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https://simpleitk.readthedocs.io/en/master/link_CppCMake_docs.html
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# A Simple C++/CMake Example¶
Building a C++ SimpleITK application is supported by using CMake to configure the build process. CMake can generate project buildsystems for Unix Makefiles, Ninja, Microsoft Visual Studio, or Mac OS Xcode.
Here is a basic CMakeLists.txt configuration file for building a SimpleITK C++ example program.
cmake_minimum_required(VERSION 3.14)
project(sitk_example)
find_package(SimpleITK)
Here is our SimpleITK C++ example file sitk_example.cxx.
// This one header will include all SimpleITK filters and external objects.
#include <SimpleITK.h>
#include <sitkImageOperators.h>
// create convenient namespace alias
namespace sitk = itk::simple;
int main ( int argc, char* argv[] ) {
sitk::PixelIDValueEnum pixelType = sitk::sitkUInt8;
std::vector<unsigned int> imageSize ( 2, 128 );
// Create a black image
sitk::Image image( imageSize, pixelType );
// Add a Gaussian blob to the image
std::vector<double> blobSize ( 2, 64.0 );
std::vector<double> blobCenter ( 2, 64.0 );
image = image + sitk::GaussianSource( pixelType, imageSize, blobSize, blobCenter );
// Write the image as a PNG file
sitk::WriteImage(image, "blob.png" );
}
Invoking cmake as below in the same directory as these files will create the project buildsystem:
cmake .
## Required CMake Variables¶
Running CMake will produce a CMakeCache.txt file that is the CMake configuration. Within that text file there are two variables needed to build a SimpleITK program, SimpleITK_DIR and ITK_DIR. SimpleITK_DIR needs to be set to the directory that contains the file SimpleITKConfig.cmake, and ITK_DIR needs to be the directory containing the file ITKConfig.cmake.
CMake attempts to find these files automatically, but it is unlikely to find them unless SimpleITK and ITK are installed into the system.
If SimpleITK has been built using the SuperBuild process, as described in Building SimpleITK, then these required directories can be found in the SuperBuild location. The SimpleITK_DIR variable should refer to the directory SimpleITK-build within the SuperBuild location, and the ITK_DIR variable should refer to the directory ITK-build in the SuperBuild location.
Here are what the settings in a CMakeCache.txt file should look like:
//The directory containing a CMake configuration file for ITK.
ITK_DIR:PATH=/home/dchen/SuperBuild/ITK-build
//The directory containing a CMake configuration file for SimpleITK.
SimpleITK_DIR:PATH=/home/dchen/SuperBuild/SimpleITK-build
## Compiling using CMake¶
A SimpleITK program can also be compiled by running CMake. Here is the command to perform the build in the current directory:
cmake --build . --config Release
The ‘config’ parameter could also be set to Debug, RelWithDebInfo or MinSizeRel. On Unix-type systems Release is the default. On Windows the config must be specified explicitly.
Of course traditional methods can be used to build a SimpleITK program (e.g. make on Unix, or Visual Studio on Windows), but it is often convenient to be able to use the same build command on all system types.
## Using CMake on Windows¶
The Windows version of CMake includes a graphical-user-interface (GUI) version of the program, in addition to the standard command-line CMake. Here is what the CMake GUI looks like:
After the user specifies the source and build directories, CMake will ask to select which compiler generator to use. The options are determined by what versions of Visual Studio are installed on the system. The user must specify x64 for the platform version, as the default option (x86) is not supported by SimpleITK.
Next are the configuration and generation process. By clicking the Configure button, cmake reads the CMakeLists.txt and sets up all the variables in the CMake cache (CMakeCache.txt). Then when the Generate button is clicked, CMake creates the Visual Studio build system.
After CMake is completed the generation process the user can actually build SimpleITK in Visual Studio. The user should select a build type in Visual Studio. We recommend choosing Release, since the performance of SimpleITK with the Debug option is several orders of magnitude slower. Also the user must select the x64 build platform, as the default one, x86, is not supported by SimpleITK. Many man hours and cpu cycles have been wasted building SimpleITK without setting the proper platform.
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2021-01-17 18:36:45
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http://www.fightfinance.com/?q=130,194,209,234,435,509,557,615,652,910
|
# Fight Finance
#### CoursesTagsRandomAllRecentScores
An 'interest rate' is the same thing as a 'yield'. or ?
A firm wishes to raise $8 million now. They will issue 7% pa semi-annual coupon bonds that will mature in 10 years and have a face value of$100 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue?
Find Piano Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Piano Bar Income Statement for year ending 30th June 2013 $m Sales 310 COGS 185 Operating expense 20 Depreciation 15 Interest expense 10 Income before tax 80 Tax at 30% 24 Net income 56 Piano Bar Balance Sheet as at 30th June 2013 2012$m $m Assets Current assets 240 230 PPE Cost 420 400 Accumul. depr. 50 35 Carrying amount 370 365 Total assets 610 595 Liabilities Current liabilities 180 190 Non-current liabilities 290 265 Owners' equity Retained earnings 90 90 Contributed equity 50 50 Total L and OE 610 595 Note: all figures are given in millions of dollars ($m).
An 'interest only' loan can also be called a:
Will the price of a call option on equity or if the standard deviation of returns (risk) of the underlying shares becomes higher?
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid annually. So there's only one coupon per year, paid in arrears every year. An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 6% pa. • Stock A has an expected return of 5% pa. • Stock B has an expected return of 10% pa. What portfolio weights should the investor have in stocks A and B respectively? You buy a house funded using a home loan. Have you or debt? An equity index is currently at 5,200 points. The 6 month futures price is 5,300 points and the total required return is 6% pa with continuous compounding. Each index point is worth$25.
What is the implied dividend yield as a continuously compounded rate per annum?
Which of the following is NOT a money market security?
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2020-09-22 17:28:48
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https://l2h.site/2019/03/01/wp-disable-jetpack-beautiful-math-solve-mathjax-conflict/
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# WP禁用Jetpack自带Beautiful Math解决MathJax冲突
Categories Wordpress建站
## 排查
If your LaTex code is broken, instead of the equation you’ll see an ugly yellow and red error message. Sorry, we can’t provide support for LaTex syntax, but there are plenty of useful guides elsewhere online. Or a quick post in our forums might find you a solution. One thing to keep in mind is that WordPress puts all of your LaTex code inside a LaTex math environment. If you try to use LaTex that doesn’t work inside the math environment (such as \begin{align} ... \end{align}), you will get an error
## 解决
• 找到“Beautiful Math”或者中文“ 美妙的数学 ”
• 禁用之
## 3 thoughts on “WP禁用Jetpack自带Beautiful Math解决MathJax冲突”
1. lambert说道：
如果用的wordpress，可以去博客后台搜索下看看，XD
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2019-06-16 11:17:37
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https://www.imrpress.com/journal/RCM/23/12/10.31083/j.rcm2312386/htm
|
NULL
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IMR Press / RCM / Volume 23 / Issue 12 / DOI: 10.31083/j.rcm2312386
Open Access Original Research
Divergent Occurrence of Carotid Intima-Media Thickness and Carotid Arteries Plaques in Stable Kidney Transplant Recipients
Show Less
1 Department of Nephrology, Transplantation and Internal Medicine, Medical University of Silesia in Katowice, 40-027 Katowice, Poland
2 Department of Internal Medicine and Oncological Chemotherapy, Medical University of Silesia in Katowice, 40-027 Katowice, Poland
*Correspondence: uryniusz@wp.pl (Aureliusz Kolonko)
Rev. Cardiovasc. Med. 2022, 23(12), 386; https://doi.org/10.31083/j.rcm2312386
Submitted: 26 July 2022 | Revised: 19 September 2022 | Accepted: 20 September 2022 | Published: 28 November 2022
This is an open access article under the CC BY 4.0 license.
Abstract
Background: Carotid atherosclerosis is one of the main cerebrovascular complications in kidney transplant recipients (KTRs). We analyzed the relationships between carotid intima-media thickness (IMT) and the occurrence and characteristics of carotid plaques in a cohort of KTRs. Methods: In 500 KTRs (aged 49.9 $\pm$ 12.0 years), IMT was measured and carotid plaques were semi-qualitatively assessed. Concomitantly, biochemical and hormonal inflammatory, vascular and calcium-phosphate metabolism parameters were also assessed. Results: In 10.2% of patients, a side-to-side IMT difference $>$0.1 mm was observed, whereas 26.8% of patients with no plaques in one carotid artery had at least one contralateral calcified plaque. Multivariate logistic regression analysis revealed that age (r${}_{\text{partial}}$ = 0.409; p $<$ 0.001), male sex (r${}_{\text{partial}}$ = 0.199; p $<$ 0.001), and coronary artery disease (r${}_{\text{partial}}$ = 0.139; p $<$ 0.01) independently increased IMT (R${}^{2}$ = 0.25). For the occurrence of calcified carotid plaques, age (r${}_{\text{partial}}$ = 0.544; p $<$ 0.001), male gender (r${}_{\text{partial}}$ = 0.127; p $<$ 0.05), and the duration of renal insufficiency prior to transplantation (r${}_{\text{partial}}$ = 0.235; p $<$ 0.001) were confirmed as independent variables. Conclusions: Substantial side-to-side differences in IMT values and carotid plaques distribution are present in a large percentage of stable KTRs. In addition, there are different clinical risk factors profiles associated with IMT and the presence of calcified plaques. Vascular and calcium-phosphate metabolism biomarkers were not associated with any carotid atherosclerosis characteristics.
Keywords
atherosclerosis
biomarkers
calcified plaques
kidney transplantation
ultrasound
1. Introduction
Carotid atherosclerosis is one of the major cardiovascular (CV) risk factors for the occurrence of an ischemic stroke [1, 2]. Traditional risk factors associated with atherosclerosis are age, male gender, smoking, dyslipidemia, hypertension, and diabetes mellitus [3]. In addition, chronic kidney disease (CKD) has also been shown to be associated with greater carotid intima-media thickness (IMT) and the occurrence of symptomatic ischemic stroke [4, 5] as well as with increased carotid artery stiffness and the presence of calcified plaques [6]. As a consequence, CV mortality is the major cause of death in CKD patients [7]. After successful kidney transplantation, despite the reduction of some risk factors (left ventricular hypertrophy, hypertension), CV disease remains a leading cause of death despite a functioning graft [8].
Quantitative evaluation of atherosclerosis with B-mode ultrasound involves the measurement of carotid intima-media thickness (IMT) and the assessment of carotid plaques. Both these measurements are biologically distinct entities and represent different phenotypes of atherosclerosis [9]. IMT is mainly reflective of hypertensive medial hypertrophy and is only weakly associated with traditional coronary risk factors whereas plaques are more strongly associated with traditional risk factors [10]. In patients with end-stage renal disease, the adverse CV consequences of hyperphosphatemia are most likely mediated via its ability to enhance the development of vascular calcifications [11]. CKD also significantly affects plaque composition [12]. Moreover, in CKD patients the rate of atherosclerotic plaque formation is a strong, independent predictor of CV events [13].
In clinical practice, we observed a significant asymmetry of carotid plaques and a substantial discrepancy between IMT values and the plaque burden in some stable kidney transplant recipients (KTRs). In our previous investigations, several clinical measures and biochemical markers were assessed in three different KTRs cohorts [14, 15, 16]. Based on our prospective kidney transplant database, we retrospectively analyzed IMT and the presence of plaques as markers of carotid atherosclerosis, as well as numerous biochemical and hormonal parameters.
2. Materials and Methods
2.1 Study Participants
This study enrolled 500 KTRs who attended our out-patient clinic from 2013 to 2017, in whom carotid artery ultrasound with IMT measurement and carotid plaque assessment were performed. Those examinations were part of the protocols of our previous clinical studies [14, 15, 16], and were approved by the Bioethics Committee of the Medical University of Silesia. All participants gave their written informed consent. The study was conducted in accordance with the Declaration of Helsinki. In addition to data retrieved from the prospective transplant center patient registry, carotid ultrasound, including the assessment of IMT and carotid plaques were performed.
Patients were identified as active smokers, when they were currently smoking or they declared the period of non-smoking as being shorter than 5 years.
2.2 Clinical and Anthropometric Measurements
Body weight and height were measured following standard procedures, and BMI was calculated in kg/m${}^{2}$.
Office arterial blood pressure (OBP) was measured three times in the sitting position in the arm without vascular access during the physical examination. Patients whose OBP was equal or above 140/90 mmHg or those who received antihypertensive medication were diagnosed as hypertensives.
Diabetes was diagnosed in accordance with the American Diabetes Association criteria [17].
The duration of renal function insufficiency was estimated, based on the data collected at the time of kidney transplantation (the period of time since the first elevated serum creatinine concentration to the kidney transplantation procedure).
2.3 Laboratory Measurements
Routine laboratory measurements were performed in the hospital laboratory (Synchron Cx-9, Beckmann Coulter Inc., Fullerton, CA, US). Plasma high-sensitivity C-reactive protein (CRP) concentration was measured by nephelometry (Siemens Healthcare Diagnostics, Deerfield, IL, US) with a limit of quantification (LoQ) of 0.02 mg/L. Intact plasma parathormon (iPTH) concentration was measured using the immunoassay method (Abbott Diagnostics, Abbott Park, IL, US) with a LoQ $<$3 pg/mL, intra-assay variation $<$6/1% and inter-assay variation $<$6.4%, whereas plasma concentrations of interleukin 6 (IL-6) and tumor necrosis factor alpha (TNF-$\alpha{}$) were measured by ELISA (R&D System, Minnesota, MN, US) with a LoQ 0.7 pg/mL and 6.23 pg/mL, intra-assay variation $<$4.2% and 3.0%, and inter-assay variation $<$6.4% and 8.4%, respectively. Plasma concentrations of osteoprotegerin (OPG) were measured with the use of an immunoassay (Microvue Bone Health; Biovendor Laboratory Medicine, Modrice, Czech Republic) with a LoQ 0.03 pmol/L, intra-assay variation $<$3.5% and inter-assay variation $<$5.8%. Plasma concentrations of asymmetric dimethylarginine (ADMA) and oxidized LDL (ox-LDL) were measured using ELISA (Immundiagnostik, AG, Bensheim, Germany), with a LoQ of 0.16 $\mathrm{\mu}$mol/L and 0.0205 U/mL, intra-assay variation $<$7.6% and $<$6.9% and inter-assay variation $<$4% and $<$14.4%, respectively. Plasma concentrations of endothelin-1 (ET-1) were measured using ELISA (USCN Life Sciences, Wuhan, People’s Republic of China), with a LoQ of 2.71 pg/mL, intra-assay variation $<$10%, inter-assay variation $<$12%.
2.4 Carotid Sonography
Carotid ultrasound was performed using a Siemens machine (Sonoline Antares, Mountain View, CA, USA), equipped with a 4.0–9.0 MHz linear transducer. Carotid arteries were examined with the patient in the supine position with the neck extended. The evaluation included the common, internal, and external carotid arteries, and the carotid bifurcation on each side. The common carotid artery intima-media thickness (IMT) was measured manually within 2 cm proximal to the carotid bulb, omitting any visible plaques. At longitudinal scans, the distal lines representing lamina intima and media were sharply visualized and the electronic calipers were placed to perform the exact IMT measurement. The accuracy of the single measurement was 0.5 mm and 3 consecutive measurements were made on each side, then the results were averaged. The highest value on both sides values was reported as the maximal IMT value. At each location, the carotid bulb and preceding common carotid artery were carefully evaluated in terms of the presence of plaques, which was classified based on the simplified scale: 0—no lesions, 1—non-calcified lesions, 2—at least one calcified lesion, 3—few calcified lesions, 4—carotid bulb heavily covered by calcified lesions. A final plaque score was equal to the highest score from both sides. All carotid sonographic examinations were performed by single investigator (AK).
2.5 Data and Statistical Analysis
Post-transplant major adverse cardio- or cerebrovascular events (MACE) were defined as the incidence of myocardial infarct, stroke, or cardiac artery stenting/surgical revascularization.
Kidney graft function was measured by the estimated glomerular filtration rate (eGFR) calculated according to the Modification of Diet in Renal Disease (MDRD) formula.
Statistical analyses were performed using the STATISTICA 13.3 PL for Windows software package (Tibco Inc., Palo Alto, CA, USA) and MedCalc 18.6 (MedCalc Software, Ostend, Belgium). Values are presented as means and 95% confidence intervals or medians with Q1–Q3 values, as appropriate, or frequencies. Comparisons were performed between 2 groups based on the mean value of maximal carotid IMT and between 3 groups, defined by the presence and type of carotid plaques. Based on the presence and type of carotid artery plaques, all study participants were assigned to the subgroup 1 (no plaques), subgroup 2 (only non-calcified plaque/plaques) or subgroup 3 (one or more calcified plaque). For these comparisons, the Student t test and the analysis of variance test (for quantitative variables) or the $\chi{}$${}^{2}$ test (for qualitative variables) were used accordingly. For variables with non-parametric distribution, the Mann-Whitney U test or the Kruskal-Wallis test was used. Receiver operator characteristics (ROC) analysis was applied to determine the cut-off values for age and the duration of renal insufficiency, associated with the presence of calcified carotid lesions. Calculation of correlations were done using the Spearman coefficient.
Multivariate backward regression analysis was performed for the variability of IMT value, including potential explanatory variables: age, sex, the presence of coronary artery disease, smoking status, hemoglobin level and the presence of calcified carotid plaques. Multivariate models included variables selected on the basis of group comparison and univariate logistic regression analyses. The stepwise selection method was used.
Multivariate backward regression analysis was also performed for the presence of calcified plaques as dependent variable, including potential explanatory variables: age, sex, BMI, the presence of hypertension, pulse pressure, IMT, coronary artery disease or MACE, smoking status, serum glucose or the presence of hyperuricemia. Another multivariate backward regression analysis was performed for the presence of calcified lesions in a subset of 319 and 146 patients, respectively, i.e., in a cohort of patients with available results of relevant biochemical markers, and included age, the number of antihypertensive drugs, CRP, OPG and sclerostin levels as potential independent variables. In all the statistical tests, the ‘p’ values below 0.05 were considered statistically significant.
3. Results
3.1 Study Group
The study group consist of 500 stable KTRs, whose clinical characteristics are presented in Table 1. Mean age at the time of the study was 49.9 $\pm$ 12.0 years. Median time after kidney transplantation was 86 (Q1–Q3, 65–117) months. The causes of end-stage renal disease were: glomerulonephritis (48%), diabetes mellitus (13.6%), pyelonephritis (11.6%), autosomal dominant polycystic kidney disease (8.4%), hypertensive nephropathy (5.6%), other and unknown (12.8%). 97.4% of patients received their organ from a deceased donor. Most of the patients received immunosuppression therapy with cyclosporine A or tacrolimus, anti-metabolic drugs (mainly mycophenolate mofetil or mycophenolate acid), and steroids.
Table 1.Clinical characteristics of study group.
Parameter Value N = 500 Age at the time of the study [years] 49.9 (48.8–51.0) Gender [M/F] 284/216 BMI [kg/m${}^{2}$] 26.3 (25.9–26.7) Dialysis vintage [months]* 25.0 (14.0–42.0) Time after transplantation [months]* 86.0 (65.0–117.0) Retransplant [n (%)] 42 (8.4) Hypertension [n (%)] 445 (89) MAP [mmHg] 100.5 (99.5–101.4) Pulse pressure [mmHg]* 50.0 (40–60) Number of antihypertensive drugs [n]* 2 (1–3) Structure of hypertensive treatment [n (%)] ACE-I/ARB 143 (28.6) Beta-blocker 348 (69.6) Ca-blocker 207 (41.4) Diuretics 142 (28.4) Diabetes [n (%)] 137 (27.4) Coronary artery disease [n (%)] 63 (12.6) Previous MACE [n (%)] 54 (10.8) Smoking status [%] 89 (17.8) eGFR [mL/min/1.73 m${}^{2}$]* 49.4 (36.4–68.0) Proteinuria $\geq$1 g/24 h [n (%)] 35 (7.0) Glucose [mmol/L]* 5.0 (4.7–5.7) Calcium [mmol/L]* 2.4 (2.3–2.5) Phosphate [mmol/L]* 1.0 (0.9–1.2) iPTH [pg/mL]* 107 (69–174) Cholesterol [mmol/L] 5.3 (5.2–5.4) Triglycerides [mmol/L]* 1.5 (1.0–2.1) Hyperlipidemia [n (%)] 265 (53) Hyperuricemia [n (%)] 286 (57.1) Hemoglobin [g%] 13.5 (13.3–13.7) Main medications [n (%)] Statins/fibrates 135 (27) Calcineurin inhibitors [CyA/Tc] 238 (48)/246 (49) Glucocorticoids 305 (61) Calcium carbonate 74 (14.8) Vitamin D 94 (18.8) Data presented as means and 95% Confidence Intervals or frequencies, except * medians and Q1–Q3 values. BMI, body mass index; MAP, mean arterial pressure; ACE-I, angiotensin converting enzyme inhibitor; ARB, angiotensin receptor blocker; MACE, major adverse cardio- and cerebrovascular event; eGFR, estimated glomerular filtration rate; iPTH, intact parathormon; CyA, cyclosporine A; Tc, tacrolimus.
3.2 Intima-Media Thickness
In the entire study group, the mean value of carotid IMT, measured with omitting the visible plaques, was 0.66 (95%: 0.64–0.67) mm, with a range 0.4–1.3 mm. The median IMT value was 0.6 (Q1–Q3: 0.6–0.7) mm. The maximum intrapatient IMT difference was 0.6 mm and was observed in 1 study participant. Generally, the degree of carotid atherosclerosis was similar in both arteries, but in 51 (10.2%) patients a the side-to-side IMT difference $>$0.1 mm was noted. Patients in this specific subgroup were older [55.6 (51.9–59.3) vs. 49.2 (48.1–50.4) years; p $<$ 0.001] and were more frequently men (72.6 vs. 55.0%; p $<$ 0.05); however, there were no differences in BMI, pulse pressure, smoking status, the cause of CKD or the duration of the period of renal insufficiency. This “asymmetric” subgroup was characterized by a significantly greater presence of carotid plaques (72.6 vs. 53.2%; $\chi{}$${}^{2}$ = 6.91, p $<$ 0.01), including calcified lesions (62.8 vs. 44.5%; $\chi{}$${}^{2}$ = 7.15, p $<$ 0.05).
All study participants were divided by using the mean value of maximal carotid IMT, i.e., 0.66 mm. Table 2 shows the comparison of patients assigned to both groups. The structure of the primary cause of CKD was similar in both groups. IMT values were strongly associated with age (R = 0.498; p $<$ 0.001). There were also positive correlations with BMI (R = 0.125; p $<$ 0.01), pulse pressure (R = 0.220; p $<$ 0.001) and the duration of renal insufficiency (R = 0.167; p $<$ 0.001). Among the laboratory parameters, IMT was positively associated with blood hemoglobin level (R = 0.153; p $<$ 0.001), OPG (R = 0.203; p $<$ 0.001), IL-6 (R = 0.146; p $<$ 0.01), serum sclerostin concentration (R = 0.198; p $<$ 0.05) and negatively associated with serum Klotho concentration (R = –0.181; p $<$ 0.05), but not with iPTH level.
Table 2.The clinical characteristics of study subgroups based on the occurrence and type of atherosclerotic plaques visualized in both carotid arteries.
Parameter Study group according to the mean IMT value p Study groups according to the carotid plaque occurrence p IMT max IMT max No plaques Non-calcified Calcified $<$0.66 mm $\geq$0.66 mm N = 224 N = 44 N = 232 N = 280 N = 220 Age [years] 45.3 (43.9–46.6) 55.8 (54.3–57.3) $<$0.001 42.7 (41.3–44.1) 49.3 (45.9–52.8)${}^{\wedge\wedge\wedge}$ 57.0 (55.7–58.3) ###${}^{\wedge\wedge\wedge}$ $<$0.001 Gender [M/F] 138/142 146/74 $<$0.001 111/113 24/20 149/83## $<$0.01 BMI [kg/m${}^{2}$] 25.8 (25.2–26.4) 26.5 (25.9–27.1) 0.12 25.8 (25.1–26.5) 25.1 (23.8–26.4) 27.0 (26.4–27.6)#${}^{\wedge}$ $<$0.05 Dialysis vintage [months]* 23 (14–42) 25 (14–41) 0.61 22 (13–40) 26 (16–38) 27 (15–45) 0.19** Time after transplant [months]* 85 (52–115) 87 (70–119) 0.20 87 (57–116) 79 (25–104) 86 (70–120) 0.44** Retransplant [n (%)] 21 (7.5) 21 (9.6) 0.40 19 (8.5) 3 (6.8) 20 (8.6) 0.92 Duration of renal insufficiency [years]* 10 (7–15) 12 (10–17) $<$0.001 9.0 (7.0–12.0) 10.0 (7.5–15.5) 14.0 (10.0–18.0) $<$0.001 Hypertension [n (%)] 240 (85.7) 200 (90.9) 0.08 190 (84.8) 37 (84.1) 218 (94.0)##${}^{\wedge}$ $<$0.01 MAP [mmHg] 101 (100–103) 99 (98–101) $<$0.05 100.0 (98–101) 100 (96–103) 101 (100–103) 0.35 Pulse pressure [mmHg]* 50 (45–60) 50 (40–59) $<$0.001 50 (40–59) 50 (45–60) 50 (45–60)### $<$0.05** Number of antihypertensive drugs [n]* 2 (1–3) 2 (1–3) 0.06 2 (1–2) 2 (1–3) 2 (1–3)### $<$0.001** Diabetes [n (%)] 80 (28.5) 83 (37.8) 0.03 49 (21.9) 9 (20.4) 79 (34.1)## $<$0.01 Coronary artery disease [n (%)] 21 (7.5) 49 (22.2) $<$0.001 11 (4.9) 3 (6.8) 49 (21.1)###${}^{\wedge}$ $<$0.001 Previous MACE [n (%)] 16 (5.7) 28 (12.7) $<$0.01 10 (4.5) 1 (2.3) 43 (18.5)###${}^{\wedge\wedge}$ $<$0.001 Smoking status [%] 48 (17.1) 41 (18.6) 0.66 43 (19.2) 10 (22.7) 36 (15.5) 0.40 eGFR [mL/min/1.73 m${}^{2}$]* 49.0 (36.1–67.7) 49.7 (37.9–68.8) 0.46 51.7 (38.4–68.3) 48.3 (38.5–63.0) 48.5 (34.8–66.9) 0.28** Proteinuria $\geq$1 g/24 h [n (%)] 15 (5.4) 18 (8.2) 0.21 16 (7.1) 4 (9.1) 15 (6.5) 0.82 Glucose [mmol/L]* 5.0 (4.6–5.5) 5.2 (4.8–5.8) $<$0.01 4.9 (4.6–5.5) 5.0 (4.4–5.4) 5.3 (4.8–5.9)# $<$0.01** Calcium [mmol/L]* 2.39 (2.30–2.50) 2.36 (2.27–2.45) $<$0.05 2.4 (2.3–2.5) 2.4 (2.3–2.4) 2.4 (2.3–2.5) 0.25** Phosphate [mmol/L]* 1.02 (0.87–1.18) 10.3 (0.87–1.15) 0.99 1.0 (0.9–1.2) 1.0 (0.8–1.2) 1.0 (0.9–1.1) 0.70** iPTH [pg/mL]* 99 (68–171) 100 (60–148) 0.39 104 (70–174) 105 (70–149) 111 (66–177) 0.74** IMT [mm]* 0.6 (0.5–0.6) 0.7 (0.7–0.8) $<$0.001 0.6 (0.5–0.7) 0.7 (0.6–0.7) 0.7 (0.6–0.8) $<$0.001** Cholesterol [mmol/L] 5.3 (5.2–5.4) 5.4 (5.2–5.5) 0.33 5.3 (5.1–5.5) 5.4 (5.0–5.7) 5.3 (5.2–5.5) 0.94 Triglycerides [mmol/L]* 1.5 (1.0–2.0) 1.5 (1.0–2.2) 0.59 1.4 (1.0–2.0) 1.5 (1.1–2.0) 1.6 (1.1–2.2) 0.49** Hyperlipidemia [n (%)] 145 (51.8) 117 (53.2) 0.76 116 (51.8) 22 (51.2) 127 (54.7) 0.75 Hyperuricemia [n (%)] 151 (52.9) 135 (61.4) 0.10 116 (51.8) 19 (43.2) 151 (65.1)##${}^{\wedge\wedge}$ $<$0.01 Hemoglobin [g%] 13.3 (13.0–13.5) 13.7 (13.5–14.0) $<$0.01 13.4 (13.1–13.6) 13.6 (12.5–14.6) 13.6 (13.3–13.8) 0.53 Data presented as means and 95 % Confidence Intervals or frequencies, except *medians and Q1–Q3 values. Statistics: ANOVA or $\chi{}$${}^{2}$ test, except **Kruskal-Wallis test. # p $<$ 0.05 vs. no plaques; ## p $<$ 0.01 vs. no plaques; ### p $<$ 0.001 vs. no plaques; ^ p $<$ 0.05 vs. non-calcified plaques; ^^ p $<$ 0.01 vs. non-calcified plaques; ^^^ p $<$ 0.001 vs. non-calcified plaques. BMI, body mass index; MAP, mean arterial pressure; MACE, major adverse cardio- and cerebrovascular event; eGFR, estimated glomerular filtration rate; iPTH, intact parathormon; IMT, intima-media thickness.
Univariate logistic regression analyses revealed that age, male sex, the presence of coronary artery disease or calcified plaques and hemoglobin level were associated with the presence of IMT $\geq$0.66 mm (Table 3).
Table 3.Results of univariate logistic regression analyses for the IMT value and for the presence of calcified carotid plaques.
IMT Calcified plaques $\beta$ $\chi$${}^{2}$ p $\beta$ $\chi$${}^{2}$ p Age [years] 0.05 10.6 $<$0.01 0.12 162.4 $<$0.001 Male sex 1.63 9.4 $<$0.01 0.57 9.8 $<$0.01 BMI [kg/m${}^{2}$] 0.04 0.63 0.43 0.06 8.6 $<$0.01 Hypertension 0.16 0.05 0.83 1.02 10.4 $<$0.01 Pulse pressure [mmHg] 0.01 0.5 0.48 0.03 16.2 $<$0.001 Number of antihypertensive drugs 0.03 0.03 0.85 0.42 30.9 $<$0.001 Coronary artery disease 1.46 4.67 $<$0.05 1.42 12.3 $<$0.001 Previous MACE –0.29 0.09 0.78 1.72 17.7 $<$0.001 Calcium [mmol/L] 0.63 0.27 0.61 0.11 0.06 0.81 Glucose [mmol/L] 0.03 0.09 0.76 0.12 5.7 $<$0.05 Hyperuricemia –0.30 0.38 0.54 0.62 10.4 $<$0.01 Hemoglobin [g%] 0.16 3.05 0.06 0.04 0.98 0.33 Duration of renal insufficiency period [years] –0.04 0.81 0.39 0.10 36.9 $<$0.001 IMT [mm] - - - 4.91 54.9 $<$0.001 Calcified plaques 1.71 12.3 $<$0.01 - - - BMI, body mass index; MACE, major adverse cardio- and cerebrovascular event; IMT, carotid intima-media thickness.
Multivariate logistic regression analysis in the entire study group showed that age (r${}_{\text{partial}}$ = 0.409; p $<$ 0.001), male sex (r${}_{\text{partial}}$ = 0.199; p $<$ 0.001), and coronary artery disease (r${}_{\text{partial}}$ = 0.139; p $<$ 0.01) independently increased IMT (R${}^{2}$ = 0.25).
3.3 Carotid Plaques
Among the 500 study patients, 276 (55.2%) had at least one plaque (non-calcified or calcified) in both carotids. Patients with the presence of plaques were significantly older [55.8 (54.5–57.0) vs. 42.7 (41.3–44.1) years; p $<$ 0.001] in comparison with KTRs with no carotid atherosclerotic lesions. They more frequently were men (62.7 vs. 49.6%; p $<$ 0.01) and were characterized by a greater BMI [26.5 (25.9–27.0) vs. 25.6 (24.9–26.3) kg/m${}^{2}$, respectively; p $<$ 0.01], IMT [Me: 0.70 (0.60–0.80) vs. Me: 0.60 (0.50–0.68) mm, respectively; p $<$ 0.001], pulse pressure [Me: 50 (45–60) vs. Me: 50 (40–59) mmHg, respectively; p $<$ 0.001] and duration of renal insufficiency [Me: 13 (10–18) vs. Me: 9 (7–12); p $<$ 0.001].
Calcified lesions were detected in 32.5% study subjects with an IMT $\leq$0.6 mm, including 17.9% of patients characterized by the presence of few calcified lesions and those with a heavily calcified carotid bulb. This finding was also present in 22.9% and 6.7% of subjects with an IMT $\leq$0.5 mm. Fig. 1 shows the distribution of calcified plaques in patients with different maximum IMT values.
Fig. 1.
The distribution of carotid calcified plaques in patients with different IMT max values.
All patients were divided into 3 groups, based on the presence and type of plaques detected in both carotid arteries. There were 224 (44.8%) patients without any plaque, 44 (8.8%) patients with only non-calcified plaque/plaques and 232 (46.4%) patients, in whom at least one calcified lesion was identified. The clinical characteristics of patients in these study groups are shown in Table 2. There were substantial differences between the study groups. However, except of age, where a significant increasing trend was noted across all 3 study groups (p $<$ 0.001), other significant differences were noted between the calcified plaques group and one or both other groups (Table 2). Of note, the statistical strength of comparison between the non-calcified and calcified plaques groups was weakened by the low number of patients in the former group. There were no differences in the proportion of patients, in whom a parathyroidectomy was performed prior to the study (7 vs. 7 vs. 8%, respectively; p = 0.96).
In general, the occurrence of calcified plaques was associated with the primary cause of CKD ($\chi{}$${}^{2}$: 12.2; p $<$ 0.05). Patients with pyelonephritis had significantly less ($\chi{}$${}^{2}$: 4.97; p $<$ 0.05), whereas patients with hypertensive nephropathy significantly more ($\chi{}$${}^{2}$: 8.3; p $<$ 0.01) carotid calcified lesions as compared with other patients. Patients with pyelonephritis were significantly younger than all other groups (46.5 vs. 50.3 years; p $<$ 0.05), but there was no age difference in case of patients with hypertensive nephropathy (52.5 vs. 50.0 years; p = 0.22).
Univariate logistic regression analyses revealed that age, male sex, BMI, IMT, pulse pressure, serum glucose level, the presence of hypertension, coronary artery disease, previous MACE, hyperuricemia, the number of antihypertensive drugs and the duration of renal insufficiency period prior to transplantation were associated with the presence of calcified carotid lesions (Table 3). There was also a significant relationship between the maximal carotid plaque score and the duration of the period of renal insufficiency ($\chi{}$${}^{2}$: 52.7; p $<$ 0.001).
Multivariate logistic regression analysis in the entire study group revealed that age (r${}_{\text{partial}}$ = 0.544; p $<$ 0.001), male sex (r${}_{\text{partial}}$ = 0.127; p $<$ 0.05), and the duration of the period of renal insufficiency (r${}_{\text{partial}}$ = 0.235; p $<$ 0.001) independently increased the risk for the presence of calcified lesions in carotid arteries (R${}^{2}$ = 0.38). Notably, if we did not include the duration of the period of renal insufficiency among the potential independent variables, only age (r${}_{\text{partial}}$ = 0.538; p $<$ 0.001) and previous MACE (r${}_{\text{partial}}$ = 0.130; p $<$ 0.01) were confirmed in multivariate analysis (R${}^{2}$ = 0.32). The ROC analysis revealed that age $>$48.5 years and the duration of the period of renal insufficiency $>$11 years increased the risk for the occurrence of calcified lesions with 81.0% and 66.2% sensitivity and 68.3% and 68.2% specificity, respectively (Fig. 2A,B).
Fig. 2.
The ROC analysis for recipient age (A) and duration of the period of renal insufficiency (B) which increased the risk for the occurrence of calcified lesions in the cohort of kidney transplant patients.
In 319 study patients, the results of circulating markers of inflammation, vascular function and calcium-phosphate metabolism were also available. In this subgroup, there were significantly higher serum CRP levels in patients with calcified plaques [Me: 3.5 (1.5–7.3) vs. Me: 2.0 (1.2–5.2) mg/L; p $<$ 0.01], whereas there were no differences in plasma OPG, ET-1, ADMA, ox-LDL, IL-6 and TNF-$\alpha{}$ levels (data not shown).
Plasma sclerostin and $\alpha{}$-Klotho concentrations were measured in a subset of 146 study patients. Sclerostin levels were significantly higher [Me: 0.9 (0.7–1.1) vs. Me: 0.7 (0.6–0.1) ng/mL; p $<$ 0.01] in patients with carotid calcified lesions, whereas the difference in $\alpha{}$-Klotho levels [Me: 470 (392–577) vs. Me: 503 (440–616) pg/mL, respectively; p = 0.053] did not reach statistical significance.
Univariate logistic regression analyses revealed that serum CRP level ($\beta{}$ = 0.05, $\chi{}$${}^{2}$ = 5.9; p $<$ 0.05), plasma OPG ($\beta{}$ = 0.17, $\chi{}$${}^{2}$ = 13.8; p $<$ 0.001) and sclerostin ($\beta{}$ = 1.82, $\chi{}$${}^{2}$ = 12.8; p $<$ 0.01) levels were associated with the presence of calcified carotid lesions. However, none of analyzed biomarkers independently influenced the occurrence of calcified plaques in the multivariate analysis.
3.4 Carotid Plaque Location
The atherosclerotic lesions distribution according to the plaque score and side involved is shown at Fig. 3.
Fig. 3.
The distribution of atherosclerotic lesions in carotid arteries according to the plaque score and side involved. 0 denoted no plaques on the given side, 1—non-calcified lesions, 2—at least one calcified lesion, 3—few calcified lesions, 4—carotid bulb heavily covered by calcified lesions.
Despite the comparable number of patients without any carotid lesion (274 vs. 279 at the right and left side, respectively), only 224 (44.8%) KTRs were free of plaques at both sides. In the group of 274 KTRs without plaques on the right side, 50 patients (18.2%) had atherosclerotic lesions on the left side, including 13 (4.7%) with uncalcified plaques and 37 (13.5%) with calcified plaques. Notably, 16 (5.8%) patients had an abundantly calcified left carotid bulb. In this group of 274 subjects, median left IMT was significantly greater in the subgroups with any ipsilateral plaques [Me: 0.6 (0.6–0.7); p $<$ 0.01], calcified plaques [Me: 0.6 (0.6–0.7); p $<$ 0.05] and abundantly calcified left carotid bulb [Me: 0.7 (0.6–0.8); p $<$ 0.01] in comparison with subgroup without plaques [Me: 0.6 (0.5–0.6)].
Out of 279 patients without plaques on the left side, 55 patients (19.7%) had at least one atherosclerotic lesion on the right side, including 18 (6.5%) with uncalcified plaques and 37 (13.3%) with calcified plaques. Twelve patients (4.3%) had a heavily calcified right carotid bulb. Median right IMT was significantly greater in the subgroups with any ipsilateral plaques [Me: 0.6 (0.6–0.7); p $<$ 0.001] and calcified plaques [Me: 0.6 (0.6–0.7); p $<$ 0.001] in comparison with subgroup without plaques [Me: 0.6 (0.5–0.6)]. In contrast, there was no difference in IMT between subgroup with abundantly calcified right carotid bulb [Me: 0.6 (0.6–0.6)] as compared with the subgroup without plaques (p = 0.11).
4. Discussion
This study analyzed the characteristics and relationship of two different ultrasound-based measures of carotid atherosclerosis, IMT and carotid plaques, in stable KTRs. In contrast to several previous studies, the IMT measurements were performed manually (to avoid IMT overestimation), which enable us to investigate the two different atherosclerotic carotid entities separately and to define their independent risk factors without overlapping bias. Age and male sex were confirmed as common independent risk factors for the occurrence of both IMT and calcified carotid plaques. Additionally, the presence of coronary artery disease was associated with increased IMT, whereas the duration of the period of pre-transplant renal insufficiency increased the risk for carotid calcified plaques in multivariate analyses. Notably, none out of the numerous analyzed biomarkers was shown to be independently associated with IMT or plaque occurrence in the study cohort. This finding preclude the use of biomarkers as surrogates of carotid atherosclerosis in daily clinical practice. Finally, we also described in detail the asymmetric distribution of carotid lesions and the relationships between IMT and the presence of calcified carotid lesions in KTRs.
IMT is recognized as an useful tool for CV risk stratification and therapy monitoring [18]. In the Carotid Atherosclerosis Progression Study during a mean follow-up of 4.2 years, IMT was highly predictive for the incidence of stroke, myocardial infarction and death [19]. Moreover, for this combined end-point, hazard ratios were considerably higher in the younger ($<$50 years) than in the older age group. IMT was also shown to increase over a 6-year follow-up period especially in patients suffering from CV events [20]. In contrast, in the Rotterdam Study, adding IMT to a model for predicting the increased CV risk did not enhance its power, however the IMT ROC area (0.71) was comparable with other well-established risk factors, including previous MACE, diabetes, smoking, systolic blood pressure and cholesterol (ROC 0.65–0.72) [21]. In the present study, age, male gender and the presence of coronary artery disease were independently associated with greater IMT values, similarly to previous findings in the general [22] and hemodialysis [23] populations, despite one contrary report based on a small KTRs cohort [24].
Decreased renal function was shown to independently increase IMT (4), even in patients with no known kidney disease and normal and/or moderately decreased eGFR values [25, 26]. Of note, in the Suita Study cohort, multivariable-adjusted carotid IMT was significantly greater in CKD patients only in a hypertensive subgroup [27]. Nevertheless, despite the previously described modest IMT regression observed after kidney transplantation [28], the range of IMT in the present study was similar to the values reported in hemodialysis patients [29], as well as in other KTRs cohorts with comparable recipient age [30, 31]. Notably, in all study subjects, IMT was measured manually by a single investigator, within the optimal location at the far wall below the bulb, omitting any visible local thickening [32]. Otherwise, both the manual and computer-based, automated methods for IMT assessment often yield higher scores, as they may not omit local intimal thickening or plaques [22, 33, 34, 35]. This study protocol was chosen to avoid the above mentioned biases and allowed us to perform a more accurate analysis of interrelationship between IMT and carotid plaque burden. In the analyzed cohort, only a small subgroup showed substantial IMT left-right asymmetry, which is in line with earlier reports [36, 37]. Interestingly, those KTRs were older and were more frequently men compared to the rest of the analyzed cohort.
The assessment of carotid plaques is a distinct measure of atherosclerosis, with advancing age as the most predominant risk factor. With aging, a decrease in fibrous plaques and an increase in atheromatous plaques is observed [38]. This discrepancy may be explained by the differences in arterial remodeling in response to plaque accumulation among the different types of arteries [39]. Carotid bulb geometry was shown to be associated with plaque volume [40, 41]. Moreover, in arteries with plaques, wall shear stress was significantly lower than in the plaque-free vessel and was linked to endothelial dysfunction [42]. This may result in partly asymmetric distribution of carotid calcified lesions that was seen in some study patients, which was also reported in another large non-CKD study [43].
Thus, the assessment of carotid plaques, particularly calcified lesions, using ultrasound may provide additional stroke risk information beyond the measurement of luminal stenosis [44]. In a population-based study, calcified carotid plaques independently increased the risk of combined vascular outcomes, including ischemic stroke, even after adjustment for IMT [45]. The presence of CKD additionally increases the total and calcified plaque burden, both in dialysis and transplant patients [46]. Moreover, the negative association between eGFR and the prevalence of carotid plaques was observed even in patients without CKD [25]. This effect is mainly mediated by phosphate retention, which constitutes an early trigger for the development of secondary hyperparathyroidism and accelerated macrovascular disease [47]. Such a mechanism is further confirmed by a consistent relationship between serum phosphate (even in the normal range) and CVD [48, 49], as well as with multiregional vascular calcification [50], and by the independent relationship between the duration of renal insufficiency (with concomitant impairment of urinary phosphate elimination) and calcified plaques found in the present study. This reciprocal relationship of vascular calcifications within different locations in CKD patients was also confirmed in other studies [22, 29, 51, 52].
In kidney transplant patients, the occurrence of carotid lesions was previously showed to be associated with age and the occurrence of arterial hypertension [53]. Importantly, despite satisfactory transplant organ function, substantial progression of carotid plaques was reported, which was associated with age, smoking, dialysis vintage and hyperphosphatemia [54]. In the present study we found that, except for age and male sex, only the duration of pre-transplant renal insufficiency independently increased the burden of calcified carotid plaques, and as we excluded the latter variable, only age and previous MACE remained significant. This is in line with another report, where the severity of the carotid plaque score was significantly higher in the MACE group than in the MACE-free group in asymptomatic CKD patients [55]. Similarly, age and coronary artery calcification score were independently associated with carotid plaques in dialysis patients [29]. Interestingly, in some reports a significantly greater plaque prevalence in dialysis patients in comparison with healthy controls was identified despite similar IMT [45, 56]. It is also worth noticing that in our cohort, there was a substantial percentage of patients with IMT $\leq$0.6 mm, in whom we found a high burden of calcified carotid plaques. More importantly, the accuracy for detecting carotid plaques was higher than for detecting abnormal IMT in examinations performed in a routine outpatient setting in a non-CKD cohort [57]. In the large Angina Prognosis Study in Stockholm, carotid IMT was a weak predictor for events, whereas carotid plaques were related to CV death or non-fatal myocardial infarction [58]. In a recent metanalysis, plaque assessment was found to be a better CV risk predictor than IMT in a non-CKD population and one large observational study provided evidence for its similar potential in CKD patients [59]. These studies suggest the greater utility of carotid plaque assessment compared to carotid IMT measurement, as a prognostic indicator for CV risk in CKD and KTRs.
Disappointingly, when we analyzed the potential associations of numerous biochemical or hormonal markers with IMT or the occurrence of carotid plaques, none of investigated inflammatory, vascular or calcium-phosphate parameters was confirmed as an independent variable in multivariate analyses. Previously, atherosclerotic plaque occurrence and progression were found to be associated with higher IL-6 levels [60] and $\alpha{}$-Klotho polymorphism [61] in CKD patients. Plaque burden was also related to ET-1, OPG and nitric oxide metabolite levels in small cohorts of dialysis patients [62, 63, 64] and with CRP level in KTRs [65]. However, a majority of biomarkers included in the present analysis have not been previously examined in kidney transplantation patients.
The main limitation of our analysis is its cross-sectional character and inclusion of 3 KTRs cohorts. However, there was a high uniformity in the methodology of these 3 cohorts, with all ultrasound examinations performed by a single investigator, using the same measurement protocol. In our study cohort, several clinical characteristics and several biochemical parameters were not available for each of study patients. Only the smoking status at the time of the study was available, whereas we have no data concerning the lifetime smoking habits. This may have explained why smoking was not found to be an independent parameter which influenced IMT or the occurrence of calcified plaques in the univariate or multivariate analyses. Finally, as calcium-phosphate metabolism before and during dialysis therapy is the crucial factor for accelerated calcification of the vessel wall, it would have been important to analyze data concerning the maximum pre-transplant phosphate and iPTH levels, as well as the use and effect of phosphate-lowering regimens, which were not available in our study group.
5. Conclusions
In this study, we described in detail the distribution of two different atherosclerotic measures—IMT and carotid plaques—in a large cohort of stable kidney transplant recipients. In addition, the profiles of different clinical risk factors associated with both those vascular entities were identified. We large side-to-side differences in IMT values and carotid plaque distribution in a substantial percentage of KTRs, which presents a high epidemiologic burden for carotid and general atherosclerosis. None of the analyzed vascular and calcium-phosphate metabolism biomarkers was associated with any of the carotid atherosclerosis measurements. Due to the high risk for CV complications and death among recipients of successful kidney transplants, the independent assessment of both IMT and calcified carotid lesions should be advocated, as it may increase the ability to identify those KTRs with the highest CV risk.
Author Contributions
AK designed the research study. AK, BS, MS and RS performed the research. RF and JC provided help and advice on statistical analysis, manuscript and figures preparation. AK analyzed the data and wrote the manuscript. AW critically reviewed the manuscript. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript.
Ethics Approval and Consent to Participate
The study research was conducted as a part of our previous clinical studies, all were accepted by the Bioethics Committee of the Medical University of Silesia (KNW/0022/KB1/81/10, KNW/0022/KB1/93/13, KNW/0022/KB1/35/I/15). All participants gave their written informed consent. The study was conducted in accordance with the Declaration of Helsinki.
Acknowledgment
Not applicable.
Funding
This research received no external funding.
Conflict of Interest
The authors declare no conflict of interest.
Publisher’s Note: IMR Press stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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# Questionbank: Permutation and Probability
You are here: Home CAT Questionbank CAT Quant Permutation and Probability Question 11
The best combinatorics questions are the ones that involve a bit of Number Systems!
## Counting 5 Digit Numbers
Q.11: How many numbers of up to 5 digits can be created using the digits 1, 2, 3 and 5 each at least once such that they are a multiple of 15?
1. 24
2. 18
3. 15
4. 12
• Correct Answer
Choice D. 12
## Explanatory Answer
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## Detailed Solution
For a number to be a multiple of 15, it has to be a multiple of 3 and of 5. So, the last digit has to be 5 and the sum of digits should be a multiple of 3.
We can have either 4–digit or 5–digit numbers. If we have a 4–digit number, sum of the digits will be 1 + 2 + 3 + 5 = 11. No 4–digit number formed with digits 1, 2, 3, 5 exactly once can be a multiple of 3. So, there is no possible 4–digit number.
Now, in any 5 digit number, we will have 1, 2, 3, 5 once and one of these 4 digits repeating once. 1 + 2 + 3 + 5 = 11. So, the digit that repeats in order for the number to be a multiple of 3 has to be 1. In this instance, sum of the digits will be 12 and this is the only possibility.
So, any 5–digit number has to have the digits 1, 1, 2, 3, 5. For the number to be a multiple of 5, it has to end in 5.
So, number should be of the form __ __ __ __ 5, with the first 4 slots taken up by 1, 1, 2, 3. These can be rearranged in ${4! \over 2!}$ = 12 ways.
There are 12 possibilities overall.
Correct Answer : 12
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## More questions from Number System
There are many ways to skin a CAT. To skin CAT, one needs to be good at finding how many ways there are.
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2019-01-18 19:41:59
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http://www.numdam.org/item/CM_1979__39_2_177_0/
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$p$-adic representations arising from descent on abelian varieties
Compositio Mathematica, Volume 39 (1979) no. 2, pp. 177-245.
@article{CM_1979__39_2_177_0,
author = {Harris, Michael},
title = {$p$-adic representations arising from descent on abelian varieties},
journal = {Compositio Mathematica},
pages = {177--245},
publisher = {Sijthoff et Noordhoff International Publishers},
volume = {39},
number = {2},
year = {1979},
zbl = {0417.14034},
mrnumber = {546966},
language = {en},
url = {http://www.numdam.org/item/CM_1979__39_2_177_0/}
}
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AU - Harris, Michael
TI - $p$-adic representations arising from descent on abelian varieties
JO - Compositio Mathematica
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DA - 1979///
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VL - 39
IS - 2
PB - Sijthoff et Noordhoff International Publishers
UR - http://www.numdam.org/item/CM_1979__39_2_177_0/
UR - https://zbmath.org/?q=an%3A0417.14034
UR - https://www.ams.org/mathscinet-getitem?mr=546966
LA - en
ID - CM_1979__39_2_177_0
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%0 Journal Article
%A Harris, Michael
%T $p$-adic representations arising from descent on abelian varieties
%J Compositio Mathematica
%D 1979
%P 177-245
%V 39
%N 2
%I Sijthoff et Noordhoff International Publishers
%G en
%F CM_1979__39_2_177_0
Harris, Michael. $p$-adic representations arising from descent on abelian varieties. Compositio Mathematica, Volume 39 (1979) no. 2, pp. 177-245. http://www.numdam.org/item/CM_1979__39_2_177_0/
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[23] N. Katz: "P-Adic Interpolation of Real Analytic Eisenstein Series," Ann. of Math., 104, 459-571, (1976). | MR | Zbl
[24] M. Lazard: "Groupes Analytiques p-Adiques," Publ. Math. I.H.E.S., 26, (1965). | Numdam | MR | Zbl
[25] S. Lang: "Algebraic Groups over Finite Fields," Ann. J. Math., 78, 553-563, (1956). | MR | Zbl
[26] S. Lang: Elliptic Functions, Reading, Mass.: Addison-Wesley, (1973). | MR | Zbl
[27] Ju. I. Manin and M.M. Vishik: "P-Adic Hecke Series of Imaginary Quadratic Fields," (Trans.) Math. U.S.S.R. Sbornik, 24, 345-371, (1974). | Zbl
[28] B. Mazur: "Rational Points of Abelian Varieties with Values in Towers of Number Fields," Inv. Math., 18, 183-266, (1972). | MR | Zbl
[29] B. Mazur: "Local Flat Duality," Am. J. Math., 92, 343-361, (1970). | MR | Zbl
[30] B. Mazur: "Rational Points of Modular Curves," in Modular Functions of One Variable V, Lecture Notes in Mathematics, 601, (1977). | MR | Zbl
[31] B. Mazur: "Trees of Rational Points of Elliptic Curves," unpublished manuscript.
