclidean nets of observables (section 2, page 12) and then reconstructs directly from that (without going through correlation functions) a Haag-Kastler local net on Minkowski space (section 3.2, p. 17).
(Currently I am somewhat perplexed by his Eulcidean locality condition on p. 12, though. But I may just need to read this in more detail and think about it…)
Posted by: Urs Schreiber on November 13, 2008 12:56 PM | Permalink | Reply to this
### Re: The Nature of Time
“your vote will have no direct effect, unless you’re a member of the Foundational Questions Institute”
The way I read it you also get the right to cast a vote that counts if you submit an essay and it gets approved, but you can’t vote for yourself.
“I can’t say I’m thrilled with most of the essays”
What in general don’t you like about them? Are they too speculative, too dull, not enough maths, not original, the crackpot index is too high/low, not crazy enough, not even wrong, something else?
I am surprised there are so few entries but perhaps there will be loads at the last minute.
Pity you are not going to give it a shot but I hope Greg Egan has time for one. The contest needs some good writers.
Posted by: PhilG on October 14, 2008 7:02 PM | Permalink | Reply to this
### Re: The Nature of Time
PhilG wrote:
What in general don’t you like about them? Are they too speculative, too dull, not enough maths, not original, the crackpot index is too high/low, not crazy enough, not even wrong, something else?
Let me just quote portions of a few of the abstracts. Can you — or anyone — guess what I think about each one?
Time is a functional entity, derived from the fundamental property of basic 3D matter particles to move at a constant linear speed. 3D matter has no ability to act or move.
We construct a relativistic theory in which time plays an active physical role in the cosmology and self-organized sustainability of the universe. ve physical role in the cosmology and self-organized sustainability of the universe. Our model, consistent with scale-invariant quantum field theory, compels an absolute unit of time. Results precisely account for current WMAP data.
By using the tools of global Lorentzian geometry I prove that, under physically reasonable conditions, the impossibility of finding a global time implies the singularity of spacetime.
The only observed relationship of time to space is a reciprocal relation, in the equation of motion. However, it seems absurd to think of space, defined as a set of points satisfying the postulates of geometry, as the inverse of time. Only when we view the observed increase of time, as the 0D inverse of the observed increase of 3D space, does it begin to make sense: The expanding universe is an expanding set of four-dimensional spacetime coordinates, just as Einstein conceived it, but this also may be its initial condition.
The following essay puts forward an academic conception of the here-and-now or “present moment.” It is then identified as coincident to both our subjective moments and Nature. We then use this conjecture to construct a scientific and objective world-view, based on our human biology, to answer the question “what is reality?”
We have recently introduced a consistent way of defining time relationally in general relativity. When quantum mechanics is formulated in terms of this new notion of time the resolution of the em measurement problem can be implemented via decoherence without the usual pitfalls. The resulting theory has the same experimental results of ordinary quantum mechanics, but every time an event is produced or a measurement happens two alternatives are possible: a) the state collapses; b) the system evolves without changing the state.
This essay takes the idea of time as a relation of distance and velocity. Then it shows how an observer traveling at a relative velocity measures the same velocity for objects that assume they are at rest, using geometry to find the affects of relativity. Then formulates a proof that v’=v so that the affects of time dilation and length contraction do not interfere with a body in motion so that Newton’s Second Law of Motion holds true without time being a force acting on a body in motion.
Posted by: John Baez on October 14, 2008 9:10 PM | Permalink | Reply to this
### Re: The Nature of Time
In general, to everybody who has serious stuff to offer:
in my humble opinion currently at FQXi they tend to have more resources to offer than there are worthwhile submissions. Serious researchers should take that as an opportunity…
Posted by: Urs Schreiber on October 14, 2008 10:48 PM | Permalink | Reply to this
### Re: The Nature of Time
Fortunately, even us non serious researchers have an opportunity to participate in this essay contest.
Posted by: Kea on October 15, 2008 1:37 AM | Permalink | Reply to this
### Re: The Nature of Time
I am again and again astonished how blind these people calling themselves serious researchers are to the deepest problems of recent day physics. The text written by these “serious” researchers about time or other profound problems is typically boring repetition of what was believed to be known already century ago.
I know that these people are not idiots as individuals: perhaps kind of institutional stupidity is in question.
Posted by: Matti Pitkänen on October 16, 2008 8:33 AM | Permalink | Reply to this
### Re: The Nature of Time
Final submissions are in and voting proper has begun.
If you didn’t like the look of the entries six weeks ago have another look now. There have been many more submissions including some “serious” stuff from the likes of Sean Carol, Julian Barbour, George Ellis, David Finklestein, Dean Rickles, David Hestenes, Paul Halpern etc.
Of course all of us authors will look forward to being judged on equal footing according to the scientific merit of our essays - or will the voting be like at the Eurovision song contest? We will see.
Coincidently for UK residents there is a Horizon program about Time on BBC2 tonight.
Posted by: PhilG on December 2, 2008 5:47 PM | Permalink | Reply to this
### Re: The Nature of Time
I found Carlo Rovelli essay lucid and brilliant as usual. However, I hit on this little text by Lee Smolin for Edge, http://www.edge.org/q2009/q09_9.html#smolin and I’m puzzled by the certainty of his sudden switch to a ‘new’ presentist and time realist viewpoint. These classical, newtonian ideas are coming back like a boomerang from some new theory? I would be grateful if anyone may offer an expert comment on this.
Posted by: Gianluca on January 5, 2009 1:46 AM | Permalink | Reply to this
Post a New Comment | 2018-01-16 09:30:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 40, "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": 0.6706807017326355, "perplexity": 953.6565175220207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886397.2/warc/CC-MAIN-20180116090056-20180116110056-00501.warc.gz"} |
https://chemistry.stackexchange.com/questions/34561/how-to-define-a-new-solvent-in-gaussian09-to-run-scrf | # How to define a new solvent in Gaussian09 to run SCRF?
I want to run computation of my system in sulphuric acid with Gaussian 09; however sulphuric acid is not among the built-in solvents in G09. I know I have to give some parameters for the new solvent I define (including static and dynamic dielectric constants). Here's an example of my input files where I try to redefine a solvent water2.
#p b3lyp/6-31G(2df,p) int(grid=ultrafine) SCRF(Solvent=Water2,Read)
Comment
0 1
...
stoichiometry=H2O1
solventname=Water2
eps=78.3553
epsinf=1.77556
G09 keeps telling me that "SCRF(Solvent=Water2,Read)" is not correct. I want to know what the correct input will be.
• try it with SCRF=(Solvent=Generic,Read), that worked for me – pH13 - Yet another Philipp Jul 31 '15 at 7:30
The correct input should be more like (replacing your Water2 with Generic)
#p b3lyp/6-31G(2df,p) int(grid=ultrafine) SCRF(Solvent=Generic,Read)
Comment
0 1
...
stoichiometry=H2O1
solventname=Water2
eps=78.3553
epsinf=1.77556
as it is stated in the Documentation under Additional Input for PCM Calculations.
Note - this only gets your calculation running. Your shown current input section might be insufficient to properly define sulphuric acid.
As Geoff noted, the Gaussian Documentary states an important fact short before showing some $\varepsilon$-values of common solvents that should be taken into account.
We list the ε values here for convenience, but be aware it is only one of many internal parameters used to define solvents. Thus, simply changing the ε value will not define a new solvent properly.
• That's probably not sufficient. As described further along in the page you link, it says " We list the $\epsilon$ values here for convenience, but be aware it is only one of many internal parameters used to define solvents. Thus, simply changing the $\epsilon$ value will not define a new solvent properly." – Geoff Hutchison Jul 31 '15 at 14:58
• Thanks! I have tested and I've found stoichiometry, static and dynamic dielectric constants are the only parameters you need to run a default SCRF calculation. If you use Solvent=formamide in the beginning and you replace the three parameters with those of water's. you will get exactly the same result with using Solvent=Water. – Chester Lin Aug 2 '15 at 16:23 | 2021-04-23 02:29:51 | {"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": 0.4720909297466278, "perplexity": 1237.8345380503254}, "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/1618039626288.96/warc/CC-MAIN-20210423011010-20210423041010-00332.warc.gz"} |
http://ideas.repec.org/a/spr/eurphb/v84y2011i3p475-490.html | # Hole dispersions for antiferromagnetic spin-\hbox{$\frac{1}{2}$}12 two-leg ladders by self-similar continuous unitary transformations
## Author Info
• S. Duffe
• G. Uhrig
()
Registered author(s):
## Abstract
No abstract is available for this item.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
File URL: http://hdl.handle.net/10.1140/epjb/e2011-20150-x
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
## Bibliographic Info
Article provided by Springer in its journal The European Physical Journal B.
Volume (Year): 84 (2011)
Issue (Month): 3 (December)
Pages: 475-490
as in new window
Handle: RePEc:spr:eurphb:v:84:y:2011:i:3:p:475-490
Contact details of provider:
Web page: http://www.springer.com/economics/journal/10051
Order Information:
Keywords:
## References
No references listed on IDEAS
You can help add them by filling out this form.
## Lists
This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.
## Corrections
When requesting a correction, please mention this item's handle: RePEc:spr:eurphb:v:84:y:2011:i:3:p:475-490. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Guenther Eichhorn) or (Christopher F Baum).
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services. | 2014-08-27 21:05:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2020770013332367, "perplexity": 2622.249928844214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829839.93/warc/CC-MAIN-20140820021349-00296-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/0863 | Lemma 70.18.2 (Étale localization and strict transform). In the situation of Definition 70.18.1. Let
$\xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B }$
be a commutative diagram of morphisms with $U$ and $V$ schemes and étale horizontal arrows. Let $V' \to V$ be the blowup of $V$ in $Z \times _ B V$. Then
1. $V' = V \times _ B B'$ and the maps $V' \to B'$ and $U \times _ V V' \to X \times _ B B'$ are étale,
2. the strict transform $U'$ of $U$ relative to $V' \to V$ is equal to $X' \times _ X U$ where $X'$ is the strict transform of $X$ relative to $B' \to B$, and
3. for a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the restriction of the strict transform $\mathcal{F}'$ to $U \times _ V V'$ is the strict transform of $\mathcal{F}|_ U$ relative to $V' \to V$.
Proof. Part (1) follows from the fact that blowup commutes with flat base change (Lemma 70.17.3), the fact that étale morphisms are flat, and that the base change of an étale morphism is étale. Part (3) then follows from the fact that taking the sheaf of sections supported on a closed commutes with pullback by étale morphisms, see Limits of Spaces, Lemma 69.14.5. Part (2) follows from (3) applied to $\mathcal{F} = \mathcal{O}_ X$. $\square$
There are also:
• 2 comment(s) on Section 70.18: Strict transform
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0863. Beware of the difference between the letter 'O' and the digit '0'. | 2023-02-04 19:36:15 | {"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9391724467277527, "perplexity": 311.55510190157594}, "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-2023-06/segments/1674764500151.93/warc/CC-MAIN-20230204173912-20230204203912-00124.warc.gz"} |
https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space | Subspace of Product Space is Homeomorphic to Factor Space
Theorem
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Suppose that $X$ is non-empty.
Then for each $i \in I$ there is a subspace $Y_i \subseteq X$ which is homeomorphic to $\struct {X_i, \tau_i}$.
Specifically, for any $z \in X$, let:
$Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$
and let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.
Then $\struct {Y_i, \upsilon_i}$ is homeomorphic to $\struct {X_i, \tau_i}$, where the homeomorphism is the restriction of the projection $\pr_i$ to $Y_i$.
Corollary
Let $T_1$ and $T_2$ be non-empty topological spaces.
Let $b \in T_2$.
Let $T_1 \times T_2$ be the product space of $T_1$ and $T_2$.
Let $T_2 \times T_1$ be the product space of $T_2$ and $T_1$.
Then:
$T_1$ is homeomorphic to the subspace $T_1 \times \set b$ of $T_1 \times T_2$
$T_1$ is homeomorphic to the subspace $\set b \times T_1$ of $T_2 \times T_1$
Proof 1
Let $z \in X$.
Let $i \in i$.
Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.
Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.
For all $j \in I$ let:
$Z_j = \begin{cases} X_i & i = j \\ \set{z_j} & j \ne i \end{cases}$
Lemma 1
$Y_i = \prod_{j \mathop \in I} Z_j$
$\Box$
From Product Space of Subspaces is Subspace of Product Space , $\struct {Y_i, \upsilon_i}$ is a product space.
Consider the projection:
$p_i: \struct {Y_i, \upsilon_i} \to \struct {X_i, \tau_i}$
From Projection from Product of Family is Surjective, $p_i$ is surjective.
From Projection from Product Topology is Open:General Result, $p_i$ is open.
Thus, by definition, we have that $p_i$ is a homeomorphism.
Consider the projection:
$\pr_i: \struct {X, \tau} \to \struct {X_i, \tau_i}$
and the restriction:
$\pr_i {\restriction_{Y_i} }: \struct {Y_i, \upsilon_i} \to \struct {X_i, \tau_i}$
Lemma 2
$\pr_i {\restriction_{Y_i} } = p_i$
$\Box$
Thus $\pr_i {\restriction_{Y_i} }: Y_i \to X_i$ is a homeomorphism.
$\blacksquare$
Proof 2
Let $z \in X$.
Let $i \in I$.
Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.
Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.
Let $p_i = \pr_i {\restriction_{Y_i}}$.
Note that by definitions of a restriction and a projection then:
$\forall y \in Y_i: \map {p_i} y = y_i$
$p_i$ is an injection
Let $x, y \in Y_i$.
Then for all $j \in I \setminus \set i$:
$x_j = z_j = y_j$
Let $\map {p_i} x = \map {p_i} y$.
Then:
$x_i = y_i$
Thus:
$x = y$
It follows that $p_i$ is an injection by definition.
$\Box$
$p_i$ is a surjection
Let $x \in X_i$.
Let $y \in Y_i$ be defined by:
$\forall j \in I: y_j = \begin{cases} z_j & j \ne i \\ x & j = i \end{cases}$
Then:
$\map {p_i} y = y_i = x$
It follows that $p_i$ is an surjection by definition.
$\Box$
$p_i$ is a continuous mapping
Let $V \in \tau_i$.
Let $\ds U = \prod_{i \mathop \in I} U_i$ where:
$U_j = \begin{cases} X_j & j \ne i \\ V & j = i \end{cases}$
From Natural Basis of Product Topology, $U$ is an element of the the natural basis.
By definition of the product topology $\tau$ on the product space $\struct {X, \tau}$ the natural basis is a basis for the product topology.
It follows that:
$U$ is open in $\struct {X, \tau}$
Let $x \in Y_i$.
Now:
$\ds x$ $\in$ $\ds \map {p_i^\gets} V$ $\ds \leadstoandfrom \ \$ $\ds \map {p_i} x$ $\in$ $\ds V$ Definition of Inverse Image Mapping of Mapping: $p_i^\gets$ $\ds \leadstoandfrom \ \$ $\ds \map {\pr_i} x$ $\in$ $\ds V$ Definition of Restriction of Mapping $p_i$ $\ds \leadstoandfrom \ \$ $\ds x$ $\in$ $\ds U$ Definition of $U$ $\ds \leadstoandfrom \ \$ $\ds x$ $\in$ $\ds U \cap Y_i$ as $x \in Y_i$
By set equality:
$\map {p_i^\gets} V = U \cap Y_i$
By definition of the subspace topology on $Y_i$:
$\map {p_i^\gets} V \in \upsilon_i$
It follows that $p_i$ is continuous by definition.
$\Box$
$p_i$ is an open mapping
Let $U \in \upsilon_i$.
Let $x \in \map {p_i^\to} U$.
Then by definition of the direct image mapping:
$\exists y \in U : x = \map {p_i} y$
By the definition of the subspace topology:
$\exists U' \in \tau: U = U' \cap Y_i$
For all $k \in I$ let $\pr_k$ denote the projection from $X$ to $X_k$.
By definition of the natural basis of the product topology $\tau$:
there exists a finite subset $J$ of $I$
and:
for each $k \in J$, there exists a $V_k \in \tau_k$
such that:
$\ds y \in \bigcap_{k \mathop \in J} \map {\pr_k^\gets} {V_k} \subseteq U'$
Then:
$\ds y \in \paren {\bigcap_{k \mathop \in J} \map{\pr_k^\gets} {V_k} } \cap Y_i \subseteq U' \cap Y_i = U$
By definition of direct image mapping:
$\ds x = \map {p_i} y \in \map {p_i^\to} {\paren {\bigcap_{k \mathop \in J} \map {\pr_k^\gets} {V_k} } \cap Y_i} \subseteq \map {p_i^\to} U$
Recall that $p_i$ is an injection.
Then:
$\ds \map {p_i^\to} {\paren {\bigcap_{k \mathop \in J} \map {\pr_k^\gets} {V_k} } \cap Y_i}$ $=$ $\ds \map {p_i^\to} {\bigcap_{k \mathop \in J} \paren {\map {\pr_k^\gets} {V_k} \cap Y_i} }$ Set Intersection is Self-Distributive $\ds$ $=$ $\ds \bigcap_{k \mathop \in J} \map {p_i^\to} {\map {\pr_k^\gets } {V_k} \cap Y_i}$ Image of Intersection under Injection
Let $k \in J$.
Lemma
$\map {p_i^\to} {\map {\pr_k^\gets} {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$
$\Box$
$\ds \bigcap_{k \mathop \in J} \map {p_i^\to} {\map{\pr_k^\gets} {V_k} \cap Y_i}$ is open in $\struct {X_i, \tau_i}$
Since $x \in \map {p_i^\to} U$ was arbitrary then it has been shown that:
$\forall x \in \map {p_i^\to} U : \exists V \in \tau_i : x \in V \subseteq \map {p_i^\to} U$
Then by definition: $\forall x \in \map {p_i^\to} U: \map {p_i^\to} U$ is a neighborhood of $x$.
$\map {p_i^\to} U$ is open in $\tau_i$.
Since $U \in \upsilon_i$ was arbitrary, $p_i$ is an open mapping.
$\blacksquare$ | 2022-08-11 14:15:07 | {"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": 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": 0.9805325865745544, "perplexity": 200.78296303111713}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00032.warc.gz"} |
https://mathspace.co/textbooks/syllabuses/Syllabus-406/topics/Topic-7200/subtopics/Subtopic-96210/?activeTab=interactive | Area of Rectangles and Distributive Property
Interactive practice questions
Find the area of the following rectangle.
$4$4 cm $7$7 cm
Easy
Less than a minute
A rectangular room has an area of $60$60 m2 and a length of $6$6 m. What is the width of the room?
Use the fact that the area of a rectangle is given by $\text{area }=\text{length }\times\text{width }$area =length ×width .
The rectangle below has been split in to two rectangles. We want to work out the area.
We want to work out the area of the following rectangle by splitting it into two smaller rectangles.
Outcomes
GM3-2
Find areas of rectangles and volumes of cuboids by applying multiplication. | 2022-01-26 17:57:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7938410639762878, "perplexity": 831.4318556878325}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304959.80/warc/CC-MAIN-20220126162115-20220126192115-00258.warc.gz"} |
https://curriculum.illustrativemathematics.org/MS/teachers/2/2/14/index.html | # Lesson 14
Four Representations
## 14.1: Which is the Bluest? (5 minutes)
### Warm-up
In this warm-up, students are asked to reason which group of blocks is the bluest and explain how they arrived at that decision. The goal is to prompt students to visualize and articulate different ways they can use ratios, equivalent ratios and proportions to support their reasoning.
### Launch
Students in groups of 2. Tell students you will show them five groups of blocks. Their job is to determine which group of blocks is the bluest. Display the image for all to see. Give students 2 minutes of quiet think time. Encourage students who have one way of supporting their decision to think about another way while they wait.
### Student Facing
1. Which group of blocks is the bluest?
2. Order the groups of blocks from least blue to bluest.
### Student Response
Student responses to this activity are available at one of our IM Certified Partners
### Activity Synthesis
Ask students to share which group of blocks is the bluest and their reasoning. Record and display student explanations for all to see. To involve more students in the conversation, consider asking some of the following questions:
• Did anyone choose the same group of blocks but would explain it differently?
• Does anyone want to add an observation to the way ____ saw the blocks?
• Do you agree or disagree? Why? Ask students to order the groups of blocks from less blue to bluest after deciding on the bluest group of blocks.
## 14.2: One Scenario, Four Representations (20 minutes)
### Activity
In this activity, students choose from different lists of things to define their own proportional and nonproportional relationships. Some of the things on the list will be familiar and others will be unfamiliar. This is a significant change from previous activities where students were always given two quantities and they had to decide if they were proportional or not. This new step gives students the opportunity to think about what quantities are related to some of the items on the lists, which is an important step of modeling with mathematics (MP4).
This activity and the next go together. Students use the work from this activity to make a visual display of their work in the next activity.
### Launch
Arrange students in groups of 2.
The names of things in the task may be unfamiliar to both English Language Learners and fluent English speakers. Before students start, take some time to ensure they know the meaning of their chosen things or ask them to do some research on their meanings. You might ask each pair of students to choose a different unfamiliar word, spend five minutes to research it, and prepare a drawing or explanation to share with the class.
It is best to approve of students’ choices before they work. For example, if students choose “legs” and “earthworms,” that will not make for a very interesting relationship.
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Conversing, Writing: MLR2 Collect and Display. While pairs are working, circulate and listen to students talk about the relationships between quantities and justify whether the relationships are proportional. Write down common or important phrases you hear students say about the relationships. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students use mathematical language in their written work and in paired and whole group discussions.
Design Principle(s): Optimize output (for justification); Support sense-making
### Student Facing
1. Select two things from different lists. Make up a situation where there is a proportional relationship between quantities that involve these things.
creatures
• starfish
• centipedes
• earthworms
• dinosaurs
length
• centimeters
• cubits
• kilometers
• parsecs
time
• nanoseconds
• minutes
• years
• millennia
volume
• milliliters
• gallons
• bushels
• cubic miles
body parts
• legs
• eyes
• neurons
• digits
area
• square microns
• acres
• hides
• square light-years
weight
• nanograms
• ounces
• deben
• metric tonnes
substance
• helium
• oobleck
• pitch
• glue
2. Select two other things from the lists, and make up a situation where there is a relationship between quantities that involve these things, but the relationship is not proportional.
3. Your teacher will give you two copies of the “One Scenario, Four Representations” sheet. For each of your situations, describe the relationships in detail. If you get stuck, consider asking your teacher for a copy of the sample response.
1. Write one or more sentences describing the relationship between the things you chose.
2. Make a table with titles in each column and at least 6 pairs of numbers relating the two things.
3. Graph the situation and label the axes.
4. Write an equation showing the relationship and explain in your own words what each number and letter in your equation means.
5. Explain how you know whether each relationship is proportional or not proportional. Give as many reasons as you can.
### Student Response
Student responses to this activity are available at one of our IM Certified Partners
### Anticipated Misconceptions
As students work, pay attention to the numbers they use in their tables. Students can be haphazard when choosing values, and their numbers may end up being unfriendly. You can ask them questions that encourage them to reason about what numbers would be friendlier. Also, depending on the things chosen, they may need to consider scales for their axes. Watch out for scales like {1, 2, 3, . . . } for the number of legs on a centipede!
Creating a relationship that is not proportional may present too significant a challenge for struggling learners. An accommodation would be to change their task to creating only a proportional relationship, or even assigning two quantities that are straightforward like “starfish legs vs. number of starfish.”
## 14.3: Make a Poster (15 minutes)
### Optional activity
In this activity, students make a visual display of their scenarios from the previous activity (after sharing their rough draft with another group and getting some feedback).
When the posters are complete and displayed around the room, students view each others' work and use sentence starters to give feedback and to critique the reasoning of others (MP3).
### Launch
Keep students in the same groups. Tell them that they should incorporate the feedback they received when making their posters.
Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, provide a task checklist which makes all the required components of the poster explicit.
Supports accessibility for: Attention; Social-emotional skills
### Student Facing
Create a visual display of your two situations that includes all the information from the previous activity.
### Student Response
Student responses to this activity are available at one of our IM Certified Partners
### Activity Synthesis
When the posters are complete, hang them around the room. Provide students with these sentence starters, and give them an opportunity to view their classmates’ work and write their responses. This provides a structured way for students to critique the reasoning of others (MP3).
The most surprising combination of things was ____________________ because ______________________________________________________________.
The group _______________________________ should check their work where they ___________________________________________________________________.
I really liked when the group ____________________________ did this __________________________________ because _______________________________________.
Representing, Reading, Writing: MLR7 Compare and Connect. Use this routine when students share their visual displays. Direct attention to the different ways pairs explained quantities and their relationships and justified whether the relationship was proportional (e.g., distance traveled in yards and time elapsed in minutes, for every one minute they traveled 5 yards, the relationship is proportional because the graph of the line passes through the origin). Emphasize the language used to describe the proportional relationships and justify whether the relationship was proportional. These exchanges strengthen students' mathematical language use and reasoning of proportional relationships.
Design Principle(s): Maximize meta-awareness
## Lesson Synthesis
### Lesson Synthesis
Reflect on the following questions.
• "Describe any part of your work today that you would do differently, if you could start over."
• "Tell me about something new you learned in this class recently."
• "Tell me about any questions you still have, or anything that is confusing you."
## 14.4: Cool-down - Explain Their Work (5 minutes)
### Cool-Down
Cool-downs for this lesson are available at one of our IM Certified Partners
## Student Lesson Summary
### Student Facing
The constant of proportionality for a proportional relationship can often be easily identified in a graph, a table, and an equation that represents it. Here is an example of all three representations for the same relationship. The constant of proportionality is circled:
On the other hand, some relationships are not proportional. If the graph of a relationship is not a straight line through the origin, if the equation cannot be expressed in the form $$y = kx$$, or if the table does not have a constant of proportionality that you can multiply by any number in the first column to get the associated number in the second column, then the relationship between the quantities is not a proportional relationship. | 2021-01-17 12:46:05 | {"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": 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": 0.40893712639808655, "perplexity": 1959.8417144476737}, "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/1610703512342.19/warc/CC-MAIN-20210117112618-20210117142618-00030.warc.gz"} |
https://math.stackexchange.com/questions/2289832/providing-an-explanation-for-trend-found-in-numbers-and-divisibility | # Providing an explanation for trend found in numbers and divisibility.
Let $A$ and $B$ be integers with the same number of digits (like 35 and 92). Why is it that $$\underbrace{AA\ldots A}_{W\ A's} \cdot \underbrace{BB\ldots B}_{Z\ B's} = \underbrace{AA\ldots A}_{Z\ A's}\cdot \underbrace{BB\ldots B}_{W\ B's}?$$ (Here the $A$'s and $B$'s are being concatenated, not multiplied.) For example A=3 and B=8, W=2 and Z=4, why would 33*8888=3333*88? Why does this equation work?
I was drinking coffee when I randomly came up with this so I was not quite successful in finding an answer. I tried explaining this trend using the logarithmic graph below. I was trying to see whether changing the difference between W and Z would have an effect on the trend and it didn't. What I noticed did have a trend was the actual smaller number between W and Z. The larger the smallest of the two was, the more the A_W/A_Z function began to behave like a regular logarithm. Can someone try to explain 1- why the expression even works in the first place and 2- why the division of AW/AN approaches a 10^n where n= the difference between W and N
The reason is that $$33\cdot8888 = (3\cdot11)(8\cdot1111) = (8\cdot11)(3\cdot1111) = 88\cdot3333.$$ In general, if $A$ and $B$ have $N$ digits, we have $$\underbrace{AA\ldots A}_{W\ A's} = A\cdot \underbrace{1\underbrace{00\ldots0}_{N-1\ 0's}1\underbrace{00\ldots0}_{N-1\ 0's}100\ldots\ldots01\underbrace{00\ldots0}_{N-1\ 0's}1}_{W\ 1's} = A\cdot X.$$ Let $X$ be the number on the right that is multiplied by $A$. Similarly we define $Y$ as the number on the right of the below equation that is multiplied by $B$. $$\underbrace{BB\ldots B}_{Z\ B's} = B\cdot \underbrace{1\underbrace{00\ldots0}_{N-1\ 0's}1\underbrace{00\ldots0}_{N-1\ 0's}100\ldots\ldots01\underbrace{00\ldots0}_{N-1\ 0's}1}_{Z\ 1's}= B\cdot Y.$$ Putting everything together, we get $$\underbrace{AA\ldots A}_{W\ A's} \cdot \underbrace{BB\ldots B}_{Z\ B's} = (A\cdot X)(B\cdot Y) = (A\cdot Y)(B\cdot X) = \underbrace{AA\ldots A}_{Z\ A's}\cdot \underbrace{BB\ldots B}_{W\ B's}.$$
It might be worth mentioning that the numbers $X$ and $Y$ are given by geometric series: $$X=10^{N(W-1)}+10^{10(W-2)}+...+10^{1N}+10^{0N}=\frac{10^{NW}-1}{10^N-1}\\ Y=10^{N(Z-1)}+10^{10(Z-2)}+...+10^{1N}+10^{0N}=\frac{10^{NZ}-1}{10^N-1}.$$
Let $A^W$ denote the numeral formed by concatenating $W$ copies of $A$, and let $N$ be the length of $A$. By inspection we have \begin{align}A^W &=A^{W-1}10^N + A \end{align} and by applying this recursively, we find \begin{align}A^W &=A^{W-1}10^N + A\\ &=(A^{W-2}10^N+A)10^N+A\\ &=A^{W-2}10^{2N}+A\,10^N+A\\ &...\\ &=A\cdot(10^{(W-1)N}+10^{(W-2)N}+...+10^{1N}+10^{0N})\\ &=A\cdot\frac{10^{NW}-1}{10^N-1} \end{align}
and similarly for $B^Z$. | 2021-12-04 19:32:19 | {"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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9191396832466125, "perplexity": 255.3897237217129}, "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/1637964363006.60/warc/CC-MAIN-20211204185021-20211204215021-00022.warc.gz"} |
http://www.zachsmithwick.com/engineered-how-long-should-you-smoke-a-pork-shoulder/ | # Engineered: How Long Should You Smoke a Pork Shoulder?
Over the holidays I helped my dad make some BBQ using his new smoker, and it turned out really well!
This wasn't ours, but it looked kind of like this!
Image Credit
Smoking a pork shoulder isn't too terribly difficult, the tricky part is getting the smoker set up so that the temperature is stable, getting the rub on the meat right, and then just being patient enough to wait however many hours it takes to cook.
During the time it was cooking, we had a temperature probe in the pork so that we could monitor its progress while cooking. At one point, about halfway through the cooking process, I was afraid that the temperature of the smoker was too hot, because I knew that it had several more hours to cook and the temperature of the meat was well over halfway to the final doneness temperature.
And that got me thinking, what should the temperature of the meat be at various points in the cooking process? I had to take a thermodynamics, fluids, and heat transfer course in college, so I decided to finally use that knowledge! As it turns out, meat cooking in a smoker can be fairly accurately modeled by basic heat transfer equations, exciting!
A Crash Course on Heat Transfer
I am by no means an expert on heat transfer, but the basic principles are pretty easy to understand.
The second law of thermodynamics says, in simple terms, that when two bodies at different temperatures interact, then heat is transferred from the warmer body to the colder body until they both reach the same temperature, known as thermal equilibrium.
In many cases of heat transfer in the real world, some object is placed into a larger environment where the temperature is different. In the case of smoking a pork shoulder, a (cold) pork shoulder is put into a (hot) smoker, and the smoker transfers heat into the pork shoulder. If the pork shoulder was left in the smoker for a long period of time, and the smoker had sufficient fuel such that it never changed temperature, then the pork shoulder would eventually reach the temperature of the smoker (or get pretty close).
The fundamental heat transfer equation is known as Newton's Law of Cooling, which is applicable for convective heat transfer (such is the case for our smoked pork shoulder example).
In terms of the temperature of the smaller body (pork shoulder) and the environmental temperature, it becomes a first order separable differential equation expressed as:
To solve this equation in order to get the temperature of the meat as a function of time, we must rearrange it a little bit.
First, multiply each side of the equation by dt and divide by $\Delta T(t)$.
Then, integrate each side:
If you recall, the integral of 1/x results in ln(x), so we have:
Solve for $\Delta T(t)$ by taking both sides to the power of e...
Substituting $T(t) - T_{env}$ for $\Delta T(t)$ and solving, we are left with:
So there we have it, a pretty simple equation with only a few variables (well, for now at least).
$T(t)$: The temperature of an object in a larger environment as a function of time elapsed in the environment. The pork shoulder, in our example.
$T_{env}$: The temperature of the environment, which would be the smoker temperature in our example.
$T_0$: The initial temperature of the object. In the case of a pork shoulder that you took out of the fridge, it would be approximately the temperature of the fridge, which is between 32 and 40 degrees Fahrenheit.
$r$: A constant that determines at what rate the object will exponentially approach the environmental temperature. This is the tricky variable that we will discuss more later.
$t$ The time elapsed in the environment. Can be any units of time as long as they are consistent with the 'r' parameter.
When you look at the solved form of the heat transfer equation, you can see that as time approaches infinity, the right portion of the equation goes to zero and the temperature of the object becomes the temperature of the environment, i.e. "thermal equilibrium". So, if you are really patient, you can set your smoker temperature to be the final desired doneness temperature of the meat, and you can't overcook it! This, however, takes a long time as we will see.
Variables That Affect How Quickly The Meat Cooks
If you were to place a metal sphere into your smoker, its temperature would increase at a different rate than a pork shoulder. This is due to a few factors:
Surface Area ($A$): The area over which heat is contacting the meat will affect how quickly that heat gets inside the meat. If the meat is sitting atop a grill grate inside a covered smoker, then you can make a reasonable approximation that the meat is being hit with convective heat across its entire physical surface area. Basically, the more of the meat's surface that is exposed to the heat, the faster it will cook.
Calculating surface area can either be a trivial matter that anyone who has taken math at above a 4th grade level should be able to do, or it can be enormously complicated and require complex modeling software and a powerful computer. Unfortunately, most shapes that we encounter in the real world (such as our pork shoulder) fall into the latter category, but in most cases we can make some approximations that they are pretty close to simpler shapes whose surface areas we can easily calculate. A pork shoulder can be approximated as either a rectangular prism, a cylinder, or a triangular prism, depending on how it is cut.
Mass ($m$): The mass of an object will determine how quickly heat is transferred into or out of it. In general, the more massive the object, the more slowly it changes temperature.
Specific Heat Capacity ($c_p$): A measure of how much heat can be transferred to an object before it changes temperature. Will depend on the physical properties of the object. I was able to find this handy table that has specific heat capacities for a variety of foods. They have pork shoulder listed at 0.59 BTU/( lb oF).
Heat Transfer Coefficient ($h$): Number that characterizes how heat is transferred to an object that includes both the properties of the object as well as the nature of the heat transfer. Convective heat transfer through a medium of air will be different than conductive heat transfer using metal, for example. This was a tricky one to research as it can get pretty complicated, but based on this paper it seems to be somewhere between 5.1 and 9.1 W/(m^2 K). My model worked best when I used 12 (granted the paper was for processed ham and not pork shoulder).
Putting these variables together, we can derive a formula for r, the rate of heat transfer from the heat transfer equation:
In this formula, the only things that will vary between pork shoulders are the mass and surface area.
Calculating Surface Area
If we make a few assumptions, then you can calculate the surface area of a pork shoulder by only knowing how much it weighs! The first assumption to make is that the pork shoulder will be of uniform density. This is not entirely accurate, but should be close enough.
The second assumption involves what shape the pork shoulder is, and its relative dimensions. The shape that seemed to work the best in my model is that of a triangular prism whose length is twice the height. Based on pork shoulders I've seen, this should be fairly close.
So, how do you determine surface area if all you know is weight (or, more precisely, mass...all of the equations are only valid on Earth...so be careful if you want to smoke a pork shoulder on the moon)? Well, if we assume constant density, $\gamma$, then we can calculate the volume:
For pork shoulder, I've assumed a density of 0.0336 lb/in^3. Once we know the volume, then we can take a few liberties to figure out the surface area. The volume of a triangular prism is equal to:
Where $b$ is the base of the triangle, $h$ is the height of the triangle, and $l$ is the length of the prism. If we assume that it is an equilateral triangle with a length that is twice the height, then the volume equation becomes:
Once we back-calculate the height of the triangle based on the volume and the above equation, we can calculate surface area as:
All right! We now can plug in all of the necessary parameters for $r$, knowing only what the mass (weight) of the pork shoulder is. Let's see what happens when we plug the numbers in.
Cooking Times and Temperatures for Pork Shoulders of Various Weight
Whew, if you've made it this far, then you have bravely traversed all the math stuff...or just skipped it. If the former, then I applaud you.
Let's first look at the scenario in which my father and I cooked our pork: in a 275 degree Fahrenheit smoker.
Fig. 1: Pork shoulder of weights between 5 and 11 pounds in a 275 degree smoker
As you can see from Fig. 1, pork shoulders of fairly standard weights can be cooked between 6 and 8 hours in a 275 degree smoker to a doneness temperature of 190 degrees (recommended for fall-apart tender pork shoulder). Another thing to note is that the meat temperature reaches the "halfway point" of doneness after only 3 hours of cooking, as the meat does not follow a linear temperature increase. So, if you are monitoring the meat temperature and early on in the cooking process feel like the temperature is increasing too rapidly, don't worry, it's supposed to do that.
Now let's look at a scenario where we cook the pork in a 225 degree smoker. Obviously, it will take longer to reach desired doneness.
Fig. 2: Pork shoulder of weights between 5 and 11 pounds in a 225 degree smoker
As you can see in Fig. 2, the time it will take the shoulder to cook at 225 degrees also varies based on weight, being between 11 and 13 hours for shoulder weights between 5 and 11 pounds.
Now, what if we dropped the smoker temperature to 190 degrees, the desired doneness temperature of the meat?
Fig. 3: Pork shoulder of weights between 5 and 11 pounds in a 190 degree smoker
An astute mathematical observer will note that the meat will actually never reach 190 degrees if the smoker is kept at exactly 190 degrees, but it will get pretty close, as you can see in Fig. 3. You can also note that it will increase in temperature fairly quickly, but after about 12 hours temperature increases become painfully slow. After 24 hours, you can pretty much call it "done".
Conclusion
Long story short, meat cooked over a relatively low temperature in a smoker will not experience a linear increase in internal temperature, so don't be worried if after a couple hours you think your meat is getting too hot too fast...that's just thermodynamics.
There ya go, if you thought your hobby (or business) of smoking pork butts was exempt from calculus and differential equations, then you were sadly mistaken.
## One thought on “Engineered: How Long Should You Smoke a Pork Shoulder?”
1. Cary Taylor says:
Amazing! So cool!
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2019-08-19 12:12:23 | {"extraction_info": {"found_math": true, "script_math_tex": 18, "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": 29, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6737145781517029, "perplexity": 618.211726390751}, "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/1566027314732.59/warc/CC-MAIN-20190819114330-20190819140330-00516.warc.gz"} |
https://www.akalin.com/intro-primality-testing | # An Introduction to Primality Testing
## Fred Akalin
### July 8, 2012
I will explain two commonly-used primality tests: Fermat and Miller-Rabin. Along the way, I will cover the basic concepts of primality testing. I won't be assuming any background in number theory, but familiarity with modular arithmetic will be helpful. I will also be providing implementations in Javascript, so familiarity with it will also be helpful. Finally, since Javascript doesn't natively support arbitrary-precision arithmetic, I wrote a simple natural number class (SNat) that represents a number as an array of decimal digits. All algorithms used are the simplest possible, except when a more efficient one is needed by the algorithms we discuss.
Primality testing is the problem of determining whether a given natural number is prime or composite. Compared to the problem of integer factorization, which is to determine the prime factors of a given natural number, primality testing turns out to be easier; integer factorization is in NP and thought to be outside P and NP-complete, whereas primality testing is now known to be in P.
Most primality tests are actually compositeness tests; they involve finding composite witnesses, which are numbers that, along with a given number to be tested, can be fed to some easily-computable function to prove that the given number is composite. (The composite witness, along with the function, is a certificate of compositeness of the given number.) A primality test can either check each possible witness or, like the Fermat and Miller-Rabin tests, it can randomly sample some number of possible witnesses and call the number prime if none turn out to be witnesses. In the latter case, there is a chance that a composite number can erroneously be called prime; ideally, this chance goes to zero quickly as the sample size increases.
The simplest possible witness type is, of course, a factor of the given number, which we'll call a factor witness. If the number to be tested is $$n$$ and the possible factor witness is $$a$$, then one can simply test whether $$a$$ divides $$n$$ (written as $$a \mid n$$) by evaluating $$n \bmod a = 0$$; that is, whether the remainder of $$n$$ divided by $$a$$ is zero. This doesn't yield a feasible deterministic primality test, though, since checking all possible witnesses is equivalent to factoring the given number. Nor does it yield a feasible probabilistic primality test, since in the worst case the given number has very few factors, which random sampling would miss.