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[33] B. Mazur and P. Swinnerton-Dyer: "Arithmetic of Weil Curves," Inv. Math., 25, 1-61, (1974). | MR | Zbl
[34] A. Néron: "Modèles Minimaux des Variétés Abéliennes sur les Corps Locaux et Globaux," Publ. Math. I.H.E.S., 21, (1964). | Numdam | MR | Zbl
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[36] F. Oort and J. Tate: "Group Schemes of Prime Order," Ann. Sci. E.N.S., 3, 1-21, (1970). | Numdam | MR | Zbl
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[41] J.-P. Serre: "Sur les Groupes de Congruence des Variétés Abéliennes II," Izv. Akad. Nauk S.S.S.R., Ser. Mat., Tom 35, 731-737, (1971). | MR | Zbl
[42] J.-P. Serre: Abelian l-Adic Representations and Elliptic Curves, New York: Benjamin, (1968). | MR | Zbl
[43] J.-P. Serre: "Groupes Analytiques p-Adiques," Sém. Bourbaki, 1963/64, Nº 270, Secretariat Mathematiques, 11 Rue Pierre Curie, Paris 6e , (1964). | Numdam | MR | Zbl
[44] J.-P. Serre and J. Tate: "Good Reduction of Abelian Varieties," Ann. of Math., 88, 492-517, (1968). | MR | Zbl
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[47] J. Tate: "Algorithm for Determining the Type of a Singular Fiber in an Elliptic Pencil," in [1], 33-52. | MR
[48] J. Tate: "Global Class Field Theory," in Algebraic Number Theory, J.W.S. Cassels and A. Fröhlich, eds., New York: Academic Press, (1967). | MR
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[51] M. Harris: "Systematic Growth of Mordell-Weil Groups of Abelian Varieties in Towers of Number Fields," to appear in Inv. Math. | MR | Zbl
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2022-12-02 19:48:34
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https://en.m.wikipedia.org/wiki/Difference_engine
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# Difference engine
The London Science Museum's difference engine, the first one actually built from Babbage's design. The design has the same precision on all columns, but when calculating polynomials, the precision on the higher-order columns could be lower.
A difference engine, first created by Charles Babbage, is an automatic mechanical calculator designed to tabulate polynomial functions. Its name is derived from the method of divided differences, a way to interpolate or tabulate functions by using a small set of polynomial coefficients. Most mathematical functions commonly used by engineers, scientists and navigators, including logarithmic and trigonometric functions, can be approximated by polynomials, so a difference engine can compute many useful tables of numbers.
The historical difficulty in producing error-free tables by teams of mathematicians and human "computers" spurred Charles Babbage's desire to build a mechanism to automate the process.
## History
Closeup of the London Science Museum's difference engine showing some of the number wheels and the sector gears between columns. The sector gears on the left show the double-high teeth very clearly. The sector gears on the middle-right are facing the back side of the engine, but the single-high teeth are clearly visible. Notice how the wheels are mirrored, with counting up from left-to-right, or counting down from left-to-right. Also notice the metal tab between "6" and "7". That tab trips the carry lever in the back when "9" passes to "0" in the front during the add steps (Step 1 and Step 3).
Per Georg Scheutz's third difference engine
The notion of a mechanical calculator for mathematical functions can be traced back to the Antikythera mechanism of the 2nd century BC, while early modern examples are attributed to Pascal and Leibniz in the 17th century. In 1784 J. H. Müller, an engineer in the Hessian army, devised and built an adding machine and described the basic principles of a difference machine in a book published in 1786 (the first written reference to a difference machine is dated to 1784), but he was unable to obtain funding to progress with the idea.[1][2][3]
### Charles Babbage's difference engines
Charles Babbage began to construct a small difference engine in c. 1819[4] and had completed it by 1822 (Difference Engine 0).[5] He announced his invention on June 14, 1822, in a paper to the Royal Astronomical Society, entitled "Note on the application of machinery to the computation of astronomical and mathematical tables".[6] This machine used the decimal number system and was powered by cranking a handle. The British government was interested, since producing tables was time-consuming and expensive and they hoped the difference engine would make the task more economical.[7]
In 1823, the British government gave Babbage £1700 to start work on the project. Although Babbage's design was feasible, the metalworking techniques of the era could not economically make parts in the precision and quantity required. Thus the implementation proved to be much more expensive and doubtful of success than the government's initial estimate. In 1832, Babbage and Joseph Clement produced a small working model (1/7 of the calculating section of Difference Engine No. 1,[5] which was intended to operate on 20-digit numbers and sixth-order differences) which operated on 6-digit numbers and second-order differences.[8][9] Lady Byron described seeing the working prototype in 1833: "We both went to see the thinking machine (for so it seems) last Monday. It raised several Nos. to the 2nd and 3rd powers, and extracted the root of a Quadratic equation."[10] Work on the larger engine was suspended in 1833.
By the time the government abandoned the project in 1842,[9][11] Babbage had received and spent over £17,000 on development, which still fell short of achieving a working engine. The government valued only the machine's output (economically produced tables), not the development (at unknown and unpredictable cost to complete) of the machine itself. Babbage did not, or was unwilling to, recognize that predicament.[7] Meanwhile, Babbage's attention had moved on to developing an analytical engine, further undermining the government's confidence in the eventual success of the difference engine. By improving the concept as an analytical engine, Babbage had made the difference engine concept obsolete, and the project to implement it an utter failure in the view of the government.[7]
The incomplete Difference Engine No. 1 was put on display to the public at the 1862 International Exhibition in South Kensington, London.[12][13]
Babbage went on to design his much more general analytical engine, but later produced an improved "Difference Engine No. 2" design (31-digit numbers and seventh-order differences),[8] between 1846 and 1849. Babbage was able to take advantage of ideas developed for the analytical engine to make the new difference engine calculate more quickly while using fewer parts.[14][15]
### Scheutzian calculation engine
Inspired by Babbage's difference engine in 1834, Per Georg Scheutz built several experimental models. In 1837 his son Edward proposed to construct a working model in metal, and in 1840 finished the calculating part, capable of calculating series with 5-digit numbers and first-order differences, which was later extended to third-order (1842). In 1843, after adding the printing part, the model was completed.
In 1851, funded by the government, construction of the larger and improved (15-digit numbers and fourth-order differences) machine began, and finished in 1853. The machine was demonstrated at the World's Fair in Paris, 1855 and then sold in 1856 to the Dudley Observatory in Albany, New York[16] (delivered in 1857).[17] In 1857 British government ordered next Scheutz's difference machine, which was built in 1859.[18][19] It had the same basic construction as the previous one. Weighed about 10 cwt (1,100 lb; 510 kg).[17]
### Others
Martin Wiberg improved Scheutz's construction (c. 1859, his machine has the same capacity as Scheutz's - 15-digit and fourth-order) but used his device only for producing and publishing printed tables (interest tables in 1860, and logarithmic tables in 1875).[20]
Alfred Deacon of London in c. 1862 produced a small difference engine (20-digit numbers and third-order differences).[16][21]
American George B. Grant started working on his calculating machine in 1869, unaware of the works of Babbage and Scheutz (Schentz). One year later (1870) he learned about difference engines and proceed to design one himself, describing his construction in 1871. In 1874 the Boston Thursday Club raised a subscription for the construction of a large-scale model, which was built in 1876. It could be expanded to enhance precision, weighed about 2,000 pounds (910 kg).[21][22][23]
Christel Hamann built one machine (16-digit numbers and second-order differences) in 1909 for the "Tables of Bauschinger and Peters" ("Logarithmic-Trigonometrical Tables with eight decimal places"), which was first published in Leipzig in 1910. Weighed about 40 kilograms (88 lb).[24][25][26]
Burroughs Corporation in about 1912 built a machine for Nautical Almanac Office which was used as a difference engine of second-order.[27]:451[28] It was later replaced in 1929 by a Burroughs Class 11 (13-digit numbers and second-order differences, or 11-digit numbers and [at least up to] fifth-order differences).[29]
Alexander John Thompson about 1927 built integrating and differencing machine (13-digit numbers and fifth-order differences) for his table of logarithms "Logarithmetica britannica". This machine was composed of four modified Triumphator calculators.[30][31][32]
Leslie Comrie in 1928 described how to use the Brunsviga-Dupla calculating machine as a difference engine of second-order (15-digit numbers).[27] He also noted in 1931 that National Accounting Machine Class 3000 could be used as a difference engine of sixth-order.[21]:137–138
### Construction of two working No. 2 difference engines
During the 1980s, Allan G. Bromley, an associate professor at the University of Sydney, Australia, studied Babbage's original drawings for the Difference and Analytical Engines at the Science Museum library in London.[33] This work led the Science Museum to construct a working calculating section of difference engine No. 2 from 1985 to 1991, under Doron Swade, the then Curator of Computing. This was to celebrate the 200th anniversary of Babbage's birth in 1999. In 2002, the printer which Babbage originally designed for the difference engine was also completed.[34] The conversion of the original design drawings into drawings suitable for engineering manufacturers' use revealed some minor errors in Babbage's design (possibly introduced as a protection in case the plans were stolen),[35] which had to be corrected. Once completed, both the engine and its printer worked flawlessly, and still do. The difference engine and printer were constructed to tolerances achievable with 19th-century technology, resolving a long-standing debate as to whether Babbage's design would actually have worked. (One of the reasons formerly advanced for the non-completion of Babbage's engines had been that engineering methods were insufficiently developed in the Victorian era.)
The printer's primary purpose is to produce stereotype plates for use in printing presses, which it does by pressing type into soft plaster to create a flong. Babbage intended that the Engine's results be conveyed directly to mass printing, having recognized that many errors in previous tables were not the result of human calculating mistakes but from error in the manual typesetting process.[7] The printer's paper output is mainly a means of checking the Engine's performance.
In addition to funding the construction of the output mechanism for the Science Museum's Difference Engine No. 2, Nathan Myhrvold commissioned the construction of a second complete Difference Engine No. 2, which was on exhibit at the Computer History Museum in Mountain View, California from 10 May 2008 until 31 January 2016.[36][37][38][39] It has since been transferred to Intellectual Ventures in Seattle where it is on display just outside the main lobby.
## Operation
The Mountain View machine in action
The difference engine consists of a number of columns, numbered from 1 to N. The machine is able to store one decimal number in each column. The machine can only add the value of a column n + 1 to column n to produce the new value of n. Column N can only store a constant, column 1 displays (and possibly prints) the value of the calculation on the current iteration.
The engine is programmed by setting initial values to the columns. Column 1 is set to the value of the polynomial at the start of computation. Column 2 is set to a value derived from the first and higher derivatives of the polynomial at the same value of X. Each of the columns from 3 to N is set to a value derived from the ${\displaystyle (n-1)}$ first and higher derivatives of the polynomial.
### Timing
In the Babbage design, one iteration (i.e., one full set of addition and carry operations) happens for each rotation of the main shaft. Odd and even columns alternately perform an addition in one cycle. The sequence of operations for column ${\displaystyle n}$ is thus:
1. Count up, receiving the value from column ${\displaystyle n+1}$ (Addition step)
2. Perform carry propagation on the counted up value
3. Count down to zero, adding to column ${\displaystyle n-1}$
4. Reset the counted-down value to its original value
Steps 1,2,3,4 occur for every odd column, while steps 3,4,1,2 occur for every even column.
While Babbage's original design placed the crank directly on the main shaft, it was later realized that the force required to crank the machine would have been too great for a human to handle comfortably. Therefore, the two models that were built incorporate a 4:1 reduction gear at the crank, and four revolutions of the crank are required to perform one full cycle.
### Steps
Each iteration creates a new result, and is accomplished in four steps corresponding to four complete turns of the handle shown at the far right in the picture below. The four steps are:
• Step 1. All even numbered columns (2,4,6,8) are added to all odd numbered columns (1,3,5,7) simultaneously. An interior sweep arm turns each even column to cause whatever number is on each wheel to count down to zero. As a wheel turns to zero, it transfers its value to a sector gear located between the odd/even columns. These values are transferred to the odd column causing them to count up. Any odd column value that passes from "9" to "0" activates a carry lever.
• Step 2. Carry propagation is accomplished by a set of spiral arms in the back that poll the carry levers in a helical manner so that a carry at any level can increment the wheel above by one. That can create a carry, which is why the arms move in a spiral. At the same time, the sector gears are returned to their original position, which causes them to increment the even column wheels back to their original values. The sector gears are double-high on one side so they can be lifted to disengage from the odd column wheels while they still remain in contact with the even column wheels.
• Step 3. This is like Step 1, except it is odd columns (3,5,7) added to even columns (2,4,6), and column one has its values transferred by a sector gear to the print mechanism on the left end of the engine. Any even column value that passes from "9" to "0" activates a carry lever. The column 1 value, the result for the polynomial, is sent to the attached printer mechanism.
• Step 4. This is like Step 2, but for doing carries on even columns, and returning odd columns to their original values.
### Subtraction
The engine represents negative numbers as ten's complements. Subtraction amounts to addition of a negative number. This works in the same manner that modern computers perform subtraction, known as two's complement.
## Method of differences
Fully operational difference engine at the Computer History Museum in Mountain View, California
The principle of a difference engine is Newton's method of divided differences. If the initial value of a polynomial (and of its finite differences) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences. For example, consider the quadratic polynomial
${\displaystyle p(x)=2x^{2}-3x+2\,}$
with the goal of tabulating the values p(0), p(1), p(2), p(3), p(4), and so forth. The table below is constructed as follows: the second column contains the values of the polynomial, the third column contains the differences of the two left neighbors in the second column, and the fourth column contains the differences of the two neighbors in the third column:
x p(x) = 2x2 − 3x + 2 diff1(x) = ( p(x + 1) − p(x) ) diff2(x) = ( diff1(x + 1) − diff1(x) )
0 2 −1 4
1 1 3 4
2 4 7 4
3 11 11
4 22
The numbers in the third values-column are constant. In fact, by starting with any polynomial of degree n, the column number n + 1 will always be constant. This is the crucial fact behind the success of the method.
This table was built from left to right, but it is possible to continue building it from right to left down a diagonal in order to compute more values. To calculate p(5) use the values from the lowest diagonal. Start with the fourth column constant value of 4 and copy it down the column. Then continue the third column by adding 4 to 11 to get 15. Next continue the second column by taking its previous value, 22 and adding the 15 from the third column. Thus p(5) is 22 + 15 = 37. In order to compute p(6), we iterate the same algorithm on the p(5) values: take 4 from the fourth column, add that to the third column's value 15 to get 19, then add that to the second column's value 37 to get 56, which is p(6). This process may be continued ad infinitum. The values of the polynomial are produced without ever having to multiply. A difference engine only needs to be able to add. From one loop to the next, it needs to store 2 numbers—in this example (the last elements in the first and second columns). To tabulate polynomials of degree n, one needs sufficient storage to hold n numbers.
Babbage's difference engine No. 2, finally built in 1991, could hold 8 numbers of 31 decimal digits each and could thus tabulate 7th degree polynomials to that precision. The best machines from Scheutz could store 4 numbers with 15 digits each.[40]
## Initial values
The initial values of columns can be calculated by first manually calculating N consecutive values of the function and by backtracking, i.e. calculating the required differences.
Col ${\displaystyle 1_{0}}$ gets the value of the function at the start of computation ${\displaystyle f(0)}$ . Col ${\displaystyle 2_{0}}$ is the difference between ${\displaystyle f(1)}$ and ${\displaystyle f(0)}$ ...[41]
If the function to be calculated is a polynomial function, expressed as
${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,}$
the initial values can be calculated directly from the constant coefficients a0, a1,a2, ..., an without calculating any data points. The initial values are thus:
• Col ${\displaystyle 1_{0}}$ = a0
• Col ${\displaystyle 2_{0}}$ = a1 + a2 + a3 + a4 + ... + an
• Col ${\displaystyle 3_{0}}$ = 2a2 + 6a3 + 14a4 + 30a5 + ...
• Col ${\displaystyle 4_{0}}$ = 6a3 + 36a4 + 150a5 + ...
• Col ${\displaystyle 5_{0}}$ = 24a4 + 240a5 + ...
• Col ${\displaystyle 6_{0}}$ = 120a5 + ...
• ${\displaystyle ...}$
### Use of derivatives
Many commonly used functions are analytic functions, which can be expressed as power series, for example as a Taylor series. The initial values can be calculated to any degree of accuracy; if done correctly the engine will give exact results for first N steps. After that, the engine will only give an approximation of the function.
The Taylor series expresses the function as a sum obtained from its derivatives at one point. For many functions the higher derivatives are trivial to obtain; for instance, the sine function at 0 has values of 0 or ${\displaystyle \pm 1}$ for all derivatives. Setting 0 as the start of computation we get the simplified Maclaurin series
${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\ x^{n}}$
The same method of calculating the initial values from the coefficients can be used as for polynomial functions. The polynomial constant coefficients will now have the value
${\displaystyle a_{n}\equiv {\frac {f^{(n)}(0)}{n!}}}$
### Curve fitting
The problem with the methods described above is that errors will accumulate and the series will tend to diverge from the true function. A solution which guarantees a constant maximum error is to use curve fitting. A minimum of N values are calculated evenly spaced along the range of the desired calculations. Using a curve fitting technique like Gaussian reduction an N−1th degree polynomial interpolation of the function is found.[41] With the optimized polynomial, the initial values can be calculated as above.
## The difference engine in other works
William Gibson and Bruce Sterling's The Difference Engine is an alternate history[42] novel that looks how society would have progressed had the difference engine worked the way Babbage envisioned it.
The story takes places in Victorian England where technological advancement is on the rise. This is due to the effect of the success of Babbage's analytical machine. The convention of steampunk where Victorian fashion is combined with the technological elements of the Industrial Revolution is seen throughout the story due to technology being so advanced in that era.
## References
1. ^ Johann Helfrich von Müller, Beschreibung seiner neu erfundenen Rechenmachine, nach ihrer Gestalt, ihrem Gebrauch und Nutzen [Description of his newly invented calculating machine, according to its form, its use and benefit] (Frankfurt and Mainz, Germany: Varrentrapp Sohn & Wenner, 1786); pages 48–50. The following Web site (in German) contains detailed photos of Müller's calculator as well as a transcription of Müller's booklet, Beschreibung …: https://www.fbi.h-da.de/fileadmin/vmi/darmstadt/objekte/rechenmaschinen/mueller/index.htm. An animated simulation of Müller's machine in operation is available on this Web site (in German): https://www.fbi.h-da.de/fileadmin/vmi/darmstadt/objekte/rechenmaschinen/mueller/simulation/index.htm.
2. ^ Michael Lindgren (Craig G. McKay, trans.), Glory and Failure: The Difference Engines of Johann Müller, Charles Babbage, and Georg and Edvard Scheutz (Cambridge, Massachusetts: MIT Press, 1990), pages 64 ff.
3. ^ Swedin, E.G.; Ferro, D.L. (2005). Computers: The Life Story of a Technology. Greenwood Press, Westport, CT. p. 14. ISBN 978-0-313-33149-7.
4. ^ Dasgupta, Subrata (2014). It Began with Babbage: The Genesis of Computer Science. Oxford University Press. p. 22. ISBN 978-0-19-930943-6.
5. ^ a b Copeland, B. Jack; Bowen, Jonathan P.; Wilson, Robin; Sprevak, Mark (2017). The Turing Guide. Oxford University Press. p. 251. ISBN 9780191065002.
6. ^ O'Connor, John J.; Robertson, Edmund F. (1998). "Charles Babbage". MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 2006-06-16. Retrieved 2006-06-14.
7. ^ a b c d Campbell-Kelly, Martin (2004). Computer: A History of the Information Machine 2nd ed. Boulder, Co: Westview Press. ISBN 978-0-8133-4264-1.
8. ^ a b O'Regan, Gerard (2012). A Brief History of Computing. Springer Science & Business Media. p. 204. ISBN 978-1-4471-2359-0.
9. ^ a b Snyder, Laura J. (2011). The Philosophical Breakfast Club: Four Remarkable Friends Who Transformed Science and Changed the World. Crown/Archetype. pp. 192, 210, 217. ISBN 978-0-307-71617-0.
10. ^ Toole, Betty Alexandra; Lovelace, Ada (1998). Ada, the Enchantress of Numbers. Mill Valley, California: Strawberry Press. p. 38. ISBN 978-0912647180. OCLC 40943907.
11. ^ Weld, Charles Richard (1848). A History of the Royal Society: With Memoirs of the Presidents. J. W. Parker. pp. 387–390.
12. ^ Tomlinson, Charles (1868). Cyclopaedia of useful arts, mechanical and chemical, manufactures, mining and engineering: in three volumes, illustrated by 63 steel engravings and 3063 wood engravings. Virtue & Co. p. 136.
13. ^ 1862, International exhibition (1862). Official catalogue of the industrial department. p. 49.
14. ^ Snyder, Laura J. (2011). The Philosophical Breakfast Club. New York: Broadway Brooks. ISBN 978-0-7679-3048-2.
15. ^ Morris, Charles R. (October 23, 2012). The Dawn of Innovation: The First American Industrial Revolution. PublicAffairs. p. 63. ISBN 9781610393577.
16. ^ a b Scheutz, George; Scheutz, Edward (1857). Specimens of Tables, Calculated, Stereomoulded, and Printed by Machinery. Whitnig. pp. VIII–XII, XIV–XV, 3.
17. ^ a b Merzbach, Uta C.; Zoology, Smithsonian Contributions To; Ripley, S. Dillon; Merzbach, Uta C. First Printing Calculator. pp. 8–9, 13, 25–26, 29–30. CiteSeerX 10.1.1.639.3286.
18. ^ Swade, Doron (2002-10-29). The Difference Engine: Charles Babbage and the Quest to Build the First Computer. Penguin Books. pp. 4, 207. ISBN 9780142001448.
19. ^ Watson, Ian (2012). The Universal Machine: From the Dawn of Computing to Digital Consciousness. Springer Science & Business Media. pp. 37–38. ISBN 978-3-642-28102-0.
20. ^ Raymond Clare Archibald: Martin Wiberg, his Table and Difference Engine, Mathematical Tables and Other Aids to Computation, 1947(2:20) 371–374. (online review) (PDF; 561 kB).
21. ^ a b c Campbell-Kelly, Martin (2003). The History of Mathematical Tables: From Sumer to Spreadsheets. OUP Oxford. pp. 132–136. ISBN 978-0-19-850841-0.
22. ^ "History of Computers and Computing, Babbage, Next differential engines, George Grant". history-computer.com. Retrieved 2017-08-29.
23. ^ Sandhurst, Phillip T. (1876). The Great Centennial Exhibition Critically Described and Illustrated. P. W. Ziegler & Company. pp. 423, 427.
24. ^ "History of Computers and Computing, Babbage, Next differential engines, Hamann". history-computer.com. Retrieved 2017-09-14.
25. ^ Bauschinger, Julius; Peters, Jean (1958). Logarithmisch-trigonometrische Tafeln mit acht Dezimalstellen, enthaltend die Logarithmen aller Zahlen von 1 bis 200000 und die Logarithmen der trigonometrischen Funktionen f"ur jede Sexagesimalsekunde des Quadranten: Bd. Tafel der achtstelligen Logarithmen aller Zahlen von 1 bis 200000. H. R. Engelmann. pp. Preface V–VI.
26. ^ Bauschinger, Julius; Peters, J. (Jean) (1910). Logarithmisch-trigonometrische Tafeln, mit acht Dezimalstellen, enthaltend die Logarithmen aller Zahlen von 1 bis 200000 und die Logarithmen der trigonometrischen Funktionen für jede Sexagesimalsekunde des Quadranten. Neu berechnet und hrsg. von J. Bauschinger und J. Peters. Stereotypausg (in German). Gerstein - University of Toronto. Leipzig W. Englemann. pp. Einleitung VI.
27. ^ a b Comrie, L. J. (1928-03-01). "On the application of the BrunsvigaDupla calculating machine to double summation with finite differences". Monthly Notices of the Royal Astronomical Society. 88 (5): 451, 453–454, 458–459. Bibcode:1928MNRAS..88..447C. doi:10.1093/mnras/88.5.447. ISSN 0035-8711 – via Astrophysics Data System.
28. ^ Horsburg, E. M. (Ellice Martin); Napier Tercentenary Exhibition (1914). Modern instruments and methods of calculation : a handbook of the Napier Tercentenary Exhibition. Gerstein - University of Toronto. London : G. Bell. pp. 127–131.
29. ^ Comrie, L. J. (1932-04-01). "The Nautical Almanac Office Burroughs machine". Monthly Notices of the Royal Astronomical Society. 92 (6): 523–524, 537–538. Bibcode:1932MNRAS..92..523C. doi:10.1093/mnras/92.6.523. ISSN 0035-8711 – via Astrophysics Data System.
30. ^ Thompson, Alexander John (1924). Logarithmetica Britannica: Being a Standard Table of Logarithms to Twenty Decimal Places. CUP Archive. pp. V/VI, XXIX, LIV–LVI, LXV (archive: pp. 7, 30, 55–59, 68). ISBN 9781001406893. Archived from the original on 2015-08-06.
31. ^ "History of Computers and Computing, Babbage, Next differential engines, Alexander John Thompson". history-computer.com. Retrieved 2017-09-22.
32. ^ Weiss, Stephan. "Publikationen". mechrech.info. Difference Engines in the 20th Century. First published in Proceedings 16th International Meeting of Collectors of Historical Calculating Instruments, Sep. 2010, Leiden. pp. 160–163. Retrieved 2017-09-22.
33. ^
34. ^ "A Modern Sequel | Babbage Engine | Computer History Museum". www.computerhistory.org.
35. ^ Babbage printer finally runs, BBC news quoting Reg Crick Accessed May 17, 2012
36. ^ "The Babbage Difference Engine No. 2 | Computer History Museum". www.computerhistory.org. Retrieved 2018-10-26.
37. ^
38. ^ Press Releases | Computer History
39. ^ Difference Engine Leaves Computer History Museum, Mark Moack, Mountain View Voice, January 29, 2016
40. ^ O'Regan, Gerard (2012). A Brief History of Computing. Springer Science & Business Media. p. 201. ISBN 978-1-4471-2359-0.
41. ^ a b Thelen, Ed (2008). "Babbage Difference Engine #2 – How to Initialize the Machine –".
42. ^ Gibson, William. The Difference Engine.
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2019-05-20 11:04:01
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https://ncatlab.org/nlab/show/parameterized+homotopy+theory
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nLab parameterized homotopy theory
Context
Bundles
bundles
fiber bundles in physics
Contents
Idea
Parameterized (stable) homotopy theory is (stable) homotopy theory of bundles of homotopy types/stable homotopy types over a given base space.
For formalizations see at
References
Last revised on April 21, 2018 at 04:37:40. See the history of this page for a list of all contributions to it.
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2019-07-22 04:07:47
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http://tjsullivan.org.uk/
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# Tim Sullivan
### Welcome!
I am Assistant Professor in Predictive Modelling in the Mathematics Institute and School of Engineering at the University of Warwick and Research Group Leader for Uncertainty Quantification at the Zuse Institute Berlin. I have wide interests in uncertainty quantification the broad sense, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. On this site you will find information about how to contact me, my research, publications, and teaching activities.
### Joining the University of Warwick
It is a pleasure to announce that I have accepted an Assistant Professorship in Predictive Modelling at the University of Warwick, to be held jointly between the Mathematics Institute and the School of Engineering, in particular the Warwick Centre for Predictive Modelling.
Published on Monday 6 April 2020 at 14:00 UTC #warwick #wcpm
### Birzhan Ayanbayev joins the UQ Group
It is a pleasure to announce that Birzhan Ayanbayev will join the UQ research group as a postdoctoral researcher with effect from 28 February 2020. He will be working on the DFG-funded project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems”.
Published on Friday 28 February 2020 at 12:00 UTC #group #job #dfg #map-estimators #ayanbayev
### Geodesic analysis in Kendall's shape space now in J. Math. Imaging Vis.
The article “Geodesic analysis in Kendall's shape space with epidemiological applications” by Esfandiar Nava-Yazdani, Christoph von Tycowicz, Hans-Christian Hege, and myself has just appeared online in the Journal of Mathematical Imaging and Vision.
E. Nava-Yazdani, H.-C. Hege, T. J. Sullivan, and C. von Tycowicz. “Geodesic analysis in Kendall's shape space with epidemiological applications.” Journal of Mathematical Imaging and Vision 1–11, 2020. doi:10.1007/s10851-020-00945-w
Abstract. We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application, we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.
Published on Tuesday 18 February 2020 at 14:00 UTC #publication #ch15 #shape-trajectories #nava-yazdani #von-tycowicz #hege
### Preprint: A rigorous theory of conditional mean embeddings
Ilja Klebanov, Ingmar Schuster, and I have just uploaded a preprint of our recent work “A rigorous theory of conditional mean embeddings” to the arXiv. In this work we take a close mathematical look at the method of conditional mean embedding. In this approach to non-parametric inference, a random variable $$Y \sim \mathbb{P}_{Y}$$ in a set $$\mathcal{Y}$$ is represented by its kernel mean embedding, the reproducing kernel Hilbert space element
$$\displaystyle \mu_{Y} = \int_{\mathcal{Y}} \psi(y) \, \mathrm{d} \mathbb{P}_{Y} (y) \in \mathcal{G},$$
and conditioning with respect to an observation $$x$$ of a related random variable $$X \sim \mathbb{P}_{X}$$ in a set $$\mathcal{X}$$ with RKHS $$\mathcal{H}$$ is performed using the Woodbury formula
$$\displaystyle \mu_{Y|X = x} = \mu_Y + (C_{XX}^{\dagger} C_{XY})^\ast \, (\varphi(x) - \mu_X) .$$
Here $$\psi \colon \mathcal{Y} \to \mathcal{G}$$ and $$\varphi \colon \mathcal{X} \to \mathcal{H}$$ are the canonical feature maps and the $$C$$'s denote the appropriate centred (cross-)covariance operators of the embedded random variables $$\psi(Y)$$ in $$\mathcal{G}$$ and $$\varphi(X)$$ in $$\mathcal{H}$$.
Our article aims to provide rigorous mathematical foundations for this attractive but apparently naïve approach to conditional probability, and hence to Bayesian inference.
Abstract. Conditional mean embeddings (CME) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embeddings of the respective probability distributions. Both centered and uncentered covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of either, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.
Published on Tuesday 3 December 2019 at 07:00 UTC #publication #preprint #mathplus #tru2 #rkhs #mean-embedding #klebanov #schuster
### Postdoc position: Analysis of MAP estimators
There is still an opening for a full-time two-year postdoctoral research position in the UQ group at the Freie Universität Berlin. This position will be associated to the project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems” funded by the DFG.
This project aims to advance the state of the art in rigorous mathematical understanding of MAP estimators in infinite-dimensional statistical inverse problems. In particular, the research in this project will connect the “small balls” approach of Dashti et al. (2013) to the calculus of variations and hence properly link the variational and fully Bayesian points of view on inverse problems. This is an exciting opportunity for someone well-versed in the calculus of variations and tools such as Γ-convergence to make an impact on fundamental questions of non-parametric statistics and inverse problems or, vice versa, for someone with a statistical inverse problems background to advance the rigorous state of the art for such methods.
Prospective candidates are encouraged to contact me with informal enquiries. Formal applications are to be sent by post or email, by 23 December 2019, under the heading MAP-Analysis, and should include a cover letter, a scientific CV including list of publications and research statement, and the contact details of two professional references.
Published on Monday 25 November 2019 at 08:00 UTC #group #job #fu-berlin #inverse-problems #dfg #map-estimators
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2020-04-08 14:14:32
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http://www.mathnet.ru/php/archive.phtml?jrnid=dm&wshow=issue&year=2020&volume=32&volume_alt=&issue=2&issue_alt=&option_lang=eng
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RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
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On the numerical semigroup generated by $\{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\}$Ze Gu 3 On the complexity of system of two monomials realization by composition circuitsS. A. Korneev 15 On complexity of realization of Boolean functions with the small number of onesN. P. Red'kin 32 On conditions of $A$-completeness for linear automata over dyadic rationalsD. V. Ronzhin 44 On the degree of restrictions of $q$-valued logic vector functions to linear manifoldsV. G. Ryabov 61 Trees with a given number of leaves and the maximum number of independent setsD. S. Taletskii, D. S. Malyshev 71 About applying binary operations for constructing a multiply transitive class of block transformationsI. V. Cherednik 85 Medial strong dependance $n$-ary operationsA. V. Cheremushkin 112 Letter to the EditorsG. A. Bakai, A. V. Shklyaev 122
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2021-01-19 08:13:36
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https://classes.golem.ph.utexas.edu/category/2017/04/gluing_together_finite_shapes.html
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## April 3, 2017
### Gluing Together Finite Shapes with Kelly
#### Posted by Emily Riehl
Guest post by David Jaz Myers
The Kan Extension Seminar continues with Kelly’s Structures defined by limits in the enriched context, I, or as I like to call it, Presenting categories whose objects are glued together out of finite shapes. This paper hit the presses in 1982 and represented both a condensation and generalization of the theory of locally finitely presentable categories. Kelly achieved both the condensing and the generalizing through his use of weighted (or, as he calls them, indexed) limits and colimits. The resulting theory is slick and highly categorical, but very dense. So, in this post, we’re going to button down a bit and skim over a few details so that we can focus on the story and the powerful results which sustain it, rather than the technicalities. I will label the propositions by where they appear in Kelly’s paper, so that you can find the technicalities if you would like.
At first, it will seem like this post is a radical departure from the previous seminar posts. But as we go on, we’ll see that gluing together finite shapes has a lot to do with the categorical approach to algebra that Evangelia wrote about in the post that kicked off our seminar.
Before I start, I’d like to thank Emily, Alexander, and Brendan for organizing this lovely seminar and inviting me to participate. I’d also like to thank my fellow students for their insight and conversation.
Throughout this paper, Kelly works in the enriched context. Where normally we like to think of there being a set of arrows between any two objects of a category, Kelly wants us to think of there being an object of arrows. This object of arrows can live in any suitably nice category, which here will mean a symmetric monoidal closed and cocomplete (and later, locally finitely presentable) category. We call this category our base category, and denote it by the letter $\mathcal{V}$.
For the rest of this blog post, we will work from the $\mathcal{V}$-point of view. When I say “category”, I will mean $\mathcal{V}$-category, and same for “functor” and “natural transformation”. For the most part, this will work smoothly. But the crucial notion of being “filtered”, which we will come to later, is only defined over the category of sets. I’ll try and be explicit about this when it comes up. For a brief intro to the $\mathcal{V}$-point of view, see the previous blog post which introduces enrichment and its limits.
#### Understanding Objects by Looking at Figures in Them
When we work with a mathematical structure $X$, we often like to do so in terms of its elements. In our base category, it suffices to look at the points of $X$, the maps $1 \to X$. But, in more general categories $\mathcal{C}$, we may need to consider figures $S \to X$ of a more general shape.
Remark For the rest of this post, I will use the term “figure in $X$” to mean a morphism $S \to X$ in a category, and “shape” to mean the domain of a figure. By an “incidence relation” between shapes, I just mean morphisms between them. A somewhat more common name for “shape” would be “test space”, but I find the word “shape” to be more cozy.
Yoneda’s lemma tells us that we lose no information if we use figures of an arbitrary shape.
Lemma (Yoneda’s Lemma) In any category, morphisms $X \to Y$ naturally correspond to functions which take figures in $X$ to figures of the same shape in $Y$ which respect the incidence relations between the shapes.
However, it will often be the case that general shapes are just too complicated to work with. After all, we are trying to study an object in a category; a general morphism into it can be no less complicated. Even worse, there are often so many general shapes that they cannot be collected into a set (if we are working in a large category). It would be very nice to have a small subcategory $\mathcal{G} \hookrightarrow \mathcal{C}$ of “tractable shapes” that we could restrict our attention to. “Tractable” is not a technical term, but more of a suggestive name like “nice”.
Such a subcategory would induce a functor $\mathcal{C}(\mathcal{G}, -) : \mathcal{C} \to \mathcal{V}^{\mathcal{G}^{\text{op}}}$ sending an object of $\mathcal{C}$ to its functor of “figures with tractable shape”. If we can glue together the objects of $\mathcal{C}$ (that is, if $\mathcal{C}$ is cocomplete), then this functor will have a left adjoint given by taking the weighted colimit of the inclusion $\mathcal{G} \hookrightarrow \mathcal{C}$. We can think of this left adjoint as gluing together a bunch of tractable shapes into an object of $\mathcal{C}$. Let’s look at a few examples of this kind of arrangement.
• As I mentioned above, it suffices to work with points (that is, figures $1 \to X$ of shape $1$) when we are in the base category. This should mean that if we take our category of “tractable shapes” to be just the subcategory containing the monoidal unit $1$, we will recover all of our base category. And indeed, $\mathcal{V}(1, -) : \mathcal{V} \to \mathcal{V}^{1^{\text{op}}}$ is an isomorphism of categories! Its inverse sends an object $W : 1^{\text{op}} \to \mathcal{V}$ of points to the copower $W \cdot 1$, the $W$-indexed sum of points.
• Let’s work over sets, and consider the category of graphs $\mathcal{C}$ (to be specific, by “graph” I mean a symmetric relation). It no longer suffices to just look at points $\ast \to X$ (figures in $X$ whose shape is a single point $\ast$), since there can be many graphs with the same set of points. Nor can we just look at figures whose shape is a single edge $\ast - \ast$, since many graphs have the same set of edges. But if we keep in mind the incidence relations between these two shapes, which is just that the point $\ast$ lies in the edge $\ast - \ast$ in two ways, then we can recover the whole of our graph $X$. We let $\mathcal{G} = \{\ast \rightrightarrows (\ast - \ast) \}$ be our subcategory of tractable shapes, and we find that $\mathcal{C}(\mathcal{G}, -) ; \mathcal{C} \to \mathcal{V}^{\mathcal{G}^{\text{op}}}$ is fully faithful. Its adjoint is the functor which glues a bunch of edges onto a bunch of points, and that it is fully faithful means that any morphism of graphs is determined by its actions on the points and edges.
• Now let’s look at an example where our object $X$ is seriously intractable. Working over sets again, let $\mathcal{C}$ be the category of topological spaces. As above, a topological space $X$ is not determined by its points. We can try to get a grip on $X$ by modeling it with triangles. Higher dimensional triangles are called simplices, and they are particularly nice spaces, so we can take our tractable objects $\mathcal{G}$ to be the subcategory of simplices and their incidence relations. Then the $\mathcal{C}(\mathcal{G}, -) : \mathcal{C} \to \mathcal{V}^{\mathcal{G}^{\text{op}}}$ sends a topological space to its singular simplicial set, and its left adjoint gives the geometric realization of an abstract simplicial set. In this case, $\mathcal{C}(\mathcal{G}, -)$ is only faithful; there are spaces (like Cantor’s space) which aren’t well studied by looking at triangles in them.
For this program of looking at “tractable shapes” to work out, we will need the objects of $\mathcal{C}$ to be suitably determined by the figures of tractable shape inside them. We will take “suitably determined” to mean that we can test if a morphism is an isomorphism by checking if it induces a bijection on figures of tractable shape.
Definition Let $\mathcal{C}$ be cocomplete and let $\mathcal{G} \hookrightarrow \mathcal{C}$ be a subcategory. We say that $\mathcal{G}$ is a strong generator of $\mathcal{C}$ if $\mathcal{C}(\mathcal{G}, -)$ is conservative (that is, if it reflects isomorphisms).
So, we are looking for a strong generator $\mathcal{G}$ consisting of “tractable” objects of $\mathcal{C}$. But what does “tractable” mean?
Let’s consider an algebraic example, say, the category $\mathcal{C} = \text{Grp}$ of groups (as a set-category). We often study algebras by looking at the equations that elements in them satisfy. For example, a group is abelian if $ab = ba$ for every two elements $a$ and $b$ in it. These finite sets of relations between elements can be seen as figures in the group; the shape of these figures is a finitely presented group, generated by the elements and quotiented by the relations in question. For example, the commuting pairs in a group are precisely the figures (homomorphisms) of shape (with domain) $\langle a, b \mid ab = ba \rangle$.
In this algebraic setting, it seems reasonable to take “tractable” to mean finitely presentable. By finitely presentable, I mean admitting a finite presentation in terms of generators and relations in the usual sense. Here is a categorical way to describe a presentation in terms of generators and relations: we have a set of generators and a set of relations together with two homomorphisms from the free algebra on the relations to the free algebra on the generators taking a relation to the left and right hand sides of the equality that represents it. Then, to be presented by such a presentation means to be a coequalizer of those two arrows. For example, the group $\langle a, b \mid ab = ba\rangle$ above is given by the coequalizer of $\langle c \rangle \rightrightarrows \langle a , b \rangle$ where the left arrow sends $c$ to $ab$ and the right sends $c$ to $ba$.
Interestingly, there is an abstract categorical definition of finite presentability which doesn’t mention generators or relations at all!
Theorem In a category of algebras $\mathcal{C}$ (say, models of a Lawvere theory), an algebra $A$ is finitely presentable if and only if the hom functor $\mathcal{C}(A, -)$ commutes with filtered colimits.
#### Finite Shapes and Gluing Them Together
Given the above abstract definition of finite presentability, we will take “tractable” to mean “finitely presentable” in the abstract categorical sense. Let’s explain precisely what this is now.
Definition A set-category $\mathcal{D}$ is filtered if every functor $\mathcal{J} \to \mathcal{D}$ with $\mathcal{J}$ finite has a cocone.
The colimit of a diagram $\mathcal{D} \to \mathcal{C}$ is filtered if $\mathcal{D}$ is.
A functor $\mathcal{C} \to \mathcal{B}$ is finitary if it commutes with filtered colimits.
An object $G$ in $\mathcal{C}$ is finitely presentable (or f.p.) if $\mathcal{C}(G, -) : \mathcal{C} \to \mathcal{V}$ is finitary. We denote by $\mathcal{C}_f$ the (full) category of f.p. objects in $\mathcal{C}$.
Finitely presentable objects are “morally finite”, relative to whatever category they are in. A good grounding example occurs in the truth value enriched setting.
Theorem Let $\Omega$ be the order of open sets in a topological space $X$, considered as a category over the truth values. If $K \subseteq X$ is a subset, then the functor $U \mapsto K \subseteq U$ is finitary if and only if $K$ is compact.
Proof Commuting with filtered colimits means $K \subseteq \bigcup_i U_i \iff \exists i (K \subseteq U_i)$ for each filtered family of open sets $(U_i)_i$. We can complete any open cover to a filtered cover by throwing in finite unions of opens in the cover; if the functor is finitary, then one such finite union contains $K$, so $K$ is compact.
If $K$ is compact, any open cover (including one that is filtered) contains a finite subcover; since the cover is filtered, there is open set in it containing the finite cover, which shows that the functor is finitary.
It’s reasonable to think of compact sets as “topologically finite”. Representing a finitary functor makes an object “finite” in the category in question. For example, a finitely presentable set is finite.
Claim A finitely presentable set is finite.
Proof If $X$ is finitely presentable, then $\text{Set}(X,-)$ commutes with filtered colimits. Every set is the filtered colimit of its finite subsets, so in particular $\text{Set}(X,X) \cong \text{colim}\text{Set}(X, n)$. But then the identity of $X$ factors through one of its finite subsets, so $X$ is finite.
It would be good to show further that every finite set is finitely presentable. We can do even better by showing that gluing together finitely many finite shapes gives us a finite shape. But first, we should make sure we know what “finitely many” means in the enriched context.
Definition A weight $W : \mathcal{D} \to \mathcal{V}$ is finite when
• $\mathcal{D}$ has finitely many objects (up to isomorphism),
• $\mathcal{D}(i, j)$ is f.p. for all $i$ and $j$ in $\mathcal{D}$, and
• $W_i$ is f.p. for all $i$ in $\mathcal{D}$.
Finite (co)limits are those whose weights are finite.
Lemma (4.41) For $\mathcal{C}$ cocomplete, $\mathcal{C}_f$ is closed under finite colimits.
Corollary A set is finite if and only if it is finitely presentable.
Proof We saw that finitely presentable sets are finite, so it remains to show that finite sets are finitely presentable. But a finite set is a finite coproduct of the set $1$, and $1$ is finitely presentable because $\text{Set}(1, X) \cong X$ commutes with filtered colimits.
Now that we have a good formal definition of “tractable”, we can define those categories whose objects are suitably determined by the tractable figures within them.
Definition A category $\mathcal{C}$ is locally finitely presentable (l.f.p.) when
• it is cocomplete, and
• there is a small subcategory $\mathcal{G} \hookrightarrow \mathcal{C}$ so that the induced $\mathcal{C}(\mathcal{G}, -)$ is finitary and conservative.
Note that $\mathcal{C}(\mathcal{G}, -)$ being finitary implies that $\mathcal{G} \hookrightarrow \mathcal{C}_f$, since we can calculate colimits in $\mathcal{V}^{\mathcal{G}^{\text{op}}}$ pointwise.
Since being finitary relies on a definition over the category of sets, there can be subtle differences between the objects which are finitely presented over sets versus objects that are finitely presented over the base $\mathcal{V}$. For example, the monoidal unit $1$ is always finitely presentable over $\mathcal{V}$, since $\mathcal{V}(1, X) \cong X$. But over sets, it might not be, since the set of points of an object in $\mathcal{V}$ might be ill-behaved. Kelly gives a good example of this: in the category of actions of a group $G$, the monoidal unit $G$ is always finitely presentable over the actions, but it is only finitely presentable over sets if $G$ is finite.
For more on these subtleties, see section 5 of the paper; from now on, we will assume that our base $\mathcal{V}$ is l.f.p. over sets and that moreover the f.p. objects in $\mathcal{V}$ are the same whether taken over $\mathcal{V}$ or over sets. This occurs if and only if the monoidal unit of $\mathcal{V}$ is f.p. over sets and the tensor product of two f.p.-over-sets objects is f.p. over sets (see 5.5 in the paper). Kelly calls this situation being “l.f.p. as a monoidal closed category”, and so we will as well.
Now, Kelly proves some serious theorems about l.f.p. categories.
Theorem (7.2) Let $\mathcal{C}$ be l.f.p. with strong generator $\mathcal{G} \hookrightarrow \mathcal{C}_f$. Then
• $\mathcal{C}_f$ is the closure of $\mathcal{G}$ under finite colimits,
• $\mathcal{C}_f$ is dense in $\mathcal{C}$, meaning that $\mathcal{C}(\mathcal{C}_f,-)$ is fully faithful.
• $\mathcal{C}(\mathcal{C}_f, -)$ is also finitary.
• $\mathcal{C}$ is complete.
This theorem shows that the finite shapes in a category whose objects are determined by some particular subcategory $\mathcal{G}$ of finite “tractable” shapes are precisely those shapes which can be glued together out of finitely many “tractable” shapes from $\mathcal{G}$. For example, working over sets, we see that the finitely presentable graphs are precisely the finite colimits of the point and the edge, that is, precisely the finite graphs. We can also deduce that the finitely presentable simplicial sets are those which have a finite number of simplices.
Rather surprisingly, we also find that $\mathcal{C}$ is complete. Perhaps this is not so surprising though; if we know how to glue our objects together out of finite shapes, then we can form families of objects by forming families of these shapes. So, l.f.p. categories are very nice categories indeed. As a corollary of this theorem, Kelly gives a few more characterizations of l.f.p. categories that show that the choice of generators $\mathcal{G}$ is not essential.
Corollary (7.3) The following are equivalent for a cocomplete category $\mathcal{C}$:
• $\mathcal{C}$ is locally finitely presentable,
• $\mathcal{C}_f$ is small and strongly generating,
• $\mathcal{C}_f$ is small and dense,
• $\mathcal{C}$ is a reflective subcategory of $\mathcal{V}^{\mathcal{G}^{\text{op}}}$ with finitary inclusion, for some $\mathcal{G}$.
Here we see that the l.f.p. categories are reflective subcategories of presheaf categories. Since presheaf categories are always fantastically nice (inheriting, as they do, most of the nice structure of the base), l.f.p. categories are fantastically nice as well. For example, in the category of sets, finite limits commute with filtered colimits. In a l.f.p. category over an l.f.p. base, we can deduce the same.
Lemma (4.9) Let $\mathcal{C}$ be a l.f.p. category over an l.f.p. (as a monoidal closed category) base. Then finite limits commute with filtered colimits in $\mathcal{C}$.
Furthermore, l.f.p. categories satisfy two really nice adjoint functor theorems. Before we can express them, we need to mention that the above discussion of filtered colimits and f.p. objects can be repeated with “finite” (that is, “less than $\aleph_0$”) being replaced by “less than $\kappa$” for any regular cardinal $\kappa$.
Theorem (7.8, 7.9) Let $\mathcal{C}$ and $\mathcal{B}$ be l.f.p. and let $F : \mathcal{C} \to \mathcal{B}$ be a functor. Then
• $F$ is a left adjoint if and only if it commutes with colimits, and
• $F$ is a right adjoint if and only if it and commutes with limits and $\kappa$-filtered colimits for some regular cardinal $\kappa$.
As an example, let $\mathcal{C}$ be the category of graphs over sets, and let $F = \mathcal{C}(\ast, -)$ be the functor of points. As a covariant representable, $F$ commutes with limits, and since colimits of graphs are constructed pointwise, it also commutes with colimits. Therefore, $F$ has both a left and a right adjoint, given by forming the discrete graph with no edges and the complete graph with all possible edges.
#### An Aside: Essentially Algebraic Theories
As an aside, let’s discuss the notion of an essentially algebraic theory. A theory is algebraic when it involves equations between functions of finitely many variables, and therefore may be interpreted in any category with finite products. A theory is essentially algebraic when it involves equations between partial functions of finitely many variables (whose domains are carved out by equations). Therefore, an essentially algebraic theory may be interpreted in any category with finite limits. The theory of categories is an example of an essentially algebraic theory which is not algebraic; composition is only defined for some pairs of morphisms.
Definition An essentially algebraic theory $\mathcal{T}$ is a small finitely complete category.
A model of an essentially algebraic theory $\mathcal{T}$ is a left exact functor $M : \mathcal{T} \to \mathcal{C}$. Being left exact (or, shortly, lex) means commuting with finite limits.
A moral of this story is that a lot of what works for Lawvere theories over sets works for enriched essentially algebraic theories. In particular, we can take the free finite completion $\mathcal{L}_{\mathcal{C}}$ of a category $\mathcal{C}$, which by definition satisfies the following universal property: $\text{Lex}(\mathcal{L}_{\mathcal{C}}, \mathcal{B}) \simeq \mathcal{V}\text{-Cat}(\mathcal{C}, \mathcal{B})$ for f.c. $\mathcal{B}$. In particular, $\mathcal{L}_1 \cong \mathcal{V}_f^{\text{op}}$, so we see that $\text{Lex}(\mathcal{V}_f^{\text{op}}, \mathcal{V}) \simeq \mathcal{V}\text{-Cat}(1, \mathcal{B}) \cong \mathcal{B}.$ Therefore, we can see $\mathcal{V}_f^{\text{op}}$ as the “theory of an object”; its models are just objects.
An algebra of an essentially algebraic theory is a model in the base category. Kelly gives the following characterization of algebras of theories.
Corollary (6.11) Let $M : \mathcal{T} \to \mathcal{V}$ be a functor from an essentially algebraic theory $\mathcal{T}$. Then the following are equivalent:
• $M$ is left exact, and hence an algebra.
• $M$ is a filtered colimit in $\mathcal{V}\text{-Cat}(\mathcal{T}, \mathcal{V})$ of representables.
• $M$ is an element of the filtered-colimit completion of $\mathcal{T}^{\text{op}}$ (as the completion of $\mathcal{T}^{\text{op}} \hookrightarrow \mathcal{V}\text{-Cat}(\mathcal{T}, \mathcal{V})$ under filtered colimits).
The appearance of filtered colimits in this theorem suggests that our brief aside into essentially algebraic theories is anything but an aside.
#### That Wasn’t an Aside!
In this section, we will see that locally finitely presentable categories and essentially algebraic theories are closely intertwined. In fact, we will show that essentially algebraic theories present l.f.p. categories, and that this gives rise to a duality between a doctrine of theories and a doctrine of l.f.p. categories. I’m using the word “doctrine” here to mean just a “theory of theories”. Thus, categories whose objects are glued together out of finite shapes are precisely the categories of algebras for essentially algebraic theories.
We begin with the following theorem.
Lemma (7.2) Let $\mathcal{C}$ be l.f.p.. Then $\mathcal{C} \simeq \text{Lex}(\mathcal{C}_f^{\text{op}}, \mathcal{V})$.
Proof By $6.11$, this is equivalent to $\mathcal{C}$ being the filtered-colimit completion of $\mathcal{C}_f^{\text{op}}$, which holds by (the rest of) $7.2$.
Since this says that $\mathcal{C}$ is the category of algebras of $\mathcal{C}_f^{\text{op}}$, we can see $\mathcal{C}_f^{\text{op}}$ as “the theory of an object of $\mathcal{C}$”. As an example, this shows that the essentially algebraic theory of groups is equivalent to the opposite of the category of finitely presentable groups. Since a finitely presentable group is a coequalizer of finitely generated free groups, left exact functors out of this category are determined by their actions on the finitely generated free groups. We ran into the fact that the opposite of the category of finitely generated free groups determines the Lawvere theory of groups before; we are seeing this fact again, but in some more generality.
Now, let’s turn out attention to the restriction and extension of scalars between theories. Let $F : \mathcal{T} \to \mathcal{T}'$ be a lex functor between theories. Since composites of lex functors are lex, we get a functor $F^\ast : \text{Lex}(\mathcal{T}', \mathcal{V}) \to \text{Lex}(\mathcal{T}, \mathcal{V})$ by precomposition. This is the restriction of scalars functor; between theories of modules of rings it would be just that. We call functors $F^{\ast} : \text{Alg}_{\mathcal{T}'} \to \text{Alg}_{\mathcal{T}}$ induced in this way algebraic functors.
Corollary (9.5, 6.12) $F^\ast$ is finitary, and therefore $\text{colim}_F : \mathcal{V}\text{-Cat}(\mathcal{T}, \mathcal{V}) \to \mathcal{V}\text{-Cat}(\mathcal{T}', \mathcal{V})$ (by which I mean left Kan extension along $F$) descends to $F_{\ast} : \text{Lex}(\mathcal{T}, \mathcal{V}) \to \text{Lex}(\mathcal{T}', \mathcal{V})$ which is left adjoint to $F^\ast$.
Adjoint to the restriction of scalars, we get our usual extension of scalars functor. As a corollary, we get the following nice result.
Corollary The category of algebras of a theory $\mathcal{T}$ is a reflective subcategory of the category $\mathcal{V}^{\mathcal{T}}$ of copresheaves on it.
Proof $\text{Alg}_{\mathcal{T}} \hookrightarrow \mathcal{V}\text{-Cat}(\mathcal{T}, \mathcal{V}) \cong \text{Lex}(\mathcal{L}_{\mathcal{T}}, \mathcal{V})$ is algebraic, given by the reflection $\mathcal{L}_{\mathcal{T}} \to \mathcal{T}$ of $\mathcal{T} \hookrightarrow \mathcal{L}_{\mathcal{T}}$ sending formal limits to the actual limits in $\mathcal{T}$.
Now, a few sections ago we proved that reflective subcategories of presheaves were locally finitely presentable. Therefore, we can turn the previous corollary into the following theorem.
Theorem (9.8) For a finitary theory $\mathcal{T}$,
• the category $\text{Alg}_{\mathcal{T}}$ of $\mathcal{T}$-algebras is l.f.p.,
• $\text{Alg}_{\mathcal{T} f}$ is the essential image of $\mathcal{T}^{\text{op}} \hookrightarrow \text{Alg}_{\mathcal{T}}$, and
• $\text{Alg}_{\mathcal{T} f} \simeq \mathcal{T}^{\text{op}}$.
This theorem intertwines l.f.p. categories and essentially algebraic theories in the following precise way.
Corollary A category $\mathcal{C}$ is l.f.p. precisely when it is the category of algebras for a finitary theory (which is then equivalent to $\mathcal{C}_f^{\text{op}}$.)
In fact, we can say something more. Since categories of algebras are l.f.p., a functor $F^{\ast} : \text{Alg}_{\mathcal{T}'} \to \text{Alg}_{\mathcal{T}}$ has a left adjoint if it is finitary and continuous. If this occurs, then its adjoint $F_{\ast}$ preserves f.p. objects by the following calculation. \begin{aligned} \text{Alg}_{\mathcal{T}'}(F_{\ast}X, \colim -) &\cong \text{Alg}_{\mathcal{T}}(X, F^{\ast}\colim -) &\qquad\text{by adjointness,}\\ &\cong \text{Alg}_{\mathcal{T}}(X, \colim F^{\ast} -) &\qquad\text{by finitariness,}\\ &\cong \colim \text{Alg}_{\mathcal{T}}(X, F^{\ast}-) &\qquad\text{by f.p.}\\ &\cong \colim \text{Alg}_{\mathcal{T}'}(F_{\ast}X, -) &\qquad\text{by adjointness.} \end{aligned}
Therefore, $F_{\ast}$ restricts to a functor $\mathcal{T}^{\text{op}} \to \mathcal{T}'^{\text{op}}$, and as a left adjoint it is cocontinuous. Therefore, $F_{\ast} : \mathcal{T} \to \mathcal{T}'$ commutes with limits. We have shown that $F^{\ast}$ is algebraic precisely when it is finitary and continuous, and we can bundle all of this up into the following general theorem.
Theorem (9.14) Let
• $\mathcal{V}\text{-Th}$ be the $\mathcal{V}\text{-Cat}$ enriched category of theories and lex functors, and
• $\mathcal{V}\text{-lfp}$ be the $\mathcal{V}\text{-Cat}$ enriched category of l.f.p. $\mathcal{V}$-categories and finitary right adjoint functors.
Then
is a bi-equivalence.
Over sets, this bi-equivalence is called the Gabriel-Ulmer duality. Following Lawvere, we could call this contravariant bi-equivalence the “structure-semantics duality”. The structure map is the one which extracts the finite objects from an l.f.p. category, and the semantics map takes a theory to its category of algebras. In the language of this post, this bi-equivalence shows that categories whose objects can be glued together out of finite shapes and the suitably finite functors between them are presented by essentially algebraic theories. On this blog, David Corfield has discussed other variants of this sort of duality.
#### Concluding Thoughts
We’ve covered a lot of ground in this blog post, and we’ve still managed to skip out on big sections of Kelly’s paper. In particular, we missed section 5 relating finite presentability over a general base to finite presentability over sets, section 6 studying flatness of functors, section 8 concerning local finite presentability over sets in more detail, and section 10 which concerns sketches. Here are some of the things we did get to see.
• Weighted (co)limits as the enriched version of (co)limits.
• Objects being glued together out of figures of “tractable shape”.
• A definition of “finite” in the general categorical context.
• A characterization of the categories whose objects are glued together out of finite shapes.
• A general notion of “algebraic theory”, general enough to include the dual of the finite shapes.
• That the models of this theory (dual to the finite shapes) are precisely the objects of the category.
Posted at April 3, 2017 1:52 AM UTC
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## 3 Comments & 0 Trackbacks
### Re: Gluing Together Finite Shapes with Kelly
Great post!
I really like your series of motivating examples, building up from points to edges to simplices. In your example of topological spaces, could we restrict to some special subcategory $\mathcal{C}' \subset \mathcal{C}$ of topological spaces under which $\mathcal{C}'(\mathcal{G}, -)$ is conservative? Would such a $\mathcal{C}'$ be something like topological manifolds?
(I was initially confused by the notation $\mathcal{C}(\mathcal{G}, -)$, because the Yoneda embedding $c \mapsto \mathcal{C}(-,c)$ would then be written $\mathcal{C}(\mathcal{C},-)$ (with $\mathcal{C}$ on the ‘other side’ of $c$). But once I got used to it, it actually works very nicely with the rest of your post.)
The characterization of l.f.p. categories in (7.2) reminds me of Hilbert spaces: every Hilbert space $H$ has a maximal orthonormal subset $B$ such that $\text{Span}(B)$ (finite linear combinations) is dense in $H$. I think of $\mathcal{G}$ as $B$ and its closure under finite colimits $\mathcal{C}_f$ as $\text{Span}(B)$. Daniel, Alexander and you had some reasons for and against this analogy; maybe you guys could raise them here.
In David Corfield’s post that you linked, he mentions this short paper by Brian Day. The upshot is that
$\text{Lex}_{\mathcal{U}}(\mathcal{C}^{op}, \mathcal{U}) \cong C\text{-Comod}$
where $C$ is a coendomorphism coalgebra given by the coend of a ‘fiber functor’ $\mathcal{C} \to \mathcal{U}_f$. We have Gabriel-Ulmer on the left, and Tannaka on the right. What’s nice about the Tannaka point of view is that structures on $\mathcal{C}$ (e.g. monoidal, rigid) can be transferred to structures on $C$ (bialgebra, antipode resp.), and vice versa. The paper doesn’t seem to use much more than the stuff in your post (all the stuff leading up to Gabriel-Ulmer duality) and Simon’s (Day convolution), but following the details is proving to be a bit harder than I had anticipated.
(Does anyone know where the names ‘end’ and ‘coend’ come from? Are they somehow motivated by ‘endomorphism algebra’ and ‘coendomorphism coalgebra’?)
Posted by: Ze on April 5, 2017 7:28 PM | Permalink | Reply to this
### Re: Gluing Together Finite Shapes with Kelly
Thanks Ze!
Topoligical spaces are not l.f.p, and indeed they are not presentable for any cardinal. This MathOverflow post lists a citation; basically, there are spaces like the Sierpinski space (which classifies open sets) that are not presentable.
I’m used to thinking about the objects of an equipment (say, the equipment of enriched cats, functors, and bimodules) as \emph{bases} of vector spaces, with bimodules/proarrows being \emph{matrices} expressed in this bases. This is because if we consider the equipment of finite sets and spans, then we can move between this and “finite dimensional $\N$-modules with a basis” quite nicely; given a span $m : M \to I \times J$, we think of $m^{-1}(i, j)$ as $M_{ij}$, and composition of spans (pullback) gives matrix multiplication. But you are saying that we should think of these objects $\mathcal{C}$ as the spaces themselves, arrows $\mathcal{J} \to \mathcal{C}$ as “lists of vectors”, with the dense ones (such as $\mathcal{C}_f \hookrightarrow \mathcal{C}$ for l.f.p. $\Ca$) being the bases.
So if we think of the categorification of sum in a vector space as being a general colimit, then I would think of a small dense subcategory as a basis for a large category. But if we were to categorify sums to be the filtered colimits, then our hilbert spaces would be l.f.p. categories which are generated by a finitary dense subcategory. Or if we used sifted colimits, then we would get categories presented by algebraic theories as our “hilbert spaces”. So I think it really depends on the doctrine.
I do wonder about categorifying concepts from spectral analysis in this setting.
That paper was very interesting! I think I need to learn more about the motivating example (recovering coalgebras from their f.d. comodules) before really getting what was going on. I’m looking forward to it.
Posted by: David Jaz Myers on April 5, 2017 8:51 PM | Permalink | Reply to this
### Re: Gluing Together Finite Shapes with Kelly
“End” is a Germanic word for where or when something stops, which is sort-of what the more-Latiny “finish” is about (also recalled in “define”: drawing the borders around an idea; and things are reasonably finite when you can finish listing them) and which is poetically evoked as a threshold or, in Latin, limina (think of sub-liminal, “under the threshold”); which is where we put (or find) limits.
Posted by: Jesse C. McKeown on April 7, 2017 3:16 AM | Permalink | Reply to this
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2023-02-04 03:26:09
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https://socratic.org/questions/how-do-you-solve-7x-4-2-x-8
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# How do you solve 7x + 4 = 2(x - 8)?
May 25, 2016
Solve for $x$ via working below
#### Explanation:
$7 x + 4 = 2 \left(x - 8\right)$
$7 x + 4 = 2 x - 16$
$5 x = 20$
$x = 4$
1. Original equation
2. Expand
3. Group like terms
4. Divide all by 5
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2021-12-03 22:58:11
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http://mathhelpforum.com/trigonometry/209121-trig-equations-print.html
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trig equations
• Dec 5th 2012, 08:08 AM
enea54
trig equations
hi guys , i am a high school student , and i have a trigonometry test tomorrow . i would appreciate every help from you.
for now i can't solve these trigonometry equations.
they are mashing up my mind
1) sin2x + sinx * cosx = 1
2)1+cosx + cos(x/2) =0
3) 1 + cos(π +x ) + cos(π/2 + x/2) =0
• Dec 5th 2012, 08:42 AM
HallsofIvy
Re: trig equations
So you just want someone else to do them for you?
• Dec 5th 2012, 09:12 AM
enea54
Re: trig equations
Quote:
Originally Posted by HallsofIvy
So you just want someone else to do them for you?
I am doing a whole summary of the trigonometry chapter at the moment , so that i'll be more prepared for tha exam. These 3 equations were the only ones i couldn't solve , i asked help from my parents , my brother but they couldn't solve it either. I wasn't left with much choices so i went online and posted them here and i think this is the most adequate place to do it. No, i'm not asking someone else to do them for me . I just want a hint, a way of doing them so that i can learn something and put it to use later.
• Dec 5th 2012, 09:29 AM
topsquark
Re: trig equations
Quote:
Originally Posted by enea54
1) sin2x + sinx * cosx = 1
$sin^2(x) + sin(x)~cos(x) = 1$
$(sin^2(x) -1) + sin(x)~cos(x) = 0$
$-cos^2(x) + sin(x)~cos(x) = 0$
I'd multiply both sides by -1, but you can do it from this too. Factor out the common cos(x) and the solution follows easily from there.
-Dan
• Dec 5th 2012, 09:35 AM
enea54
Re: trig equations
thanks a lot , that was what i needed, thanks
• Dec 5th 2012, 09:47 AM
ebaines
Re: trig equations
For the second one you can replace 'x' with a new variable 'w' where x = 2w. This gives
$1 + \cos(2w) + \cos(w) = 0$
Now use the identity $cos(2w)= 2cos^2w-1$ and solve for cos(w), and from that determine values for w, then x.
• Dec 5th 2012, 09:49 AM
topsquark
Re: trig equations
Ya beat me ebaines! Almost word for word. (Nod)
-Dan
• Dec 5th 2012, 09:54 AM
Plato
Re: trig equations
Notice that $sin^2(x) + sin(x)~cos(x) = 1$
can be written as $\cos(x)[\sin(x)-\cos(x)]=0$
• Dec 5th 2012, 10:26 AM
Soroban
Re: trig equations
Hello, enea54!
Here's the last one . . .
Quote:
$(3)\;1 + \cos(x + \pi) + \cos\left(\tfrac{x}{2}+\tfrac{\pi}{2}\right) \:=\:0$
We have: . $1 + \cos(x + \pi) + \cos\left(\tfrac{x+\pi}{2}\right) \:=\:0$
Let $\theta \:=\:x + \pi$
We have: . $1 + \cos\theta + \cos\tfrac{\theta}{2} \:=\:0$
n . $1 + \left(2\cos^2\!\tfrac{\theta}{2} - 1\right) + \cos\tfrac{\theta}{2} \:=\:0$
. . . . . . . . . $2\cos^2\!\tfrac{\theta}{2} + \cos\tfrac{\theta}{2} \:=\:0$
n . . . . . $\cos\tfrac{\theta}{2}\left(2\cos\tfrac{\theta}{2} + 1\right) \:=\:0$
$\cos\tfrac{\theta}{2} \:=\:0 \quad\Rightarrow\quad \tfrac{\theta}{2} \:=\:\begin{Bmatrix}\frac{\pi}{2} \\ \\[-4mm] \frac{3\pi}{2} \end{Bmatrix} \quad\Rightarrow\quad \theta \:=\:\begin{Bmatrix} \pi \\ 3\pi \end{Bmatrix}$
. . . . $x + \pi \:=\:\begin{Bmatrix}\pi \\ 3\pi\end{Bmatrix} \quad\Rightarrow\quad \boxed{x \:=\:\begin{Bmatrix}0 \\ 2\pi\end{Bmatrix}}$
$2\cos\tfrac{\theta}{2} + 1 \:=\:0 \quad\Rightarrow\quad \cos\tfrac{\theta}{2} \:=\:-\tfrac{1}{2} \quad\Rightarrow\quad \tfrac{\theta}{2} \:=\:\begin{Bmatrix}\frac{2\pi}{3} \\ \\[-4mm] \frac{4\pi}{3} \end{Bmatrix} \quad\Rightarrow\quad \theta \:=\:\begin{Bmatrix}\frac{4\pi}{3} \\ \\[-4mm] \frac{8\pi}{3} \end{Bmatrix}$
. . . . $x + \pi \:=\:\begin{Bmatrix}\frac{4\pi}{3} \\ \\[-4mm] \frac{8\pi}{3} \end{Bmatrix} \quad\Rightarrow\quad \boxed{x \:=\:\begin{Bmatrix} \frac{\pi}{3} \\ \\[-4mm] \frac{5\pi}{3} \end{Bmatrix}}$
• Dec 5th 2012, 11:39 AM
enea54
Re: trig equations
(Bow)(Bow)thanks a lot guys, you are the best of the best , thanks again(Bow)(Bow)(Bow)
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2016-12-03 23:02:22
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https://tutorme.com/tutors/197089/interview/
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# Tutor profile: Elijah W.
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Elijah W.
Tutor for six years
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## Questions
### Subject:Writing
TutorMe
Question:
Write about your favorite aspect(s) of business (numbers, marketing, policy making, etc.) and create an essay that encompasses both informative information and entertaining qualities.
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Elijah W.
The Shrek Franchise From princesses to pirates, Disney seems to portray every role imaginable and that would include a green ogre and a foul mouthed donkey. Shrek was first released in May of 2001. Produced and distributed by DreamWorks, Shrek was made with a 60 million dollar budget. At the end of Shrek’s reign at the box office, it had raked in 484.4 million dollars worldwide but that was just the start of the franchise. Shrek 2 (2004), Shrek the Third (2007), and Shrek Forever After (2010) were produced along with spin-offs such as Puss and Boots (2011). All in all, the Shrek franchise has raked in 3.5 billion dollars at the box office alone. What made Shrek so popular has been its ability to cater to young and adult audiences. Shrek features childish and whimsical humor and also crude adult humor. Also, Shrek makes several cultural references to other Disney movies such as Snow White and the Seven Dwarves, Tinkerbell, The Three Little Pigs, Peter Pan, and many more. These references introduce the younger audience to more Disney franchises while providing nostalgia for older viewers. Shrek also features new age pop music such as “All Star” by Smash Mouth and oldies like “Try a Little Tenderness.” Marketing for Shrek started shortly before the movie’s release date in May, Burger King promoted Shrek with character shaped candies that would be included in kid meals. Baskin Robbins sold Shrek themed ice cream. Shortly after the major success of Shrek, Disney began to strike up licensing deals to sell Shrek products. Clothing manufacturers began to create Shrek themed clothing, toy companies such as Hasbro began to produce Shrek toys, and Conagra Foods, owners of Kid Cuisine, began to incorporate Shrek into foods. After each Shrek movie, the sells of Shrek merchandise skyrocketed which produced billions of dollars in revenue for Disney. The products were mainly targeted towards children and priced reasonably to ensure more sells. Shrek would be further promoted in publications such as Rolling Stone magazine and Entertainment Weekly and this fueled the Shrek mania. Many video games were produced following the release of Shrek such as Shrek: Hassle at The Castle and Shrek 2: The Game. In 2006, the total combined sales of Shrek console games reached 2.5 million units. All in all, the Shrek franchise has been and will continue to be successful.
### Subject:Anatomy
TutorMe
Question:
In your own words, list and briefly describe the importance of the 11 major organ systems in the human body.
Inactive
Elijah W.
### Subject:Algebra
TutorMe
Question:
Find (x) using both factoring and the quadratic formula: x$$^{2}$$ + 5x - 6
Inactive
Elijah W.
To begin, we will solve by factoring or determine if the expression can be factored in this format. If the original expression can not be factored, perfect square factorization may need to be utilized. Firstly, we must identify the numbers and signs of importance, which are the middle and last term, and find which numbers will multiply together that equal the sign and number of the last term of the expression. To do this, we have to find the factors of 6 which are: 1, 2, 3, and 6. From there, we must determine which factors of 6 can be multiplied together to equal negative 6. To answer that, we must know the ways in which the product of aforementioned factors can be multiplied together to equal negative six. So, following the rules of multiplication, one factor must be positive and one factor must be negative in order to have a negative six. So, the multiplication of factors could be: (-1(6)), (1,(-6)), (-2,(3)), or (2(3)). Next, we must determine (when added together) if any of those products will create the sum of positive five. Looking at the factors, we can see that (-1(6)) will equal negative six and add together to equal positive 5. So, we would then know the factors of the expression are (x+6) and (x-1). Finally, we set the factors of the expression equal to zero and solve for x. $$x-1=0$$ $$x+6=0$$ $$x=1$$ $$x=-6$$ Next, we must find x by using the quadratic formula. The quadratic formula is: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ The variable (a) is equal to the first term, the variable (b) is equal to the second term, and the variable (c) is equal to the third term. We would simply solve this by entering in the numbers. $$x=\frac{-5\pm\sqrt{5^2-4(1)(-6)}}{2(-6)}$$ We would then work out the formula for both the addition and subtraction of the square root. $$x=\frac{-5+\sqrt{5^2-4(1)(-6)}}{2(-6)}$$ and $$x=\frac{-5-\sqrt{5^2-4(1)(-6)}}{2(-6)}$$ Should the number under the square root found to be negative, we must reject using the quadratic formula for finding any real solutions for the expression since the square root of a negative number does not exist. We would only include those solutions when looking for imaginary solutions. After completing the mathematical steps to solve the above quadratic formulas, we find that $$x=-1$$ and $$x=6$$ are the real solutions to the expression.
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2020-08-08 09:10:13
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https://physics.stackexchange.com/questions/230622/understanding-of-dipole-moment-and-its-vector-property/232073
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# Understanding of dipole moment and its vector property
I have a trouble understanding the electric dipole moment.
The electric dipole moment formula is $${\bf p}= \int {\bf r}' \rho({\bf r}')d\tau '$$ I'm interested in the coordinate, the origin of which is changed into $\bf a$. $${\bf r}' = {\bar {\bf r}}' + {\bf a}$$ Now calculate dipole moment in the new coordinate \begin{align} {\bar {\bf p}} &= \int {\bar {\bf r}}' \rho ({\bar {\bf r}}') d\bar\tau' \\ &= \int ({\bf r}'-{\bf a}) \rho ({\bar {\bf r}'}) d\bar\tau' \end{align} In Griffiths, it says $$\rho ({\bar {\bf r}}') = \rho ({{\bf r}}')$$ so that yields ${\bar {\bf p}}= {\bf p} - Q {\bf a}$.
I don't understand how to verify $\rho ({\bar {\bf r}}') = \rho ({{\bf r}}')$.
• I think, to get $\bar{\bf{p}}$, you need to work with $\bar{\rho}(\bf{r})=\rho(\bf{r}-\bf{a})$. Then you substitute $\bar{\bf{r}}=\bf{r}-\bf{a}$ in order to do the integral. Just using your old $\rho$ and shifting the integration variable will give you $\bf{p}$ again rather than $\bar{\bf{p}}$. – Photon Jan 20 '16 at 11:26
I think you are confused with the new coordinates system.
You sould correct the above equations like this:
\begin{align} \bar{\mathbf p} &= \int \bar{\mathbf r} \bar\rho(\bar{\mathbf r})d\bar\tau \\ &=\int ({\mathbf r}-\mathbf a) \rho(\mathbf r)d\tau\\ &=\mathbf p - \mathbf a\int\rho(\mathbf r)d\tau = \mathbf p - Q\mathbf a \end{align}
In other words, you should consider a new density function $\bar\rho$.
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2021-05-09 08:35:04
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http://ionoale.it/ccwd/fluids-physics-problems-and-solutions.html
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6 Problem-Solving Strategies; 4. Reddy) SOLUTION MANUAL:: An Introduction to Thermal Physics by Schroeder, Daniel V SOLUTION MANUAL:: An Introduction to Thermodynamics and Statistical Mechanics (2nd Ed, Keith Stowe). We speak of the pressure ata point within a volume of fluid, meaning thatwe imagine the point to be surrounded by a small container and divide the total force exerted by the fluid against the walls of the container by the area of the container. Free practice questions for AP Physics 2 - Fluid Dynamics. 1 Fluid Statics Fluids, i. Sir some topics of 11th like mechanical properties of solid,fluid and. If you don't find your favourite problems here, please nominate! Mechanics (incl. Assume that a constant pressure P. 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According to the Thinkquest website, large industries including those that make chemicals and plastics dump a large amount of waste into the water. Fluid Mechanics Problems for Qualifying Exam (Fall 2014) 1. Chapter 2: Pressure and Fluid Statics Pressure For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion. their ability to apply these principles in the solution of problems. Shed the societal and cultural narratives holding you back and let free step-by-step Physics: Principles and Problems textbook solutions reorient your old paradigms. Chapter 12: Many fluids are flowing in this scene. Kinematic equations relate the variables of motion to one another. Engineers Institute of India is Top Ranked GATE Coaching Institute with Highest Results. 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Wanted : The speed of fluid at the smaller pipe (v 2) Solution : The equation of continuity : A 1 v 1 = A 2 v 2. Download it once and read it on your Kindle device, PC, phones or tablets. Morozov 1 Solutions for Problem 1 a) Q= 0 for adiabatic processes, and thus the rst law of thermodynamics becomes: U+ A= 0; (1) where Ais the work done by gas, and U is its internal energy. Chapter 13 - Fluid Mechanics solutions from HC Verma Solutions for Class 11 Physics Part 1. For example, the speed of sound in air at sea level at a temperature of 59°F (15°C) is about 760 miles per hour (340 meters per second). I had a lot of fun solving I E Irodov "Problems in General Physics" when I was preparing for the IIT-JEE. 2• The net work done on the fluid is therefore equal to. 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The buoyant force, which equals the weight of the fluid displaced, is thus greater than the weight of the object. If you have any doubt, please visit to DISCUSSION FORUM to ask your questions and reply to the questions already asked by your friends and other users. their ability to apply these principles in the solution of problems. Calculate the percentage of the sphere volume above the water surface, if the rope breaks and the sphere rises to the surface coming to rest. ρ = 1000 kg/m3 g = 9. DOWNLOAD ANY SOLUTION MANUAL FOR FREE > 186-Problems In General Physics ,2ed,by Irodov > 222- Fluid Mechanics With Engineering Applications,10ed,by E. Initially, a ball has a speed of 5. Answer & Solution : FLUIDS. Explain why the liquid moves upward, against gravity, into your mouth Solution: To draw a liquid up a straw, we expand our lungs This reduces the air pressure inside the mouth to less than […]. This is called the buoyant force. 5 cm is lowered below the surface of water reservoir by a rope of tension 1. The Most Beautiful Equation in Math. For more analyses of compressible flow through nozzles and diffusers, click the topic “Gas Dynamics” please: Gas Dynamics. Cimbala McGraw-Hill, 2013 CHAPTER 3 PRESSURE AND FLUID STATICS PROPRIETARY AND CONFIDENTIAL This Manual is the proprietary property of The McGraw-Hill Companies, Inc. All the solutions of Pressure in Fluids and Atmospheric Pressure - Physics explained in detail by experts to help students prepare for their ICSE exams. NOW is the time to make today the first day of the rest of your life. Fins are especially important for situations where the convecting medium is air or some gas (with lower h) and the surface area of the object that needs to lose (or gain) heat is (relatively) small. The book contains about 200 problems worked out in detail. W ater is pumped with a 120 kP a compressor entering the lower pipe (1) and flows upward at. Givens/ conversions: diameter ≡ d = 3. It also helps support the weight of this swimmer. Dželalija, Physics The International System of Units To report the result of a measurement of a certain physical quantity, a unit for the quantity must be defined. Answer & Solution : FLUIDS. When traveling in any moving vehicle, you rarely maintain the same velocity throughout an entire trip. Specific gravity = ρ/ρ water. NCERT Exemplar Class 11 Physics is very important resource for students preparing for XI Board Examination. Mastering Physics Solutions Chapter 15 Fluids Mastering Physics Solutions Chapter 15 Fluids Q. 1216 rev/min 0. In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. Figures are not included. You may have a calculator. 0002 meter radius fixed horizontally at its bottom. We drink them, breathe them, swim in them; they circulate. Be very clear which 3 you want graded (see below). In an isolated system, given heat is always equal to taken heat or heat change in the system is equal to zero. Density is the property of a body which measures how its mass is distributed over its volume. At SeeTheSolutions. In contrast, the viscosity of non-Newtonian fluids is not constant, but rather varies greatly depending on the force applied. In each case consider the top of the blue “box” to be the surface of the water in which the plate is submerged. Weight is a downwards force due to gravity. 1 Calculate the pressure created by a 30 m depth of water given the density of water is 1000 kg/m 3 and gravity 9. You pull out your old geology text and look up gold in the mineral table, and read that its density is 19. The reynolds transport theorem the, extensibility of the hindrance resulting from a fluid oldroyd. CAPA solutions from previous sets can be found by logging onto CAPA and selecting “View Previous Set” instead of “Try Current Set” Any special exam requests – should contact Daniel. Normal stresses are referred to as pressure p. Understanding Multivariable Calculus: Problems, Solutions, and Tips, taught by award-winning Professor Bruce H. Physics Van The Illinois Physics Van is a traveling science show for kids! By performing and explaining exciting physics demonstrations, we show our audiences that science is fun and worthwhile for people who wonder about why the world acts the way it does. Physics problems with solutions and tutorials with full explanations are included. In your new program, you’ll find: • Example problems that build reasoning and problem-solving skills. Brown,Instructor Duke University PhysicsDepartment A few of the problems have rather detailed solutions (due to Prof. The book is ideal as a supplement or exam review for undergraduate and graduate courses in fluid dynamics, continuum mechanics, turbulence, ocean and atmospheric sciences, and related areas. This course is the next step for students and professionals to expand their knowledge for work or study in. Chapter 2: Pressure and Fluid Statics Pressure For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion. PROBLEM STATEMENT: An air-hockey puck has a mass of 50 g and is 9 cm in diameter. Physics problems: fluids and elasticity ; Problem 6. The caps have negligible mass, are watertight, and can freely slide up and down the tube. Calculate the buoyant force and weight. Serway Chapter 9. This page serves as a source for worked problems and materials from past Physics 1250 courses at Ohio State. , Weak solutions of an initial boundary value problem for an incompressible viscous fluids (1987) SIAM J. I am happy to answer your email. Fluids can flow steadily, or be turbulent. 1 Calculate the pressure created by a 30 m depth of water given the density of water is 1000 kg/m 3 and gravity 9. Physics JEE Main fluids Paper Solutions. 0×10 4 V/m using equation 28. If the equilibrium configuration of the tube is as shown in Figure (14. Update the question so it's on-topic for Physics Stack Exchange. Let's review these basic definitions and concepts working some problems along the way. The solutions here are original and so they are freely available information. Navier-Stokes (NS) equations are the mass, momentum and energy conservation expressions for Newtonian-fluids, i. contents chapter previous next prep find. A diffuser is a device which slows down fluid. 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We have invited the following speakers to the Laser Analytics Group: Christophe Leterrier 3 December 2019 Christophe Leterrier has been working on the organization of the axon since his PhD, where he studied the axonal targeting of the CB1 cannabinoid receptor. Category:Fluid mechanics | Physics: Problems and Solutions | Fandom FANDOM Physics: Problems and Solutions is a FANDOM Lifestyle Community. Download NEET AIPMT Physics MCQ on Mechanical Properties of Solids and Fluids Practice Sample Papers / Problems with Solution 2017-2018. Estimate the density of sea water. Physics Assignment Help With Stability of Floating body. Assume the hinge at A is frictionless. A method of obtaining solutions to nonlinear flow problems by integrating the rate of change of the solution with respect to a suitable parameter is described. Based on the author’s many years of lectures and tutorials at Novosibirsk State University and the University of Manchester, Physics of Continuous Media: Problems and Solutions in Electromagnetism, Fluid Mechanics and MHD, Second Edition takes a problems-based approach to teaching continuous media. Predict the weight of a completely or partially submerged object of known mass and volume. What Is a Fluid? • State the common phases of matter. Chapter 4: Friction. chapter 02: fluid statics. Qualifying Exam Solutions: Thermal Physics and Statistical Mechanics Alexandre V. Permalink: https://lib. The AP Physics 1 and 2 Exams include a new multiple-select, multiple-choice question type with two correct answers, both of which students must identify to earn credit. Topics to be covered:. A typical 1 hour exam would contain 3-4 problems similar to those in the practice exams. Find the density of the paperweight. Hydrostatics 5 1/9 The vehicle is filled with oil. Parallel conducting tracks separated by 3. Mechanics is a broad area of physics, and these problems are taken from a broad range of experiences that arise naturally in day-to-day life. Introduction to Fluid Dynamics* T. This chapter talks of hydrostatic i. Physics Fluid Flow (1 of 7) Bernoulli's Equation. This book will offer problems both for applied situations and those of a theoretical nature. All rights reserved. Fluids Practice Problems PSI AP Physics B Name_____ Multiple Choice Questions 1. 00 cm, determine. Chapter 4: Friction. To understand Bernoulli's Equation and its application. The world has become obsessed with the Western notions of progress, development, and globalization, the latter a form of human and economic homogenization. Cem KAHRUMAN. You may have a calculator. During this process, velocity of fluid increases with decreasing pressure. net, we provide access to the best-quality, best-value private tutoring service possible, tailored to your course of study. Read Book Physics Fluids Problems And Solutions Baisonore Fluid dynamics - problems and solutions - Basic Physics Physics problems: fluids and elasticity. I am actually philosophically opposed to. Giancoli Answers is not affiliated with the textbook publisher. Mechanics is a broad area of physics, and these problems are taken from a broad range of experiences that arise naturally in day-to-day life. NCERT Exemplar Class 11 Physics is very important resource for students preparing for XI Board Examination. Computational fluid dynamics (CFD) is a tool with amazing flexibility, accuracy and breadth of application. Gases and liquids are fluids, although sometimes the dividing line between liquids and solids is not always clear. In each case consider the top of the blue “box” to be the surface of the water in which the plate is submerged. NAME _____ DATE _____ CLASS _____ 88 Holt Physics Problem Workbook Copyright © by Holt, Rinehart and Winston. 3) Problems in the aerodynamics of radiant gases. Consider a steady, incompressible boundary layer with thickness, δ(x), that de-velops on a flat plate with leading edge at x = 0. Insert this value into the equation for specific gravity to find. Pressure and buoyancy. SUMMARY: The basic equations of fluid mechanics are stated, with enough derivation to make them plausible but with-out rigour. Without making any assumptions about the form of the body force f, the final equation for an incompressible Newtonian fluid would be ρDv Dt = − ∇p + μ∇2v + f. A hypodermic syringe filled with normal saline solution has an inner barrel diameter of 10. 505) Description. Fluid Dynamics via Examples and Solutions provides a substantial set of example problems and detailed model solutions covering various phenomena and effects in fluids. time and the acceleration vs. January 29: Chapter One, Problems 6, 17, 34, 49, 53 and 85. In the theory of pattern formation in continuous systems (e. 12 FLUID DYNAMICS AND ITS BIOLOGICAL AND MEDICAL APPLICATIONS Figure 12. The physics-controlled mesh functionality in the CFD Module accounts for boundary conditions in fluid flow problems in order to compute accurate solutions. • • Write and apply Bernoulli's equation s equation for the general case and apply for (a) a fluid at rest, (b) a fluid at constant pressure, and (c) flow through a horizontal pipe. Includes full solutions and score reporting. 4) Problems in the aerodynamics of rarefied gases. is acting on a surface with area A. Physics Fluid Flow (1 of 7) Bernoulli's Equation. Permalink: https://lib. Fluids: Hydrostatics Hydrostatics #2 Hydrodynamics Mixed Fluids: Gravitation: Gravitational Fields and Forces Satellites: Heat: Heat Transfer Methods Phase Diagrams Thermal Expansion: Ideal Gases and Thermodynamics: Ideal Gas Law Heat Cycle Confined Gases: Kinematics: Uniformly Accelerated Motion (non-freefall). Reactor Engineering and Safety (Formerly. Fairman - August 1996 Hydraulic systems use a incompressible fluid, such as oil or water, to transmit forces from one location to another within the fluid. Each problem and its solution is followed by one or more exercises on the same topic, the exercises corresponding to problems that have been solved in the text are assigned the same number. Two objects m1 and m2 each with a mass of 6 kg and 9 kg separated by a distance of 5 Parabolic motion, work and kinetic energy, linear momentum, linear and angular motion – problems and solutions. Get this from a library! Physics of continuous media : problems and solutions in electromagnetism, fluid mechanics and MHD. Physics problems: fluids and elasticity ; Problem 6. Most test questions can be answered on the basis of a mastery of the first three years of under-graduate physics. This dif-. The aim of the test is to determine the extent of the examinees' grasp of fundamental principles and their ability to apply these principles in the solution of problems. Explore all types of flow, such as visible, implied, turbulent, laminar, and so on, present in this scene. ANSYS CFD goes beyond qualitative. Melissa Axelsson • 4 years, 2 months ago • login to reply We do not recommend a book for this course. Physics problems with solutions and tutorials with full explanations are included. Sir some topics of 11th like mechanical properties of solid,fluid and. SOLUTION MANUAL:: An Introduction to Signals and Systems by John Stuller SOLUTION MANUAL:: An Introduction to the Finite Element Method (3rd Ed. 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Fluids at rest questions If you're seeing this message, it means we're having trouble loading external resources on our website. How estimate mass? 2. The physics-controlled mesh functionality in the CFD Module accounts for boundary conditions in fluid flow problems in order to compute accurate solutions. Fluid Dynamics Multiple Choice Questions & Answers (MCQs), fluid dynamics quiz answers pdf 1 to learn online physics certificate course. The Archimedes' Principle is introduced and demonstrated through a number of problems. The caps have negligible mass, are watertight, and can freely slide up and down the tube. () The tube is cylindrical, and the right arm has twice the radius of the left arm. P1A1s1 - P. Differential Pressure = (1. A nozzle is a device which accelerates fluid. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. 4 Problem 9. Fluid dynamics - problems and solutions. 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Subtopic : (a) Pressure (b) Streamline flow (c) Bernoulli's principle (d) Viscosity (e) Reynolds number (f) Surface tension (g) Summary (h) Points to ponder (i) Exercises (j) Additional exercises (k) Appendix (l) Elastic behaviour of solids (m) Stress and. Some of these cookies are essential to the operation of the site, while others help to improve your experience by providing insights into how the site is being used. NCERT Solutions Class 11 Physics Chapter 10 Mechanical Properties of Fluids – Here are all the NCERT solutions for Class 11 Physics Chapter 10. Feel free to come to the session, or work the problems on your own. Michel van Biezen 808,549 views. Intermediate Physics for Medicine and Biology http://www. The AP Physics 1 and 2 Exams include a new multiple-select, multiple-choice question type with two correct answers, both of which students must identify to earn credit. , and Lightfoot, E. 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This text can therefore help you gain a complete understanding of fluid mechanics and draw on your own practical experiences to tackle. Physics 05-01 Fluids and Density. Calculations of Heat Transfer. You should then try to work out the problem on your own. 0002 meter radius fixed horizontally. This chapter addresses these problems, possible solutions, and, in some cases, preventive measures. Here we have provided NCERT Exemplar Problems Solutions along with NCERT Exemplar Problems Class 11. Click on date to download homework solutions. contents: fluid mechanics chapter 01: fluid properties. The height of the fluid in the manometer is measured relative to the spinal column, and the manometer is open to the atmosphere. So consequently the physics of liquids and gases is not just another topic but is basic to life itselfanother topic but is basic to life itself. College"Physics" Student"Solutions"Manual" Chapter"12" " 95" " ( ), (1000) 2000 or 2 / 2 2 πη ρ πη ρ η ρ π Q r r Q r r Q = ≤ ≥ " so "that em in u d am ter. 185 November 29, 1999, revised October 31, 2001, November 1, 2002, and November 5, 2003 This outlines the methodology for solving fluid dynamics problems as presented in this class, from start to finish. You pull out your old geology text and look up gold in the mineral table, and read that its density is 19. Givens/ conversions: diameter ≡ d = 3. Units , Dimensions and Errors Practical Physics Kinematics Laws of Motion Work ,Energy And Power Centre of Mass, Momentum & Impulse Rotational Motion Gravitation Properties of Matter Heat And Thermodynamics Simple Harmonic Motion Waves Electrostatics Current Electricity. 0002 meter radius fixed horizontally at its bottom. 1 Development of Force Concept; 4. Non-Newtonian Fluids. 0 x 105 Pa is cooled so that its volume decreases from 1. 6 g/cm³), so the object will float in mercury. In this article, we are providing HC Verma Solutions PDF which can be downloaded for free. To keep our discussion under control, we will restrict our discussion to air resistance forces proportional to v and v 2. • Define the rate of flow rate of flow for a fluid and solve problems using velocity and cross- - section. edu is a platform for academics to share research papers. College"Physics" Student"Solutions"Manual" Chapter"11" " 81" " Solution" Using"the"equation" A F P = , we"can"solve"for"the"pressure:" ( ) 7. Problem: An incompressible fluid is flowing through a horizontal tube which, at some point has a constriction such that the area of the tube becomes much smaller. CAPA solutions from previous sets can be found by logging onto CAPA and selecting “View Previous Set” instead of “Try Current Set” Any special exam requests – should contact Daniel. Review solution method of first order ordinary differential equations Applications in fluid dynamics - Design of containers and funnels Applications in heat conduction analysis - Design of heat spreaders in microelectronics Applications in combined heat conduction and convection - Design of heating and cooling chambers. com,1999:blog. That means, velocity of fluid decreases with increasing pressure. A typical 1 hour exam would contain 3-4 problems similar to those in the practice exams. Branches of Physics include Classical Physics, Mechanics, Astrodynamics, Nuclear Physics, Electromagnetism, Fluid Mechanics, Geophysics, Thermodynamics and more. In each case consider the top of the blue “box” to be the surface of the water in which the plate is submerged. 5 m up the incline, the ball has a speed of 1. • Solve problems due to momentum changes. The solutions to the problems are initially hidden, and can be shown in gray boxes or hidden again by clicking "Show/hide solution. onset of convection in a fluid heated from below) one finds that even distant boundaries can have surprisingly important effects. Pascal's Principle and Hydraulics SUBJECT: Physics TOPIC: Hydraulics DESCRIPTION: A set of mathematics problems dealing with hydraulics. Fluid Mechanics FE Review These slides contain some notes, thoughts about what to study, and some practice problems. 1 meter is filled with water to a height of 0. Explain why the liquid moves upward, against gravity, into your mouth Solution: To draw a liquid up a straw, we expand our lungs This reduces the air pressure inside the mouth to less than […]. I had a lot of fun solving I E Irodov "Problems in General Physics" when I was preparing for the IIT-JEE. This equation provides very useful information about the. • Fluid pressure is independent of the shape or area of its container. This is one of over 2,200 courses on OCW. Kazhikov, A. Introduction to Vectors. A boundary layer mesh is automatically generated in order to resolve the gradients in velocity that usually arise at the surfaces where wall conditions are applied. 7 FLUID FORCES When you have completed this tutorial you should be able to • Solve forces due to pressure difference. Lightbulbs and heat output As mentioned above, the energy supplied to a lightbulb from an electrical power source is converted into light and heat. When we talk about density it's usually mass density we're referring to. At any point within a static fluid, the pressure on all sides must be equal—otherwise, the fluid at that point would react to a net force and accelerate. Nuclear Physics. Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. Fluid Mechanics is intended to provide a comprehensive guide to a full understanding of the theory and many applications of fluid mechanics. Principles of Fluid Mechanics Laminar flow - for N Re < 2,000 Turbulent flow - for N Re > 4,000 Example 4-1 : A ventilation shaft of diameter 5 m passes an airflow of 200 m 3 /sec at a mean density of 1. The weight of. In a previous article, we have shared HC Verma Concept of Physics PDF for both volume 1 and volume 2. The material is elaborated in special applications sections by more than 200 exercises and separately listed solutions. A vertical tube of radius 1 cm, open at the top to the atmosphere, contains 2 cm of oil (‰oil =0:82‰water)°oatingon3 cmofwater. Homework, Quizzes, and Examinations. Physics is one of the oldest academic disciplines of science. Pressure and buoyancy. Concepts of Physics Part 1, Numerical Problems with their solutions, Short Answer Solutions for Chapter 13 - Fluid Mechanics from the latest edition of HC Verma Book. 4 CHAPTER 14. Biomedical engineering deals with the application of ones knowledge of engineering to medicine and biology to help with healthcare. This booklet includes the solutions relevant to the EXERCISES & PROBLEMS sections of the 6th edition of Fundamentals of Physics, by Halliday, Resnick, and Walker. The concepts of displacement, distance, velocity, speed, acceleration are thoroughly discussed. Part 1 Problem 1. edu is a platform for academics to share research papers. Normal stresses are referred to as pressure p. PEDLEY Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver St. A hollow plastic sphere of radius 3. Mixing a trace of an additional element into a material used in electronics could. Fluid dynamics and Bernoulli's equation. About HC Verma Solutions PDF. Chapter 3: Motion In Two Dimensions. W ater is pumped with a 120 kP a compressor entering the lower pipe (1) and flows upward at. Chapter 1: Vectors. However, most problems do not have included solutions and never will have. The AP Physics 1 and 2 Exams include a new multiple-select, multiple-choice question type with two correct answers, both of which students must identify to earn credit. Part 1 Problem 1. Sometimes you want the light and no heat (as in a torch); other times you want heat and light (in a bathroom heat lamp or the hot-food display at a take-away shop). Reckless Racing HD ($0. 036m3/s Calculate Ns at point A 11. Practice Problems: Capacitors Solutions. Morozov 1 Solutions for Problem 1 a) Q= 0 for adiabatic processes, and thus the rst law of thermodynamics becomes: U+ A= 0; (1) where Ais the work done by gas, and U is its internal energy. Physics Chap—1 0-HW SOLUTIONS Page 1 SOLUTIONS to Homework Questions and Problems Chapter 10 Fluids Homework. 9 Moving fluids. 5 cm is lowered below the surface of water reservoir by a rope of tension 1. 0008 m 3 Same as question 4 Based on continuity, less area means more speed and based on Bernoulli, more speed means less pressure. 505) Description. Free Response Problems 1. All of the fundamental problems have complete solutions and answers. One Hour Per Lesson, 24 Lessons Per Course. Acceleration due to gravity (g) = 10 m/s 2. Another friction problem. The important principles in step 2 will help recognize this.
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2020-05-27 15:38:55
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https://math.stackexchange.com/questions/3503918/evaluate-int-02-pix-sin6x-cos5x-dx
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# Evaluate $\int_0^{2\pi}x\sin^6x.\cos^5x.dx$
Evaluate $$\int_0^{2\pi}x\sin^6x.\cos^5x.dx$$
Set $$t=\sin x\implies dt=\cos x.dx$$ $$I=\int_0^{2\pi}(2\pi-x)\sin^6x.\cos^4x.\cos x.dx\\ 2I=2\pi\int_0^{2\pi}\sin^6x.(1-\sin^2x)^2.\cos x.dx\\ I=\pi\int_0^{2\pi}\sin^6x.(1-2\sin^2x+\sin^4x).\cos x.dx=\pi\int_0^{\color{red}{?}}[t^6-2t^8+t^{10}]dt$$ The solution given in my reference is $$\dfrac{32\pi}{693}$$ but if I set $$x:0\to2\pi\implies t:0\to 0$$, integral becomes zero, so what exactly should be the uper limit of the definite integral ?
Thanx @José Carlos Santos, $$I=\pi\bigg[\int_0^{\pi/2}f(x)dx+\int_{\pi/2}^{3\pi/2}f(x)dx+\int_{\pi/2}^{2\pi}f(x)dx\bigg]\\ =\pi\Big[\int_0^{1}\big[\frac{t^7}{7}-\frac{2t^9}{9}+\frac{t^{11}}{11}\big]dt+\int_{1}^{-1}\big[\frac{t^7}{7}-\frac{2t^9}{9}+\frac{t^{11}}{11}\big]dt+\int_{-1}^{0}\big[\frac{t^7}{7}-\frac{2t^9}{9}+\frac{t^{11}}{11}\big]dt\Big]=0$$ Even If I split the limits, seems like it still gives me $$0$$, how do I deal with it properly ?
• Let $I(t)=\int_0^t f(x) dx$ with $f(x)$ to be the integrand. Then $I(t)-\frac{32 \pi}{693}$ is negative on $[0,2\pi]$! – Math-fun Jan 10 at 11:27
• wolframalpha.com/input/… – lab bhattacharjee Jan 10 at 11:34
• – lab bhattacharjee Jan 10 at 11:35
• @labbhattacharjee The function that is being integrated here is $x\sin^6(x)\cos^5(x)$, not $\sin^6(x)\cos^5(x)$. – José Carlos Santos Jan 10 at 11:42
• @Santos, we can remove $x$ using the link supplied above – lab bhattacharjee Jan 10 at 11:47
Note that your substitution is not working because the substitution rule works only for one-one functions, whereas $$\sin x$$ is not a one-one function in $$(0,2\pi)$$. Instead, use the fact that $$\int_0^{2a}f(x)dx=\int_0^af(x)+f(2a-x)dx$$ $$\implies I=\int_0^{2\pi}x\sin^6x\cos^5xdx$$ $$=\int_0^\pi(x+2\pi-x)\sin^6x\cos^5xdx$$ $$=2\pi\int_0^\pi\sin^6x\cos^5xdx$$ Again apply the same property and get $$I=2\pi\int_0^{\pi/2}\sin^6x\cos^5x-\sin^6x\cos^5xdx$$ $$=0$$
• what difference it makes, still if I take $t=\sin x$ I will still get $0$,right ? – ss1729 Jan 10 at 12:12
• @ss1729, You cannot substitute $t = \sin x$ in the original integral because $\sin x$ is not a one-one function. – Martund Jan 10 at 12:18
• breaking the domain and integrating separately 'd work ? – ss1729 Jan 10 at 12:22
• @ss1729, Yeah, that would work. But the method I am suggesting in my answer is almost mental arithmetic (if you practice this rule), so that would be a lot easier, because you'll need to do integration by parts in each of the term separately. – Martund Jan 10 at 12:26
• actually I didnt get what you are suggesting ?. $0$ is not the solution right ? – ss1729 Jan 10 at 12:27
Hint
Applying the rule
$$\int_a^bf(x)dx = \int_a^bf(a+b-x)dx$$
$$I = \int_0^{2\pi}x\sin^6x\cos^5xdx = \int_0^{2\pi}(2\pi-x)\sin^6(2\pi-x)\cos^5(2\pi-x)dx$$
Using the fact that $$\sin(2\pi-x) = -\sin x$$ and $$\cos(2\pi-x) = \cos x$$
$$I = \pi\int_0^{2\pi}\sin^6x \cos^5x dx$$
• That is not the function that you were supposed to integrate here. – José Carlos Santos Jan 10 at 11:43
• This is the question, no? – Dhanvi Sreenivasan Jan 10 at 11:47
• No. The question is about $\int_0^{2\pi}x\sin^6(x)\cos^5(x)\,\mathrm dx$. – José Carlos Santos Jan 10 at 11:50
• Yeah, so I simplified it to remove the x term – Dhanvi Sreenivasan Jan 10 at 11:51
• this is something I forgot to put when I typed the question here. The real problem I am confused is about the limit. ie, after your last step. – ss1729 Jan 10 at 12:09
In order to compute$$\int_0^{\pi/2}x\sin^x(x)\cos^5(x)\,\mathrm dx\tag1$$using the substitution that you mentioned, then, since, when $$x$$ goes from $$0$$ to $$\frac\pi2$$, $$\sin(x)$$ goes from $$0$$ to $$1$$, you get$$\int_0^1\arcsin(t)(t^6-2t^8+t^{10})\,\mathrm dt=\frac{4(-5\,156+3\,465\pi)}{2\,401\,245}.$$So, $$(1)=\frac{4(-5\,156+3\,465\pi)}{2\,401\,245}$$.
Now, using the same idea in order to compute$$\int_{\pi/2}^{3\pi/2}x\sin^x(x)\cos^5(x)\,\mathrm dx,$$what you get is$$\int_1^{-1}\bigl(\pi-\arcsin(t)\bigr)(t^6-2t^8+t^{10})\,\mathrm dt=-\frac{16\pi}{693}.$$Can you take it from here?
• Please check I have modified OP to include attempt to solve by slitting the limits as you showed, but I think it is still giving $0$ ? – ss1729 Jan 10 at 12:56
• Yes, the answer is $0$. – José Carlos Santos Jan 10 at 12:58
• actually my reference gives the solution $\dfrac{32\pi}{693}$, so it must be wrong ? – ss1729 Jan 10 at 12:59
• Yes, it is wrong. – José Carlos Santos Jan 10 at 13:07
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2020-09-26 17:13:09
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https://blog.supplysideliberal.com/post/87081177244/three-revolutions
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# Three Revolutions
Today it has been two years since my first post: “What is a Supply-Side Liberal?" My first anniversary post, ”A Year in the Life of a Supply-Side Liberal,“ provides an introduction to this blog and tells of the exhilarating experience of my first year of blogging. Today, I wanted to talk about some of the pictures in my mind of possible futures that keep me going.
As for the blog itself, one of my standards of excellence for an independent economics blog is Tyler Cowen and Alex Tabarrok's Marginal Revolution blog. Day after day, Tyler and Alex give people reason to come back and learn more. Their tagline to explain their title "Marginal Revolution” is “Small Steps Toward a Much Better World.” Although there would have to be many small steps along the way to each of these, I tend to think of the revolutions I want to see happen in a more discrete way. Let me talk about three revolutions I hope to see, in order of how fast I think they could happen.