The simplest useful witness type is a Fermat witness, which relies on the following theorem of Fermat:
(Fermat's little theorem.) If $$n$$ is prime and $$a$$ is not a multiple of $$n$$, then $a^{n-1} \equiv 1 \pmod{n}\text{.}$
Thus, a Fermat witness is a number $$1 \lt a \lt n$$ such that $$a^{n-1} \not\equiv 1 \pmod{n}$$. Conversely, if $$n$$ is composite and $$a^{n-1} \equiv 1 \pmod{n}$$, then $$a$$ is a Fermat liar.
Let n = and a = .
If $$n$$ has at least one Fermat witness that is relatively prime, then we can show that at least half of all possible witnesses are Fermat witnesses. (Roughly, if $$a$$ is the Fermat witness and $$a_1, a_2, \dotsc, a_s$$ are Fermat liars, then all $$a \cdot a_i$$ are also Fermat witnesses.) Therefore, for a sample of $$k$$ possible witnesses of $$n$$, the probability of all of them being Fermat liars is $$\le 2^{-k}$$, which goes to zero quickly enough to be practical.
However, there is the possibility that $$n$$ is a composite number with no relatively prime Fermat witnesses. These are called Carmichael numbers. Even though Carmichael numbers are rare, their existence still makes the Fermat primality test unsuitable for some situations, as when the numbers to be tested are provided by some adversary.
Here is the Fermat compositeness test implemented in Javascript:
// Runs the Fermat compositeness test given n > 2 and 1 < a < n.
// Calculates r = a^{n-1} mod n and whether a is a Fermat witness to n
// (i.e., r != 1, which means n is composite). Returns a dictionary
// with a, n, r, and isCompositeByFermat, which is true iff a is a
// Fermat witness to n.
function testCompositenessByFermat(n, a) {
n = SNat.cast(n);
a = SNat.cast(a);
if (n.le(2)) {
throw new RangeError('n must be > 2');
}
if (a.le(1) || a.ge(n)) {
throw new RangeError('a must satisfy 1 < a < n');
}
var r = a.powMod(n.minus(1), n);
var isCompositeByFermat = r.ne(1);
return {
a: a,
n: n,
r: r,
isCompositeByFermat: isCompositeByFermat
};
}
Note that the algorithm depends on the efficiency of modular exponentiation when calculating $$a^{n-1} \pmod{n}$$. The naive method is unsuitable since it requires $$Θ(n)$$ $$b$$-bit multiplications, where $$b = \lceil \lg n \rceil$$. SNat uses repeated squaring, which requires only $$Θ(\lg n)$$ $$b$$-bit multiplications.
Another useful witness type is a non-trivial square root of unity $$\operatorname{mod} n$$; that is, a number $$a ≠ \pm 1 \pmod{n}$$ such that $$a^2 \equiv 1 \pmod{n}$$. It is a theorem of number theory that if $$n$$ is prime, there are no non-trivial square roots of unity $$\operatorname{mod} n$$. Therefore, if we do find one, that means $$n$$ is composite. In fact, finding one leads directly to factors of $$n$$. By definition, a non-trivial square root of unity $$a$$ satisfies $$a \pm 1 ≠ 0 \pmod{n}$$ and $$a^2 - 1 \equiv 0 \pmod{n}$$. Factoring the latter leads to $$(a+1)(a-1) \equiv 0 \pmod{n}$$, which means that $$n$$ divides $$(a+1)(a-1)$$. But the first condition says that $$n$$ divides neither $$a+1$$ nor $$a-1$$, so it must be a product of two numbers $$p \mid a+1$$ and $$q \mid a-1$$. Then $$\gcd(a+1, n)$$[1] and $$\gcd(a-1, n)$$ are factors of $$n$$.
Finding non-trivial square roots of unity by itself doesn't give a useful primality testing algorithm, but combining it with the Fermat primality test does. $$a^{n-1} \bmod n$$ either equals $$1$$ or not. If it doesn't, you're done since you have a Fermat witness. If it does equal $$1$$, and $$n-1$$ is even, then consider the square root of $$a^{n-1}$$, i.e. $$a^{(n-1)/2}$$. If it is not $$\pm 1$$, then it is a non-trivial square root of unity. If it is $$-1$$, then you can't do anything else. But if it is $$1$$, and $$(n-1)/2$$ is even, you can then take another square root and repeat the test, stopping when the exponent of $$a$$ becomes odd or when you get a result not equal to $$1$$.
To turn this into an algorithm, you simply start from the bottom up: find the greatest odd factor of $$n-1$$, call it $$t$$, and keep squaring $$a^t$$ mod $$n$$ until you find a non-trivial square root of $$n$$ or until you can deduce the value of $$a^{n-1}$$. In fact, this is almost as fast as the original Fermat primality test, since the exponentiation by $$n-1$$ has to do the same sort of squaring, and we're just adding comparisons to $$±1$$ in between squarings.
The original idea for the test above is from Artjuhov, although it is usually credited to Miller. Therefore, we call $$a$$ an Artjuhov witness[2] of $$n$$ if it shows $$n$$ composite by the above test.
Let n = and a = .
If $$n$$ is an odd composite, then it can be shown (originally by Rabin) that at least three quarters of all possible witnesses are Artjuhov witnesses. Therefore, for a sample of $$k$$ possible witnesses of $$n$$, the probability of all of them being Artjuhov liars is $$\le 4^{-k}$$, which is stronger than the bound for the Fermat primality test. Furthermore, this bound is unconditional; there is nothing like Carmichael numbers for the Artjuhov test.
Here is the Artjuhov compositeness test, implemented in Javascript:
// Runs the Artjuhov compositeness test given n > 2 and 1 < a < n-1.
// Finds the largest s such that n-1 = t*2^s, calculates r = a^t mod
// n, then repeatedly squares r (mod n) up to s times until r is
// congruent to -1, 0, or 1 (mod n). Then, based on the value of s
// and the final value of r and i (the number of squarings),
// determines whether a is an Artjuhov witness to n (i.e., n is
// composite).
//
// Returns a dictionary with, a, n, s, t, i, r, rSqrt = sqrt(r) if i >
// 0 and null otherwise, and isCompositeByArtjuhov, which is true iff
// a is an Artjuhov witness to n.
function testCompositenessByArtjuhov(n, a) {
n = SNat.cast(n);
a = SNat.cast(a);
if (n.le(2)) {
throw new RangeError('n must be > 2');
}
if (a.le(1) || a.ge(n)) {
throw new RangeError('a must satisfy 1 < a < n');
}
var nMinusOne = n.minus(1);
// Find the largest s and t such that n-1 = t*2^s.
var t = nMinusOne;
var s = new SNat(0);
while (t.isEven()) {
t = t.div(2);
s = s.plus(1);
}
// Find the smallest 0 <= i < s such that a^{t*2^i} = 0/-1/+1 (mod
// n).
var i = new SNat(0);
var rSqrt = null;
var r = a.powMod(t, n);
while (i.lt(s) && r.gt(1) && r.lt(nMinusOne)) {
i = i.plus(1);
rSqrt = r;
r = r.times(r).mod(n);
}
var isCompositeByArtjuhov = false;
if (s.isZero()) {
// If 0 = i = s, then this reduces to the Fermat primality test.
isCompositeByArtjuhov = r.ne(1);
} else if (i.isZero()) {
// If 0 = i < s, then:
//
// * r = 0 (mod n) -> a^{n-1} = 0 (mod n), and
// * r = +/-1 (mod n) -> a^{n-1} = 1 (mod n).
isCompositeByArtjuhov = r.isZero();
} else if (i.lt(s)) {
// If 0 < i < s, then:
//
// * r = 0 (mod n) -> a^{n-1} = 0 (mod n),
// * r = +1 (mod n) -> a^{t*2^{i-1}} is a non-trivial square root of
// unity mod n, and
// * r = -1 (mod n) -> a^{n-1} = 1 (mod n).
//
// Note that the last case means r = n - 1 > 1.
isCompositeByArtjuhov = r.le(1);
} else {
// If 0 < i = s, then:
//
// * r = 0 (mod n) can't happen,
// * r = +1 (mod n) -> a^{t*2^{i-1}} is a non-trivial square root of
// unity mod n, and
// * r > +1 (mod n) -> failure of the Fermat primality test.
isCompositeByArtjuhov = true;
}
return {
a: a,
n: n,
t: t,
s: s,
i: i,
r: r,
rSqrt: rSqrt,
isCompositeByArtjuhov: isCompositeByArtjuhov
};
}
With the two compositeness tests above, we can now write a probabilistic primality test:
// Returns true iff a is a Fermat witness to n, and thus n is
// composite. a and n must satisfy the same conditions as in
// testCompositenessByFermat.
function hasFermatWitness(n, a) {
return testCompositenessByFermat(n, a).isCompositeByFermat;
}
// Returns true iff a is an Arjuhov witness to n, and thus n is
// composite. a and n must satisfy the same conditions as in
// testCompositenessByArtjuhov.
function hasArtjuhovWitness(n, a) {
return testCompositenessByArtjuhov(n, a).isCompositeByArtjuhov;
}
// Returns true if n is probably prime, based on sampling the given
// number of possible witnesses and testing them against n. If false
// is returned, then n is definitely composite.
//
// By default, uses the Artjuhov test for witnesses with 20 samples
// and Math.random for the random number generator. This gives an
// error bound of 4^-20 if true is returned.
function isProbablePrime(n, hasWitness, numSamples, rng) {
n = SNat.cast(n);
hasWitness = hasWitness || hasArtjuhovWitness;
rng = rng || Math.random;
numSamples = numSamples || 20;
if (n.le(1)) {
return false;
}
if (n.le(3)) {
return true;
}
if (n.isEven()) {
return false;
}
for (var i = 0; i < numSamples; ++i) {
var a = SNat.random(2, n.minus(2), rng);
if (hasWitness(n, a)) {
return false;
}
}
return true;
}
isProbablePrime called with hasFermatWitness is the Fermat primality test, and isProbablePrime called with hasArtjuhovWitness is the Miller-Rabin primality test. The latter is the current general primality test of choice, replacing the Solovay-Strassen primality test.
We can also use isProbablePrime to randomly generate probable primes, which is useful for cryptographic applications:
// Returns a probable b-bit prime that is at least 2^b. All
// parameters but b are passed to isProbablePrime.
function findProbablePrime(b, hasWitness, rng, numSamples) {
b = SNat.cast(b);
var lb = (new SNat(2)).pow(b.minus(1));
var ub = lb.times(2);
while (true) {
var n = SNat.random(lb, ub);
if (isProbablePrime(n, hasWitness, rng, numSamples)) {
return n;
}
}
}
In this case, for sufficiently large $$b$$, the Fermat primality test is acceptable, since Carmichael numbers are so rare and we're the ones generating the possible primes to be tested.[3]
There are other primality tests, but they're less often used in practice because they're either less efficient or more sophisticated than the algorithms above, or they require $$n$$ to have special properties. Perhaps the most interesting of these tests is the AKS primality test, which proved once and for all that primality testing is in P.
Like this post? Subscribe to my feed or follow me on Twitter .
## Footnotes
[1] $$\gcd$$ is the greatest common divisor function.
[2] “Artjuhov witness” is an idiosyncratic name on my part; a more common name is strong witness, which I don't like.
[3] According to Wikipedia, PGP uses the Fermat primality test. | 2020-03-29 08:55:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6774837374687195, "perplexity": 1096.5593353332642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370494064.21/warc/CC-MAIN-20200329074745-20200329104745-00244.warc.gz"} |
http://www.interactiveassessment.com/question/193/ | DIRECTIONS: Drag the two ordered pairs to the answer zones.
Which two points would form a slope of $\frac{-2}{3}$?
$(1,2)$
$(3,5)$
$(4,-6)$
$(2,3)$
$(-1,2)$
$(2,0)$ | 2017-12-11 04:11:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.755610466003418, "perplexity": 2167.1880533001477}, "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/1512948512121.15/warc/CC-MAIN-20171211033436-20171211053436-00493.warc.gz"} |
https://azatolblackboard.wordpress.com/2011/10/25/pondering-higher-order-functions-and-derivatives/ | # Pondering higher order functions and derivatives
So I started thinking about other possible alien math systems the past few days and that got me on the topic of higher order functions. These have many names, and a lot of the complexities of things like functional analysis are beyond me, but I do understand a few of the basics, and I understand higher order functions from the perspective of programming.
A higher order function does the same thing with functions that functions do with numbers. It is a function that takes functions as its parameters instead of numbers. The derivative is a classic example:
$Limit_h \rightarrow 0 \frac{f(x+h) - f(x)}{h}$
If we consider x to be a general variable rather than a specific value to be calculated on, than this definition is a higher order function. We provide a function and get another function back. Just like with sin(x) we provide a number and get a number back.
There are tons of higher order functions, but I’m mostly interested in the ones related to the derivative and integral. Of course the integral doesn’t give us back a single value, so I turn it into a higher order function by taking the integral from 0 to x.
You could also have functions that received higher order functions themselves as parameters, such as the following:
$N(O(f(x)), r) = O(O(O(....f(x))$ –r iterations–
$N(D(x^3), 3) = 6$
That’s just a function that iterates a higher order function a given number of times.
The higher order functions I was thinking of both are similar to infinite series in a way. They bring up the same questions of convergence, but its harder for me to say in what sense a function can converge?
$G(f(x), n) = N(D(f(x)), n), n \geq 0$
$G(f(x), n) = N(I(f(x)), -n), n < 0$
So G is a generalized derivative or integrator that gives the nth derivative for positive n, or the -nth integral for negative n
$P(f(x)) = \sum_{i=0} G(f(X), i)$
$M(f(x)) = \sum_{i=0} G(f(x), -i)$
P represents the sum of all derivatives of a function, and M is the sum of all integrals of a function. P only exists where f(x) is infinitely differentiable, and M exists wherever f(x) is continuous. P seems to “converge” for a lot more functions than M converges for.
Examples of P converging:
1. Any polynomial converges because the derivative will eventually be 0
ex. $x^3 \rightarrow x^3 + 3 x^2 + 6 x + 6$
2. $a^x$ for a < e converges because the derivative decreases like a geometric series
Examples of M converging:
1. $a^x$ for a > e converges in the same kind of geometric series except the multiplier is 1 / log(a) instead of log(a).
The question that I’m thinking about now is whether any function has the property that both P and M converge for it.
Sin(x) might seem to converge for both P and M, since its derivatives cancel each other out. Unfortunately, Sin(x) does not converge to any one value at infinity.
If P and M both converge for a function I think it would be a rather special function. The thing is that, in order for P to converge, the derivatives have to be getting smaller in magnitude. So Abs(G(f(x), n)) has to be decreasing, but that seems to imply that the Integrals are generally increasing in magnitude.
It seems to me that In order for both P and M to converge, there has to be a function that’s the “max” in some sense, so that the derivative and integral of that function are both smaller everywhere than the function itself. That function if found would define a whole range of functions which have converging P and M values.
Does there exist a function f(x) such that for all x > 0 $0 < \int_{0} f(x) < f(x) \land 0 < D(f(x)) < f(x)$ ?
We can think of this, in a way, as a differential inequality. I don’t know if much is known about this sort of thing, but I certainly don’t know how to proceed with it that far. If we consider the integral to be the function g(x) and focus on that we get:
g’ = h
$x>0 \rightarrow g' > g > 0$
$x>0 \rightarrow 0 < h' < h$
Functions that grow very rapidly have larger derivatives and even larger 2nd derivatives and so on. Whereas other functions such as polynomials have their integrals grow such that at some positive a, the higher order terms will dominate and make them larger.
So our f(x) would have to be some strange function, small enough that its derivatives are not any larger, but also strange enough for its integrals to always be smaller. I can imagine this being true on a small interval but not the entire positive real line.
Getting a better idea of what functions have smaller derivatives for all positive x, and which functions have smaller integrals would go a long way to figuring it out, so I will think on that some more. | 2017-11-23 22:10:23 | {"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": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8717326521873474, "perplexity": 291.07220972669285}, "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/1510934806979.99/warc/CC-MAIN-20171123214752-20171123234752-00255.warc.gz"} |
https://homework.cpm.org/category/CCI_CT/textbook/calc/chapter/2/lesson/2.4.1/problem/2-139 | Home > CALC > Chapter 2 > Lesson 2.4.1 > Problem2-139
2-139.
Write an expression that will calculate the slope between (a, f(a)) and (b, f(b)). A sketch may help. Homework Help ✎
$\text{slope }=\frac{\Delta y}{\Delta x}$ | 2020-03-30 13:14:10 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.6347965598106384, "perplexity": 9369.19080394927}, "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-2020-16/segments/1585370497042.33/warc/CC-MAIN-20200330120036-20200330150036-00463.warc.gz"} |
https://www.scienceforums.net/topic/121246-dead-galaxy-questions/ | ## Recommended Posts
So I have noticed it’s about a week now everyday there is a new article from a different source about this. Not sure why it keeps making the news day after day as though there is something up with it we should know about.
From what I have been told the constant of 1/137 may have changed and if it fell below 1 it can put the universe in universe in a vacuum decay state
##### Share on other sites
From what I have been told the constant of 1/137 may have changed and if it fell below 1 it can put the universe in universe in a vacuum decay state with this data found
Is this true
if that is the case how long before it hits us
Edited by Bmpbmp1975
##### Share on other sites
Where did you hear that this is the result of a change in the fine structure constant ?
A Hubble bubble due to change in vacuum would propogate at c and as a result we would get no warning.
##### Share on other sites
47 minutes ago, Mordred said:
Where did you hear that this is the result of a change in the fine structure constant ?
A Hubble bubble due to change in vacuum would propogate at c and as a result we would get no warning.
So someone in another forum told me that the Newley found dead galaxy can mean the 1/137 has changed?
so if it did change then it can put the universe constant less than one which would move the universe out of metastable state
( Well I would question such a claim unless they could provide a decent peer review article on it. There have been numerous studies of a potential varying fine structure constant. However deep field studies show no variation outside error measurement bars.) ?
has the 1/137 changed?
has the métastable state changed?
Edited by Bmpbmp1975
##### Share on other sites
Well I would question such a claim unless they could provide a decent peer review article on it. There have been numerous studies of a potential varying fine structure constant. However deep field studies show no variation outside error measurement bars.
##### Share on other sites
Fine-structure constant (alpha) is used in equation of ionisation energy of Hydrogen. So, if alpha changes, ionisation energy must change.. or it would have to be compensated by change of other constant (which would not be considered constant anymore obviously)..
4 hours ago, Bmpbmp1975 said:
From what I have been told the constant of 1/137 may have changed and if it fell below 1 it can put the universe in universe in a vacuum decay state
Fine-structure constant is already below 1. It's 1/137 (less than 0.0073) already ;)
##### Share on other sites
8 hours ago, Bmpbmp1975 said:
I noticed that there is not a single word about fine-structure constant nor it's alleged variability in the above article which you provided in OP..
##### Share on other sites
9 hours ago, Bmpbmp1975 said:
So someone in another forum told me that the Newley found dead galaxy can mean the 1/137 has changed?
Can mean. As in, one of many possible explanations, and without any kind of justification. I would surmise the argument is that if the fine structure constant changed then electromagnetic interactions would change characteristics or even "turn off" and we'd stop getting the light we're used to seeing. But there would probably be other ramifications and we'd have to investigate further.
Mordred has stated that a Hubble bubble would travel at c. It would already be here if we can see this galaxy's weird state.
##### Share on other sites
Even with the propogation speed aside, the ramifications of a change in the fine structure constant should also affect spectrography measurements for example the 21 cm Hydrogen line.
Something like that would be bigger news than the article itself I would think.
Anyways here is the arxiv article for the above study. You will note that a changing fine structure constant is not on the list of possible causes.
Edited by Mordred
##### Share on other sites
The article says nothing about the light coming from it, just that it's not forming new stars. The fine structure constant wouldn't be the culprit — that's for electromagnetic interactions. You'd have to have a nuclear coupling change if that kind of explanation had any merit.
##### Share on other sites
My apologies I am still new and limitied to 5 posts per day. I saw your answers and that helped my understanding. Question
can a change in the fine-structure constant lead to vacuum decay.
Edited by Bmpbmp1975
##### Share on other sites
Hrrm vacuum decay isn't a very accurate descriptive of a quantum vacuum. A more accurate descriptive would be a different vacuum state.
A change in the fine structure can affect the vacuum impedance value. Through the equivalent relation
$\alpha=\frac{e^2}{4\pi}\frac{Z_o}{\hbar}$
##### Share on other sites
15 hours ago, swansont said:
The article says nothing about the light coming from it, just that it's not forming new stars. The fine structure constant wouldn't be the culprit — that's for electromagnetic interactions. You'd have to have a nuclear coupling change if that kind of explanation had any merit.
Where the he*l did I put my magnifying glass...
##### Share on other sites
44 minutes ago, Mordred said:
Hrrm vacuum decay isn't a very accurate descriptive of a quantum vacuum. A more accurate descriptive would be a different vacuum state.
A change in the fine structure can affect the vacuum impedance value. Through the equivalent relation
α=e24πZo
So that’s a yes then, a change could lead to vacuum collapse
##### Share on other sites
How do you collapse a vacuum ? A vacuum can have a wide range of energy states however a true vacuum is impossible under QM.
Edited by Mordred
##### Share on other sites
7 hours ago, Bmpbmp1975 said:
My apologies I am still new and limitied to 5 posts per day. I saw your answers and that helped my understanding. Question
can a change in the fine-structure constant lead to vacuum decay.
People have looked to see if the fine structure constant (or any other fundamental constants) have changed over time. There is no evidence that they have or can change. You might as well worry about whether gravity might stop working tomorrow.
##### Share on other sites
1 hour ago, Mordred said:
How do you collapse a vacuum ? A vacuum can have a wide range of energy states however a true vacuum is impossible under QM.
QM?
##### Share on other sites
43 minutes ago, Bmpbmp1975 said:
QM?
Quantum mechanics
##### Share on other sites
4 hours ago, MigL said:
Where the he*l did I put my magnifying glass...
Press ctrl - +...
##### Share on other sites
2 hours ago, Strange said:
Quantum mechanics
I didn’t know it was impossible all the articles I have read and the works of Katie Mack state it is possible?
##### Share on other sites
17 hours ago, Mordred said:
Hrrm vacuum decay isn't a very accurate descriptive of a quantum vacuum. A more accurate descriptive would be a different vacuum state.
A change in the fine structure can affect the vacuum impedance value. Through the equivalent relation
α=e24πZo
And you also stated it can be an change in the vacuum colloque theory
##### Share on other sites
What is vacuum colloque theory ?
##### Share on other sites
Sorry vacuum collapse stupid spell check
##### Share on other sites
Ah. Well I stated vaccum collapse isn't accurate. Not in the circumstances your describing. You have to understand a vacuum describes an energy state with associations to pressure.
For example the electroweak symmetry breaking resulted in a phase transition from the false vacuum to true vacuum. The false vaccuum describes a higher potential. This is one of the more likely causes of inflation.
Now ask yourself this question if the fine structure does change then some other of the following relations must also change.
$\alpha=\frac{1}{4\pi\epsilon^2}\frac{e^2}{\hbar c}$
Where $\epsilon$ is the electric constant or permittability of free space.
e is the electric charge, c is the speed of light. So if the fine structure changes then another constant must have also changed.
##### Share on other sites
This topic is now closed to further replies.
× | 2020-07-05 08:10:31 | {"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": 0.23606330156326294, "perplexity": 1515.2621325209577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655887046.62/warc/CC-MAIN-20200705055259-20200705085259-00038.warc.gz"} |
https://omaclaren.com/ | # Linear or nonlinear with respect to what?
Overview
I’m teaching a partial differential equations (PDEs) course in the mathematics department at the moment. A typical ‘gimme’ question for assignments and tests is to get the students to classify a given equation as linear or nonlinear (most of the theory we develop in the course is for linear equations so we need to know what this means). Since we aim to introduce the students to a bit of operator theory we often switch back and forward between talking about linear/nonlinear PDEs and linear/nonlinear operators.
One of the students noticed that this introduced some ambiguity into our classification problem and asked a great question. I think it illustrates a useful general point about terminology like linear vs nonlinear and how these terms can be misleading or ambiguous. So here’s the question and my attempt at clarifying the ambiguity.
The question
It’s my understanding that a PDE is linear if we can write it in the form $Lu = f(x,t)$, where L is a linear differential operator.
If we are given a PDE that looks like $Au = 0$ for some differential operator A and asked to show that the PDE is nonlinear, I can (probably) show that A is not a linear differential operator. However this doesn’t necessarily imply that you cannot rearrange the equation in such a way to make it linear.
For example the operator A defined by
$Au = (u^2+1)u_t+(u^2+1)u_{xx}$
is not a linear differential operator. However the equation Au = 0 is the same as $u_t + u_{xx} = 0$, and the differential operator B defined by $Bu = u_t + u_{xx}$ is linear.
So (I believe I’m correct in saying this), the original PDE is linear, because it can be rewritten in this form Lu = f(x,t) for some linear differential operator L and function f(x,t).
My question is what sort of working are we expected to show, if we aim to prove the PDE Au=0 is not linear? For the purposes of the assignment does it suffice to prove that A is not linear?
My response
Here was my response and attempt to clarify (corrections/comments welcome!).
Great question!
As you’ve noticed there is some ambiguity when we move back and forward between talking about equations and operators. This is to be expected since a function (e.g. an operator) is a different type of mathematical object to an equation
For example the function $f:x \mapsto x^2$ is a different ‘object’ to the equation $x^2 = 0$.
You’ve correctly noticed that if we can write a differential equation as Lu = f where L is some linear operator then the differential equation is also called linear. Unfortunately, again as you’ve noticed, this definition makes it hard to decide when an equation is nonlinear as you may be able to write a linear equation in terms of a nonlinear operator with the right choice of f. This is because the negation of ‘there exists’ a linear operator is ‘there doesn’t exist a linear operator’.
So proving that an equation is linear is easy using the operator definition – we just find any linear operator that works.
On the other hand, proving that an equation is nonlinear is harder using this definition – it would require showing all operators for which Au = f are nonlinear.
This seems too hard to do directly, so let’s reformulate it in an equivalent but easier-to-use way.
We want to keep our definitions of linear and nonlinear as close as possible for the two cases of operators and equations.
Improved definitions
An operator acting on u is linear iff L(au+bv) = aL(u) + bL(v) for any u and v in the operator’s domain and constants a, b.
and
Given an equation written in the from Au = f for some operator A and forcing function f, the equation is linear iff A(au+bv) = aA(u) + bA(v) for any two solutions u, v to the equation Au = f.
I think this definition should cover your example (try it! Note that it is slightly subtle how this makes a difference! But, basically, we get to use the f = 0 in the equation case now).
Also note that:
The function definition now explicitly talks about linearity with respect to how it operates on objects in its domain while the equation definition talks explicitly about behaviour with respect to solutions to that equation. This seems natural given the different ‘nature’ of ‘functions’ and ‘equations’.
Does that make sense?
Morally speaking
I think the broader lesson is that terms like linear/nonlinear are relative to the specific mathematical representation chosen and how we interact with that representation. A ‘system’ is not really intrinsically linear or nonlinear, rather an ‘action’ (or function or operator or process) is linear or nonlinear with respect to a specific set of ‘objects’ or ‘measurements’ or ‘perturbations’ or whatever. This needs to be made explicit for an unambiguous classification to be carried out.
Generalisation
Perhaps generalising too far, something like this came up in some recent ‘philosophical’ discussions I’ve been having over at Mayo’s blog (and was also at the heart of another scientific disagreement I once had with an experimentalist about interpreting aquaporin knockout experiments…).
For example, it has been pointed out that while ‘chaos’ is typically associated with (usually finite-dimensional) nonlinear systems, there are examples of infinite-dimensional linear systems that exhibit all the hallmarks of chaos – see e.g. ‘Linear vs nonlinear and infinite vs finite: An interpretation of chaos‘ by Protopopescu for just one example. So, changing the underlying ‘objects’ used in the representation changes the classification as ‘linear’ or ‘nonlinear’ or, as Protopopescu states
Linear and nonlinear are somewhat interchangeable features, depending on scale and representation…chaotic behavior occurs… when we have to deal with infinite amounts of information at a finite level of operability. In this sense, even the most deterministic system will behave stochastically due to unavoidable and unknown truncations of information.
This theme appears again and again at various levels of abstraction – e.g. we saw it in a high-school math problem where a singularity (a type of ‘lack of regularity’) arose (which we interpreted as) due to an incompatibility between a regular higher-dimensional system and a constraint restricting that system to a lower-dimensional space. (Compare the abstract operator itself with the operator + equating it to zero to get an equation.) We were faced with the choice of a regular but underdetermined system that required additional information for a unique solution or a ‘unique’ but singular (effectively overdetermined) system. Similarly other ‘irregular’ behaviour like ‘irreversibility’ can often be thought of as arising due to a combination of ‘reversible’ (symmetric/regular etc) microscopic laws + asymmetric boundary conditions/incomplete measurement constraints. Similar connections between ‘low/high dimensional’ systems and ‘stable/unstable’ systems are discussed by Kuehn in ‘The curse of instability‘.
To me this presents a helpful heuristic decomposition of models of the world into two-level decompositions like ‘irregular nature’ -> ‘regular, high-dimensional nature’ + ‘limited accessibility to nature’ (h/t Plato) or ‘internal dynamics’ + ‘boundary conditions’, ‘reversible laws’ + ‘irreversible reductions/coarse-graining’ etc. Note also that, on this view, ‘infinite’ and ‘finite’ are effectively ‘relative’, ‘structural’ concepts – if our ‘access’ to the ‘real world’ is always and instrinsically limited it leads us to perceive the world as effectively infinite (in some sense) regardless of whether the world is ‘actually’ infinite. You still can’t really avoid ‘structural infinities’ – e.g. continuous transformations – though.
It seems clear that this also inevitably introduces ‘measurement problems’ that aren’t that dissimilar to those considered to be intrinsic to quantum mechanics into even ‘classical’ systems, and leads to ideas like conceiving of ‘stochastic’ models as ‘chaotic deterministic’ systems and vice-versa.
# Recent reading: a miscellany of slightly obscure things
Sometimes I forget which things I’m currently reading (i.e. dipping in and out of). So, here are a few notes, mainly to myself and mainly about books and more obscure sources than the usual current research papers.
A couple of things on category theory: Category Theory for the Sciences by Spivak and Sets for Mathematics by Lawvere and Rosebrugh. (Also Mathematical Physics by Geroch, but that is more of a broad coverage of essential mathematics using category theory than a book introducing/studying category theory itself.) Really enjoying both. Would like to code up some of the content of Spivak to illustrate the main ideas.
A few things on mathematical biology/physiology etc (mainly for work/background I should know but have either forgotten or not learned). Mathematical Physiology by Keener and Sneyd (the latter being my old PhD supervisor). Free Energy Transduction and Biochemical Cycle Kinetics by Hill (as well as the older, longer version). An underrated book, I need to summarise the best bits at some point. Basic Principles of Membrane Transport by Schultz. Another great classic, helped me a lot during my PhD. Both a bit old but the main thing that seems to have changed is that we have actually identified a lot of the proteins behind the mechanisms originally predicted on based on coarse information and largely theoretical modelling!
Stochastic Modelling for Systems Biology by Wilkinson, Chemical Biophysics by Qian and Beard and Stochastic Process for Physics and Chemistry by van Kampen. Good complements to the above books, generally more focused on stochastic aspects, but still similar concepts. See also the papers Entropy Production in Mesoscopic Stochastic Thermodynamics: Nonequilibrium Kinetic Cycles Driven by Chemical Potentials, Temperatures, and Mechanical Forces by Qian et al. as well as Contact Geometry of Mesoscopic Thermodynamics and Dynamics by Grmela. Also, the book Statistical Thermodynamics of Nonequilibrium processes by Keizer. Should summarise the various key concepts and how to think about ‘mesoscopic’ processes in biology.
A few references on mechanics: some point particle stuff (want to use in some applications), also differential geometry, symmetry etc. Introduction to Physical Modelling by Wellstead (mainly interested in the ‘mobility analogy’). The Variational Principles of Mechanics by Lanczos (a classic!). Analytical Dynamics by Udwadia and Kalaba. Nonholonomic Mechanics and Control by Bloch et al. First Steps in Differential Geometry: Riemannian, Contact, Symplectic by McInerney. Discrete Differential Geometry: An Applied Introduction by Grinspun et al. Foundations of Mechanics by Abraham and Marsden. Introduction to Mechanics and Symmetry by Ratiu and Marsden. Mathematical Foundations of Elasticity by Marsden and Hughes. Also the paper: ‘On the Nature of Constraints for Continua Undergoing Dissipative Processes’ by Rajagopal and Srinivasa.
Dynamical systems (research and teaching – solution and analysis methods): Numerical Continuation Methods for Dynamical Systems by Krauskopf, Osinga and Galan-Vioque. Recipes for Continuation by Dankowicz and Schilder. Stability, Instability and Chaos by Glendinning. Nonlinear Systems by Drazin. Elements of Applied Bifurcation Theory by Kuznetsov. Applications of Lie Groups to Differential Equations by Olver. Scaling by Barenblatt. Renormalization Methods: A Guide For Beginners by McComb. Multiple Time Scale Dynamics by Kuehn.
Measure, Integral and Probability by Capinski and Kopp, Integral, Measure and Derivative by Shilov and Gurevich and Hilbert Space Methods in Probability and Statistical Inference by Small and McLeish (see also Functional Analysis by Muscat). Probability via Expectation By Whittle. Functional Analysis for Probability and Stochastic Processes: An Introduction by Bobrowski. Trying to decide on my preferred abstract framework for thinking about these topics. Each presents a slightly different perspective, each has its strengths and weaknesses. Will have to write a ‘compare and contrast’ to help me decide. I’ve pretty well decided on the functional analysis point of view. Update: see also Differential Geometry and Statistics by Amari and Differential Geometry and Statistics by Murray and Rice. So basically: functional analysis + differential geometry seems to be the way to go. Same as for mechanics.
Related to the above, a few books (and a paper or two) on inverse problems, parameter estimation, Bayesian inference and numerical approximation. Data Assimilation: A Mathematical Introduction by Law, Stuart and Zygalakis. Inverse Problems: A Bayesian Perspective by Stuart. Mapping Of Probabilities by Tarantola (as well as his classic book Inverse Problem Theory). Statistical and Computational Inverse Problems by Kaipio and Somersalo. PTLoS by Jaynes (Ch. 18; I keep reinventing something similar to this but don’t quite understand it. I think it might correspond to reinventing the functional analysis approach?). Data Analysis and Approximate Models by Davies. Moore, Kearfott and Cloud Introduction to Interval Analysis. Measuring Statistical Evidence Using Relative Belief by Evans. Theoretical Numerical Analysis: A Functional Analysis Framework by Atkinson and Han. Moore and Cloud Computational Functional Analysis. Discrete and Continuous Boundary Problems by Atkinson. Fletcher Computational Galerkin Methods. Functional Data Analysis by Ramsay Silverman.
Teaching PDEs: Partial Differential Equations for Scientists and Engineers by Farlow. Applied Mathematics by Logan. Partial Differential Equations by Evans. Advanced Engineering Mathematics by Greenberg. Green’s functions and boundary value problems by Stakgold. Principles and Techniques of Applied Mathematics by Friedman. Partial Differential Equations of Applied Mathematics by Zauderer. A First Course in Continuum Mechanics by Gonzalez and Stuart. Physical Foundations of Continuum Mechanics By Murdoch. Nonlinear Partial Differential Equations by Debnath. Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods by Tang.Methods of Mathematical Physics II by Courant and Hilbert. Ames Nonlinear PDEs in Engineering.Ern and Guermond Theory and Practice of Finite Elements. Still need to find a book I really like that balances mathematical, numerical and physical concepts at the right level. The short article Generalized Solutions by Tao is nice.
# Conditional probability as the basic notion of probability theory
Overview
A number of conceptual debates in both applications and philosophy of statistics and probability implicitly or explicitly depend on which concept of conditional probability is used. In particular there are two main conceptions floating about – ‘ratio’ based, which takes unconditional probability as basic and conditional probability as derived (and corresponds to Kolmogorov’s approach), and the reverse case which takes conditional probability as basic. I will call this the ‘conditionalist’ view. I point to a few arguments in favor of this latter view, how it relates to ‘model closure’ and a hierarchical/structural view of theories, and why it is popular among certain Bayesians as a resolution of the ‘catchall’ problem.
Disclaimer
Obviously I am only one of many to make this point. I still find it useful to record my agreement with the ‘conditionalists’. Very rough for now. Many more examples to come. Version 0.2
[Edit: I have a new appreciation for Kolmogorov’s approach after teaching it recently. If we consider a Kolmogorov ‘probability model’ to be a full probability space/probability triple, rather than just the measure, then we effectively get the same thing as advocated here. We have to imagine that we have – in principle – a sufficiently large (and generally ‘inaccessible’) background probability space to work ‘within’. Each concrete probability space, i.e. particular model, is then a restriction of this ‘global universe’. This makes the fact explicit that we are always using at least some restriction conditions, and forces us to give the form these restriction conditions take (e.g. orthogonality conditions?).
A rough idea occurs to me – we trade-off the fact that our ‘background universe’ grows exponentially as we relax closure assumptions (i.e. make less details irrelevant and hence more details relevant and hence more unique possibilities) with the expectation that as we include more details our models will become more deterministic. Hence our distributions ‘shrink’ relative to the new domains even if they are ‘bigger’ than their restriction to the old domains. So each ‘point’ in a higher dimension contains a whole universe of lower dimension. Think power sets. An interesting starting point for thinking more about the mathematical ‘universe(s)’ we work in (and how we get a hierarchy of sub-universes) is https://ncatlab.org/nlab/show/universe. See also https://en.wikipedia.org/wiki/Universe_mathematics]. See also the more recent blog post on the meaning of the terms linear/nonlinear and infinite/finite.
What conditional probability could not be
The above heading is the title of a paper by Alan Hájek (2003, Synthese) see here.
…the ratio analysis of conditional probability…has become so entrenched that it is often referred to as the definition of conditional probability. I argue that it is not even an adequate analysis of that concept…I marshal many examples from scientific and philosophical practice against the ratio analysis. I conclude more positively: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it.
The article is a good read and is in agreement with the positions of a number of Bayesians, as well as my own sort-of/occasional-Bayesian-but-there-are-probably-deeper-issues-to-worry-about view.
In and of itself I find the above article fairly convincing; the point has been reinforced to me however by reading a number of similar arguments for and against, as well as my own thinking about the nature of mathematical modelling.
I will collect some of these below, and then present my own main motivations for adopting a conditionalist view, which are somewhat independent of the Bayesian/Frequentist divide.
A collection of examples
[To fill in]
Bayesian classics
– de Finetti
– Jaynes
– ?
Internet arguments
– Gelman, Mayo, Wasserman and other characters
– Pearl, causality and conditioning
My motivation – hierarchies, closure, contradiction, expansion and invariant structure
Catchall vs conditional closure
My first post used conditional probability statements to formulate the basic idea of ‘model closure’ and argue against needing a ‘catchall’ (see the post for details). You’ll notice, however, that this argument only makes sense if you accept (as I did somewhat implicitly) that conditional probability is a basic notion and can be defined even in the absence of a joint distribution.
So, for the record, I take conditional probability as basic and definable even in the absence of unconditional distributions. Thus a ‘catchall’ unconditional distribution is not required for closure.
[Another disclaimer re: the following – these view of mine have been motivated by a number of authors, from Jaynes to Gelman, to my own lecturers in mathematical modeling and physics. So while I present it as my own perspective, it is inevitably strongly derivative of a number of others’ views. Perhaps the most original part is relating this view to the ideas of structural invariance, but this concept has itself been advocated by many.]
Conditional contradiction, hierarchies, regularisation, model expansion and invariant structure
Models always use temporary, approximate closures.
We generally need to begin work by ‘fixing’ (conditioning on) ‘external’ variables and working ‘within’ a system. As illustrated in the previous post, however, we often (inevitably?) reach contradictions or inconsistencies within our models as we approach the ‘boundary’ of our model closures. This leads to the idea (for one example) of ‘singular limits’.
Again as illustrated in the previous post, the way (or one way) to resolve this inconsistency is to ‘expand’ our model by embedding it in a larger model which relaxes a constraint implicit in the smaller model. This naturally leads to greater undetermination due to the additional degrees of freedom. This larger model is also often structurally isomorphic to the original model (at least in some respects), however, and thus gives us a ‘hierarchical’ and ‘structural’ – if not absolutely fixed – foundation to reason from. [Shades of Godel.]
So my perspective is thus ‘conditionalist’, ‘hierarchicalist’ and ‘structuralist’.
A (slightly) more concrete example
Consider a model of the form
p(a|c) = ∫ p(a|b)p(b|c) db
Where we have used the closure condition p(a|b,c) = p(a|b) to make p(a|b) ‘internal’ (invariant) relative to the ‘external’ variable (last conditioning variable) c.
We reason as follows – we want a model (directly) independent of our controlled variables c, with only boundary values b depending in a known manner on c.
IF we reach an internal contradiction – identified for example by p(a|b,c) != p(a|b) – we can (hopefully) expand our model to resolve this by moving previously controlled or ignored variables into the set of explanatory variables (ie expanding the state space) and then rewriting things so as to recover a model of the same schematic/structural ‘causal’ form via the redefinitions
p(a|c”) = ∫ p(a|b,c’)p(b,c’|c”) dbdc’
Equiv.
p(a|c”) = ∫ p(a|b’)p(b’|c”) db’
Where we have split c into (c’,c”) and defined b’ as (b,c’).
We now have an expanded theory having a different partition of variable classes. This leads to greater indeterminacy in the (internal/explanatory) variables, but gives a corresponding theory which possesses the same (invariant) structure as before. By prioritising the theory form I am taking a structuralist view of the essence of mathematical and scientific theories. Variable indeterminacy is the price we pay for removing inconsistency and maintaining structure at a higher level, but it is very often worth it (and exciting) – it corresponds in many cases to ‘new’ or ‘novel’ phenomena appearing. [Bifurcations].
Again, see the previous post for a simple example of expanding a model to remove a singularity and hence introducing indeterminacy.
Observations
Note that we make crucial use of a ‘conditionalist’ and hierarchical view of model structure. Yet another reason to take conditional probability (and conditional thinking) as basic, instead of unconditional probability.
Note also that what was previously a non-probabilistic variable can always become probabilistic as we ‘shift’ where we are in the hierarchy. The position of a variable in the structure is more important than the nature of the variable itself. Another reason to not dismiss Bayesian modelling for allowing us to treat variables as probabilistic (internal to the theory) if and when we choose to – or are forced to.
A possible point of agreement with the frequentist view, however, is that we always maintain some ‘conditioned on’ but non-probabilistic variables (controlled or ‘external’ variables) as temporary scaffolding.
# Is this high-school mathematics problem well-posed?
Overview and background
A brief discussion of well-posedness, singular problems and invariance, in the context of a high-school mathematics problem. Promoted by my return to NZ for a bit and catching up with family – my Dad is doing a PhD in mathematics education (more on that one day) and asked me to have a go at a problem he is using in a demonstration. I present my first naive solution and subsequent refinement. My Dad and I argue and then possibly agree. I was hospitalized shortly after but our discussion (probably) had nothing to do with this. Version 0.5.
The problem
No, not this one.
Instead consider the following ‘ladder problem’ as posed in an NCEA Level 3 mathematics exam (final year of high school in NZ):
A naive solution
Under ‘exam conditions’ – drinking my obligatory daily flat white and having a maths problem suddenly handed to me by my Dad – this was (roughly) my approach. In sketchy, narrative form.
1. Read problem definition. Derivatives. Constraint.
2. Chain rule, implicit differentiation, or something.
So
x’ given. y’ desired. c(x, y)=0 given.
(1) x^2 + y^2 = 25
Differentiate. Drop constants.
(2) xx’ + yy’ = 0
ie
(2′) y’ = -xx’/y
Need x. Use (1) again for x:
(1): x = sqrt(25-y^2)
Into (2′):
(3) y’ = -sqrt(25-y^2)x’/y
All RHS quantities known. Plug in.
Ans: y’ = 0.8 m/s
Assuming no outrageous errors, I think this is what they were after.
What he was getting at was this – looking at (3) clearly the problem is ill-defined, or singular, as y approaches zero. This can’t really be saved by any sensible, obvious or consistent dominant balance involving x or x’ going to zero at the same time.
This presents a nice toy model for thinking about regularisation (see also here, though the examples there are less directly relevant to the current problem) – I often find it a good principle to think about exactly how singularities arise and think of ways to remove them and hence ‘regularise’ a problem. This often points to a better conceptual understanding of a given problem.
As I have said again and again elsewhere on this blog, this sort of process concerns finding, testing and modifying different ‘model closures’.
A ‘resolution’
Let’s look at one resolution, that is not in itself incorrect but I don’t find especially illuminating. This was what my Dad pointed me to at some point. We argued a bit about whether this captured the essence of the ‘paradox’ and its resolution. My preferred – but, ultimately complementary – solution is given in the following section.
The solution my Dad preferred is presented in the link here and is described as follows:
Using results from related rate problems, some calculus books suggest that a ladder leaning against a wall and sliding under the influence of gravity will reach speeds that approach infinity. This Demonstration is built from the actual equations that govern the motion of the ladder as determined by the theory of rigid body mechanics. It shows that a sliding ladder never reaches very high speeds. The motion can be followed in two contrasting situations, with the top of the ladder either free to move away from the wall or constrained to be in contact with the wall. The forces are calculated for the falling ladder just before the top hits the floor.
The problem I have with this resolution is that, while likely correct (I haven’t checked all the details), it seems to obscure the key issues. It jumps straight to forces and gravity and Newton. But how exactly does the purely ‘geometric’ problem breakdown? Does it? When do we, if ever, need to move from kinematics to dynamics? What are the key/minimal conservation relations required for a well-posed problem?
(In other words, due to my undergrad education and for better or worse, I’ve been somewhat influenced by the spirit of Rational Mechanics [a la Truesdell, Noll], and would quite like a more axiomatic breakdown.)
An alternative perspective
Note: I don’t think the modification here contradicts the sort of solution proposed in the previous section. It is simply another perspective aimed at conceptual clarification.
Again, l’ll adopt a sketchy, narrative description.
Singular problems often result from an incorrect reduction of dimension and hence can be regularised by reintroducing additional scales, dimensions, quantities or cutoffs.
The ‘physical’ resolution noted that the ladder can detach from the wall. A tension between the wall constraint and the motion constraints appears to produce the singularity.
Consider a perfectly horizontal ladder lying on the ground. If it stays attached and the other end continues to move according to the given kinematic condition then the only possibility is that the ladder is being stretched. This violates the (presumably valid) assumption that the ladder is a rigid object (but see later for more on this!).
In fact, this shows up in the Wolfram example. The simulation allows you to (requires you to?) solve two different problems – the ladder able to detach and the kinematic constraint (given horizontal rate of motion for the bottom of the ladder) satisfied (I think) OR the ladder not able to detach and the horizontal (kinematic motion) constraint dropped in favour of a rate determined by angular and linear momentum conservation for a rigid rod falling under gravity.
Let’s consider the first case – i.e. a detachable ladder with the constraint of a fixed horizontal rate of motion for the bottom of the ladder satisfied. (This is presumably just as physically realisable in an experimental setup as a freely-falling ladder, e.g. by connecting it to a controlled pulling mechanism, and closer to the original problem specification.)
In this case we can remove the contradiction between the model and constraints (which generates the singularity) by simply introducing a moving coordinate system. This is implicitly fixed in the original solution. The key invariant is still the ladder length. See the figure below
Now, for convenience, let’s continue to fix the y coordinate origin at 0, but allow the x coordinate origin to be variable. Call this x0, but note this is not in general constant.
Redo the calculations. Keep the same numbering.
(1) (x-x0)^2 + y^2 = 25
Differentiate. Note x0 varies in time! Drop constants.
(2) (x-x0)(x’-x0′) + yy’ = 0
This expresses the key problem invariant – the ladder length. As expected, the price of an enlarged, non-singular problem is greater underdetermination. The original problem has x0, x0′ = 0, but if the ladder detaches then these are not true in general.
Note y=0 now implies x0 = 0 and/or x0′ = x’. This latter case, with x0 unknown, allows a rigid sliding of the ladder along the ground. In general, we can maintain sensible dominant balances so as to define the behaviour for small y and in the limit as y goes to zero.
In general, preservation of the key invariant (ladder length) plus special boundary constraints (touching the wall and/or floor) now allows the solution of particular cases. So we now have two well-posed (or better-posed) problems – touching the wall and touching the floor, respectively – with an underdetermined but non-singular problem in-between. We can’t, for example, say exactly when the ladder might be expected to detach from the wall, on the basis of the given info. The detachment point is unknown. For the sliding problem the initial x0 is also unknown in general. (Relevant exercise for the reader: Google ‘matched asymptotic expansion’).
So no, the problem is not fully well-posed, though it is soluble by making special assumptions. It is also (to me) clearer now where the additional information should come from – for example (a bound on) the rotation rate required to keep the bar in contact with the wall, given the kinematic condition (staying as close as possible to the problem as posed). This is of course determined by angular and linear momentum conservation, as in the Wolfram simulation.
It also raises other, equally realistic, possibilities though – violation of the rigid body assumption leading to deformation (stretching/strain, where x0=0 say but x’>x0′) or fracture (similar to the detachment case).
So, at some point one may need to introduce additional information – eg conservation of linear/angular momentum but also maybe material properties – to solve the expanded problem, but this shouldn’t obscure the key invariants and assumptions used, why they are required and at what point they are introduced.
This leads to a more general lesson.
Morally speaking
The key lesson to me is this:
The price of removing a singularity by embedding a problem in a higher dimensional problem is typically greater undetermination requiring additional information to solve in full generality. Regardless, it is helpful to view the original problem as a particular limit of an expanded problem.
# Asymptotics, renormalization and scientific theories
Overview
In lieu of a post with original material and/or updates on the other posts, here is a nice quote relating to some of the key themes that I’ve started exploring on this blog. Specifically a quote about asymptotics and renormalization (and, by implication, model closure, approximation and invariance), and how these can illuminate some aspects of the nature of scientific theories.
On renormalization
From ‘Intermediate Asymptotics and Renormalization Group Theory’ by Goldenfeld, Martin, Oono (1989).
[a] macroscopic phenomenological description…consists of two parts: the universal structure, i.e., the structure of the equation itself, and phenomenological parameters sensitive to the specific microscopic physics of the system. Any good phenomenological description of a system always has this structure: a universal part and a few detail-sensitive parameters…In this sense, it is [also] possible that there is no good macroscopic phenomenology [for a given system of interest].
Thus if we consider a set of transformations that alters only the microscopic parameters of a model…the macroscopic universal features should remain unchanged. Therefore, if we can absorb the changes caused by modification of microscopic parameters into a few phenomenological parameters, we can obtain universal relations between phenomenological parameters.
If this is possible by introducing a finite number of phenomenological parameters, we say that the model (or the system) is renormalizable. This is the standard method of formulating the problem of extracting macroscopic phenomenology with RG. RG seeks the microscopic detail sensitive parts in the theory and tries to absorb them into macroscopic phenomenological parameters.
…Suppose that the macroscopic phenomenology of a system can be described successfully with a renormalizable microscopic model. The phenomenological parameters must be provided from either experiment or from a description valid at a smaller length scale. Is this a fundamental limitation of the renormalizable theory? If one is a reductionist, the answer is probably yes. However, another point of view is that microscopic models are not more fundamental than macroscopic phenomenology.
In fact, it is inevitable that in constructing models of physical systems, phenomena beyond some energy scale (or on length scales below a threshold) are neglected. In this sense, all present-day theoretical physics is macroscopic phenomenology.
Renormalization group theory has taught us how to extract definite macroscopic conclusions from this vague description. Of course, this is not always possible…However, we clearly recognize general macroscopic features of the world in our daily lives as macroscopic creatures! Thus, we may believe that for many important aspects of the macroscopic world there must be renormalizability. We may say that renormalizability makes physics possible.
# Closure: objective and subjective, truth and approximation
Overview
A sketch of a few thoughts on ‘objective’ vs ‘subjective’ and ‘truth’ vs ‘approximation’ in the context of what I’ve been calling ‘model closure‘. Taking a roughly/informally category theory perspective. Includes more discussion of how the data space is idealised/closed as well as the parameter/theory space, as well as issues of invariance, multiple scales, intermediate asymptotics and renormalization.
Disclaimer
Still very rough. I have included some handwritten notes for now – will convert to typeset later. [Version: 0.3]
Orientation: objective and subjective, truth and approximation
First, I want to set the basic conceptual picture. I’ve mentioned this perspective a few times but I think it’s good to re-emphasise using some visualisation. Consider the following conceptual pictures, all making similar points:
even in ‘model world’ (c.f. the ‘real’ world) we need to distinguish between the ‘objective, external’ world and the ‘subjective, internal’ world. In particular, this distinction is drawn relative to the boundary defining the model closure, and applies to both ‘data’ and ‘parameters’.
As I have discussed in other posts, closure is what delinates the boundary between estimating parameters within a model structure and testing the model adequacy with respect to external reality. We have essentially already considered the parameter closure, i.e. discarding ‘irrelevant’ parameters (theoretical constructs). The same idea applies, however, to the data space closure. Some do not distinguish ‘within’ and ‘without’ in the way done here for various reasons – from ‘all models are wrong and therefore subjective’ to leaving ‘lumps of probability‘ to keep the ‘options open’ somewhat. There is some truth in these general ideas; after all, all closures are provisional. I still prefer to explicitly introduce and distinguish ‘inside’ and ‘outside’ a model and ‘objective’ and ‘subjective’ constructs, however – even when both are (and really, can only be) imagined.
‘Intermediate’ structure and multiple scales
On the other hand, a subtle issue emerges in a similar way to in the ‘tacking paradox’ post – the distinction between predictive irrelevance and more ‘complete’ irrelevance, i.e. the presence or absence and nature of further internal degrees of freedom. We need to find a way to follow the advice to
Rule out the accidental features
And you will see: the world is marvellous
– Alexander Block (translated by Sir James Lighthill)
This ‘intermediate’ perspective is described in Barenblatt’s ‘Scaling‘ which quotes the above and also give the following painting as a conceptual example:
Figure 3
“Lincoln in Dalivision, Salvador Dali Lincoln in Dalivision Print, Lincoln in Dalivision”. One (relatively) small scale depicts ‘Gala’ gazing at the sea, which in turn ‘merges into’, at an ‘intermediate’ scale, a portrait of Abraham Lincoln. The ‘frame’ of the full painting ends our ‘boundary of interest’. If we stand much much further back, we no longer recognise any interesting features – our ‘largest’ observation scale determines the largest scale features we wish to perceive.
Related to the (applied mathematics) concepts of intermediate asymptotics and renormalization scaling is another set of concepts that I will (loosely) draw on below – the (thermodynamic) concepts of ‘external variables’, ‘internal variables’ and ‘internal coordinates’. Roughly speaking, the external variables determine the overall ‘shape’ of the closure as determined by ‘background’ conditions and connect our invariant theories (see next) to external measurements, the internal variables are intermediate variables that form (approximately, at least) an invariant and predictively complete set for a (scale-free) phenomenon of interest, while the internal coordinates index a finer set of internal degrees of freedom. In general the internal variables are determined from integrals over internal degrees of freedom/internal coordinates. So we have (at least) three scales – ‘external’, ‘intermediate’ and ‘small’.
This enables us [or will eventually] to compare theories that are a priori distinct, e.g. have different parameter domains and definitions, but seem similar when looked at in the right way. That is, it may be possible to find a common, scale-free predictive theory with a (relatively) invariant set of internal variables that serve as a common target mapping for the variables of distinct theories to enable consistent comparison. To connect back to reality requires ‘boundary closures’ on ‘either side’ of the intermediate, invariant theory – i.e. data space closure via a notion of measurement and parameter space closure via a notion of stability under manipulation/variation in other degrees of freedom (and relates to the formulation of priors).
A basic theme emerges:
‘causality’ and ‘mechanistic’ understanding are about invariant structures under the scales and controls of interest; probability enters into consideration in a somewhat secondary manner: to capture uncertainty within and between structural relationships, and in determining the resolution of control and measurement accuracy.
For now, here are some (very quickly sketched) handwritten notes.
0.0 A first attempt at a ‘closure functor’
0.1 A first/another attempt at relating model closure to ideas of invariance, intermediate asymptotics etc
Further notes
Besides properly tidying these ideas up, I also want to connect them to Laurie Davies’ ‘Approximate models‘ approach.
# Causal recipes
From Cakes, Custards and Category Theory by Eugenia Cheng:
The idea of maths is to look for similarities between things so that you only need one ‘recipe’ for many different situations. The key is that when you ignore some details, the situations become easier to understand, and you can fill in the variables later…
…once you’ve made the abstract ‘recipe’ you will find that you won’t be able to apply it to everything. But you are at least in a position to try, and sometimes surprising things turn out to work in the same recipe.
This connects with my earlier post on what the domain of the ‘for all’ is in the closure conditions – we are taking a rather structuralist view of causal theories (or model closure schema). That is, we are saying what the structure, expressed in terms of relationships between a collection of objects, of an idealised causal theory looks like without worrying too much (for now) about the nature of objects to be ‘filled in’.
Obviously more needs to be said on the crucial ideas of idealisation and approximation (though I’ve touched on these somewhat) and hence the process of slotting objects in. This is what I’d like to focus on next, hopefully, before further linking to some of the other causal literature.
Postscript
This idea of focusing on the essence of the recipe rather than the details of the objects is of course quite generally applicable (get it!) and, I feel, has a lot of pedagogical value. For example I recently read a nice article on improving the teaching of simple significance testing here. The author takes a quite similar ‘structuralist’ (in my view) and ‘abstract recipe’ perspective. Which is somewhat ironic since, without meaning to nitpick a nice article, claims
When statistics is taught by mathematicians, I can see the temptation. In mathematical terms, the differences between tests are the interesting part. This is where mathematicians show their chops, and it’s where they do the difficult and important job of inventing new recipes to cook reliable results from new ingredients in new situations. Users of statistics, though, would be happy to stipulate that mathematicians have been clever, and that we’re all grateful to them, so we can get onto the job of doing the statistics we need to do
Ironically, as argued above, a mathematician (or at least one who likes the ‘abstract nonsense’ of category theory) would probably prefer the view expressed earlier in the same article:
Every significance test works exactly the same way. We should teach this first, teach it often, and teach it loudly; but we don’t. Instead, we make a huge mistake: we whiz by it and begin teaching test after test, bombarding students with derivations of test statistics and distributions and paying more attention to differences among tests than to their crucial, underlying identity. No wonder students resent statistics.
# The ‘tacking paradox’: model closure and irrelevant hypotheses
Overview
I. The tacking paradox in philosophy of science
Interlude – Bayesian or not?
II. A resolution
Interlude – severe tests and tracking truth?
III. Implications for mathematical/computational models in practice (sort of)
Disclaimer
This is the one of (what should be) a few posts which aim to connect some basic puzzles in the philosophy and methodology of science to the practice of mathematical and computational modelling. They are not intended to be particularly deep philosophically or to be (directly) practical scientifically. Nor are they fully complete expositions. Still, I find thinking about these puzzles in this context to be an interesting exercise which might provide a conceptual guide for better understanding (and perhaps improving?) the practice of mathematical and computational modelling. These are written by a mathematical modeller grappling with philosophical questions, rather than by a philosopher, so bear that in mind! Comments, criticisms and feedback of course welcome! [Current version: 3.0.]
I. The tacking paradox in philosophy of science (or, the problem of irrelevant hypotheses)
The so-called tacking paradox (or at least one instance) can be described in minimal terms as follows. More detail is given on Philosopher Deborah Mayo’s blog here, (along with some responses in the comments section that don’t seem too far from the resolution given here, though they are a little unclear to me in places). As I noted in my other posts, I will prefer to think of ‘hypotheses’ h as parameters within mathematical model structures predicting data y (search this blog for more). The basic perspective from which I will try to resolve this problem is that of schematic ‘model closure’ assumptions.
Firstly, we need to define what it means to ‘Bayesian confirm’ (really, ‘Likelihood confirm’) a hypothesis h given data y. Let’s take the following statement to capture this idea, in terms of ‘predictive confirmation’:
(1) p(y|h,b) > p(y|b)
That is, if the hypothesis h makes the data more likely (under a given model p) then this is taken to mean ‘y confirms h’. Note that we have included a controlled/given background context b.
Also note that ‘confirmation’ as defined here is thus a change in probability rather than a probability itself. There are a number of different positions on this topic (see e.g. Mayo’s discussion) but I prefer to think of ‘confirmation’/’evidence’ given new data as a change in belief/probability induced by that data (to do – further references) to a new state of belief/probability. This is the difference between ‘state variables’ and ‘fluxes’ in physics/dynamical systems – or ‘stocks’ and ‘flows’ to use an equivalent terminology (which I really dislike!).
So we ‘confirm’ a hypothesis when it makes a ‘successful prediction’ of newly observed data. This seems a fairly non-controversial assumption in the sense that a minimal measure of the ‘quality’ of a theory ought to mean that one can predict observations better (to at least some degree) than not having that theory. Note that this is relative to and requires the existence of a prior predictive distribution p(y|b), and this should exist in a standard Bayesian account (more on this one day). I will think of this as ‘predictive relevance’.
Now, suppose that the scheme (1) is true for a hypothesis h1 and data y0, i.e. p(y0|h1,b) > p(y0|b). Say y0 represents some planetary observations and h1 some aspect of (parameter in) Newton’s theory.
Next, ‘irrelevance’ of a hypothesis h” is usually defined in this context as:
(2) p(y|h’,h”,b) = p(y|h’,b)
which is clearly relative to y, h’, p and b. Note that in my terminology we have ‘predictive irrelevance’ of h” here.
This leads to the following argument. Let h2 be a typical ‘irrelevant theory’ e.g. a theory about the colour of my hat, that is (for example) a parameter representing possible ‘colour values’ my hat could take. Then we have
p(y0|h1,b) > p(y0|b) {given that y0 Bayesian/Likelihood confirms h1}
p(y0|h1,h2,b) = p(y0|h1,b) {assuming irrelevance of h2}
So
p(y0|h1,h2,b) > p(y0|b)
Therefore y0 Bayesian/Likelihood confirms (h1&h2) (with respect to model p and background b).
So what’s the ‘paradox’? The (allegedly) troubling thing is that h2 is supposed to be ‘irrelevant’ and yet it seems to be confirmed along with h1. So planetary observations seem to be able to confirm something like ‘Newton’s theory is true and my hat is red’.
More concretely, one might try to argue as follows: since (h1&h2) is confirmed and since the joint proposition/logical conjunction (h1&h2) logically entails h2, then h2 is confirmed {confirmation/epistemic closure principle}.
So, according to the above argument, ‘my hat is red’ could be confirmed by planetary observations. This argument scheme captures a notion of ‘knowledge is closed under deductive entailment’ or ‘epistemic closure’ in the epistemological literature. Note, however, that this does not follow from any of the main model closure axioms that we have put forward thus far – the ‘model closure’ we refer to is not ‘epistemic closure’. In fact, the approach we follow has more in common with those taken to deny epistemic closure, such as Nozick and/or Dretske (see here).
Interlude – Bayesian or not?
Before I give my preferred resolution of the paradox, there are a couple of points to distinguish here – first, is the Bayesian/Likelihoodist language appropriate to express a resolution of this problem? Second, are the concepts involved in the resolution inherently part of or extrinsic to the Bayesian/Likelihoodist approach? This second point is, I take it, what led Clark Glymour to write ‘Why I am not a Bayesian’ (1981) (see also Pearl’s ‘What I am only a half-Bayesian’) – my interpretation of his point being that the resolution to ‘paradoxes’ such as these may or may not be expressible within the Bayesian language but the underlying concepts driving what we translate into Bayesian language are additional to and not a part of the basic Bayesian account.
I basically agree with Glymour on this general point, but use the Bayesian language to express the concepts required to resolve the ‘paradox’. My view, as expressed elsewhere on this blog, is that these are additional ‘closure’ assumptions. As pointed out above, these are not ‘epistemic closure’ assumptions but rather schematic model structure closure assumptions (see here and here). The need for assumptions such as these, whether considered ‘pure’ Bayesian or not, are, however, explicitly and/or implicitly acknowledged by many Bayesians (e.g. Jaynes, Gelman etc).
II. A resolution
Firstly, consider whether we have really captured the notion of ‘irrelevance’. We seem to have predictive irrelevance but what about ‘boundary/background irrelevance’ – i.e if the variable is ‘truly’ irrelevant then we could imagine moving it into the ‘boundary/controlled’ or ‘background’ variables and varying it without affecting the variables that matter. Thus I argue that we have more information available (knowledge of possible relationships) in the problem specification than we have used.
In particular, based on the closure conditions I gave in the first post on this blog, I would argue that applying the model closure assumptions (1-3) in that post to p(y0|h1,h2,b) requires us to specify both p(y0|h1,h2,b) = p(y0|h1,b) {predictive irrelevance}, as well as an expression for p(h1|h2,b). That is
We are obligated, according to our model closure principles, to say how varying h2 affects h1 in order to have a well-posed problem. It either has a relevant affect – varying h2 by experimental control affects h1 – or it is a ‘fully irrelevant’ background variable.
In light of the above discussion, we will take ‘h2 is an irrelevant hypothesis’ to further mean
(3) p(h1|h2,b) = p(h1|b) for all h1, h2, b
i.e. h2 falls into the ‘truly irrelevant background variables’, rather than the ‘controlled and controlling boundary values’ b. This means that varying h2 cannot control h1: the parameters of Newton’s theory are not manipulable by changing the colour of my hat. It is a truly ‘passive cog’ capable of no explanatory work.
Note also that we are actually speaking at the schematic/structural level here – i.e. for any value h1, h2 take – and hence counterfactually about particular instances conceived as members of a set of possible values.
So in this context I can vary (or imagine varying) my hat colour and how this affects other variables. Though perhaps unfamiliar to many, this is actually a common way of framing theories in the physical sciences, even in classical mechanics, e.g. D’Alembert’s principle and related ideas, which require ‘virtual’ (counterfactual) displacements.
This leads to [to do – proper latex in wordpress]
p(y0|h2,b) = ∫ p(y0|h1,h2,b)p(h1|h2,b) dh1
= ∫ p(y0|h1,b)p(h1|h2,b) dh1 {by ‘predictive irrelevance’ of h2}
= ∫ p(y0|h1,b)p(h1|b) dh1 {by ‘h1 is not manipulable by h2’}
= p(y0|b)
Therefore
p(y0|h2,b) = p(y0|b)
and so h2 is not confirmed by y0 at all! Note again that, in defining our original closure conditions, we required some assumption to be made on p(h2|h1,b) – the one chosen in the particular context here seems to best represent the concept of ‘irrelevance’ intended. Thus when we include both predictive irrelevance and boundary irrelevance/non-manipulability closure assumptions then there is no paradox.
Until some p(h1|h2) is given we have an ill-posed problem – or, at best we can find a class of solutions and require boundary conditions to further pick out solutions capturing our particular circumstances.
For example, one could also imagine an h2 which is simply a ‘duplicate’ of h1 – this satisfies predictive irrelevance in that it adds no predictive ability to know the same thing twice, but may be considered the opposite (singular/delta) limit of p(h1|h2).
Interlude – severe tests and tracking truth?
Since I motivated this problem with reference to Mayo’s blog, how might the ‘severe testing’ concept relate? For now, a quick thought: if by ‘test’ we mean ‘behaviour under specified experimental manipulations‘ then we see some similarity. In particular, one might imagaine that the ‘testing’ aspect refers to defining boundary conditions and related behaviour under possible (‘counterfactual’ or ‘virtual’) boundary manipulations, which is a crucial part of the ‘model closure’ account here.
Similarly, Nozick’s ‘truth tracking’ account in epistemology is relativised to methods and, if we equate ‘methods’ to ‘model structures’ – which seems appropriate since a ‘model structure’ is really a functional recipe – then it also has much in common with the ‘model closure’ (again, as opposed to epistemic closure) account given here. Furthermore, I think Kripke’s supposed ‘red barn’ counterexample (see here) to Nozick’s theory seems to fail for similar reasons of being an ill-posed problem: the solution depends on how the ‘boundary of the problem’ is closed.
I will (hopefully) have more to say on these topics at some point.
III. Implications for the everyday ‘mathematical/computational modeller’
What does this mean for people building mathematical and/or computational models of complex phenomena such as those of biology? As all of us ‘mathematical modellers’ who have tried to do something even resembling ‘real science’ know, we almost always face the ‘simple model/complex model’ and ‘modelling for understanding/modelling for prediction’ trade-offs.
Consider this common experience: you present a slightly too complicated model (all of them, and none of them, basically) and show it ‘predicting’ some experimental result ‘correctly’. The first question is, of course, so what? Why should I trust your model? Followed by ‘I could fit an elephant (with a wiggly trunk) with that model’ and/or ‘most of those parameters appear completely irrelevant – what are the most important parameters. Have you done a sensitivity analysis?’.
You see the parallel with the tacking paradox – with all those (presumably) extraneous, irrelevant parameters (hypotheses) ‘tacked onto’ your model, how can you possibly say that it is ‘confirmed’ by the fact that it predicts some experiment? Which of your parameters really capture the ‘true mechanism’ and which are ‘irrelevant’?
The resolution is of course that
a) the model as a whole can be ‘confirmed’ (that is, made more probable to some degree by fitting the data/avoiding being falsified etc)
but
b) we don’t know which parts are confirmed and by how much, unless we know how the parameters (hypotheses) within the model relate to each other.
In order to further reduce the model to ‘minimal’ or ‘mechanistic’ form, we need to define behaviour under (possible) manipulation (boundary conditions). Predictively irrelevant variables either have ‘boundary condition’ effects or no effects, but we need to say which is the case.
One problem then in practice is that, without going further and investigating relations between parameters (via direct manipulation and/or varying boundary/contextual assumptions, say), we are restricted in our ability to generalise to new situations – without being able to identify ‘modular’ or ‘invariant’ model components (more on this one day, hopefully) and the context within which this invariance applies, we don’t know which can be used to build models of similar but differing situations.
From a ‘machine learning’ point of view this could be considered a form of bias-variance trade-off – without stable (invariant) sub-components that apply to other contexts we are at risk of ‘overfitting’. So ‘bias’ is really (a form of) ‘knowledge external to this particular dataset‘.
To put it another way, Newton’s law of universal gravitation is a whole lot more useful as a force model than Maclaren’s law of forces between these two particular objects in this particular context, precisely because it is an invariant feature of nature valid for a wide range (e.g. inertial) of frames of reference. Thus mere prediction on one dataset is not enough to be scientifically interesting. Which we all know of course but – let’s be honest! – can often forget in the day-to-day grind.
To me, these ‘extra-statistical’ closure assumptions are often guided by balancing the competing goals of prediction and understanding. I have some thoughts on how this balance can be clarified, and how some related areas of research bear on this, but this post is getting long and the margins of this blog are too small to..
# Model schema and the ‘structuralist’ interpretation of ‘for all’: the uniformity of what?
Disclaimer
A short note mentioning an update of a previous post, as well as an additional comment.
Overview
I have re-written a number of parts of my first post ‘For all’ is not ‘catch all’: closure, model schema and how a Bayesian can be a Falsificationist. I’ve added some brief references to Jaynes, for one, but also have tried to clarify the nature of the ‘for all’ statement and its domain of application. This came up in the comment section. I’ve copied this section below (in blue), as well as added an additional comment after.
What is the domain of the ‘for all’?
A further clarification is needed [see the comment section for the origins of this]: the closure conditions are schematic/structural and only implicitly determine the domain of validity B for a given theory. That is, in the general scheme, b and B are placeholders; for a particular proposed theory we need to find particular b and B such that the closure conditions are satisfied. This has an affinity with the ideas of mathematical structuralism (without necessarily committing to endorsing the entire position, at least for now). For example, Awodey (2004, An Answer to Hellman’s Question), describes:
the idea of specifying, for a given…theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or determination of the ‘objects’ involved…The statement of the inferential machinery involved thus becomes a…part of the mathematics…the methods of reasoning involved in different parts of mathematics are not ‘global’ and uniform across fields…but are themselves ‘local’ or relative…[we make] schematic statement[s] about a structure…which can have various instances
This lack of specificity or determination is not an accidental feature of mathematics, to be described as universal quantification over all particular instances in a specific foundational system as the foundationalist would have it…rather it is characteristic of mathematical statements that the particular nature of the entities involved plays no role, but rather their relations, operations, etc. – the ‘structures’ that they bear – are related, connected, and described in the statements and proofs of the theorems.
This can be seen as following in the (in this case, algebraic) ‘structuralist’ tradition of Hilbert (1899, in a letter to Frege):
it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes…
…the application of a theory to the world of appearances always requires a certain measure of good will and tactfulness: e.g., that we substitute the smallest possible bodies for points and the longest possible ones, e.g., light-rays, for lines. At the same time, the further a theory has been developed and the more finely articulated its structure, the more obvious the kind of application it has to the world of appearances
So, here we are defining a model schema capturing the idea of the ‘closure of a model’ or, alternatively, a ‘closed model structure’, and meant to capture some notion of induction ‘within’ a model structure and falsification ‘outside’ a model structure. Hilbert’s last paragraph captures this second point.
Suppose we have a background of interest for which we want to create a theory. It may be/almost certainly is the case that there are (many) possible contexts/backgrounds for which we cannot find ‘good’ theories satisfying the closure conditions – e.g. the theories are either much too general or much too specific. This is why psychology is in some ways ‘harder’ than physics – it is very difficult to partition the large number of possibly relevant variables for predicting ‘target’ variables y into a small number of invariant theoretical contructs x, a small set of ‘controllable’ variables b’ and a large set of ‘irrelevant’ variables b”. If we wish to retain an ability to ‘realistically represent’ the phenomenon of interest captured by y, then most things will be ‘explanatory variables’ needing to be placed in x and/or controlled in b’. That is, we will have a very descriptive theory, as opposed to a very ‘causal’ theory. Note that the division (3) into ‘controlled’ and ‘irrelevant’ variables b’ and b”, respectively, tries to help with this, to some extent, but means that controlled lab experiments can be both quite reproducible within a lab but can fail to generalise outside it.
The closure conditions mean that we still know what a theory should look like, if it exists, though and this helps with the search.
A further comment – the uniformity of what?
It’s also worth noting (as I did in the comment section on the original post) that when confronted with complex phenomena we have a choice of
– developing a (probably ad hoc) theory for unusual events by moving some b” variables into b’ or x and having a more complicated theory structure (in terms of number of theoretically-relevant variables)
– having no theory for unusual events (for the moment) and focusing on those which satisfy the closure. These are the ‘simple but general’ theories like the ideal gas.
Thus different theories have different divisions of x, b’ and b”.
We are hence attempting to avoid the problem of requiring an assumption about the uniformity of nature by only making assumptions about the uniformity of our models and accepting that they may not ‘cover’ the entire real world (see also here). This explains why we are ‘inductive’ ‘inside’ the model – uniformity applies here – but ‘falsificationist’ ‘outside’ our model, i.e. in assessing whether its assumed uniformity holds when held up against the real world.
Whether the closure conditions are satisfied depends on your willingness to accept the closure conditions in a given situation which depends on how you define your observable y (for example). I have more to say on this measurement issue at some point in the future, but suffice to say a ‘coarse’ y makes it easier to accept that the conditions are satisfied, as Hilbert implied.
# Model closure and formalism in economics, and a topological metaphor for model closure
Disclaimer
This is the one of (what should be) a few posts which aim to connect some basic puzzles in the philosophy and methodology of science to the practice of mathematical and computational modelling. They are not intended to be particularly deep philosophically or to be (directly) practical scientifically. Nor are they fully complete expositions. Still, I find thinking about these puzzles in this context to be an interesting exercise which might provide a conceptual guide for better understanding (and perhaps improving?) the practice of mathematical and computational modelling. These are written by a mathematical modeller grappling with philosophical questions, rather than by a philosopher, so bear that in mind! Comments, criticisms and feedback of course welcome! [Current version: 2.0]
Overview
I. Model closure and formalism in economics
II. A topological metaphor for model closure – manifolds, charts and atlases
I. Model closure and formalism in economics
Lars P. Syll gives a nice quote here from Shelia Dow, an expert in economic methodology (who I haven’t encountered before), on what is required for obtaining model closure in the context of economics:
…structures with fixed (or at least predictably random) interrelations between separable parts (e.g., economic agents) and predictable (or at least predictably random) outside influences…
…Any formal model is a closed system. Variables are specified and identified as endogenous or exogenous, and relations are specified between them. This is a mechanism for separating off some aspect of an open-system reality for analysis. But, for consistency with the subject matter, any analytical closure needs to be justified on the grounds that, for the purposes of the analysis, it is not unreasonable to treat the variables as having a stable identity, for them to have stable interrelations and not to be subject to unanticipated influences from outside … But in applying such an analysis it is important then to consider what has been assumed away…
I take the quote to make quite similar points to my previous post about the need not just for all formal models to have a closure but also where these closure assumptions come in. This post tried to connect the issue of model closure to debates about ‘catchall’ hypotheses in Bayesian inference.
The point I argued was that the appropriate ‘model closure’ for Bayesian inference (and of course all formal models have a closure, as Shelia argues) occurs (or should occur) at the level of model structure and ‘boundary conditions’ (priors) and does not require a probability distribution over the ‘background’ or ‘external’ variables.
Rather, closure occurs via a collection of ‘structural’ conditional probability statements with some variables only appearing on the right-hand side of the conditioning (and hence not possessing/requiring a probability distribution). These provide an assumed separation into ‘inside the system’, a ‘closed boundary’ based on experimentally-controlled variables and the external/irrelevant ‘outside the system’ variables. Once this closure is established, Bayesian inference can be carried out within this boundary, where probability distributions can be normalised, but not outside. This closure is always temporary and falsifiable, however, and requires qualitatively different inference methods for assessing its validity, such as ‘pure significance’ tests, unless again embedded in a further higher-level model. This is why there can be ‘falsificationist Bayesians’ (e.g. Andrew Gelman).
Note the subtle point that we are constructing a sort of ‘meta-model’ of the process of inference itself, into which we embed particular models of interest.
As I stated in the comments on Lars’ blog, in some ways I’m more optimistic about closure – I think the search for model closure is the search for interesting theories and is part of the ‘stupendous beauty of closure’ referenced in my post. I agree, however, that we may not always be able to find it (especially for ‘messy’ subjects like biology, psychology, economics etc). This came up in my exchanges with the philosopher Greg Gandenberger and is something I need to elaborate on at some point.
This latter issue, as I see it, has its clarification through the roles of idealisation and approximation and the separation of the ‘formal/mathematical’ and ‘actual/possible’ worlds. I’ll return to the topic of formalising (to some extent) the processes of idealisation, measurement and ‘seeing’ vs ‘doing’ at some point in the future. For now just keep in mind that every closure establishes a formal model by separating it off from the real world at some point and hence, as the cliche goes, ‘all models are wrong’. They are ‘wrong’ because of the closure; however, the ‘but some are useful‘ part comes in when we find useful closures.
Useful/beautiful closures may or may not exist – that’s the fun and challenge of doing science!
II. A topological metaphor for model closure – manifolds, charts and atlases
A metaphor for the role of limited (closed) theories can be found in topology and geometry: we might imagine a collection of possibilities of the ‘real world’ as a sort of Platonic abstract manifold of some sort to which we have finite access (to do – Plato’s cave…, possible worlds). Our (closed) models form a patchwork of charts, each of which only cover a small part of the possible world manifold. As the Wikipedia page states
It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map’s boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure.
As mathematical modellers using idealisations (closures) we inevitably use limited models (charts) only covering some part of the target world. Furthermore we may/will never be able to fully cover the entire ‘possible world manifold’. Our best hope is a collection of charts – a so-called atlas – covering as much as we can and with certain mutual consistency properties. In the Bayesian account, each chart would be analogous to a probability distribution over possible parameter values/’true’ states of the world. Note that in this interpretation of the level of model closure, the Bayesian account appears to require some sort of ‘possible world‘ interpretation.
Returning to the topological metaphor, it is again helpful for guiding us on understanding consistency properties – to compare two models of our state of knowledge (charts) we require the analogy of a transition map (see the atlas page) between the charts (models). Without these we cannot compare models of our state of knowledge/information (charts).
This corresponds to the intuitive Bayesian and Likelihoodist constraint that many advocate: one should not compare parameter values between different models unless embedded into a larger model or a mapping between models is provided.
An interesting connection to explore further in the epistemology literature is whether Susan Haack’s ‘crossword puzzle’ metaphor relates to the topological metaphor given here. Her crossword puzzle metaphor for epistemology involves the search for a collection of words (think models/knowledge) satisfying both external empirical ‘clues’ (data) and mutual consistency of ‘intersecting words’ (coherence/invariance properties). | 2016-08-28 08:41:23 | {"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": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7210352420806885, "perplexity": 1336.61973589406}, "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-2016-36/segments/1471982935910.69/warc/CC-MAIN-20160823200855-00152-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://xxicla.dm.uba.ar/viewAbstract.php?code=1444 | Conference abstracts
Session S07 - Finite Fields
No date set
## Similarity between the algebraic structure associated with projective space and combinatorial design via Hasse diagram
### CPAq/UFMS - FEEC/UNICAMP, Brasil - leandro.lima@ufms.br
Combinatorial design is an important combinatorial structure having a high degree of regularity and which is related to the existence and construction of systems of sets with finite cardinality, [1]. As examples we mention the existing relationship between error-correcting codes in the Hamming space and combinatorial design, where the codewords of weight 3 of the Hamming code form a triple Steiner system STS(7), a projective plane of order 2, known as the Fano plane, [2], as well as $q$-analogs of a code whose codewords have constant Hamming weight in the Hamming space, a code belonging to a Grassmannian in the projective space, [3,4]. Projective space of order $m$ over a finite field $\mathbb{F}_p$, denoted by $\mathcal{P}(\mathbb{F}_{p}^{m})$, (note that $\mathbb{F}_{p^m}$ is isomorphic to $\mathbb{F}_{p}^{m}$), is the set of all the subspaces in the vector space $\mathbb{F}_{p}^{m}$. The projective space endowed with the subspace distance $d(X,Y)=dim(X)+dim(Y)-2dim(X \cap Y)$ is a metric space. Hence, the subspace code $\mathcal{C}$ with parameters $(n,M,d)$ in the projective space is a subset of $\mathcal{P}(\mathbb{F}_{p}^{m})$ with cardinality $M$ with a subspace distance at least $d$ between any two codewords, [5]. In this paper we show the existing similarity between the Hasse diagram of an Abelian group consisting of the product of multiplicative finite Abelian groups $\mathbb{Z}_p^m$ and the Hasse diagram of the projective space $\mathcal{P}(\mathbb{F}_{p}^{m})$, with the aim to provide the elements that may be useful in the identification and in the construction of good subspaces codes, [6].