1. The Electronic Money Revolution. The world’s attempts at economic stabilization since 2008 have left much to be desired. The main reason has been the partial crippling of monetary policy due to the difficulties of making interest rates negative when paper currency guarantees to all an interest rate of at last zero (minus storage costs). This difficulty is called the zero lower bound. Since I published “How Subordinating Paper Currency to Electronic Money Can End Recessions and End Inflation” in November 2012, I have been writing and traveling the world speaking to spread the word that the zero lower bound is a policy choice, not a law of nature. I argue that it is a bad policy choice. The benefits of economic stabilization without needing to have long-run inflation far outweigh the inconveniences of dealing with negative interest rates and an exchange rate between paper currency and electronic money that is sometimes away from 1-for-1.
I have collected links to everything I have written about eliminating the zero lower bound in my post “How and Why to Eliminate the Zero Lower Bound: A Reader’s Guide.” On why to eliminate the zero lower bound, let me recommend
On how to eliminate the zero lower bound, let me recommend this presentation that I have given in various versions at the Bank of England, the Bank of Japan, Japan’s Ministry of Finance, Danmarks Nationalbank, the Banque de France, the Federal Reserve Board, and the US Treasury:
I have seminars scheduled in July at the ECB, the Bundesbank, the Banca D'Italia and the Swiss National Bank. In addition to the posts above, this presentation relies on what I say in these fairly technical posts:
I believe a transition to a monetary system based on electronic money that avoids creating a zero lower bound is almost inevitable. The electronic money revolution will happen. The question is when. The more people there are who understand the principles and reasoning involved, the quicker that day will come. Some countries may lag behind, but some country will lead the way.
2. The Supply-Side Liberal Revolution. Posts about policies to foster economic growth, while taking care of the poor, are the heart of this blog, as you can see from looking down my list of most popular columns and posts. For economic growth, beyond the basics I wrote about in “The Government and the Mob,” the key policies are those I wrote about with Noah Smith in “One of the Biggest Threats to America’s Future Has the Easiest Fix” (followed up by “Capital Budgeting: The Powerpoint File”), the kind of individual effort Noah and I recommended in “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t” and blocking attempts to squash the kind of disruptive innovation that Clay Christensen talks about. (See my post on Monday: “Saint Clay.”) It also doesn’t hurt to understand the key role that knowledge plays in economic growth, something I talk about in my post “Two Types of Knowledge: Human Capital and Information."
For taking care of the poor, many of the key issues are political. First, as I argue in "Inequality Aversion Utility Functions: Would $1000 Mean More to a Poorer Family than$4000 to One Twice as Rich?” it is crucial not to be distracted by a fascination with the division of wealth and income between the middle class and the rich from the primary task of taking care of the poor. Besides a safety net focused on helping the poor rather than unsustainably trying to give large amounts of money to the middle class, key policies to help the poor are 1. more open immigration, 2. job freedom, and 3. school reform:
One key to sustainably getting resources for helping the poor is to do it in a way that causes the fewest economic distortions. In addition to focusing on the right kinds of taxes, to the extent that there must be taxes, I believe that there is great potential in the kind of public contribution system that I talk about in the links in my post “The Red Banker on Supply-Side Liberalism." People often hate taxes, so they try to avoid them. Those efforts at tax avoidance are a social waste. So it makes sense to get many of the resources for helping the poor from public contributions that people won’t want to avoid as much as taxes, and that allow those contributing to be creative in making the world a better place. The creativity and flexibility fostered by a public contribution system are also bound to lead to technological progress in ways to help the poor.
3. The Heroic Revolution. By making the right choices, anyone can be a hero in the sense of making the world of the future a significantly better place than it otherwise would have been. For many, the objective of making the world a better place takes on a religious flavor, as it does for me (though for me in a resolutely non-supernaturalist way). See for example my sermons
and see Noah’s wonderful religion guest post
But regardless of one’s views on religion, hope and faith that one can make things better is the key to actually making things better. This is a principle I write about in
There are reasons to have hope that one can make the world a better place. The most basic is the argument that all it takes is to durably convince the younger generation that there is a better way:
But there also the power of gratitude, as I write of in
However, in the end, our success at making the world a better place will depend crucially on our ability to see clearly what is better and what is worse. Whatever its flaws, and despite all the ways it goes astray, religion has something to say about this. But so does the economics of happiness–in particular the work drawing on the intuitions of many people about what "better” means that I write of in
Summing Up. I believe in the potential of the blogosphere to change the world. I hope my view of the good of our noble species and the rest of the universe is clear enough that I am pushing in the right direction rather than in the wrong direction. My gut-level reaction to partisan politics in the United States is that enormous time and effort is wasted by Republicans and Democrats as they cancel each other out in opposition to one another. A key source of this wasted effort is that people are much too quick to assume they know the right direction to go. Many partisans assume they know the right direction to go, despite failing to undertake thoroughgoing discussions according to the principles of open, heated, but respectful discussion laid out by John Stuart Mill. (Those principles are familiar to those of you who follow my every-other-week series of posts on John’s book On Liberty, such as "John Stuart Mill’s Brief for Freedom of Speech.“) The blogosphere can help forward that kind of discussion, and get us a little closer to the truth.
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2019-09-19 06:13:14
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https://www.physicsforums.com/threads/vector-and-tensor-problems.498986/
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# Vector and Tensor problems
1. May 15, 2011
### slashragnarok
1. Show that curl v is twice the local angular velocity (w), where v is the velocity vector of the fluid.
2. Prove that I:v= div v where I is a unit tensor.
3. Explain why alternating unit tensor is very important in order to describe the cross product of two vectors.
4. If div E=0, div H=0, curl E=- $$\partial$$H/$$\partial$$ t , curl H= $$\partial$$E /$$\partial$$t then show that E and H satisfy $$\nabla$$2u= $$\partial$$2u/$$\partial$$t2
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2016-07-25 11:52:50
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https://math.stackexchange.com/questions/1100142/inequality-involving-power-sums
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# inequality involving power sums
Let $x_1, x_2, ... ,x_n$ be positive real numbers and define $S(k)$ to be the power sum $S(k) = x_1^k + x_2^k +... + x_n^k$ . It is given that $S(3) = 3$ and that $S(5) = 5$. Find the best lower bound for $S(1)$.
Remarks:
(1) An application of Cauchy Schwarz shows that $S(1) > 9/5$ is an admissible lower bound.
(2) The problem appears (with a smaller bound) in the book "Inequalities" by Cvetkovski (p. 360).
Thanks.
Since the domain of admissible values of $x$ is compact, the sum $S(1)$ attains its minimum. To find it we apply the method of Lagrangian multiplies. Put $$\mathcal L=S(1)+\lambda(S(3)-3)+\mu(S(5)-5).$$
Then $$\frac{\partial{\mathcal L}}{\partial{x_i}}=1+3\lambda x_i^2+5\mu x_i^4.$$
Hence the minimum is attained when some of $x_i$ are at the relative boundary of the domain (that is equals zero), and the others are roots of the equation $1+3\lambda x^2+5\mu x^4$. Hence $k$ of $x_i$ equals $a>0$ and $l$ of $x_i$ equals $b>0$. Then $ka^3+lb^3=3$ and $ka^5+lb^5=5$. Solving this system numerically (by Mathcad), suggests that the minimum value of $S(1)=2.1367\dots$ is attained when $k=l=1$ and $a=1.3642\dots.$
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2019-08-17 21:39:33
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https://www.isa-afp.org/entries/IsaGeoCoq.html
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# Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid
Title: Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid Author: Roland Coghetto (roland_coghetto /at/ hotmail /dot/ com) Submission date: 2021-01-31 Abstract: The GeoCoq library contains a formalization of geometry using the Coq proof assistant. It contains both proofs about the foundations of geometry and high-level proofs in the same style as in high school. We port a part of the GeoCoq 2.4.0 library to Isabelle/HOL: more precisely, the files Chap02.v to Chap13_3.v, suma.v as well as the associated definitions and some useful files for the demonstration of certain parallel postulates. The synthetic approach of the demonstrations is directly inspired by those contained in GeoCoq. The names of the lemmas and theorems used are kept as far as possible as well as the definitions. It should be noted that T.J.M. Makarios has done some proofs in Tarski's Geometry. It uses a definition that does not quite coincide with the definition used in Geocoq and here. Furthermore, corresponding definitions in the Poincaré Disc Model development are not identical to those defined in GeoCoq. In the last part, it is formalized that, in the neutral/absolute space, the axiom of the parallels of Tarski's system implies the Playfair axiom, the 5th postulate of Euclid and Euclid's original parallel postulate. These proofs, which are not constructive, are directly inspired by Pierre Boutry, Charly Gries, Julien Narboux and Pascal Schreck. BibTeX: @article{IsaGeoCoq-AFP, author = {Roland Coghetto}, title = {Tarski's Parallel Postulate implies the 5th Postulate of Euclid, the Postulate of Playfair and the original Parallel Postulate of Euclid}, journal = {Archive of Formal Proofs}, month = jan, year = 2021, note = {\url{https://isa-afp.org/entries/IsaGeoCoq.html}, Formal proof development}, ISSN = {2150-914x}, } License: GNU Lesser General Public License (LGPL)
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2021-05-16 06:30:23
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https://gking.harvard.edu/home?sv_list_box_delta=og-3633-c57fdf3d&pager_id=1&destination=home&page=4%2C1%2C0%2C11
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Gary King is the Weatherhead University Professor at Harvard University. He also serves as Director of the Institute for Quantitative Social Science. He and his research group develop and apply empirical methods in many areas of social science research. Full bio and CV
## Research Areas
• Evaluating Social Security Forecasts
The accuracy of U.S. Social Security Administration (SSA) demographic and financial forecasts is crucial for the solvency of its Trust Funds, government programs comprising greater than 50% of all federal government expenditures, industry decision making, and the evidence base of many scholarly articles. Forecasts are also essential for scoring policy proposals, put forward by both political parties. Because SSA makes public little replication information, and uses ad hoc, qualitative, and antiquated statistical forecasting methods, no one in or out of government has been able to produce fully independent alternative forecasts or policy scorings. Yet, no systematic evaluation of SSA forecasts has ever been published by SSA or anyone else. We show that SSA's forecasting errors were approximately unbiased until about 2000, but then began to grow quickly, with increasingly overconfident uncertainty intervals. Moreover, the errors all turn out to be in the same potentially dangerous direction, each making the Social Security Trust Funds look healthier than they actually are. We also discover the cause of these findings with evidence from a large number of interviews we conducted with participants at every level of the forecasting and policy processes. We show that SSA's forecasting procedures meet all the conditions the modern social-psychology and statistical literatures demonstrate make bias likely. When those conditions mixed with potent new political forces trying to change Social Security and influence the forecasts, SSA's actuaries hunkered down trying hard to insulate themselves from the intense political pressures. Unfortunately, this otherwise laudable resistance to undue influence, along with their ad hoc qualitative forecasting models, led them to also miss important changes in the input data such as retirees living longer lives, and drawing more benefits, than predicted by simple extrapolations. We explain that solving this problem involves using (a) removing human judgment where possible, by using formal statistical methods -- via the revolution in data science and big data; (b) instituting formal structural procedures when human judgment is required -- via the revolution in social psychological research; and (c) requiring transparency and data sharing to catch errors that slip through -- via the revolution in data sharing & replication.An article at Barron's about our work.
Proof that previously used estimators of electoral incumbency advantage were biased, and a new unbiased estimator. Also, the first systematic demonstration that constituency service by legislators increases the incumbency advantage.
• Information Control by Authoritarian Governments
• Mexican Health Care Evaluation
An evaluation of the Mexican Seguro Popular program (designed to extend health insurance and regular and preventive medical care, pharmaceuticals, and health facilities to 50 million uninsured Mexicans), one of the world's largest health policy reforms of the last two decades. Our evaluation features a new design for field experiments that is more robust to the political interventions and implementation errors that have ruined many similar previous efforts; new statistical methods that produce more reliable and efficient results using fewer resources, assumptions, and data; and an implementation of these methods in the largest randomized health policy experiment to date. (See the Harvard Gazette story on this project.)
• Presidency Research; Voting Behavior
Resolution of the paradox of why polls are so variable over time during presidential campaigns even though the vote outcome is easily predictable before it starts. Also, a resolution of a key controversy over absentee ballots during the 2000 presidential election; and the methodology of small-n research on executives.
• Informatics and Data Sharing
Replication Standards New standards, protocols, and software for citing, sharing, analyzing, archiving, preserving, distributing, cataloging, translating, disseminating, naming, verifying, and replicating scholarly research data and analyses. Also includes proposals to improve the norms of data sharing and replication in science.
• International Conflict
Methods for coding, analyzing, and forecasting international conflict and state failure. Evidence that the causes of conflict, theorized to be important but often found to be small or ephemeral, are indeed tiny for the vast majority of dyads, but are large, stable, and replicable wherever the ex ante probability of conflict is large.
• Legislative Redistricting
The definition of partisan symmetry as a standard for fairness in redistricting; methods and software for measuring partisan bias and electoral responsiveness; discussion of U.S. Supreme Court rulings about this work. Evidence that U.S. redistricting reduces bias and increases responsiveness, and that the electoral college is fair; applications to legislatures, primaries, and multiparty systems.
• Mortality Studies
Methods for forecasting mortality rates (overall or for time series data cross-classified by age, sex, country, and cause); estimating mortality rates in areas without vital registration; measuring inequality in risk of death; applications to US mortality, the future of the Social Security, armed conflict, heart failure, and human security.
Publications and other projects designed to improve teaching, learning, and university administration, as well as broader writings on the future of the social sciences.
• Anchoring Vignettes (for interpersonal incomparability)
Methods for interpersonal incomparability, when respondents (from different cultures, genders, countries, or ethnic groups) understand survey questions in different ways; for developing theoretical definitions of complicated concepts apparently definable only by example (i.e., "you know it when you see it").
• Automated Text Analysis
Automated and computer-assisted methods of extracting, organizing, understanding, conceptualizing, and consuming knowledge from massive quantities of unstructured text.
• Causal Inference
Methods for detecting and reducing model dependence (i.e., when minor model changes produce substantively different inferences) in inferring causal effects and other counterfactuals. Matching methods; "politically robust" and cluster-randomized experimental designs; causal bias decompositions.
• Event Counts and Durations
Statistical models to explain or predict how many events occur for each fixed time period, or the time between events. An application to cabinet dissolution in parliamentary democracies which united two previously warring scholarly literature. Other applications to international relations and U.S. Supreme Court appointments.
• Ecological Inference
Inferring individual behavior from group-level data: The first approach to incorporate both unit-level deterministic bounds and cross-unit statistical information, methods for 2x2 and larger tables, Bayesian model averaging, applications to elections, software.
• Missing Data & Measurement Error
Statistical methods to accommodate missing information in data sets due to scattered unit nonresponse, missing variables, or values or variables measured with error. Easy-to-use algorithms and software for multiple imputation and multiple overimputation for surveys, time series, and time series cross-sectional data. Applications to electoral, and other compositional, data.
• Qualitative Research
How the same unified theory of inference underlies quantitative and qualitative research alike; scientific inference when quantification is difficult or impossible; research design; empirical research in legal scholarship.
• Rare Events
How to save 99% of your data collection costs; bias corrections for logistic regression in estimating probabilities and causal effects in rare events data; estimating base probabilities or any quantity from case-control data; automated coding of events.
• Survey Research
How surveys work and a variety of methods to use with surveys. Surveys for estimating death rates, why election polls are so variable when the vote is so predictable, and health inequality.
• Unifying Statistical Analysis
Development of a unified approach to statistical modeling, inference, interpretation, presentation, analysis, and software; integrated with most of the other projects listed here.
# Computer-Assisted Keyword and Document Set Discovery from Unstructured Text
Gary King, Patrick Lam, and Margaret Roberts. In Press. “Computer-Assisted Keyword and Document Set Discovery from Unstructured Text.” American Journal of Political Science.Abstract
The (unheralded) first step in many applications of automated text analysis involves selecting keywords to choose documents from a large text corpus for further study. Although all substantive results depend on this choice, researchers usually pick keywords in ad hoc ways that are far from optimal and usually biased. Paradoxically, this often means that the validity of the most sophisticated text analysis methods depend in practice on the inadequate keyword counting or matching methods they are designed to replace. Improved methods of keyword selection would also be valuable in many other areas, such as following conversations that rapidly innovate language to evade authorities, seek political advantage, or express creativity; generic web searching; eDiscovery; look-alike modeling; intelligence analysis; and sentiment and topic analysis. We develop a computer-assisted (as opposed to fully automated) statistical approach that suggests keywords from available text without needing structured data as inputs. This framing poses the statistical problem in a new way, which leads to a widely applicable algorithm. Our specific approach is based on training classifiers, extracting information from (rather than correcting) their mistakes, and summarizing results with Boolean search strings. We illustrate how the technique works with analyses of English texts about the Boston Marathon Bombings, Chinese social media posts designed to evade censorship, among others.
# A Unified Approach to Measurement Error and Missing Data: Details and Extensions
Matthew Blackwell, James Honaker, and Gary King. 2017. “A Unified Approach to Measurement Error and Missing Data: Details and Extensions.” Sociological Methods and Research, 46, 3, Pp. 342-369. Publisher's VersionAbstract
We extend a unified and easy-to-use approach to measurement error and missing data. In our companion article, Blackwell, Honaker, and King give an intuitive overview of the new technique, along with practical suggestions and empirical applications. Here, we offer more precise technical details, more sophisticated measurement error model specifications and estimation procedures, and analyses to assess the approach’s robustness to correlated measurement errors and to errors in categorical variables. These results support using the technique to reduce bias and increase efficiency in a wide variety of empirical research.
# A Unified Approach to Measurement Error and Missing Data: Overview and Applications
Matthew Blackwell, James Honaker, and Gary King. 2017. “A Unified Approach to Measurement Error and Missing Data: Overview and Applications.” Sociological Methods and Research, 46, 3, Pp. 303-341. Publisher's VersionAbstract
Although social scientists devote considerable effort to mitigating measurement error during data collection, they often ignore the issue during data analysis. And although many statistical methods have been proposed for reducing measurement error-induced biases, few have been widely used because of implausible assumptions, high levels of model dependence, difficult computation, or inapplicability with multiple mismeasured variables. We develop an easy-to-use alternative without these problems; it generalizes the popular multiple imputation (MI) framework by treating missing data problems as a limiting special case of extreme measurement error, and corrects for both. Like MI, the proposed framework is a simple two-step procedure, so that in the second step researchers can use whatever statistical method they would have if there had been no problem in the first place. We also offer empirical illustrations, open source software that implements all the methods described herein, and a companion paper with technical details and extensions (Blackwell, Honaker, and King, 2017b).
# Google Flu Trends Still Appears Sick: An Evaluation of the 2013‐2014 Flu Season
David Lazer, Ryan Kennedy, Gary King, and Alessandro Vespignani. 2014. “Google Flu Trends Still Appears Sick: An Evaluation of the 2013‐2014 Flu Season”.Abstract
Last year was difficult for Google Flu Trends (GFT). In early 2013, Nature reported that GFT was estimating more than double the percentage of doctor visits for influenza like illness than the Centers for Disease Control and Prevention s (CDC) sentinel reports during the 2012 2013 flu season (1). Given that GFT was designed to forecast upcoming CDC reports, this was a problematic finding. In March 2014, our report in Science found that the overestimation problem in GFT was also present in the 2011 2012 flu season (2). The report also found strong evidence of autocorrelation and seasonality in the GFT errors, and presented evidence that the issues were likely, at least in part, due to modifications made by Google s search algorithm and the decision by GFT engineers not to use previous CDC reports or seasonality estimates in their models what the article labeled algorithm dynamics and big data hubris respectively. Moreover, the report and the supporting online materials detailed how difficult/impossible it is to replicate the GFT results, undermining independent efforts to explore the source of GFT errors and formulate improvements.
# The Parable of Google Flu: Traps in Big Data Analysis
David Lazer, Ryan Kennedy, Gary King, and Alessandro Vespignani. 2014. “The Parable of Google Flu: Traps in Big Data Analysis.” Science, 343, 14 March, Pp. 1203-1205.Abstract
Large errors in flu prediction were largely avoidable, which offers lessons for the use of big data.
In February 2013, Google Flu Trends (GFT) made headlines but not for a reason that Google executives or the creators of the flu tracking system would have hoped. Nature reported that GFT was predicting more than double the proportion of doctor visits for influenza-like illness (ILI) than the Centers for Disease Control and Prevention (CDC), which bases its estimates on surveillance reports from laboratories across the United States ( 1, 2). This happened despite the fact that GFT was built to predict CDC reports. Given that GFT is often held up as an exemplary use of big data ( 3, 4), what lessons can we draw from this error?
# Restructuring the Social Sciences: Reflections from Harvard's Institute for Quantitative Social Science
Gary King. 2014. “Restructuring the Social Sciences: Reflections from Harvard's Institute for Quantitative Social Science.” PS: Political Science and Politics, 47, 1, Pp. 165-172. Cambridge University Press versionAbstract
The social sciences are undergoing a dramatic transformation from studying problems to solving them; from making do with a small number of sparse data sets to analyzing increasing quantities of diverse, highly informative data; from isolated scholars toiling away on their own to larger scale, collaborative, interdisciplinary, lab-style research teams; and from a purely academic pursuit to having a major impact on the world. To facilitate these important developments, universities, funding agencies, and governments need to shore up and adapt the infrastructure that supports social science research. We discuss some of these developments here, as well as a new type of organization we created at Harvard to help encourage them -- the Institute for Quantitative Social Science. An increasing number of universities are beginning efforts to respond with similar institutions. This paper provides some suggestions for how individual universities might respond and how we might work together to advance social science more generally.
# Book Review of Forecasting Presidential Elections'
Gary King. 1985. “Book Review of Forecasting Presidential Elections'.” American Political Science Review, 79, 3, Pp. 855.Abstract
This is a book review of Steven J. Rosenstone, Forecasting Presidential Elections, New Haven: Yale University Press, 1983.
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2018-05-25 20:20:46
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https://ltwork.net/occurring-in-the-presence-of-oxygen-anaerobic-the-stage--2518700
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# Occurring in the presence of oxygen anaerobic The stage of respiration that occurs in anaerobic condition In animal cells aerobic
###### Question:
Occurring in the presence of oxygen
anaerobic
The stage of respiration that occurs in anaerobic condition In
animal cells
aerobic
3. The first step in the chemical breakdown of glucose
alcohol fermentation
lactic acid fermentation
4. Occurring in the absence of oxygen
The stage of respiration that occurs in anaerobic conditions
In less complex organisms
glycolysis
$Occurring in the presence of oxygen anaerobic The stage of respiration that occurs in anaerobic cond$
### 1. An elementary reaction A B is conducted using a spherical pellet impregnated with platinum throughout
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### Write two or more sentences each about the following words or phrases.Federalism, sovereignty, limited government, separation of
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### Solve 6x + 6 = 2x + 13
Solve 6x + 6 = 2x + 13...
Suppose a group of volunteers is planning to build a park near a local lake. The lake is known to contain low levels of arsenic (As). Therefore, prior to starting construction, the group decides to measure the current level of arsenic in the lake. A) If a 15.7 cm3 sample of lake water is found to ha...
### What mass of nickel (Ni) is in a 2.4 Kg sample of propanol if the concentration is 20 ppb ? (atomic
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### Which sentence best supports the inference that Johnson was not unanimously appreciated during his tenure as president? Lyndon Baines
Which sentence best supports the inference that Johnson was not unanimously appreciated during his tenure as president? Lyndon Baines Johnson Lyndon B. Johnson was the 36th President of the United States. He was born on a farm near Stonewall, Texas, in the hill country west of Austin. Johnson was st...
### In isosceles △HAM, m∡A =32°, . What is m∠H?
In isosceles △HAM, m∡A =32°, . What is m∠H?...
### Sheri studies to get good grades because this is what her parents expect of her. This is an
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### Find the radius and the coordinates of the center of the circle: (x+6)^2+(y-1)^2=25
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### What is Charlemagne Péralte's opinion of the American presence in Haiti? What kind of action Haitian
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### Which of the following is equivalent to 17g – 2g?Group of answer choices
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### Apatient’s uvula rises midline when she says ''ahh'' and she has a positive gag reflex. the nurse
Apatient’s uvula rises midline when she says ""ahh"" and she has a positive gag reflex. the nurse has just tested which cranial nerves?...
### - 3x + 2(5x-7)9- 3(2x - 4)8x - 613 - 2x)-5+5(x + 4)4(6n +9) - 10n14 - 3(4n-1)-8n-8(-4-2n)7k
- 3x + 2(5x-7) 9- 3(2x - 4) 8x - 613 - 2x) -5+5(x + 4) 4(6n +9) - 10n 14 - 3(4n-1) -8n-8(-4-2n) 7k - 2(3k + 1) - 9 -6+5(8-k) - 8k k+1 - 4(2k-9) -10k - 3+2(5 + 6k) 8 + 9x + 4(11 - 2x) -4(-2x - 7) + 6x - 7 9-3(-4 + 3x) + 12x 5(2y - 4) + 2(y + 9) -4(3u - 1) + 7(3-2u) 6(-5u+1) - 3(4u - 12) 3(-u-5)+8(2u...
### World geoHelp pls Pic
World geo Help pls Pic $World geo Help pls Pic$...
### María y javier volvieron muy tarde del cine porque fueron a la de las once. a. sesión c. pantalla b. taquilla d. butaca
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### In kessy's old kitchen, he could bake 10 cookies or mix 15 glasses of lemonade in one day. now kessy
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### 'We can't fight our way through this, Robinson. We've got no army. There's virtually nobody on our side. No owners, no umpires,
"We can't fight our way through this, Robinson. We've got no army. There's virtually nobody on our side. No owners, no umpires, very few newspapermen. And I'm afraid that many fans will be hostile. We'll be in a tough position. We can win only if we can convince the world that I'm doing this because...
### Is Washington D. C Hotter or colder than London England
Is Washington D. C Hotter or colder than London England...
-- 0.013597--
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2022-12-08 16:13:51
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https://westsideelectronics.com/using-the-3-cent-microcontroller/
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# Using the 3 cent microcontroller as a encoder counter
Making a rotatory encoder counter in under a dime.
Even in the world of microcontrollers, a $0.03 microcontroller in single units is an oddity. It is very cheap. The cheapest microcontrollers you can find in single units among online distributors like Digikey or Mouser are in the region of$1. To push it below the $1 mark, your order usually has to be in the thousands. Putting it into perspective, it is a no-name smartphone for$6 in a world of \$600 smartphones.
Is it great? Heck no. It uses an esoteric implementation of C called mini-C, documentation is all over the place, and the ICs are hard to source. But is it fun to use? That depends on your idea of fun. I like constraints and asking myself: "Now what can I do with this?"
I looked at these ICs a few years ago and poked around a little, exploring what it can be used. Since then, there is better documentation in the IDE, the ICs cost 4x more(!) and the open source toolchain (SDCC) has become more mature.
This time around I want to use this IC as an encoder counter.
# Counting Signals
Keeping track of encoder counts is an important activity. Encoders are frequently used to keep track of the angular position, and that information is used to keep track of things like how fast a wheel is spinning or how far the print head of a printer has advanced.
However, encoder counting is also notoriously busy. Imagine that you had to stare at a screen that displayed two variables of either one or zero. By keeping track of how often the one or zeros appear and in which order, you can tell the position of the encoder and the direction it is turning. You can stare at the screen all day, which means you can't do anything else... or you could set up a notification to tell you when each variable has changed.
Instead of notifications, microcontrollers use interrupts to make sure that they don't miss counts. But with hundreds of counts a second, you might not be able to do anything else other than to service the interrupt, which is why some (read: expensive) chips have dedicated encoder modules, which freeing up the main controller to do other things like reading a control signal or sending out data.
Not all microcontrollers have this module and this is where the Padauk microcontroller comes in.
# I/O
The concept is quite simple: use the Padauk microcontroller to do the busywork of reading the encoder, and then provide an easy means for another microcontroller to access the counts.
I decided on having two ways to read the count. SPI to get the exact count from the microcontroller, as well as a 2 pin count/dir control scheme for less position-critical applications: the Padauk microcontroller will output a pulse over one pin when it detects the encoder turning, and indicate the direction of the encoder using the other pin.
## How a Rotary Encoder works
Rotary encoders work by outputting gray code. It is a pair of signals that indicate the number of turns, as well as the direction of the knob.
By reading when each signal changes and keeping track of which signal changes first, the count on the encoder can be accurately incremented or decremented.
The wiring for the encoder is also fairly simple, with pull-up resistors on each channel.
# Coding
## Emulating the IC
I tested the code on the emulator from Padauk. I found the debugging experience to be quite useful in fixing a few errors in my logic. The Padauk IDE automatically displays all the registers and their current state as well as the values of variables in each part of the code as you step through.
Because the instruction set for the Padauk ICs is so small, it is possible to display everything on the IDE. One thing I didn't like was that the documentation for Mini-C is split across the Application Note in the IDE as well as in the datasheet of the IC.
In the code I set up two interrupts to read the gray code from the encoder and to store it to a global variable. This global variable can be incremented and decremented.
## Two's complement
Mini-C doesn't have the concept of negative numbers which is usually implemented as two's complement.
This method is consistent with the logic of binary and arithmetic operations so it is a really useful way of representing negative numbers. However in lieu of not needing to have negative values since I'm more interested in the relative position of the encoder, another way to achieve this is to initialize the value in the middle of the range.
If we start at zero and don't implement two's compliment, the value will roll around from 0 back to 100, which is undesirable. Instead, I started the value in the middle of the range, and this allows it to decrement and increment normally. All I have to do is to keep track of the initial value and reset it before it rolls over.
## Programming the IC
When you hit Program, Padauk's programmer saves the binary to the memory of the programmer so you can use it as a standalone device or trigger the download process from the computer.
The PFS173 that I was using supports Multiple Time Programming (MTP) and it does what it says on the lid, it allows it to be programmed multiple times. There are other Padauk ICs can only be programmed once such as the PMC150.
# Putting everything together
Along the way, I added a few other functionalities, such as the ability for the board to serve as the main controller instead of just interpreting the encoder signals as well as being able to save the last set PWM value when it turned off or on. When I was reasonably happy with the result, I moved it to a dedicated board on my main project.
# Thoughts
In total it took me about two days to familiarize myself with the functionality and to debug the code. With regular use, programming the IC and soldering it could be shortened to a few hours.
For me, having this ultra-cheap microcontroller around is pretty useful as a general purpose IC that can take the place of more specialized components or devices. I like it because it is cheap, easy to solder, and is easy to understand and program.
## Documentation
The documentation has improved since the last time I looked at it. Now there are proper sections and categories that are clearly segmented and there aren't any broken links. The language has also been cleaned up since the last time. I checked the Mandarin version of the software and documentation and I didn't find any differences. I do still wish for better documentation on Mini-C however.
## SDCC
Small Device C Compiler is now more mature and supports a wide variety of Padauk ICs. However, you'll still need to build the programmer yourself and there is no emulator. The benefit is that you get access to modern C, which is definitely better than the esoteric mini-C that Padauk implements. If you don't have a Windows device, that is probably also the only way you can program these ICs.
I still personally like using Padauk's IDE and mini-C because I find that they work well enough for my purposes. Being able to step through the code is still enormously useful for me and I don't fancy swapping the microcontroller from the board to the debugger and vice-versa for each update of the code
## Cost
Padauk ICs now cost almost 4x as much from the time I bought them in 2019.
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2022-05-18 12:12:14
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http://physics.stackexchange.com/questions/27924/particles-for-all-forces-how-do-they-know-where-to-go-and-what-to-avoid
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# Particles for all forces: how do they know where to go, and what to avoid?
Here's an intuitive problem which I can't get around, can someone please explain it?
Consider a proton P and an electron E moving through the electromagnetic field (or other particles for other forces, same argument). They exert a force upon one another. In classical mechanics this is expressed as their contributing to the field and the field exerts a force back upon them in turn. In quantum mechanics the model is the exchange of a particle.
Let's say one such particle X is emitted from P and heads towards E. In the basic scenario, E absorbs it and changes its momentum accordingly. Fine.
How does X know where E is going to be by the time it arrives? What's to stop E dodging it, or having some other particle intercept X en route?
Are P and E emitting a constant stream of force-carrying particles towards every other non-force-carrying particle in the universe? Doesn't this imply a vast amount of radiation all over the place?
I am tempted to shrug of the entire particle exchange as a mere numerical convenience; a discretization of the Maxwell equations perhaps. I am reluctant to say "virtual particle" because I suspect that term means something different to what I think it means.
Or is it a kind of observer effect: E "observes" X in the act of absorbing it, all non-intercepting paths have zero probability when the waveform collapses?
Or have I missed the point entirely?
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The particle exchange model is a convenient story that helps us remember how to draw Feynman diagrams, which, in turn, help us remember terms in the perturbation series of interacting QFTs. But I think that it's unwise to take it too seriously--fundamentally, you still have particles interacting with local quantum fields. It's just that we can, in a certain limit, make weakly interacting quantized fields look like they're interacting via an infinite series of particle exchanges, with only the few lowest-order ones important. – Jerry Schirmer May 7 '12 at 12:19
You can alleviate much of the confusion by a)discarding the classical notion of a particle being a point like object and more importantly b)thinking in terms of fields interacting with each other, i.e an electron field/proton field coupling with other fields. +1 for a good question. – Antillar Maximus May 7 '12 at 12:27
@JerrySchirmer: It is not "just a story" as there are Feynman diagrams in string theory where you don't have quantum fields. Besides, the particle picture is mathematically equivalent to other formulations, so any paradox must have a resolution. – Ron Maimon May 7 '12 at 17:20
@RonMaimon: but you would have to admit that the Feynman diagrams are simply bookkeeping techniques for keeping track of terms in a perturbation theory in string theory. In any case, the physical thing is the sum of the perturbation theory, not the individual terms. – Jerry Schirmer May 7 '12 at 18:24
@JerrySchirmer: Sure, that's true, but you can measure intermediate photon states in principle, by measuring the quantum field, and see that an electron is producing virtual photons. The different histories interfere together to produce the usual perturbation expansion, but the resulting quantum mechanical story is correct, in the sense that the particle emission and absorption (in old-fasioned perturbation theory, where you don't have particles going back in time) is compatible with what you would see if you measure the instantaneous quantum field at two times. – Ron Maimon May 7 '12 at 18:44
This choice is closest to the the correct one.
I am tempted to shrug of the entire particle exchange as a mere numerical convenience; a discretization of the Maxwell equations perhaps. I am reluctant to say "virtual particle" because I suspect that term means something different to what I think it means.
And virtual exchange is a correct description, because during the interaction the exchanged particle is not on mass shell.
Keep in mind that in the microcosm of particles nature is quantum mechanical. The particle scattering on another particle and the momentum and energy and quantum number exchanges between them are all described by one wave function, one mathematical formula that gives the probability for the interaction to take place in the way it has been ( will be ) observed.. Thus it is not a matter for "knowing" but a matter of "being".
The Feynman diagrams that give rise to the "particle exchange" framework are just a mathematical algorithm for the calculations and help in understanding how to proceed with them.
To see how classical fields are built up by the substructure of quantum mechanics see the essay here.
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"Thus it is not a matter for "knowing" but a matter of "being"". Well said :) – Manishearth May 7 '12 at 12:37
In a non relativistic Classical Mechanics (CM) there is an interaction potential involving both coordinates: $U(\vec{r}_1-\vec{r}_2)$ and the corresponding force present in either particle equation. There is no need in "exchange" interpretation here. Same for non relativistic QM.
In relativistic case the potential becomes "retarded". Its time evolution may be expanded in a Fourier series and each plane wave can be called a "longitudinal virtual photon". You see, it it nearly the same interaction potential (force) as in the non relativistic CM, acting between charged particles.
Apart from retarded "longitudinal" potential, there is also "transversal" vector potential that may include real electromagnetic waves propagating in all directions, not only between charged particles in question. The real photons are not absorbed but scattered so they do not contribute into the charge "attraction". The latter is described with those "virtual photons".
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As Jerry Schirmer points out, it is not really a discretization of the Maxwell equations as you say, but rather a series expansion of the quantum mechanical cross section for interaction. Thus you put in an electron and a proton with some momenta and you want to calculate the probability of them coming out with some other momenta, which you can express as something like $${}_\textrm{out}\langle p^+,q_1;e^-,q_2|p^+,p_1;e^-,p_2\rangle_\textrm{in}=\lim_{T\rightarrow\infty}\langle p^+,q_1;e^-,q_2|e^{iH(2T)}|p^+,p_1;e^-,p_2\rangle.$$ You then make a series expansion of this quantity in the interaction hamiltonian (or more exactly in the interaction strength $\alpha=e^2/\hbar c$). Feynman's contribution (one of them, anyway) was to give a graphical way of constructing each of the terms in the series (most of which involve pretty ugly integrals and will in fact diverge if not treated properly using renormalization) so that each term gets interpreted as a physical process where, say, the electron and proton interchange a virtual photon.
The truth is of course that these virtual photon exchanges are not physical: only the whole scattering process is physical and you cannot observe what happens in the middle.
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You can observe what is in the middle by making a measurement using classical probes you electrically polarize at the right moment. You can actually do this for long enough scales. The Feynman expansion is physical, it is not a mathematical trick. – Ron Maimon May 7 '12 at 17:44
In the particle exchange picture, the particles are emitted in all directions and only the ones going from P in the direction of E that hit E are intercepted and have an effect. The other particles interfere themselves out of existence, as there is no on-shell state they can enter while conserving energy, or else return to P, giving the self-energy modification to P's mass. In fact, most return to P, since the self-energy is divergent, while only a small fraction make it to E by comparison.
This process is virtual, so that it is defined by temporary intermediate states which only can stick around until their phase randomizes them away. For the case of a classical force, you need to use particles that go every which way, forward and backward in time.
Consider two classical objects interacting with a (free) quantum field according to this Lagrangian:
$$\int |\partial_\phi|^2 + \phi(x) s(x)$$
where the source is two delta functions $s(x) = g\delta(x-x_0) + g\delta(x-x_1)$. Each of these classical sources is steadily spitting out and absorbing particles per unit time at a steady rate g, as you can see by the added source term in the Hamitlonian:
$$g\phi(x_0) = g\int {d^3k\over 2E_k} e^{ikx_0} \alpha_k + e^{-ikx_0}\alpha^\dagger_k$$
the g term is multiplying a creation operator and an annihilation operator, so the Hamiltonian has a steady amplitude g per unit time to emit any on-shell particle, and the same amplitude to absorb one. If you have no other source, the particles that are absorbed are those emitted by the source, and you just get an (infinite) self-energy renormalization of the mass.
This description is the on-shell old-fasioned perturbation theory, in which the intermediate states are k-states and the description is Hamiltonian in time. This is not covariant, but it shows you that particles are spat out and absorbed, and the two sources only interact to the extent that some of the particles spat out by one are absorbed by the other. The old-fasioned picture is useless for actual computations, but it reveals the particle processes most clearly, because it follows the annihilation and creation of physical particles in detail in time.
The result of the interaction when there are two sources is altered by those particles produced by one, absorbed by the other later. The covariant Schwinger/Feynman form of this introduces particles that meander around in space and time both. Those that do not get absorbed by the other make a field around the particle.
The fact that you are doing things by loop order means that you are not considering the process of a particle emitted by one source absorbed by itself, since this is a loop. The loop order separation of terms makes the scattering process look weird, since it looks like the emitted particle knew where to go to find the other particle. It didn't. If it came back to the first particle, we would include it as part of the next order of Feynman diagram as part of the self-energy graph.
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The following is very loose and sloppy; I wanted to try explaining it on a pop level.
Remember that reality is not what our intuition suggests it is. A "particle" is not just a particle: it is also a wave; it is also an excitation of a field. It makes sense in some situations to predict results by thinking of "particles" as having a specific position and momentum, but that's not their essential nature (and is actually in conflict with reality as verified by numerous experiments).
In your example, when the proton P emits a photon X, you really ought to think of it as emitting a light wave which doesn't have a nailed-down orientation. Rather, it has a probability (calculable, with difficulty) of being at any number of different points, until it "interacts" with the electron and the "wavefunction collapses". I'm putting these events in quotation marks to emphasize that there are different possible interpretations, but they do accurately describe the underlying reality.
To address your specific questions, I'm going to be a bit sloppy.
• How does X know where E is going to be by the time it arrives? What's to stop E dodging it, or having some other particle intercept X en route?
P doesn't "know" where E is going to be and neither does X, but they don't need to. X doesn't start out as a particle, it starts out as a spherical wavefront, and whenever the X wavefront impacts E, suddenly the whole X-wave "pops" into an X-particle just at the right time and location to interact with E. E can't dodge the wavefront unless it's going faster than light. It's certainly possible for some other particle (F) to get hit by the X-wavefront and interact with it instead, but that means F is going to be sending out wavefronts too and we have a three-particle interaction instead of a two-particle one, and remember P is continually sending out wavefronts and some of them are going to hit X. (It's perfectly possible for one particle to shield another from the action of a third; that's how a Faraday cage works)
• Are P and E emitting a constant stream of force-carrying particles towards every other non-force-carrying particle in the universe? Doesn't this imply a vast amount of radiation all over the place?
You could think of it like that, but it's not like P and E know the positions of all other particles; they just send out their fields/wavefronts "everywhere" and it just so happens that the right amount of energy appears at the right place and the right time to make everything work. At placetimes where there's no particle to interact with, well, if a tree falls in the forest and so forth, it doesn't matter.
If you really want to blow your mind, consider that the equations about photon emission also admit a solution where the photon moves backwards in time. So, if you naively interpret that literally, anyone time-travelling to the past would be blinded because all the photons from all times in the future would be constantly bombarding them.
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## protected by Qmechanic♦Mar 12 at 22:55
Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.
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2014-10-25 21:43:34
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https://matheducators.stackexchange.com/questions/26069/what-is-the-justification-to-teach-the-redundant-use-of-parentheses-in-multipl
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What is the justification to teach the (redundant) use of parentheses in multiplications?
Example: 5 x 18 = (5 x 10) + (5 x 8) instead of 5 x 10 + 5 x 8?
• The teacher must have been offended by dear Aunt Sally and refused to excuse her. Jan 28 at 15:56
• If I see 5 x 10 + 5 x 8 then very first thing I need to do is think about order of precedence and mentally map that to (5 x 10) + (5 x 8) anyway. So certainly presenting it as such up front saves me a step. What would be the justification for not doing it? You think that the extra step is worthwhile in itself? Or just that you personally find doing it so intuitive you think everyone else will too? Jan 28 at 18:19
• It's actually required if you're brave enough to use Windows Calculator in standard mode. :-/ Jan 28 at 19:57
• @RobertColumbia or somebody forgot to email the teacher's dad a shark. Jan 29 at 4:25
• The parentheses in (5 x 10) + (5 x 8) are only redundant because we have a convention that says 5 x 10 + 5 x 8 should be parsed as (5 x 10) + (5 x 8) rather than as 5 x ((10 + 5) x 8) or any of the other possible ways. You need to teach that convention before you can rely on that convention. Using parentheses is pretty convenient to teach that convention.
– Stef
Jan 29 at 20:02
• or have machine-like reading ability for complicated logical expressions. In my own writing I've mostly ignored each approach, and freely use different bracketing symbols for better comprehension and short-cut terms (e.g. "(line 3)" rather than writing out what "line 3" is). For example, see the style used in this MSE answer. I think it would be better to spend less time testing the evaluation of "artificial interpretation problems" like $5 \times 10 + 5 \times 8$ (continued) Jan 28 at 19:08
• @anjama Especially when dealing with operators which aren’t handled uniformly across languages. For example exponentiation is left-associative in some languages (Python for example) and right-associative in others (Wolfram Alpha for example), so 4^3^2 is ambiguous without parenthesis, and possibly misunderstood without a proper knowledge of the language in the absence of parenthesis. Jan 29 at 2:55
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2023-03-21 23:18:26
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http://emporiazoo.com/vi1fv/55e14a-cloudbeds-company-reviews
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The Young’s modulus or Modulus of elasticity is a numerical constant for the material. The units of Young’s modulus in the English system are pounds per square inch (psi), and in the metric system newtons per square metre (N/m 2). Modulus of elasticity definition, any of several coefficients of elasticity of a body, expressing the ratio between a stress or force per unit area that acts to deform the body and the corresponding fractional deformation caused by the stress. Young’s modulus of elasticity is ratio between stress and strain. Young's modulus definition, a coefficient of elasticity of a substance, expressing the ratio between a stress that acts to change the length of a body and the fractional change in … Depending upon the three of strain, there are three types of elastic moduli, viz. Find the young’s modulus of elasticity for the material which is 200 cm long, 7.5 cm wide and 15 cm deep. G = Modulus of Rigidity. Modulus of elasticity measures the resistance of a material to non-permanent or elastic deformation when a ratio of the stress is applied to its body. The SI unit of modulus of elasticity E is the pascal (Pa). It can also be defined as the change of stress with respect to the elastic strain and may be computed by the following relation. This is a specific form of Hooke’s law of elasticity. If you have any query regarding or if you need any other information related to elastic constant, ask by commenting. This is a very useful parameter in material science. Elastic Modulus in Physics | Definition, Formulas, Symbol, Units – Elasticity Due to the highly overall non linear behavior of rubber I believe Young’s modulus for rubber is usually quoted forces small loads. When under stress, materials will primarily expose their elastic properties. See more. Young’s Modulus or Elastic Modulus or Tensile Modulus, is the measurement of mechanical properties of linear elastic solids like rods, wires, etc. non-permanently) when a stress is applied to it, defined as the ratio of stress to strain. Some of these are Bulk modulus and Shear modulus etc. So it has no significance beyond the proportional limit in the stress-strain graph. Modulus of elasticity will have same unit as that for stress because strain is unit less parameter and therefore Modulus of elasticity unit in S.I. The SI unit of modulus of elasticity E is the pascal (Pa). Modulus of elasticity definition: the ratio of the stress applied to a body or substance to the resulting strain within the... | Meaning, pronunciation, translations and examples Young's modulus of elasticity synonyms, Young's modulus of elasticity pronunciation, Young's modulus of elasticity translation, English dictionary definition of Young's modulus of elasticity. Syn. (if the stress and deformation are along an axis) Young's modulus; Synonyms . coefficient of elasticity; Young's modulus of elasticity. E = Young's modulus (Modulus of Elasticity) (Pa , (N/m 2), psi (lb f /in 2)) Young's modulus can be used to predict the elongation or compression of an object when exposed to a force; Note that strain is a dimensionless unit since it is the ratio of two lengths. 5. modulus of elasticity synonyms, modulus of elasticity pronunciation, modulus of elasticity translation, English dictionary definition of modulus of elasticity. K = Bulk Modulus . Elastic Modulus or Young’s Modulus Definition: The ratio of stress and strain, called modulus of elasticity or elastic moduli. Elasticity: Elasticity defines a property of an object that has the ability to regain its original shape after being stretched or compressed. It is defined as the ratio of the stress along an axis over the strain along that axis in the range of elastic soil behaviour. 80 to 90: Aluminium : 60 to 80. Modulus of elasticity = unit stress/unit strain . Definition & Formula. TYPES OF MODULI OF ELASTICITY. coefficient of elasticity; Young's modulus of elasticity. Stress is applied to force per unit area, and strain is proportional change in length. The formula which is used to calculate the modulus of elasticity of the bone or any other solid is the ratio of stress that has been applied on the bone to the strain that bone undergoes. Elastic modulus of the bone measures the resistance offered by the bone to change its shape elastically or to undergo deformation elastically when some amount of stress is applied to it. Modulus of Elasticity also referred to as Elastic Modulus or just Modulus is the quantification of the ratio of a material's elasticity. *Response times vary by subject and question complexity. fcu,28: concrete compressive strength at 28 days. Wrought iron. The deformation modulus is Young’s modulus (E) for a rock mass (E rm) or a soil mass; it is the ratio of principal stress in one direction (σ x) to corresponding strain in the elastic range in the same direction (ε x).It is not easily measured, so estimation methods based on geophysical measurements or rock mass classification schemes tend to be used. By a material per unit volume, the maximum amount of energy that can be absorbed without creating any permanent deformation in the elastic limit is known as modulus of resilience. Young Modulus Instead of drawing a force - extension graph, if you plot stress against strain for an object showing (linear) elastic behaviour, you get a straight line. Define modulus of elasticity. Material: Modulus of elasticity (E) in GPa i.e. For this reason, given the above general definition of elasticity, Young’s modulus is also called the modulus of elasticity. What is a Hooke’s law and how it is applicable for the concept of elasticity. Young’s Modulus. Young’s modulus is a numerical constant, named after the 18th-century English physician and physicist Thomas Young. The greater the modulus, the stiffer the material, or the smaller the elastic strain that results from the application of a given stress. Image source: Stress–strain curve - Wikipedia. Define Young's modulus of elasticity. Young’s modulus, also known as modulus of elasticity or elasticity modulus is named after the British physicist Thomas Young. 190 to 200. These are all most useful relations between all elastic constant which are used to solve any engineering problem related to them. I've learnt that the Young's modulus of elasticity is defined as the ratio of stress and strain when the material obeys Hooke's law. The value of Elastic modulus at 28 days of concrete age is given in BS 8110: Part II 1985: Where: ko: is 20 KN per square millimeter for normal weight concrete. modulus of compression Quick Reference The ratio of mechanical stress to strain in an elastic material when that material is being compressed; it is the modulus of elasticity applied to a material under compression:modulus of compression = compressive force per … modulus of elasticity (plural moduli of elasticity) A quantity that measures an object or substance's resistance to being deformed elastically (i.e. The gradient of the straight-line graph is the Young's modulus, E E is constant and does not change for a given material. n the ratio of the stress applied to a body or substance to the resulting strain within the elastic limit. Young’s modulus specifies the measure of the ability of a material to withstand length changes under tensile or compressive forces. Modulus of elasticity (or also referred to as Young’s modulus) is the ratio of stress to strain in elastic range of deformation. Q: The shaft consists of two sections that are rigidly connected. GN/m 2 or kN/mm 2. The modulus of elasticity, also known as Young's modulus, is a material property and a measure of its stiffness under compression or tension. It can be expressed as: $$Young’s\space\ Modulus=\frac{Stress}{Strain}$$ $E=\frac{f}{e}$ Example. Cast iron: 100 to 160. Definition of Modulus of Elasticity: It can be defined as the slope of the relation between the stress and strain. Young’s modulus is a measure of the ability of a material to withstand changes in length under lengthwise tension or compression. The modulus of elasticity formula is simply stress divided by strain. But it also common practice to state it as the ratio of two length units - like m/m or in/in. Elastic modulus is also known as modulus of elasticity and is sometimes referred to as Young’s modulus. Steel and Nickel. Young’s modulus is also termed the modulus of elasticity. a) Young’s Modulus(Y) or modulus corresponding to longitudinal strain b) Bulk modulus(K) or modulus corresponding to the volume strain c) Rigidity modulus(N) or modulus … Learn about the deforming force applied on an elastic object and how the stress and strain works on an object. Examples: the modulus of elasticity of a PMMA contact lens is about 3000 MPa, it is around 0.4 to 1.5 MPa for silicone hydrogel lenses and 0.3 to 0.5 MPa for hydrogel lenses. This is because stress is proportional to strain. It's an one of a most important functions in strength of materials, frequently used to analyse the stiffness of a solid material. Young's Modulus, often represented by the Greek symbol Ε, also known as elasticity modulus, is a physical quantity to express the elasticity (ratio of stress & strain) of material. 90 to 110: Brass. We have listed youngs modulus for some of the materials. For typical metals, modulus of elasticity is in the range between 45 GPa (6.5 x 10 6 psi) to 407 GPa (59 x 10 6 psi). It is denoted by symbol E. It is also known as modulus of elasticity or elastic modulus. Median response time is 34 minutes and may be longer for new subjects. The elastic modulus is often used for estimation of soil settement and elastic deformation analysis. The modulus is an important design parameter used for computing elastic deflections. Soil Young's modulus (E), commonly reffred to as soil elastic modulus, is an elastic soil parameter and a measure of soil stiffness. ADVERTISEMENTS: It is a measure of stiffness or resistance to deformation of a material. Where E is basically Modulus of elasticity and we can define modulus of elasticity as the ratio of stress to strain. Syn. Young's modulus definition: a modulus of elasticity , applicable to the stretching of a wire etc, equal to the ratio... | Meaning, pronunciation, translations and examples We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. Young’s modulus of elasticity. But the value of Young’s Modulus is mostly used. Examples: the modulus of elasticity of a PMMA contact lens is about 3000 MPa, it is around 0.4 to 1.5 MPa for silicone hydrogel lenses and 0.3 to 0.5 MPa for hydrogel lenses. n the ratio of the stress applied to a body or substance to the resulting strain within the elastic limit. There are some other numbers exists which provide us a measure of elastic properties of a material. 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2021-12-07 17:47:19
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https://octeract.gg/docs/blog/domain-reduction/feasibility-based-bounds-tightening-fbbt/
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1. Home
2. Docs
3. Math blog
4. Domain Reduction
5. Feasibility based bounds tightening (FBBT)
# Feasibility based bounds tightening (FBBT)
The basic idea of FBBT is to find intervals for the variables where no feasible point can be found. Consider the intervals $$I_1 = [0,0.75]$$ and $$I_2 = [0.75,1.5]$$ for the variable $$x_1$$ for problem $$P$$. We know that there are feasible points $$x_1, x_2$$ for $$P$$ if $$x_1 \in I_2$$, and therefore we can not discard this interval.