[1] - D.R. Stinson, \textit{Combinatorial Designs: Constructions and Analysis}, Springer Verlag, New York, USA, 2004.
[2]-T.Etzion and N. Silberstein, "Error-Correcting codes in projective spaces via rank metric codes and Ferrers diagrams," \textit{ IEEE Trans. Inform. Theory}, vol. 55, n.º7, pp.2909-2919, Jul. 2009.
[3]-M.Braun, T. Etzion, P.R.J. Ostergard, A. Vardy, and A. Warssermann, "Existence of q-analogs of Steiner sstems," arxiv.org/abs/1304.1462, Apr. 2013.
[4]-T. Etzion and A. Vardy, "Error-Correcting codes in projective space," \textit{IEEE Intl. Symp. on Inform. Theory - ISIT-08}, pp. 871-875, Toronto, Canada, Jul. 2008.
[5]-A. Khaleghi, D. Silva, and F.R. Kschischang, "Subspace codes," \textit{Lecture Notes in Computer Science}, vol. 5921, pp. 1-21, 2009.
[6]-C.H.A. Costa e M. Guerreiro, "Automorphisms of finite Abelian groups,", MS thesis, Mathematics Dept, UFV, Vi\c{c}osa, Minas Gerais, 2014. (in Portuguese)
Joint work with Reginaldo Palazzo Jr. (FEEC/UNICAMP) e-mail: palazzo@dt.fee.unicamp.br. | 2017-05-27 02:19:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8350623846054077, "perplexity": 1432.4182491850186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608765.79/warc/CC-MAIN-20170527021224-20170527041224-00402.warc.gz"} |
http://electronics.stackexchange.com/questions/28813/doesnt-using-resistors-serial-to-leds-all-the-time-waste-a-lot-of-energy?answertab=votes | # Doesn't using resistors serial to LEDs all the time waste a lot of energy?
I was wondering a long time why in every scheme there is always a resistor to go with the LED, and finally it seems like the answer in this question explains why.
But still, isn't this a big problem? Don't those resistors waste a lot of power and isn't there really any other practical solution?
UPD: A reasonable update for the question given all the good answers I've received is to maybe provide some numbers to show just how much power is lost to heat from resistors in a typical lighting application? (Most answers say that the power loss i so small that it doesn't matter. I think it would be good if anyone could get the real numbers, to solidify that answer, then I could accept that answer and keep it on top for future interested people.)
-
Here's a related question electronics.stackexchange.com/q/23974/3552 – sharptooth Mar 28 '12 at 5:48
As a response to your update: it's impossible to define the typical lighting application (even though the most common until now is as indicator); is instead easier to point the most critical applications, which I think to be power lighting and low low power operation (the second can be bypassed with flashing) – clabacchio Mar 28 '12 at 14:52
Alright, but certainly, providing a way to calculate the lost power given the configuration of a specific system and perhaps showing an example calculation for a typical system is good enough, just to give a rough estimate, how much power are we talking about... – Cray Mar 28 '12 at 15:01
Resistor: I^2 * R --- Dc/Dc: Pin(1-efficiency) – clabacchio Mar 29 '12 at 15:01
Yes, it wastes power, but most of the time it's not enough power to matter.
In cases where effeciency matters, you use other more complicated means. For example, take a look at the schematic for my KnurdLight example project. This is battery operated and just about all the power is going into the LEDs. In this case I used a boost converter that directly regulated LED current instead of a normal power supply that regulates voltage. There is no series resistor to make a fixed voltage supply look at least partially like a current source because the power supply is a current source in the first place. R6 is in series with the LED string, but is only 30 Ω and is for sensing the current so that the boost converter can regulate it.
-
When an LED is driven with a resistor, it's necessary that the supply voltage be higher than the forward drop of the LED; the current drawn from the supply will be equal to the current through the LED. The percentage of supply power that goes to the LED will correspond to the ratio of the LED forward voltage to the supply voltage.
There are other ways of driving LEDs which will work with supply voltages below the forward drop of the LED, or which will draw less current from the supply than they put through the LED. Such techniques may e.g. reduce by half the current drawn from a 5-volt supply to feed 20mA through a 2-volt LED, but the circuitry required will almost certainly be more expensive than a resistor. In many situations, even when running from batteries, the power consumed by an LED will represent a tiny fraction of overall energy usage; even if one could reduce LED-related power consumption by 99% using only $0.05 worth of extra circuitry, the savings wouldn't be worth the cost when compared with simply using a resistor and accepting the sub-optimal efficiency. - # Why resistors? The reason we use resistors to set LED current is that an LED is a diode, and like most diodes, it just looks like a voltage drop when forward biased. There is very little to control current if hooked up to a voltage source; the V/I graph's slope is so steep that a 0.1 V change in diode voltage could mean a 10X change in current. Thus a direct connection to a supply without a workable current limiting mechanism will likely destroy the LED. So we put a resistor in there to make the slope shallow enough to control the current. Typically, you figure out how much current you want in the LED based on some brightness measurement from the data sheet, or buy one and guess. For typical indicator LEDs, I start with 2 mA for normal or 0.5 mA for high-efficiency LEDs, and usually have to reduce the current further. Once you pick a current, you take that, the voltage of your source (VS), and the forward voltage of your LED at your current (VF, try to get this from the graph in the data sheet rather than the table, which typically is characterized at 10 mA or more), and plug them into the following equation to get your resistance: R = (VS - VF) / I Derivation: Given that the voltage drop across the resistor is VR = I * R (Ohm's Law), that the current in the loop is constant (Kirchoff's Current Law), and that the source voltage is equal to VF + VR (Kirchoff's Voltage Law): VS = VF + VR = VF + I * R; VS - VF = I * R; R = (VS - VF) / I # High Power LEDs For applications where the power waste is a problem, such as in large-scale lighting applications, you don't use a resistor but instead use a current regulator to set the LED's current. These current regulators work like switching voltage regulators, except instead of dividing down the output voltage and comparing to a reference and adjusting the output, they use a current-sensing element (current-sense transformer or low-value resistor) to generate the voltage that is compared to the reference. This can get you lots of efficiency, depending by switching element loss and switching frequency. (Higher frequencies react faster and use smaller components but are less efficient.) - Yes, it wastes power. On the other hand, in volume production a resistor will cost a fraction of a penny (US$0.01 for our international people). When the cost/benefit/difficulty analysis is done a simple resistor starts to look really nice.
-
The wasted power is often very small (tens of milliwatts) if you're driving the LED with 5 V or an equally small voltage.
Sure, it is a problem in systems where you have limited-capacity batteries, but then other schemes (like LED drivers using PWM) are used.
-
There's no such thing as "wasted current". The current through the resistor is the same current that makes the LED light up, so it's not wasted at all. What you mean is "wasted power/energy". – stevenvh Mar 28 '12 at 14:46
@steventh indeed. Will edit my post. – Renan Mar 28 '12 at 16:02
You wanted a calculation. Here is the basic form of the calculation.
A typical red LED has a forward voltage drop of 1.8 V, and a maximum continuous current of around 20 mA.
Now what's our voltage? Lets say we want to use a 3.3 V source.
So we will have a voltage drop of 3.3 V - 2.1 V = 1.2 V over our resistor. The current through the resistor will be 20 mA, so our power is 1.2 V * 20 mA = 24 mW. That is not really a lot of power, although it is a significant fraction of power consumption of the LED. The LED itself uses 1.8V * 20mA = 36 mW.
-
In other words, for every 3 watts of power usefully delivered to the LED, you are wasting 2 watts of power as heat. – rjmunro Mar 28 '12 at 23:31
In that particular scenario yes. In other scenarios, you could have far less of the energy wasted. If your voltage is high enough, you can also put more than one LED in series, which lets them share a single resistor (but requires that they both be on or off at the same time). In that case your waste percentage can drop substantially. Also keep in mind that the lumen per watt of LEDs even when including the resistor loss is still far better than most alternatives. The choice is not resistor or no resistor, but (LED+resistor) vs (no LED) vs (other light source). – Kevin Cathcart Mar 28 '12 at 23:44
Nice example, upvoted you. – Cray Mar 29 '12 at 15:16
Please remember that the current quoted on the data sheet is typically the most the part will handle. This will generate a large amount of light, usually for illumination (i.e. flashlight) purposes, but hard to look at if you want a display or indicator. (Display currents are typically much, much lower. In this example, I would bet that 1-2 mA would be plenty.) Also, like on other semiconductor data sheets, the maximum current may be valid only with an infinite heat sink (i.e. water bath), and may generate enough heat to destroy the part without a heat sink. – Mike DeSimone Apr 2 '12 at 13:08
For LEDs, the max continous current normally does not assume a heatsink, since the plastic that surounds the LED is a poor heat conductor, making a heatsink rather ineffective. But you are correct that in circuit design you really should target a lower current. However, I frequenty see designs running 20mA LEDs at 15mA or more even for indicator purposes. I would be better in general to chose the current resulting in the desired brightness. – Kevin Cathcart Apr 3 '12 at 16:40
Yes, and no. When current passes through the resistor, it generates heat and therefore wastes energy. However, if you took the resistor out (and therefore drove the LED at a higher voltage) you'd be driving more current through the circuit and thus actually burning more power than with the resister in place.
Remember that with constant voltage, current is inversely proportional to resistance. The more resistance you inline into the circuit, the less current you pass, and therefore the less power you consume. So while the resistor itself plays a part in generating heat in the circuit, its presence there actually means that less heat will be generated overall.
-
Well as I understand now, the problem with no resistors is that there is not good cheap way to control the voltage and that without it the LEDs will simply burn. Resistors are not used in those circuits to limit current in order to save power, but in order to keep the circuit from burning. – Cray Mar 28 '12 at 14:58 @Cray: It's not the voltage drop that kills LEDs, it's the heat from the current flow. Solving for the appropriate voltage drop makes the math simpler, but is's not the critical factor in the physics of it all. The internal resistance of an LED decreases as the voltage across it increases, so trying to work out the math without a traditional resistor in the circuit is difficult (and yes, of course an LED has internal resistance; it's not a superconductor). – tylerl Mar 28 '12 at 22:58 | 2013-05-22 08:19:43 | {"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": 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": 0.6154690384864807, "perplexity": 801.0335812472354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701508530/warc/CC-MAIN-20130516105148-00099-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://openstudy.com/updates/558fc81be4b0bba2a1175d8c | ## anonymous one year ago Bag C and Bag D each contain 70 marbles. All the marbles inside both bags are red, white or blue. In bag C, R:W = 2:3 and W:B = 3:5 In bag D, R:W = 2:3 and W:B = 4:5 What is the total number of white marbles in BOTH bags?
1. amoodarya
first I o for bag C w+r+b=70 $\frac{r}{w}=\frac{2}{3} \rightarrow r=\frac{2}{3}w\\\frac{w}{b}=\frac{3}{5} \rightarrow b=\frac{5}{3}w$ now put on above equation $w+r+b=70\\ww+\frac{2}{3}w+\frac{5}{3}w=70\\w=21$ so white marbles in bag c=21 can you solve like this for bag D ?
2. amoodarya
i type ww , w +2/3w+5/3w=70 is correct
3. mathmate
Alternatively, we can combine the ratio to include all three colours. For Bag C r:$$\color{blue}{w}$$=2:$$\color{blue}{3}$$ $$\color{blue}{w}$$:b=and $$\color{blue}{3}$$ :5 Since w has a matching value of 3 in both ratios, we can say r:$$\color{blue}{w}$$:b=2:$$\color{blue}{3}$$ :5 So $$\color{blue}{w}$$ = $$\color{blue}{3}$$/(2+$$\color{blue}{3}$$ +5)=3/70=21 For bag D r:$$\color{blue}{w}$$=2:3 = 8:$$\color{blue}{12}$$ $$\color{blue}{w}$$:b=4:5=$$\color{blue}{12}$$ :15 we need to match the value for white in both ratios to combine to a single ratio (using LCM of 3 and 4=$$\color{blue}{12}$$ ) We can then say r:$$\color{blue}{w}$$:b=8:$$\color{blue}{12}$$ :15 and the number of while marbles can be calculated in a similar way to bag C. | 2016-10-22 13:55:38 | {"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": 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": 0.6370365619659424, "perplexity": 2473.5108990243643}, "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/1476988718987.23/warc/CC-MAIN-20161020183838-00511-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/differentiating-the-integral-form-of-the-continuity-equation-for-fluids.464655/ | Differentiating the Integral Form of the Continuity Equation for Fluids
Homework Statement
I am working on a problem that asks to use the integral form of the continuity equation (for a steady flow) and show that it can equal this (by taking the derivative of it): dr/r + dV/V + dA/A = 0 where V is Velocity and r is the density.
Homework Equations
What would the derivative be with respect to?
The Attempt at a Solution
I was able to bring it down to: rVA=0 but I am unaware how to differentiate this so that it looks like the equation above.
Thanks. | 2021-04-20 08:12:21 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9532513618469238, "perplexity": 281.4858270496041}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039379601.74/warc/CC-MAIN-20210420060507-20210420090507-00478.warc.gz"} |
https://www.rdocumentation.org/packages/ape/versions/3.0-2/topics/mantel.test | # mantel.test
From ape v3.0-2
0th
Percentile
##### Mantel Test for Similarity of Two Matrices
This function computes Mantel's permutation test for similarity of two matrices. It permutes the rows and columns of the two matrices randomly and calculates a $Z$-statistic.
Keywords
multivariate
##### Usage
mantel.test(m1, m2, nperm = 999, graph = FALSE,
alternative = "two.sided", ...)
##### Details
The function calculates a $Z$-statistic for the Mantel test, equal to the sum of the pairwise product of the lower triangles of the permuted matrices, for each permutation of rows and columns. It compares the permuted distribution with the $Z$-statistic observed for the actual data.
If graph = TRUE, the functions plots the density estimate of the permutation distribution along with the observed $Z$-statistic as a vertical line.
The ... argument allows the user to give further options to the plot function: the title main be changed with main=, the axis labels with xlab =, and ylab =, and so on.
##### Value
z.statthe $Z$-statistic (sum of rows*columns of lower triangle) of the data matrices.p$P$-value (quantile of the observed $Z$-statistic in the permutation distribution).alternativethe alternative hypothesis.
##### References
Mantel, N. (1967) The detection of disease clustering and a generalized regression approach. Cancer Research, 27, 209--220.
Manly, B. F. J. (1986) Multivariate statistical methods: a primer. London: Chapman & Hall.
##### Examples
q1 <- matrix(runif(36), nrow = 6)
q2 <- matrix(runif(36), nrow = 6)
mantel.test(q1, q2, graph = TRUE,
main = "Mantel test: a random example with 6 X 6 matrices",
xlab = "z-statistic", ylab = "Density",
sub = "The vertical line shows the observed z-statistic")
Documentation reproduced from package ape, version 3.0-2, License: GPL (>= 2)
### Community examples
Looks like there are no examples yet. | 2019-10-23 21:17:39 | {"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": 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": 0.4908747375011444, "perplexity": 6100.6427694069425}, "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/1570987836295.98/warc/CC-MAIN-20191023201520-20191023225020-00316.warc.gz"} |
http://ieee.ns02.info/xpl/tocresult.jsp?reload=true&isnumber=6930788 | # IEEE Transactions on Network Science and Engineering
### Early Access Articles
Early Access articles are made available in advance of the final electronic or print versions. Early Access articles are peer reviewed but may not be fully edited. They are fully citable from the moment they appear in IEEE Xplore.
## Filter Results
Displaying Results 1 - 25 of 41
• ### On the convergence of message passing computation of harmonic influence in social networks
Publication Year: 2018, Page(s): 1
| | PDF (1432 KB)
The harmonic influence is a measure of node influence in social networks that quantifies the ability of a leader node to alter the network's average opinion, acting against an adversary field node. The definition of harmonic influence assumes linear interactions between the nodes described by an undirected weighted graph; its computation is equivalent to solve, for every node, a discrete Di... View full abstract»
• ### Graph Theoretical Analysis on Distributed Line Graphs for Peer-to-Peer Networks
Publication Year: 2018, Page(s): 1
| | PDF (3889 KB)
Distributed line graphs were introduced by Zhang and Liu as an overlay for Peer-to-Peer networks. Distributed line graphs have some useful properties as a network topology, such as out-regular and small diameter, where the former implies that each user possesses a constant size of routing table and the latter means that a reasonably small number of hops is necessary to reach a target user. In this... View full abstract»
• ### Scalable Privacy-Preserving Participant Selection for Mobile Crowdsensing Systems: Participant Grouping and Secure Group Bidding
Publication Year: 2018, Page(s): 1
| | PDF (1739 KB)
Mobile crowdsensing (MCS) has been emerging as a new sensing paradigm where vast numbers of mobile devices are used for sensing and collecting data in various applications. Auction based participant selection has been widely used for current MCS systems to achieve user incentive and task assignment optimization. However, participant selection problems solved with auction-based approaches usually i... View full abstract»
• ### Contract Mechanism and Performance Analysis for Data Transaction in Mobile Social Networks
Publication Year: 2017, Page(s): 1
| | PDF (1062 KB)
In this paper, a novel mobile data offloading method is proposed based on an external-infrastructure-free approach. Specifically, through the hotspot function of smartphones, data demands from some mobile users, who have used out the data with their monthly data plans, can be offloaded by some other mobile users who still have redundant unused data. In order to model this data transaction among mo... View full abstract»
• ### A General Framework for Sensor Placement in Source Localization
Publication Year: 2018, Page(s): 1
| | PDF (1286 KB) | Media
When an epidemic spreads in a given network of individuals or communities, can we detect its source using only the information provided by a small set of nodes? We propose a general framework that incorporates two dimensions. First, we can either rely exclusively on a set of selected nodes (i.e., sensors) which always reveal their state independently of any particular epidemic (these are called st... View full abstract»
• ### An Efficient Randomized Algorithm for Rumor Blocking in Online Social Networks
Publication Year: 2017, Page(s): 1
| | PDF (940 KB) | Media
Social networks allow rapid spread of ideas and innovations while negative information can also propagate widely. When a user receives two opposing opinions, they tend to believe the one arrives first. Therefore, once misinformation or rumor is detected, one containment method is to introduce a positive cascade competing against the rumor. Given a budget $k$, the rumo... View full abstract»
• ### Threshold Models of Cascades in Large-Scale Networks
Publication Year: 2017, Page(s): 1
| | PDF (1776 KB)
The spread of new beliefs, behaviors, and technologies in social and economic networks are often driven by cascading mechanisms. Global behaviors emerge from the interplay between the interconnections structure and the local agents interactions. We focus on the Threshold Model (TM) of cascades that can be interpreted as the best response dynamics in a network game. Each agent is equipped with an i... View full abstract»
• ### Shotgun assembly of labeled graphs
Publication Year: 2017, Page(s): 1
| | PDF (761 KB)
We consider the problem of reconstructing graphs or labeled graphs from neighborhoods of a given radius r. Special instances of this problem include DNA shotgun assembly, neural network reconstruction, and assembling random jigsaw puzzles. We provide some necessary and some sufficient conditions for correct recovery both in combinatorial terms and for some generative models including random labeli... View full abstract»
• ### Identifying Influential Spreaders in Complex Multilayer Networks: A centrality perspective
Publication Year: 2017, Page(s): 1
| | PDF (251 KB) | Media
Identifying influential spreaders in complex networks is of paramount importance for understanding and controlling the spreading dynamics. A challenging and yet inadequately explored task is to detect such influential nodes in multilayer networks, i.e., networks that encompass different types of connections (e.g., different relationships) among the nodes, hence facilitating a multilayer structure.... View full abstract»
• ### Modeling Spread of Preferences in Social Networks for Sampling-based Preference Aggregation
Publication Year: 2017, Page(s): 1
| | PDF (616 KB) | Media
Given a large population, it is an intensive task to gather individual preferences over a set of alternatives and arrive at an aggregate or collective preference of the population. We show that social network underlying the population can be harnessed to accomplish this task effectively, by sampling preferences of a small subset of representative nodes. We first develop a Facebook app to create a ... View full abstract»
• ### Time to extinction for the SIS epidemic model: new bounds on the tail probabilities
Publication Year: 2017, Page(s): 1
| | PDF (462 KB)
In this paper, we present a novel estimation of the time to extinction of a Susceptible-Infected-Susceptible (SIS) epidemic model over a general network of interactions. Specifically, we prove that, for an effective infection rate above a threshold depending on the topology of the network, the time to extinction grows exponentially in the size of the population, with probability converging to 1 as... View full abstract»
• ### Passivity Analysis and Pinning Control of Multi-Weighted Complex Dynamical Networks
Publication Year: 2017, Page(s): 1
| | PDF (3612 KB)
This paper studies a multi-weighted network model with different dimensions of output and input vectors. Firstly, we analyze passivity of proposed network model by employing some inequality techniques and Lyapunov functional method, and give a synchronization condition for output-strictly passive complex dynamical network with multi-weights (CDNMWs). Furthermore, by using pinning adaptive strategi... View full abstract»
• ### A General Method to Determine Asymptotic Capacity Upper Bounds for Wireless Networks
Publication Year: 2017, Page(s): 1
| | PDF (520 KB)
Capacity scaling laws offer fundamental understanding on the trend of user throughput behavior when the network size increases. Since the seminal work of Gupta and Kumar, there have been active research efforts in developing capacity scaling laws for ad hoc networks under various advanced physical (PHY) layer technologies. These efforts led to many custom-designed solutions, most of which were mat... View full abstract»
• ### Contact Adaption during Epidemics: A Multilayer Network Formulation Approach
Publication Year: 2017, Page(s): 1
| | PDF (6956 KB) | Media
People change their physical contacts as a preventive response to infectious disease propagations. Yet, only a few mathematical models consider the coupled dynamics of the disease propagation and the contact adaptation process. This paper presents a model where each agent has a default contact neighborhood set, and switches to a different contact set once she becomes alert about infection among he... View full abstract»
• ### Minimizing Social Cost of Vaccinating Network SIS Epidemics
Publication Year: 2017, Page(s): 1
| | PDF (2088 KB)
Reducing the economic costs (losses) as much as possible is one of the main goals of controlling virus spreading and worm propagation on complex networks. Taking into account the interactions and conflicts of interests among egoistic individuals (nodes) in a network, we introduce the zero-determinant (ZD) strategy into our proposed non-cooperative networking vaccination game with the economic ince... View full abstract»
• ### Cascading Edge Failures: A Dynamic Network Process
Publication Year: 2017, Page(s): 1
| | PDF (741 KB)
This paper studies a network process that can potentially be used to model cascading failures in networks. The Dynamic Bond Percolation (DBP) process models, through stochastic local rules, the failure or recovery of an edge (i, j) in a network. The probability that a working link fails or a failed link recovers may be independent of the state of other links or may be dependent locally on the stat... View full abstract»
• ### Modelling Spreading Process Induced by Agent Mobility in Complex Networks
Publication Year: 2017, Page(s): 1
| | PDF (35642 KB)
Most conventional epidemic models assume contact-based contagion process. We depart from this assumption and study epidemic spreading process in networks caused by agents acting as carrier of infection. These agents traverse from origins to destinations following specific paths in a network and in the process, infecting the sites they travel across. We focus our work on the Susceptible-Infected-Re... View full abstract»
• ### Detecting Cascades from Weak Signatures
Publication Year: 2017, Page(s): 1
| | PDF (2697 KB)
Inspired by cyber-security applications, we consider the problem of detecting an infection process in a network when the indication that any particular node is infected is extremely noisy. Instead of waiting for a single node to provide sufficient evidence that it is indeed infected, we take advantage of the graph structure to detect cascades of weak indications of failures. We view the detection ... View full abstract»
• ### A Resource Allocation Mechanism for Cloud Radio Access Network Based on Cell Differentiation And Integration Concept
Publication Year: 2017, Page(s): 1
| | PDF (3456 KB)
A Self-Organising Cloud Radio Access Network (C-RAN) is proposed, which dynamically adapt to varying capacity demands. The Base Band Units and Remote Radio Heads are scaled semi-statically based on the concept of cell differentiation and integration (CDI) while a dynamic load balancing is formulated as an integer-based optimisation problem with constraints. A Discrete Particle Swarm Optimisation (... View full abstract»
• ### Comparing the Effects of Failures in Power Grids under the AC and DC Power Flow Models
Publication Year: 2017, Page(s): 1
| | PDF (1542 KB) | Media
In this paper, we compare the effects of failures in power grids under the nonlinear AC and linearized DC power flow models. First, we numerically demonstrate that when there are no failures and the assumptions underlying the DC model are valid, the DC model approximates the AC model well in four considered test networks. Then, to evaluate the validity of the DC approximation upon failures, we num... View full abstract»
• ### Belief Dynamics in Social Networks: A Fluid-Based Analysis
Publication Year: 2017, Page(s): 1
| | PDF (436 KB) | Media
The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into collective dominant social beliefs and into the impact of different components of the system, such as users' interactions, while being able to predict users... View full abstract»
• ### Recovering asymmetric communities in the stochastic block model
Publication Year: 2017, Page(s): 1
| | PDF (1823 KB) | Media
We consider the sparse stochastic block model in the case where the degrees are uninformative. The case where the two communities have approximately the same size has been extensively studied and we concentrate here on the community detection problem in the case of unbalanced communities. In this setting, spectral algorithms based on the non-backtracking matrix are known to solve the community det... View full abstract»
• ### Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs
Publication Year: 2017, Page(s): 1
| | PDF (2277 KB)
We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivit... View full abstract»
• ### Provision of Public Goods on Networks: On Existence, Uniqueness, and Centralities
Publication Year: 2017, Page(s): 1
| | PDF (417 KB)
We consider the provision of public goods on networks of strategic agents. We study different effort outcomes of these network games, namely, the Nash equilibria, Pareto efficient effort profiles, and semi-cooperative equilibria (resulting from interactions among coalitions of agents). We identify necessary and sufficient conditions on the structure of the network for the uniqueness of the Nash eq... View full abstract»
• ### Weighted Bearing-Compass Dynamics: Edge and Leader Selection
Publication Year: 2017, Page(s): 1
| | PDF (3409 KB)
This paper considers the design and effective interfaces of a distributed robotic formation running planar weighted bearing-compass dynamics. We present results which support methodologies to construct formation topologies using submodular optimization techniques. Further, a convex optimization framework is developed for the selection of edge weights which increase performance. We explore a method... View full abstract»
## Aims & Scope
The IEEE Transactions on Network Science and Engineering is committed to timely publishing of peer-reviewed technical articles that deal with the theory and applications of network science and the interconnections among the elements in a system that form a network. In particular, the IEEE Transactions on Network Science and Engineering publishes articles on understanding, prediction, and control of structures and behaviors of networks at the fundamental level.
Full Aims & Scope
## Meet Our Editors
Editor-in-Chief
Dapeng Oliver Wu
University of Florida
Dept. of Electrical & Computer Engineering
P. O. Box 116130
Gainesville, FL 32611
Email: dpwu@ufl.edu | 2018-01-17 01:16:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.332402765750885, "perplexity": 2440.5503124882744}, "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-2018-05/segments/1516084886792.7/warc/CC-MAIN-20180117003801-20180117023801-00024.warc.gz"} |
http://www.ctan.org/tex-archive/support/bm2ltx | # Directory tex-archive/support/bm2ltx
BM2LTX - (c) 1995 claas bontus
email: bontus@al6000.physik.uni-siegen.de
bontus@141.99.131.151
---------
BM2LTX is on the public domain. I used some functions of
Ian Ashdown's PCX library. See file PCX_LIB.DOC for further
information.
You are free to
* distibute BM2LTX
* modify the source code, as long as you document all changes
* transform BM2LTX to other operating systems.
Files
-----
bm2ltx.exe - the executable for DOS and OS/2
pcx_fmt.doc - explanation of the PCX format by Ian Ashdown
pcx_lib.doc - explanation of Ian Ashdown's PCX library
b2l_src.zip - ZIP-file containing the source code of BM2LTX
Explanation
-----------
BM2LTX converts PCX graphics files to a LaTeX readable format.
If you want to convert the file TEST.PCX to TEST.PIC just type
BM2LTX TEST.PCX TEST.PIC
You can include TEST.PIC in your LaTeX documents using the
following commands:
\begin{figure}
\begin{center}
\input{TEST.PIC}
\end{center}
\caption{Hello world!}
\end{figure}
The picture will be stored in the DVI file, so that it can be
processed independently of drivers or platforms.
Installation
------------
OS/2: You need to copy EMX.DLL to a directory listed in your
LIBPATH environment variable. If you have already EMX.DLL
replace it only if the one you got now is newer.
DOS: BM2LTX is bound with emx.exe. Therefore it should be able
to work with DOS as is. Problems can occur if the processor
----------
Files created with BM2LTX can be used with LaTeX on different
platforms, no matter which graphics formats are supported by
-------------
Files created with BM2LTX can become very large in size. A picture
consisting of 100x100 pixels can yield to a file with up to 5000
lines of code. TeX can slow down processing these files or it
can even run out of memory.
Things someone might do
-----------------------
* Add options to resize the pictures
* Find an algorithm which yields to smaller files
* Add the possibiliy to process graphics formats other than PCX
## Files
Name Size Date Notes
EMX.DLL 60990 1994-12-21 01:00
b2l_src.zip 34561 1995-02-03 01:00
bm2ltx.exe 87853 1995-02-02 01:00
pcx_fmt.doc 27951 1991-12-01 01:00
pcx_lib.doc 9816 1991-12-01 01:00 | 2014-09-30 19:55:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6033385396003723, "perplexity": 12138.808971217784}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663135.34/warc/CC-MAIN-20140930004103-00249-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://deepai.org/publication/parameter-uniform-fitted-mesh-higher-order-finite-difference-scheme-for-singularly-perturbed-problem-with-an-interior-turning-point | # Parameter-uniform fitted mesh higher order finite difference scheme for singularly perturbed problem with an interior turning point
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow exponential boundary layers. Parameter-explicit theoretical bounds on the derivatives of the analytical solution are given, which are used in the error analysis of the proposed scheme. The problem is discretized by a hybrid finite difference scheme comprises of midpoint-upwind and central difference operator on an appropriate piecewise-uniform fitted mesh. An error analysis has been carried out for the proposed scheme by splitting the solution into regular and singular components and the method has been shown second order uniform convergent except for a logarithmic factor with respect to the singular perturbation parameter. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects. Numerical experiments show that the proposed method gives competitive results in comparison to those of other methods exist in the literature.
## Authors
• 4 publications
• 1 publication
• 1 publication
06/04/2019
### Parameter uniform essentially first order convergence of a fitted mesh method for a class of parabolic singularly perturbed Robin problem for a system of reaction-diffusion equ
In this paper, a class of linear parabolic systems of singularly perturb...
06/17/2022
### Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem
A singularly perturbed convection-diffusion problem posed on the unit sq...
08/18/2021
### Hybrid high-order method for singularly perturbed fourth-order problems on curved domains
We propose a novel hybrid high-order method (HHO) to approximate singula...
09/14/2020
### On construction of a global numerical solution for a semilinear singularly–perturbed reaction diffusion boundary value problem
A class of different schemes for the numerical solving of semilinear sin...
10/19/2020
### Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition
A singularly perturbed parabolic problem of convection-diffusion type wi...
07/27/2021
### Parameter-uniform numerical methods for singularly perturbed linear transport problems
Pointwise accurate numerical methods are constructed and analysed for th...
03/11/2020
### A Non-Standard Finite Difference Scheme for MHD Boundary Layer Fluid Flow
This paper deals with a non-standard finite difference scheme defined on...
##### This week in AI
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.
## 1 Introduction
Singularly perturbed problems arise often in the modeling of various modern complicated processes, such as viscous flow problems with large Reynolds numbers [9], convective heat transport problems with large Péclet numbers [10], drift diffusion equation of semiconductor device modelling [21], electromagnetic field problems in moving media [8], financial modelling [5] and turbulence models [15] etc. Most of the singularly perturbed problems cannot be completely solved by analytical techniques. Consequently, numerical techniques are getting much attention to get some useful insights on the solutions of singularly perturbed problems. In general, two classes of methods, namely, fitted operator methods and fitted mesh methods have been used to solve such problems.
Those singularly perturbed convection-diffusion problems, in which the convection coefficient vanishes at some points of the domain of the problem, are called singularly perturbed turning point problems (SPTPPs), and zeros of the convection coefficient are said to be turning points. Here, we consider the following class of singularly perturbed two-point boundary value problems with an interior turning point at [12]:
{Lu(x)≡εu′′(x)+a(x)u′(x)−b(x)u(x)=f(x),x∈Ω=(−1,1),u(−1)=A,u(1)=B, (1.1)
where is a small perturbation parameter satisfying , and are given constants, and are sufficiently smooth functions. We impose the following restriction to ensure that the solution of Eq. (1.1) exhibits twin boundary layers
a(0)=0,a′(0)<0. (1.2)
Moreover, for some constant there exists a positive constant , such that
|a(x)|≥α>0,δ≤|x|≤1. (1.3)
Also is required to be bounded below by some positive constant , ,
b(x)≥β>0,x∈¯Ω=[−1,1], (1.4)
to guarantee that the operator is inverse monotone on and to exclude the so-called resonance phenomena [2]. We also impose the following restriction to ensure that there are no other turning points in the interval :
|a′(x)|≥∣∣∣a′(0)2∣∣∣,x∈¯Ω=[−1,1]. (1.5)
This class of singularly perturbed turning point problem (SPTPP) (1.1) has a unique solution possess twin outflow boundary layers of exponential type at both end points , under the assumptions (1.2)- (1.5).
It is very difficult to deal singularly perturbed turning point problems analytically. The study of these problems received much attention in the literature due to the complexity involved in finding uniformly valid asymptotic expansions unlike non-turning problems. Some authors, such as, Jingde [7], O’Malley [16, 17], Wasow [24] studied qualitative aspects of these problems, namely, existence, uniqueness and asymptotic behavior of the solution.
In general, since the convection coefficient has zero inside the domain therefore numerical treatment of singularly perturbed turning point problem becomes more difficult than the singularly perturbed non-turning point problems. Abrahamsson [1], Berger et al. [3] and Farrell [6] establish a priori bounds for interior turning point problems; in particular it is shown that a bound is independent of singular perturbation parameter if and only if reaction coefficient is greater than zero at the turning point. It is also shown there how the ratio of reaction coefficient and first derivative of convection coefficient , , at the turning point plays a key role in determining the behavior of the solution [3]. It is shown that for , the solution is smooth near turning point and two outflow boundary layers of exponential type exhibits at both the endpoints of the domain. In this case the turning point is sometimes called a diverging flow or expansion turning point. On the other hand, if , there is in general no boundary layers exhibited and an interior layer appears at the turning point, the nature of which depends in a fundamental way on . For , the interior layer is called cusp layer because it can be approximately modelled by a cusp-like function. Interior layer turning point is sometimes called a converging flow or compression turning point. In inviscid fluid dynamics, the diverging flow turning point corresponds to a sonic point and converging flow turning point to a shock point. For the case at the turning point, the solution exhibits a very interesting phenomenon called Ackerberg-O’Malley resonance phenomenon [2].
Berger et al. [3] also show that the modified version of El Mistikawy Werle scheme is uniformly convergent of in the norm using the analytic bounds obtained in [3]. Farrell [6] obtained a set of sufficient conditions for uniform convergence in the discrete norm on uniform mesh, not only for exponentially fitted schemes, but also for a large class of schemes of upwinded type. Kadalbajoo and Patidar [11] gave a numerical scheme based on cubic spline approximation with nonuniform mesh for SPTPP (1.1)-(1.5) and established second order -uniform convergence. Natesan et al. [20] proposed a numerical method based on the classical upwind finite difference scheme on a Shishkin mesh and proved that the proposed scheme is uniformly convergent of almost order one. In [12], Kadalbajoo and Gupta derived asymptotic bounds for the derivatives of the analytical solution of SPTPP (1.1)-(1.5) and proposed a computational method comprises B-spline collocation scheme on a non-uniform Shiskin mesh. They shown that this scheme is second order accurate in the maximum norm. Kadalbajoo et al. [13] also suggested B-spline collocation with artificial viscosity on uniform mesh for the same class of SPTPP (1.1)-(1.5). In [18], Munyakazi and Patidar conclude that convergence acceleration Richardson extrapolation technique on existing numerical schemes for the above class of turning point problem does not improve the rate of convergence. However, Becher and Roos [4] show that Richardson extrapolation on upwind scheme with piecewise-uniform Shishkin mesh works fine and improves the accuracy to under the assumption . Recently, Munyakazi et al. [19] proposed a fitted operator finite difference scheme for singularly perturbed turning point problem having an interior layer and also shown that with Richardson extrapolation technique, accuracy and order of convergence of the scheme can be improved upto two. For a general review of existing literature on asymptotic and numerical analysis of turning point problems, one can see [22].
In this paper, we focus to devise a second order uniformly convergent finite difference scheme for SPTPP (1.1) on piecewise uniform mesh of Shishkin type without using any convergence acceleration technique like Richardson extrapolation. The proposed method combines the midpoint upwind difference scheme and classical central finite difference scheme on piecewise uniform mesh. The requirements of higher order truncation error and monotonicity play a vital role in the construction of this scheme. One can observe the fact that the classical central difference scheme is monotone if is relatively large than the convection coefficient , if , where is the mesh width and has second order truncation error on uniform mesh. On the other hand, midpoint upwind difference operator is monotone for all value of and for relatively large convection coefficient than the reaction coefficient such that . Moreover, midpoint upwind operator possess second order truncation error away from the boundary layer region. Also, Shiskin mesh equally distribute the number of mesh points inside and outside the boundary layers, therefore one can gets a coarse mesh region outside the boundary layer and fine mesh region inside the boundary layer. Utilizing these facts, we employ midpoint upwind difference scheme in coarse mesh region and central difference operator in fine mesh region of Shishkin mesh. Since, central difference operator yields first order truncation error at transition points, we use midpoint upwind operator on transition points. Such type of higher order scheme for singularly perturbed non-turning convection-diffusion problem was introduced by Stynes and Roos [23]. To analyze the proposed scheme theoretically, we split the numerical solution into regular and singular components and analyze them separately by using tools such as truncation error bounds, discrete minimum principle and appropriate choices of barrier functions.
Notation. Throughout the paper we use as a generic positive constant independent of and mesh parameters. For any given function ( a non-negative integer), is a global maximum norm over the domain defined by
||g||=max¯Ω|g(x)|.
## 2 A-priori Estimates for Continuous Problem
In this section some bounds of the exact solution and its derivatives are discussed. These bounds will be needed for error analysis of proposed numerical scheme in later sections. Derivation of these bounds are well known and can be found in [12]. Systematically, we use minimum principle to derive these bounds.
###### Lemma 2.1.
([12].) (Minimum Principle) Let and . Then implies that
Since the concerned SPTPP (1.1)-(1.5) is linear, minimum principle ensure the existence and uniqueness of the classical solution. Using the above minimum principle, one can easily prove the following uniform stability estimate for the differential operator .
###### Lemma 2.2.
([12].) (Uniform Stability Estimate) , solution of the SPTPP (1.1)-(1.5), satisfies the following stability estimate:
||u(x)||≤||f||β+max(|A|,|B|),∀x∈¯Ω.
To exclude the turning point and to obtain the bounds for the solution and its derivatives in the non-turning point region of the domain, we divide the domain into three subdomains as , and such that , where . Further, following theorem gives bounds for the derivatives of in the subintervals and individually.
###### Theorem 2.1.
([12].) If and , then solution of the SPTPP (1.1)-(1.2) satisfies the following bounds for any :
|uj(x)|≤C(1+ε−jexp(−α(1+x)ε)),j=1,⋯,m+1,x∈Ω1,
|uj(x)|≤C(1+ε−jexp(−α(1−x)ε)),j=1,⋯,m+1,x∈Ω3,
Next, we state a theorem, which gives the bounds for the derivatives of the solution in the turning point region and deduce that the solution is smooth in subdomain .