If we repeat this procedure for $$I_2$$, e.g., divide the interval $$[0.75,1.5]$$ into $$[0.75,1.125]$$ and $$[1.125,1.5]$$, we will see that the interval $$[1.125,1.5]$$ does not have any feasible points, and therefore we can discard it and update the upper bound for $$x_1$$ from 1.5 to 1.125 (see Figure 3).
The question is how can we know when to discard an interval. FBBT uses interval arithmetic to answer this question.
Consider the function $$f(x_1) = 2x_1 + 3$$ with $$x_1 \in [1,2]$$. Given that (f) is an increasing function on this domain, we can bound it by taking the minimum and maximum of the interval $$[1,2]: f(1)=5 \leq f(x_1) \leq 7= f(2)$$.
Similarly, the function $$g(x_1,x_2) = x_1x_2\$$ with $$x_1 \in [1.125,1.5]$$ and $$x_2 \in [1,4]$$ is also increasing with respect to both variables in the domain, and therefore it can be bounded using the extremes of the intervals, i.e., $$g(1.125,1)=1.125\leq g(x_1,x_2) \leq 6 = g(1.5,4)$$. Interval arithmetic can be used for other types of structures too.
Taking the constraint $$x_1x_2 \leq 1$$ in problem (P) and the bounds found using interval arithmetic for the function $$g(x_1,x_2)$$, we notice that $$g$$ is bounded below by 1.125 if we restrict $$x_1$$ to the interval $$I_2 = [1.125,1.5]$$.
Therefore, the constraint is infeasible in the interval (as the upper bound of the constraint is 1), $$I_2$$ can be discarded, and we can reduce the domain of $$x_1$$ in $$\Omega$$ from $$[0,1.5]$$ to $$[0,1.125]$$.
## Putting this into Practice
Let $${x_i}\in{[l_i,u_i]}$$ ($$i=1,\dots,n$$), $$f(x_1,\dots,x_n)$$ and $$UB$$ be the variables, the objective function and a known upper bound of a problem respectively.
Similarly, w.l.o.g. let $${c_j(x_1,\dots,x_n)}\leq{b_j}$$ for $$j=1,\dots,m$$ be the constraints of the problem (all the arguments that will follow can be generalised for greater than inequalities as well as equalities in the constraints). Then, if $${I}\subset{[l_i,u_i]}$$, let $$g^{i,I}_L$$ and $$g^{i,I}_U$$ be the lower and upper bounds for any function $$g(x_1,\dots,x_n)$$ calculated using interval arithmetic on $${x_i}\in{I}$$.
If lower and upper bounds are calculated for every function using $${x_i}\in{I}$$, then the subset $${x_i}\in{I}$$ can be discarded from the feasible set if:
The lower bound of the objective function is larger than the known upper bound, i.e, $${f^{i,I}_L}>UB$$ (in this case, we know that the value of $$x_i$$ in an optimal solution cannot be contained in the subset $$I$$).
The lower bound of a constraint is larger than the right hand side, i.e., $${{c_j}_{L}^{i,I}}>b_j$$ (in this case, the smallest value of the constraint for $${x_i}\in{I}$$ is larger than $$b_j$$, which implies that the problem is infeasible for any point with $${x_i}\in{I}$$).
Octeract Engine applies the previous methodology by evaluating different subsets of the intervals for the variables and updating the lower and upper bounds of the variable. In particular, for every variable $${x_i}\in{[l_i,u_i]}$$, the lower bound $$l_i$$ is updated by splitting the original interval in two subintervals $$I_1=[l_i,M]$$ and $$I_2=[M,u_i]$$, and determining if they can be eliminated.
If both intervals can be eliminated, then the problem is infeasible. If $$I_1$$ can be eliminated but not $$I_2$$, then the lower bound $$l_i$$ is set equal to $$M$$, and the process is repeated. Similarly, if $$I_1$$ cannot be eliminated the process is repeated splitting $$I_1$$ in two. A similar process is also done to update the upper bound $$u_i$$.
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2022-12-08 08:59:27
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https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/GammaZ?action=diff&version=2
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# Changes between Version 1 and Version 2 of GammaZ
Ignore:
Timestamp:
04/06/12 16:33:02 (8 years ago)
Comment:
--
### Legend:
Unmodified
v1 == Gamma/Z interference in e+e- -> mu+ mu- == * Calculate analytically the cross section for %$e^+ e^- \to \mu^+ \mu^-$% as function of the CMS energy, including * Calculate analytically the cross section for $e^+ e^- \to \mu^+ \mu^-$ as function of the CMS energy, including both exchange of a photon and a Z boson. Neglect masses of the external fermions. Compare your result with Software.MadGraph. * The Forward-Backward asymmetry is %$A_{FB}(\theta)=\frac{ d\sigma (\theta) - d\sigma(\pi-\theta)}{ d\sigma (\theta) + d\sigma(\pi-\theta)}$%. $A_{FB}(\theta)=\frac{ d\sigma (\theta) - d\sigma(\pi-\theta)}{ d\sigma (\theta) + d\sigma(\pi-\theta)}$. Plot it as a function of the CMS energy.
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2019-12-10 13:37:02
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https://www.technicalfeeder.com/2021/11/typescript-pick-nested-object/
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# TypeScript Pick nested object
There might be some cases where we don’t want to create a new type/interface that is actually used in a different interface. Assume that we have the following interface.
export interface Person {
name: string;
age: number;
lang: string;
postalCode: number;
city: string;
street: string;
}
}
How can we pick some of these definitions?
## Utility Type Pick
TypeScript offers Utility type called “Pick”. This type picks specified properties’ types from a type/interface.
type PickPerson = Pick<Person, "name" | "age" | "address">;
const person: PickPerson = {
name: "Yuto",
age: 34,
// Type '{ name: string; age: number; lang: string; }' is not assignable to type 'PickPerson'.
// Object literal may only specify known properties, and 'lang' does not exist in type 'PickPerson'.
// lang: "ja",
city: "hoge-city",
postalCode: 12345,
street: "super-street",
},
};
There is an error on “lang” property because it is not specified in Pick type even though it exists in Person interface. This is useful when we want to pick top-level properties but it can’t be used for second-level properties.
“address” property is an object. When we want to get the definition of address property, we might want to try the following code first.
type AddressType = Pick<Person, "address">;
city: "hoge-city",
postalCode: 12345,
street: "super-street",
},
};
But this is not what we want. We want only the 3 properties without “address” property. Dot chain is not available in the string.
// Type '"address.postalCode"' does not satisfy the constraint 'keyof Person'.ts(2344)
type PostalCodeType = Pick<Person, "address.postalCode">;
## Access the property name with brackets
A nested property in interface/type can be accessed with brackets.
type AddressType = Person["address"];
city: "hoge-city",
postalCode: 12345,
street: "super-street",
};
It gets the definition of address property. If we want to get one of the types in the address property, just add additional brackets.
type PostalCodeType = Person["address"]["postalCode"];
const postalCode: PostalCodeType = 123;
// 'errorPostalCode' is declared but its value is never read.ts(6133)
// Type 'string' is not assignable to type 'number'.ts(2322)
const errorPostalCode: PostalCodeType = "123";
The PostalCodeType is number. Therefore, string can’t be assigned.
## Get data type of an Array
Let’s try the same thing to array property.
export interface Department1 {
name: string;
members: {
name: string;
age: number;
}[];
}
type PickNestedPersons = Pick<Department1, "members">;
const person: PickNestedPersons = {
members: [{
name: "Yuto",
age: 34,
}],
};
// Type '"members.name"' does not satisfy the constraint 'keyof Department1'.ts(2344)
// type PickNestedPerson = Pick<Department1, "members.name">;
If using Pick<>, we have to add members property. The basic solution would be to extract the definition into a different type/interface.
export interface Department2 {
name: string;
members: Person[];
}
const person: Person = {
name: "Yuto",
age: 34,
lang: "ja",
city: "hoge-city",
postalCode: 12345,
street: "super-street",
}
};
The Person interface has already been defined and its definition has extra properties that are not necessary. If we want only name/age in Department1, access it with brackets and pass 0 to the index.
type PickNestedPerson = Department1["members"][0];
const person: PickNestedPerson = {
name: "Yuto",
age: 34,
};
We can define Company interface like this. If the Company interface requires all properties in Department interface, we should write in the second way, otherwise, first way.
export interface Company1 {
name: string;
departments: {
name: Department1["name"];
members: Department1["members"];
}[];
}
export interface Company2 {
name: string;
departments: Department2[];
}
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2023-02-06 04:04:18
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https://handwiki.org/wiki/Affine_differential_geometry
|
# Affine differential geometry
Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. The name affine differential geometry follows from Klein's Erlangen program. The basic difference between affine and Riemannian differential geometry is that affine differential geometry studies manifolds equipped with a volume form rather than a metric.
## Preliminaries
Here we consider the simplest case, i.e. manifolds of codimension one. Let MRn+1 be an n-dimensional manifold, and let ξ be a vector field on Rn+1 transverse to M such that TpRn+1 = TpM ⊕ Span(ξ) for all pM, where ⊕ denotes the direct sum and Span the linear span.
For a smooth manifold, say N, let Ψ(N) denote the module of smooth vector fields over N. Let D : Ψ(Rn+1)×Ψ(Rn+1) → Ψ(Rn+1) be the standard covariant derivative on Rn+1 where D(X, Y) = DXY. We can decompose DXY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: DXY = ∇XY + h(X,Y)ξ, where ∇ : Ψ(M)×Ψ(M) → Ψ(M) is the induced connexion on M and h : Ψ(M)×Ψ(M) → R is a bilinear form. Notice that ∇ and h depend upon the choice of transverse vector field ξ. We consider only those hypersurfaces for which h is non-degenerate. This is a property of the hypersurface M and does not depend upon the choice of transverse vector field ξ.[1] If h is non-degenerate then we say that M is non-degenerate. In the case of curves in the plane, the non-degenerate curves are those without inflexions. In the case of surfaces in 3-space, the non-degenerate surfaces are those without parabolic points.
We may also consider the derivative of ξ in some tangent direction, say X. This quantity, DXξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten equation: DXξ = −SX + τ(X)ξ. The type-(1,1)-tensor S : Ψ(M) → Ψ(M) is called the affine shape operator, the differential one-form τ : Ψ(M) → R is called the transverse connexion form. Again, both S and τ depend upon the choice of transverse vector field ξ.
## The first induced volume form
Let Ω : Ψ(Rn+1)n+1R be a volume form defined on Rn+1. We can induce a volume form on M given by ω : Ψ(M)nR given by ω(X1,...,Xn) := Ω(X1,...,Xn,ξ). This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,...,Xn is always equal to ω(X1,...,Xn). Notice that ω depends on the choice of transverse vector field ξ.
## The second induced volume form
For tangent vectors X1,...,Xn let H := (hi,j) be the n × n matrix given by hi,j := h(Xi,Xj). We define a second volume form on M given by ν : Ψ(M)nR, where ν(X1,...,Xn) := |det(H)|12. Again, this is a natural definition to make. If M = Rn and h is the Euclidean scalar product then ν(X1,...,Xn) is always the standard Euclidean volume spanned by the vectors X1,...,Xn. Since h depends on the choice of transverse vector field ξ it follows that ν does too.
## Two natural conditions
We impose two natural conditions. The first is that the induced connexion ∇ and the induced volume form ω be compatible, i.e. ∇ω ≡ 0. This means that Xω = 0 for all X ∈ Ψ(M). In other words, if we parallel transport the vectors X1,...,Xn along some curve in M, with respect to the connexion ∇, then the volume spanned by X1,...,Xn, with respect to the volume form ω, does not change. A direct calculation[1] shows that Xω = τ(X and so Xω = 0 for all X ∈ Ψ(M) if, and only if, τ ≡ 0, i.e. DXξ ∈ Ψ(M) for all X ∈ Ψ(M). This means that the derivative of ξ, in a tangent direction X, with respect to D always yields a, possibly zero, tangent vector to M. The second condition is that the two volume forms ω and ν coincide, i.e. ω ≡ ν.
## The conclusion
It can be shown[1] that there is, up to sign, a unique choice of transverse vector field ξ for which the two conditions that ∇ω ≡ 0 and ω ≡ ν are both satisfied. These two special transverse vector fields are called affine normal vector fields, or sometimes called Blaschke normal fields.[2] From its dependence on volume forms for its definition we see that the affine normal vector field is invariant under volume preserving affine transformations. These transformations are given by SL(n+1,R) ⋉ Rn+1, where SL(n+1,R) denotes the special linear group of (n+1) × (n+1) matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product. SL(n+1,R) ⋉ Rn+1 forms a Lie group.
## The affine normal line
The affine normal line at a point pM is the line passing through p and parallel to ξ.
### Plane curves
Affine normal line for the curve γ(t) = (t + 2t2,t2) at t = 0.
The affine normal vector field for a curve in the plane has a nice geometrical interpretation.[2] Let IR be an open interval and let γ : IR2 be a smooth parametrisation of a plane curve. We assume that γ(I) is a non-degenerate curve (in the sense of Nomizu and Sasaki[1]), i.e. is without inflexion points. Consider a point p = γ(t0) on the plane curve. Since γ(I) is without inflexion points it follows that γ(t0) is not an inflexion point and so the curve will be locally convex,[3] i.e. all of the points γ(t) with t0 − ε < t < t0 + ε, for sufficiently small ε, will lie on the same side of the tangent line to γ(I) at γ(t0).
Consider the tangent line to γ(I) at γ(t0), and consider near-by parallel lines on the side of the tangent line containing the piece of curve P := {γ(t) ∈ R2 : t0 − ε < t < t0 + ε}. For parallel lines sufficiently close to the tangent line they will intersect P in exactly two points. On each parallel line we mark the midpoint of the line segment joining these two intersection points. For each parallel line we get a midpoint, and so the locus of midpoints traces out a curve starting at p. The limiting tangent line to the locus of midpoints as we approach p is exactly the affine normal line, i.e. the line containing the affine normal vector to γ(I) at γ(t0). Notice that this is an affine invariant construction since parallelism and midpoints are invariant under affine transformations.
Consider the parabola given by the parametrisation γ(t) = (t + 2t2,t2). This has the equation x2 + 4y2 − 4xyy = 0. The tangent line at γ(0) has the equation y = 0 and so the parallel lines are given by y = k for sufficiently small k ≥ 0. The line y = k intersects the curve at x = 2k ± k. The locus of midpoints is given by {(2k,k) : k ≥ 0}. These form a line segment, and so the limiting tangent line to this line segment as we tend to γ(0) is just the line containing this line segment, i.e. the line x = 2y. In that case the affine normal line to the curve at γ(0) has the equation x = 2y. In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by ξ(0) = 213·(2,1).[4] In the figure the red curve is the curve γ, the black lines are the tangent line and some near-by tangent lines, the black dots are the midpoints on the displayed lines, and the blue line is the locus of midpoints.
### Surfaces in 3-space
A similar analogue exists for finding the affine normal line at elliptic points of smooth surfaces in 3-space. This time one takes planes parallel to the tangent plane. These, for planes sufficiently close to the tangent plane, intersect the surface to make convex plane curves. Each convex plane curve has a centre of mass. The locus of centres of mass trace out a curve in 3-space. The limiting tangent line to this locus as one tends to the original surface point is the affine normal line, i.e. the line containing the affine normal vector.
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2023-01-27 07:36:45
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https://kitchingroup.cheme.cmu.edu/blog/2015/11/17/New-publication-in-Surface-Science/
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## New publication in Surface Science
| categories: | tags:
Some time ago we published this paper on the coverage dependence of the adsorption energies of atomic adsorbates on different sites xu-2014-probin-cover. One of the concerns in that work was whether van der Waal forces were significant for some adsorbates. Well, now we have addressed that concern in a new paper in Surface Science! In this new work we use the BEEF functional to simultaneously access the impact of van der Waal forces on the adsorption energy trends, as well as do some error analysis on the significance of the coverage dependence. I won't ruin the surprise too much; the good news is that yes van der Waals do influence adsorption of atomic adsorbates on metal surfaces, but the trends are mostly the same! See the paper for more details. Congratulations Hari!
@article{thirumalai-2015-pt-pd,
author = "Hari Thirumalai and John R. Kitchin",
title = {The Role of Vdw Interactions in Coverage Dependent Adsorption
Energies of Atomic Adsorbates on Pt(111) and Pd(111)},
journal = "Surface Science ",
pages = " - ",
year = 2015,
doi = {10.1016/j.susc.2015.10.001},
url =
"http://www.sciencedirect.com/science/article/pii/S0039602815003052",
issn = "0039-6028",
}
org-mode source
Org-mode version = 8.2.10
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2021-07-24 06:42:03
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https://gamedev.stackexchange.com/questions/132162/how-to-check-mouse-coordinates-and-mouse-events
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# How to check mouse coordinates and mouse events
I'm building a simple game where if the mouse's X coordinate is greater than 200, and you click, the screen will turn red. Here's the code:
var mouseX = event.clientX;
var mouseY = event.clientY;
function startGame(){
if(mouseX > 200){
document.body.style.backgroundColor = "red";
}
}
Am I using an incorect way of finding coordinates or am i using an incorrect way of finding mouse down or what? I can't use Jquery btw.
The event should be "mousedown", not "onmousedown". Also, you need to put an event parameter inside your startGame function and declare your variables inside it. Try this:
function startGame(event){
var mouseX = event.clientX;
var mouseY = event.clientY;
if(mouseX > 200){
document.body.style.backgroundColor = "red";
}
}
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2020-01-25 19:31:18
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http://mathhelpforum.com/algebra/105035-rational-expressions-print.html
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# Rational expressions
• September 29th 2009, 12:13 PM
Rheanna
Rational expressions
I've been looking at the examples in the book and can't find any examples on similar to this problem and I'm trying to figure out how I would go about solving it. Thank You
r- r^2-1/r / 1- r-1/r everything is all divided but I don't know how to write the tags in format.
• September 29th 2009, 12:38 PM
masters
Quote:
Originally Posted by Rheanna
I've been looking at the examples in the book and can't find any examples on similar to this problem and I'm trying to figure out how I would go about solving it. Thank You
r- r^2-1/r / 1- r-1/r everything is all divided but I don't know how to write the tags in format.
Hi Rheanna,
Parentheses might help. Is this your rational expression?
$\dfrac{r-r^2-\dfrac{1}{r}}{1-r-\dfrac{1}{r}}$
• September 29th 2009, 12:46 PM
Rheanna
Quote:
Originally Posted by masters
Hi Rheanna,
Parentheses might help. Is this your rational expression?
$\dfrac{r-r^2-\dfrac{1}{r}}{1-r-\dfrac{1}{r}}$
top one is r square 2-1/r
bottom one is r-1/r
it comes out to 1 as the answer but i'm trying to figure out how to come up with that.
• September 29th 2009, 12:50 PM
masters
Quote:
Originally Posted by masters
Hi Rheanna,
Parentheses might help. Is this your rational expression?
$\dfrac{r-r^2-\dfrac{1}{r}}{1-r-\dfrac{1}{r}}$
Quote:
Originally Posted by Rheanna
top one is r square 2-1/r
bottom one is r-1/r
You mean like this:
$\dfrac{r-\dfrac{r^2-1}{r}}{1-\dfrac{r-1}{r}}$
• September 29th 2009, 01:00 PM
Rheanna
yes :)
• September 29th 2009, 01:11 PM
masters
Quote:
Originally Posted by masters
You mean like this:
$\dfrac{r-\dfrac{r^2-1}{r}}{1-\dfrac{r-1}{r}}$
Alrighty then,
$\dfrac{r-\dfrac{r^2-1}{r}}{1-\dfrac{r-1}{r}}=\dfrac{\dfrac{r^2-(r^2-1)}{r}}{\dfrac{r-(r-1)}{r}}=\dfrac{\dfrac{r^2-r^2+1}{r}}{\dfrac{r-r+1}{r}}=\dfrac{\dfrac{1}{r}}{\dfrac{1}{r}}=1$
• September 29th 2009, 01:17 PM
Rheanna
lol, even staring at the problem i'm still lost. :)(Thinking)
• September 29th 2009, 01:38 PM
masters
Quote:
Originally Posted by masters
Alrighty then,
$\dfrac{r-\dfrac{r^2-1}{r}}{1-\dfrac{r-1}{r}}=\dfrac{\dfrac{r^2-(r^2-1)}{r}}{\dfrac{r-(r-1)}{r}}=\dfrac{\dfrac{r^2-r^2+1}{r}}{\dfrac{r-r+1}{r}}=\dfrac{\dfrac{1}{r}}{\dfrac{1}{r}}=1$
Let me do it another way. See if this helps.
Multiply numerator and denominator by r.
$\dfrac{r-\dfrac{r^2-1}{r}}{1-\dfrac{r-1}{r}}$
$\dfrac{r\left(r-\dfrac{r^2-1}{r}\right)}{r\left(1-\dfrac{r-1}{r}\right)}=\dfrac{r^2-r^2+1}{r-r+1}=\frac{1}{1}=1$
• September 29th 2009, 03:51 PM
Rheanna
yeah that r square was confusing me. (Itwasntme) ugh I know I got 0 on this pre test and Thursday is the test test. (Crying)
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2015-10-09 16:26:04
|
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|
https://sunpech.com/2014/06/converting-your-vhs-and-betamax-tapes-to-digital/
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# Converting Your VHS and Betamax Tapes to Digital
I’ve started a new initiative to help my parents convert all their VHS and Betamax tapes over into the digital world. These tapes are those precious family home videos. These are mostly tapes from the 1970s, 80s, 90s, and even part of the early 2000s.
In today’s times we should all be capturing everything in digital format already. From DSLRs, HD camcorders, to even our smartphones. We just need a way to bring the past thirty plus years of our lives into the modern age.
A few years ago, we had a VHS to DVD recorder for the house. But these were and are expensive– $150-200!! Plus, they save to DVD, which may as well join the VHS/Betamax family as being old and outdated! The world is digital now and the future isn’t in CDs, DVDs, or even Blu-ray discs. The present and future is in local hard drives in many of our devices and PCs, which should be syncing to another backup or cloud solution. All these choices are cheaper and better than ever. ## What You Need • A computer running Windows or Mac OS X. • A VCR or Betamax player that works still. • Elgato Video Capture Device (~$80).
• Enough hard disk space on your computer.
Below we have both a VHS and Betamax players.
Uses the standard RCA cables as input/output.
The video capture device. It’s USB.
## Extras
So here’s a quick list of what I recommend as extra if you got the cash or really care about backing up your stuff.
• Drobo 5D (~$575) - For backup and redundancy for your data. • Seagate Barracuda 3 TB HDD (~$100) - I recommend 3 or more, depending on how much you need. Or…
• WD Red 3 TB NAS Hard Drive (~\$125)- Even better, get the NAS optimized hard drives. Just avoid purchasing any green drives as they don’t perform well and even Drobo doesn’t recommend them.
• Handbrake - Software to convert DVDs, CDs, or other digital formats.
The Drobo 5D.
The three internal hard drives, and two unused bays.
## Do It Yourself
So these are my recommendations. I do not recommend you take them to CostCo or some other store to have them convert your tapes to DVDs/online. No! I personally consider the tapes to be too valuable to leave with a store or stranger to convert. You can easily do this yourself with the peace of mind that your valuable memories won’t get accidentally lost or destroyed.
## How it Works
Here’s how the conversion process works for video tapes. You put the tape into the VCR and play them as you would normally. The video/audio feed then connects via the capture card and goes into your computer. Your computer’s software will record all this digitally. The space required varies depending on how long the videos are, but I find that it’s between 500MB for 30 minutes, and 1.7GB for 2 hours of video. The software allows you to trim the beginning and end in case there’s blank or empty space.
A screenshot of the video capture software running on Mac OS X (Mavericks)– pretty straightforward guide.
If you have some DVDs that you want to convert, there’s Handbrake software. What it does is detect the video/audio source files needed to convert into a single video file output.
I recommend naming the files something that makes sense to you. I personally use the following format:
YEAR-MONTH-DAY - DESCRIPTION.mp4
So a video of my Halloween in 1985 would be named: “1985-10-31 - Steven’s Halloween.mp4”
If the month or day is unknown, it’s okay. Not all our videos are timestamped that well. But year and description are the most important. So something like, “1989 - Steven playing in the backyard.mp4” is still good.
This is similar to how I organize my photos in Lightroom. The date is first so it gives you a sense of when it happened. They also align by date when your file manager like Windows Explorer or Mac’s Finder organizes by filename.
The purpose of the Drobo 5D is so you can backup your files and have redundancy. Imagine if you only had an external hard drive to backup your work– and if that failed and you didn’t have another copy anywhere. You might be devastated! Hard drives do fail. It’s not a matter of it, but when.
So with Drobo, if you have two or more hard drives attached, one of the drives could fail and your data would still be safe. Or if there’s a red light, it’ll be warning sign that something is wrong and to replace that hard drive as soon as possible!
Yes, you pay a lot for the Drobo enclosure and proprietary software, but the internal disks are getting cheaper and bigger every year. You can easily add more drives to it and it will reconfigure itself automatically– little to no work on your part!
I hope you found this post useful. As a final note, I’m teaching my parents how to do this– and it’s a bit of a struggle for them since they are computer illiterate. But it should be very doable for you if you born after 1970– i.e you know how to open a program, copy&paste, rename files, etc… then you should be good to go.
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2022-09-24 22:36:31
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https://www.gradesaver.com/textbooks/engineering/computer-science/invitation-to-computer-science/chapter-7-7-3-communication-protocols-practice-problems-page-362/6
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Invitation to Computer Science 8th Edition
Sending a $100-$ character message, 800 bits, across a 100 million bit per second (100 Mpbs) Ethernet takes about $8 \times 10^{-6}$ seconds, or 8 $\mu s$. The the likelihood that one of the other 19 users on the system would transmit during that tiny window of time is quite small, so the probability of a the collision would be small.
Sending a $100-$ character message, 800 bits, across a 100 million bit per second (100 Mpbs) Ethernet takes about $8 \times 10^{-6}$ seconds, or 8 \musec. The the likelihood that one of the other 19 users on the system would transmit during that tiny window of time is quite small, so the probability of a the collision would be small.
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2019-09-20 15:59:11
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http://askmetips.com/standard-error/standard-error-from-standard-deviation.php
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Home > Standard Error > Standard Error From Standard Deviation
Standard Error From Standard Deviation
Contents
The standard error is the standard deviation of the Student t-distribution. All three terms mean the extent to which values in a distribution differ from one another. In a World Where Gods Exist Why Wouldn't Every Nation Be Theocratic? If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively. my review here
Bootstrapping is an option to derive confidence intervals in cases when you are doubting the normality of your data. Related To leave a comment for the author, please The graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16. BMJ 1994;309: 996. [PMC free article] [PubMed]4. However, the sample standard deviation, s, is an estimate of σ. https://en.wikipedia.org/wiki/Standard_error
How To Calculate Standard Error In Excel
Review of the use of statistics in Infection and Immunity. Why does Deep Space Nine spin? Math Calculators All Math Categories Statistics Calculators Number Conversions Matrix Calculators Algebra Calculators Geometry Calculators Area & Volume Calculators Time & Date Calculators Multiplication Table Unit Conversions Electronics Calculators Electrical Calculators As the standard error is a type of standard deviation, confusion is understandable.
National Center for Health Statistics (24). The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. Open topic with navigation Variance, Standard Deviation and Spread The standard deviation of the mean (SD) is the most commonly used measure of the spread of values in a distribution. Standard Error Definition doi:10.4103/2229-3485.100662. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample".
What would you call "razor blade"? In this scenario, the 2000 voters are a sample from all the actual voters. Standard error of the mean Further information: Variance §Sum of uncorrelated variables (Bienaymé formula) The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a http://handbook.cochrane.org/chapter_7/7_7_3_2_obtaining_standard_deviations_from_standard_errors_and.htm Standard error From Wikipedia, the free encyclopedia Jump to: navigation, search For the computer programming concept, see standard error stream.
Retrieved 17 July 2014. Standard Error Of Estimate The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate. The 95% confidence interval for the average effect of the drug is that it lowers cholesterol by 18 to 22 units. The margin of error of 2% is a quantitative measure of the uncertainty – the possible difference between the true proportion who will vote for candidate A and the estimate of
Standard Error Calculator
A natural way to describe the variation of these sample means around the true population mean is the standard deviation of the distribution of the sample means. https://www.r-bloggers.com/standard-deviation-vs-standard-error/ However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and How To Calculate Standard Error In Excel The standard error is most useful as a means of calculating a confidence interval. Standard Error In R The next graph shows the sampling distribution of the mean (the distribution of the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women.
Note: The Student's probability distribution is a good approximation of the Gaussian when the sample size is over 100. this page Whether or not that formula is appropriate depends on what statistic we are talking about. Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?". Standard Error Formula Statistics
set.seed(20151204) #generate some random data x<-rnorm(10) #compute the standard deviation sd(x) 1.144105 For normally distributed data the standard deviation has some extra information, namely the 68-95-99.7 rule which tells us the v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Moments Correction for finite population The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered http://askmetips.com/standard-error/standard-error-of-measurement-refers-to-the-standard-deviation-of.php The standard deviation is computed solely from sample attributes.
As an example of the use of the relative standard error, consider two surveys of household income that both result in a sample mean of \$50,000. Convert Standard Deviation To Standard Error In Excel The standard deviation of the age was 9.27 years. The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean.
Note: The Student's probability distribution is a good approximation of the Gaussian when the sample size is over 100.
The mean of all possible sample means is equal to the population mean. This often leads to confusion about their interchangeability. Recent popular posts Election 2016: Tracking Emotions with R and Python The new R Graph Gallery Paper published: mlr - Machine Learning in R Most visited articles of the week How Standard Deviation Of The Mean Scenario 1.
In fact, data organizations often set reliability standards that their data must reach before publication. The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years. In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the useful reference The standard error is computed from known sample statistics.
Calculations for the control group are performed in a similar way. Warning: The NCBI web site requires JavaScript to function. Relative standard error See also: Relative standard deviation The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage. For the purpose of this example, the 9,732 runners who completed the 2012 run are the entire population of interest.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above Journal of the Royal Statistical Society. Because the 9,732 runners are the entire population, 33.88 years is the population mean, μ {\displaystyle \mu } , and 9.27 years is the population standard deviation, σ.
more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed All journals should follow this practice.NotesCompeting interests: None declared.References1. Similar Worksheets Calculate Standard Deviation from Standard Error How to Calculate Standard Deviation from Probability & Samples Worksheet for how to Calculate Antilog Worksheet for how to Calculate Permutations nPr and For illustration, the graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16.
This often leads to confusion about their interchangeability. Is it Possible to Write Straight Eights in 12/8 Why was Washington State an attractive site for aluminum production during World War II?
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Why Win2PDF? Trusted for over 18 years, Win2PDF is a supported, safe, reliable, and easy to use Print to PDF solution. > Check uni and college websites to see what skills. – paper size (typically a4paper or letterpaper), – if having the text on both sides of the page (twoside) or only on the front (oneside), – if placing the chapter titles only on right pages (openright) or any (openany). WF ED 540, CLASS MEETING 7, RMarkdown, 2016 1. 875 inches wide on 10pt documents. Join the web’s most supportive community of creators and get high-quality tools for hosting, sharing, and streaming videos in gorgeous HD and 4K with no ads. I recommend this engine because: You write in Markdown, a plain text formatting system. ∗ = Not available in the OT1 font encoding. I searched web a lot but didn't see anything that'd help. Opacity = 0. Hi, I have been trying to change the font size in a PDF report written in Rmarkdown but in vain. Turn your analyses into high quality documents, reports, presentations and dashboards with R Markdown. class: center, middle, inverse, title-slide # RMarkdown Basics ### Haley Jeppson, Sam Tyner ### 2018/06/08 --- class: primary # What do you use to write? - Microsoft Word - LaTeX. This Document is incorporated by reference into the following. Some document elements that we are counting on fail to appear. The first reason is that one or more fonts are stored inside PDF document. I can specify font, size, color, border, but there is missing any option how to switch. It will retain all graphics, fonts and formatting. com logo is Elliot Swonger by Jason Ramirez The fonts presented on this website are their authors' property, and are either freeware, shareware, demo versions or public domain. You won't be able to override these field font settings with free tools. An example of the reports produced by these files can be found here. Title: 0172_QuickRef_2. I also read that footers can be added (if included in the file definition 4. R Markdown files permit you to interweave R code with ordinary text to produce well-formatted data analysis reports that are easy to modify. Type Display in the search box, touch or click on Settings, and then touch or click Display. of font types is crucial in the text design process, empirical analyses of reading performance of people with dyslexia has focused more on font size [23, 26] rather than on font type. Download 803 Free Calligraphy Fonts. If you are not familiar with Markdown. If you have any question, please leave the message here:. If you are seeing this message, you probably have an ad blocker turned on. Convert multiple PDFs at once, design workflow automation, and use your current Dropbox folders as input and output location. pdf_document: latex_engine: xelatex can you post this as an issue on the. It don't need OneDrive to sync notes among PC, Android, IPhone. Text = "My Form" ' Set the opacity to 75%. The family spans 42 designs drawn to excel in a variety of print and digital applications. bib file from a collection in the repo folder, and cite using [@xx] (I guess that's pandoc + knitr). The mathematics is done using a version of $$\LaTeX$$, the premiere mathematics typesetting program. This service enables you to rotate PDF documents permanently, free of charge. A significant ideological transition has taken place in the discipline of architecture in the last few years. With R Markdown, you can easily create reproducible data analysis reports, presentations, dashboards, interactive applications, books, dissertations, websites, and journal articles, while enjoying the simplicity of Markdown and the great power of. RENDER & SAVE OUTPUT 6. Arrange and view a large number of fonts which are installed on your computer, applying their various styles to text of your choice Font Viewer renders all installed font families in a scrollable grid view, and allows you to change the size, color, background-color and displayed text, all at the touch of a button. Note the options background and size only apply to. Workflow R Markdown is a format for writing reproducible, dynamic reports with R. TTF) format for Windows & Mac. text offset of text from bar extend extend bar height align logical barsize size of bar fontsize size of text angle angle of text geom one of ’text’ or ’label’ hjust justify text horizontally color color for clade & label, of length 1 or 2 fill fill label background, only work with. Markdown Cheat Sheet. Free PDF Reader supports multi view mode, page rotation, text copy, page snapshot, PDF printing and several advanced features. 26378> not available (Font) size <24. PDFelement is your all-in-one PDF solution. Change the default font size for all text. Upload PDF or design online. You can also change the font size for desktop icons by right-clicking on your desktop and choosing Show View Options. Generally applicable transcriptome-wide analysis of translation using anota2seq. Can't export a pdf for printing? Hi, I'm working in Adobe Indesign CS6 and I can't export a print pdf, just interactive. One possibility to make it smaller is to reduce the font size (e. The usage is rmarkdown. Better BibTeX offers an easy way to export bibliographic records from Zotero as plain. PDF Expert is the best PDF writer for Mac. Is there a way to specify the size of the grid in inches/cm/pixels?. Rmarkdown, landscape, portrait, Landscape in Rmarkdown, portrait and landscape in rmarkdown, rstudio. Select the document you want to convert. How about HTML5 slides? You can actually also try HTML slides, which smell fancy and modern. Rmd files to users. Resources such as publications, research materials, forms, and other tools related to the activities of the California Energy Commission are available. The free version of PDF Architect already allows you to view, rotate, delete and rearrange pages as well as merge multiple documents. How can I make the text in the pdf download larger? It is hard to read in the tiny size it is now. For example, if you set the size of a ggplot figure to large, then fonts etc. Fast and high-quality result. 5 Markdown syntax. Math inside RMarkdown. I searched web a lot but didn't see anything that'd help. The following code example demonstrates how to use the Font constructor and the Size, SizeInPoints, and Unit properties. At the top left, click. All text in the PDF Flip Book will be visible to the search engines after the conversion, and it is crucial if you want your domain to have a higher ranking on the result list. As such, I've had to try and learn how to change the font within RMarkdown. It proceeded to do more harm than good — for all sorts of reasons — but mainly because it is a hugely inefficient way to format text. HTML Click to copy. Its easy-to-use interface helps you to create PDF files by simply selecting the "print" command from any application, creating documents which can be viewed on any computer with a PDF viewer. This wikiHow teaches you how to compress the size of a PDF file, so it takes up less storage space, using Smallpdf. cairo_pdf and (on macOS only) quartz for other devices that can produce PDF. I'm very pleased to finally see font size increasing generically across the web. > (or of the entire document, if it can't be changed for just the table) > > With my actual data, the table is too wide to fit on a page in the pdf output; > perhaps if I reduce the font size I can get it to fit. png, the file venndiagram2-pdf-converted-to. PDF is often a better format to present PowerPoint presentations electronically. Create and enter text into the desired annotation format. Chapter 7 - Fonts This chapter describes basic font information including font characteristics. See the help for a specific high level plotting function (e. download the PDF-file (updated, 0. Here you can find all information about one of the most popular eBook readers on the market called the jetBook. This is an introduction to the use of Markdown with embedded R code to create dynamic documents in multiple formats, e. 6 MB) and fast. A significant ideological transition has taken place in the discipline of architecture in the last few years. ,Nice Gift @ Only £4. This Document is incorporated by reference into the following. Change the color of the text with the fill() function. Changing the font size in a document makes it more attractive to readers. This is an introduction to the use of Markdown with embedded R code to create dynamic documents in multiple formats, e. More details of font families and encodings and especially handling text in a non-Latin-1 encoding and embedding fonts can be found in Paul Murrell and Brian Ripley (2006) Non-standard fonts in PostScript and PDF graphics. XIII and biblical bifolia that serves as cover, focusing especially on the description of this second fragment (= S2). Markdown is a very simple way to format text in such a way that it can be converted automatically into many different formats, including. If the font is not available on the reviewer's computer, font substitution results, even if the fonts are embedded. Here you can find the list with the standard set of fonts common to all versions of Windows and their Mac substitutes, referred sometimes as "browser safe fonts". The rmarkdown file is called by the rscript one time for each unique car name in the subset of the mtcars data. (You may also look into telling rmarkdown to use. md file with knitr. Watchmen FULL TEXT. We let you use a free and simple online file converter that works with a wide range of file types. tex file that gets included in_header , but I get 12pt CMR. With geom_text or annotate in ggplot2, you can set a number of properties of the text. There are two steps to this process I use the knitr package to create a. If the font is not available on the reviewer’s computer, font substitution results, even if the fonts are embedded. The model was iterated 50 times to generate a total of 25 000. syntax, can also use ctrl+alt+i keyboard shortcut. Access, tutorials for quick insight into the most important functions of the current software, downloads to the most current versions of software, patient marketing resoureces and more. I am not sure whether this issue is related to this one from last year. Principles of Design The principles of design describe the ways that artists use the elements of art in a work of art. The choice of font would depend on the type of book as well. VIEW PDFs • Quickly open and view PDF documents. 75 ' Size the form to be 300 pixels in height and width. 1 * SweaveListingUtils + more). You can create a slide show broken up into sections by using the # and ## heading tags (you can also create a new slide without a header using a horizontal rule (---). Consider the size of the font family when selecting a typeface for text. After the image file is generated, you can download it to your hard disk. Invented by Adobe, PDF is now an open standard maintained by the International Organization for Standardization (ISO). This online PDF editor will help you to edit PDF text font size and font color online. 95p !,Martavis Bryant Signed Wilson NFL Full Size Football (TSE COA),New Boston Bruins Mens Size Small S Black/Grey Distressed Print Hoodie MSRP. PDF Architect is the affordable alternative to expensive PDF software. NOTE: The PDF Reflow feature in Word 2013 works best with documents that are mostly text. I made a list of the fonts on my desktop and what they look like. How is a report generated from R code Regardless of which format you use, the basic idea is the same: knitr extracts R code in the input document, evaluates it and writes the results to. class: center, middle, inverse, title-slide # Advanced R Markdown ## Behind the Knit Button ### Yihui Xie, RStudio ### 2017/01/14. Portable Document Format (PDF) is a file format used to present and exchange documents reliably, independent of software, hardware, or operating system. R uses recycling of vectors in this situation to determine the attributes for each point, i. # R Markdown. Download Demo GitHub Project ©Mozilla and individual contributors. Auto-Fill PDF Forms From Data Files AutoMailMerge™ for Adobe® Acrobat® Easily create, secure and email personalized PDF forms. Alternatively, you can right-click the table on Github and choose Inspect Element to view the actual HTML tags for that table. When you create a new document, there is always the need to change font size in PDF. In order to do that we need to use the `{r }. Convert files like Word to PDF. As such, I've had to try and learn how to change the font within RMarkdown. It is run by Sid Steward, author of PDF Hacks and the popular PDF Toolkit. You control the display of the document; formatting words as bold or italic, adding images, and creating lists are just a few of the things we can do with Markdown. And now, it's connected to the Adobe Document Cloud − making it easier than ever to work across computers and mobile devices. The following entries may be used as part or all of the text. The most useful thing about Rmarkdown for us as researchers / data analysists is the ability to embed R code in markdown. This is what I used to control font size and color in an R-markdown file. The PDF to Image converter will convert your PDF file to image file automatically if upload is completed. United States District Court Western District of Washington Honorable Ricardo S. Standard effects are available, including font type, color, and size. If I don't run rmarkdown::render but instead use the "Knit" button in RStudio, the PDF was also produced just fine. Martinez, Chief District Judge William M. > (or of the entire document, if it can't be changed for just the table) > > With my actual data, the table is too wide to fit on a page in the pdf output; > perhaps if I reduce the font size I can get it to fit. This is a quick and easy way to tune the appearance of your document, yet with the price of a large file size (> 700KB) since the whole Bootstrap library needs to be packed in. text 1 ## Tab 2 text 2 ### End tabset Tabset text 1 End tabset Tab 1 Tab 2--- name: My Template — sub-option description citation_package The LaTeX package to process citations, natbib, biblatex or none X X X code_folding Let readers to toggle the display of R code, "none", "hide", or "show" X colortheme Beamer color theme to use X. Formatting text in Markdown has a very gentle learning curve. The PDF is rendered in Helvetica, but at 10 point. eval=c(1, 3, 4) or eval=-(4:5) Text Results. Enter any data, customize the chart's colors, fonts and other details, then download it or easily share it with a shortened url | Meta-Chart. The font CMR10. Loading Unsubscribe from MAC PDF SOLUTIONS? Printing PDFs with best font size - Duration: 3:18. Are you looking for the best pdf editor for mac that functions flawlessly on the new MacOS? This guide list some of the top options available in the market. texi -o example19. 5 centimeter leading. Flipsnack is the easiest way to make interactive brochures, magazines & catalogs. Auto-Fill PDF Forms From Data Files AutoMailMerge™ for Adobe® Acrobat® Easily create, secure and email personalized PDF forms. Online and as Download. Markdown is a very simple way to format text in such a way that it can be converted automatically into many different formats, including. German, French, Spanish. com e-mail address as an attachment and type "convert" (without the quotation marks) in the subject line, the document will be converted to Kindle format and you should be able to use the Aa button to change the font size. PDF (Portable Document Format) is a formatting language developed by Adobe, an extensible page-description protocol that implements the native file format based on PostScript language, uses standard compression algorithm, documents can contain text, graphics, multimedia, custom data types and more. A book created for a 3 hour workshop on rmarkdown. text 1 ## Tab 2 text 2 ### End tabset Tabset text 1 End tabset Tab 1 Tab 2--- name: My Template — sub-option description citation_package The LaTeX package to process citations, natbib, biblatex or none X X X code_folding Let readers to toggle the display of R code, "none", "hide", or "show" X colortheme Beamer color theme to use X. Guidance Document(s):. Have you installed Adobe Acrobat? If yes, try following: Run Adobe Acrobat>Open file>Edit PDF>Use the toolbar to change PDF font size. How about HTML5 slides? You can actually also try HTML slides, which smell fancy and modern. Rnw file, the following code generates:. We don’t ask for your email address, just upload a PDF file from a computer or the cloud and the conversion will take place on our servers. You can create a slide show broken up into sections by using the # and ## heading tags (you can also create a new slide without a header using a horizontal rule (---). We will use it in every homework assignment. This example shows how to use the smallest available font (tiny) in L a T e X and the small caps style. Readdle Team enables technology for millions of people, making them more productive, solving real problems and changing what’s possible with iPhone, iPad and Mac. The converted text may have line breaks in places you don’t want. Maintained by @jdorfman, @mervinej & @XhmikosR. There are some important things to have in mind. The Markdown Guide is a free and open-source reference guide that explains how to use Markdown, the simple and easy-to-use markup language you can use to format virtually any document. 15 Catalina. It focuses on the communication strategies through which this ficció sentimental ’s text (a point of convergence for thematic, metaphorical and figural weaving, anchored to the Tristanian, lyrical-amorous, ausiasmarquiana, cancioneril and Passionist imagery of planctus virginis ) transforms memory and reuses it into an invention. Revised 1995, amended October 1999, May 2000, January, 2005, July, 2005, and November 24, 2015. ← Code Externalization. PDF is a document file format that contains text, images, data etc. The following entries may be used as part or all of the text. Use multiple languages including R, Python, and SQL. In the toolbar at the top of the page, change any of the following options: Font - Click the current font's name, then click the font that you want to use in the drop-down menu. Try fonts from selection of high quality & professional desktop and web fonts. And more text. But the title maintains it's font. Fonts so good you’ll want to send yourself a letter. Also, the html version provides additional features such as changing text size, font, and colors. We use cookies and similar technologies to ensure the correct work of the website, improve your browsing experience and analyze site traffic. Overpass Try it Try Mono 2 00 3 00 4 00 5 00 6 00 7 00 8 00 9 00 italic. If you want to save to a personal computer, you can download this image in full size. Is it possible to reduce the font size of my code markdown here on SO to match that of Programmers. After this figure there exists a space between the figure and the text. And my knowledge of Latex is very limited. When you create a new document, there is always the need to change font size in PDF. The blue underlined text remains, but they are no longer clickable. This wikiHow teaches you how to turn a text, photo, Microsoft Office, or XPS file into a PDF (Portable Document Format) file. Entering text in one column and the image in another would create a similar effect. The following code example demonstrates how to use the Font constructor and the Size, SizeInPoints, and Unit properties. Try it: Recipes, Wikipedia, News. You can also save individual PDFs to other file formats, including text, XML, HTML, and Microsoft Word. And more text. 1001 Free Fonts offers a huge selection of free Calligraphy Fonts for Windows and Macintosh. If you have any question, please leave the message here:. Rmarkdown, landscape, portrait, Landscape in Rmarkdown, portrait and landscape in rmarkdown, rstudio. Now, you should have kra. Starting with R Markdown flexdashboard By Jonathan Scholtes on June 22, 2016 • ( 4) This is a rather straightforward example using R Markdown, flexdashboard and an Excel data source. XIII and biblical bifolia that serves as cover, focusing especially on the description of this second fragment (= S2). Stop hunting down missing icons you need, combining from multiple sets, or finding that company's official logo in a dirty corner of the internet. Unlike other free readers, it enables you to create PDFs, participate in a shared review, and more. You can create a slide show broken up into sections by using the # and ## heading tags (you can also create a new slide without a header using a horizontal rule (---). Below we show how to resize PDF pages online. Some examples of using $$\LaTeX$$ in R Markdown documents. In addition, the font defaults to Arial. By default, text is black and the background grey. INTRODUCTION. The real power and "Ah-ha" moment is when I pull up PDF Annotator in a video conference and my PDF becomes a virtual whiteboard. Notice - font size in preview window depends on hardware and internet browser you use. The file size is relatively small, distracting slide transitions are usually removed, and everyone has a PDF reader. It splits a pdf given a range of page numbers (range mode), every page is saved as a pdf (burst mode), or (odd-even mode), where odd and even pages are saved as two separate pdf files. Author: Evangelos Vlachogiannis Created Date: 2/23/2007 5:56:37 PM. The goal of this tutorial is to introduce you to kableExtra, which you can use to build common complex tables and manipulate table styles. You can print to the PDF Printer driver just as you would any other printer. This is the reference I use when making web pages and I expect you will find it useful too. Blank printable calendar 2019 or other years. Music Paper. In current versions of Adobe symbol fonts it is character 160, so text(x, y, "\xA0", font = 5) may work. RMarkdown is a framework that provides a literate programming format for data science. Font Size: 12 (but this may change based on your paper size and also on the font that you have selected) Line spacing: 1. Text and Symbol Size. – Make sure that URL is selected in the Type drop-down list and write the URL in the URL text field. Most of them are free, while some require a small donation, and some can be purchased in the SimplytheBest Marketplace. Citation management tools allow a user to organize and retrieve information, such as citations for books, articles, and Web sites, by interfacing with library databases. Upload your PDF to us and we'll automatically calculate the price to make you a bound book, in the size and binding of your choice, based on the number of pages in your PDF. Details Whoops! There was a problem previewing Watchmen FULL TEXT. The More Actions button is the three dots on the Microsoft Edge browser at the top right hand corner of the screen. Use it to embed R code and results into slideshows, pdfs, html documents, Word files and more. Boring, zzzzz. If you find any issues or have a suggestion for something we can add, please create an issue, or just make a comment on the Gist. I have been trying to change the font size. Click Export in the blue vertical menu that runs along the left side of your screen. Browser Extension. Convert PDF to Word. Select the text and then use the options in the Properties Toolbar to edit the size, style and font: 3. Welcome to the Simply the Best Fonts archive where you can find a whole lot of wonderful fonts. We believe free and open source data analysis software is a foundation for innovative and important work in science, education, and industry. In this paper we present the rst study that measures the impact of the font type on the reading performance of 48. add these lines to your RMD main text area and see the effects. From time to time you may want to use your Open Document Text (ODT) files in a PDF format. The total allowable width of the text area is 6:5 inches wide by 8:75 inches high. Originating in a displeasure with the ‘starchitecture’ system and the focus on aesthetic innovation, a growing number of architects, emboldened by the 2007–8 economic crisis, have staged a rebellion against the dominant mode of architectural production. Music Paper. 1em) relative to the normal text. Partial Unmodified GNU Ghostscript Source Code (5. This comment has been minimized. One can also cut out the middle-man (Rmd) and generate the exact same HTML, PDF and Word reports using native R scripts. From time to time, users need to copy text from an Adobe PDF. Rmarkdown cheatsheet-2. This Document is incorporated by reference into the following. BibleWorks is a Bible software program for exegesis and Bible study, with extensive Greek, Hebrew, LXX (Septuagint), and English resources. (In other words, the author needs to go back and spend some time working on the pdf formatting. We use cookies and similar technologies to ensure the correct work of the website, improve your browsing experience and analyze site traffic. R Markdown is an authoring format that enables easy creation of dynamic documents, presentations, and reports from R. Or copy & paste this link into an email or IM:. Image filters and changes in their size specified in the PDF are not applied. Table 4: Greek Letters α \alpha θ \theta o o τ \tau β \beta ϑ \vartheta π \pi υ \upsilon γ \gamma ι \iota \$ \varpi φ \phi δ \delta κ \kappa ρ \rho ϕ \varphi \epsilon λ \lambda % \varrho χ \chi. pdf file - just for use in the Virtual Mechanics tutorials. Choose a size from the list, or specify your own. R Markdown is an authoring format that enables easy creation of dynamic documents, presentations, and reports from R. This should not be a hustle in any way. In this post, I will outline several approaches I found searching through Stack Overflow, then explain a new approach using Latex. 6, a single global option size will control the font size of the whole chunk, and this trick is no longer needed. R Markdown supports a reproducible workflow for dozens of static and dynamic output formats including HTML, PDF, MS Word, Beamer, HTML5 slides, Tufte-style handouts, books. texi makeinfo --no-validate --force example19. tex file that gets included in_header , but I get 12pt CMR. Free pdf world maps to download, physical world maps, political world maps, all on PDF format in A/4 size. Opacity = 0. Use a productive notebook interface to weave together narrative text and code to produce elegantly formatted output. If you omit pkg, the default is assumed to be rmarkdown. There are many flavors of Markdown invented by different people, and Pandoc’s flavor is the most comprehensive one to our knowledge. Flinders2, Vernon C. In its simplest form, it looks something like this: h3 {text-shadow: 0. Best Resume. Font support in R is generally not very good. You can select the font, size, text position on the image and add different text for specific frames. If you have questions, comments, or suggestions about our website, department, your personal information, or an individual within the EDD, there are several ways to contact us:. To change the font size of text elements, use cex (short for character expansion ratio). If size() is not used, the window will be given a default size of 100 x 100 pixels. Experts in PDF editing - desktop / server / web. Select Print Dialog Box. A default font and font-size are required for mPDF to function. If the text within the footnote is very long, LaTeX may split the footnote over several pages. Boring, zzzzz. Or copy & paste this link into an email or IM:. Formats for converting from R Markdown to a PDF or LaTeX document. Actually, that drop-down menu seems to just change what’s shown in the header at the top of the file!. (doc, xls, ppt. The PDF is rendered in Helvetica, but at 10 point. Local font files will affect the way websites are displayed on your machine. Free PDF Reader is a free windows application for reading and viewing pdf documents. paste text from a PDF document into another software format. Rnw documents. > (or of the entire document, if it can't be changed for just the table) > > With my actual data, the table is too wide to fit on a page in the pdf output; > perhaps if I reduce the font size I can get it to fit. The font CMR10. This post was produced with R Markdown. With geom_text or annotate in ggplot2, you can set a number of properties of the text. in Basic > Sans serif 5,384,214 downloads (3,526 yesterday) 131 comments 100% Free - 16 On snot and fonts. There's a whole range of rules out there on file sizes depending on what back-office system the court is using to manage the filings it receives. And more text. Customize your PDF Not only can you convert your JPG to PDF, but here are some optional settings to customize your PDF, including page size, margin and page orientation. asm-1 7:42 PM 1. You can use its API or transform images, XML, XSL-FO, HTML, SVG, PCL, XPS, EPUB or simple text files to PDF documents. com provides instant access to powerful support tools, anytime. RMarkdown Basics. txt -s -o example19. 5 centimeter leading. You can correct typos, change fonts and typeface size, adjust alignment, add superscripts or subscripts, and resize text or paragraphs. TinyTex is good one working with Rmarkdown. Foxit Reader. I understand that the size of font is large when you open PDF's with form in Edge. We will use it in every homework assignment. Remove all images or individual images. Is it possible to reduce the font size of my code markdown here on SO to match that of Programmers. convert - similarly to pandoc function in knitr. Both my advisor and I are quite happy with it. Font Size Guide - convert points to inches Converting Points to Inches: 72 points is equal to 1 inch… usually When creating and designing documents for things like custom label printing , which is what we do here at LabelValue, the standard unit of measurement for text is the point (pt). Since pander aims at rendering R objects into Pandoc's markdown all four (multiline, simple, grid, rmarkdown) of Pandoc's formats are supported. Create stunning HTML5 flipbooks online for free. com on 16/11/19. This is the standard for book margins. Page insertion and deletion: Delete unwanted content before the PDF is created. File converter converts files ZIP, PDF, OCR, TXT, FB2, EPUB, DOC, DOCX, RTF, DJVU, HTM, HTML, TIF, TIFF, BMP, JPG, JPEG, JFIF, PNG, GIF, ICO online quickly. Reduce PDF Size a free PDF document compression software, it can quickly compress PDF documents and reduce PDF files size, it supports five different compression settings, you can choose different compression ratio according your own requirements. Form 7 - Notice of appearance ( PDF 9k) ( RTF 31k) Form 8 - Submitting appearance ( PDF 9k) ( RTF 32k) Form 9 - Conditional appearance (PDF 9k) ( RTF 32k) Form 12 - Application for order to show cause ( PDF 12k) ( RTF 43k) Form 17 - Application for removal ( PDF 14k). Administrative Tools Manage user passwords Status Server & Licensing information Ask a question.
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2019-12-12 03:53:49
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https://math.stackexchange.com/questions/3039934/how-can-i-find-a-and-b-if-i-have-half-of-the-matrix-2-times-2-one-eigen
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# How can I find $a$ and $b$ if I have half of the matrix ($2\times 2$), one eigenvector, and one eigenvalue?
How can I find $$a$$ and $$b$$ if I have half of the matrix ($$2\times 2$$), one eigenvector, and one eigenvalue? The matrix is $$2\times 2$$:
$$A=\left( \begin{array}{cc} 6 & a \\ 5 & b \end{array} \right)\,,$$
and the eigenvector is $$(-4,-6)$$ associated to this eigenvalue $$\lambda=6$$.
Sorry for my English. I'm Chilean.
• By definition, what does it mean for $6$ to be an eigenvalue? – Shubham Johri Dec 14 '18 at 22:11
• the 6 is the eigenvalue of the eigenvector (-4,6) – Nelson Aguilera Dec 14 '18 at 22:12
• Use the definition of $\lambda$ i.e. $Av=\lambda v$. You are given $\lambda$ and $v$. – Yadati Kiran Dec 14 '18 at 22:13
Since $$\lambda = 6$$ is an eigenvalue and $$v = [-4 \;| -6]^\mathbf{T}$$ is an eigenvector, then , by definition, it must be :
$$Av = \lambda v \implies \begin{pmatrix}6 & a \\ 5 & b \end{pmatrix}\begin{pmatrix} -4\\-6\end{pmatrix}= 6\begin{pmatrix} -4\\-6\end{pmatrix} \implies \begin{cases} -24 -6a = -24 \\ -20 - 6b = -36 \end{cases}$$
Consider $$u=\begin{pmatrix} 6 \\ 5 \end{pmatrix} \qquad x=\begin{pmatrix} a \\ b \end{pmatrix} \qquad v=\begin{pmatrix} -4 \\ -6 \end{pmatrix}$$ Then your data can be rewritten as $$-4u-6x=6v$$ and therefore $$x=-\frac{2}{3}u-v$$
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2019-06-27 10:44:09
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https://mathoverflow.net/questions/338332/is-there-any-edge-but-not-vertex-transitive-polytope-in-d-ge-4-dimensions
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# Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $$P\subset\Bbb R^d$$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking for polytopes which are edge- but not vertex-transitive. There are infinitely many of these for $$d=2$$, and exactly two for $$d=3$$ (rhombic dodecahedron and rhombic tricontrahedron, see below).
$$\quad\quad$$ $$\quad\quad$$
I do not know a single example for $$d\ge 4$$.
I believe it is easy to see that the edge-graph of such a polytope must be bipartite, and thus, zonotopes might be a good place to start looking. But my constructions fail for $$d\ge 4$$.
• To echo your bipartite point: Wikipedia says "Every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular." – Joseph O'Rourke Aug 14 '19 at 13:51
• @JosephO'Rourke That was my thought. But one has to be careful: the symmetry group of the polytope might be smaller than the one of its edge-graph. "Bipartite" should still hold, but I do not immediately know about "semi-symmetric or biregular". Update: I have read the definition of these terms, and they should probably still hold. – M. Winter Aug 14 '19 at 13:54
• Both your examples in d=3 are the convex hull of the union of a regular polyhedron and its dual (appropriately scaled). Have you tried this construction in d=4? – Yoav Kallus Aug 14 '19 at 14:02
• @YoavKallus That's a great idea. Thank you. What else comes to my mind now is to let a vertex grow out of each facet of a (regular) polytope and at some point the original edges will vanish in the inside of a facet. This might give examples. This corresponds to your idea of looking for the "appropriate scaling". And no, have not tried this, but will do so now. – M. Winter Aug 14 '19 at 14:08
• @YoavKallus At least the construction using the 4-cube and the 4-crosspolytope does not work (no matter the scaling, their are at least two orbits of edges). – M. Winter Aug 15 '19 at 7:24
The answer is No, there are no other such polytopes, as I was able to show in this recent preprint.
Theorem. In dimension $$d\ge 4$$, an edge-transitive polytope is vertex-transitive.
The idea is as follows: first, show that every edge-transitive polytope $$P$$ that is not vertex-transitive has the following three properties:
1. all edges of $$P$$ are of the same length,
2. $$P$$ has an edge in-sphere, and
3. the edge-graph of $$P$$ is bipartite.
Call a polytope with these three properties bipartite. One then tries to classify these polytopes instead. This is easier, because every face of a bipartite polytope is again bipartite (not true for edge- or vertex-transitive polytopes).
The second step is to deal with all inscribed bipartite polytopes. It is not hard to see that these are zonotopes. By a result from another preprint of mine (see also this question), inscribed zonotopes with all edges of the same length are vertex-transitive. We can therefore exclude all the inscribed bipartite polytopes.
In the third step one classifies all the 3-dimensional non-inscribed bipartite polyhedra. This is quite tedious. Here is one example of a polyhedron which satisfies 2. and 3., but fails to have all edges of the same length. The deviation is so miniscule, that it cannot be spotted visually.
The result is then that there are only two such polyhedra: exactly those that I already mentioned in the question.
The final step is then to show that no 4-dimensional non-inscribed bipartite polytope can be built if we can use only these two polyhedra as facets. This uses a straight-forward argument on dihedral angles (see also Nick's answer).
If you consider a tiling of 3-space to be a 4-dimensional polytope, then the Rhombic dodecahedral honeycomb would work.
Other possibilities are limited by the potential 3-faces. Because every edge has one endpoint in each of two vertex orbits, the 2-faces must all have evenly many sides.
If the edge-transitivity descends to the 3-faces, then the 3-faces must be among the 9 isotoxal polyhedra: the five Platonic solids, the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, or the rhombic triacontahedron. The only ones of these with only even-length faces are the 3-cube, the rhombic dodecahedron, and the rhombic triacontahedron. With dihedral angles of 90°, 120°, and 144° respectively, these can only build up the 4-cube, the cubic tiling of 3-space, and the above-mentioned rhombic dodecahedral honeycomb. (A subgroup of the 4-cube's symmetry group acts in an edge-transitive but not vertex-transitive manner; you can color alternate vertices in two colors).
On the other hand, perhaps the 3-faces are not isotoxal: this occurs if, for some pair of edges $$e$$ and $$e'$$ of a 3-face $$G$$, every symmetry mapping $$e$$ to $$e'$$ also maps $$G$$ to a different 3-face incident to $$e'$$. In this case we can still say that the 3-cells are equilateral polyhedra with all even-length faces. There's also fairly strong requirements on the vertex figures, which must be vertex-transitive.
You can 1-subdivide any regular polytope to obtain edge-transitive but not vertex-transitive bipartite graph. More generally, you can take an incidence graph between k-faces and r-faces in a regular polytope. By incidence I mean putting an edge if one contains the other.
If you don't want a geometric property, e.g., convexity, then you can easily do so by taking any favourite finite group of yours. Namely, take a group G, take two subgroups H1 and H2 that have different indices, and consider the incidence graph with cosets, i.e., gH1 and gH2 are adjacent. The group G acts transitively on each side and on edges, but one cannot map a vertex to the other side.
The face-incidence example is indeed a special case of this algebraic construction, by taking two parabolic subgroups that correspond to faces.
• Sorry, but I do not understand how any of these gives a convex edge-transitive polytope. E.g. the first part: I cannot just subdivide the edges of a regular polytope, I have to change the geometry for that too, and by that, probably create other edges. – M. Winter Aug 20 '19 at 15:30
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2020-10-30 20:25:18
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https://pt.overleaf.com/learn/how-to/How_to_jump_to_the_source_code_from_the_typeset_PDF_(SyncTeX)
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Overleaf enables you to go from a location in the typeset PDF to the corresponding location in your LaTeX source code, and vice-versa. Note: This feature will not work if your project’s main .tex file is stored inside a folder (i.e., it must be placed at the top (root) level of your project). Several other possible causes are listed here. You can either: double-click on the PDF which causes Overleaf’s editor to scroll to the corresponding location in the source code, or use the arrow keys that sit on the divider between the source code panel and the preview panel, to jump from source-to-preview, or preview-to-source: If you don’t see those arrows, or double-clicking the PDF doesn’t work, you need to make sure your project’s PDF viewer is set to “Built-in” and not “Native”. Here is a short video showing how to set your project’s PDF viewer and demonstrating Overleaf’s source/PDF synchronization features. SyncTeX: The TeXnology behind this Overleaf feature Overleaf’s “jump to source code location” and “jump to PDF location” capability is implemented using SyncTeX, a standard feature that is incorporated in all TeX engines. Readers interested in a brief introduction to SyncTeX can find out more in this Overleaf article.
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2021-08-02 10:16:04
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https://accountingexplained.com/financial/equity/retained-earnings
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# Retained Earnings
Retained earnings (also known as accumulated earnings) is a component of shareholders equity which represents the amount of net income left-over with the company since its incorporation after periodic distribution to shareholders in the form of dividends. Retained earnings belong to the shareholders of the company and the company’s board of directors can decide to pay them out as dividends, in part or whole.
For a company that is continuously making losses, retained earnings is replaced by accumulated losses, which is an equity component representing the total amount of loss made by the company since its incorporation. However, it is quite possible that a company may have retained earnings in spite of a net loss for a particular period. This is because retained earnings is sum of net income or loss over more than one periods.
Opening retained earnings are adjusted for any changes in accounting policies and accounting errors. It is because changes in accounting policies and rectification of accounting errors are required to be reflected in the financial statements retrospectively, which means that historical asset, liability, equity, revenue and expense figures are changed as if the new policy has been always applicable or the accounting error never took place. Since comparative income statement is presented for only one year, changes to prior period revenue and expenses are reflected in opening retained earnings.
## Formula
Opening retained earnings Add/(Less) Adjustment due to changes in accounting policies and errors Add/(Less) Net income (loss) for a period Less Dividends Equals Closing retained earnings
## Statement of retained earnings
Statement of retained earnings is a report that reconciles the retained earnings of a company at the start of an accounting period to retained earnings at the end of the accounting period. It reports figures for any adjustment to opening retained earnings, net income or net loss for the period and cash dividends or stock dividends (i.e. bonus shares).
## Example & journal entries
GJ Coffees, Inc. retained earnings as at 1 January 2014 were $20 million. During the year, the company generated net income of$8 million and declared dividends of $5 million. The external auditors of the company identified an accounting error dating back to 2007. Closing inventories were overvalued as at 31 December 2007 by$2 million. Applicable tax rate is 30%. Prepare statement of retained earnings for GJ Coffees, Inc. as at 31 December 2014.
Solution
Retained earnings as at 1 January 2014 $20.0 million Less: adjustment on account of rectification of accounting error$1.6 million Add: net income for the period $8.0 million Less: dividends$5.0 million Retained earnings as at 31 December 2014 $21.4 million At the end of financial year 2007, inventories were overvalued by$2 million, which means that the cost of goods sold was simultaneously understated. Undervaluation of cost of goods sold in turn overstated the profit before taxes by $2 million, overstated taxes by$0.6 million ($2 million * 0.3) and overstated both net income and closing retained earnings for 2007 by$1.4 million. In order to rectify the error, the opening retained earnings must be reduced by this amount.
Adjustment for rectification of error is journalized as follows:
Retained earnings $1.4 million Income tax payable$0.6 million Inventories $2 million Transfer of net income to retained earnings during the closing process involves the income and expense summary account: Income & expense summary$8 million Retained earnings $8 million Assuming the company generated net loss of equal amount, the journal entry would be exactly opposite: Retained earnings$8 million Income and expense summary $8 million Dividends are recognized as follows: Retained earnings$5 million Dividends payable \$5 million
Written by Obaidullah Jan, ACA, CFAhire me at
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2017-11-22 16:33:02
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http://susam.in/notes/long-non-text-breaking-float-in-ie/
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## Long non-text breaking float in IE
If there are two HTML div elements placed side-by-side using float: left and float: right, a long HTML element in one of the div elements breaks the floating property of the other in Internet Explorer 6.
Here is a sample page that demonstrates this issue: http://susam.in/files/notes/css-hacks/long-nontext-and-sidebar/long-nontext-and-sidebar.html. Here is the CSS used for both the HTML div elements.
#long {
width: 154%;
height: 20px;
border: 1px solid #000000;
background: #008000;
}
#id1 {
float: left;
width: 65%;
border: 1px solid #000080;
background: #ffcc00;
}
#id2 {
float: right;
width: 30%;
border: 1px solid #008000;
background: #ccccff;
}
In each output shown below, the box on the left has its ID as id1 and the one on the right has its ID as id2. The green box within the first box has its ID as long.
Output in Firefox 3.6
The sidebar with its ID as id2 is intact in Firefox even though the green div element has spilled outside the id1 div element. However, the layout breaks in Internet Explorer 6 as can be seen below.
Output in Internet Explorer 6
This issue may be caused by any type of long non-text content in one of the div elements. The long green div element here is only an example. Another such examples could be a very long image.
The fix is simple. Overflowing content should be clipped with overflow: hidden style in the div element where such long non-text content is expected.
Here is a page that demonstrates the fix: http://susam.in/files/notes/css-hacks/long-nontext-and-sidebar/long-nontext-and-sidebar-fixed.html.
Output in Firefox 3.6
Output in Internet Explorer 6
[css]
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2017-09-26 18:17:04
|
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http://www.openproblemgarden.org/category/noy_marc
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# Noy, Marc
## 3-Colourability of Arrangements of Great Circles ★★
Author(s): Felsner; Hurtado; Noy; Streinu
Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
\begin{conjecture} Every arrangement graph of a set of great circles is $3$-colourable. \end{conjecture}
Keywords: arrangement graph; graph coloring
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2017-06-28 03:51:15
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http://math.stackexchange.com/questions/121274/finding-y-from-y-sqrt5x2y-3
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# Finding $y$ from $y'=\sqrt{5x+2y-3}$.
I'd love your help with finding the function $y$ from the following differential equation:
$y'=\sqrt{5x+2y-3}$.
I tried to use $z=5x+2-3$, so $z'=5+2y'$ , and $y'=\frac{z'}{2}-2.5$
and from the equation $\frac{z'}{2}-2.5=y'=\sqrt z$, and then $z'=2 \sqrt z+5$, so $\frac{dz}{2 \sqrt z +5} = dx$, but using integration here is difficult and won't lead me to $y$.
I tries to use substitution in other ways like $z=\sqrt{5x+2y-3}$, or $z^2= 5x+2y-3$ but then again I got stuck in the middle.. Any suggestion?
Thanks a lot
-
Integration here is simple $$\int\frac{dz}{2\sqrt{z}+5}=\{t=\sqrt{z}\}=\int\frac{2t dt}{2t+5}=\int\left(1-\frac{5}{2t+5}\right)dt=t-\frac{5}{2}\int\frac{d(2t+5)}{2t+5}=$$ $$t-\frac{5}{2}\ln(2t+5)+C=\sqrt{z}-\frac{5}{2}\ln(2\sqrt{z}+5)+C$$
Yeah, but then I'll have a problem with expressing $y$. – Jozef Mar 17 '12 at 11:39
In the theory of differential equtions it is not always possible to express $y$ in terms of $x$, so it is not required and by convention relation between $x$ and $y$ that you have obtained is called solution of differential equation – Norbert Mar 17 '12 at 12:02
@Jozef the last expression can then be converted to $x$ terms like $\sqrt{5x-1}-\frac{5}{2}\ln(2\sqrt{5x-1}+5)+C$, so you are getting the final result in the form $y=f(x)$ – Kirthi Raman Mar 17 '12 at 13:17
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2016-05-24 20:15:24
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http://codeforces.com/blog/entry/71594
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### I_love_tigersugar's blog
By I_love_tigersugar, 11 months ago,
1255A - Changing Volume
Author: UncleGrandpa925. Prepared by UncleGrandpa925
Tutorial
1255B - Fridge Lockers
Author: prof.PVH ft. MikeMirzayanov. Prepared by UncleGrandpa925
Tutorial
1255C - League of Leesins
Author: MikeMirzayanov. Prepared by UncleGrandpa925
Tutorial
Author: Prof.PVH. Prepared by Prof.PVH
Tutorial
Source code
1254B1 - Send Boxes to Alice (Easy Version)
Author: MofK. Prepared by UncleGrandpa925
Tutorial
Source code
1254B2 - Send Boxes to Alice (Hard Version)
Author: MofK. Prepared by MofK and UncleGrandpa925
Tutorial
Source code
1254C - Point Ordering
Author: ngk. Prepared by ngk
Tutorial
Source code
1254D - Tree Queries
Author: prof.PVH Prepared by prof.PVH
Tutorial
Source code
1254E - Send Tree to Charlie
Author: MofK Prepared by: MofK
Tutorial
Source code
Tutorial of Math and Games
• +144
» 11 months ago, # | 0 Auto comment: topic has been updated by prof.PVH (previous revision, new revision, compare).
• » » 11 months ago, # ^ | -40 when 601 round will be rated
» 11 months ago, # | -12 Why my submissions are skipped? :/
• » » 11 months ago, # ^ | -14 Same code was submitted by other handle.
• » » » 11 months ago, # ^ | 0 How can I know who did it?
• » » » » 11 months ago, # ^ | 0 cf will send an email did you use Ideone ?
» 11 months ago, # | -27 orz
» 11 months ago, # | +10 For Problem D I just sorted the adjacency list by size of the subtree (then flatten it with euler tour). That way even if nodes are heavy I can update all the subtrees in $sqrt(N)log(N)$ since it will have at most $sqrt(N)$ different value in sizes and I can update several subtrees of the same size in one single range update (because they are in one complete range in the euler tour) and you want to add the same value $\frac{1}{n}d(n-sz)$ to all of them.
• » » 11 months ago, # ^ | +9 Hacked. Each update for a subtree size should call only 1 Fenwick Tree update instead of 2. Or you can do this.
• » » 11 months ago, # ^ | 0 I did the same thing and was getting TLE, but then I realized that I can optimize the worst case if I batch update queries for the same node, so if there is 1 node that has sqrt(2*n) different values, worst case will be Q/2 modifies which should reduce the constant by 2 and make it work. It now passes.https://codeforces.com/contest/1254/submission/65470890
• » » 11 months ago, # ^ | 0 I use similar idea about different subtree size. But I also divede the queries into sqrt(Q) parts then get a O(NsqrtN) off-line algorithm: We use an array to modify the updates and calculate the presum after each part. Then for each query with type 2, we just need to calculate the updates from the queries with type 1 in the same part. For all Nsqrt(N) such pairs of queries, we sort them by dfs order using bucket sorting. Then we insert the pairs into corresponding vertex's vector, then for each vertex we calculation the values of all the queries corresponding to it one by one with the vector containing its subtrees with different size and their dfs orders, just like a merge sorting.Code:65993177
» 11 months ago, # | +9 prof.PVHFor editorial of Div2E2/Div1B2, shouldn't it be min(Si%k, k - Si%k) instead min(Si%k, (k-Si)%k)?And btw, anyone has the link for the markups used for blogs/comments on CodeForces?
• » » 11 months ago, # ^ | +9 It was fixed, thank you!
• » » 11 months ago, # ^ | 0 does the idea of grouping k chocolates into their coordinates meridian work? I did it on contest, but got wa on pretests of E1 (but not on pretests of E2) and not sure if it is a bug or wrong solution.
• » » » 11 months ago, # ^ | 0 I think it will work, but then the hard part here is you don't know the number of candies in each part to determine the median.
• » » » » 11 months ago, # ^ | +8 I did it in the contest (65365892). Basically, you just divide all candies into groups of $K$ and move each group into its median box. Just need to be careful because the number of candies at a time can exceed $K$.
• » » » » » 8 months ago, # ^ | +5 So E1 editorial solution can be done for E2 also with some modifications (or a lot of modifications :D). Am I right?
• » » » » » 2 months ago, # ^ | 0 How can we prove that median will give optimal answer
» 11 months ago, # | 0 Notice that we can map the given rectangle to an array. In other words, there exists a path in the rectangle that goes through every cell exactly once. can someone explain this part
• » » 11 months ago, # ^ | 0 Sorry for the word choice. I have rewritten the tutorial for easier reading.
• » » » 11 months ago, # ^ | 0 Thanks ,I get it now
» 11 months ago, # | ← Rev. 2 → +3 I don't understand why the round was rated for me. On what basis did you decide that the mistake in problem B did not affect me ? This mistake completely ruined my contest. I wasted 100 min in B. And when I up-solved C and D I could solve them quickly. this could have been a very good contest for me but now I lost 50 points. and yet you refuse to make round unrated
• » » 11 months ago, # ^ | +9 did you submit the form?
• » » » 11 months ago, # ^ | 0 I am exactly in his position. Wasted a lot of time for b. I filled the form,but with no effect.
• » » » » 11 months ago, # ^ | 0 Me too.
• » » » 11 months ago, # ^ | 0 Yes. I did. and I explained in detail what happens even sent a message to MikeMirzayanov
» 11 months ago, # | ← Rev. 4 → +8 How I did D:Call a vertex heavy if it has more than $T$ neighbours. For updates at $u$, first update the complement of the subtree rooted at $u$. Then, if $u$ is heavy, just keep track of the update at $u$. Otherwise, update the subtrees rooted at the children of $u$. This is $O(T\log(N))$.For queries at $u$, first query the value at $u$. Then, jump upwards to the next heavy vertex until we reach the root, and update the answer with the values at these heavy vertices. This is $O(\log(N)+N/T)$. Setting $T=\sqrt{N/\log(N)}\approx93$ gives $O\left(Q\sqrt{N\log(N)}\right)$ total. Runs in under half a second on system tests but I found an input that makes it run in 2.6 seconds.(seems like a less intelligent version of this $O(N\log(N))$ solution: https://codeforces.com/blog/entry/71534?#comment-559222)
» 11 months ago, # | +122 An alternative solution for D. Tree Queries.It's obvious that for each "modify" operation we can spend $O(degree_x \log n)$ time naively updating the answer using BIT in the tree-dfs order. And now we can get some improvements. For each node we initially choose one of its sons of the largest size. For a "modify" we only update its father's "subtree" and the heaviest son's subtree. And for each query, we not only calculate the answer in the BIT, but also jump up to the query node's ancestors that haven't updated the query node yet and calculate the extra answer.It can be shown that there are at most $O(\log n)$ such ancestors to deal with. So the total complexity is $O((n+q) \log n)$, with quite a small constant.Code: 65403262
• » » 11 months ago, # ^ | +26 Beautiful solution!
» 11 months ago, # | +8 My solution to Tree Queries: For a set of updates, we can easily calculate their contributions to each node in $O(n)$ time -- for each node $u$ we get the sum of $d$ of all operations that $v=u$, then iterate all neighbours of $u$. Since we do not need to support online query, the updates can be done with difference in linear complexity. Also it's easy to tell an update's contribution to a specific node $u$: we just need to get the size of the subtree of $u$ when we root at $v$. This can be done in $O(n\log n) - O(1)$ complexity. We calculate the contribution to each node every $\sqrt n$ operations, and when query on a node we consider the latest $\sqrt n$ updates' contribution to it. This works in $O(n\log n+n\sqrt n)$ time.
» 11 months ago, # | 0 Can someone explain in Div2E2, how can we assume that if we are taking k-S[i]%k from i+1 th element then it will not become negative ( does it have enough candy to give). The author has said something related to this doubt as 'S[i]>S[i+1]'. Can someone give an intuitive proof of th this. Thanks in advance.
• » » 4 months ago, # ^ | 0 lucifer_1502 Did you get it? If yes, then can you please explain? I've spent 4hrs on this problem, but couldn't come to a conclusion.
» 11 months ago, # | ← Rev. 6 → 0 In div2 E1 or (div1 B1) how to prove that maximum number of divisors of a number less than 10^5 is 240 (It is given in editorial but no proof has been provided) .Thanks !
• » » 11 months ago, # ^ | +10 Hi. To prove that, you can either factorise all intergers from $1$ to $10^5$ to count the number of divisors it has, or just google search for the list of highly composite numbers. For estimation purposes, the number of divisors a number $n$ has is about $O(n^{1/3})$
» 11 months ago, # | +8 In problem B1, it suffices to iterate through only the prime divisors (taking as k) of the total number of chocolates.
• » » 11 months ago, # ^ | 0 Absolutely
» 11 months ago, # | 0 Can anyone explain this for me :"A number ≤10^5 has at most 240 divisors". I tried to create number that can have most divisors but all I found was 128 divisors (120120). Thanks in advance.
• » » 11 months ago, # ^ | +1 So that's why they said "at most" XD... Moreover 120120 is greater than 10^5 XD...
• » » 11 months ago, # ^ | +1 I checked... it's actually 83160 with 128 factors.
• » » » 11 months ago, # ^ | 0 okay, got it, thanks!
• » » » 11 months ago, # ^ | 0 In div2 E1 or (div1 B1) how to prove that maximum number of divisors of a number less than 10^5 is 240 (It is given in editorial but no proof has been provided) ?
• » » 11 months ago, # ^ | +1 Sorry, it is in 83160 with 128 divisors. I mistook this problem for the original problem(which has a constraint of 1e6)
• » » » 11 months ago, # ^ | 0 thanks, but I still wonder how to get maximum number of divisors for number with value that does not exceed some specified boundary.
• » » » » 11 months ago, # ^ | ← Rev. 3 → +1 Please see my comment above. There also exists a way of backtracking. You just need to backtrack for a sequence of $k_1, k_2,...k_n$ such that $(k_1+1)*(k_2+1)*...*(k_n+1)$ is maximum, ${p_1}^{k_1}*{p_2}^{k_2}*...*{p_n}^{k_n}$ $\le N$ and $k_1 \ge k_2 \ge k_3 ... \ge k_n$ But on practical I think referring to the highly-composite number list is a more convenient way
• » » » » » 11 months ago, # ^ | 0 thanks, I will note it for later use.
» 11 months ago, # | 0 @prof.PVHpls explain this..STATEMENT : "Now it is clear that we will divide the array into segments, such that sum of each segment is divisible by K, and move all the chocolates pieces in that segment into one common box."lets say my input array is : [ 6 7 9 7 6 7 7 ] now the sum of the array is 49 ... and optimal divisor(k) for this array would be 7 ... Why do I need to move all the candies into common box ?we can see that we need to pass one candy from index '2' to index '0' and one candy from index '2' to index '4'... and that would 4 seconds... and we are not moving all the candies to any common box..so can please explain that statement
• » » 11 months ago, # ^ | +3 That statement is about the easy version, where each element is 0 or 1.
• » » » 11 months ago, # ^ | 0 sam721I realised that I after posting the comment :p ... my bad, I opened both questions, and I thougth I was reading E1, but I was actually reading the E2 :p .
» 11 months ago, # | 0 I still don't understand tutorial for feeding chicken? Does anyone have better explanation?
• » » 11 months ago, # ^ | 0 You can solve the 1xn problem assigning equal or almost equal quantities of rice fields to each chicken, for example, if there are 32 rice fields and 6 chickens, you go 6, 6, 5, 5, 5, 5, giving the non-rice fields in between the rice fields to the corresponding chicken.Now "cut" and "stretch" the original rectangle into one long strip or array of 1x(r*c). You can do that in several ways, like zig-zag, spiral, etc, and solve like above.
• » » » 11 months ago, # ^ | 0 I guess the hardest trick for me was to understand that it doesn't matter how many fields belong to one chicken. It is all about rice per chicken.Zig Zag was another trick.
» 11 months ago, # | 0 There is a way that cost $O((n+q)\log n)$ for problem D.Here it is.
» 11 months ago, # | 0 please help me in solving problem B1 SEND BOXES TO ALICE(EASY VERSION). it's giving me wrong ans in test case 7...
» 11 months ago, # | +10 Problem E2 is very nice ! Thank you!
» 11 months ago, # | +10 Div2 Task E2 is awesome.
» 10 months ago, # | 0 fridge locker is completely a case of circular rotation
» 8 months ago, # | +3 In problem E2, is there a proof (or at least help me understand) for this fortunately true statement : " The only concern is that: is there any scenario when there exists some i such that $S_i > S_{i+1}$, which is indeed a violation? Fortunately, the answer is no. ".? Thanks <3.
• » » 4 months ago, # ^ | 0 _LNHTD_ Did you get it? If yes, then can you please explain? I've spent 4hrs on this problem, but couldn't come to a conclusion.
» 6 months ago, # | ← Rev. 3 → 0 Div2. E2/ Div1. B2:"But if you have solved the problem to this phase already, you can easily realize that we only need to consider prime k, because if k is composite then picking any prime divisors of it will lead to either a better result or an equal result."Why and how?UPD: Understood. If all $a_i$ are to be made divisible by some composite say p*q (p and q are primes), then they are also made to be divisible by p and q, at (in the worst case) the same cost. So it is sufficient to only consider the primes (as they guarantee an equal, if not better, answer).
• » » 4 months ago, # ^ | 0 Can you please explain in Div2E2, how can we assume that if we are taking k-S[i]%k from i+1 th element then it will not become negative ( does it have enough candy to give). The author has said something related to this doubt as 'S[i]>S[i+1]'. Can you give an intuitive proof of this? Thanks in advance.I've already spent hours but couldn't get to any conclusion. the_hyp0cr1t3
• » » » 4 months ago, # ^ | 0 If $S_i \% k \le (k - S_i \% k)$, then it is optimal to move $S_i \% k$ chocolates from $S_i$ to $S_{i+1}$.Otherwise $(k - S_i \% k) < S_i \% k$. It is optimal to move $(k - S_i \% k)$ chocolates from $S_{i+1}$ to $S_i$. We know that $S_{i+1} \geq S_i$, so there are at least $S_i \% k$ chocolates in $S_{i+1}$. And $(k - S_i \% k) < S_i \% k$. So there are always enough.
• » » » » 4 months ago, # ^ | 0 the_hyp0cr1t3 I tried to prove it this way, I do not understand where am I getting wrong
• » » » » 4 months ago, # ^ | 0 How does Si+1>=Si implies that there are at least Si%k chocolates in Si+1?
• » » » » » 4 months ago, # ^ | 0 $S_i$ has at least $S_i \% k$ chocolates, so $S_{i+1}$ also has at least that many.
• » » » » » » 7 weeks ago, # ^ | ← Rev. 2 → 0 Si+1 is greater than Si%k but how can we prove that it is greater than Si+k-Si%k because will be taking chocolates out of ai+1 i.e. si+1-si?
» 5 months ago, # | 0 In Div1 B1 why moving all the ones to $temp[(i+i+p-1)/2]$ is optimal than to moving ones to $(temp[i]+temp[i+p-1])/2$ ??
» 4 months ago, # | 0 Hi, MofK, question regarding following statements:The only concern is that: is there any scenario when there exists some i such that Si>Si+1, which is indeed a violation?Since S[i]=sum(a[0]..a[i]), thus S[i+1]-S[i]=a[i+1]. If S[i]>S[i+1], then a[i+1]=S[i+1]-S[i]<0. To my understanding, all a[i], either before or after any operation, should be positive. If I were correct, then there should be no case that S[i]>S[i+1].Anything I am missing here?
• » » 4 months ago, # ^ | ← Rev. 2 → 0 happy15 But how do we make sure that ai+1 will remain positive after the most optimum choice of operation for that case when Si%k>k/2
» 4 months ago, # | 0 We can enforce the solution to iterate each Si from 1 to n−1 and always choose to decrease if there is a tieIn fact, even if we choose to increase for every tie, we can enforce a non-decreasing sequence. We just have to fix and follow it.
• » » 4 months ago, # ^ | 0 Can you please explain in Div2E2, how can we assume that if we are taking k-S[i]%k from i+1 th element then it will not become negative ( does it have enough candy to give). The author has said something related to this doubt as 'S[i]>S[i+1]'. Can you give an intuitive proof of this? I've already spent hours but couldn't get to any conclusion. Thanks in advance. mshiladityam
» 4 months ago, # | ← Rev. 2 → 0 I am getting WA on test case 7 for the problem 1254B1 - Send Boxes to Alice (Easy Version), the code seems to be logically correct to me, where am I getting wrong?: 86285300UPD: Fixed
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2020-10-29 14:01:13
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|
https://trac-hacks.org/ticket/2927
|
Opened 9 years ago
Closed 8 years ago
# import tool for wiki-based blogs (or documentation to help in development of one)
Reported by: Owned by: ckcin pacopablo normal FullBlogPlugin trivial pacopablo 0.11
### Description
I think that many users of the TracBlogPlugin and SimpleBlogPlugin may be discouraged from switching based on the lack of an import tool. I personally feel that FullBlogPlugin is a much nicer tool with many features to offer. I personally plan on using FullBlogPlugin when I upgrade my server to 0.11 but I would also like to carry along my wiki blog post with out having to repost them all. So basically from my point of view, either a tool or documentation to help build a tool would be great.
Also while I've only been involved with Trac as a User, I'd be happy to help develop and test any tool. I am currently only using 0.11 as a testing system as we upgrade our website to use Trac as the main engine. So any testing won't currently affect my operations.
### comment:1 follow-up: ↓ 2 Changed 9 years ago by osimons
Thanks for positive feedback on the plugin - glad you like it!
As for importing, others have also hinted at it at IRC but no firm request before now. I haven't had the need to import myself, so that is why I have no code for it. Should be simple enough to do, and would perhaps just need to supplement with some properties to be able to copy all information from the wiki to full blog posts.
It won't be part of the running plugin itself anyway, so I'm fine with either attaching it to the plugin home page or adding it to a new /contrib directory inside the plugin - the latter perhaps being the best idea.
If you want to pick this up and develop a basic import tool, I'll be happy to help you out.
### comment:2 in reply to: ↑ 1 Changed 9 years ago by pacopablo
• Cc pacopablo added; anonymous removed
OK, I think I've admitted defeat wrt TracBlogPlugin. As such, I'm trying to migrate http://pacopablo.com and am working on this migration script.
The migration does look fairly straight forward, though #2956 needs resolving.
I think that putting the script in a contrib directory and then putting instructions on the wiki is the best way forward.
I'll attach the script when I'm done with it
### Changed 9 years ago by pacopablo
Script for migrating TracBlogPlugin to FullBlogPlugin
### comment:3 follow-ups: ↓ 4 ↓ 6 Changed 9 years ago by pacopablo
Attached the first rev of the migration script. It worked for my site. Would like feedback regarding features.
Also, can we get this into /contrib ?
### comment:4 in reply to: ↑ 3 ; follow-up: ↓ 5 Changed 9 years ago by ckcin
First, Thanks! From what I can tell, everything worked great.
All my posts were migrated and my tags were nicely converted to categories but as expected still are accessible through the Tags cloud. Delete-only worked perfect as well.
This all said, for the most part my blog entries with TracBlogPlugin have only been been simple Twitter-like one line entries. I'll try to make some more interesting blog posts and test the tool again a little later.
### comment:5 in reply to: ↑ 4 Changed 9 years ago by ckcin
Continuing ckcin:
I've been playing some new blog entries, the only issues I've found is that attachments don't transfer. This said, I think it is more of a warning statement to users vs an actual issue.
### comment:6 in reply to: ↑ 3 ; follow-up: ↓ 7 Changed 9 years ago by guyer@…
Attached the first rev of the migration script. It worked for my site. Would like feedback regarding features.
When I attempt to run migrate-tracblog.py, I get
Unable to insert 2007_10_16_08.54 into the FullBlog
Unable to insert 2007_12_13_09.13 into the FullBlog
The script quits after this, without comment, even though I have a lot more blog entries.
Oddly, enough, if I examine the database with sqlite3 after this, I do find
INSERT INTO "fullblog_posts" VALUES('2007_12_13_09.13' ...
so the second item is inserted, even though it says it wasn't.
If I have the exception handler in insert_blog_post() print the error, I get
cannot commit transaction - SQL statements in progress
googling for this error message turned up a number of hits, including http://mail.python.org/pipermail/python-list/2005-May/323328.html, but that's about pysqlite 2.0.2, and I'm running 2.4.1:
>>> import trac.db.sqlite_backend as test
>>> test._ver
(3, 4, 0)
>>> test.have_pysqlite
2
>>> test.sqlite.version
'2.4.1'
Even though that message seems obsolete, on a whim I tried commenting out cnx.commit() and adding some diagnostic print statements. I can see that script then "handles" all of my blog entries without further error, but obviously nothing actually gets inserted into the database. I'm completely ignorant wrt SQL, so I have no idea where to move the .commit() that's more appropriate. I also have no idea what a "DML statement" is, but it appears that "INSERT" is one, so that message may be a complete red herring.
I'm running Python 2.5, Trac 0.11rc2, Apache 2.2.8 under Mac OS X 10.5.3.
### comment:7 in reply to: ↑ 6 ; follow-up: ↓ 8 Changed 9 years ago by pacopablo
• Owner changed from osimons to pacopablo
• Status changed from new to assigned
Even though that message seems obsolete, on a whim I tried commenting out cnx.commit() and adding some diagnostic print statements. I can see that script then "handles" all of my blog entries without further error, but obviously nothing actually gets inserted into the database. I'm completely ignorant wrt SQL, so I have no idea where to move the .commit() that's more appropriate. I also have no idea what a "DML statement" is, but it appears that "INSERT" is one, so that message may be a complete red herring.
I'm running Python 2.5, Trac 0.11rc2, Apache 2.2.8 under Mac OS X 10.5.3.
I can't really see why that statement would be causing issues. All it's doing is committing the INSERT.
You could try the following diff:
• ## migrate-tracblog.py
old (name, version, title, body, epochtime(publish_time), epochtime(version_time), version_comment, version_author, author, categories)) cnx.commit() except: print("Unable to insert %s into the FullBlog" % name) cnx.rollback() cnx.close() return cnx.commit() cnx.close() def Main(opts):
Should be functionally equivalent, though.
I mostly use PostgreSQL, would you be able to provide me with a sample sqlite db that fails?
### comment:8 in reply to: ↑ 7 Changed 9 years ago by guyer@…
I can't really see why that statement would be causing issues. All it's doing is committing the INSERT.
Agreed. I was hacking around pretty blindly.
You could try the following diff:
Now it spews
2008-06-11 14:46:09,353 Trac[migrate-tracblog] DEBUG: Error loading wiki page 2007/10/16/08.54
Traceback (most recent call last):
File "research5/migrate-tracblog.py", line 119, in Main
categories)
File "research5/migrate-tracblog.py", line 79, in insert_blog_post
cnx.commit()
OperationalError: cannot commit transaction - SQL statements in progress
to trac.log for seemingly all of my blog entries (not just the two that got reported in the original version).
For what it's worth, in both the original and patched versions, the trac.log entries ends with
2008-06-11 14:46:09,449 Trac[ticket] DEBUG: SELECT * FROM (SELECT id, keywords, COALESCE(keywords, '') AS fields FROM ticket WHERE status != 'closed') s WHERE (fields LIKE %s) ORDER BY id
2008-06-11 14:46:09,450 Trac[tags] DEBUG: SELECT bp1.name, bp1.categories, bp1.version FROM fullblog_posts bp1,(SELECT name, max(version) AS ver FROM fullblog_posts GROUP BY name) bp2 WHERE bp1.version = bp2.ver AND bp1.name = bp2.name AND (bp1.categories LIKE %s) ORDER BY bp1.name
I mostly use PostgreSQL, would you be able to provide me with a sample sqlite db that fails?
I'll mail it to you privately.
### Changed 9 years ago by mfisk@…
With bugfixes to avoid SQL errors
### comment:9 Changed 9 years ago by mfisk@…
I just attached a second version of the script with some fixes to the database code so that it actually works for me. This successfully migrated a pretty large blog for m.
### comment:10 follow-up: ↓ 14 Changed 9 years ago by risto.kankkunen@…
Using Python2.4.4 and pysqlite2.3.2 in Debian Etch I got the same error message when trying to migrate blog posts:
OperationalError: SQL logic error or missing database
I could repro the problem just using pysqlite: if you loop with one cursor and then try to commit changes to the same connection inside the loop (while the original cursor is still active), you get the same error:
cur1 = con.cursor()
for r in cur1:
cur2 = con.cursor()
cur2.execute(...)
con.commit() # <-- this fails
I made the following changes to get the script working:
@@ -56,11 +56,10 @@
""" Return seconds from epoch from a datetime object """
return int(time.mktime(t.timetuple()))
-def insert_blog_post(env, name, version, title, body, publish_time,
+def insert_blog_post(cnx, name, version, title, body, publish_time,
version_time, version_comment, version_author,
author, categories):
""" Insert the post into the FullBlog tables """
- cnx = env.get_db_cnx()
cur = cnx.cursor()
try:
cur.execute("INSERT INTO fullblog_posts "
@@ -70,11 +69,9 @@
(name, version, title, body, epochtime(publish_time),
epochtime(version_time), version_comment, version_author,
author, categories))
- cnx.commit()
- except:
- print("Unable to insert %s into the FullBlog" % name)
- cnx.rollback()
- cnx.close()
+ except Exception, e:
+ print("Unable to insert %s into the FullBlog: %s" % (name, e))
+ raise
def Main(opts):
""" Cross your fingers and pray """
@@ -87,6 +84,7 @@
req = Mock(perm=MockPerm())
blog = tags.query(req, ' '.join(tlist + ['realm:wiki']))
+ cnx = env.get_db_cnx()
for resource, page_tags in blog:
try:
page = WikiPage(env, version=1, name=resource.id)
@@ -110,7 +108,8 @@
else:
title = name
body = fulltext
- insert_blog_post(env, name, version, title, body,
+ print "Adding post %s, v%s: %s" % (name, version, title)
+ insert_blog_post(cnx, name, version, title, body,
publish_time, version_time,
version_comment, version_author, author,
categories)
@@ -121,7 +120,13 @@
except:
exc_info=True)
- continue
+ print "Failed to add post %s, v%s: %s" % (name, version, title)
+ print "Undoing back all changes"
+ cnx.rollback()
+ return 1
+
+ cnx.commit()
+ cnx.close()
return 0
I'll attach the fixed version of the script here.
### comment:11 Changed 9 years ago by risto.kankkunen@…
Bummer, I hadn't refreshed this page and hadn't noticed that mfisk@… had pretty much solved the problem the same way. I added some diagnostic output to the script, so I'll attach it anyway, but the version by mfisk@… should work as well.
### Changed 9 years ago by risto.kankkunen@…
Avoid 'SQL logic error or missing database'
### comment:12 Changed 9 years ago by risto.kankkunen@…
I just found out that neither migrate-tracblog.2.py nor my version of the script works with the -d option:
Traceback (most recent call last):
File "migrate-tracblog.2.py", line 168, in ?
rc = Main(options)
File "migrate-tracblog.2.py", line 122, in Main
page.delete()
File "/tmp/python-stage/Trac-0.11-py2.4.egg/trac/wiki/model.py", line 111, in delete
db.commit()
pysqlite2.dbapi2.OperationalError: SQL logic error or missing database
I fixed the problem by collecting the matching resources into a list, thus completing the SELECT before there is an attempt to commit.
### Changed 9 years ago by risto.kankkunen@…
Avoid 'SQL logic error or missing database' even with -d option
### Changed 9 years ago by risto.kankkunen@…
Migrate also the blog attachments
### comment:13 Changed 9 years ago by anonymous
• Priority changed from lowest to normal
### comment:14 in reply to: ↑ 10 ; follow-up: ↓ 17 Changed 8 years ago by pacopablo
I could repro the problem just using pysqlite: if you loop with one cursor and then try to commit changes to the same connection inside the loop (while the original cursor is still active), you get the same error:
OK, I understand this, however, I don't see how my original script would be doing this, as I am opening and closing a connection every time I migrate a page. I guess this could possibly be related to the trac connection pooling. Perhaps it doesn't close the cursor and when I get call get_db_cnx() I am getting back a connection with an already open cursor.
I'll commit the latest version of the migration script that you provided as it does work. However, I would really like to understand why my version doesn't work with SQLite.
### comment:15 follow-up: ↓ 16 Changed 8 years ago by pacopablo
• Resolution set to fixed
• Status changed from assigned to closed
(In [4062]) Added migrate-tracblog.py script. Closes #2927
### comment:16 in reply to: ↑ 15 Changed 8 years ago by guyer@…
(In [4062]) Added migrate-tracblog.py script. Closes #2927
This script resolve the problems I was having earlier. Thanks.
### comment:17 in reply to: ↑ 14 Changed 8 years ago by anonymous
OK, I understand this, however, I don't see how my original script would be doing this, as I am opening and closing a connection every time I migrate a page. I guess this could possibly be related to the trac connection pooling.
Sorry for not responding earlier... I'm replying based on my vague recollection, but I think the problem was that tags.query() returns a generator that is actually an open cursor. And because of the connection pooling your cnx refers to the same connection the tags module got. The same phenomenon happened with the WikiPage class.
### comment:18 follow-up: ↓ 19 Changed 8 years ago by osimons
Hmm. Interesting. I haven't been involved in this at all, but that is an interesting topic with 2 open cursors - one an open generator and the other updates/changes data. According to the pool code, as this executes in the same thread, you will get the same connection returned for both.
I suppose the behaviour for each db type will be different depending on what amount of caching is done in cursor, and what it needs to keep fetching from data as it iterates - and that changes while iterating. Postgres, SQLite and others may all give different behaviour.
Does it work if you consume the generator first, and then work on the data? Try wrapping the use of the generator in a list(), like list(generator_func(stuff)).
### comment:19 in reply to: ↑ 18 Changed 8 years ago by anonymous
Does it work if you consume the generator first, and then work on the data? Try wrapping the use of the generator in a list(), like list(generator_func(stuff)).
That is exactly the change I made between migrate-tracblog3.py and migrate-tracblog4.py.
### comment:20 Changed 8 years ago by risto.kankkunen@…
Sorry for writing anonymously by accident, it was me who wrote comment:17 and comment:19.
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2017-01-18 04:55:50
|
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|
https://brilliant.org/problems/ambiguity-of-small-and-large/
|
# Ambiguity of Small and Large
Algebra Level 4
Let the sum of the zeros of $$5x^5=x^4$$ be $$r$$.
Evaluate $$p+q$$, when $$\dfrac pq=3r^2+4r^3+5r^4+6r^5+\cdots$$ and $$p,q$$ are coprime positive integers.
×
|
2018-09-22 02:19:45
|
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|
https://dave.thehorners.com/tech-talk/programming/314-greasemonkey
|
Dave Horner's Website - Yet another perspective on things...
54 guests
Rough Hits : 4157013
how did u find my site?
nature of God
There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.
--Charles Robert Darwin
\begin{bmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & 0 & \vdots \\ \vdots & 0 & \ddots & 0\\ 0 & \ldots & 0 & 1_{n} \end{bmatrix}
# Greasemonkey
Wednesday, 27 June 2007 18:46
Dive Into Greasemonkey
< Prev Next >
|
2020-06-03 12:10:20
|
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|
http://community.boredofstudies.org/238/extracurricular-higher-level/380749/i-forgot-clever-way-doing-these-integrals.html
|
# Thread: I forgot the clever way of doing these integrals...
1. ## I forgot the clever way of doing these integrals...
$\int \frac{\sin^2 x - 4\sin x \cos x + 3\cos^2 x}{\sin x +\cos x}dx$
2. ## Re: I forgot the clever way of doing these integrals...
Originally Posted by leehuan
$\int \frac{\sin^2 x - 4\sin x \cos x + 3\cos^2 x}{\sin x +\cos x}dx$
$\noindent 4 + \sin^2{x} - 4\sin{x}\cos{x} + 3\cos^2{x} - 4 \equiv 4 - 3\sin^2{x} - 4\sin{x}\cos{x} - \cos^2{x} \\\\ \equiv 4-(\sin{x}+\cos{x})(3\sin{x}+\cos{x})$
3. ## Re: I forgot the clever way of doing these integrals...
Originally Posted by leehuan
$\int \frac{\sin^2 x - 4\sin x \cos x + 3\cos^2 x}{\sin x +\cos x}dx$
If numerator were: sin2x + 4sinxcosx + 3cos2x things would be much easier.
In this case numerator = (sinx + 3cosx)(sinx + cosx). But as it stands, I have not figured out how. Alternatively, were the denominator (sinx - cos x) it'd also be simple.
The numerator as it stands = (sinx - cosx)(sinx - 3cosx)
4. ## Re: I forgot the clever way of doing these integrals...
There are currently 1 users browsing this thread. (0 members and 1 guests)
#### Posting Permissions
• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•
|
2018-08-19 11:15:54
|
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|
https://lists.nongnu.org/archive/html/help-texinfo/2015-12/msg00005.html
|
help-texinfo
[Top][All Lists]
Re: [help-texinfo] trying to make a unicode math char appear in all form
From: Paul Jakma Subject: Re: [help-texinfo] trying to make a unicode math char appear in all formats Date: Fri, 11 Dec 2015 10:23:52 +0000 (GMT) User-agent: Alpine 2.20 (LFD 67 2015-01-07)
```On Thu, 10 Dec 2015, Gavin Smith wrote:
```
```The following works
\input texinfo
@c @tracingmacros=1
@iftex
@macro precb{}
@math{\\prec}
@end macro
@end iftex
@ifnottex
@macro precb{}
≺
@end macro
@end ifnottex
@bye
```
Note the backslash before prec in the macro body has to be escaped with another backslash. I inverted the nesting of the macro and conditional here.
```
Aha :)
```
```Note also that you can't do @macro prec because @prec is already
defined as a control sequence.
```
```
Ah..
```
```The "@tex" and "@end tex" lines have to be on lines by themselves. Any
backslashes in a macro body should be doubled, as well.
```
```
```
Just adding newline to make that happen seems to break the inline format macro. So..
```
a @precb{} b
```
Seems to do the trick, just /one/ little issue is that it doesn't seem to allow you to format the whole thing as math in one go, e.g.:
```
@math{a @precb{} b}
Makes stuff unhappy.
Is there a way to make that work?
Thanks very much! :)
regards,
--
Paul Jakma address@hidden @pjakma Key ID: 64A2FF6A
Fortune:
Anybody can win, unless there happens to be a second entry.```
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2021-06-18 23:33:54
|
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|
https://www.rdocumentation.org/packages/Rcmdr/versions/2.5-3/topics/AuxiliarySoftware
|
# AuxiliarySoftware
0th
Percentile
##### Install R Commander Auxiliary Software
Install Pandoc and LaTeX to increase the capabilities of the R Commander.
Keywords
misc
##### Details
The capabilities of the R Commander can be enhanced by installing additional software. The R Commander will work without this software but some features will not be activated. The following addiitional software can conveniently be installed via the R Commander Tools > Install auxiliary software menu. The resulting dialog box will take you to websites where the software can be downloaded. This menu item will only be displayed if one or more of these software packages are missing.