###### Theorem 2.2.
([3].) Let be the solution of SPTPP defined from the equations(1.1)-(1.5), and . Then for and sufficiently small , there exists a positive constant such that
|u(j)(x)|≤C,j=1,2,…,m,∀x∈Ω2.
It turns out that the bounds for continuous solution given in Theorem 2.1 and Theorem 2.2 are not adequate to obtain -uniform error estimate for the proposed scheme. Therefore, to analyze the proposed scheme correctly, we need to derive more precise bounds on these derivatives by decomposing the solution into regular component and singular component as
u(x)=v(x)+w(x),∀x∈¯Ω,
where the smooth component satisfies homogeneous problem and singular component satisfies homogeneous problem with appropriate boundary conditions. Using the technique given in [12], we get the following bounds for smooth and singular components in the region :
|v(j)(x)|≤C(1+ε((m−1)−j)e−α(1+x)/ε),∀x∈Ω1,
|w(j)(x)|≤Cε−je−α(1+x)/ε.∀x∈Ω1.
In the same manner, we can obtain analogous estimates for subinterval , while the solution and its derivatives are smooth in the subinterval . Hence, on the whole domain , the bounds on and , and their derivatives are given in the following theorem:
###### Theorem 2.3.
([12].) Let and then for all the smooth component satisfies
|v(j)(x)|≤C(1+ε((m−1)−j)(exp(−α(1+x)ε)+exp(−α(1−x)ε))),∀x∈¯Ω,
and the singular component satisfies
|w(i)(x)|≤Cε−i(exp(−α(1+x)ε)+exp(−α(1−x)ε)),∀x∈¯Ω.
## 3 Fitted Mesh Higher-Order Scheme
In this section, first we construct fitted piecewise-uniform mesh of Shishkin type to discretize the domain and then employ a specially designed finite difference scheme on this mesh to discretize the SPTPP (1.1)-(1.2). The fitted mesh is constructed by dividing into three subintervals and such that . For be an integer, divides each of the subintervals and into mesh intervals and with mesh intervals such that . Here, the transition parameter is obtained by taking
τ=min{14,τ0εlnN}.
The constant is independent of the parameter and the number of mesh points and will be chosen later on during the analysis of proposed scheme. This mesh is coarse on and fine on and on . If and are fine and coarse mesh width respectively, then mesh width is defined as
hi=⎧⎪⎨⎪⎩h=4τ/N, i=1,2,…,N/4,H=4(1−τ)/N, i=N/4+1,…,3N/4,h=4τ/N, i=3N/4+1,…,N.
One can easily observe that
N−1≤H≤4N−1,h=4τ0εN−1lnN
Since, convection coefficient changes its sign at the turning point , therefore, we construct a finite difference scheme to discretize the SPTPP (1.1) in the following manner
LNU(xi)≡⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩LNcU≡εδ2Ui+aiD0Ui−biUi=fi,i=1,2,…,N/4−1,LNmpU≡εδ2Ui+ai±1/2D±Ui−(bU)i±1/2=fi±1/2,i=N/4,…,3N/4,LNcU≡εδ2Ui+aiD0Ui−biUi=fi,i=3N/4+1,…,N−1,U0=A,UN=B, (3.1)
where,
LNmpU≡{εδ2Ui+ai+1/2D+Ui−(bU)i+1/2=fi+1/2,if ai>0εδ2Ui+ai−1/2D−Ui−(bU)i−1/2=fi−1/2,% if ai<0.
Here, we used the following definition to construct above scheme
vi=v(xi),vi+1/2=vi+vi+12,vi−1/2=vi−1+vi2,ˆhi=hi+hi+12,
D+vi=vi+1−vihi+1, ,D−vi=vi−vi−1hi, D0vi=vi+1−vi−12ˆhi, δ2vi=(D+vi−D−vi)ˆhi.
It is clear that proposed finite difference operator in scheme (3.1) is a combination of central difference operator and midpoint upwind difference operator , which is constructed by using knowledge judiciously about the sign of the convection term, location of the turning point and truncation error behavior of these operators. After simplifying the terms in (3.1), the difference scheme takes the form , where the coefficients are given by
pli=(εhiˆhi−ai2ˆhi),pci=(−pli−pri−bi), pri=(εhi+1ˆhi+ai2ˆhi), i=1,2,…,N/4−1, 3N/4+1…,N−1, pli=(εhiˆhi), pci=(−pli−pri−bi+1/2), pri=(εhi+1ˆhi+ai+1/2hi+1−bi+12), if ai>0 pli=(εhiˆhi−ai−1/2hi−bi−12), pci=(−pli−pri−bi−1/2), pri=(εhi+1ˆhi), if ai<0, i=N/4,…,3N/4.
## 4 Uniform Convergence
Here, in this section first we shall establish the consistency and stability estimate through discrete minimum principle and then analyze proposed numerical method (3.1) for -uniform convergence by analogous decomposition of discrete solution into smooth and singular components as of continuous solution.
###### Lemma 4.1.
( Discrete Minimum Principle) Let us suppose that , where
h||a||2ε<1, i.e., 2τ0||a||
Then the operator defined by (3.1) satisfies a discrete minimum principle, if is a mesh function that satisfies and for , then for .
Proof. In order to establish the discrete minimum principle, We simply check that the associated system matrix is -matrix with the choice of the midpoint upwind and central difference operator used in the definition of the difference scheme (3.1). It allow us to establish the following inequalities on the coefficients of the difference operator :
pli>0,pri>0,pli+pci+pri<0,i=1,2,…,N−1. (4.2)
In the case of central difference operator , conditions in (4.2) are satisfied if if , then one can check and for and . For the case of midpoint upwind operator , the conditions in (4.2) are satisfied if if From these sign patterns on the coefficients of associated system matrix, one can deduce that operator is of negative type and therefore satisfies a discrete minimum principle. Moreover, it ensures that the operator is uniformly stable in the maximum norm.
###### Lemma 4.2.
Let be any mesh function such that Then for all we have
|ZNi|≤1βmax1≤j≤N−1|LNZNj|.
Proof. Let us introduce two comparison functions defined by
Ψ±i=1βmax1≤j≤N−1|LNZNj|±ZNi.
Clearly one can notice that since Furthermore, for we have
LNΨ±i=−bβmax1≤j≤N−1|LNZNj|±LNZNi≤0,
as Therefore, discrete minimum principle (4.1) implies that , which gives desired result.
Further, using the valid Taylor’s series expansion, we obtained the following truncation error estimates for different finite difference operator employed in the operator : On a uniform mesh with step size , we have
|LNcui−(Lu)(xi)|≤C(ε~h2|u(iv)|+~h2|u(iii)|).
On an arbitrary non-uniform mesh, we have
|LNcui−(Lu)(xi)|≤C(ε(hi+hi+1)|u(iii)|+(hi+hi+1)|u(ii)|).
Here, one can notice that order of truncation error is reduced to one only if the central difference operator is employed on arbitrary non-uniform mesh instead of uniform mesh. Moreover, We have the following truncation error bounds corresponding to the midpoint upwind difference operator, which are valid for both uniform and non-uniform mesh:
|LNmpui−(Lu)(xi−1/2)|≤{C(ε(hi+hi+1)|u(iii)|+h2i+1(|u(iii)|+|u(ii)|+|ui|)),if a(x)>0,C(ε(hi+hi+1)|u(iii)|+h2i(|u(iii)|+|u(ii)|+|ui|)),if a(x)<0.
Note that the order of truncation error is higher by one in the convection term for midpoint upwind operator than the centered difference operator on a non-uniform mesh. This is the reason to apply midpoint upwind scheme at the transition points and of proposed mesh.
Further the solution of the discrete problem can be decomposed in an analogous manner as that of the continuous solution into the following sum
U=V+W, (4.3a) where, LNV=f,V(−1)=v(−1),V(1)=v(1), (4.3b) LNW=0,W(−1)=w(−1),W(1)=w(1). (4.3c)
Therefore, the error can be written in the form
U−u=(V−v)+(W−w),
so the errors in the smooth and singular components of the solution can be estimated separately.
###### Lemma 4.3.
(Error in smooth component) Assume that satsifies the assumption (4.1). Then the regular component of the error satisfies the following error bound
|(V−v)(xi)|≤{CN−2,∀i=0,1,…,N/4−1,3N/4+1,…,N,CN−1(ε+N−1)∀i=N/4,N/4+1,…,3N/4.
Proof. Using the usual truncation error estimates given above and bounds for the smooth component given in Theorem (2.3), we have
|LN(V−v)(xi)| ≤{CN−2(ε|v(iv)|+|v(iii)|),∀i=0,1,…,N/4−1,3N/4+1,…,N,CN−1(ε|v(iii)|+N−1(|v(iii)|+|v(ii)|+|vi|)),∀i=N/4,N/4+1,…,3N/4. ≤{CN−2,∀i=0,1,…,N/4−1,3N/4+1,…,N,CN−1(ε+N−1),∀i=N/4,N/4+1,…,3N/4,
and applying Lemma 4.2, we obtain the required result.
Since and , we consider both the region and individually to get the error estimates for the layer component . Therefore, we consider the following barrier functions for a positive constant :
ΦLi=⎧⎪⎨⎪⎩∏ij=1(1+γhjε)−1,i=1,…N/2,1,i=0.ΦRi=⎧⎪⎨⎪⎩∏Nj=i+1(1+γhjε)−1,i=N/2,…N−1,1,i=N. (4.4)
First we prove the following technical result.
###### Lemma 4.4.
If , the barrier functions satsisfy the inequalities
LNΦLi≤0,∀i=1,2,…,N/2,LNΦRi≤0,∀i=N/2,…,N−1.
Proof. We begin with the left hand barrier function and analyze each of the different discretizations used in the definition of the operator . First, in the case of midpoint upwind operator with , we have . Using the properties, and , and with the condition , one can easily observe that
LNmpΦLi =(γ2εhi+1ˆhi−ai+1/2γε−bi+12)ΦLi+1−bi2ΦLi =(2γ2ε(hi+12ˆhi−1)+(2γ2ε−ai+1/2γε−bi+12)−bi2(1+γhi+1ε))ΦLi+1≤0.
In the case of central difference operator with , we have
LNcΦLi =(2γ2ε(hi+12ˆhi−1)+(2γ2ε−aiγεhi+12ˆhi))ΦLi+1−(aiγεhi2ˆhi+bi)ΦLi≤0.
Similarly, applying the midpoint upwind operator for the case , we have
LNmpΦRi=(2γ2ε(hi2ˆhi−1)+(2γ2ε+ai−1/2γε−bi−12)−bi2(1+γhiε))ΦLi−1≤0.
In the same manner if we use central difference operator with , we also get It completes the proof.
###### Lemma 4.5.
The barrier functions and and layer component satisfy
|Wi|≤CΦLi,∀i=0,1,…N/2,|Wi|≤CΦRi,∀i=N/2,…,N.
Moreover, following bounds are valid for the layer component in no layer region
|Wi|≤CN−2,∀i=N/4,…,3N/4.
Proof. Construct the barrier functions . By Lemma 4.4, we have . Now using the discrete minimum principle we obtain the requred bound. Furthermore, to obtain the bound for in no layer region , we have for :
ΦLi≤ΦLN4 =N/4∏j=1(1+γhjε)−1=(1+γhε)−N/4=(1+4γτεN)−N/4=(1+4γτ0N−1lnN)−N/4 =(1+8N−1lnN)−N/4=((1+8N−1lnN)−N/8)2≤CN−2,
for the choice of . Here, we have used the inequality with to prove . Using similar argument for barrier function , we obtain desired bounds for in the domain .
###### Lemma 4.6.
(Error in singular component) Assume that satsifies the assumption (4.1) and . Then the singular component of the error satisfies the following error estimates
|(W−w)(xi)|≤{CN−2(lnN)2,∀i=0,1,…,N/4−1,3N/4+1,…,N,CN−2,∀i=N/4,N/4+1,…,3N/4.
Proof. We split our discussion into the two cases of boundary layer region and no boundary layer region to analyze the singular component of the error. Since , it is sufficient to consider only the subinterval and using same argument one can get similar estimate for the subinterval . Both and are small in , therefore we will use triangle inequlaity, Theorem 2.3, Lemma 4.5 instead of the usual truncaton error argument, to get the required error bounds on layer component in . For , using triangle inequality, we have
|(W−w)(xi)| ≤|W(xi)|+|w(xi)| ≤Ci∏j=1(1+γhjε)−1+Cexp(−α(1+xi)ε) ≤Ci∏j=1(1+γhjε)−1(since e−α(1+xi)/ε≤ΦLi) ≤CN−2(Using Lemma~{}???). (4.5)
Proceeding in a similar manner in subinterval , one can prove
|(W−w)(xi)|≤CN−2,∀i=N/2,…,3N/4. (4.6)
We now consider the boundary layer region to estimate the singular component of the error. In this case, we obtain the following singular component of the local truncation error estimates for :
|LN(W−w)(xi)| ≤Ch2(ε|w(iv)|+|w(iii)|) =16CN−2τ2(ε|w(iv)|+|w(iii)|) ≤CN−2ε2(lnN)2(ε−3exp(−α(1+xi)ε)) =C(N−2(lnN)2εexp(−α(1+xi)ε)) ≤C(N−2(lnN)2εΦLi). (4.7)
From the Eq. (4), , also we have . Therefore, if we choose
Ψ±(xi)=CN−2(1+(lnN)2ΦLi)±(W−w)(xi),∀i=0,1,… | 2022-06-26 00:00:32 | {"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": 0.807281494140625, "perplexity": 592.1121480389241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036176.7/warc/CC-MAIN-20220625220543-20220626010543-00724.warc.gz"} |
https://esp.mit.edu/teach/teachers/ddhyy/bio.html | ESP Biography
DAVID YANG, ESP Teacher
Major: 18
College/Employer: MIT
Year of Graduation: 2018
Not Available.
Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
M12701: Neuron Models in Splash 2018 (Nov. 17 - 18, 2018)
BRAINS (now with math!)
M12711: Beyond P vs NP: Hardness of Approximation in Splash 2018 (Nov. 17 - 18, 2018)
This summer, I want to go to each of the million biggest cities in the world. I want to save money though, so I need to know in what order I should go to these cities to minimize my total airplane ticket cost. Please help! Unfortunately, if you believe that the computational complexity classes P and NP are different, I am out of luck. There is no way I could possibly compute the best path in time. But luckily, I have some room in my budget - I'm willing to settle for any route whose cost is within 50% of the price of the best route. Can I quickly find a decent plan then? What if I need within 25%? Such is the domain of the PCP theorem, the most important theorem in computational complexity proved in the last 30 years. We will talk about approximate solutions to hard (NP-complete) problems, and our best ways of showing that they don't exist.
S12028: The Rite of Righting Your Writing in HSSP Spring 2018 (Feb. 24, 2018)
This class will focus on the development of each individuals students' writing style. Specifically, we will work on making our writing pieces clear, concise and effective at communicating their intended meaning(s). To facilitate the development of our own writing, we will read and analyze essays and short stories of notable authors. We will practice editing our own writing and the writing of other students. Finally, we will cover technical aspects of writing including formatting, annotations and citation styles. This class will be writing intensive, students should expect to write at least one double-spaced page and read one short piece in preparation for each meeting.
M12113: Three Dimensional Geometry in HSSP Spring 2018 (Feb. 24, 2018)
WHAT IS THE SHAPE OF THE UNIVERSE? IN 1982, MATHEMATICIAN WILLIAM PAUL THURSTON REALIZED THAT ALL THREE DIMENSIONAL SHAPES CAN BE BUILT OUT OF JUST 8 BASIC GEOMETRIES. WHAT CAN THIS DO FOR US? IN 2010, A MAJOR FURTHER BREAKTHROUGH HAPPENED. FASHION DESIGNER DAI FUJIWARA DISCOVERED THAT HE COULD MAKE CLOTHES OUT OF THESE GEOMETRIES, FINALLY GIVING US ACCESS TO THE TRUE NATURE OF THE UNIVERSE. WHERE CAN I LEARN MORE? WWW.AMS.ORG/PUBLICOUTREACH/AMS-NEWS/RELEASES/THURSTON-MIYAKE AND ALSO THIS CLASS
S10027: Hydrogen in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of hydrogen!
S10029: Helium in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of helium!
S10031: Lithium in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of lithium!
S10032: Beryllium in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of beryllium!
S10034: Boron in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of boron!
S10035: Carbon in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of carbon!
S10036: Nitrogen in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of nitrogen!
S10037: Oxygen in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of oxygen!
S10038: Fluorine in Splash 2015 (Nov. 21 - 22, 2015)
Learn about the chemistry of fluorine!
M10042: Points on a Plane in Splash 2015 (Nov. 21 - 22, 2015)
A Mobius transformation of the complex plane sends z to $$\frac{az+b}{cz+d}.$$ Any collection of three points can be sent to any other collection of three points by a Mobius transformation. Two collections of four points can be sent to each other only if they have the same cross ratio. What happens with more points?
M10043: Division Algebras and Topology in Splash 2015 (Nov. 21 - 22, 2015)
There are the real numbers, the complex numbers, the quaternions, and finally, the octonions. All these systems have multiplication and inverses. We'll show that they're the only such systems using topology.
M10046: Division Algebras and Symmetry in Splash 2015 (Nov. 21 - 22, 2015)
R C Q O Guess what the above letters mean. Now can you guess what these mean? E6,E7,E8 F4, G2 Well, we got bored of rotations, so we're going to use division algebras to give us new weird kinds of symmetries. Anyways, can you guess the meaning of this square? A1 A2 C3 F4 A2 A2xA2 A5 E6 C3 A5 D6 E7 F4 E6 E7 E8
M10047: Symmetry and Topology in Splash 2015 (Nov. 21 - 22, 2015)
Three dimensions. Rotations. A sphere. Ten dimensions. Rotations. Two OGres. We are going to cut the OGres into pieces. By doing so, we will get a good understanding of the most symmetrical kind of geometry, the geometry of OGres. Maybe even affine OGres. AKA: "What do really symmetrical high-dimensional objects look like?"
M10048: An action of the W-algebra on the equivariant intersection cohomology of moduli spaces of instantons in Splash 2015 (Nov. 21 - 22, 2015)
The goal of this class is to expose students to a topic on the cutting edge of algebra. String theorists predict the existence of an 6-dimensional physical theory called Theory X (that's actually the name.) It turns out that Theory X leads to lots of interesting mathematics (very relevant here is the fact that 6=2+4.) Come learn about it.
M7555: Quivers in Splash! 2013 (Nov. 23 - 24, 2013)
A quiver is one of the most accessible objects which have an interesting "representation theory". A quiver is just a graph where each edge has a direction. Imagine that you have a vector space for each vertex (if you don't know what a vector space is, you'll be taught) and a map for each edge. What happens????
M7556: Counting Curves and Quantum Cohomology in Splash! 2013 (Nov. 23 - 24, 2013)
There is 1 line through 2 points. There is 1 conic through 5 points. There is 1 conic tangent to 5 lines. Boring, huh? How many conics are tangent to 5 conics? (3264.) If you have a degree three polynomial in three dimensional space and you look at the surface where it's zero, how many lines lie on this surface? (27.) We will solve these problems and then explain how this is related to something called "quantum cohomology."
M7557: Quadratic Forms in Splash! 2013 (Nov. 23 - 24, 2013)
A quadratic form is just a degree 2 polynomial (in more than one variable). What interesting things can you say about these? Well... | 2021-06-23 09:03:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.40549129247665405, "perplexity": 1393.831749938042}, "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-2021-25/segments/1623488536512.90/warc/CC-MAIN-20210623073050-20210623103050-00241.warc.gz"} |
https://physics.stackexchange.com/questions/384008/why-is-k-taken-as-1-in-the-derivation-of-f-kma | # Why is $k$ taken as 1 in the derivation of $F=kma$? [duplicate]
This question already has an answer here:
In the derivation of F=ma, when we reach the point F=kma, we take k=1. Why can't we take 'k' as some other value?
## marked as duplicate by John Rennie, stafusa, Chris♦, valerio, Qmechanic♦Feb 3 '18 at 10:37
We make use of S.I. units. The statement $F=ma$ defines the newton (N) as the force required to accelerate a mass of $1kg$ by $1ms^{-2}$ in the direction of the force. This means that $k=1$. | 2019-11-13 14:32:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8338384032249451, "perplexity": 434.0682129650863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667262.54/warc/CC-MAIN-20191113140725-20191113164725-00033.warc.gz"} |
https://www.nag.com/numeric/nl/nagdoc_27cpp/clhtml/f12/f12acc.html | # NAG CL Interfacef12acc (real_proc)
Note: this function uses optional parameters to define choices in the problem specification. If you wish to use default settings for all of the optional parameters, then the option setting function f12adc need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adc for a detailed description of the specification of the optional parameters.
## 1Purpose
f12acc is a post-processing function that must be called following a final exit from f12abc. These are part of a suite of functions for the solution of real sparse eigensystems. The suite also includes f12aac, f12adc and f12aec.
## 2Specification
#include
void f12acc (Integer *nconv, double dr[], double di[], double z[], double sigmar, double sigmai, const double resid[], double v[], double comm[], Integer icomm[], NagError *fail)
The function may be called by the names: f12acc, nag_sparseig_real_proc or nag_real_sparse_eigensystem_sol.
## 3Description
The suite of functions is designed to calculate some of the eigenvalues, $\lambda$, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.
Following a call to f12abc, f12acc returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real nonsymmetric matrices. There is negligible additional cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
f12acc is based on the function dneupd from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12acc, is a post-processing function that must be called following a successful final exit from f12abc. f12acc uses data returned from f12abc and options, set either by default or explicitly by calling f12adc, to return the converged approximations to selected eigenvalues and (optionally):
• the corresponding approximate eigenvectors;
• an orthonormal basis for the associated approximate invariant subspace;
• both.
## 4References
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
## 5Arguments
1: $\mathbf{nconv}$Integer * Output
On exit: the number of converged eigenvalues as found by f12abc.
2: $\mathbf{dr}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array dr must be at least ${\mathbf{nev}}+1$ (see f12aac).
On exit: the first nconv locations of the array dr contain the real parts of the converged approximate eigenvalues.
3: $\mathbf{di}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array di must be at least ${\mathbf{nev}}+1$ (see f12aac).
On exit: the first nconv locations of the array di contain the imaginary parts of the converged approximate eigenvalues.
4: $\mathbf{z}\left[{\mathbf{n}}×\left({\mathbf{nev}}+1\right)\right]$double Output
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12adc) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in dr and di. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive array segments. The first segment holds the real part of the eigenvector and the second holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.
For example, the first eigenvector has real parts stored in locations ${\mathbf{z}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and imaginary parts stored in ${\mathbf{z}}\left[\mathit{i}-1\right]$, for $\mathit{i}={\mathbf{n}}+1,2{\mathbf{n}}$.
5: $\mathbf{sigmar}$double Input
On entry: if one of the ${\mathbf{Shifted Inverse Real}}$ modes have been selected then sigmar contains the real part of the shift used; otherwise sigmar is not referenced.
6: $\mathbf{sigmai}$double Input
On entry: if one of the ${\mathbf{Shifted Inverse Real}}$ modes have been selected then sigmai contains the imaginary part of the shift used; otherwise sigmai is not referenced.
7: $\mathbf{resid}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array resid must be at least ${\mathbf{n}}$ (see f12aac).
On entry: must not be modified following a call to f12abc since it contains data required by f12acc.
8: $\mathbf{v}\left[{\mathbf{n}}×{\mathbf{ncv}}\right]$double Input/Output
The $\mathit{i}$th element of the $\mathit{j}$th basis vector is stored in location ${\mathbf{v}}\left[{\mathbf{n}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{ncv}}$.
On entry: the ncv sections of v, of length $n$, contain the Arnoldi basis vectors for $\mathrm{OP}$ as constructed by f12abc.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ has been set, or the option ${\mathbf{Vectors}}=\mathrm{RITZ}$ has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.
9: $\mathbf{comm}\left[\mathit{dim}\right]$double Communication Array
Note: the actual argument supplied must be the array comm supplied to the initialization routine f12aac.
On initial entry: must remain unchanged from the prior call to f12abc.
On exit: contains data on the current state of the solution.
10: $\mathbf{icomm}\left[\mathit{dim}\right]$Integer Communication Array
Note: the actual argument supplied must be the array icomm supplied to the initialization routine f12aac.
On initial entry: must remain unchanged from the prior call to f12abc.
On exit: contains data on the current state of the solution.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INITIALIZATION
Either the solver function has not been called prior to the call of this function or a communication array has become corrupted.
NE_INTERNAL_EIGVEC_FAIL
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_OPTION
On entry, ${\mathbf{Vectors}}=\mathrm{SELECT}$, but this is not yet implemented.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_RITZ_COUNT
Got a different count of the number of converged Ritz values than the value passed to it through the argument icomm: number counted $=〈\mathit{\text{value}}〉$, number expected $=〈\mathit{\text{value}}〉$. This usually indicates that a communication array has been altered or has become corrupted between calls to f12abc and f12acc.
NE_SCHUR_EIG_FAIL
During calculation of a real Schur form, there was a failure to compute $〈\mathit{\text{value}}〉$ eigenvalues in a total of $〈\mathit{\text{value}}〉$ iterations.
NE_SCHUR_REORDER
The computed Schur form could not be reordered by an internal call. This function returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=〈\mathit{\text{value}}〉$. Please contact NAG.
NE_ZERO_EIGS_FOUND
The number of eigenvalues found to sufficient accuracy, as communicated through the argument icomm, is zero. You should experiment with different values of nev and ncv, or select a different computational mode or increase the maximum number of iterations prior to calling f12abc.
## 7Accuracy
The relative accuracy of a Ritz value, $\lambda$, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}×\left|\lambda \right|$. The default ${\mathbf{Tolerance}}$ used is the machine precision given by X02AJC.
## 8Parallelism and Performance
f12acc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
None.
## 10Example
This example solves $Ax=\lambda Bx$ in regular-invert mode, where $A$ and $B$ are obtained from the standard central difference discretization of the one-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.
### 10.1Program Text
Program Text (f12acce.c)
### 10.2Program Data
Program Data (f12acce.d)
### 10.3Program Results
Program Results (f12acce.r) | 2021-06-23 21:14:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 54, "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": 0.8456243276596069, "perplexity": 1518.1887714720492}, "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/1623488540235.72/warc/CC-MAIN-20210623195636-20210623225636-00264.warc.gz"} |
https://blog.mbedded.ninja/electronics/teardowns/belkin-wemo-teardown/ | TEARDOWNS
# Belkin WeMo Teardown
## Overview
The Belkin WeMo is a home automation product that works as a smartphone controlled power interrupter for standard electrical outlets.
Following are some brief stats about the product:
Model Num: F7C027au WeMo ID WeMo.781 Serial Number 221307K11000781 Driver MAC EC1A59799F61 Input 240V, 10A Output 240V, 10A
The screws are triangle (anti-tamper), 2.3mm. 2 of the four are hidden underneath the back sticker, which has to be cut before you can open it.
There are two PCBs, with a 3-wire connector between them. One can bet that the three wires are logic power, ground, and relay on/off control. 1 contains all the power circuitry, one the low-voltage logic.
## Power PCB
They use a mechanical relay to switch the mains to the outlet. The relay is the HF3FA (005-HTF, SM47C), and is rated for 250VAC at 10A. As you would expect, it switches the live wire, ground and neutral are connected permanetly to the output.
The low-voltage rails for the logic board look to be generated by a transformer, diode (only a single one, not a bridge!) and capacitor arrangement from the 240VAC. The transformer has the number OEM-EPC130305001 GP on it.
For protection, there looks to be a thermal PTC fuse (VR1), with something that looks like an LED taped to the side of it. This kind of looks like a post-PCB design hack, maybe they added the ability for the LED to go on if the PTC fuse triggers?
There is also an inductory looking thingy (L3), which I think could be EMI filtering for the logic level power rail (it is right next to the connector).
## Logic PCB
The main chip on this board is the Ralink RT5350F (other numbers on the IC are TPMX249409, 1246PTA1). This is a combined wireless front-end (1T1R 2.4G CMOS RF), stack (802.11n 1T1R MAC/BBP) and embedded CPU. I had a quick look on on AliExpress.com, and they seem to sell for about US$2.90, in quantities of 5 (Jan 2014). This is cheap (really cheap)! DigiKey nor any other non-Chinese supplier (try www.findchips.com) did not seem to stock these. Notice in the above picture that there is a PCB antenna in the upper-right corner of the PCB (when looking from the front of an assembled device, so upper-left in the image). On the other side of the board are two more ICs, the Winbond W9825G6JH-61 1302P 62267700003 (166MHz 256MB SDRAM TSOP 54-pin 0.80mm pitch IC), which was available on DigiKey for US$3.23 (1) and US$2.33 (1000). The other IC is a Macronix (MXIC) MX25L12835EMI-10G, which is 128Mbit, 104MHz SPI flash memory. This sells for US$3.35 (1) and US2.42 (1000) on DigiKey.
There is also a miniature switch on the reverse-side of the PCB (inaccessable to the user).
Another tear down for the Belkin WeMo exists at OpenWrt. | 2019-10-14 17:55:52 | {"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": 0.3003840446472168, "perplexity": 7478.905720461527}, "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/1570986654086.1/warc/CC-MAIN-20191014173924-20191014201424-00204.warc.gz"} |
https://www.prepanywhere.com/prep/textbooks/10-principles-of-mathematics-nelson/chapters/chapter-1-system-of-linear-equations/materials/1-5-equivalent-linear-systems/videos/q19 | 19. Q19
Save videos to My Cheatsheet for later, for easy studying.
Video Solution
Q1
Q2
Q3
L1
L2
L3
Similar Question 1
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle x+4 y=8\\y+2 x=0 </code></p>
Similar Question 2
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle 2 y=3 x-1\\8 y-4=12 x </code></p>
Similar Question 3
<p> Write an equation that forms a system of equations with <code class='latex inline'> x+y=4 </code> , so that the system has</p><p>a) no solution</p><p>b) infinitely many solutions</p><p>c) one solution</p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Solve the following systems of equations, and explain the nature of each intersection:</p><p><code class='latex inline'>\displaystyle \begin{array}{lllll} &2x + y &= 3 \\ &2x + y &= 4 \\ \end{array} </code></p>
<p> Consider this system of linear equations:<br> <code class='latex inline'>\displaystyle \begin{array}{cccc} & 2x + y = 7\\ &8x + 4y = 10\\ \end{array} </code> </p><p><strong>a)</strong> Can you create an equivalent system that contains only one variable? </p><p><strong>b)</strong> What does your result for part a) suggest about the solution to the original system? </p><p><strong>c)</strong> What does your result for part a) suggest about the graphs of both lines? </p>
<p>Explain how you know that this system of equations has no solution.</p><p><code class='latex inline'>\displaystyle \begin{array}{lllll} &15 -6y = 9x\\ &3x + 2y = 8 \end{array} </code></p>
<p>For the first equation in the system of linear equations below, write an equivalent equation without denominators. Then solve the system.</p><p><code class='latex inline'>\displaystyle \left\{\begin{array}{l}\frac{x}{5}+\frac{y}{3}=6 \\ x-2 y=8\end{array}\right. </code></p>
<p>Use the equation <code class='latex inline'>3x + 4y= 2</code></p><p>Write another equation that will create a linear system with each number of solutions.</p><p>i) none</p><p>ii) one</p><p>iii) infinitely many</p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle x+4 y=8\\y+2 x=0 </code></p>
<p>Solve the following systems of equations, and explain the nature of each intersection:</p><p><code class='latex inline'>\displaystyle \begin{array}{lllll} &7x -3y &= 9 \\ &35x - 15y &= 45 \end{array} </code></p>
<p>a) Can you solve the linear system <code class='latex inline'>y=3x-2</code> and <code class='latex inline'>6x-2y-4=0</code>? Explain your reasoning.</p><p>b) Can you solve the linear system <code class='latex inline'>y=4x-3</code> and <code class='latex inline'>8x-2y+ 5 =0</code>?</p><p>Explain your reasoning.</p><p>c) Explain how you can tell, without solving, how many solutions a linear system has.</p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle 2 y-x-4=0\\3 x-6 y-12=0 </code></p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle \begin{aligned} & 3 c+d=4 \\ & 6 c+2 d=8 \end{aligned} </code></p>
<p>Solve each system of equations by elimination. Check each solution.</p><p><code class='latex inline'>\displaystyle 2 x+5 y=3 </code></p><p><code class='latex inline'>\displaystyle 4 x+10 y-6=0 </code></p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle x+y=2\\3 x=6-3 y </code></p>
<p> Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle 3 x-y=0\\6 x-2 y=3 </code></p>
<p> Write an equation that forms a system of equations with <code class='latex inline'> x+y=4 </code> , so that the system has</p><p>a) no solution</p><p>b) infinitely many solutions</p><p>c) one solution</p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle x+2 y-7=0\\3 x+6 y-14=0 </code></p>
<p> Write a system of equations that has the point <code class='latex inline'> (3,2) </code> as</p><p>a) the only solution</p><p>b) one of infinitely many solutions</p>
<p>Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle 2 y=3 x-1\\8 y-4=12 x </code></p>
<p>Solve each system of equations by elimination. Check each solution.</p><p><code class='latex inline'>\displaystyle 2 x-3 y=8 </code></p><p><code class='latex inline'>\displaystyle 4 x-6 y=10 </code></p>
<p>Without graphing, determine whether each system has one solution no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle 4 x-2 y=0\\2 x-y=3 </code></p>
How did you do?
Found an error or missing video? We'll update it within the hour! 👉
Save videos to My Cheatsheet for later, for easy studying. | 2021-09-21 10:36:36 | {"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": 0.9882668852806091, "perplexity": 8824.264590164803}, "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/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00224.warc.gz"} |
http://www.thefullwiki.org/Hydrology | # Hydrology: Wikis
Advertisements
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
### Did you know ...
More interesting facts on Hydrology
Include this on your site/blog:
# Encyclopedia
### From Wikipedia, the free encyclopedia
Water covers 70% of the Earth's surface.
Hydrology is the study of the movement, distribution, and quality of water throughout Earth, and thus addresses both the hydrologic cycle and water resources. A practitioner of hydrology is a hydrologist, working within the fields of either earth or environmental science, physical geography, geology or civil and environmental engineering.
Domains of hydrology include hydrometeorology, surface hydrology, hydrogeology, drainage basin management and water chemistry, where water plays the central role. Oceanography and meteorology are not included because water is only one of many important aspects.
Hydrological research is useful as it allows us to better understand the world in which we live, and also provides insight for environmental engineering, policy and planning.
The term hydrology is from Greek: ὕδωρ, hydōr, "water"; and λόγος, logos, "study".
## History of hydrology
Hydrology has been a subject of investigation and engineering for millennia. For example, about 4000 B.C. the Nile was dammed to improve agricultural productivity of previously barren lands. Mesopotamian towns were protected from flooding with high earthen walls. Aqueducts were built by the Greeks and Ancient Romans, while the History of China shows they built irrigation and flood control works. The ancient Sinhalese used hydrology to build complex irrigation Works in Sri Lanka, also known for invention of the Valve Pit which allowed construction of large reservoirs, anicuts and canals which still function.
Marcus Vitruvius, in the first century B.C., described a philosophical theory of the hydrologic cycle, in which precipitation falling in the mountains infiltrated the Earth's surface and led to streams and springs in the lowlands. With adoption of a more scientific approach, Leonardo da Vinci and Bernard Palissy independently reached an accurate representation of the hydrologic cycle. It was not until the 17th century that hydrologic variables began to be quantified.
Pioneers of the modern science of hydrology include Pierre Perrault, Edme Mariotte and Edmund Halley. By measuring rainfall, runoff, and drainage area, Perrault showed that rainfall was sufficient to account for flow of the Seine. Marriotte combined velocity and river cross-section measurements to obtain discharge, again in the Seine. Halley showed that the evaporation from the Mediterranean Sea was sufficient to account for the outflow of rivers flowing into the sea.
Advances in the 18th century included the Bernoulli piezometer and Bernoulli's equation, by Daniel Bernoulli, the Pitot tube. The 19th century saw development in groundwater hydrology, including Darcy's law, the Dupuit-Thiem well formula, and Hagen-Poiseuille's capillary flow equation.
Rational analyses began to replace empiricism in the 20th century, while governmental agencies began their own hydrological research programs. Of particular importance were Leroy Sherman's unit hydrograph, the infiltration theory of Robert E. Horton, and C.V. Theis's Aquifer test/equation describing well hydraulics.
Since the 1950s, hydrology has been approached with a more theoretical basis than in the past, facilitated by advances in the physical understanding of hydrological processes and by the advent of computers and especially Geographic Information Systems (GIS).
## Hydrologic cycle
The central theme of hydrology is that water moves throughout the Earth through different pathways and at different rates. The most vivid image of this is in the evaporation of water from the ocean, which forms clouds. These clouds drift over the land and produce rain. The rainwater flows into lakes, rivers, or aquifers. The water in lakes, rivers, and aquifers then either evaporates back to the atmosphere or eventually flows back to the ocean, completing a cycle.
## Branches of hydrology
Chemical hydrology is the study of the chemical characteristics of water.
Ecohydrology is the study of interactions between organisms and the hydrologic cycle.
Hydrogeology is the study of the presence and movement of ground water.
Hydroinformatics is the adaptation of information technology to hydrology and water resources applications.
Hydrometeorology is the study of the transfer of water and energy between land and water body surfaces and the lower atmosphere.
Isotope hydrology is the study of the isotopic signatures of water.
Surface hydrology is the study of hydrologic processes that operate at or near Earth's surface.
Advertisements
## Hydrologic measurements
Measurement is fundamental for assessing water resources and understanding the processes involved in the hydrologic cycle. Because the hydrologic cycle is so diverse, hydrologic measurement methods span many disciplines: including soils, oceanography, atmospheric science, geology, geophysics and limnology, to name a few. Here, hydrologic measurement methods are organized by hydrologic sub-disciplines. Each of these subdisciplines is addressed briefly with a practical discussion of the methods used to date and a bibliography of background information.
Quantifying groundwater flow and transport
• Aquifer characterization
• Flow direction
• Piezometer - groundwater pressure and, by inference, groundwater depth (see: aquifer test)
• Conductivity, storativity, transmisivity
• Geophysical methods
Quantifying surface water flow and transport
• Direct and indirect discharge measurements
Quantifying hydrologic exchange at the land-atmospheric boundary
• Precipitation
• Bulk rain events
• Disdrometer - precipitation characteristics
• Radar - cloud properties, rain rate estimation, hail and snow detection
• Rain gauge - rain and snowfall
• Satellite - rainy area identification, rain rate estimation, land-cover/land-use, soil moisture
• Sling psychrometer - humidity
• Snow, hail and ice
• Dew, mist and fog
• Evaporation
• Transpiration
• Natural ecosystems
• Agronomic ecosystems
• Momentum
• Heat flux
• Energy budgets
Uncertainty analyses
Remote sensing of hydrologic processes
• Land based sensors
• Airborne Sensors
• Satellite sensors
Water quality
• Sample collection
• In-situ methods
• Physical measurements (includes sediment concentration)
• Collection of samples to quantify Organic Compounds
• Collection of samples to quantify Inorganic Compounds
• Analysis of aqueous Organic Compounds
• Analysis of aqueous Inorganic Compounds
• Microbiological sampling and analysis
Integrating measurement and modeling
• Budget analyses
• Parameter estimation
• Scaling in time and space
• Data assimilation
• Quality control of data — see for example Double mass analysis
## Hydrologic prediction
Observations of hydrologic processes are used to make predictions of the future behaviour of hydrologic systems (water flow, water quality). One of the major current concerns in hydrologic research is the Prediction in Ungauged Basins (PUB), i.e. in basins where no or only very few data exist.
### Statistical hydrology
By analysing the statistical properties of hydrologic records, such as rainfall or river flow, hydrologists can estimate future hydrologic phenomena, assuming the characteristics of the processes remain unchanged.
These estimates are important for engineers and economists so that proper risk analysis can be performed to influence investment decisions in future infrastructure and to determine the yield reliability characteristics of water supply systems. Statistical information is utilised to formulate operating rules for large dams forming part of systems which include agricultural, industrial and residential demands.
See: return period.
### Hydrologic modeling
Hydrologic models are simplified, conceptual representations of a part of the hydrologic cycle. They are primarily used for hydrologic prediction and for understanding hydrologic processes. Two major types of hydrologic models can be distinguished:
• Models based on data. These models are black box systems, using mathematical and statistical concepts to link a certain input (for instance rainfall) to the model output (for instance runoff). Commonly used techniques are regression, transfer functions, and system identification. The simplest of these models may be linear models, but it is common to deploy non-linear components to represent some general aspects of a catchment's response without going deeply into the real physical processes involved. An example of such an aspect is the well-known behaviour that a catchment will respond much more quickly and strongly when it is already wet than when it is dry..