Here are the details:
• Pandoc: The Pandoc documentation-conversion software is used by the R Commander to generate HTML (web), PDF, and Word files from the editable R Markdown document that is created by default during an R Commander interactive session. Pandoc is required by the rmarkdown package, which, along with the knitr package, performs these conversions. In the absence of Pandoc, R Markdown documents in the R Commander are processed by the older markdown package, which is capable only of producing HTML output. Pandoc is available from http://johnmacfarlane.net/pandoc/installing.html.
On Windows systems, Pandoc installs into a non-standard location in your user directory, typically C:\Users\<your user id>\AppData\Local\Pandoc\, and then places this subdirectory on your system path. You may have to reboot for the change to your path to take effect, and I have found it necessary on two Windows 10 systems to re-run the Pandoc installer, first to uninstall Pandoc, and then to re-install it before it would be work.
• LaTeX: The LaTeX technical-typesetting system is required by the R Commander for PDF output from R Markdown or knitr documents produced during interactive R Commander sessions. In the absence of LaTeX, direct PDF output is unavailable. Complete LaTeX systems are available for the various platforms that support R and the R Commander, including MikTeX from http://miktex.org/download for Windows systems; MacTeX from http://www.tug.org/mactex/ for MacOSX; and various sources (see http://latex-project.org/ftp.html) for Linux/Unix systems.
Commander
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2020-07-08 10:57:03
|
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https://crunchnumbers.net/zero-coupon-bond-value/
|
A zero-coupon bond is a type of bond that doesn’t make coupon payments. This type of bond is issued with a big discount to its face value.
At the time of maturity, the bondholder receives the face value of the bond, which means that the current price has to be lower than the face price. The investor’s earnings come entirely from the gain on redemption because there are no coupon payments.
Bonds can be held until maturity date or they can be sold on the secondary markets (markets where previously issued financial instruments (including bonds) are sold and bought). Because the only payment is made at the maturity, prices of zero-coupon bonds fluctuate more significantly that prices of coupon bonds.
The more an investor has to wait until the maturity date, the lower the price of the bond should be. This happens because of the time value of money, according to which, the same amount of money available at the present time is worth more than the same amount available in the future because money available at the present time can be used to create more money.
### How to calculate the price of a zero-coupon bond?
Price of the zero-coupon bond is calculated much easier than a coupon bond price since there are no coupon payments. It is calculated as:
$P = \frac{M}{(1 + r)^{n}}$
Where P is the current price of a bond, M is the face or nominal value, r is the required rate of interest, n is the number of years until maturity.
### Example of price of a zero-coupon bond calculation
Let’s assume an investor wants to make 10% of return on a bond. The face value of the bond is $10,000. The bond is redeemed in 5 years. What price the investor would pay for this bond? • M =$10,000
• r = 10%
• n = 5
$P = \frac{10,000}{(1 + 0.10)^{5}} = \6,211.18$
The price of the bond should be $6,211.18 if the investor wants to make 10% of return. ### How to calculate YTM of a zero-coupon bond? YTM (yield to maturity) is an annual income level or profitability, which investors gain by buying a bond or other fixed-interest security at the current market price and holding it in their portfolio until maturity. Yield to maturity for zero-coupon bonds is calculated as: $YTM = \sqrt[n]{ \frac{Face\;value}{Current\;price} } - 1$ ### Example of YTM of a zero-coupon bond calculation Let’s assume an investor wants to buy a zero-coupon bond and wants to evaluate what YTM of this bond would be. • The face value of the bond is$10,000.
• The price of the bond is \$9,100.
• There are 2 years until maturity.
$YTM = \sqrt[2]{ \frac{10,000}{9,100} } - 1 = 4.83\%$
YTM of this bond is 4.83%. Which means the investor would receive 4.83% of the nominal value every year until maturity.
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2020-06-05 04:39:52
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https://tz.hisvoicetoday.org/2215-led-lifespan-light-emitting-diode-lifetime-amp-mtbf.html
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# LED Lifespan: light emitting diode lifetime & MTBF
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Unlike many other semiconductor devices, LEDs have a limited operation lifespan or lifetime.
The LED lifespan or LED lifetime is not infinite and performance degrades over time with light levels falling gradually.
In view of their long lifespan, LEDs are considered as reliable light sources, both as indicators and for lighting, but their lifespan as well as other factors like failures and the MTBF need to be considered when using them in any design.
## LED MTBF
For any component or system, the MTBF is the Mean Time Between Failures. The MTBF is the elapsed time which is predicted between inherent failures of a component or system during its operation.
The MTBF is a figure used in calculations for the reliability of items of equipment. In order to be able to calculate the MTBF of the equipment, it is necessary to know the MTBF of the individual components, e.g. the LED MTBF in this case.
The failure rate for a component, and the MTBF are linked. MTBF can be calculated as the inverse of the failure rate if it is assumed that there is a constant failure rate, which is not unreasonable as a first order assumption.
$\mathrm{MTBF}=\frac{\mathrm{Hours of operation}}{\mathrm{Number of failures}}$
The MTBF figures are often quoted in the manufacturers data sheets. However the MTBF can be considerably reduced by operating components close to their rated limits. Hostile environments such as high temperature and vibration also reduce the MTBF.
However when run within their limits, the LED lamps have a long lifetime, and do not fail very often.
LED light reduces over time. This form of LED lifespan or LED lifetime is particularly important for applications such as lighting where it is important that the light is maintained above a certain level.
A term called lumen deprecation is used to describe this aspect of the LED performance.
The LED lifetime or LED lifespan is the time to when the light output falls to a given level. The LED life expectancy may be quoted in the format L70 or L50, for the life to when the light output falls to 70% or 50% of the original value respectively.
The LED lifetime, i.e. the LED operational life is generally defined in the following terms:
L70% = Time to 70% of illumination (lumen maintenance)
L50% = Time to 50% of illumination (lumen maintenance)
The L70 value was chosen because a power LED industry group called the Alliance for Solid-State Illumination Systems and Technologies, ASSIST, undertook tests which demonstrated people generally did not notice a gradually diminishing LED light output until it had dropped by 30% of its original brightness, i.e. to 70% of initial light output. This then gave rise to the L70 figure. However for non-critical areas the L50 figure may also be used.
As a rough guide, most LEDs intended for lighting applications offer L70 values of 50,000 to 60,000 hours, although with improved manufacture of the high brightness LEDs used for lighting, some manufacturers are now quoting figures of 100 000 hours.
## Factors affecting LED lifespan
There are a number of factors that affect the useful LED lifetime. By ensuring that the LED is protected from adverse conditions it is possible to ensure the maximum lifetime is maintained.
• Temperature: One of the major issues in ensuring the maximum life is obtained from a LED is keeping the temperature down. Excess temperature will considerably shorten the life. To prevent the LED chip running over temperature there are a number of elements that can be included within the design
• Good thermal path from LED chip to mount: It is necessary to ensure that the heat can be effectively removed from the LED semiconductor itself. This is the first step in ensuring the LED junction temperature does not rise to high and adversely affect the LED lifetime.
• Good bonding between LED and external mount: It is necessary to ensure that the LED package is effectively bonded to the element on which it is mounted. The thermal resistance should be as low as possible, possibly using thermal mounting grease t ensure complete contact.
• Good heatsink: In order that heat is removed effectively from the overall assembly the heatsink on which the LED is mounted should have a low thermal resistance. It should also be located so that heat will flow away from the heatsink. For LED lighting, this is particularly important because often lamps will be located within small light fittings and this will not aid cooling, and hence the LED lifetime will be reduced.
• LED drive level: To obtain the best LED lifetime, the LED should be driven well within its ratings. Overdriving a LED will drastically reduce its lifetime, although it will increase the light output.
• Power supply: The power supply needs to match the light emitting diode for optimum LED life expectancy. Not only should the voltage be regulated, but the current also needs to be closely controlled to ensure the LED does not run outside its ratings, or even too close to its maximum ratings.
• Environment: General conditions such as vibration, and temperature extremes - even when not operating - place mechanical stresses on the diode which will reduce the LED lifetime. Ideally, a LED should be operated within a stable dry environment. When this is not possible, a shorter LED lifetime should be anticipated.
Although it may appear obvious at first sight that the LED life should be as long as possible, this may not always be the main requirement. It is possible that in some cases light output is more important than LED lifetime, and in this case it may be permissible to overdrive the LED to obtain the additional light. Additionally budgetary constraints may limit the inclusion of more effective thermal management, and in this case a decision can be made to balance LED life expectancy against the cost.
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2022-05-21 16:05:07
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https://www.physicsforums.com/threads/socket-problem-in-python.922561/
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# Python socket problem in python
Tags:
1. Aug 12, 2017
Hi everyone, I have just started learning sockets in python and I have made the following code:
Code (Python):
import socket
HOST = ''
PORT = 8888
def main():
while 1:
s = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
s.bind((HOST, PORT))
s.listen(1)
while 1:
data = conn.recv(1024)
if ( data == "blabla"):
conn.send("works")
else:
conn.send("nope")
conn.close()
s.close()
if __name__ == '__main__':
main()
so obviously, when I connect to that server and send "blabla", I should normally get "works" as an answer, but somehow, I always get the "nope" like if the if condition was not respected. Can you please help me to solve this ??
PS : I use python 3.6 in Linux.
2. Aug 12, 2017
### Ibix
What does the server think it's getting? A debugger is the Right Way to do that, but adding a print instruction would probably be a start.
Hint: Don't just print the string, add a "$" at the beginning and end so you see any whitespace characters. 3. Aug 12, 2017 ### ShaddollDa9u I have tried to put print(data) before the if condition. It seems that the server is getting ---> b'quit\r\n' How do I fix it ?? PS: if I add$, the following code wouldn't be considered as a comment ??
4. Aug 12, 2017
### Ibix
I'd guess your client program isn't doing what you think it's doing. Interesting that the first character in the output is a b - what if you type something else? Is it just sending the first character? You haven't posted your client code, though, so that's only a guess.
I meant print("#"+data+"#"). Doesn't have to be # characters if you want to use something else. It doesn't matter anyway, as you've obviously got something other than a trailing newline problem.
5. Aug 12, 2017
All the data seems to be received as " b'*my data*\r\n' ". I didn't make a client code, I connect to the server by the Linux command "telnet localhost 8888".
The weirdest thing is that when I put print("#"+data+"#") as you said, I have the following error :
Traceback (most recent call last):
File "ip.py", line 37, in <module>
main()
File "ip.py", line 22, in main
print("#"+data+"#")
TypeError: must be str, not bytes
This is getting really weird lol
6. Aug 12, 2017
### Ibix
I missed the apostrophes in there. The initial b is telling you that what follows between the apostrophes is raw bytes, not a string.
Convert it to a string using str() and strip the newline with the strip() method.
7. Aug 12, 2017
I have converted now in str() but still getting " b'*my data*\r\n' ". What arguments do I have to put in strip to keep only my data?
8. Aug 12, 2017
### Ibix
9. Aug 12, 2017
Now I have tried to put data = data.decode() and it seems that I don't have the " b'*my data*\r\n' " thing anymore, but still, the if condition is note satisfied. Here is my code now :
Code (Python):
import socket
HOST = ''
PORT = 8888
def main():
while 1:
s = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
s.bind((HOST, PORT))
s.listen(1)
while 1:
data = conn.recv(1024)
data = str(data.decode())
print(data)
if(data == "blabla"):
conn.send("works")
else:
conn.send("nope".encode()) #It seems that bytes are required for conn.send, not str
conn.close()
s.close()
if __name__ == '__main__':
main()
10. Aug 12, 2017
### NTL2009
I think you need to post the client code. We are only seeing half of what we need.
I've written socket code in python, but I'd have to look to refresh myself on the details. One thing though, I see you are coding " if ( data == "blabla"): ", that may be your problem. What is coming back looks to have leading/trailing chars, in addition to your data (" b'*my data*\r\n'"). So change your code to see if your string contains "blabla", instead of is equal to "blabla". Or if the placement of your string is predictable (I think it will be), look only at that sub-string for a match.
edit - just saw your recent post - instead of telling us "condition not satisfied", you need to tell us what was found. That's the key to why it wasn't satisfied. You (and we) are in the dark w/o that info.
What was the output of " print(data) "?
11. Aug 12, 2017
### Ibix
Right. Because the data you're sending includes trailing newlines, so "blabla\r\n" doesn't match "blabla". The strip() method of str removes trailing newlines without any arguments (at least it does in python 2) add a strip() and you should be good.
12. Aug 12, 2017
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2017-08-22 13:48:20
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https://library.kiwix.org/bitcoin.stackexchange.com_en_all_2021-04/A/question/69177.html
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## What is the meaning and application of the apostrophes in derivation paths of BIP32 / BIP44?
1
Can we please clarify if there is any difference (because apparently some technical documentation explicitly requests to include apostrophes) between a derivation from extended private key (xprv) or public key (xpub) of a child subkey, where m = root (xprv or xpub).
m/44/0/0/0 and m/44'/0'/0'/0
What are the meaning of the apostrophes and how / when should they be applied? Do they change anything in a derivation path -- will the derived child subkey be different with or without them?
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2022-01-17 10:18:08
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http://www.biodiversity-science.net/EN/10.3724/SP.J.2010.547
|
Biodiversity Science ›› 2010, Vol. 18 ›› Issue (6): 547-558.
Special Issue: Biological Invasions: Mechanisms Impacts and Management;
• Special Issue •
### On the niche stasis of intercontinental invasive plants
Li Zhu; Keping Ma*
1. State Key Laboratory of Vegetation and Environmental Change, Institute of Botany, Chinese Academy of Sciences, Beijing 100093
• Received:2010-09-02 Online:2011-01-31
• Keping Ma
Biological invasions represent a growing threat to biodiversity. The movement of organisms among continents by humans has caused profound changes in structure and function of the ecosystem to which they have been introduced. By testing the differences in the relative contributions of the various origins of the invasive plants to each region, we found intercontinental invasions were more prevalent than intracontinental invasions, primarily including the exchange of species among Eastern Asia–North America, Eastern Asia–South America, Europe–South Africa, Europe–North America, Europe–Eastern Asia, and North America–Oceania. They have posed a higher threat than intracontinental invasions. Thus, preventing future invasions is the most cost-effective form of management. Species distribution models (SDMs) are increasingly used to estimate risks of biological invasions. Niche stasis, the tendency of a species niche to remain unchanged across space and time, is often assumed when applying these models to predict and explain biogeographical patterns. Yet, both niche change and conservatism have occurred for intercontinental invasive exotic plants, which were severely disconnected from their source populations and often adapt rapidly to conditions in the new range. To further understand the niche characteristics of invasive species, it is therefore necessary to consider which factors limit range expansion in the native region. After comparing the similarities and differences of invasive species expansion across continents to within continents, we propose that the probabilities of niche shifts occurring depended primarily on the ecological and evolutionary processes limiting the species in its native range such as dispersal limit, species interaction, adapted evolution, ecological plasticity and population characteristics. Most limited factors of species niche properties are more consistent with there being a niche shift than niche stasis in the new range. Therefore, we suggest the following areas for future research: (1) multi-scales studies on niche attributes across spatial, temporal, environmental and phylogenetic investigations; (2) comparative studies that identify both the groups of species that are characterized by environmental niche stasis or shifts, and the traits that the species are more prone to niche change; and (3) niche dynamics over time to estimate the propensity, historical rate, and magnitude of niche shift. Such understanding will improve our confidence in SDM-based predictions of the impacts of climate change and species invasions on species distributions and biodiversity.
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2020-08-04 23:15:38
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https://www.physicsforums.com/threads/weighted-average-of-weights-themselves.435772/
|
# Weighted average of weights themselves
Hi,
I have a list of 50 companies and their market capitalizations. (The market cap of a company is the collective value of all the shares of the company. So if a company’s share price is $10 and there exists 200 shares of the company, then the market cap of the company is$2000.)
My question is about calculating the weighted average market cap of the group. The weights I want use are the market caps themselves. So basically I am squaring each market cap value then adding them and then dividing the result with the sum of the market caps of the 50 companies.
Wt. Avg Mkt. Cap = ( M1 * M1 + M2*M2 +…………M50* M50 ) / (M1 + M2 +…..M50)
Can I do that ? I mean is it correct technically? Of course I understand that it doesn't give the central tendancy of the group.
Thanks,
MG.
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2020-10-27 17:31:06
|
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http://crypto.stackexchange.com/questions/14398/how-can-i-simulate-and-measure-brute-force-hacking-using-rsa/14402
|
# How can I simulate and measure brute force hacking using RSA?
I want to implement brute force hacking using RSA. How can I simulate and measure time duration? What software (or software-based simulator) would be needed?
-
First, you shoudl specify, what you actually want to brute-force. Is it factoring $n$, or finding $d$, or finding $m$ to a certain ciphertext $c$? Anyway, brute force in general is a bad idea here, since the "try out everything" approach has a MUCH higher complexity than factoring with the GNFS (very rough approximation: $2^{n}$ vs $2^{n/3}$) – tylo Feb 10 '14 at 13:38
Maybe you meant RSA hacking using brute force? – catpnosis Feb 11 '14 at 3:23
Firstly, I assume we are talking about classical computers
Implementing a brute force attack on a RSA may not be the most sensible thing, unless of course the security parameter of your target system is small.. (160 bit numbers! )
Even then you may not want to implement a brute force here.. try using Fermat's Factoring or Pollards $\rho$ method.
If you just want to measure time try to do a more theoretical research, i.e. use the time take per multiplication op. and the no. of operations your computer can perform in a sec to determine exactly how long will it take you to brute force a $n-$bit security.
To sum it up, YOU CANNOT BRUTE FORCE THE (properly implemented) MODERN DAY RSA (~4096 bit numbers) even if you use all computers in your state.
-
Did you mean 160 digits (about 530 bits) rather than 160 bit? That's more like the limit of brute-forcing RSA for an individual. – fgrieu Feb 9 '14 at 17:00
160 bits sound about right if by "brute force" one means literally iterating through all odd numbers from $\sqrt(N)$ and checking if the value divides $N$. – Henrick Hellström Feb 10 '14 at 1:33
@fgrieu 530 bit is in the ballpark of brute force, however, it will take some time (months) to do that with a modern day PC (I assume the asker does not have access to supercomputers or clusters). – Subhayan Feb 11 '14 at 3:18
I don't understand what you mean simulate, but if you need a toy example in python the code is at the end of the message. If you need to show the performance of more advanced algorithms use factor(x) in pari/gp. See here for details.
import gmpy, time, random, math
def genprime(bits):
p=1
while(gmpy.is_prime(p)==0):
p = random.randrange(math.pow(2,bits-1),math.pow(2,bits))
return p
BITS=20
found=False
p=genprime(BITS)
q=genprime(BITS)
n=p*q
cnt=2
s_time=time.time()
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2016-02-11 21:24:21
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https://www.jithinjp.in/2018/Dynamic-Time-wrapping
|
# DTW- Dynamic Time Wrapping
DTW is a family of algorithms which compute the local stretch or compression to apply to the time axes of two time series in order to optimally map one onto the other. DTW outputs the remaining cumulative distance between the two and, if desired, the mapping itself (warping function). In another word they can compute the similarity between time series which may vary in time (i.e. wrap in time).
DTW is widely used e.g. for classification and clustering tasks in Econometrics, Chemometrics and general time series mining. Originally designed for speech recognition, Basically DTW finds the optical global alignment between two time series by exploiting temporal distortion between them.
Multiple studies as confirmed that for the problem involving time series data for example a classification task, algorithm exploiting the DTW to wrap the invariance among the signal is hard to beat. [1][2]
##### Things to be considered while implementing or using DTW:
1. Z Normalizing
2. just-in-time normalization
3. early abandoning techniques
4. Applying endpoint constrain
5. setting up wrapping window
##### Computing DTW,
Compute the distance matrix(cost matrix) between every possible pair of points between the two time series. Any possible wrapping between two time series will be a path through the computed cost matrix.
$DTW(q,c) = min \Bigg( \frac{\sqrt{\sum_{k=1}^{K}w_k)}}{K} \Bigg)$, here w is the wrapping constant.
The optimal (minimum) path or wrapping between two time series will provide us with the DTW, which can be obtained using the below recursive function.
$\gamma(i,j) = distance(q_i,c_i) + min( \gamma(i-1,j-1), \gamma(i-1,j), \gamma(1,j-1))$
Its important to note that DTW is symmetric and has constancy of self similarity but does not follow positivity (separation) and triangular inequality. This would be the best way to say DTW is a distance measure and not a metric. Now interesting if we increase the sample space to an enormous level, we would be able to find lots of A,B, and C which would follow triangular inequality(just read this over the internet try to find the source). Mathematically in the limit, as w tends to zero DTW is a metric, interesting isn’t it?
##### Generalizing DTW to Multidimensional Data:
This can be achieved in two way, independent (calculate DTW separately for each dimension and sum up) and dependent DTW computation
#### Reference
[1] Bagnall, A., Lines, J., Bostrom, A., Large, J., & Keogh, E. (2017). The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances. Data Mining and Knowledge Discovery, 31(3), 606-660.
[2] Bagnall, A., Lines, J., Bostrom, A., Large, J., & Keogh, E. (2017). The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances. Data Mining and Knowledge Discovery, 31(3), 606-660.
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2019-05-21 13:00:12
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|
https://www.jobilize.com/online/course/0-2-practice-tests-1-4-and-final-exams-by-openstax?qcr=www.quizover.com&page=15
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# 0.2 Practice tests (1-4) and final exams (Page 16/36)
Page 16 / 36
83 . $\Sigma X\sim N\left(n{\mu }_{x},\left(\sqrt{n}\right)\left({\sigma }_{x}\right)\right)$ so $\Sigma X\sim N\left(4000,28.3\right)$
84 .The probability is 0.50, because 5,000 is the mean of the sampling distribution of sums of size 40 from this population. Sums of random variables computed from a sample of sufficient size are normally distributed, and in a normal distribution, half the values lie below the mean.
85 . Using the empirical rule, you would expect 95 percent of the values to be within two standard deviations of the mean. Using the formula for the standard deviation is for a sample sum: $\left(\sqrt{n}\right)\left({\sigma }_{x}\right)=\left(\sqrt{40}\right)\left(7\right)=44.3$ so you would expect 95 percent of the values to be between 5,000 + (2)(44.3) and 5,000 – (2)(44.3), or between 4,911.4 and 588.6.
86 . $\mu -\left(\sqrt{n}\right)\left({\sigma }_{x}\right)=5000-\left(\sqrt{40}\right)\left(7\right)=4955.7$
87 . $5000+\left(2.2\right)\left(\sqrt{40}\right)\left(7\right)=5097.4$
## 7.3: using the central limit theorem
88 . The law of large numbers says that as sample size increases, the sample mean tends to get nearer and nearer to the population mean.
89 . You would expect the mean from a sample of size 100 to be nearer to the population mean, because the law of large numbers says that as sample size increases, the sample mean tends to approach the population mea.
90 . X ~ N (0.10, 0.20)
91 . $\overline{X}\sim N\left({\mu }_{x},\frac{{\sigma }_{x}}{\sqrt{n}}\right)$ and the standard deviation of a uniform distribution is $\frac{b-a}{\sqrt{12}}$ . In this example, the standard deviation of the distribution is $\frac{b-a}{\sqrt{12}}=\frac{0.10}{\sqrt{12}}=0.03$
so $\overline{X}\sim N\left(0.15,0.003\right)$
92 .
## 8.1: confidence interval, single population mean, population standard deviation known, normal
Use the following information to answer the next seven exercises. You draw a sample of size 30 from a normally distributed population with a standard deviation of four.
1 . What is the standard error of the sample mean in this scenario, rounded to two decimal places?
2 . What is the distribution of the sample mean?
3 . If you want to construct a two-sided 95% confidence interval, how much probability will be in each tail of the distribution?
4 . What is the appropriate z -score and error bound or margin of error ( EBM ) for a 95% confidence interval for this data?
5 . Rounding to two decimal places, what is the 95% confidence interval if the sample mean is 41?
6 . What is the 90% confidence interval if the sample mean is 41? Round to two decimal places
7 . Suppose the sample size in this study had been 50, rather than 30. What would the 95% confidence interval be if the sample mean is 41? Round your answer to two decimal places.
8 . For any given data set and sampling situation, which would you expect to be wider: a 95% confidence interval or a 99% confidence interval?
## 8.2: confidence interval, single population mean, standard deviation unknown, student’s t
9 . Comparing graphs of the standard normal distribution ( z -distribution) and a t -distribution with 15 degrees of freedom ( df ), how do they differ?
10 . Comparing graphs of the standard normal distribution ( z -distribution) and a t -distribution with 15 degrees of freedom ( df ), how are they similar?
Use the following information to answer the next five exercises. Body temperature is known to be distributed normally among healthy adults. Because you do not know the population standard deviation, you use the t-distribution to study body temperature. You collect data from a random sample of 20 healthy adults and find that your sample temperatures have a mean of 98.4 and a sample standard deviation of 0.3 (both in degrees Fahrenheit).
what is standard deviation?
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Dominic
Yeah....the simplest one
IRFAN
what is the number of x
10
Elicia
Javed Arif
Jawed
how will you know if a group of data set is a sample or population
population is the whole set and the sample is the subset of population.
umair
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
Bhavika
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Alexander
Pls I hv a problem on t test is there anyone who can help?
Peggy
Dominic
Bhavika is right
Dominic
what is the problem peggy?
Bhavika
hi
Sandeep
Hello
hi
Bhavika
hii Bhavika
Dar
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
Hoshyar
three coins are tossed. find the probability of no head
three coins are tossed consecutively or what ?
umair
umair
or .125 is the probability of getting no head when 3 coins are tossed
umair
🤣🤣🤣
Simone
what is two tailed test
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
umair
in this test you are worried on both the ends
umair
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
umair
In order to apply weddles rule for numerical integration what is minimum number of ordinates
excuse me?
Gabriel
why?
didn't understand the question though.
Gabriel
which question? ?
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
John
geometric mean of two numbers 4 and 16 is:
10
umair
really
iphone
quartile deviation of 8 8 8 is:
iphone
sorry 8 is the geometric mean of 4,16
umair
quartile deviation of 8 8 8 is
iphone
can you please expalin the whole question ?
umair
mcq
iphone
h
iphone
can you please post the picture of that ?
umair
how
iphone
hello
John
10 now
John
how to find out the value
can you be more specific ?
umair
yes
KrishnaReddy
what is the difference between inferential and descriptive statistics
descriptive statistics gives you the result on the the data like you can calculate various things like variance,mean,median etc. however, inferential stats is involved in prediction of future trends using the previous stored data.
umair
if you need more help i am up for the help.
umair
Thanks a lot
Anjali
Inferential Statistics involves drawing conclusions on a population based on analysis of a sample. Descriptive statistics summarises or describes your current data as numerical calculations or graphs.
fred
my pleasure😊. Helping others offers me satisfaction 😊
umair
for poisson distribution mean............variance.
both are equal to mu
Faizan
mean=variance
Faizan
what is a variable
something that changes
Festus
why we only calculate 4 moment of mean? asked in papers.
why we only 4 moment of mean ? asked in BA exam
Faizan
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
Refiloe
for normal distribution mean is 2 & variance is 4 find mu 4?
repeat quastion again
Yusuf
find mu 4. it can be wrong but want to prove how.
Faizan
for a normal distribution if mu 4 is 12 then find mu 3?
Question hi wrong ha
Tahir
ye BA mcqs me aya he teen he. 2dafa aya he
Faizan
if X is normally distributed. (n,b). then its mean deviation is?
Faizan
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
nikita
which question is zero
Faizan
sorry it is (5,16) in place of (n,b)
Faizan
I got. thanks. it is zero.
Faizan
a random variable having binomial distribution is?
Bokaho
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2019-08-24 02:40:43
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|
https://electronics.stackexchange.com/questions/432605/identify-the-problems-in-this-verilog-snippet
|
# Identify the problems in this Verilog snippet
I think that the code below is wrong because it uses blocking (=) instead of non-blocking (<=) assignments but since there is only one statement in the always block is this an issue?
Also, it is sensitive to op but it also updates op within the always block, will this create some sort of loop condition?
Are there other problems with it?
always @ (a or op)
begin
op = a + b;
end
op should not be in the sensitivity list but it does not hurt.
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2019-06-26 20:53:12
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https://www.cp2k.org/exercises:2017_ethz_mmm:single_point_calculation?do=
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# Open SourceMolecular Dynamics
### Sidebar
#### For Developers
exercises:2017_ethz_mmm:single_point_calculation
# Computation of the Lennard Jones curve
In this exercise you will compute the Lennard-Jones energy curve for a system of two Krypton (Kr) atoms.
In Part I you find the instructions for computing the energy of two Kr atoms at a distance $r=4.00 Å$.
In Part II you find the instructions for getting the energy profile as a function of $r$.
Additonal parameters for Neon (Ne) and combination rules to obtain new parameters are provided in Part III and IV.
## Part I: Single Point (Energy) calculation
In this section a commented CP2K input example for a single point calculation is provided. Comments are added and signaled with '!'.
#### 1. Step
Save the following input to a file named energy.inp
energy.inp
&GLOBAL ! section to select the kind of calculation
RUN_TYPE ENERGY ! select type of calculation. In this case: ENERGY (=Single point calculation)
&END GLOBAL
&FORCE_EVAL ! section with parameters and system description
METHOD FIST ! Molecular Mechanics method
&MM ! specification of MM parameters
&FORCEFIELD ! parameters needed to describe the potential
&SPLINE
EMAX_SPLINE 10000 ! numeric parameter to ensure calculation stability. Should not be changed
&END
&NONBONDED ! parameters for the non bonded interactions
&LENNARD-JONES ! Lennard-Jones parameters
atoms Kr Kr
EPSILON [K_e] 164.56
SIGMA [angstrom] 3.601
RCUT [angstrom] 25.0
&END LENNARD-JONES
&END NONBONDED
&CHARGE
ATOM Kr
CHARGE 0.0
&END CHARGE
&END FORCEFIELD
&POISSON ! solver for non periodic calculations
PERIODIC NONE
&EWALD
EWALD_TYPE none
&END EWALD
&END POISSON
&END MM
&SUBSYS ! system description
&CELL
ABC [angstrom] 10 10 10
PERIODIC NONE
&END CELL
&COORD
UNIT angstrom
Kr 0 0 0
Kr 4 0 0
&END COORD
&END SUBSYS
&END FORCE_EVAL
#### 2. Step
Submit a CP2K calculation with the following commands:
bsub -n 1 mpirun cp2k.popt -i energy.inp -o energy.out
#### 3. Step
Afterwards the file energy.out will look like this:
**** **** ****** ** PROGRAM STARTED AT 2014-01-20 11:32:08.142
***** ** *** *** ** PROGRAM STARTED ON some_server.ethz.ch
** **** ****** PROGRAM STARTED BY you
***** ** ** ** ** PROGRAM PROCESS ID 24183
**** ** ******* ** PROGRAM STARTED IN /home/you/XERCISES
...
some stufff
...
ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.): 0.003617048870059
...
some other stuff
...
**** **** ****** ** PROGRAM ENDED AT 2014-01-20 12:24:18.154
***** ** *** *** ** PROGRAM RAN ON some_server.ethz.ch
** **** ****** PROGRAM RAN BY you
***** ** ** ** ** PROGRAM PROCESS ID 24993
**** ** ******* ** PROGRAM STOPPED IN /home/you/EXERCISES
If you get the closing Banner you know that cp2k worked. The following line tells you the result:
ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.): 0.003617048870059
This is the energy (in Hartree) for a system of 2 Kr atoms at distance $r=4.00 Å$
Note, that in the input-file EPSILON is given in units of Kelvin, whereas in the output the energy is printed in Hartree, which is the unit of energy in the system of atomic units (a.u.). To convert from Kelvin to Hartree you have to multiply with the Boltzmann constant $k_\text{b} = 3.1668154 \cdot 10^{-6} \frac{E_\text{H}}{\text{K}}$ .
## Part II: Computation of the LJ energy curve
In this section a few scripts to get the LJ energy profiles are presented.
#### 1. Step
In order to get a good profile, a set of energy values as a function of the interatomic distance is needed. You can use the energy.inp input file and change the Kr coordinates in order to get different starting distances.
The output file will be rewritten every time you run a calculation, unless you change its name.
To do so:
$mv energy.out energy_dist4A.out If you run multiple calculations, it is always good to keep track of what you have done by producing an input and an output for every distance you are planning to run. For doing so: $ cp energy.inp energy_dist2A.inp
then edit the input file with the new coordinates (e.g. 2 Å). you can now run CP2K and produce the output file:
$cp2k.popt -i energy_dist2A.inp -o energy_dist2A.out #### 2. Step When you have tested a few distances, you can produce a table looking like: Input file Distance (Å) Energy (Eh) energy_dist1A.inp 1 energy_dist2A.inp 2 energy_dist3A.inp 3 This is the Lennard Jones energy curve for two Kr atoms. By using any plotting program you can now get a representation of the energy profile. #### 3. Step Here are reported the LJ parameters for Ne atoms. Those are to replace the Kr parameters in the input file, along with your Ne coordinates that have to replace the Kr coordinates. A new LJ curve for Ne atoms can be now generated. &NONBONDED &LENNARD-JONES ! Lennard-Jones Ne parameters atoms Ne Ne EPSILON [K_e] 36.831 SIGMA [angstrom] 2.775 RCUT [angstrom] 25.0 &END LENNARD-JONES &END NONBONDED &CHARGE ATOM Ne CHARGE 0.0 &END CHARGE #### 4. Step Here are reported the combination rules for pairs unlike pairs, i.e. for pairs of non identical atoms. Once generated the ε and σ parameters for the couple Kr/Ne, generate once more the LJ dissociation curve. Compare the “mixed” curve to the two “pure” curves and report the position and depth of the minimum. $$\sigma_{ij}= \sqrt{\sigma_i\sigma_j}$$ $$\epsilon_{ij}= \sqrt{\epsilon_i\epsilon_j}$$ Remember that you are running with two different atom types. For this reason some of the input sections MUST BE REPLICATED for the two kinds of atoms present • The “ LENNARD-JONES ” section must be present for all the three possible couples: Kr-Kr, Ne-Ne and Ne-Kr. Example: &LENNARD-JONES ! Lennard-Jones parameters for Ar-Ar interaction atoms Kr Kr EPSILON [K_e] 164.56 SIGMA [angstrom] 3.601 RCUT [angstrom] 25.0 &END LENNARD-JONES &LENNARD-JONES ! Lennard-Jones Ne-Ne parameters atoms Ne Ne EPSILON [K_e] 36.831 SIGMA [angstrom] 2.775 RCUT [angstrom] 25.0 &END LENNARD-JONES &LENNARD-JONES ! Lennard-Jones parameters for Kr-Ne interaction atoms Kr Ne EPSILON [K_e] YOUR EPSILON SIGMA [angstrom] YOUR SIGMA RCUT [angstrom] 25.0 &END LENNARD-JONES • The “ CHARGE ” section must be also duplicated: &CHARGE ATOM Ne CHARGE 0.0 &END CHARGE &CHARGE ATOM Kr CHARGE 0.0 &END CHARGE ## Questions • Sketch the LJ energy curve for the two set of parameters ($\sigma$and$\epsilon\$) provided.
• Report, for both curves, the minimum energy distance and the depth of the minimum.
• What are the major differences between the curves? How do they relate to the sets of parameters provided?
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2019-11-21 05:40:41
|
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https://handwiki.org/wiki/Finance:Stock_market_crash
|
# Finance:Stock market crash
Short description
Sudden dramatic decline of stock prices across a significant cross-section of a stock market
Stock price graph illustrating the 2020 stock market crash, showing a sharp drop in stock price, followed by a recovery.
A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often follow speculation and economic bubbles.
A stock market crash is a social phenomenon where external economic events combine with crowd psychology in a positive feedback loop where selling by some market participants drives more market participants to sell. Generally speaking, crashes usually occur under the following conditions: a prolonged period of rising stock prices (a bull market) and excessive economic optimism, a market where price–earnings ratios exceed long-term averages, and extensive use of margin debt and leverage by market participants. Other aspects such as wars, large corporate hacks, changes in federal laws and regulations, and natural disasters within economically productive areas may also influence a significant decline in the stock market value of a wide range of stocks. Stock prices for corporations competing against the affected corporations may rise despite the crash.[1]
There is no numerically specific definition of a stock market crash but the term commonly applies to declines of over 10% in a stock market index over a period of several days. Crashes are often distinguished from bear markets (periods of declining stock market prices that are measured in months or years) as crashes include panic selling and abrupt, dramatic price declines. Crashes are often associated with bear markets; however, they do not necessarily occur simultaneously. Black Monday (1987), for example, did not lead to a bear market. Likewise, the bursting of the Japanese asset price bubble occurred over several years without any notable crashes.
Crashes are generally unexpected. As Niall Ferguson stated, "Before the crash, our world seems almost stationary, deceptively so, balanced, at a set point. So that when the crash finally hits — as inevitably it will — everyone seems surprised. And our brains keep telling us it’s not time for a crash."[2]
## Historical background
Courtyard of the Amsterdam Stock Exchange (or Beurs van Hendrick de Keyser in Dutch), the world's first formal stock exchange. It was in the 17th-century Dutch Republic that the fundamental elements of a formal stock market (as we know it today) were 'invented'.
Business ventures with multiple shareholders became popular with commenda contracts in medieval Italy and shareholder companies date back to ancient Rome.
The world's first stock market was that of 17th-century Amsterdam, where an active secondary market in company shares emerged. The two major companies were the Dutch East India Company and the Dutch West India Company, founded in 1602 and 1621. Other companies existed, but they were not as large and constituted a small portion of the stock market.[4]
It is important to note that from about the early 1600s to about the mid-18th century, the Dutch Republic's economic, business and financial systems were the most advanced and sophisticated ever seen in history.[5][6][7][8] In its Golden Age, the Dutch Republic was responsible for many pioneering innovations in economic, business and financial history of the world.[9] Like the first well-recorded asset price bubble in history (in the 1630s), early stock market bubbles and crashes also had their roots in socio-politico-economic activities of the 17th-century Dutch Republic (the birthplace of the world's first formal stock exchange and stock market),[10][6][11][12] the Dutch East India Company (the world's first formally listed public company), and the Dutch West India Company in particular.
## Examples
### Tulip Mania
Tulip Mania (1634-1637), in which some single tulip bulbs allegedly sold for more than 10 times the annual income of a skilled artisan, is often considered to be the first recorded economic bubble.
### Panic of 1907
Main page: Finance:Panic of 1907
In 1907 and in 1908, stock prices fell by nearly 50% due to a variety of factors, led by the manipulation of copper stocks by the Knickerbocker Trust Company.[13] Shares of United Copper rose gradually up to October, and thereafter crashed, leading to panic.[14][15] Several investment trusts and banks that had invested their money in the stock market fell and started to close down. Further bank runs were prevented due to the intervention of J. P. Morgan.[16] The panic continued to 1908 and led to the formation of the Federal Reserve in 1913.[17]
### Wall Street Crash of 1929
Main page: Finance:Wall Street Crash of 1929
Crowd gathering on Wall Street the day after the 1929 crash.
The economy grew for most of the Roaring Twenties. It was a technological golden age, as innovations such as the radio, automobile, aviation, telephone, and the electric power transmission grid were deployed and adopted. Companies that had pioneered these advances, including Radio Corporation of America (RCA) and General Motors, saw their stocks soar. Financial corporations also did well, as Wall Street bankers floated mutual fund companies (then known as investment trusts) like the Goldman Sachs Trading Corporation. Investors were infatuated with the returns available in the stock market, especially by the use of leverage through margin debt.
On August 24, 1921, the Dow Jones Industrial Average (DJIA) was at 63.9. By September 3, 1929, it had risen more than sixfold to 381.2. It did not regain this level for another 25 years. By the summer of 1929, it was clear that the economy was contracting, and the stock market went through a series of unsettling price declines. These declines fed investor anxiety, and events came to a head on October 24, 28, and 29 (known respectively as Black Thursday, Black Monday, and Black Tuesday).
On Black Monday, the DJIA fell 38.33 points to 260, a drop of 12.8%. The deluge of selling overwhelmed the ticker tape system that normally gave investors the current prices of their shares. Telephone lines and telegraphs were clogged and were unable to cope. This information vacuum only led to more fear and panic. The technology of the New Era, previously much celebrated by investors, now served to deepen their suffering.
The following day, Black Tuesday, was a day of chaos. Forced to liquidate their stocks because of margin calls, overextended investors flooded the exchange with sell orders. The Dow fell 30.57 points to close at 230.07 on that day. The glamour stocks of the age saw their values plummet. Across the two days, the DJIA fell 23%.
By the end of the weekend of November 11, 1929, the index stood at 228, a cumulative drop of 40% from the September high. The markets rallied in succeeding months, but it was a temporary recovery that led unsuspecting investors into further losses. The DJIA lost 89% of its value before finally bottoming out in July 1932. The crash was followed by the Great Depression, the worst economic crisis of modern times, which plagued the stock market and Wall Street throughout the 1930s.
### October 19, 1987
Main page: Finance:Black Monday (1987)
DJIA (19 July 1987 through 19 January 1988).
The mid-1980s were a time of strong economic optimism. From August 1982 to its peak in August 1987, the Dow Jones Industrial Average (DJIA) rose from 776 to 2722. The rise in market indices for the 19 largest markets in the world averaged 296% during this period. The average number of shares traded on the New York Stock Exchange rose from 65 million shares to 181 million shares.[18]
The crash on October 19, 1987, Black Monday, was the climatic culmination of a market decline that had begun five days before on October 14. The DJIA fell 3.81% on October 14, followed by another 4.60% drop on Friday, October 16. On Black Monday, the DJIA plummeted 508 points, losing 22.6% of its value in one day. The S&P 500 Index dropped 20.4%, falling from 282.7 to 225.06. The NASDAQ Composite lost only 11.3%, not because of restraint on the part of sellers, but because the NASDAQ market system failed. Deluged with sell orders, many stocks on the NYSE faced trading halts and delays. Of the 2,257 NYSE-listed stocks, there were 195 trading delays and halts during the day.[19] The NASDAQ market fared much worse. Because of its reliance on a "market making" system that allowed market makers to withdraw from trading, liquidity in NASDAQ stocks dried up. Trading in many stocks encountered a pathological condition where the bid price for a stock exceeded the ask price. These "locked" conditions severely curtailed trading. On October 19, trading in Microsoft shares on the NASDAQ lasted a total of 54 minutes.
The crash was the greatest single-day loss that Wall Street had ever suffered in continuous trading up to that point. Between the start of trading on October 14 to the close on October 19, the DJIA lost 760 points, a decline of over 31%.
In October 1987, all major world markets crashed or declined substantially. The FTSE 100 Index lost 10.8% on that Monday and a further 12.2% the following day. The least affected was Austria (a fall of 11.4%) while the most affected was Hong Kong with a drop of 45.8%. Out of 23 major industrial countries, 19 had a decline greater than 20%.[20]
Despite fears of a repeat of the Great Depression, the market rallied immediately after the crash, posting a record one-day gain of 102.27 the very next day and 186.64 points on Thursday October 22. It took only two years for the Dow to recover completely; by September 1989, the market had regained all of the value it had lost in the 1987 crash. The DJIA gained 0.6% during calendar year 1987.
No definitive conclusions have been reached on the reasons behind the 1987 Crash. Stocks had been in a multi-year bull run and market price–earnings ratios in the U.S. were above the post-war average. The S&P 500 was trading at 23 times earnings, a postwar high and well above the average of 14.5 times earnings.[21] Herd behavior and psychological feedback loops play a critical part in all stock market crashes but analysts have also tried to look for external triggering events. Aside from the general worries of stock market overvaluation, blame for the collapse has been apportioned to such factors as program trading, portfolio insurance and derivatives, and prior news of worsening economic indicators (i.e. a large U.S. merchandise trade deficit and a falling United States dollar , which seemed to imply future interest rate hikes).[22]
One of the consequences of the 1987 Crash was the introduction of the circuit breaker or trading curb on the NYSE. Based upon the idea that a cooling off period would help dissipate panic selling, these mandatory market shutdowns are triggered whenever a large pre-defined market decline occurs during the trading day.
### Crash of 2008–2009
Main page: Finance:Financial crisis of 2007–2008
The collapse of Lehman Brothers was a symbol of the Crash of 2008
OMX Iceland 15 closing prices during the five trading weeks from September 29, 2008 to October 31, 2008.
On September 15, 2008, the bankruptcy of Lehman Brothers and the collapse of Merrill Lynch along with a liquidity crisis of American International Group, all primarily due to exposure to packaged subprime loans and credit default swaps issued to insure these loans and their issuers, rapidly devolved into a global crisis. This resulted in several bank failures in Europe and sharp reductions in the value of stocks and commodities worldwide. The failure of banks in Iceland resulted in a devaluation of the Icelandic króna and threatened the government with bankruptcy. Iceland obtained an emergency loan from the International Monetary Fund in November.[23] In the United States, 15 banks failed in 2008, while several others were rescued through government intervention or acquisitions by other banks.[24] On October 11, 2008, the head of the International Monetary Fund (IMF) warned that the world financial system was teetering on the "brink of systemic meltdown".[25]
The economic crisis caused countries to close their markets temporarily.
On October 8, the Indonesian stock market halted trading, after a 10% drop in one day.
The Times of London reported that the meltdown was being called the Crash of 2008, and older traders were comparing it with Black Monday in 1987. The fall that week of 21% compared to a 28.3% fall 21 years earlier, but some traders were saying it was worse. "At least then it was a short, sharp, shock on one day. This has been relentless all week."[26] Other media also referred to the events as the "Crash of 2008".[27]
From October 6–10, 2008, the Dow Jones Industrial Average (DJIA) closed lower in all five sessions. Volume levels were record-breaking. The DJIA fell over 1,874 points, or 18%, in its worst weekly decline ever on both a points and percentage basis. The S&P 500 fell more than 20%.[28] The week also set 3 top ten NYSE Group Volume Records with October 8 at #5, October 9 at #10, and October 10 at #1.
Having been suspended for three successive trading days (October 9, 10, and 13), the Icelandic stock market reopened on 14 October, with the main index, the OMX Iceland 15, closing at 678.4, which was about 77% lower than the 3,004.6 at the close on October 8. This reflected that the value of the three big banks, which had formed 73.2% of the value of the OMX Iceland 15, had been set to zero.
On October 24, 2008, many of the world's stock exchanges experienced the worst declines in their history, with drops of around 10% in most indices.[29] In the U.S., the DJIA fell 3.6%, although not as much as other markets.[30] The United States dollar and Japanese yen soared against other major currencies, particularly the British pound and Canadian dollar, as world investors sought safe havens. Later that day, the deputy governor of the Bank of England, Charlie Bean, suggested that "This is a once in a lifetime crisis, and possibly the largest financial crisis of its kind in human history."[31]
By March 6, 2009 the DJIA had dropped 54% to 6,469 from its peak of 14,164 on October 9, 2007, over a span of 17 months, before beginning to recover.[32]
### 2010 flash crash
Main page: Finance:2010 flash crash
On May 6, 2010, stock market indices in the U.S. fell 9% within a few minutes primarily as a result of automated high-frequency traders.
### 2015–16 Chinese stock market turbulence
Main page: Finance:2015–16 Chinese stock market turbulence
In July 2015, most stocks on the Shanghai Stock Exchange fell 30% within a few weeks due to concerns about a slowing economy.
### 2020 coronavirus stock market crash
During the week of February 24–28, 2020, stock markets dropped as the COVID-19 pandemic spread globally. The FTSE 100 dropped 13%, while the DJIA and S&P 500 Index dropped 11-12% in the biggest downward weekly drop since the financial crisis of 2007-2008.
On Monday, March 9, 2020, after the launch of the 2020 Russia–Saudi Arabia oil price war, the FTSE and other major European stock market indices fell by nearly 8%. Asian markets fell sharply and the S&P 500 Index dropped 7.60%.[33] The Italian FTSE MIB fell 2,323.98 points, or 11.17%.[34]
On March 12, 2020, a day after President Donald Trump announced a travel ban from Europe, stock prices again fell sharply. The DJIA declined 9.99% — the largest daily decline since Black Monday (1987) — despite the Federal Reserve announcing it would inject $1.5 trillion into money markets.[35] The S&P 500 and the Nasdaq each dropped by approximately 9.5%. The major European stock market indexes all fell over 10%.[36] On March 16, 2020, after it became clear that a recession was inevitable, the DJIA dropped 12.93%, or 2,997 points, the largest point drop since Black Monday (1987), surpassing the drop in the prior week, the Nasdaq Composite dropped 12.32%, and the S&P 500 Index dropped 11.98%.[37] By the end of May 2020, the stock market indices briefly recovered to their levels at the end of February 2020.(citation?) In June 2020 the Nasdaq surpassed its pre-crash high followed by the S&P 500 in August and the Dow in November. ## Mathematical theory ### Random walk theory The conventional assumption is that stock markets behave according to a random log-normal distribution.[38] Among others, mathematician Benoit Mandelbrot suggested as early as 1963 that the statistics prove this assumption incorrect.[39] Mandelbrot observed that large movements in prices (i.e. crashes) are much more common than would be predicted from a log-normal distribution. Mandelbrot and others suggested that the nature of market moves is generally much better explained using non-linear analysis and concepts of chaos theory.[40] This has been expressed in non-mathematical terms by George Soros in his discussions of what he calls reflexivity of markets and their non-linear movement.[41] George Soros said in late October 1987, 'Mr. Robert Prechter's reversal proved to be the crack that started the avalanche'.[42][43] ### Self-organized criticality Research at the Massachusetts Institute of Technology suggests that there is evidence that the frequency of stock market crashes follows an inverse cubic power law.[44] This and other studies such as Didier Sornette's work suggest that stock market crashes are a sign of self-organized criticality in financial markets.[45] ### Lévy flight In 1963, Mandelbrot proposed that instead of following a strict random walk, stock price variations executed a Lévy flight.[46] A Lévy flight is a random walk that is occasionally disrupted by large movements. In 1995, Rosario Mantegna and Gene Stanley analyzed a million records of the S&P 500 Index, calculating the returns over a five-year period.[47] Researchers continue to study this theory, particularly using computer simulation of crowd behavior, and the applicability of models to reproduce crash-like phenomena. ### Result of investor imitation In 2011, using statistical analysis tools of complex systems, research at the New England Complex Systems Institute found that the panics that lead to crashes come from a dramatic increase in imitation among investors, which always occurred during the year before each market crash. When investors closely follow each other's cues, it is easier for panic to take hold and affect the market. This work is a mathematical demonstration of a significant advance warning sign of impending market crashes.[48][49] ## Trading curbs and trading halts One mitigation strategy has been the introduction of trading curbs, also known as "circuit breakers", which are a trading halt in the cash market and the corresponding trading halt in the derivative markets triggered by the halt in the cash market, all of which are affected based on substantial movements in a broad market indicator. Since their inception after Black Monday (1987), trading curbs have been modified to prevent both speculative gains and dramatic losses within a small time frame. ### United States There are three thresholds, which represent different levels of decline in the S&P 500 Index: 7% (Level 1), 13% (Level 2), and 20% (Level 3).[50] • If Threshold Level 1 (a 7% drop) is breached before 3:25pm, trading halts for a minimum of 15 minutes. At or after 3:25 pm, trading continues unless there is a Level 3 halt. • If Threshold Level 2 (a 13% drop) is breached before 1 pm, the market closes for two hours. If such a decline occurs between 1 pm and 2 pm, there is a one-hour pause. The market would close for the day if stocks sank to that level after 2 pm • If Threshold Level 3 (a 20% drop) is breached, the market would close for the day, regardless of the time.[51] ### France For the CAC 40 stock market index in France, daily price limits are implemented in cash and derivative markets. Securities traded on the markets are divided into three categories according to the number and volume of daily transactions. Price limits for each security vary by category. For instance, for the most liquid category, when the price movement of a security from the previous day's closing price exceeds 10%, trading is suspended for 15 minutes. If the price then goes up or down by more than 5%, transactions are again suspended for 15 minutes. The 5% threshold may apply once more before transactions are halted for the rest of the day. When such a suspension occurs, transactions on options based on the underlying security are also suspended. Further, when stocks representing more than 35% of the capitalization of the CAC40 Index are halted, the calculation of the CAC40 Index is suspended and the index is replaced by a trend indicator. When stocks representing less than 25% of the capitalization of the CAC40 Index are halted, trading on the derivative markets are suspended for half an hour or one hour, and additional margin deposits are requested. ## See also • List of stock market crashes and bear markets • VIX, Chicago Board Options Exchange Market Volatility Index ## References 1. Galbraith, J. The Great Crash 1929, 1988 edition, Houghton Mifflin Co. Boston, p.xii-xvii 2. Farrell, Paul B. (February 19, 2014). "Crash of 2014: Like 1929, you'll never hear it coming". Marketwatch. 3. De la Vega, Joseph: Confusión de confusiones (1688): Portions Descriptive of the Amsterdam Stock Exchange. Selected and translated by Hermann Kellenbenz. (Cambridge, MA: Baker Library, Harvard Graduate School of Business Administration, 1957) 4. Stringham, Edward Peter; Curott, Nicholas A. (2015), 'On the Origins of Stock Markets,' [Chapter 14, Part IV: Institutions and Organizations]; in The Oxford Handbook of Austrian Economics, edited by Peter J. Boettke and Christopher J. Coyne. (Oxford University Press, 2015, ISBN:978-0199811762), pp. 324–344 5. "The Keynes Conundrum by David P. Goldman". First Things (firstthings.com). 1 October 2010. "Reuven Brenner & David P. Goldman (2010): "Western societies developed the institutions that support entrepreneurship only through a long and fitful process of trial and error. Stock and commodity exchanges, investment banks, mutual funds, deposit banking, securitization, and other markets have their roots in the Dutch innovations of the seventeenth century but reached maturity, in many cases, only during the past quarter of a century."" 6. Goetzmann, William N.; Rouwenhorst, K. Geert (2005). The Origins of Value: The Financial Innovations that Created Modern Capital Markets. Oxford University Press. pp. 165–175. ISBN 9780195175714. 7. Goetzmann, William N.; Rouwenhorst, K. Geert (2008). The History of Financial Innovation, in Carbon Finance, Environmental Market Solutions to Climate Change. (Yale School of Forestry and Environmental Studies, chapter 1, pp. 18–43). Goetzmann & Rouwenhorst (2008): "The 17th and 18th centuries in the Netherlands were a remarkable time for finance. Many of the financial products or instruments that we see today emerged during a relatively short period. In particular, merchants and bankers developed what we would today call securitization. Mutual funds and various other forms of structured finance that still exist today emerged in the 17th and 18th centuries in Holland." 8. Soll, Jacob (2014). The Reckoning: Financial Accountability and the Making and Breaking of Nations. New York: Basic Books. "Jacob Soll (2014): "With the complexity of the stock exchange, [17th-century] Dutch merchants' knowledge of finance became more sophisticated than that of their Italian predecessors or German neighbors."" 9. Johannessen, Jon-Arild: Innovations Lead to Economic Crises: Explaining the Bubble Economy. (London: Palgrave Macmillan, 2017) 10. Macaulay, Catherine R. (2015). "Capitalism's renaissance? The potential of repositioning the financial 'meta-economy'". (Futures, Volume 68, April 2015, p. 5–18) 11. "AMERICAN BANKS "IN THE JUNGLE".". The Advertiser (Adelaide): p. 8. March 16, 1933. 12. "Born of a Panic: Forming the Fed System". Federal Reserve Bank of Minneapolis. 13. Tucker, Abigail (October 9, 2008). "The Financial Panic of 1907: Running from History". Smithsonian Magazine. 14. Nocholas, Tom. Stock market swings and the value of innovation. Boston: Harvard Business School. 15. "Preliminary Observations on the October 1987 Crash". Government Accountability Office. January 1988. 16. U.S. GAO op. cit. p.55 17. Sornette, Didier Sornette (2003). "Critical Market Crashes". Physics Reports 378 (1): 1–98. doi:10.1016/S0370-1573(02)00634-8. Bibcode2003PhR...378....1S. 18. U.S. GAO op. cit. p.37 19. What caused the Stock Market Crash of 1987? 20. "BBC NEWS – Business – IMF approves$2.1bn Iceland loan". BBC News. November 20, 2008.