• Models based on process descriptions. These models try to represent the physical processes observed in the real world. Typically, such models contain representations of surface runoff, subsurface flow, evapotranspiration, and channel flow, but they can be far more complicated. These models are known as deterministic hydrology models. Deterministic hydrology models can be subdivided into single-event models and continuous simulation models.
Recent research in hydrologic modeling tries to have a more global approach to the understanding of the behaviour of hydrologic systems to make better predictions and to face the major challenges in water resources management.
## Hydrologic transport
See main article: Hydrologic transport model
Water movement is a significant means by which other material, such as soil or pollutants, are transported from place to place. Initial input to receiving waters may arise from a point source discharge or a line source or area source, such as surface runoff. Since the 1960s rather complex mathematical models have been developed, facilitated by the availability of high speed computers. The most common pollutant classes analyzed are nutrients, pesticides, total dissolved solids and sediment.
## Further reading
• Introduction to Hydrology, 4e. Viessman and Lewis, 1996. ISBN 0-673-99337-X
• Handbook of Hydrology. ISBN 0-07-039732-5
• Encyclopedia of Hydrological Sciences. ISBN 0-471-49103-9
• Hydrological Processes, ISSN: 1099-1085 (electronic) 0885-6087 (paper), John Wiley & Sons
• Journal of Hydroinformatics, ISSN: 1464-7141, IWA Publishing
• Hydrology Research (formerly Nordic Hydrology), ISSN: 0029-1277, IWA Publishing
• Journal of Hydrologic Engineering, ISSN: 0733-9496, ASCE Publication
• Hydrologic Analysis and Design. McCuen, Third Edition, 2005. ISBN 0-13-142424-6
# Study guide
Up to date as of January 14, 2010
### From Wikiversity
Rain near the village Lunde, The north of Funen, Denmark.
## Content summary
Hydrology is the study of the water cycle. Natural and man-made processes guide water in its varied forms to a variety of ends. Use as irrigation, drinking water and process water by man is very common.
## Learning Materials
...
### Hydrology Lessons
#### Diffusion
For a source contaminant concentration $\ C_0$ entering a flow of velocity $\vec U$ at a distance $\ x$ upstream from a point, the downstream concentration $\ C$ at that point is determined by the ratio...
$\frac{C}{C_0} = e^{\frac{x*\vec U}{D}}$
Where... $\ D$ is the local dispersion coefficient determined by $\ D = 0.067 * depth * V_f$
and where friction velocity is $V_f = \sqrt{g*depth*ChannelSlope}$
#### Velocity Distribution in an Open Channel (River)
Velocity distribution within a river follows a standard velocity profile for a confined space (pipe-flow)with the exception that the vertical distribution is truncated at the surface due to reduced friction with atmospheric gases versus the high friction against riverbed materials.
Boundary Layer thickness, also known as the displacement thickness, is defined by:
$\delta_d = \int_0^H \frac{1}{H} \left( 1-\frac{\vec u}{\vec U} \right) dz$
where $\vec U$ is the average velocity and $\vec u(x)$ is the velocity distribution in a channel of uniform depth $\ H$.
$\bar U = \frac{\int_A \left( \rho \vec V \cdot \hat n \right) dA}{\rho A}$
So in the case of this river, where the current is always normal to the cross-sectional area of the river (idealized)...
$\bar U = \frac{\int_x \int_y \left( \rho \vec V \right) dydx}{\rho A}$
## Assignments
• ...
• ...
### Study guide:
• Physical Properties of the Oceans
1. Wikipedia article:Temperature
2. Wikipedia article:Density
## References
Additional helpful readings include:
## Active participants
Please sign up!! Active participants in this Learning Group
• ...
# Simple English
covers 70% of the Earth's surface.]]
Hydrology (from Greek: Yδωρ, hudōr, "water"; and λόγος, logos, "study") is the study of the movement, distribution, and quality of water throughout the Earth. A person who studies hydrology is called a hydrologist.
Advertisements
Got something to say? Make a comment. Your name Your email address Message | 2018-12-09 19:04:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40942180156707764, "perplexity": 5657.256093121107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823009.19/warc/CC-MAIN-20181209185547-20181209211547-00052.warc.gz"} |
https://www.physicsforums.com/threads/relation-between-entropy-and-temperature.954206/ | # Relation Between Entropy and Temperature
## Homework Statement
[/B]
Strap in, this one's kind of long. (This problem is from 'Six Ideas That Shaped Physics, Unit T' by Thomas A Moore, 2nd edition. Problem T6R2.)
Imagine that aliens deliver into your hands two identical objects made of substances whose multiplicities increase linearly with thermal energy, something like
$$\Omega = \frac{aNU}{\epsilon}$$
where ε is some energy unit and a is some constant. Answer the following questions about these objects.
a) Do they have a well-defined temperature? If so, how does this temperature depend on the objects' thermal energy?
b) If these objects are placed in thermal contact, will energy spontaneously flow from hot to cold? Will the objects eventually come into equilibrium at a certain common temperature? (Hint: I suggest drawing a graph of ΩAB versus macropartition. This will also help with the next part.)
c) How will the size of the random fluctuations in the energies of these objects compare to those for two normal objects placed in thermal contact?
## Homework Equations
[/B]
$$\Omega = \frac{aNU}{\epsilon}$$ From the problem statement
$$\frac{1}{T} = \frac{\partial{S}}{\partial{U}}$$ (definition of temperature)
$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$ (q = U/ε) (definition of multiplicity given in text)
3. The Attempt at a Solution
Using the equation for the definition of temperature gives me T = U/k. (I can reproduce this if needed but don't want to clutter things too much. I want to focus on the information in the table.) In a 'normal' object, the temperature and thermal energy are related by T = U/(3Nk). (The book uses 'Einstein solids' as its model for these things. So that's what I'm comparing these 'alien' objects to.)
Let's just say N = 1 for simplicity from here on. So this just tells me that the temperature increase for these 'alien' objects, in response to a given increase of thermal energy, is different from that of an Einstein solid by a factor of 1/3. So fundamentally, these objects should behave much like normal ones. I.e. they would have well-defined temperatures, they would come to equilibrium by exchanging heat energy, energy would spontaneously flow from the warmer one to the colder one, etc.
The author hints that I should make a graph or table of macropartition vs ΩAB. Unfortunately I have no idea how to present a table in this post, so that will make things difficult. But the basic idea is that if I assume each object consists of 1 atom (N = 1, again for simplicity) and there are 6 'atoms' of energy ε, so that U = 6ε, then for each macro-distribution of those 'atoms' (in the left column), I would list the multiplicity ΩAB of the macrostate as a whole. And this will be the product of the individual multiplicities of objects A and B, as computed from the formula for multiplicity given in the problem statement. So for example, for the distribution 1:5 (where 1 bit of energy is in object A and 5 are in object B), I would compute the multiplicity of A, the multiplicity of B, then multiply them to get the multiplicity of this macrostate.
(Hopefully that's clear. I wish I could just make a table, but I don't know how.)
But right away I run into a problem. Consider the macropartition 0:6, where object A has none of the energy atoms and object B has all 6 of them. If I use the expression for multiplicity given in the problem, then ΩA is zero, thus making the multiplicity of the whole macropartition zero. This would mean there is no way for one of the objects to have all the energy and the other one to have none. But I don't see any reason why that should be the case here any more than for an Einstein solid.
So I figure maybe I'm using the wrong multiplicity equation, but if I use the other one (the one given in the text, for an Einstein solid), then I just get the same table I would for an Einstein solid, which doesn't seem right either.
Is my problem maybe that I'm only using 1 atom, and the difference between the two kinds of object ('alien' vs 'normal') only comes out when using multiple atoms? But that seems wrong, because even with only one atom, there is still the factor of 1/3 that I mentioned above.
So I guess my question is:
Why is it that when I calculate the temperature/thermal energy relationship, I'm getting a straightforward answer that seems physically plausible (U = kT, versus U = 3kT for a 'normal' solid), but when I try to calculate the multiplicity of the 0:6 macropartition, I get an answer that makes no physical sense? Am I just using the wrong multiplicity equation? Or is this just to show that the given multiplicity equation makes no sense physically? Or something else?
UPDATE: I'm now seeing that I need to use the multiplicity equation given in the text, for both objects, in each macropartition. But now my trouble is that this simply reproduces the macropartition vs ΩAB table of an Einstein solid, in which case the problem becomes profoundly uninteresting. But I presume there's something I'm supposed to learn here. I don't see how the multiplicity expression given in the problem statement
$$\Omega = \frac{aNU}{\epsilon}$$
comes into play viz a viz the table. I see that I can derive an expression for temperature from it. But I don't see how temperature and the table are related, but the author seems to want us to spot that connection.
Last edited:
Related Calculus and Beyond Homework Help News on Phys.org
stevendaryl
Staff Emeritus
## Homework Equations
[/B]
$$\Omega = \frac{aNU}{\epsilon}$$ From the problem statement
$$\frac{1}{T} = \frac{\partial{S}}{\partial{U}}$$ (definition of temperature)
$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$ (q = U/ε) (definition of multiplicity given in text)
For this homework assignment, they are telling you NOT to use the second formula for ##\Omega##, but instead to use the formula ##\Omega = \frac{aNU}{\epsilon}##.
The relationship that you're missing is this: ##S = k log(\Omega)##. (I'm assuming that your ##\Omega## means the density of states for a given value of ##U##)
For this homework assignment, they are telling you NOT to use the second formula for ##\Omega##, but instead to use the formula ##\Omega = \frac{aNU}{\epsilon}##.
The relationship that you're missing is this: ##S = k log(\Omega)##. (I'm assuming that your ##\Omega## means the density of states for a given value of ##U##)
So using ##S = kln(\Omega)## seems to still leave me in a quandary, because for the macropartition ##0:6## (where object A has no bits of energy, object B has all 6), ##\Omega_{A} = 0## and therefore ##\Omega_{AB} = \Omega_{A} \Omega_{B} = 0##, which gives me an undefined/unphysical entropy, since there is no ##ln(0)##.
As to which expression I use for multiplicity, my understanding was that the expression
$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$
is essentially just a combinatorics result. In fact, we derived it as part of the homework, and that's exactly what it was: how many ways to distribute q bits of energy amongst 3N oscillators of an Einstein solid. So I was (am) confused by this problem, because I don't see the point of redefining multiplicity and then seeing what happens, since multiplicity just is what it is. It is math, not physics (though it does give rise to physics). I mean, it makes no sense, purely from a combinatorics standpoint, that there is not exactly 1 way of distributing 0 bits of energy among the oscillators of object A. Yet that's what the problem's definition of 'multiplicity' gives me. That just seems absurd and not worth thinking about, but at the same time I trust the textbook author to give me meaningful problems, so I conclude I must be missing something.
Anyway, thus far I have two conclusions about these 'alien objects':
1) They are normal in the sense that they have well-defined temperatures, etc. (which I suspect is incorrect, but if I graph S vs U, I essentially just get an ln function with decreasing slope, i.e. decreasing temperature, as for a normal object
2) The system of the two objects in contact has no entropy for the macropartitions where one of them has zero energy, which is different from Einstein solids. | 2021-03-06 05:30:22 | {"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": 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": 0.8308183550834656, "perplexity": 533.7159058168287}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374391.90/warc/CC-MAIN-20210306035529-20210306065529-00637.warc.gz"} |
https://mulloverthing.com/is-gradient-descent-calculus/ | MullOverThing
Useful tips for everyday
# Is gradient descent calculus?
## Is gradient descent calculus?
Gradient Descent is an optimization algorithm for finding a local minimum of a differentiable function. You start by defining the initial parameter’s values and from there gradient descent uses calculus to iteratively adjust the values so they minimize the given cost-function.
What is meant by calculus of variation?
calculus of variations, branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible.
What is calculus of variations and what are its applications?
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Many important problems involve functions of several variables.
### Who found the calculus of variations?
The calculus of variations goes back to the 17th century and Isaac Newton. Newton de- veloped the theory to solve the minimal resis- tance problem and later the brachistochrome problem.
What is gradient descent example?
Gradient descent will find different ones depending on our initial guess and our step size. If we choose x 0 = 6 x_0 = 6 x0=6x, start subscript, 0, end subscript, equals, 6 and α = 0.2 \alpha = 0.2 α=0. 2alpha, equals, 0, point, 2, for example, gradient descent moves as shown in the graph below.
What is the formula of gradient descent?
The equation of this straight line would be Y = mX + b where m is the slope and b is its intercept on the Y-axis.
## Who invented calculus?
Isaac Newton
Researchers in England may have finally settled the centuries-old debate over who gets credit for the creation of calculus. For years, English scientist Isaac Newton and German philosopher Gottfried Leibniz both claimed credit for inventing the mathematical system sometime around the end of the seventeenth century.
Is calculus of variations functional analysis?
Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis.
How do you use gradient descent?
Gradient descent is an iterative optimization algorithm for finding the local minimum of a function. To find the local minimum of a function using gradient descent, we must take steps proportional to the negative of the gradient (move away from the gradient) of the function at the current point.
### Who is the true father of calculus?
Gottfried Leibniz
The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways.
Who is the real father of calculus?
Sir Isaac Newton
Sir Isaac Newton was a mathematician and scientist, and he was the first person who is credited with developing calculus. It is is an incremental development, as many other mathematicians had part of the idea.
What is the difference between calculus and functions?
The word function in calculus refers to something like f(x)=x2+2×3 or f(x)=sin(x) etc…. In linear algebra, the word function is used like- A linear transformation is a function from V→W. And the functions of calculus like f(x)=x2+2×3 or f(x)=sin(x) etc.
## How to calculate gradient in gradient descent?
How to understand Gradient Descent algorithm Initialize the weights (a & b) with random values and calculate Error (SSE) Calculate the gradient i.e. change in SSE when the weights (a & b) are changed by a very small value from their original randomly initialized value. Adjust the weights with the gradients to reach the optimal values where SSE is minimized
Why do we use gradient descent in linear regression?
The main reason why gradient descent is used for linear regression is the computational complexity: it’s computationally cheaper (faster) to find the solution using the gradient descent in some cases.
What are the weaknesses of gradient descent?
Weaknesses of Gradient Descent: The learning rate can affect which minimum you reach and how quickly you reach it. If learning rate is too high (misses the minima) or too low (time consuming) Can…
### What is steepest descent algorithm?
Steepest Descent. The steepest descent algorithm is an old mathematical tool for numerically finding the minimum value of a function, based on the gradient of that function. | 2022-10-01 04:57:04 | {"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": 0.8013766407966614, "perplexity": 687.6958468113065}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335530.56/warc/CC-MAIN-20221001035148-20221001065148-00158.warc.gz"} |
https://civilnoteppt.com/switch-angle-in-railway/ | # Switch Angle in Railway
22nd December 2019
## Switch Angle in Railway
Switch angle or angle of switch divergence is defined as the angle formed between the gauge lines of the stock rail and tongue rail(Switch rails).
If the switch angle is more, the entry of the train will not be smooth and consequently, the speed of trains will have to be reduced. On the other hand, a small switch angle will increase the overall length of the turnout.
Hence, small switch angles are provided in case of fast-moving trains. But, in case of slow-moving train or station yards, greater switch angle is recommended.
Switch angle depends upon the length of tongue rails and heel divergence, which is given below:
Case:1 When the thickness of tongue rails at toe = 0
Let,
• d = Heel divergence
• D = Length of tongue rail
• $\theta$ = Switch Angle
Now, from the above fig.
$\sin \theta$ = (heel divergence / Length of tongue rail) = $\frac{d}{D}$
Switch Angle = $\theta$ = $\sin ^{-1}\frac{d}{D}$
Case:2 When the thickness of tongue rails at toe = t
Let,
• t = Thickness of tongue rail at toe
• $D_{1}$ = Actual length of tongue rail
• $D_{2}$ = Theoritical length of tongue rail
• x = Difference between $D_{1}$ and $D_{2}$
From the above equation and fig, we can write
• Actual Length of tongue rails $D_{1}$= $\frac{ ( d - t ) }{ \sin \theta }$
• Theoretical Length of tongue rails $D_{2}$ = $\frac{d}{ \sin \theta }$
Now, x = ( $D_{2}$$D_{1}$ ) = $\frac{t}{ \sin \theta }$ | 2020-03-28 11:26:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 32, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7006489038467407, "perplexity": 4374.655803218947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370491857.4/warc/CC-MAIN-20200328104722-20200328134722-00308.warc.gz"} |
http://www.mif.vu.lt/~garunkstis/ | Ramūnas Garunkštis Professor at the Department of Probability Theory and Number Theory Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania
Research interests: Analytic Number Theory, Zeta Functions
### Papers
1. On the Lerch zeta-function, Lith. Math. J. 36 (1996), 337-346 (with A. Laurinčikas).
2. The universality theorem with weight for the Lerch zeta-function, in: New Trends in Probability and Statistics. V.4: Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996. Eds. A.Laurinčikas, E.Manstavičius and V.Stakenas, Vilnius: TEV, Utrecht: VSP, (1997), 59-67.
3. An explicit form of the limit distribution with weight for the Lerch zeta-function in the space of analytic functions, Lith. Math. J. 37 (1997), 230-242.
4. On one Hilbert's problem for the Lerch zeta-function, Publ. Inst. Math., 65 (79) (1999), 63-69 (with A. Laurinčikas).
5. On zeros of the Lerch zeta-function, in: Number Theory and Its Applications, S.Kanemitsu and K.Gyory (eds.), Kluwer Academic Publishers, (1999), 129-143 (with A. Laurinčikas). PDF
6. On zeros of the Lerch zeta-function.II, in: Probability Theory and Mathematical Statistics, Proceedings of the Seventh Vilnius Conf. 1998, B.Grigelionis et al. (Eds.), TEV/Vilnius, VSP/Utrecht, (1999), 267-276. PDF
7. On zeros of the Lerch zeta function. III, Scient. works of Lith. Math. Soc.: supl. to "Liet. Matem. Rink.", Vilnius: Technika, 1999, pp. 24-30. PDF
8. A note on the Riemann $\xi$-function, Liet. Matem. Rink., 40 (Special Issue) (2000), 18-20 (Lithuanian).
9. The Lerch zeta-function, Integral Transforms and Special Functions, 10 (2000), 211-226 (with A. Laurinčikas).
10. A note on the zeros of the Lerch zeta-function, Liet. Matem. Rink. 41 (Special Issue) (2001), 53-57 (Lithuanian).
11. Twists of Lerch zeta-functions, Liet. matem. rink. 41 (2001), 172-182 (with J. Steuding). PDF
12. On the zeros of the derivative of the Lerch zeta-function, Liet. Matem. Rink. 42 (Special Issue) (2002), 47-49 (Lithuanian).
13. On the zero distributions of Lerch zeta-functions, Analysis 22 (2002), 1-12 (with J. Steuding). PDF
14. On a positivity property of the Riemann $\xi$-function, Liet. matem. rink. 42 (2002), 179-184. PDF
15. On the universality of Estermann zeta-functions, Analysis 22 (2002), 285-296 (with A. Laurinčikas, R. Šleževičienė and J. Steuding). PDF
16. On some inequalities concerning $\pi (x)$, Exp. Math. 11 (2002), 297-301. PDF
17. The Lerch zeta-function, Kluwer Academic Publishers, 2002, 197 pp. (with A.Laurinčikas).
18. Do Lerch zeta-functions satisfy the Lindelof hypothesis?, in: Analytic and Probabilistic Methods in Number Theory, Proceedings of the Third Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 24-28 September 2001, (eds. A. Dubickas, A. Laurinčikas and E. Manstavičius), TEV, Vilnius, (2002), 61-74 (with J. Steuding). PDF
19. On the mean square of Lerch zeta-functions, Arch. Math. 80 (2003), 47-60 (with A. Laurinčikas and J. Steuding). PDF
20. On the Chebyshev function $\psi(x)$, Liet. matem. rink. 43 (2003), 487-496 = Lith. Math. J. 401-409.
21. The effective universality theorem for the Riemann zeta function, in: Proceedings of the session in analytic number theory and Diophantine equations, MPI-Bonn, January - June 2002, Ed. by D. R. Heath-Brown, B. Z. Moroz, Bonner mathematische Schriften, 360 (2003), 21 pp. PDF
22. An approximation of the Hurwitz zeta-function by a finite sum, Liet. Matem. Rink. 43 (Special Issue) (2003), 32-34 (Lithuanian).
23. On the Voronin's universality theorem for the Riemann zeta-function, Proceedings of Scientific Seminar of the Faculty of Physics and Mathematics, Šiauliai University 6 (2003), 29-33. PDF
24. An approximate functional equation for the Lerch zeta-function, Math. Notes 74 (2003), 469-476 (with A. Laurinčikas and J.Steuding). PDF
25. Approximation of the Lerch zeta-function, Liet. matem. rink. 44 (2004), 176-180 = Lith. Math. J. 140-144. PDF
26. Universality of Dirichlet L-functions with shifted characters, Liet. Matem. Rink. 44 (Special Issue) (2004), 48-50.
27. Growth of the Lerch zeta-function, Liet. matem. rink. 45 (2005), 45-56 = Lith. Math. J. 34-43. PDF
28. Note on the zeros of the Hurwitz zeta-function, in: Voronoi's impact on modern science. Book 3: proceedings of the third Voronoi Conference on Number Theory and Spatial Tessellations. Mathematics and its Applications, 55 (2005), 10-12. PDF
29. Simple zeros and discrete moments of the derivative of the Riemann zeta-function, J. Number Theory 115 (2005), 310-321 (with J. Steuding). PDF
30. On the distribution of zeros of the Hurwitz zeta-function, Math. Comp. 76 (2007), 323-337 (with J. Steuding). PDF
31. On the Backlund equivalent for the Lindelof hypothesis, Adv. Stud. Pure Math. 49 (2007), 91-104. PDF
32. Sum of the periodic zeta-function over the nontrivial zeros of the Riemann zeta-function, Analysis, München, 28 (2008), 209-217 (with J. Kalpokas). PDF
33. Note on zeros of the derivative of the Selberg zeta-function, Arch. Math. 91 (2008), 238-246. PDF . Corrigendum, Arch. Math. 93 (2009), page 143. PDF
34. Selberg's Central Limit Theorem on the Critical Line and the Lerch Zeta-Function, in: Proceedings of the conference New Directions in the Theory of Universal Zeta- and L-Functions, Würzburg, Germany, October 6-10, 2008, Shaker Verlag, (2009), 57-64 (with A. Grigutis and A. Laurinčikas). PDF
35. Effective uniform approximation by the Riemann zeta-function, Publ. Mat. 54 (2010), 209-219 (with A. Laurinčikas, K. Matsumoto, J. Steuding and R. Steuding). PDF
36. Sum of the Dirichlet L-function over nontrivial zeros of another Dirichlet L-function, Acta Math. Hungar., 128 (2010), 287-298 (with J. Kalpokas and J. Steuding). PDF
37. Self-approximation of Dirichlet L-functions, J. Number Theory, 131(7) (2011), 1286-1295. arXiv:1006.1507
38. Questions around the nontrivial zeros of the Riemann zeta-function - computations and classifications, Math. Model. Anal., 16(1) (2011), 72-81. (with J. Steuding). PDF
39. Uniqueness theorems for L-functions, Comment. Math. Univ. St. Pauli, 60, No. 1,2 (2011), 15-35. (with J. Grahl and J. Steuding). PDF
40. Zeros of the Lerch transcendent function, Mathematical Modelling and Analysis, 17, No. 2 (2012), 245-250. (with A. Grigutis). PDF
41. The a-values of the Selberg zeta-function, Lith. Math. J., 52, No. 2 (2012), 145-154. (with R. Šimenas). PDF
42. Zeros of the periodic zeta-function, Šiauliai Mathematical Seminar, 8(16) (2013), 49-62. (with R. Tamošiūnas). PDF
43. Zeros of the Estermann zeta-function, Journal of the Australian Mathematical Society 94 (2013), 38-49, doi:10.1017/S1446788712000419 (with A. Dubickas, J. Steuding, and R. Steuding). PDF
44. Complex B-splines and Hurwitz zeta functions, LMS Journal of Computation and Mathematics 16 (2013), 61-77, (with B. Forster, P. Massopust, and J. Steuding). PDF
45. The discrete mean square of the Dirichlet L-function at nontrivial zeros of another Dirichlet L-function, International Journal of Number Theory 9(4) (2013), 945-963, (with J. Kalpokas). PDF
46. Universality of the Selberg zeta-function for the modular group, Forum Mathematicum 25(3) (2013), 533-564, (with P. Drungilas and A. Kačėnas). PDF
47. On the roots of the equation $\zeta(s)=a$, Abh. Math. Semin. Univ. Hambg. 84 (2014), 1-15, (with J. Steuding). arXiv:1011.5339
48. Self-approximation of Hurwitz zeta-functions, Functiones et Approximatio 51(1) (2014), 181-188, (with E. Karikovas). PDF
49. The a-points of the Selberg zeta-function are uniformly distributed modulo one, Illinois J. Math. 58(1) (2014), 207–218, (with J. Steuding and R. Šimėnas). PDF
50. On the Speiser equivalent for the Riemann hypothesis, European Journal of Mathematics DOI 10.1007/s40879-014-0033-1, (with R. Šimėnas). PDF
### International conferences, visits
1. The II International Conference “Analytic and Probabilistic Methods in Number Theory”, Palanga, Lithuania, September 23-27, 1996.
2. Renyi Institute, Budapest, Hungary, November 1-30, 1997.
3. 22nd European Meeting of Statisticians, 7th Vilnius Conf. on Probability Theory and Mathematical Statistics, Vilnius, Lithuania, August 12-18, 1998.
4. Paul Erdos and his Mathematics, Budapest, Hungary, July 4-11, 1999.
5. J.W. Goethe University, Frankfurt on Main, Germany, November 2-30, 1999.
6. Colloquium on Number Theory in honour of the 60th birthday of Professors K. Gyory and A. Sarkozy, Debrecen, Hungary, July 3-7, 2000.
7. IV International Conference “Modern Problems of Number Theory and Its Applications” dedicated to 180th anniversary of P.L. Chebysheff and 110th anniversary of I.M. Vinogradov, Tula, Russia, September 10-15, 2001.
8. The III International Conference “Analytic and Probabilistic Methods in Number Theory”, Palanga, Lithuania, September 24-28, 2001.
9. J.W. Goethe University, Frankfurt on Main, Germany, June 5-July 5, 2002.
10. Freiburg University, Germany, June 7, 2002.
11. Workshop “Special Activity in Analytic Number Theory and Diophantine equations”, Max Planck Institute for Mathematics, Bonn, Germany, June 24-28, 2002.
12. Voronoi conference on analytic number theory and spatial tessellations, Kyiv, Ukraine, September 22-28, 2003.
13. J.W. Goethe University, Frankfurt on Main, Germany, January 27-February 26, 2004.
14. Nagoya University, Nagoya, Japan, June 11-19, 2005.
15. International Conference on Probability and Number Theory (P&NT 05), Kanazawa, Japan, June 20-24, 2005.
16. The 17-th Czech and Slovak International Conference on Number Theory, Malenovice, Czech Republic, September 5-10, 2005.
17. The 30th birthday of Voronin's universality theorem. Colloquium: value distribution of Dirichlet series, Madrid, Spain, December 5-9, 2005.
18. Conference “Advances in Number Theory and Random Matrix Theory”, University of Rochester, Rochester, NY, USA, , June 5-8, 2006.
19. The IV International Conference “Analytic and Probabilistic Methods in Number Theory”, Palanga, Lithuania, September 24-30, 2006.
20. “West Coast Number Theory Conference”, Ensenada, Baja California, Mexico, December 17-21, 2006.
21. “International Conference on Number Theory” dedicated to the 60th birthday of Professor Antanas Laurinčikas, Šiauliai, Lithuania, August 11-15, 2008.
22. “New Directions in the Theory of Universal Zeta- and L-Functions”, Würzburg, Germany, October 6-10, 2008.
23. Würzburg University, Würzburg, Germany, December 9-20, 2009.
24. “15th International Conference Mathematical Modelling and Analysis”, Druskininkai, Lithuania, May 26-29, 2010.
25. Satellite Conference to ICM 2010 “Analytic and Combinatorial Number Theory”, Institute of Mathematical Sciences, Chennai, India, August 29-September 3, 2010.
26. Conference “Functions in Number Theory and Their Probabilistic Aspects”, RIMS, Kyoto University Kyoto, Japan, December 13-17, 2010.
27. “Workshop on various zeta functions and related topics”, The University of Tokyo, Tokyo, Japan, December 21 and 22, 2010.
28. “27th Journées Arithmétiques”, Vilnius University, Vilnius, Lithuania, 27th June - 1st July 2011.
29. The V International Conference “Analytic and Probabilistic Methods in Number Theory”, Palanga, Lithuania, September 4-10, 2011.
30. Würzburg University, Würzburg, Germany, October 1-30, 2011.
31. Conference “Synergies And Vistas in Analytic Number Theory”, University of Oxford, Oxford, UK, September 3-7, 2012. | 2015-04-28 03:41:31 | {"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": 0.9187895655632019, "perplexity": 7268.667211644395}, "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-2015-18/segments/1429246660628.16/warc/CC-MAIN-20150417045740-00090-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://brilliant.org/problems/should-you-switch/ | # Should You Switch
Logic Level 1
Given that you don't know what $$X$$ is (but only what is in your envelope), should you switch?
× | 2017-05-24 10:00:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5918311476707458, "perplexity": 3292.336016930134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607811.15/warc/CC-MAIN-20170524093528-20170524113528-00173.warc.gz"} |
https://www.jiskha.com/questions/2955/a-certain-diet-requires-no-more-than-60-units-of-carbohydrates-at-least-45-units-of | # math
A certain diet requires no more than 60 units of carbohydrates, at least 45 units of protein, and at least 30 units of fat each day. Each ounce of Supplement A provides 5 units of carbs, 3 units of protein, and 4 units of fat. Each ounce of Supplement B iprovides 2 units of carbs, 2 units of protein, and 1 unit of fat. If Supplemnet A cost $1.50 per once and Supplemnt B costs$1.00 per ounce, how many ounces of each upplement should be taken daily to minimize the cost of diet?
i can't come up with system of
equations for this thingg..
Let C,P and F denote carbohydrates, protein and fat. Let D denote what the diet requires. Then we're told
60C<=D
D<=45P
D<=30F
A = 5C + 3P + 4F
B = 2C + 2P + 1F
A=1.50
B=1.00
We want to maximize
xA+yB=D
where x is the number of Aoz and y is Boz.
and minimize
1.50x+1.00y=cost
There are a number of ways to solve this, but considering the size of the numbers you might just make a table that starts at x=0 and y=30 and find the feasible set of values, then find which has the smallest cost in the feasible set.
After I reviewed my preious post I realized I could be a little more helpful.
If you use x and y for the number of oz of each supplement, then we want to know how many of each is the ideal amount for the diet and cost. We know they provide
1oz A = 5c + 3p + 4f
ioz B = 2c + 2p + 1f
In terms of c, p and f
5x+2y<=60 carb. requirement
3x+2y=>45 prot. requirement
4x+y=>30 fat requirement
We also know
x=>0 and y=>0 we are only interested in values in the first quadrant.
If you graph that set of lines and examine where they intersect you'll find vertices to test in the cost equation
1.50x+1.00y=Cost
I graphed those lines and there should be three vertices to test.
I hope this is more helpful.
find the solution to the system by graphing x+y=-3 and 3x-y=7
1. 👍
2. 👎
3. 👁
1. infinity
1. 👍
2. 👎
## Similar Questions
1. ### Mathematics in the modern world
.) rina needs at least 48 units of protein, 60 units of carbohydrates, and 50 units of fat each month. from each kilogram of food a, she receives 2 units of protein, 4 units of carbohydrates, and 5 units of fats. food b contains 3
2. ### Geometry
Describe in words the translation of X represented by the translation rule T < -7, -8 > (X). A. 7 units to the right and 8 units up B. 8 units to the left and 7 units up C. 7 units to the right and 8 units down D. 7 units to the
3. ### math
A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 36.12 units of niacin and 30,080 units of retinol. She uses
4. ### Accounting
Below is budgeted production and sales information for Flushing Company for the month of December: Product XXX Product ZZZ Estimated beginning inventory 30,000 units 18,000 units Desired ending inventory 34,000 units 17,000 units
Which answer describes the transformation of g(x)=log4(x−2)+4 from the parent function f(x)=log4x? It is the graph of f(x) shifted 2 units right and 4 units up. It is the graph of f(x) shifted 2 units left and 4 units down. It
1. ### Math
Rectangles ABCD and EFGH are similar figures because angles A, B, C, and D are congruent to angles E, F, G, and H, respectively. If DA equals 19 units, AB equals 8 units, and HE equals 133 units, what does EF equal? A. 56 units B.
2. ### phy
The angle an airplane propeller makes with the horizontal as a function of time is given by \theta = (115 rad/s)t + (45.0 rad/s^2)t^2. calculate the angular acceleration from 0 to 1.o sec
3. ### Physics
Consider a pendulum of length 4.557 m. The acceleration of gravity is 9.832 m/s2 . a) What is its period at the North Pole? Answer in units of s. b) What is its frequency? Answer in units of Hz. c) What is its period in Chicago,
4. ### Calculus
In a right triangle, the hypotenuse is of fixed length of 15 units, one side is increasing in length by 4 units per second while the third side is decreasing in size. At a certain instant the increasing side is of length 9 units. | 2021-12-06 09:26:41 | {"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": 0.636346161365509, "perplexity": 1549.5740488717988}, "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/1637964363290.59/warc/CC-MAIN-20211206072825-20211206102825-00214.warc.gz"} |
https://www.freemathhelp.com/forum/threads/dirichlet-boundary-value-on-disk-problem.44311/ | # Dirichlet Boundary Value on Disk Problem
G
#### Guest
##### Guest
Hi Everyone:
I am having problems with a Dirichlet boundary value for a disk question. They questions gives the conditions of:
0 <= r < 2, -pi <= theta <= pi
u(2,theta) = f(theta), -pi <= theta <= pi
The problem is to solve given f(theta) = cos^2 (x) (cosine squared of x)
This is in the "Fundamentals of Differential Equations" textbook p. 649 (section 10.7) problem # 8.
I was thinking I should use: u(r,theta) = a0/2 + SUM(an cos(n*theta) + bn sin(n*theta) and solve for an and bn.
I tried this for #7 (which has the answer in the back) but I didn't come close to the answer.
Anyone have any ideas?
Thanks!
#### royhaas
##### Full Member
If the coefficients of the sine terms are not identically zero, check your integration since you are approximating an even function. i.e., $$\displaystyle cos^2(x) = 0.5(1+cos(2x))$$. | 2021-06-15 00:38:52 | {"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": 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": 0.8445935249328613, "perplexity": 3164.0910388231773}, "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/1623487614006.8/warc/CC-MAIN-20210614232115-20210615022115-00581.warc.gz"} |
https://tex.stackexchange.com/questions/168340/adjusting-spacing-before-and-after-subparagraph-with-sectsty | # Adjusting spacing before and after subparagraph with sectsty
I am helping a friend that is using sectsty to format the titles in his document. How can I adjust the spacing before and after the titles, in particular the subparagraph one?
(I know that titlesec is more powerful. But for now we just need either a quick fix or the full translation of the following to titlesec.)
This is what we have now:
\usepackage{sectsty}
\allsectionsfont{\centering\normalfont\normalsize}
\subsectionfont{\noindent\normalfont\normalsize\emph}
%\subsubsectionfont{\indent\normalfont\normalsize\itshape}
\subparagraphfont{\indent\normalfont\normalsize\itshape}
As usual this is because of thesis requirements made by ignorant people.
• Always post complete examples rather than code snippets, please! – Werner Mar 29 '14 at 5:50
• Which document class do you use? – Mico Mar 29 '14 at 6:23
• apa6, I'll post a complete example when I have access to the original document. – alfC Mar 29 '14 at 7:50
sectsty only updates only the sectional unit font-related setting but still use the traditional \@startsection construction:
\renewcommand\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
% {\normalfont\Large\bfseries}}
{\normalfont\Large\bfseries\SS@sectfont}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
% {\normalfont\large\bfseries}}
{\normalfont\large\bfseries\SS@subsectfont}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subsubsectfont}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
{3.25ex \@plus1ex \@minus.2ex}%
{-1em}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@parafont}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{3.25ex \@plus1ex \@minus .2ex}%
{-1em}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subparafont}}
The normal font-selections are commented out and substituted with the sectsty formatting \SS@.... Read up on the meaning of each component of \@startsection in Where can I find help files or documentation for commands like \@startsection for LaTeX? and focus on arguments 4 (before skip) & 5 (after skip).
Here's an example that adjust the space before and after the \subparagraph unit:
\documentclass{article}
\usepackage{sectsty}
\allsectionsfont{\centering\normalfont\normalsize}
\subsectionfont{\noindent\normalfont\normalsize\emph}
%\subsubsectionfont{\indent\normalfont\normalsize\itshape}
\subparagraphfont{\indent\normalfont\normalsize\itshape}
\begin{document}
\section{A section}
Some text
\subsection{A subsection}
Some text
\subsubsection{A subsubsection}
Some text
\paragraph{A paragraph}
Some text
\subparagraph{A subparagraph}
Some text
\makeatletter
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{2\baselineskip \@plus1ex \@minus .2ex}%
{\baselineskip}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subparafont}}
\makeatother
\subparagraph{A subparagraph}
Some text
\end{document}
Note that sectsty is assuming that your working with a document class for which the sectional units use \@startsection without modification from the default classes, with the exclusion of KOMA-Script (from the sectsty documentation; section 1 Introduction, p 2):
The secsty package provides a set of commands for changing the font used for the various sectional headings in the standard LaTeX 2e document classes: article, book, and report. This package also works with the KOMA-Script classes scrartcl, scrbook, and scrreprt.
While it "will work" with other document classes, the effect is merely to overwrite the sectional units' definitions with that of the default document classes mentioned above. For example, apa6 defines
\renewcommand{\section}{\@startsection {section}{1}{\z@}%
{\b@level@one@skip}{\e@level@one@skip}%
{\centering\normalfont\normalsize\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{\z@}%
{\b@level@two@skip}{\e@level@two@skip}%
{\normalfont\normalsize\bfseries}}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{\parindent}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-1em}%
\renewcommand{\paragraph}{\@startsection{paragraph}{4}{\parindent}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-1em}%
\renewcommand{\subparagraph}[1]{\@startsection{subparagraph}{5}{1em}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-\z@\relax}%
{\normalfont\normalsize\itshape\hspace{\parindent}{#1}\textit{.}}{\relax}}
In order to maintain the sectional unit structure, you're better off redefining the above set manually* (separate from sectsty, as was done in the first example). The same approach would hold (including a redefinition to your liking within a \makeatletter...\makeatother pair) as apa6 also uses \@startsection to define the sectional units.
* Of course, other packages could be used as well, if they are compatible with apa6, but that seems to be out of the question here.
• Since you have to redefine \subparagragraph anyway, what's the point in using sectsty? – egreg Mar 29 '14 at 9:37
• @egreg: The OP wants to update the \subparagraph spacing, but maintain some consistency/ease in terms of the modification for all other sectional unit fonts. Either way, apa6 (the document class in use) is "not compatible" with sectsty, so I've added some more detail. – Werner Mar 29 '14 at 17:21
• @Werner, "apa6 is not compatible with sectsty". Good to know, I wish we didn't use sectsty to start with, it's being a pain so far. – alfC Mar 30 '14 at 5:39 | 2019-05-21 05:15:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8614755868911743, "perplexity": 5144.993940073556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256227.38/warc/CC-MAIN-20190521042221-20190521064221-00309.warc.gz"} |
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0855.34055 | Language: Search: Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.
Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0855.34055
Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 27, No.4, 369-386 (1996). ISSN 0362-546X
Linear autonomous ordinary differential equations of the form (1) $dx/dt = Ax(t) + f(t) + \mu G (x(t),t)$, where $A$ is a continuous or an almost periodic square matrix, $f(t)$ is an almost periodic function, $G(t)$ is a Green function and $\mu$ is a small parameter are considered.\par The theory of almost periodic functions (a.p.f.) and almost periodic solutions of the ordinary differential equations (1) is well known. In the article a generalization of a.p.f., first, to a so-called pseudo almost periodic function (p.a.p.f.) and, second, to generalized pseudo almost periodic functions (g.p.a.p.f.) is given.\par Definition 1. A function $f$ which can be written as a sum $f = g + h$, where $g$ is a.p.f. and $h$ is a continuous bounded function with $M (|h |) = 0$, $(M (\cdot)$ is the asymptotic mean value, defined by $M(w) = \lim_{\tau \to \infty} ({1 \over 2 \tau} \int^r_{- \tau} w(s) ds))$ is called p.a.p.f.\par Definition 2. A function $f$ is called g.a.p.f. if in contrary to Definition 1 we assume that the function $f$ is neither continuous nor bounded.\par Under the assumption that the linear system of differential equations has an exponential dichotomy a theorem of existence of the pseudo almost periodic solutions of (1) is proved.
[S.G.Zhuravlev (Moskva)]
MSC 2000:
*34D05 Asymptotic stability of ODE
34C27 Almost periodic solutions of ODE
Keywords: linear autonomous ordinary differential equations; asymptotic mean value; exponential dichotomy; pseudo almost periodic solutions
Highlights
Master Server | 2013-05-19 05:21:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9196035861968994, "perplexity": 598.1468018874834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696383263/warc/CC-MAIN-20130516092623-00018-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-maths-chapter-14-exercise-14-3 | # NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning (Ex 14.3) Exercise 14.3
## NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning (Ex 14.3) Exercise 14.3
Free PDF download of NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.3 (Ex 14.3) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 11 Maths Chapter 14 Mathematical Reasoning Exercise 14.3 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all exercise solutions in your emails.
Do you need help with your Homework? Are you preparing for Exams?
Study without Internet (Offline)
## Access NCERT Solutions for Class 11 Maths Chapter 14 – Mathematical Reasoning
Exercise 14.3
1. For each of the following compound statements first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings.
Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.
Consider the given statement, ‘All rational numbers are real and all real numbers are not complex’.
The connecting word in the above compound statement is ‘AND’.
We will now determine the component statements.
Let the first component statement be P.
P: All rational numbers are real.
Let the second component statement be q.
q: All real numbers are not complex.
Therefore, the connecting word is ‘AND’.
The component statements for the given compound statement are,
P: All rational numbers are real.
q: All real numbers are not complex.
(ii) Square of an integer is positive or negative.
Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings.
Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.
Consider the given statement, ‘Square of an integer is positive or negative’.
The connecting word in the above compound statement is ‘OR’.
We will now determine the component statements.
Let the first component statement be P.
P : Square of an integer is positive.
Let the second component statement be q.
q: Square of an integer is negative.
Therefore, the connecting word is ‘AND’.
The component statements for the given compound statement are,
p: Square of an integer is positive.
q: Square of an integer is negative.
(iii) The sand heats up quickly in the Sun and does not cool down fast at night.
Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings.
Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.
Consider the given statement, ‘The sand heats up quickly in the Sun and does not cool down fast at night’.
The connecting word in the above compound statement is ‘AND’.
We will now determine the component statements.
Let the first component statement be P.
P: The sand heats up quickly in the Sun.
Let the second component statement be q.
q: The sand does not cool down fast at night.
Therefore, the connecting word is ‘AND’.
The component statements for the given compound statement are,
p: The sand heats up quickly in the Sun.
q: The sand does not cool down fast at night.
(iv) $x=2$ and $x=3$ are the roots of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.
Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings.
Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.
Consider the given statement, ‘$x=2$ and $x=3$ are the roots of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$’.
The connecting word in the above compound statement is ‘AND’.
We will now determine the component statements.
Let the first component statement be P.
$\text{p}$: $x=2$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.
Let the second component statement be q.
$\text{q}$: $x=3$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.
Therefore, the connecting word is ‘AND’.
The component statements for the given compound statement are,
$\text{p}$: $x=2$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.
$\text{q}$: $x=3$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.
2. Identify the quantifier in the following statements and write the negation of the statements.
(i) There exists a number which is equal to its square.
Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.
Consider the given statement, ‘There exists a number which is equal to its square’.
The quantifier in the above statement is, ‘There exists’.
The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.
The given statement does not carry the word ‘not’.
To write the negation of the given statement, we will add the word ‘not’ to it.
Therefore, the negation of the given statement is, ‘There exists a number which is not equal to its square’.
(ii) For every real number $x$, $x$ is less than $x+1$.
Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.
Consider the given statement, ‘For every real number $x$, $x$ is less than $x+1$’.
The quantifier in the above statement is, ‘For every’.
The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.
The given statement does not carry the word ‘not’.
To write the negation of the given statement, we will add the word ‘not’ to it.
Therefore, the negation of the given statement is, ‘For there exists a real number $x$ such that $x$ is not less than $x+1$’.
(iii) There exists a capital for every state in India.
Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.
Consider the given statement, ‘There exists a capital for every state in India’.
The quantifier in the above statement is, ‘There exists’.
The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.
The given statement does not carry the word ‘not’.
To write the negation of the given statement, we will add the word ‘not’ to it.
Therefore, the negation of the given statement is, ‘There exists a state in India which does not have a capital’.
3. Check whether the following pair of statements is a negation of each other. Give reasons for the answer.
(i) $x+y=y+x$ is true for every real number $x$ and $y$.
(ii) There exists a real number $x$ and $y$ for which $x+y=y+x$
Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a statement is false then its negation is true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’.
Consider the first statement.
The given statement is that, ‘$x+y=y+x$ is true for every real number $x$ and $y$’.
The given statement carries the word ‘not’.
To write the negation of the given statement, we will add the word ‘not’ to it.
Thus, the negation of the given statement is, ‘$x+y=y+x$ is not true for every real numbers $x$ and $y$’.
Now this statement is not similar to the second statement.
Therefore, the given pair of statements are not the negations of each other.
4. State whether the ‘Or’ used in the following statements is exclusive ‘or’ inclusive. Give reasons for your answer.
(i) Sun rises or Moon sets.
Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously.
Consider the given statement, ‘Sun rises or Moon sets’.
The ‘or’ in the above statement is exclusive.
This is because the Sun cannot rise and the Moon cannot set together.
(ii) To apply for a driving license, you should have a ration card or a passport.
Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously.
Consider the given statement, ‘To apply for a driving license, you should have a ration card or a passport’.
The ‘or’ in the above statement is inclusive.
This is because a person can have both a passport and ration card to apply for the driver’s license.
(iii) All integers are positive or negative.
Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously.
Consider the given statement, ‘All integers are positive or negative’.
The ‘or’ in the above statement is exclusive.
This is because all integers cannot be both positive and negative.
## NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning
Opting for the NCERT solutions for Ex 14.3 Class 11 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 14.3 Class 11 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.
Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 11 students who are thorough with all the concepts from the Subject Mathematical Reasoning textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 11 Maths Chapter 14 Exercise 14.3 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.
Besides these NCERT solutions for Class 11 Maths Chapter 14 Exercise 14.3, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it.
Do not delay any more. Download the NCERT solutions for Class 11 Maths Chapter 14 Exercise 14.3 from Vedantu website now for better exam preparation. If you have the Vedantu app in your phone, you can download the same through the app as well. The best part of these solutions is these can be accessed both online and offline as well. | 2022-09-28 22:15:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3503642678260803, "perplexity": 691.4854900115301}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335286.15/warc/CC-MAIN-20220928212030-20220929002030-00560.warc.gz"} |
https://codereview.stackexchange.com/questions/57658/read-table-row-with-multiple-options | # Read table row with multiple options
I developing an algorithm to read a row from a table (specifically, I'm reading from an Excel file).
• When an empty cell is found (recognized by an ElementNotFoundException
• When a cell with errors is found (which corresponds to a null value)
• Both the two previous conditions
This is the current algorithm:
public IEnumerable<string> ReadRow(string spreadsheet, int column, int row, ReadOptions readOptions = ReadOptions.ReadUntilEmpty)
{
var values = new List<string>();
{
try
{
for (var currentColumn = column; ; currentColumn++)
{
// Cell has errors, so we exit from the loop
if (value == null) break;
}
}
catch (ElementNotFoundException)
{
// The row is terminated
// so we exit the loop withouth loggin any error
}
}
{
for (var currentColumn = column; ; currentColumn++)
{
try
{
// Cell has errors, so we exit from the loop
if (value == null) break;
}
catch (ElementNotFoundException)
{
// so we just ignore empty cells
}
}
}
{
try
{
for (var currentColumn = column; ; currentColumn++)
{
}
}
catch (ElementNotFoundException)
{
// The row is terminated
// so we exit the loop withouth loggin any error
}
}
return values;
}
And just for reference, the enumeration used:
[Flags]
{
}
How can I improve it?
Also, any idea on how I can reuse this code to read a column instead a row?
• You always terminate the reading of the row if the element isn't found. This means you can have just one Try-Catch block that wraps your If statement.
• Instead of checking for && in your first statement, use || (or instead of and). You don't really care if ReadUntilEmpty is set at that point; only ReadUntilCellError matters. This eliminates an entire block from the If statement.
• You could go one step further and only check the flag just before, or with, the null value check.
public IEnumerable<string> ReadRow(string spreadsheet, int column, int row, ReadOptions readOptions = ReadOptions.ReadUntilEmpty)
{
var values = new List<string>();
try
{
for (var currentColumn = column; ; currentColumn++)
{
{
// Cell has errors, so we exit from the loop
if (value == null) break;
}
}
}
catch (ElementNotFoundException)
{
// The row is terminated
// so we exit the loop withouth loggin any
}
return values;
}
Ultimately I realized there's never actually a need to check the ReadUntilEmpty flag, so you could probably just drop the Enum and change ReadUntilError to a Boolean with a default value of false.
Abstracting this to work on columns instead of rows is easy. The logic doesn't change at all. You just call it with the row number passed into the column argument and the column number passed into the row argument. So, really, naming the hard part and I'm drawing a blank on any actually useful names, but for the sake of giving an example...
public IEnumerable<string> ReadRow(string spreadsheet, int iteratorIndex, int secondaryIndex, ReadOptions readOptions = ReadOptions.ReadUntilEmpty)
{
var values = new List<string>();
try
{
for (var index = iteratorIndex; ; index++)
{
{
// Cell has errors, so we exit from the loop
if (value == null) break;
}
}
}
catch (ElementNotFoundException)
{
// The row is terminated
// so we exit the loop withouth loggin any
}
return values;
}
One final note: I'm not entirely comfortable with the fact that the catch block doesn't actually contain any code. It feels a bit hacky.
As @ckuhn203 hinted at the end of his answer, the empty catch block smells.
The problem is that you're catching an exception that should be avoidable: reading an empty cell isn't something that should be exceptional in Excel after all.
I think if ReadCell returned a struct that represented a cell's value, you wouldn't need to catch that exception.
Something like this:
public struct CellValueInfo
{
public bool IsEmpty { get { return _isEmpty; } }
public bool IsError { get { return _isError; } }
public object Value { get { return _value; } }
public CellValueInfo(object value, bool isEmpty, bool isError)
{
_value = value;
_isEmpty = isEmpty;
_isError = isError;
}
}
Granted, this could be greatly improved, but it's a starting point: an immutable structure that contains the information you need - the idea is to store the information that lets you determine whether a cell has a valid value, an error, or is empty.
Don't rely on exceptions when you can avoid them, especially with Excel interop.
This is the current version, developed based also on @ckuhn203 answer:
[Flags]
{
}
var values = new List<string>();
for (var currentColumn = column; currentColumn < CellLocation.MaxColumnIndex; currentColumn++)
{
string value;
{
// An empty cell has been found
// so we exit the loop if requested by the options
}
// Cell has errors, so we exit from the loop if requested by the options | 2021-11-28 00:15:24 | {"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": 0.18794763088226318, "perplexity": 4747.977506609568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358323.91/warc/CC-MAIN-20211127223710-20211128013710-00227.warc.gz"} |
https://www.lesswrong.com/posts/bzt3gebSTTqE8hXSm/ocd-proofs-forgetting-and-an-eldritch-god | # 7
Frontpage
This post made me realize something very important, and in hindsight very obvious about myself. I'm leaving it up for future reference. I don't think it's actually very well written.
Originally, it was tagged [Math], not [Health] and [Math].
I envy computers sometimes for their memory.
My basic intuition from learning signals and Fourier transforms in my EE major is that any limited-space memory system that has to adapt to new surroundings, also has to have some factor by which old memories decay over time -- otherwise the system would become saturated with information content. (I am not being precise with these terms. Look where I point.)
But computers don't usually have as ruthless a "use-it-or-lose-it" forgetting mechanism as human brains do built into them. Their decay is mostly due to (i) other conscious agents going in and rearranging their memories; (ii) the slow encroaching wave of entropy on their hard disks. (i) can be dealt with by locking the computer in a room away from human hands, and praying the Poincare recurrence theorem combined with thermodynamic noise doesn't imply it will eventually birth an AGI all on its own. But that's a symptom of (ii), which can be mostly dealt with via RAID-5 and an influx of hard drives. With those two in place, on a human timescale, computers essentially never have to forget something once it's encoded in memory.
Human beings, we aren't like that. We forget. We forget so easily. And we warp the memories we do have to fit our twisted little narratives in the moment. I'm not a terribly existential person, but the fact that we ship of Theseus ourselves according to the ridiculous whims of future us, people just like us now but ever-so-slightly slower, crazier, and more tied down to the local flora for their survival, you have to admit -- it's a little unsettling.
I've been having a bit of a crisis over the last few days. It's about math, as all my crises are.
The problem is simple: I prove $X$ once. I walk away for a few days. When I go back to look at $X$ again, I suddenly realize the steps of the proof don't immediately spring to mind any more. I have to prove it again, don't I; or accept it on faith that $X$ is a fact, that former me proved $X$ to be a fact, that no, there's really nothing much to be gained from re-proving something in roughly the same way that I did before.
The cognition is contrarian: Proofs take effort. I don't like wasting effort. And just because I could prove $X$ once is no guarantee at all that I could prove $X$ again right now -- memories decay, I only have limited vision and limited CPU cycles, maybe I actually can't prove $X$ again, and that means I don't really understand $X$ in some sense, and I won't know unless I try, right? And math is supposed to be this tower of logic, where -- in theory -- we could break everything all the way down to set/category theory and build it back up, right? That's what good mathematicians do, right? Aren't we supposed to be the true Scotsmen of deductive reasoning?
---
I once ran a campaign where there were 4 deities. One of them was called the White Noise; the Noise derived its power from all friction generated in the environment. When you and a stranger are walking down a street and don't want to bump into each other, but you step in the same direction they step, and then you step the other way at the same time that they do -- the White Noise feeds. When you're short 2 cents at the convenience store and the clerk says "Don't worry about it", but then you leave and worry that maybe the clerk will get fired because of the disparity at the end of the day. When lovers quarrel. When soldiers don't fight as hard as they should because they don't fully believe in their cause -- but also (scholars posit) the whole friction generated by war itself, the ultimate (pacifists say) zero-sum waste of resources. Eventually all becomes friction, and time stops.
If I had to choose a deity out of the 4 to follow, it would be the White Noise in a heartbeat. To me, it represents the ultimate decision that pointless human bickering is far preferable to brutal, paperclip-maximizing, mechanized efficiency.
Which is why it's so fucking stressful to me that the one time I do want mechanized efficiency, I can't seem to acquire it. I just want to be able to prove something once, and then store the memory of that proof in my head so that I don't have to keep doing it over and over again, man. Is that too much to ask?
---
Yes, it is. But let's go back to what we said before.
Mathematicians (human ones at least, not Coq or something) don't actually work in a strictly deductive fashion. Even if they wanted to, they realistically couldn't. We all have limits to what we can store in our short term memory, and the limits vary, but they presumably don't vary by orders of magnitude; similarly, we all have limits to how fast we can think through the logical implications of a thing, but again, this doesn't vary by orders of magnitude. And the subset of math we care about here is math-as-group-enterprise; so the fact that mathematicians are all competing for roughly the same rewards suggests that they would find ways to work around those fundamental limits in order to outdo each other. So it's certainly not the case that every other good mathematician on the planet does a proof once, memoizes exactly how to do it, and then just keeps that memory fresh.
What does happen, then? Probably what we would expect from a common sense view of human nature: They forget. They forget almost everything they learn at the undergraduate and graduate level, just like anyone else would. Oh, they probably don't forget as much relevant to their field of interest -- but I highly doubt there's any tenured, non-set theory professor out there who consciously reruns through proving Zorn's lemma from scratch once or twice a year, just to make sure they can still do it. It's a difficult, convoluted proof, and they have more important work to be done. New work.
Now, I don't think any working mathematician will admit this. They'll probably say something along the lines of, "Well of course I can't prove it on sight right now, but give me a couple days and I can probably get back to you with one. Just need to refresh my memory." That might even be true. But they're not just refreshing their memory -- they're turning a highly-tuned neural network, full of a lifetime of proving difficult propositions and engaging in clever logical tricks, back towards a problem they already understood the solution to once when they were less well-attuned.
Maybe, someday, we'll be able to fuck around with human memory enough that nobody will forget how to do a proof after the first time they spend energy on figuring out how to do it. But that day isn't today. Today, the White Noise scores a point for human imperfection. And while that might be locally frustrating -- think how much easier I would have it if that were the case! -- I wouldn't actually want to live in a world where everyone, universally, had those powers of memory. The competition would just adapt, quickly, to the newfound power, and I wouldn't end up with an obvious improvement in the group effort as before. I want it for me, and me alone; and that's a sign that I'm not actually upset for a good reason. I'm just being selfish. Even my OCD channels human bias.
# 7
New Comment
Be careful what you wish for. The latest research on memory and learning indicates that forgetting is an important part of our neural architecture, helping us to discern general rules and principles, filtering the important from the mundane, and thus preventing us from overfitting to specific situations. It's entirely possible that if you removed your ability to forget, you'd only be able to prove at a particular time of day, holding a particular piece of chalk, while standing in front of a particular blackboard while wearing the sweater you wore when you originally proved .
I usually solve issues like this by writing the thing down, carefully showing my work, and filing the resulting notes in a central, trusted place with good indexes and lots of links to sources and related ideas where available. If I go back later and have to try to understand my notes or find that my mind has changed, I figure out what I was going on about and why, then make copious annotations. This way, I export large portions of my mind to the environment. If I need to demonstrate to myself or somebody else that, for example, I can prove X, I can just pull the relevant notes and work them as needed into some sort of document.
I'm just being selfish. Even my OCD channels human bias.
Eh. While this might not apply to everyone, I'd say many people have at one time or another, wished they (personally) knew "everything". That might not be achievable, it might not make sense to pursue as goal, but it seems more like an "illogical desire"* (that might follows from enjoying the knowledge you have/missing the knowledge you don't), than a matter of selfishness.
*You can also think of it as "your brain" thinking you can learn everything, as it believing in you; believing you can do anything.
and an eldritch god
I didn't understand this bit.
Aaaaaaaaaaaaaaaaaaaaand now I'm thinking I know what's wrong with me.
https://deponysum.com/2019/04/28/ocd-what-i-learned-fighting-mind-cancer/ | 2022-08-19 05:05:12 | {"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": 0.5452189445495605, "perplexity": 864.2190821201342}, "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/1659882573623.4/warc/CC-MAIN-20220819035957-20220819065957-00219.warc.gz"} |
https://physics.stackexchange.com/questions/549458/deriving-the-momentum-operator-hatp-with-wavefunction-psi-using-the-tran | # Deriving the momentum operator $\hat{p}$ with wavefunction $\Psi$ using the translation operator
The momentum operator $$\hat{p}$$ can be written as $$\hat{p} = i\hbar \lim_{a \rightarrow 0} \frac{\hat{T}(a\hat{r})- \hat{\mathbb{I}}}{a}$$ where $$\mathbb{\hat{I}}$$ is the identity operator and $$\hat{r}$$ being the unit vector in a given direction. For the case of a single pasrticle with wavefunction $$\psi(r)$$ it can be written as $$( \hat{p} \psi )(r) = i\hbar \lim_{a \rightarrow 0} \frac{(\hat{T}\psi)(r)- \psi(r)}{a} = i\hbar \lim_{a \rightarrow 0} \frac{\psi(r-a)- \psi(r)}{a} \, .$$
Now I am not sure how to proceed. My thought was to use L'Hôpital's rule: $$= i\hbar \lim_{a \rightarrow 0} \frac{\frac{d}{da}[\psi(r-a) - \psi(r)]}{\frac{d}{da}a} = i\hbar \lim_{a \rightarrow 0 } \frac{\partial \psi(r-a)}{\partial a} = i \hbar \frac{\partial \psi(r)}{\partial a} = 0 \, .$$ However, it is stated on Wikipedia that the result should be $$(\hat{p}\psi )(r) = -i \hbar \frac{\partial \psi(r)}{\partial r} \, .$$ I am not sure as to why the wavefuction is being derived with respect to $$r$$ and not $$a$$.
• Where is the negative sign for the derivative? May 5 '20 at 17:08
As you already mentioned $$(\hat p \psi)(r) = i \hbar \lim_{a\to 0}\frac{\psi(r - a) - \psi (r)}{a}$$ but this is already the derivative of $$\psi(r)$$ with respect to $$r$$ so $$i \hbar \lim_{a\to 0}\frac{\psi(r - a) - \psi (r)}{a} = - i \hbar \frac{\partial\psi}{\partial r}(r)$$
• I think the numerator should be either $f(x+a)-f(x)$ or $f(x)-f(x-a)$. May 5 '20 at 16:59
You already have an answer but I'll show you why your method is actually consistent. When you take the derivative with respect to something in the argument you have to be careful. I define $$y=r-a$$ as a helping tool. That means that \begin{align}\frac d{da}\psi(x-a)&=\frac d{da}\psi(y(a))\\ &=\frac {dy}{da}\frac {d\psi(y)}{dy}\\ &=-\psi'(y) \end{align} After taking the limit this becomes $$-\lim_{a\rightarrow 0}\psi'(r-a)=-\psi'(r)=-\frac{\partial\psi(r)}{\partial r}$$ You have to do this because a derivative with respect to $$a$$, $$\lim_{a\rightarrow 0}\frac{\partial}{\partial a}$$, makes no sense when you take the limit to zero. You might be wondering about the minus sign in my answer. The derivative is defined as $$\lim_{a\rightarrow 0}\frac{f(x+a)-f(x)}{a}$$ so with a plus sign. The momentum operator is $$\hat p=-i\hbar\frac \partial{\partial x}$$ so this checks out. | 2021-11-30 05:21:20 | {"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": 21, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9745092391967773, "perplexity": 116.36047155688793}, "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-2021-49/segments/1637964358953.29/warc/CC-MAIN-20211130050047-20211130080047-00115.warc.gz"} |
https://www.thejournal.club/c/paper/353313/ | #### An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL
##### Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, Dimitrios M. Thilikos
In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{{\phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${\phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and ${\phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence ${\phi}$, the Graph $\boxtimes$-Modification to Planarity and ${\phi}$ is solvable in $f(k,|{\phi}|)\cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman's Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.
arrow_drop_up | 2021-10-16 20:48:19 | {"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": 0.6716099381446838, "perplexity": 535.4657467066606}, "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/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00388.warc.gz"} |
https://thetutorverse.com/services/practice-test/isee-practice-test/ | # ISEE Practice Test
##### $110.00 –$160.00
Looking for a proctored group test for the ISEE Upper, Middle, and Lower levels? Look no further!
Proctored and self-administered, online options available below. (Click More Information to read about both options!)
Email hello@thetutorverse.com to request a specific date and time for an exam!
Not ready to take a practice test yet? Don’t worry – The Tutorverse has you covered.
• Click this link to find our ISEE workbooks with over 1,000 practice test questions.
• We also have one-on-one Tutoring for those arriving late to the party – find out more here.
More InformationLess Information
#### ISEE Practice Test
PROCTORED
Group practice tests are fully proctored and are hosted online via Zoom.us.
Students will receive:
• a copy of one of our proprietary practice tests
• a personalized score report
• fully proctored test-taking experience
SELF-ADMINSTERED
We also offer self-administered tests for at-home administration.
Students will receive:
• a copy of one of our proprietary practice tests
• a personalized score report
• a set of administration instructions
Students must supply their own pencils, quality erasers, and snack (optional). We strongly recommend that all students
• get a full night’s sleep
• eat a healthy, well-balanced breakfast
• log in around 10 minutes early
• use the restroom before arriving
Note: Testing dates, locations, and times are contingent on minimum registration requirements. If we are unable to meet the enrollment requirement, we will reach out with other options for testing.
###### Class Course Schedule
Middle / Upper Jun 6, 2021
Sunday, 9:00 AM-12:30 PM
###### Student Information
• Your student's safety is important. This authorization will let us know who your student may leave the building with after class. Unfortunately, if the person's name is not listed on this form, a parent/guardian must be contacted for permission before the student can travel. If other, please provide a complete list of persons authorized to pick up the student.
• Please provide first and last names as well as any necessary contact information.
• If so, please list below.
• If so, please specify below.
###### Primary Contact Information
5 seats are still available
Middle / Upper Oct 17, 2021
Sunday, 9:00 AM-12:30 PM
###### Student Information
• Your student's safety is important. This authorization will let us know who your student may leave the building with after class. Unfortunately, if the person's name is not listed on this form, a parent/guardian must be contacted for permission before the student can travel. If other, please provide a complete list of persons authorized to pick up the student.
• Please provide first and last names as well as any necessary contact information.
• If so, please list below.
• If so, please specify below.
###### Primary Contact Information
5 seats are still available
Middle / Upper Nov 14, 2021
Sunday, 9:00 AM-12:30 PM
###### Student Information
• Your student's safety is important. This authorization will let us know who your student may leave the building with after class. Unfortunately, if the person's name is not listed on this form, a parent/guardian must be contacted for permission before the student can travel. If other, please provide a complete list of persons authorized to pick up the student.
• Please provide first and last names as well as any necessary contact information.
• If so, please list below.
• If so, please specify below.
###### Primary Contact Information
5 seats are still available
Self-Administered
Date & Time of your choosing
At Home $160.00$110.00
###### Student Information
• Your student's safety is important. This authorization will let us know who your student may leave the building with after class. Unfortunately, if the person's name is not listed on this form, a parent/guardian must be contacted for permission before the student can travel. If other, please provide a complete list of persons authorized to pick up the student.
• Please provide first and last names as well as any necessary contact information.
• If so, please list below.
• If so, please specify below. | 2021-04-10 19:13:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17891262471675873, "perplexity": 5061.287331781793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038057476.6/warc/CC-MAIN-20210410181215-20210410211215-00366.warc.gz"} |
https://math.stackexchange.com/questions/2499983/on-different-definitions-of-groups/2500081 | On different definitions of groups
Let a group $G$ be defined as a $4$-tuple $G=(|G|,\mu_G,i_g,e_G)$ satisfying the following axioms:
1) $\mu_G(\mu_G(x,y),z)=\mu_G(x,\mu_G(y,z))$
2) $\mu_G(x,e_G)=x=\mu_G(e_G,x)$
3) $\mu_G(x,i_G(x))=e_G=\mu_G(i_G(x),x)$
To each group $G$, we associate a pair $G'=(|G|,\delta_G)$, with $\delta_G(x,y):=\mu_G(x,i_G(y))$. This pair characterises the group, in the sense that, if $G_1,G_2$ are two groups with $G_1'=G_2'$ then $G_1=G_2$.
Now the question: Suppose $|X|$ is any set and $\delta\colon|X|\times|X| \to |X|$ any map. I want to write down a set of axioms for the pair $X = (|X|, \delta)$, which will be necessary and sufficient for it to arise from a group $G$ in the manner described above. That is, assuming $|X|$ and $\delta$ given, find convenient necessary and sufficient conditions for there to exist a group $G$ such that $G'=(|X|,\delta)$.
My work: since $\delta$ will have to be the $\delta_G$ of some group $G$, I tried to express the group multiplication using $\delta$:
$\mu_G(x,y)=\delta(x,\delta(\delta(y,y),y))$
Then I required for this multiplication the usual axioms identifying groups, which assume the following form:
associativity:
$\delta(\delta(x,\delta(\delta(y,y),y)), \delta(\delta(z,z),z)=\delta(x,\delta(\delta(\delta(y,\delta(\delta(z,z),z)),\delta(y,\delta(\delta(z,z),z))),\delta(y,\delta(\delta(z,z),z)))$
unit:
$\delta(x,\delta(\delta(\delta(x,x),\delta(x,x)),\delta(x,x)))=x=\delta(\delta(x,x),\delta(\delta(x,x),x))$
inverse:
$\delta(y,y)=\delta(x,x)$
First question: does it make sense?
Second question: is there a better choice of axioms?
• The question makes good sense, and I'm pretty sure I've seen answers that involved shorter axioms than the ones in the question, but I don't remember where I saw them. (That's why this is a comment and not an answer.) Nov 1, 2017 at 16:58
1 Answer
You don't need to include the $i_g$ and $e_G$ in $G$, they can be defined in terms of $\delta$ but you will need that $|G|$ is not empty. After axiom $\delta(x,x) = \delta(y,y)$, define $e_G =\delta(x,x)$ and $x^{-1}= \delta(e_G,x)$. Then the rest of the axioms become much shorter. For example the associativity axiom becomes $\delta (x,(\delta (y,z^{-1})^{-1}) =\delta (\delta(x,y^{-1}),z^{-1}).$ | 2022-07-04 22:17:22 | {"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": 0.9545496702194214, "perplexity": 106.38278869404031}, "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-2022-27/segments/1656104496688.78/warc/CC-MAIN-20220704202455-20220704232455-00536.warc.gz"} |
https://brilliant.org/problems/primes-inverse/ | # Primes reciprocated
Find the sum of all primes $$P$$ such that the decimal expansion of $$\frac{1}{P}$$ has a fundamental period $$5$$.
Details and Assumptions:
• As an explicit example, $$\frac{1}{3}=0.33333\ldots$$ has fundamental period $$1$$ and $$\frac{1}{101}=0.00990099\ldots$$ has a fundamental period $$4$$.
× | 2017-12-11 09:48:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.9951283931732178, "perplexity": 255.90104295799796}, "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/1512948513330.14/warc/CC-MAIN-20171211090353-20171211110353-00463.warc.gz"} |
https://zbmath.org/?q=an:0603.68035 | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
An algorithm for solving second order linear homogeneous differential equations. (English) Zbl 0603.68035
Summary: In this paper we present an algorithm for finding a “closed-form” solution of the differential equation $y''+ay'+by$, where $a$ and $b$ are rational functions of a complex variable $x$, provided a “closed-form” solution exists. The algorithm is so arranged that if no solution is found, then no solution can exist.
##### MSC:
68W30 Symbolic computation and algebraic computation 34A05 Methods of solution of ODE
##### Keywords:
closed-form solution
Full Text:
##### References:
[1] Baldassarri, F.: On second order linear differential equations with algebraic solutions on algebraic curves. Am. J. Math. 102, 517-535 (1980) · Zbl 0438.34007 [2] Baldassarri, F.; Dwork, B.: On second order linear differential equations with algebraic solutions. Am. J. Math. 101, 42-76 (1979) · Zbl 0425.34007 [3] Borel, A.: Groupes lin6aires alg briques. Attn. math. 64, 20-82 (1956) · Zbl 0070.26104 [4] Burnside, W.: Theory of groups of finite order. (1955) · Zbl 0064.25105 [5] Char, B. W.; Fee, G. J.; Geddes, K. O.; G,h; Gonnet; Monagan, M. B.: A tutorial introduction to Maple. J. symbolic computation 2 (1986) [6] Kaplansky, I.: An introduction to differential algebra. (1957) · Zbl 0083.03301 [7] Kolchin, E. R.: Differential algebra and algebraic groups. (1973) · Zbl 0264.12102 [8] Pavelle, R.; Wang, P. S.: MACSYMA from F to G. J. symbolic computation 1, 69-100 (1985) · Zbl 0573.68007 [9] Saunders, B. D.: A11 implementation of kovaeic’s algorithm for solving second order linear homogeneous differential equations. Proc. SYMSAC ’81, 105-108 (1981) [10] Singer, M. F.: Algebraic solutions of nth order linear differential equations. 54 (1979) [11] Singer, M. F.: Liouvillian solutions ofnth order homogeneous linear differential equations. Ant. J. Math. 103, 661-682 (1981) · Zbl 0477.12026 [12] Singer, M. F.: Solving homogeneous linear differential equations in terms of second order linear differential equations. Amer. J. Math. (1985) · Zbl 0564.12022 [13] Smith, C.: A discussion and implementation of kovacie’s algorithm for ordinary differential equations. (1984) | 2016-04-30 15:12:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7901774048805237, "perplexity": 6272.6133252016625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111868.79/warc/CC-MAIN-20160428161511-00110-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Space_vector_modulation | Space vector modulation
Space vector modulation (SVM) is an algorithm for the control of pulse width modulation (PWM).[1] It is used for the creation of alternating current (AC) waveforms; most commonly to drive 3 phase AC powered motors at varying speeds from DC using multiple class-D amplifiers. There are variations of SVM that result in different quality and computational requirements. One active area of development is in the reduction of total harmonic distortion (THD) created by the rapid switching inherent to these algorithms.
Principle
Topology of a basic three phase inverter.
A three-phase inverter as shown to the right converts a DC supply, via a series of switches, to three output legs which could be connected to a three-phase motor.
The switches must be controlled so that at no time are both switches in the same leg turned on or else the DC supply would be shorted. This requirement may be met by the complementary operation of the switches within a leg. i.e. if A+ is on then A is off and vice versa. This leads to eight possible switching vectors for the inverter, V0 through V7 with six active switching vectors and two zero vectors.
Vector A+ B+ C+ A B C VAB VBC VCA
V0 = {000} OFF OFF OFF ON ON ON 0 0 0 zero vector
V1 = {100} ON OFF OFF OFF ON ON +Vdc 0 −Vdc active vector
V2 = {110} ON ON OFF OFF OFF ON 0 +Vdc −Vdc active vector
V3 = {010} OFF ON OFF ON OFF ON −Vdc +Vdc 0 active vector
V4 = {011} OFF ON ON ON OFF OFF −Vdc 0 +Vdc active vector
V5 = {001} OFF OFF ON ON ON OFF 0 −Vdc +Vdc active vector
V6 = {101} ON OFF ON OFF ON OFF +Vdc −Vdc 0 active vector
V7 = {111} ON ON ON OFF OFF OFF 0 0 0 zero vector
Note that looking down the columns for the active switching vectors V1-6, the output voltages vary as a pulsed sinusoid, with each leg offset by 120 degrees of phase angle.
To implement space vector modulation, a reference signal Vref is sampled with a frequency fs (Ts = 1/fs). The reference signal may be generated from three separate phase references using the ${\displaystyle \alpha \beta \gamma }$ transform. The reference vector is then synthesized using a combination of the two adjacent active switching vectors and one or both of the zero vectors. Various strategies of selecting the order of the vectors and which zero vector(s) to use exist. Strategy selection will affect the harmonic content and the switching losses.
All eight possible switching vectors for a three-leg inverter using space vector modulation. An example Vref is shown in the first sector. Vref_MAX is the maximum amplitude of Vref before non-linear overmodulation is reached.
More complicated SVM strategies for the unbalanced operation of four-leg three-phase inverters do exist. In these strategies the switching vectors define a 3D shape (a hexagonal prism in ${\displaystyle \alpha \beta \gamma }$ coordinates[2] or a dodecahedron in abc coordinates[3]) rather than a 2D hexagon. | 2018-09-20 14:12:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.28572696447372437, "perplexity": 3339.689603526603}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156513.14/warc/CC-MAIN-20180920140359-20180920160759-00263.warc.gz"} |
https://www.techwhiff.com/issue/the-points-1-4-and-2-1-are-collinear-the-points-3-19--252399 | # The points (1,-4) and (2,-1) are collinear. The points (3,19) and (5,25) are also collinear. What can you say about the lines that cross these points?
###### Question:
the points (1,-4) and (2,-1) are collinear. The points (3,19) and (5,25) are also collinear. What can you say about the lines that cross these points?
### Starting with light energy, carbon dioxide, and water, which of the following sequences of biological steps explains how Usain Bolt was able to run the 100 meter dash in record time?
Starting with light energy, carbon dioxide, and water, which of the following sequences of biological steps explains how Usain Bolt was able to run the 100 meter dash in record time?...
### At the beginning of a week, a stock’s price is $9.40 per share. For that entire week, the stock’s price decreased at a steady rate of$0.20/day. Christine graphs the price of the stock versus time. She plots the price of the stock along the vertical axis, and she plots time in days along the horizontal axis. What is the y-intercept of the line?
At the beginning of a week, a stock’s price is $9.40 per share. For that entire week, the stock’s price decreased at a steady rate of$0.20/day. Christine graphs the price of the stock versus time. She plots the price of the stock along the vertical axis, and she plots time in days along the hor...
### What was one result related to the advent of the cotton gin? A) There was an increased demand for slave labor B) It took twice as long to gin cotton C) Cotton was no longer used by the textile industry D) This enabled cotton to be grown in the cities like New York
What was one result related to the advent of the cotton gin? A) There was an increased demand for slave labor B) It took twice as long to gin cotton C) Cotton was no longer used by the textile industry D) This enabled cotton to be grown in the cities like New York...
### The phenomenon of marble cancer is due to (a) soot particles (c) fog (b) CFCs (d) acid rain
The phenomenon of marble cancer is due to (a) soot particles (c) fog (b) CFCs (d) acid rain...
### Q3. A state college charges $8,792 tuition per year for Texas residents and$13,134 tuition per year for out-of-state students. Over 4 years, how much more would an out-of-state student pay? 0 $4,342.00 O$21,926.00 O $17,368.00 O$87,704.00
Q3. A state college charges $8,792 tuition per year for Texas residents and$13,134 tuition per year for out-of-state students. Over 4 years, how much more would an out-of-state student pay? 0 $4,342.00 O$21,926.00 O $17,368.00 O$87,704.00...
### 1. Firms generally choose to finance temporary current operating assets with short-term debt because a. short-term interest rates have traditionally been more stable than long-term interest rates. b. a firm that borrows heavily on a long-term basis is more apt to be unable to repay the debt than a firm that borrows short term. c. the yield curve is normally downward sloping. d. short-term debt has a higher cost than equity capital. e. matching the maturities of assets and liabilities reduces ris
1. Firms generally choose to finance temporary current operating assets with short-term debt because a. short-term interest rates have traditionally been more stable than long-term interest rates. b. a firm that borrows heavily on a long-term basis is more apt to be unable to repay the debt than a f...
### Sophia had pieces of rope that were 3 1/5 feet long 4.5 feet long and 5 3/10 feet long 4.5 feet long and 5 3/10 feet long what was the total number of feert of rope that sophia had
sophia had pieces of rope that were 3 1/5 feet long 4.5 feet long and 5 3/10 feet long 4.5 feet long and 5 3/10 feet long what was the total number of feert of rope that sophia had...
### $\frac{-23}{30} + \frac{5}{48}$
$\frac{-23}{30} + \frac{5}{48}$...
### What is the formula for a line that passes through point (6,5) and runs perpindicular to a line with a slope of 1/2
What is the formula for a line that passes through point (6,5) and runs perpindicular to a line with a slope of 1/2...
### What is common factor for 30 and 50 100
what is common factor for 30 and 50 100...
### A motorcycle is traveling at a constant speed of 75miles per hour. How many feet does it travel in 7seconds? Remember that 1 mile is 5280 feet.
A motorcycle is traveling at a constant speed of 75miles per hour. How many feet does it travel in 7seconds? Remember that 1 mile is 5280 feet....
### What was a motivation for German settlers to come to Louisiana? A.They believed that trading would be easier in Louisiana. B.They wanted to flee war in their country. C.They wanted to learn how to farm. D.They wanted to live in an environment similar to their homeland’s.
What was a motivation for German settlers to come to Louisiana? A.They believed that trading would be easier in Louisiana. B.They wanted to flee war in their country. C.They wanted to learn how to farm. D.They wanted to live in an environment similar to their homeland’s....
### Divide p(x)=x^4+6x^3+7x^2-6x-8 by x+4. What is the remainder
Divide p(x)=x^4+6x^3+7x^2-6x-8 by x+4. What is the remainder...
### Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 2 seconds. Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling. (This model for free fall neglects air resistance.) If the distance fallen after t seconds is denoted by s ( t ) and measured in meters, then Galileo's
Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 2 seconds. Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional...
### 2. What evidence do you have from the sim? Be sure to include temperature of samples. My evidence from the sun is that Sample A... 3. Explain what you discovered in the sim about why food coloring spreads faster in warmer water. This can explain why the food coloring spread faster in warm water because ...
2. What evidence do you have from the sim? Be sure to include temperature of samples. My evidence from the sun is that Sample A... 3. Explain what you discovered in the sim about why food coloring spreads faster in warmer water. This can explain why the food coloring spread faster in warm water beca...
### Tell me a joke. and make it good.
tell me a joke. and make it good....
### What is the area of a rectangle with a length of 4 1/8 ft and a width of 2 ft
what is the area of a rectangle with a length of 4 1/8 ft and a width of 2 ft...
### Plzzzzzzzzzzzzzzzzzzzzzzzzzz
plzzzzzzzzzzzzzzzzzzzzzzzzzz...
### Baby Jay accidently sucked his thumb, now seems to love to suck his thumb, and has learned to do it over and over again. This best exemplifies the concept of a
Baby Jay accidently sucked his thumb, now seems to love to suck his thumb, and has learned to do it over and over again. This best exemplifies the concept of a...
### Ma Question 1: Identify the organelle above * Ribosome O Mitochondria O Nucleus Chloroplast
ma Question 1: Identify the organelle above * Ribosome O Mitochondria O Nucleus Chloroplast...
-- 0.045810-- | 2023-02-07 12:17:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3407805562019348, "perplexity": 1640.2943415537484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500456.61/warc/CC-MAIN-20230207102930-20230207132930-00186.warc.gz"} |
https://www.waldhessen-trophy.de/120ton/Jul/Tue_4358/ | ## Block Diagram method for a simple mechanical system 422016,
Development of the equations of motion and block diagrams for a simple mechanical system.Block Diagrams: Modeling and Simulation,Block Diagram Modeling: Analogy Approach Physical laws are used to predict the behavior (both static and dynamic) of systems. Electrical engineering relies on Ohm’s and Kirchoff’s laws Mechanical engineering on Newton’s law Electromagnetics on Faradays and Lenz’s laws Fluids on continuity and Bernoulli’s law Based on electrical analogies, we can derive the fundamentalBlock Diagram Representation - uml.edu,22.451 Dynamic Systems – Block Diagrams Example - Block Diagram Representation Represent the block diagram as a single block relating X(s) and Y(s), and then obtain the overall transfer function s 5 s 2 s 3 1 + + + + X(s) s 3 1 + ∑ s 5 s 2 + + s 1 3 + U2(s) U1(s) U(s) Y(s) X(s) U(s) Solution First, the parallel combination is simplified and,Mechanical Block - Hewlett Packard Enterprise Community,Mechanical Block Code 8D 07. The shipping lock has been removed - which is the only suggestion I've been able to find. I am very frustrated with HP "support" and am hoping that someone on this forum can help me. I tried calling the HP support line (which says it is available 24/7) and got a message telling me to call back during normal business,On Teaching the Simplification of Block Diagrams*,On Teaching the Simplification of Block Diagrams* C. MEI Department of Mechanical Engineering, The University of Michigan—Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA. E-mail: [email protected] Due to their simplicity and versatility, block diagrams are widely used by control engineers to model all types of dynamic systems.Block Diagrams | Mechanical Engineering | Engineering,,Block diagrams solution extends ConceptDraw DIAGRAM software with templates, samples and libraries of vector stencils for drawing the block diagrams. Mechanical System Design Block DiagramBlock Diagrams: Modeling and Simulation,Block Diagram Modeling: Analogy Approach Physical laws are used to predict the behavior (both static and dynamic) of systems. Electrical engineering relies on Ohm’s and Kirchoff’s laws Mechanical engineering on Newton’s law Electromagnetics on Faradays and Lenz’s laws Fluids on continuity and Bernoulli’s law Based on electrical analogies, we can derive the fundamentalMechanical Block - Hewlett Packard Enterprise Community,Mechanical Block Code 8D 07. The shipping lock has been removed - which is the only suggestion I've been able to find. I am very frustrated with HP "support" and am hoping that someone on this forum can help me. I tried calling the HP support line (which says it is available 24/7) and got a message telling me to call back during normal business,Section 2 Block Diagrams & Signal Flow Graphs,K. Webb MAE 4421 3 Block Diagrams In the introductory section we saw examples of block diagrams to represent systems, e.g.: Block diagrams consist of Blocks–these represent subsystems – typically modeled by, and labeled with, a transfer function Signals–On Teaching the Simplification of Block Diagrams*,On Teaching the Simplification of Block Diagrams* C. MEI Department of Mechanical Engineering, The University of Michigan—Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA. E-mail: [email protected] Due to their simplicity and versatility, block diagrams are widely used by control engineers to model all types of dynamic systems.
## Mechanical Layout of Scripture | The Mechanical Layout of,
The mechanical layout of Scripture is a valuable skill to aid students or preachers in determining the author’s meaning within the Scriptures. The mechanical layout of Scripture outlines a verse or verses of Scripture in order to expose the flow of thought in a visual way.3.2.3 Block Diagram of Di fferential Equation Models,a block diagram for this state-space model. Example 14 Block diagram for a second order differential equation Given the differential equation m¨y = −Dy˙ −K fy +F (3.36) (which is a model of the mass-spring-damper system, cf. Example 4 on page 26). We will draw a block diagramModelling of Mechanical Systems - Tutorialspoint,Rotational mechanical systems move about a fixed axis. These systems mainly consist of three basic elements. Those are moment of inertia, torsional spring and dashpot. If a torque is applied to a rotational mechanical system, then it is opposed by opposing torques due to moment of inertia, elasticity and friction of the system.EXAMPLE PROBLEMS AND SOLUTIONS,EXAMPLE PROBLEMS AND SOLUTIONS A-3-1. Simplify the block diagram shown in Figure 3-42. Solution. First, move the branch point of the path involving HI outside the loop involving H,, as shown in Figure 3-43(a).Then eliminating two loops results in Figure 3-43(b).Combining twoControl Systems - Block Diagram Algebra - Tutorialspoint,The equivalent block diagram is shown below. Similarly, you can represent the positive feedback connection of two blocks with a single block. The transfer function of this single block is the closed loop transfer function of the positive feedback, i.e., $\frac{G(s)}{1-G(s)H(s)}$ Block Diagram Algebra forMechanical Engineering Drawing Symbols Pdf Free Download,,All the best Mechanical Engineering Drawing Symbols Pdf Free Download 36+ collected on this page. Feel free to explore, study and enjoy paintings with PaintingValleyMechanical Engineering: Block Diagram of Transmission,,Apr 6, 2016 - Mechanical Engineering: Block Diagram of Transmission system!!Mechanical Engineering | Mechanical Drawing Software,,This solution extends ConceptDraw DIAGRAM.9 mechanical drawing software (or later) with samples of mechanical drawing symbols, templates and libraries of design elements, for help when drafting mechanical engineering drawings, or parts, assembly, pneumatic, Mechanical Block DiagramTypes and Applications of Engineering Drawings,ASME Y14.24-2012 [Revision of ASME Y14.24-1999 (R2009)] Types and Applications of Engineering Drawings Engineering Drawing and Related Documentation PracticesBlock Diagrams | Mechanical Engineering | Engineering,,Block diagrams solution extends ConceptDraw DIAGRAM software with templates, samples and libraries of vector stencils for drawing the block diagrams. Mechanical System Design Block Diagram
## Mechanical Engineering | Mechanical Drawing Software,
This solution extends ConceptDraw DIAGRAM.9 mechanical drawing software (or later) with samples of mechanical drawing symbols, templates and libraries of design elements, for help when drafting mechanical engineering drawings, or parts, assembly, pneumatic, Mechanical Block DiagramOn Teaching the Simplification of Block Diagrams*,On Teaching the Simplification of Block Diagrams* C. MEI Department of Mechanical Engineering, The University of Michigan—Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA. E-mail: [email protected] Due to their simplicity and versatility, block diagrams are widely used by control engineers to model all types of dynamic systems.Mechanical Engineering: Block Diagram of Transmission,,Apr 6, 2016 - Mechanical Engineering: Block Diagram of Transmission system!!Mechanical Engineering Projects Ideas for College Students,,The following is a brief description of mechanical work. COMPLETE BLOCK DIAGRAM WITH DESCRIPTION: The block diagram of a computer-controlled drilling machine is shown below which consists of the following parts.They are Microcontroller,stepper motors,AC motor,limit switches,relay,PC,MAX232 etc.Types and Applications of Engineering Drawings,ASME Y14.24-2012 [Revision of ASME Y14.24-1999 (R2009)] Types and Applications of Engineering Drawings Engineering Drawing and Related Documentation PracticesBlock Diagram Reduction,Block Diagram Algebra for Branch Point Figure 8: Branch Points . Block Diagram Reduction Rules Table 1: Block Diagram Reduction Rules Table 2: Basic rules with block diagram transformation . Example 1: Example 2: Example 3: Example 4: Example5: ECE 680 Modern Automatic Control Routh’s Stability Criterion June 13, 2007 1Control Engineering Problems with Solutions,Block diagrams are used by engineers to show how the possibly large number of components, which are present in many systems, are interconnected. The information in a block may be purely descriptive, such as that shown in Figure 2.1, which describes the components of a typical measurement system, orMechanical Engineering Drawing Symbols Pdf Free Download,,All the best Mechanical Engineering Drawing Symbols Pdf Free Download 36+ collected on this page. Feel free to explore, study and enjoy paintings with PaintingValleyMathematical Modelling of Control System | Mechanical,,Oct 25, 2020· Mathematical modeling of a control system is the process of drawing the block diagrams for these types of systems in order to determine their performance and transfer functions. Now let us describe the mechanical and electrical type of systems in detail.ME451: Control Systems,Block diagram with feedbackBlock diagram with feedback Next StabilityStability of linear control systems, one of the most of linear control systems, one of the most important topics in feedback control Exercises Read Section 2.7, 2.8. Go over the derivation for DC motor transfer functions by yourself. Obtain T(s)/Ea(s).
## Mechanical Fuel Pumps - Basic Information And Troubleshooting
Jan 23, 2020· Common GM Mechanical Fuel Pumps How To Replace Mechanical Fuel Pumps: STEP-1. So, There are two holes in the passenger side front of the engine. These holes were for engines that used front engine mounts. The top bolt hole aligns with theBlock Diagram Templates - SmartDraw,Browse block diagram templates and examples you can make with SmartDraw.BANG & OLUFSEN BEOLAB 2 6861 REPAIR MANUAL Pdf,Page 10 Block diagram Block diagram BLOCK DIAGRAM, Measuring points MEASURING POINTS STATIC ELECTRICITY MAY DESTROY THE PRODUCT The mechanical limit for the driver is easily reached when the vacuum tightness in WARNING! BeoLab 2 is broken. Note! All measurements have to be done with the NTC module connected !,,,,,,, | 2021-08-02 18:17:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3370365500450134, "perplexity": 3015.4028154757243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154356.39/warc/CC-MAIN-20210802172339-20210802202339-00551.warc.gz"} |
https://www.greencarcongress.com/2007/06/nevis_engine_co.html | ## NEVIS Engine Company Closes Seed Round; New Two-Stroke Concept Engine Claims to Nearly Double Fuel Efficiency
##### 14 June 2007
The Bortone cycle. Click to enlarge.
The NEVIS Engine Company Ltd. has successfully closed its first round of seed funding to help it further test and develop its current two-cylinder prototype engine as well as to develop awareness of the technology. The Italian company has been developing the prototype with €2.5 million (US$3.3 million) in government grant money. The NEVIS (New Exhaust Valve & Intake System) engine embodies a new two-stroke combustion cycle (the “Bortone” cycle after its inventor) and a new design than combines, among other features, annular (i.e., doughnut-shaped) pistons, modular cylinder construction and a sinusoidal camshaft similar to those adopted in engines with cylinders arranged co-axially around the shaft. A NEVIS module. Click to enlarge. The company claims that the engine offers efficient combustion at all levels of power demands, and that it can nearly double the fuel efficiency obtained by conventional internal combustion engine technologies. Although the NEVIS engine is lighter and smaller than a conventional ICE engine of comparable cylinder displacement, it offers more power, due in part, the company says, to the engine offering six times the number of power strokes per revolution than a traditional four-stroke engine. The two-cylinder prototype was put through a bench test with engine ignition for the first time in Lecce, Italy in early 2006 as part of the review process by MUIR (Ministry of Education, University and Research) for the government grants that funded the development of the prototype. First NEVIS prototype Cylinders 2 Displacement 1,000 cc Bore 80mm int., 178mm ext. Power kW (hp) Est. 187 (250) @ 2,000rpm Average piston velocity 7.5 m/sec Engine block Steel/aluminum Weight 80 kg Power/Weight ratio (kW/kg) 2.38 Compression ratio 7:1 to 38:1 Injectors per cylinder 3 Sparkplugs per cylinder 3 This first test confirmed the correct functioning of the Bortone Cycle and the basic mechanical operation of the NEVIS engine. Preliminary testing without ignition also confirmed nearly 50% less friction/pumping resistance for an engine with a comparable displacement. The five key design concepts of the engine are: • Obtaining optimized scavenging without the need for an additional turbine or compressor. • Development of a new two-stroke cycle that allows partial loads to have an expansion stroke greater than the compression stroke (like the Miller cycle in a four-stroke engine, but in a two-stroke cycle). Enabling that is a controlled annular exhaust valve that allows for variable duration and phasing of its opening. • The use of a sinusoidal camshaft to transform the alternate motion of the pistons into rotary motion. The shaft in the NEVIS provides the ability to complete three combustion cycles within a single shaft revolution. • A variable compression ratio made possible at all rpm and power loads by the regulation of an annular screw element within the shaft. • The use of annular pistons to enhance thermal efficiency, allow for a light structure, and to integrate the engine’s other key concepts. Within all that, the principal innovation of the NEVIS engine, according to the company, is the system used to vary load needs. The extended opening of the exhaust valve allows the piston to expel the air that has replaced the exhaust gases through the cleaning phase. The longer the valve stays open, the smaller the amount of air available for the combustion that follows, and as the compression ratio may be varied as desired, it is possible to have very small quantities of air charge. In other words, the load is reduced but not the efficiency of the engine that totally utilizes the expansion of the combustion with an expansion stroke that is inversely longer in relation to the load entity. If the opening time of the exhaust valve is short and the air is prevented from flowing out of the exhaust duct, the compression ratio can turn back to the initial proportion and significant loads can be achieved, especially if the inertia of the air in the intake duct causes some supercharging. For the sake of efficiency, it is better not to run the engine under pressures that are too high as this requires greater depressions in the combustion chamber that can only be obtained using the kinetic energy of the exhaust gases which, flowing out at high velocity, cause depressions. The more these are intense and durable the higher the velocity and the quantity of the exhaust gases. Thus, anticipating the opening of the exhaust valve when there is still a certain pressure in the combustion chamber improves the filling process, but causes a loss of efficiency as expansion doesn’t take complete advantage of the pressure given by the combusted gases. The NEVIS engine overcomes a number of issues with traditional two-stroke engines: • The increased size of the intake and exhaust ports enable higher flow. • The scavenging of the NEVIS engine differs from that of the traditional two-stroke and it can be compared to four-stroke efficiency. It can be complete at all rpm and at all loads with an optimal expulsion of all combustion residues due to the new cycle, to the particular unidirectional outflow and to the longer scavenging phase. • The variable lifting law of the exhaust valve and its regulating timing system, in combination with the new “loading” method, avoids the losses of any quantity of fuel at the end of the scavenging phase; in fact, the exhaust valve during this precise phase will always be closed. The NEVIS engine also provides a time for vaporization that is 2.6 times longer than in the normal two-stroke engine: this allows the use of direct injection operating with relatively modest pressure and ensures good injection even at high rpm and high loads. • There is a longer-lasting pressure at the end of the stroke due to the modest height of the intake ports and that the opening of the exhaust begins at a point similar to that of a four-stroke engine (55-60 degrees of angle of crank) with respect to the bottom dead point. The company is also exploring adapting the NEVIS for HCCI (Homogeneous Charge Compression Ignition). The NEVIS engine adapted for HCCI would not only increase efficiency further but would also do away with NOx without the PM (particulate matter) emissions of a diesel, according to the company. Resources: ### Comments If this engine can truly deliver the goods, then the big three better figure out a way to get a piece of the pie. Here is why: With such a high power density, you could probably make an engine of about 50 Hp with less of 30 Kg of mass. Why would this be important? Because if this engine can brake the 50% thermal efficiency barrier, that would make it the best engine for a genset in a series electric dominant hybrid. As with all new engine proposals, it only becomes relevant after they can produce a working example at power outputs usable in the real world that meets all requirements. Yeah there have been people trying the reinvent the ICE for over a hundred years now, almost all of them have come out with unsuccessful products. I remember the quiseturbine craze just a few years ago, what a load that was: they have not made a working gasoline burning version yet, works great with air and steam though. Does this design solve the pollution problem associated with 2 stroke engines? I thought the 2 stroke is a dying engine (except for chainsaws and unique applications) due to it's high pollution of burning oil mixed in the gas? yeah, no info above on pollution However, a 1000cc 80Kg engine with 250hp@2000rpm *IS* a usable size and power output, if they could succesfully apply HCCI that should help with any potential pollution issues. A very well-thought out engine concept that has inherently much lower friction than trational piston-crank 4-stroke cycle engine. Coupled with the inherent variable compression mechanism and Kadenacy uniflow scavenging mechanism taking advantage of the inertia of the exhaust gas, along with the variable exhaust valve timing in the compression stroke in order to throttle the air intake charge without require a power-robbing negative-pressure throttle plate, partial-load efficiency will be much better than traditional gasoline engine and even diesel engine, given the much lower internal friction level. Peak efficiency will probably be no more than about ~15-20% higher than of the best of current optimized 4-stroke cycle engine due entirely to reduction in friction, BUT, partial-load efficiency can easily DOUBLE the best of current engines, given the lower friction level, variable compression, and the extended time in the cycle for exhaust, scavenging, and combustion. Conceptually, this engine operates similarly to the 2-stroke Detroit Diesel or Orbital 2-stroke GDI, but it has much improved scavenging mechanism, much improved flow, and even lower internal friction due to the use of one large annular piston replacing 2-3 smaller pistons having much larger side-wall surface subjected to friction loss. However, some oil loss from the piston rings to the intake air on the upward compression stroke may be inevitable, and unless this oil can be burned completely within one combustion cycle, higher HC emission may be possible. The longer combustion duration when the piston is at TDC may help in this respect. The high heat in the combustion chamber may impair lubrication of the "Guillotine" exhaust valve sliding up and down on the inner surface of the cylinder head. Perhaps some advance tribology method may be required to overcome this issue. Of course, more problems may surface during further development process, but so far, this concept looks very promising. darwin, No oil and gas mixing required here. Lubricating oil can be sprayed on the lower cylinder when the piston is near TDC. Series hybrids certainly would profit by a more efficient engine, but the cost better not be much since greater all-electric ranges are becoming greater as battery prices decline, making greater engine economies more and more irrelevant. The series hybrid is only a transitional vehicle anyway. Kerry, If one can make a H2-ICE-HEV that can match the efficiency of electrical generation and transmission via the grid, then why bother with plug-in charging nitely and expensive and heavy battery? Instead of burning the coal, natural gas or waste biomass to generate electricity, one will just need to convert these directly into H2 via gasification process at even higher efficiency than electrical generation. Then, if the FC-HEV or super-efficient ICE-HEV can utilize the H2 with an overall efficiency from-source-to-wheel higher than BEV, one can forget about low-cost battery development. A 1.5kwh of battery capacity per HEV will be all that will be needed. What to do with solar and wind electricity? Just feed it directly to the grid, while conserving coal, natural gas, or waste biomass to generate H2. Do not make H2 from electricy using normal-temp electrolysis of water, because this is an expensive and inefficient way to produce H2. 4kg of H2 energy can be stored in mobile tanks 160-liter size at a BEV-equivalent capacity of 115 kwh, with the cost the storage tank of ~$2000 USD, for a range of 300 miles. Fuelcell costing too much? Possible! Now, do you see the beauty of a low-cost superefficient ICE? Try to calculate how much it would cost for a BEV with that much battery capacity?
Posted by: Kerry Buehrt | Jun 15, 2007 6:40:34 PM
Kerry,
If one can make a H2-ICE-HEV that can match the efficiency of electrical generation and transmission via the grid, then why bother with plug-in charging nitely and expensive and heavy battery?
Why bother?
Because what you are saying is not physically POSSIBLE, it defies the laws of physics.
http://greyfalcon.net/hydrogen
http://greyfalcon.net/hydrogen.png
http://greyfalcon.net/hydrogen4.png
Furthermore, hydrogen infrastructure does not exist.
So why couldn't we just make electric quickcharge infrastructure?
Can fill up for about a hundred mile drive in 1 minute.
http://greyfalcon.net/quickcharge3
http://greyfalcon.net/quickcharge
GreyFalcon,
I have debunked Ulf Bossel's defamation of the Hydrogen Economy many times here in GCC.
His assumption is that H2 will be produced by room-temp electrolysis of water using electricity from either the grid or wind or solar. He is correct to say that H2 produced this way will give H2-Vehicle 1/2-1/3 the efficiency of electric BEV.
But, commercially, H2 is not produced by electrolysis of water. It is produced by steam reformation of natural gas with about 70% efficiency, or even higher when the resultant heat will be recycled into electricity production. Gasification of coal or waste biomass can produce H2 at comparable efficiency as from natural gas.
In contrast, a Combined-cycle natural gas power plant is rated at 55-60% efficiency. Coal-fired powerplants has 35-40% efficiency. Distributon of this electricity through the grid cost 8% of the energy in the electricity.
If the H2 is produced locally, distribution of the H2 locally within a 15-mile radius only cost 3% of the energy within the Hydrogen. If the H2-Vehicle is fueled right where the Hydrogen is produced within the city limit, the cost of distribution will be even less. Compression of the H2 will cost ~7-9% of the energy of the H2, but a substantial portion of this energy is recoverable to do useful work, for example, to run accessories in the vehicle. See theaircar.com website, in which a car is run entirely on compressed air.
Finally, a FCV like the latest Honda FCX was quoted by Honda to have 60% efficiency tank to wheel. A BEV is recognized to have about ~70% efficiency grid to wheel, factoring in losses in the charger, battery's internal resistance, heat production in power semiconductors and in the motor, requiring liquid cooling, and battery self-discharge.
So, for H2-V: 70% x 94% (distr+compressn)x60%= 39% source-to-wheel efficiency!
For BEV: 55% x 92%(distribution)x70%= 35% source-to-wheel efficiency! Note that coal-fired electricity will result in even lower overall efficiency for BEV, but I am a kind person.
What about wind and solar electricity? Don't bother to make H2 with them. Just feed them to the grid and save the coal or biomass for H2 generation. There will be plenty of waste biomass to serve as H2 source to meet all transportation need.
In the future, high-temp electrolysis for H2 production, having twice the electrical efficiency of normal temp can be used, using the waste heat from biomass or coal gasification, will make wind and solar electricity H2 equally efficient as BEV from renewable electricity.
Roger Pham,
That a very good run down, and I like it, only one problem: how much is a FCV going to cost? Surely PEM Fuel Cells with composite fuel tanks with hydrogen absorber, decompression generator, electric batteries and motor will cost more then just larger electric batteries and motor (as in a EV). Also add in the cost of a new infrastructure for transporting and making hydrogen, which EV and biofuels need to a much lesser extent. Electric grids already exist and off-peak power is just begging to be used, all that needed is the cars. Many biofuels can run in existing cars and can us existing infrastructure for gasoline and diesel, all that needed is the production. Hydrogen needs new cars, new infrastructure and new production!
Ben,
Good question. That's why a very efficient ICE but low cost, with peak efficiency approaching that of the Fuel Cell would be a great alternative. Don't dismiss the ICE as yet, that's the gist of this discussion. Development on the FC front is also promosing, with prediction that cost of the FC stack can be reduced to $4000 USD/vehicle eventually. GM and Honda have plan to put FCV in limited production by 2008. Infrastructure for H2? Very simple. Put a gasification plant for every 10 x 10 =100 square miles of urban area, and then each vehicle will have to travel but 5-7 mile one-way trip to fuel up, initially. No need to put H2 into tankers to travel down the road. Definitely not a station every street corner like we have for gas station now, but such is a waste and not really necessary. When more H2 vehicle will show up, there will be sufficient commercial justification for additional pipelines to conduct H2 from these plants to the more local stations. Honda has a device whereby you can make H2 from home from natural gas, if you happen to live too far away from a H2-filling station. This device also provide heat for your hot water and home heating also, I would presume, from the exothermic reaction of NG to H2 conversion, thus higher efficiency of conversion than calculated when waste heat can be utilized. Expect that H2 will cost the same as gasoline now for an equivalent energy content, but H2-Vehicles can travel twice as far as gasoline vehicles, so you can bet that you'll pay 1/2 as much for energy cost to travel the same distance. This energy cost saving will pay for the higher initial cost of the H2-capable vehicles. a ICE with near Carnot effiecency would be nice, but I doubt its viable. There have been many claims from many companies over the years of new kinds of engines but none work out as good as claimed. Still the amount of change need for hydrogen is far more extensive then EV or biofuels, time for change is very short if peak oil is a consideration. Also what happens when natural gas peaks (Soon after oil by most estimates) and coal is restricted by greenhouse gas initiatives? Water electrolysis my be the last viable option for hydrogen, even so high efficiency electrolysis of water is under development. Why hydrogen, why not zinc paste? Zinc can be recycled at high efficiencies (~70%) at room temperature and can be recycled at the fuel station. Zinc paste is also not explosive or combustible (as long is does not dry out) and as a silvery toothpaste consistancy like paste it’s far easier to store then hydrogen. Zinc-air fuel cells are also more efficient then PEM. hydrogen powered turbines with high speed alternator/generators are the best bet. anyone with water and electricity can make their own fuel. my company will be developing hybrid turbogensets for vehicles in the next year or so, but 500 mile range systems will cost about$50k. roughly 80% efficient with a 90% efficient turbine, 95-99% efficient mechanical to electric conversion, and 96-98% efficient power conditioning. The main advance will be in ultra efficient liquid hydrogen and liquid oxygen production.
With all the claims of 50%improvements in this and 30% improvement in that, and so on and so forth, at the end of the day there is 100% B.S., if half of these exagerated claims were true- when you opened your gas cap- gas would gush out of it. Where does it say the engine even runs- but I guess it will spin a darn good yarn. I'm not just talking about this engine but I read about countless engine technologies every week and research galore but come on, wolf boy, sooner or later no one will listen if the truth is thrown about sooo loosely. Sorry if I snapped but these claims are friggin rediculous-I Digress I WANT FACTS AND I WANT EM NOW -is this turning into politics?
The comments to this entry are closed. | 2022-12-01 10:40:52 | {"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": 0.349325954914093, "perplexity": 3283.8066133488446}, "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/1669446710808.72/warc/CC-MAIN-20221201085558-20221201115558-00001.warc.gz"} |
https://stats.stackexchange.com/questions/197477/how-to-get-the-regressed-output/361964#361964 | # How to get the regressed output?
I have a model Y = slope1*variable1 + slope2*variable2 + Intercept.
I used lm in R to get slope1, slope2 and Intercept.
In this case, variable1 is my main effect and I want to remove the effect of variable2. The goal is to see if there is any association between Y controlled for variable 2 (regressed Y) and variable 1.
In order to plot these :
(1) Should I subtract observed_Y-slope2*variable2-Intercept-Residuals and whatever is remaining (I call this regressed Y) is actually the value that is due to variable 1 or
(2) Can I use model_fitted_dot_values from R and plot that as a function of variable1 and claim that model_fitted_dot_values are the regressed values of my dependent variable?
Any help is greatly appreciated. | 2021-09-25 06:46:51 | {"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": 0.5241879224777222, "perplexity": 1857.1032271435517}, "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-2021-39/segments/1631780057598.98/warc/CC-MAIN-20210925052020-20210925082020-00364.warc.gz"} |
https://www.jpe.ac.cn/EN/10.1093/jpe/rtw106 | J Plant Ecol ›› 2018, Vol. 11 ›› Issue (2): 169-179.
• Research Articles •
### Relative importance of hydrological variables in predicting the habitat suitability of Euryale ferox Salisb.
Ji Yoon Kim1, Gu-Yeon Kim1, Yuno Do1, Hee-Sun Park2 and Gea-Jae Joo1,*
1. 1 Department of Integrated Biological Science, College of Natural Sciences, Pusan National University, Busan 46241, Republic of Korea; 2 Nakdong Estuary Eco Center, Busan 49435, Republic of Korea
• Received:2015-12-21 Accepted:2016-09-27 Published:2018-02-06
• Contact: Joo, Gea-Jae
Abstract: Aims Aquatic ecosystems are a priority for conservation as they have become rapidly degraded with land-use changes. Predicting the habitat range of an endangered species provides crucial information for biodiversity conservation in such rapidly changing environments. However, the complex network structure of aquatic ecosystems restricts spatial prediction variables and has hitherto limited the use of habitat models to predict species occurrence in aquatic ecosystems. We used the maximum entropy model to evaluate the potential distribution of an endangered aquatic species, Euryale ferox Salisb. We tested the relative influence of (i) climatic variables, (ii) topographic variables, and (iii) hydrological variables derived from remote sensing data to improve the prediction of occurrence of aquatic plant species.
Methods We considered the southern part of the Korean Peninsula as the modeling extent for the potential distribution of E. ferox. Occurrence records for E. ferox were collected from the literature and field surveys. We applied maximum entropy modeling using remotely sensed environmental variables and evaluated their relative importance as prediction variables with variation partitioning.
Important findings The species distribution model predicted potential habitats of E. ferox that matched the actual distribution well. Floodplain wetlands and shallow reservoirs were the favored habitats of E. ferox. Quantitative loss and fragmentation of wetland habitats appeared to be a major reason for the decrease of E. ferox populations. Our results also imply that hydrological variables (i.e. normalized difference water index) derived from remote sensing data greatly increased model prediction (relative contribution: 10.5–37.0%) in the aquatic ecosystem. However, interspecific competition within a similar niche environment should be considered to increase the accuracy of the distribution model.
Aims Aquatic ecosystems are a priority for conservation as they have become rapidly degraded with land-use changes. Predicting the habitat range of an endangered species provides crucial information for biodiversity conservation in such rapidly changing environments. However, the complex network structure of aquatic ecosystems restricts spatial prediction variables and has hitherto limited the use of habitat models to predict species occurrence in aquatic ecosystems. We used the maximum entropy model to evaluate the potential distribution of an endangered aquatic species, Euryale ferox Salisb. We tested the relative influence of (i) climatic variables, (ii) topographic variables, and (iii) hydrological variables derived from remote sensing data to improve the prediction of occurrence of aquatic plant species.
Methods We considered the southern part of the Korean Peninsula as the modeling extent for the potential distribution of E. ferox. Occurrence records for E. ferox were collected from the literature and field surveys. We applied maximum entropy modeling using remotely sensed environmental variables and evaluated their relative importance as prediction variables with variation partitioning.
Important findings The species distribution model predicted potential habitats of E. ferox that matched the actual distribution well. Floodplain wetlands and shallow reservoirs were the favored habitats of E. ferox. Quantitative loss and fragmentation of wetland habitats appeared to be a major reason for the decrease of E. ferox populations. Our results also imply that hydrological variables (i.e. normalized difference water index) derived from remote sensing data greatly increased model prediction (relative contribution: 10.5–37.0%) in the aquatic ecosystem. However, interspecific competition within a similar niche environment should be considered to increase the accuracy of the distribution model. | 2022-09-29 16:32:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4209754168987274, "perplexity": 7187.084225297664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335362.18/warc/CC-MAIN-20220929163117-20220929193117-00689.warc.gz"} |
https://physics.aps.org/articles/v13/s115 | Synopsis
Hints of Dark Bosons
Physics 13, s115
A signal predicted for a type of dark matter appears in the spectra of ytterbium isotopes.
Excitement is brewing among dark-matter researchers over signs that they may have detected the first signals of this elusive substance. In June, the XENON Collaboration reported seeing flashes of light that match those expected from a type of dark matter called dark bosons. Now two teams of researchers report results from another set of experiments also seeking dark bosons, with one of the teams finding hints consistent with this particle [1, 2].
Researchers have proposed a plethora of dark-matter candidates that explain astrophysical observations while conforming to the results of previous experiments. One of those candidates is the dark-matter boson, a particle that is predicted to interact weakly with ordinary matter. These dark bosons would be “virtually” exchanged between an atom’s electrons and neutrons and induce tiny forces between them, thereby changing the atom’s transition frequencies. Detecting this change was the aim of the two teams.
In their experiments, both teams measured the so-called isotope shift—the change in atomic spectra exhibited by isotopes of the same element. A group led by Vladan Vuletić of the Massachusetts Institute of Technology measured this shift between five ytterbium istotopes, while Michael Drewsen of Aarhus University in Denmark and colleagues measured the shift between five calcium ones.
The standard model predicts that, to first order, the isotope shifts should lie on a straight line on a “King” plot, a common method for representing the phenomenon. The Drewsen group’s measurements fit with this prediction. The Vuletić group, however, reports a departure from predictions, finding a deviation from linearity with a statistical significance of 3 sigma. The researchers say that the deviation is consistent with a leading correction to the standard model’s prediction, but it could also indicate the existence of dark bosons.
–Katherine Wright
Katherine Wright is the Deputy Editor of Physics.
References
1. I. Counts et al., “Evidence for nonlinear isotope shift in ${\text{Yb}}^{+}$search for new boson,” Phys. Rev. Lett. 125, 123002 (2020).
2. C. Solaro et al., “Improved isotope-shift-based bounds on bosons beyond the standard model through measurements of the ${}^{2}{D}_{3∕2}{-}^{2}{D}_{5∕2}$ interval in ${\text{Ca}}^{+}$,” Phys. Rev. Lett. 125, 123003 (2020).
Subject Areas
Particles and Fields
Related Articles
Particles and Fields
Sizing up Exotic Nuclei with Radioactive Molecules
New research shows that radioactive molecules can be used to study the variation in the shapes and sizes of exotic nuclei that are particularly sensitive to fundamental symmetry violations. Read More »
Particles and Fields
Generating Antineutrons and Hyperons with Existing and Future Facilities
Antineutrons and hyperons are challenging to produce and study, but researchers propose a new approach that would generate these particles using existing and planned accelerators. Read More »
Particles and Fields
Neutrinos Rising from the Floor
A neutrino background that could confound dark matter searches is now becoming an opportunity for probing new physics. Read More » | 2021-07-31 03:36:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5138028860092163, "perplexity": 2388.3344824710703}, "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/1627046154042.23/warc/CC-MAIN-20210731011529-20210731041529-00575.warc.gz"} |
https://www.physicsforums.com/threads/blackbody-radiation.720161/ | 1. Nov 1, 2013
sayebms
consider energy for a damped electric oscillator . ("$f$" indicates the dipole moment of the oscillator)
in the absence of the damping force
$U= \frac{1}{2}kx^2 +1/2 (\frac{d^2x}{dt^2}) ^2$
and the energy conservation tells us $dU=0$.
but if there is damping force we get the following using larmor formula and energy of a dipole in an electric field, for the conservation of energy:
$\int_t^τ ( \frac{dU}{dt} + \frac{2}{3c^2} (\frac{d^2f}{dt^2})^2 -E \frac{df}{dt} )$
and here is i don't understand: using the fact that $\frac{4π^2 v_0}{3c^3 L} = σ$
and the above conseravtion of energy formula we get to
$Kf+L \frac{d^2f}{dt^2} -2/(3c^3) \frac{d^3f}{dt^3}=E$
i dont really know how we got to this last formula using the above equations. any help is appreciated. and for those who have access to the book The question is from page 184 of the book "Planck's Columbia Lectures".
2. Nov 1, 2013
Simon Bridge
Check that first equation.
You can't do it with just those equations - you also need to know how that $\sigma$ comes in.
3. Nov 1, 2013
sayebms
Thank you very much for your time.
actually that is the part i didn't understand. because i don't see any relation between the formula for the conseravation of energy and $σ$ , in the book it just appears suddenly and then the author concludes the final equation (the differential equation) by saying that the constant $σ$ is small.
4. Nov 1, 2013
Simon Bridge
That sounds... odd.
Either the author has introduced it someplace else that he expects you to have read (maybe quite early in the book) or you are expected to crunch your way through the math and notice a bunch of constants making a mess ... rearrange them into the ratio that he labels with a sigma, realize that it is small...
5. Nov 2, 2013
sayebms
It is indeed odd. But even more surprisingly i found what it refers to after reading the next chapter.
After some pages it is indicated that the constant sigma actually refers to the damping constant (as the system is considered as damped oscillator since it emits radiation) but still i don't get the relation between this constant beeing small and the last formula which is given:
$Kf+L \frac{d^2f}{dt^2}−2/(3c^3)\frac{d^3f}{dt^3}=E$
but i suspect that it is somehow used to approximate the following:
$2/(3c^3)\frac{d^2f}{dt^2}= −2/(3c^3)\frac{d^3f}{dt^3}$
could it be right??
6. Nov 2, 2013
Simon Bridge
The approach to take is to consider what the author wants to calculate and where the author starts from.
If this is unclear, then use a different text book.
7. Nov 3, 2013
Jano L.
To derive the equation, we may argue this way. Plug-in the expression for $U$ into the integral, differentiate term by term and express $\ddot{f}^2$ as
$$\frac{d}{dt}\left(\ddot{f}\dot{f}\right) - \dddot{f} \dot{f}.$$
Collect the terms containing $\dot{f}$. We arrive at
$$\int_t^{t+\tau} \left( Kf + L\ddot{f} - \frac{2}{3c^3}\dddot{f} - E_z \right)\dot{f} + \frac{2}{3c^3}\frac{d}{dt}\left(\ddot{f} \dot{f}\right) \,dt.$$
Now since the last terms is total time derivative, its integral is easy to do. It is equal to
$$\ddot{f} \dot{f} (t + \tau) - \ddot{f} \dot{f} (t).$$
If we assume that the system oscillates with low damping (low $\sigma$), the values of its amplitude, velocity and acceleration are almost the same as in the previous period, hence this term is small (with respect to $\ddot{f} \dot{f}$.
If we neglect it, we have the condition
$$\int_t^{t+\tau} \left( Kf + L\ddot{f} - \frac{2}{3c^3}\dddot{f} - E_z \right)\dot{f}\,dt = 0.$$
One possible way to satisfy this condition is to postulate
$$Kf + L\ddot{f} - \frac{2}{3c^3}\dddot{f} - E_z = 0.$$
I think this should be regarded as an heuristic procedure to arrive at some new interesting equation. It should not be regarded as rigorous derivation, since there are those two arbitrary steps.
The equation works approximately (to first approximation) for periodic motions. However, nowadays we know that this equation is not very satisfactory in general, because it leads to all sorts of problems (runaways, acceleration before force is applied, and alike). | 2017-12-11 11:33:41 | {"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9327472448348999, "perplexity": 396.0372842480745}, "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-2017-51/segments/1512948513478.11/warc/CC-MAIN-20171211105804-20171211125804-00463.warc.gz"} |
https://stats.stackexchange.com/questions/156182/untangling-lump-samp-panel-data | Untangling lump samp panel data?
Edit: Context: I am estimating persistence analog Heckman (1981) based on firm level data. The endogenous variable is the distribution amount of profits directed to the owners. To estimate persistence, I plan to regress the endogenous variable against its past realization. To circumvent the initial conditions problem, I plan on using the Wooldridge method ("Simple solution to the initial conditions problem", 2005). So far, I dummied the endogenous variable and I thus apply a dynamic random effects discrete choice model.
I am using Stata 13 to estimate a simple probit model. I use panel data of secondary nature. The data is conducted via an annual survey. Unfortunately, the endogenous variable of interest is conducted for the past three years respectively. That is, each year the survey taker is asked to indicate the sum of the endogenous variable for the past two years and the current year.
Let's have an example. The picture below shows the data structure. The first row depicts the periods, the second the endogenous variable and its annual value. The following rows beneath show the past individual annual values and the three year sum - that is the observed value. So we have 12 observed periods in this example. Period t0 contains the sum of periods t-1; t-2; t-3 - in this case 3 + 6 + 7 = 16. For period t1 the survey taker discloses 16 yet again - though this time as the sum of periods t-2; t-3; t-0 and thus 6 + 7 + 3 = 16 . Each observed value thus consists of the sum of the two previous years and the current year. The picture below tries to show that via the "red" and "blue" shadows.
The problem being, I only see the three year sum - that is the "diagonal, bold values". I do not have the annual data. Given the outset of my research question, having the annual data would however be highly beneficial. Two questions arise:
(1) Is there any smart way how I can untangle the lump sum data? (2) If not, what can I do?
• Would you possibly have some starting data, such as the data for period $-2$ and the sum for periods $-1$ and $-2$? Two such pieces of information are needed to obtain a unique deconvolution of these windowed (or moving-average) sums. – whuber Jun 9 '15 at 17:28
• @whuber, unfortunately I don't. I personally do not see a arithmetic way to arrive at the unique and thus deconvolunized (cool word, thanks!) value. – Rachel Jun 9 '15 at 17:30
• OK, then we need to seek ways to avoid using that missing information. It is possible that you could obtain needed estimates without it, but that would depend on details of your model and what you're trying to do with it. There are many ways to approach this, but any edits you can make to the question to provide more context would help guide your readers and would-be respondents. – whuber Jun 9 '15 at 17:33
• Sure, no problem: I am estimating persistence analog Heckman (1981) based on firm level data. The endogenous variable is the distribution amount of profits directed to the owners. To estimate persistence, I plan to regress the one year lag of the endogenous variable on the endogenous variable. To circumvent the initial conditions problem, I plan on using the Wooldridge method ("Simple solution to the initial conditions problem", 2005). So far, I dummied the endogenous variable and I thus apply a dynamic random effects probit model – Rachel Jun 9 '15 at 18:06 | 2019-08-18 14:02:38 | {"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": 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": 0.5701783299446106, "perplexity": 1003.7882372401898}, "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-2019-35/segments/1566027313889.29/warc/CC-MAIN-20190818124516-20190818150516-00370.warc.gz"} |