21. LETZING, JOHN (October 10, 2008). "Two banks fold, bringing total to 15 failures this year". MarketWatch.
22. Wroughton, Lesley; Murphy, Francois (October 11, 2008). "IMF warns of financial meltdown". Reuters.
23. Wighton, David (October 11, 2008). "'It's irrational fear. The markets will trash anything that walks'". The Times.
24. Surowiecki, James (January 5, 2009). "WHAT PRECIPITATED THE STOCK MARKET CRASH OF 2008?". The New Yorker.
25. KUMAR, V. PHANI (October 24, 2008). "Indexes fall hard on bloody Friday". Marketwatch.
26. Twin, Alexandra (October 24, 2008). "Stocks can't shake the blues". CNN.
27. Dempsey, Alison L. (September 8, 2017). Evolutions in Corporate Governance: Towards an Ethical Framework for Business Conduct. Routledge. ISBN 9781351277389.
28. "Global shares plunge in worst day since financial crisis". BBC News. March 9, 2020.
29. "Coronavirus: Milan bourse plunges, spread soars". Agenzia Nazionale Stampa Associata. 9 March 2020.
30. Lynch, David J.; Heath, Thomas; Telford, Taylor; Long, Heather (March 12, 2020). "U.S. stock market suffers worst crash since 1987, as Americans wake up to a new normal of life". The Washington Post.
31. Wearden, Graeme; Jolly (now), Jasper (March 12, 2020). "Wall Street and FTSE 100 plunge on worst day since 1987 – as it happened". The Guardian.
32. Malkiel, Burton G. (1973). A Random Walk Down Wall Street (6th ed.). W.W. Norton & Company, Inc.. ISBN 978-0-393-06245-8.
33. Gammon, Katharine Stoel (November 16, 2006). "'Father of Fractals' takes on the stock market". Massachusetts Institute of Technology.
34. Soros, George (1988). Alchemy of Finance. Simon & Schuster. ISBN 9780671662387.
35. Hulbert, Mark (November 25, 2009). "Short term vs. long-term". Marketwatch.
36. Thomas Jr, Landon (October 13, 2007). "The Man Who Won as Others Lost". The New York Times.
37. Gavaix, Xavier (May 14, 2003). "Stock trade patterns could predict financial earthquakes". Massachusetts Institute of Technology.
38. Sornette, Didier (29 June 1999). "Self-Organized Percolation Model for Stock Market Fluctuations". Physica A: Statistical Mechanics and Its Applications 271 (3–4): 496–506. doi:10.1016/S0378-4371(99)00290-3. Bibcode1999PhyA..271..496S.
39. Mandelbrot, Benoit (1 January 1963). "The Variation of Certain Speculative Prices". The Journal of Business 36 ([object Attr]): 394. doi:10.1086/294632.
40. Mantegna, Rosario N.; Stanley, H. Eugene (6 July 1995). "Scaling behaviour in the dynamics of an economic index". Nature 376 (6535): 46–49. doi:10.1038/376046a0. Bibcode1995Natur.376...46M.
41. Harmon, Dion; de Aguiar, Marcus A. M.; Chinellato, David D.; Braha, Dan; Epstein, Irving R.; Bar-Yam, Yaneer (13 February 2011). "Predicting economic market crises using measures of collective panic". arXiv:1102.2620 [q-fin.ST].
42. Keim, Brandon (March 18, 2011). "Possible Early Warning Sign for Market Crashes". Wired.
43. Guzman, Zack; Koba, Mark (January 7, 2016). "When do circuit breakers kick in? CNBC Explains". CNBC.
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2021-10-16 00:50:46
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http://books.duhnnae.com/2017/jun8/149825897099-Remarks-on-Einstein-unit-tangent-bundles-Mathematics-Differential-Geometry.php
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# Remarks on $η$-Einstein unit tangent bundles - Mathematics > Differential Geometry
Remarks on $η$-Einstein unit tangent bundles - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.
Abstract: We study the geometric properties of the base manifold for the unit tangentbundle satisfying the $\eta$-Einstein condition with the standard contactmetric structure. One of the main theorems is that the unit tangent bundle of4-dimensional Einstein manifold, equipped with the canonical contact metricstructure, is $\eta$-Einstein manifold if and only if base manifold is thespace of constant sectional curvature 1 or 2.
Author: Y. D. Chai, S. H. Chun, J. H. Park, K. Sekigawa
Source: https://arxiv.org/
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2017-10-24 09:50:52
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https://math.libretexts.org/Courses/Monroe_Community_College/MTH_104_Intermediate_Algebra/6%3A_Factoring/6.4%3A_Factor_Special_Products
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# 6.3: Factor Special Products
##### Learning Objectives
By the end of this section, you will be able to:
• Factor perfect square trinomials
• Factor differences of squares
• Factor sums and differences of cubes
Before you get started, take this readiness quiz.
1. Simplify: $$(3x^2)^3$$.
2. Multiply: $$(m+4)^2$$.
3. Multiply: $$(x−3)(x+3)$$.
We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.
## Factor Perfect Square Trinomials
Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.
The trinomial $$9x^2+24x+16$$ is called a perfect square trinomial. It is the square of the binomial $$3x+4$$.
In this chapter, you will start with a perfect square trinomial and factor it into its prime factors. You could factor this trinomial using the methods described in the last section, since it is of the form $$ax^2+bx+c$$. But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work. Here is the pattern—the reverse of the binomial squares pattern.
##### PERFECT SQUARE TRINOMIALS PATTERN
If $$a$$ and $$b$$ are real numbers
$a^2+2ab+b^2=(a+b)^2\nonumber$
$a^2−2ab+b^2=(a−b)^2\nonumber$
To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, $$a^2$$. Next check that the last term is a perfect square, $$b^2$$. Then check the middle term—is it the product, $$2ab$$? If everything checks, you can easily write the factors.
##### Example $$\PageIndex{1}$$: How to Factor Perfect Square Trinomials
Factor: $$9x^2+12x+4$$.
##### Example $$\PageIndex{2}$$
Factor: $$4x^2+12x+9$$.
$$(2x+3)^2$$
##### Example $$\PageIndex{3}$$
Factor: $$9y^2+24y+16$$.
$$(3y+4)^2$$
The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern $$a^2−2ab+b^2$$, which factors to $$(a−b)^2$$.
The steps are summarized here.
##### FACTOR PERFECT SQUARE TRINOMIALS
$$\begin{array} {lllll} \textbf{Step 1.} &\text{Does the trinomial fit the pattern?} &\quad &\hspace{7mm} a^2+2ab+b^2 &\hspace{7mm} a^2−2ab+b^2 \\ &\text{Are the first and last terms perfect squares?} &\quad & &\\ &\text{Write them as squares.} &\quad &\hspace{5mm}(a)^2\hspace{16mm} (b)^2 &\hspace{6mm}(a)^2\hspace{16mm} (b)^2 \\ &\text{Check the middle term. Is it }2ab? &\quad &\hspace{12mm} {\,}^{\searrow}{\,}_{2·a·b}{\,}^{\swarrow} &\hspace{12mm} {\,}^{\searrow}{\,}_{2·a·b}{\,}^{\swarrow} \\ \textbf{Step 2.} &\text{Write the square of the binomial.} &\quad &\hspace{13mm} (a+b)^2 &\hspace{13mm} (a−b)^2 \\ \textbf{Step 3.} &\text{Check by multiplying.} & & & \end{array}$$
We’ll work one now where the middle term is negative.
##### Example $$\PageIndex{4}$$
Factor: $$81y^2−72y+16$$.
The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be $$(a−b)^2$$.
$$81 y^{2}-72 y+16$$ Are the first and last terms perfect squares? Check the middle term. Does it match $$(a−b)^2$$? Yes. Write as the square of a binomial. $$(9 y-4)^{2}$$ Check by multiplying: $(9y−4)^2\nonumber$$(9y)^2−2·9y·4+4^2\nonumber$$81y^2−72y+16\checkmark\nonumber$
##### Example $$\PageIndex{5}$$
Factor: $$64y^2−80y+25$$.
$$(8y−5)^2$$
##### Example $$\PageIndex{6}$$
Factor: $$16z^2−72z+81$$.
$$(4z−9)^2$$
The next example will be a perfect square trinomial with two variables.
##### Example $$\PageIndex{7}$$
Factor: $$36x^2+84xy+49y^2$$.
$$36 x^{2}+84 x y+49 y^{2}$$ Test each term to verify the pattern. Factor. $$(6 x+7 y)^{2}$$ Check by multiplying. $(6x+7y)^2\nonumber$$(6x)^2+2·6x·7y+(7y)^2\nonumber$$36x^2+84xy+49y^2\checkmark\nonumber$
##### Example $$\PageIndex{8}$$
Factor: $$49x^2+84xy+36y^2$$.
$$(7x+6y)^2$$
##### Example $$\PageIndex{9}$$
Factor: $$64m^2+112mn+49n^2$$.
$$(8m+7n)^2$$
Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.
##### Example $$\PageIndex{10}$$
Factor: $$100x^2y−80xy+16y$$.
$$100 x^{2} y-80 x y+16 y$$ Is there a GCF? Yes, $$4y$$, so factor it out. $$4 y\left(25 x^{2}-20 x+4\right)$$ Is this a perfect square trinomial? Verify the pattern. Factor. $$4 y(5 x-2)^{2}$$ Remember: Keep the factor 4y in the final product. Check: $4y(5x−2)^2\nonumber$$4y[(5x)2−2·5x·2+22]\nonumber$ $4y(25x2−20x+4)\nonumber$100x2y−80xy+16y\checkmark\]
##### Example $$\PageIndex{11}$$
Factor: $$8x^2y−24xy+18y$$.
$$2y(2x−3)^2$$
##### Example $$\PageIndex{12}$$
Factor: $$27p^2q+90pq+75q$$.
$$3q(3p+5)^2$$
## Factor Differences of Squares
The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:
A difference of squares factors to a product of conjugates.
##### DIFFERENCE OF SQUARES PATTERN
If $$a$$ and $$b$$ are real numbers,
Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.
##### Example $$\PageIndex{13}$$: How to Factor a Trinomial Using the Difference of Squares
Factor: $$64y^2−1$$.
##### Example $$\PageIndex{14}$$
Factor: $$121m^2−1$$.
$$(11m−1)(11m+1)$$
##### Example $$\PageIndex{15}$$
Factor: $$81y^2−1$$.
$$(9y−1)(9y+1)$$
##### FACTOR DIFFERENCES OF SQUARES.
$$\begin{array} {llll} \textbf{Step 1.} &\text{Does the binomial fit the pattern?} &\qquad &\hspace{5mm} a^2−b^2 \\ &\text{Is this a difference?} &\qquad &\hspace{2mm} \text{____−____} \\ &\text{Are the first and last terms perfect squares?} & & \\ \textbf{Step 2.} &\text{Write them as squares.} &\qquad &\hspace{3mm} (a)^2−(b)^2 \\ \textbf{Step 3.} &\text{Write the product of conjugates.} &\qquad &(a−b)(a+b) \\ \textbf{Step 4.} &\text{Check by multiplying.} & & \end{array}$$
It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression $$a^2+b^2$$ is prime!
The next example shows variables in both terms.
##### Example $$\PageIndex{16}$$
Factor: $$144x^2−49y^2$$.
$$\begin{array} {lll} &\quad &144x^2−49y^2 \\ \text{Is this a difference of squares? Yes.} &\quad &(12x)^2−(7y)^2 \\ \text{Factor as the product of conjugates.} &\quad &(12x−7y)(12x+7y) \\ \text{Check by multiplying.} &\quad &(12x−7y)(12x+7y) \\ \text{Check by multiplying.} &\quad & \\ &\quad & \\ &\quad & \\ \hspace{14mm} (12x−7y)(12x+7y) &\quad & \\ \hspace{21mm} 144x^2−49y^2\checkmark &\quad & \end{array}$$
##### Example $$\PageIndex{17}$$
Factor: $$196m^2−25n^2$$.
$$(14m−5n)(14m+5n)$$
##### Example $$\PageIndex{18}$$
Factor: $$121p^2−9q^2$$.
$$(11p−3q)(11p+3q)$$
As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.
Also, to completely factor the binomial in the next example, we’ll factor a difference of squares twice!
##### Example $$\PageIndex{19}$$
Factor: $$48x^4y^2−243y^2$$.
$$\begin{array} {ll} &48x^4y^2−243y^2 \\ \text{Is there a GCF? Yes, }3y^2\text{—factor it out!} &3y^2(16x^4−81) \\ \text{Is the binomial a difference of squares? Yes.} &3y^2\left((4x^2)^2−(9)^2\right) \\ \text{Factor as a product of conjugates.} &3y^2(4x^2−9)(4x^2+9) \\ \text{Notice the first binomial is also a difference of squares!} &3y^2((2x)^2−(3)^2)(4x^2+9) \\ \text{Factor it as the product of conjugates.} &3y^2(2x−3)(2x+3)(4x^2+9) \end{array}$$
The last factor, the sum of squares, cannot be factored.
$$\begin{array} {l} \text{Check by multiplying:} \\ \hspace{10mm} 3y^2(2x−3)(2x+3)(4x^2+9) \\ \\ \\ \hspace{15mm} 3y^2(4x^2−9)(4x^2+9) \\ \hspace{20mm} 3y^2(16x^4−81) \\ \hspace{19mm} 48x^4y^2−243y^2\checkmark\end{array}$$
##### Example $$\PageIndex{20}$$
Factor: $$2x^4y^2−32y^2$$.
$$2y^2(x−2)(x+2)(x^2+4)$$
##### Example $$\PageIndex{21}$$
Factor: $$7a^4c^2−7b^4c^2$$.
$$7c^2(a−b)(a+b)(a^2+b^2)$$
The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.
##### Example $$\PageIndex{22}$$
Factor: $$x^2−6x+9−y^2$$.
Notice that the first three terms form a perfect square trinomial.
$$x^{2}-6 x+9-y^{2}$$ Factor by grouping the first three terms. $$\underbrace{x^{2}-6 x+9} - y^{2}$$ Use the perfect square trinomial pattern. $$(x-3)^{2}-y^{2}$$ Is this a difference of squares? Yes. Yes—write them as squares. Factor as the product of conjugates. $$(x-3-y)(x-3+y)$$
You may want to rewrite the solution as $$(x−y−3)(x+y−3)$$.
##### Example $$\PageIndex{23}$$
Factor: $$x^2−10x+25−y^2$$.
$$(x−5−y)(x−5+y)$$
##### Example $$\PageIndex{24}$$
Factor: $$x^2+6x+9−4y^2$$.
$$(x+3−2y)(x+3+2y)$$
## Factor Sums and Differences of Cubes
There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.
$a^3+b^3=(a+b)(a^2−ab+b^2\nonumber$
$a^3−b^3=(a−b)(a^2+ab+b^2)\nonumber$
We’ll check the first pattern and leave the second to you.
$$\color{red}(a+b) \color{black} \left(a^{2}-a b+b^{2}\right)$$ Distribute. $$\color{red}a \color{black}\left(a^{2}-a b+b^{2}\right)+ \color{red}b \color{black}\left(a^{2}-a b+b^{2}\right)$$ Multiply. $$a^{3}-a^{2} b+a b^{2}+a^{2} b-a b^{2}+b^{3}$$ Combine like terms. $$a^{3}+b^{3}$$
##### SUM AND DIFFERENCE OF CUBES PATTERN
$a^3+b^3=(a+b)(a^2−ab+b^2\nonumber$$a^3−b^3=(a−b)(a^2+ab+b^2)\nonumber$
The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.
The trinomial factor in the sum and difference of cubes pattern cannot be factored.
It be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in Table.
n 1 2 3 4 5 6 7 8 9 10
$$n^3$$ 1 8 27 64 125 216 343 512 729 1000
##### Example $$\PageIndex{25}$$: How to Factor the Sum or Difference of Cubes
Factor: $$x^3+64$$.
##### Example $$\PageIndex{26}$$
Factor: $$x^3+27$$.
$$(x+3)(x^2−3x+9)$$
##### Example $$\PageIndex{27}$$
Factor: $$y^3+8$$.
$$(y+2)(y^2−2y+4)$$
##### FACTOR THE SUM OR DIFFERENCE OF CUBES.
1. Does the binomial fit the sum or difference of cubes pattern?
Is it a sum or difference?
Are the first and last terms perfect cubes?
2. Write them as cubes.
3. Use either the sum or difference of cubes pattern.
4. Simplify inside the parentheses.
5. Check by multiplying the factors.
##### Example $$\PageIndex{28}$$
Factor: $$27u^3−125v^3$$.
$$27 u^{3}-125 v^{3}$$ This binomial is a difference. The first and last terms are perfect cubes. Write the terms as cubes. Use the difference of cubes pattern. Simplify. Check by multiplying. We’ll leave the check to you.
##### Example $$\PageIndex{29}$$
Factor: $$8x^3−27y^3$$.
$$(2x−3y)(4x^2+6xy+9y^2)$$
##### Example $$\PageIndex{30}$$
Factor: $$1000m^3−125n^3$$.
$$(10m−5n)(100m^2+50mn+25n^2)$$
In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.
##### Example $$\PageIndex{31}$$
Factor: $$6x^3y+48y^4$$.
$$6 x^{3} y+48 y^{4}$$ Factor the common factor. $$6 y\left(x^{3}+8 y^{3}\right)$$ This binomial is a sum The first and last terms are perfect cubes. Write the terms as cubes. Use the sum of cubes pattern. Simplify.
Check:
To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by 6y.6y. We’ll leave the multiplication for you.
##### Example $$\PageIndex{32}$$
Factor: $$500p^3+4q^3$$.
$$4(5p+q)(25p^2−5pq+q^2)$$
##### Example $$\PageIndex{33}$$
Factor: $$432c^3+686d^3$$.
$$2(6c+7d)(36c^2−42cd+49d^2)$$
The first term in the next example is a binomial cubed.
##### Example $$\PageIndex{34}$$
Factor: $$(x+5)^3−64x^3$$.
$$(x+5)^{3}-64 x^{3}$$ This binomial is a difference. The first and last terms are perfect cubes. Write the terms as cubes. Use the difference of cubes pattern. Simplify. $$(x+5-4 x)\left(x^{2}+10 x+25+4 x^{2}+20 x+16 x^{2}\right)$$ $$(-3 x+5)\left(21 x^{2}+30 x+25\right)$$ Check by multiplying. We’ll leave the check to you.
##### Example $$\PageIndex{35}$$
Factor: $$(y+1)^3−27y^3$$.
$$(−2y+1)(13y^2+5y+1)$$
##### Example $$\PageIndex{36}$$
Factor: $$(n+3)^3−125n^3$$.
$$(−4n+3)(31n^2+21n+9)$$
Access this online resource for additional instruction and practice with factoring special products.
## Key Concepts
• Perfect Square Trinomials Pattern: If a and b are real numbers,
$\begin{array} {l} a^2+2ab+b^2=(a+b)^2 \\ a^2−2ab+b^2=(a−b)^2\end{array} \nonumber$
• How to factor perfect square trinomials.
$$\begin{array} {lllll} \textbf{Step 1.} &\text{Does the trinomial fit the pattern?} &\quad &\hspace{7mm} a^2+2ab+b^2 &\hspace{7mm} a^2−2ab+b^2 \\ &\text{Are the first and last terms perfect squares?} &\quad & &\\ &\text{Write them as squares.} &\quad &\hspace{5mm}(a)^2\hspace{16mm} (b)^2 &\hspace{6mm}(a)^2\hspace{16mm} (b)^2 \\ &\text{Check the middle term. Is it }2ab? &\quad &\hspace{12mm} {\,}^{\searrow}{\,}_{2·a·b}{\,}^{\swarrow} &\hspace{12mm} {\,}^{\searrow}{\,}_{2·a·b}{\,}^{\swarrow} \\ \textbf{Step 2.} &\text{Write the square of the binomial.} &\quad &\hspace{13mm} (a+b)^2 &\hspace{13mm} (a−b)^2 \\ \textbf{Step 3.} &\text{Check by multiplying.} & & & \end{array}$$
• Difference of Squares Pattern: If a,ba,b are real numbers,
• How to factor differences of squares.
$$\begin{array} {llll} \textbf{Step 1.} &\text{Does the binomial fit the pattern?} &\qquad &\hspace{5mm} a^2−b^2 \\ &\text{Is this a difference?} &\qquad &\hspace{2mm} \text{____−____} \\ &\text{Are the first and last terms perfect squares?} & & \\ \textbf{Step 2.} &\text{Write them as squares.} &\qquad &\hspace{3mm} (a)^2−(b)^2 \\ \textbf{Step 3.} &\text{Write the product of conjugates.} &\qquad &(a−b)(a+b) \\ \textbf{Step 4.} &\text{Check by multiplying.} & & \end{array}$$
• Sum and Difference of Cubes Pattern
$$\begin{array} {l} a^3+b3=(a+b)(a^2−ab+b^2) \\ a^3−b^3=(a−b)(a^2+ab+b^2) \end{array}$$
• How to factor the sum or difference of cubes.
1. Does the binomial fit the sum or difference of cubes pattern?
Is it a sum or difference?
Are the first and last terms perfect cubes?
2. Write them as cubes.
3. Use either the sum or difference of cubes pattern.
4. Simplify inside the parentheses
5. Check by multiplying the factors.
This page titled 6.3: Factor Special Products is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.
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# Telescoping Series Calculator
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A telescoping series is a series where each term u k u_k u k can be written as u k = t k − t k + 1 u_k = t_{k} - t_{k+1} u k = t k − t k + 1 for some series t k t_{k} t k. is a geometric sequence with the common factor 2. Car research starts at Edmunds! Get detailed expert vehicle reviews and ratings for every car on the market. Advanced Math - Series Convergence Calculator, Telescoping Series Test Last blog post, we went over what an alternating series is and how to determine if it converges using the alternating series test. By using this website, you agree to our Cookie Policy. The sequence of partial sums of a series sometimes tends to a real limit. The Stage V and Tier 4 Final compliant machine combines the lift height and reach of a regular telescopic loader with the material handling capacity of a classic wheeled loader. Serving drivers near Wyoming MI , Kentwood & Walker MI. Battery Size: I have this (app) application and I need it to run for (X) hours, what size battery will I need? This calculator will tell you what battery size you need. The formula is a. Since |r| < 1, the series is convergent. Central Autohaus - Dallas, Dallas, TX, 214-377-9010. Does the nth term approach 0? if not, series diverges. New 2020 BMW 3 Series 330i xDrive Sedan North America 4dr Car for sale - only$45,270. EXAMPLE 1 Determine whether the series converges. Product Series Search Enter product name or part number. Tutorial on introductory material on series. Calculate the weight needed for proper tractor set-up based on tractor type. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC. Browse other questions tagged calculus sequences-and-series telescopic-series or ask your own question. Large Display. Savings Distribution Calculator: Use this calculator to determine savings balance after a series monthly, quarterly or annual withdrawals. 7] Which gives us: [7. Explore the features and specifications of the 2021 BMW 3 Series Sedan. 2021 Super Cub C125 ABS OVERVIEW - Honda It’s the machine that changed motorcycling in America, and that put the whole world on two wheels. Given here is an online Sum of series calculator to perform summation of sequences calculation. The original formula is from Fred Puhn's book on handling called "How to make your car handle". a upper inf lower k=1 (1/k+2)-(1/k+3) b upper inf lower k=0 (1/(3k+1)(3k+4)). Our first example from above is a geometric series: (The ratio between each term is ½) And, as promised, we can show you why that series equals 1 using Algebra:. Patio doors have come a long way in their ability to enhance the look and style of your home. Sum of a Telescoping Series (II) Soledad Mª Sáez Martínez and Félix Martínez de la Rosa; The P-Series Theorem Patrick W. By using this website, you agree to our Cookie Policy. 5 for cylindrical I. Beautiful live math expression display. When you find what you think might be a telescoping series, write out some terms until you see a pattern. Custom Hoists, Inc®. Our products will be your support for any light conditions – from dawn till twilight. Python Series. We will examine Geometric Series, Telescoping Series, and Harmonic Series. Design a LPad (Driver Attenuation Circuit) to lower the response of one of your drivers. Series Geometric Senes Telescoping Series Limaçons and Lemniscates Parametric Equations—Second Derivatives and Tangent Lines Partial Fractions Convergence and Divergence Series Indexing Arithmetic of Series Integration by Parts Il Vector Functions Implicit Differentiation Il Infinite Limits of Integration Partial Fractions Ill p-Series. An optional, fully-enclosed, Metal Winter Cab, available for 700 and 900 Series Grasshopper power units, provides protection from wintery conditions for maximum comfort and productivity. Just like an extended telescope getting compressed into a smaller size, the partial sums get compressed into a. Professor Bruce H. 400 series refrigerators A fully integrated, built-in modular landscape of refrigerators, freezers, fridge-freezers and wine climate cabinets can be combined to great effect. The PRO version of Electrodoc doesn't have any ads, and it has more features than the free version. The first step is to create a plan. Looking for a primer on how to solve a telescoping series using partial fractions? See how it's done with this free video college algebra lesson. It's quite simple: You provide the calculator with some basic information about the loan, and it does the math and spits out your monthly payment. (for example 2 series in the picture) 1 Comment. Free online scientific notation calculator. B-Line series cable ladder reduces total installed cost. b) P 1 n=1 1 +1 1 +2. Consider the series. Select the first 3 or 4 bands for 20%, 10% or 5% resistors and all 5 bands for precision (2% or less), 5-band resistors. Then read the interest-only initial periods tutorial – #14. L with a telescoping design, this single curtain rod has a black finish to enhance your interior decor while making it a. Then, once you have computed the payment, click on the "Create Amortization Schedule" button to create a chart you can print out. let do cube root of 9. Instructions: Create a scatter plot using the form below. Math Help Boards: Sum Calculator. A compact design with a very powerful motor. Free Summation Calculator. Available in fully automatic (Model S4000 series - push a button to open/close the door panel and fold/unfold the sections) for ceiling heights up to 13’ 6”. Gehl brand construction, agriculture and industrial compact equipment and parts. Design a LPad (Driver Attenuation Circuit) to lower the response of one of your drivers. WEIGHT is 22. , vineland, NJ, 856-692-2448. Be able to determine the nth partial sum of any geometric series. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. p-Series Convergence The p-series is given by 1/n p = 1/1 p + 1/2 p + 1/3 p + where p > 0 by definition. In 1945, Joseph Cyril Bamford began using his engineering flair and innovative thinking to invent new construction equipment. 3 Infinite Series 19. The online tool used solve the given infinite geometric series is called as infinite geometric series. lim 8Ä_ 8 "#8 " # œ Á! _ è By the way, it is quite possible for the sum of a series to be infinite even if the terms get smaller and smaller. Algebra rules for convergent series are given below. Calculate the weight needed for proper tractor set-up based on tractor type. Our calculator can create literally hundreds of different feeding chart recipes designed to maximize peak performance and yield in your plants. Used 2017 Ford C-Max Hybrid from Jim ONeal Ford Inc in Sellersburg, IN, 47172. It is rugged and reliable, featuring heavy duty doors and toolboxes, along with a commercial grade ladder rack. 6, we get: [7. You need Hussey Seating Company MAXAM retractable telescopic gym bleachers. Volume Calculator - Wedge Enclosures 2 Measure the outside dimensions of your enclosure and the material thickness, then enter the numbers below to calculate interior volume. Estates Inc. Sum of a Telescoping Series (II) Soledad Mª Sáez Martínez and Félix Martínez de la Rosa; The P-Series Theorem Patrick W. Same app, with the same and even more features. In fact this series like the one for Achilles and the tortoise s race is convergent i. Use our simple Gear Ratio Calculator to quickly find the correct gear ratio for your ring and pinion gear set. In this section, we discuss the sum of infinite Geometric Series only. The geometric series and the telescoping series make their appearance in this chapter. Lux to Lumens Calculator The lux to lumens calculator below will help you choose the illumination solution that best meets the specific requirements of your application. The drawings in this article are based on Rohn H Series Masts. Find the value of the convergent in nite series X1 n=3 4 n2 + 6n+ 8 by using partial fractions to rewrite it as a telescoping series. King Cobra Series. The Makito™ X decoder delivers dual channel HD digital video to 3G-SDI and HDMI interfaces, supporting all HD standards up to 1080p60. At the same time, the ToolMaster Series offers the styling and finish that only a fiberglass truck cap can provide. Our mission at Metaltech is to develop, manufacture and sell safe, reliable, innovative products that make work easier and faster to execute with built in safety and sturdiness. Optionally, you can add a title a name to the axes. Since it is a geometric series, we know that it converges when \eqalign{ |x+2|/3& 1\cr |x+2|& 3\cr -3 x+2 & 3\cr -5 x& 1. Passive low pass 1st order. Research the 2020 Hyundai Elantra GT in Florence, SC at BMW of Florence. The n-th partial sum of a series is the sum of the first n terms. "Telescoping Sum. 648 - too high. 1 + 2 + … ∞), which would be impossible to evaluate without some kind of cutoff point. When you find what you think might be a telescoping series, write out some terms until you see a pattern. This is the distinction between absolute and conditional convergence, which we explore in this section. It is capable of computing sums over finite, infinite and parameterized sequences. Learn more about the 2020 BMW 3 Series. where Sk is the partial sum using the first k values of n. 3 Geometric Series; 19. This calculator will perform addition, subtraction, division, or multiplication on two given scientific notations SN, also referred to as exponential notation. 6 and ranked #5. If the ratio r lies between -1< r <1 then the series converges or else it is a diverging series. New 2020 BMW 3 Series 330i xDrive Sedan North America 4dr Car for sale - only $45,270. Since |r| < 1, the series is convergent. Consider the following series: 1 / 2 + 1 / 6 + 1 / 12 + 1 / 20 + 1 / 30 + 1 / 42 + 1 / 56 + 1 / 72 + 1 / 90 + 1 / 110 This looks rather intimidating to calculate if you don't have a computer or calculator to do the work for you; that's going to have one very large least common. and Schorn, M. Enter the sequence, the start value and end value from sigma notation and get a numerical sum. Gehl brand construction, agriculture and industrial compact equipment and parts. Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). It is capable of computing sums over finite, infinite and parameterized sequences. A sum in which subsequent terms cancel each other, leaving only initial and final terms. Parker's Cylinder Division designs and manufactures the largest selection of actuation in North America. Interactive Javascript module on generating a table of values for geometric series. The denotation for the terms in a sequence is: a1, a2, a3, a4, an. This type of series can be easily calculated since all but a few terms are cancelled out. Mobile cranes. Spacious, Suave, and stylish, the M45 offers a unique blend of performance and comfort. Some applicants, though, have no score due to lack of credit history. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your. Read here for an explanation plus a handy calculator which will convert dB values into watts and vice versa. Sequence, arithmetic series, geometric series : this page updated 19. The main purpose of this calculator is to find expression for the n th term of a given sequence. A sum in which subsequent terms cancel each other, leaving only initial and final terms. " Online Integral Calculator ». The number and variety of exercises where the student must determine the appropriate series test necessary to determine convergence of a given series has been increased. If p is equal or less than 1, it. See the 2020 BMW 5 Series price range, expert review, consumer reviews, safety ratings, and listings near you. This calculator will compute a loan's payment amount at various payment intervals -- based on the principal amount borrowed, the length of the loan and the annual interest rate. The free tool below will allow you to calculate the summation of an expression. The 2020 BMW 7-Series has a much bigger grille, but behind it is a posh, powerful, thirsty luxury sedan that’s loaded with tech. This second course in the calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented. 2 Year / 2,000 Hour Warranty. X-Series Water / Trans / Oil Temperature Gauge 0 ~ 300F / 40 ~ 150C X-Series Exhaust Gas Temperature Gauge 0 ~ 1800F / 0 ~ 1000C X-Series Voltmeter Gauge 8 ~ 18V. Partial fractions decomposition is the opposite of adding fractions, we are trying to break a rational expression Read More. It will calculate the spring rate for solid and hallow sway bars. Learn how to reduce structural supports in cable ladder installations. (These are geometric or telescoping series. Shop now and Clean Like a Pro. Main 757-838-5000; Schedule Test Drive; 2301 W Mercury Blvd Hampton, VA 23666. 1 An Introduction to Infinite Series; 19. So: 2500sin(10°) = 434 lb. Telescoping Series. Sequence, arithmetic series, geometric series : this page updated 19. In fact this series like the one for Achilles and the tortoise s race is convergent i. Results are based on CO2 emissions, fuel and taxable price. ) The first term of the sequence is a = –6. Series that converge may or may not have a known sum, however. for the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. The SXF115 is a passive subwoofer designed to extend the performance of Martin Audio WPM and MLA Mini systems down to 42Hz. The original formula is from Fred Puhn's book on handling called "How to make your car handle". Telescoping Series and Harmonic Series EXPLORE Using a Calculator to Find the Sum of a Series The Ratio and Root Tests 519 519 522 527 529 532 533 535 535 537 539. 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Simply enter your target output voltage Vout and chosen value for R1 (100 to 1000 Ohms), click Calculate, and the required value of R2 will be displayed. 3 was replaced with telescoping series that are not geometric and that require re-indexing. Objective: The student will be able to find the slope of a curve at a point, emphasizing points at which there are vertical tangents, and no tangents; find the tangent line to a curve at a point and use it for local linear approximation; understand the instantaneous rate of change as the limit of average rate of change; and approximate rate of change from. See the instructions within the documentation for more details on performing this analysis. All these terms now collapse, or telescope. Pinnacle Plus+ Process-Proven, Pulsed DC for Single-Magnetron Reactive Sputtering of Thin Dielectric Films. If then we write If the sequence s n is not convergent then we say that the series is divergent. n th-Term Test for Divergence If the sequence {a n} does not converge to zero, then the series a n diverges. where Sk is the partial sum using the first k values of n. Home equity loans are limited to$100,000 or the amount of equity you have in your home. Etiquette dictates flags must be illuminated if they remain outside at night. This calculator will determine the gage class and tolerance per ANSI ASME B89. Sometimes, people mistakenly use the terms series and sequence. Since 1973 we have partnered with our customers to design cylinders that work in several robust applications. A compilation of free financial calculators involving mortgages, loans, investments, debt, credit cards, retirement, and more, each with related information to gain in-depth knowledge. This calculator will perform addition, subtraction, division, or multiplication on two given scientific notations SN, also referred to as exponential notation. Integral Test – In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. 0L TwinPower Turbo 4-Cylinder 16V DOHC Transmission: Automatic. The 89 Series is a high-performance axial heatsinkable resistor. The drawings in this article are based on Rohn H Series Masts. all work unless otherwise indicated. Online calculator to quickly determine Water Flow Rate through an Orifice. Example: Determine whether the given series converge. You may use an approved calculator for this quiz. X-Series Water / Trans / Oil Temperature Gauge 0 ~ 300F / 40 ~ 150C X-Series Exhaust Gas Temperature Gauge 0 ~ 1800F / 0 ~ 1000C X-Series Voltmeter Gauge 8 ~ 18V. Compute eA for each of the following examples a A 0 1 0 0 b A 1 1 0 1 c A a b 0 a 7. 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Telescoping Series and Harmonic Series EXPLORE Using a Calculator to Find the Sum of a Series The Ratio and Root Tests 519 519 522 527 529 532 533 535 535 537 539. Effective January 1, 2020, the warranty period has increased for all Mustang by Manitou U. See the 2020 BMW 5 Series price range, expert review, consumer reviews, safety ratings, and listings near you. 9] Factorising gives us the formula for the series of natural numbers from n 1 to n 2: Ken. Expression: Variable. We have ensured that our track loaders provide the power and features necessary to get the job done. 6 out of 5 stars 100. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided. The n-th partial sum of a series is the sum of the first n terms. 6 and ranked #5. A compact design with a very powerful motor. It is capable of computing sums over finite, infinite and parameterized sequences. For material handling, Liebherr offers a wide variety of machines and technologies that are custom-engineered for these applications. Standard features include an ergonomically designed helm station, spacious seating options with custom dedicated storage, Fusion Satellite Ready AM/FM Stereo, as well as a wet bar with solid surface top, sink, and faucet. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2. The Slim Jim, designed by the late Fred Judd, G2BCX, can be a great portable ‘roll up’ antenna, if made out of 300Ω or 450Ω ladder line / twin feeder. In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. 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Hi-Lo is a line of telescoping travel trailers which feature a hydraulic lift system for easy transport & storage. Browse other questions tagged calculus sequences-and-series telescopic-series or ask your own question. 8œ" _ 8 #8 " SOLUTION Since , this series diverges to. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {S,}. Series Code Search A B C D E F G H I J K L M N O P Q R S T U V. 6529 for more information. Getting A USDA Loan With No Credit Score. By using this website, you agree to our Cookie Policy. PHD Optimax® brand repairable round body pneumatic cylinders are available in six bore sizes and with 10 standard travels lengths. New 2020 Ford F-150 from Metro Ford Sales Inc. Seed spacing down the row then is: 125 cm / 17. Now let's try to separate the term #1/(n(n+1)# in the sum of two terms:. View pictures, specs, and pricing & schedule a test drive today. This heavy duty series of Telescoping Masts uses 1-1/4" 16 gauge galvanized tubing for top section. com specializes in supplying the finest basketball equipment at the best prices. In the cold, in the heat and in rough terrain. The Telescoping Series! This type of infinite series utilizes the technique of Partial Fractions which is a way for us to express a rational function (algebraic fraction) as a sum of simpler fractions. by Xtend & Climb. If the ratio r lies between -1< r <1 then the series converges or else it is a diverging series. Example: Determine whether the given series converge. Used 2017 BMW 4-Series 430i in. We will now develop yet another technique which applies to convergent alternating series. 264 1080p60 with an extremely low end-to-end latency. A variety of sizes, shapes and safety features. Browse other questions tagged calculus sequences-and-series telescopic-series or ask your own question. Pinnacle Plus+ generators deliver DC power in a pulsing configuration enabling reactive sputtering of extremely uniform, high-quality dielectric films. This is a challenging sub-section of algebra that requires the solver to look for patterns in a series of fractions and use lots of logical thinking. Certified Used 2019 BMW 4 Series 2dr Car for sale - only$43,900. Find home improvement and building products for your business at Lowe's For Pros. Call (812) 246-4441 for more information. Telescoping Series. By contrast, the sequence 2, 3, 5, 8, 14, 22. Call (215) 643-3322 for more information. 2 The nth-Term Test for Divergence; 19. Solve a wide array of problems in the physical, biological, and social sciences, engineering, economics, and other areas with the skills you learn in Understanding Calculus II: Problems, Solutions, and Tips. The is no direct evidence to support the first documented use of pulleys and pulleys systems however, it is believed that primitive pulley systems were first used around 1500 BCE by the people of Mesopotamia to hoist and move water. Here's an example. Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{(2n - 1)(2n + 1)}$ is convergent or divergent. On the flip side, if you need to bring an infinite geometric series, you may use this geometric series calculator. Not for use in California** Easy Transport- Built-in wheels and telescoping carry handle, light enough to be transported by one person Lasting Power- Run time up to 14 hours(at 25% load) Safety In Mind- Low oil shutdown protects generator from damage Proven Reliability- 2-Year Briggs & Stratton Warranty. Expression: Variable. For example, the series 1, 2, 4, 8, 16. Section 11. VA loan funding fees & service requirements. Geometric Series A pure geometric series or geometric progression is one where the ratio, r, between successive terms is a constant. Relations to other figurate numbers. Course Index Area Under the Curve (Example 1). It comes collapsed at 9 ft. Clients using a TDD/TTY device: 1-800-539-8336. We offer an extensive product lineup high-load capability, and excellent dust-resistant performance to meet the needs of a variety of industries, such as machine tools, semiconductor manufacturing equipment, general industrial devices, and medical equipment. Not for use in California** Easy Transport- Built-in wheels and telescoping carry handle, light enough to be transported by one person Lasting Power- Run time up to 14 hours(at 25% load) Safety In Mind- Low oil shutdown protects generator from damage Proven Reliability- 2-Year Briggs & Stratton Warranty. Note: For an example of a telescoping sums question, see question #2 in the Additional Examples section below. 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If the infinite series of An converges, then the limit as n approaches infinity of Bn exists. Infinite geometric series is an infinite numbered series which has a common ratio ‘r’ between any two consecutive numbers in the series. However, this calculator can create a loan amortization schedule given only three of four necessary variables. Find more Mathematics widgets in Wolfram|Alpha. View pictures, specs, and pricing on our huge selection of vehicles. This calculator will help you make sure the desks you are considering will work for your height and it helps you know where to set it when you are getting started. Call (256) 536-3800 for more information. Starting their long-continuing production of camping trailers since the company’s inception, travel trailers, fifth wheel and motorhomes have. Our calculator limits your interest deduction to the interest payment that would be paid on a$1,000,000 mortgage. You may use an approved calculator for this quiz. KMHH35LE2LU119689. Instructions: Create a scatter plot using the form below. M4 Magpul Series. The first order low pass filter consists of a resistor and a capacitor connected in series. The 1/2s cancel, the 1/3s cancel, the 1/4s cancel, and so on. This is a very versatile calculator that will output sequences and allow you to calculate the sum of a sequence between a starting item and an n-th term, as well as tell you the value of the n-th term of interest. Liebherr has launched its first telescopic wheeled loader, the L 509 Tele, which is available globally. The LM317 / LM338 / LM350 series of voltage regulators can also be configured to regulate current in a circuit. Electrodoc Pro is a simple and powerful collection of electronics tools and references. The main purpose of this calculator is to find expression for the n th term of a given sequence. The first step is to create a plan. All rights belong to the owner! Sum of series. 2021 Super Cub C125 ABS OVERVIEW - Honda It’s the machine that changed motorcycling in America, and that put the whole world on two wheels. Series Rules. In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. Math Help Boards: Sum Calculator. 2 - Assessment Methods: Daily questioning, graded homework assignments, quizzes and/or tests. Telescoping series is a series where all terms cancel out except for the first and last one. ? What are some examples about Ramanujan’s hypothesis given by a(n) <0? Calculate S3,S4, and S5 and then find the sum for the telescoping series?. The calculator above will display the value, the tolerance and performs a simple check to verify if the calculated resistance matches one of the EIA standard values. Call (734) 529-5521 for more information. 13 - 5 Sums of Infinite Series. Any interest paid on first or second mortgages over this amount is not tax deductible. 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Calculate the weight needed for proper tractor set-up based on tractor type. Shots are off by Shooting from a distance of Gun Model Glock 17 Glock 17L Glock 18 Glock 19 Glock 26 Glock 34 Glock 20 Glock 29 Glock 31 Glock 32 Glock 33 Glock 25 Glock 28 Glock 22 Glock 22C Glock 23 Glock 24 Glock 24C Glock 27. The LM317 / LM338 / LM350 series of voltage regulators can also be configured to regulate current in a circuit. Hydraulic Cylinders. Ceiling heights up to 16’ can be accommodated on the semi-automatic Model S3000 series. Compute the radius of convergence and interval of convergence of a power series. Online calculator to quickly determine Water Flow Rate through an Orifice. 17Calculus - You CAN ace calculus. This calculator will compute a loan's payment amount at various payment intervals -- based on the principal amount borrowed, the length of the loan and the annual interest rate. (These are geometric or telescoping series. Determine the convergence or divergence of the series and nd the exact sum if the series converges. It will calculate the spring rate for solid and hallow sway bars. We will now develop yet another technique which applies to convergent alternating series. Previous Next Consistently the first to market with new innovations in venting systems, DuraVent has captured a leadership position in emerging markets. At the same time, the ToolMaster Series offers the styling and finish that only a fiberglass truck cap can provide. and Riese, A. The Unexposed Secret of Series Calculator. Answer: Let a n = 1=(n 3), for n 4. ) A telescoping series. HOW TO CHOOSE YOUR FRONT SIGHT HEIGHT. Python Series. com allows you to find the sum of a series online. Series – user3812887 Dec 4 '16 at 11:40 If n = 5; I need to have 1 + 3 + 7 + 13 + 21 = 45; the Matlab function can take only one input; and that is n. 4 Taylor series:. 8] Collecting like terms: [7. So, the sum of the series, which is the limit of. 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In 1953, the JCB backhoe loader revolutionised the construction industry. WEIGHT is 22. Sum of a Telescoping Series (II) Soledad Mª Sáez Martínez and Félix Martínez de la Rosa; The P-Series Theorem Patrick W. Try the given examples, or. In addition, the performance of existing systems with lower pressures can be upgraded with a simple retrofit of the pre-existing high flow nozzles. It may contain constants, variables, certain "well-known" operations (e. Starting their long-continuing production of camping trailers since the company’s inception, travel trailers, fifth wheel and motorhomes have. For any series, if there is a formula for the nth partial sum, s n, the exact sum is n n s s o f lim (2) if the limit exists. Series OCG Series OCG Round Body Pneumatic Cylinder. In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. See the instructions within the documentation for more details on performing this analysis. Pneumatic Cylinders from AutomationDirect - the best value in industrial automation. Nov 08 2013 Convergence amp Divergence Geometric Series Telescoping Series Harmonic Series Divergence Test Duration 50 43. Find home improvement and building products for your business at Lowe's For Pros. New 2020 Ford F-150 from Chapman Ford Lancaster, PA in Lancaster, PA, 17601. Toll Free: (800) 478-3790. If the ratio r lies between -1< r <1 then the series converges or else it is a diverging series. Click here to find the nearest Unger distributor near you. These come in a wider range of styles and colors and mostly use metal build materials. Trim: 320i Sedan Color: Pearl White Interior: Leather Engine: 2. " I wanted to use the telescoping proof, which shows that if An exists then the limit of the partial sums exist. Standard cushions reduce end of travel impact. Infinite geometric series is an infinite numbered series which has a common ratio ‘r’ between any two consecutive numbers in the series. 6 out of 5 stars 100. and Riese, A. Be able to determine the nth partial sum of any geometric series. See the instructions within the documentation for more details on performing this analysis. Featured Listing. Hiking gets you out into the great outdoors, as well as being a good workout. Large Display. Results are based on CO2 emissions, fuel and taxable price. Large Display. Usually, a given power series will converge (that is, approach a finite sum) for all values of x within a certain interval around zero—in particular, whenever the absolute value of x is less than some positive number r, known as the radius. Telescoping. Added Apr 14, 2011 by HighOPS in Mathematics. Show Hide all comments. The 1/2s cancel, the 1/3s cancel, the 1/4s cancel, and so on. All you have to do is type your X and Y data and the scatterplot maker will do the rest. On the flip side, if you need to bring an infinite geometric series, you may use this geometric series calculator. See full list on tutorial. Determine if each of the following series converges and, for the convergent series, compute the sum of the series. The original formula is from Fred Puhn's book on handling called "How to make your car handle". Solve Calculus problems stepwise using the Ti-Nspire Calculator. Compare Two CDs Calculator. Low prices, fast shipping, satisfaction guaranteed. Taylor series [11/3/1994] Please describe the Taylor series. Questions separated by topic from Further Pure 1 Maths A-level past papers. How to Use an Online Curve PlotterContents1 How to Use an Online Curve Plotter1. A telescoping series does not have a set form, like the geometric and p-series do. For example, using partial fractions and cancelling a bunch of terms, we find that; An infinite series that arises from Parseval’s theorem in Fourier analysis. 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2020-10-22 03:16:38
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https://mathhelpboards.com/threads/linear-recurrence-sequences.231/
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Linear recurrence sequences
Casio
Member
I have a linear recurrence sequence,
3, -1.5, 0.75, -375
x = a
a = 3
x2, = -1.5, x3, = 0.75, x4 = -375...
x2 = rx1+d
x3 = rx2+d
-1.5 = 3r + d
0.75 = -1.5 + d
-1.5 - 0.75 = (3r + d) - (-1.5 + d)
r = - 0.5
Sub in equation (2)
d = -1.5 - 3r = -1.5 - 3(-0.5)
d = 0
x4 = -0.5 x 0.75 + 0 = -0.375
Anyone see where I am going wrong!
P.S. I will practice the new form latex here as soon as I can.
Alexmahone
Active member
-1.5 = 3r + d
0.75 = -1.5 + d
Anyone see where I am going wrong!
Shouldn't that 2nd equation be 0.75 = -1.5r + d ?
Casio
Member
Shouldn't that 2nd equation be 0.75 = -1.5r + d ?
Thank you for that, typo error on my part, the result there = 0.75
But that is not helping me see where I am making a mistake?
Alexmahone
Active member
Thank you for that, typo error on my part, the result there = 0.75
But that is not helping me see where I am making a mistake?
So your sequence is a geometric progression with a=3 and r=-0.5. So I guess your answer of -0.375 is correct.
Last edited:
Casio
Member
So your sequence is a geometric progression with a=3 and r=-0.5. So I guess your answer of 0.375 is correct.
It seems my misunderstanding here is the answer = -0.375
I thought the answer should have been -375
These solutions are not the same?
soroban
Well-known member
Hello, Casio!
I have a linear recurrence sequence: .$3,\;\text{-}1.5,\;0.75,\;\text{-}375$ .??
There is NO WAY that the sequence is: .$3,\;-1\!\!\frac{1}{2},\;\frac{3}{4},\;\color{red}{-375}\;\cdots$
. . That last term must be -0.375 . . .
We have a geometric sequence with first term $a = 3$ and common ratio $r = -0.5$
The recurrence is: .$a_{n+1} \:=\:-0.5a_n$
Casio
Member
Hello, Casio!
There is NO WAY that the sequence is: .$3,\;-1\!\!\frac{1}{2},\;\frac{3}{4},\;\color{red}{-375}\;\cdots$
. . That last term must be -0.375 . . .
We have a geometric sequence with first term $a = 3$ and common ratio $r = -0.5$
The recurrence is: .$a_{n+1} \:=\:-0.5a_n$
Very sorry, you are right I copied the sequence incorrectly?
Ackbach
Indicium Physicus
Staff member
Very sorry, you are right I copied the sequence incorrectly?
Is that a statement or a question?
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2021-01-23 05:19:36
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https://community.babycenter.com/post/a76882838/same-due-date
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# Same due date
Posted 12/28/2020
Isn’t it crazy how i got pregnant at the same time last year #RIH nyasr mommys lovee you always.. well here i am pregnant again and my bbg this time has the same due dat as my son 🥰 this is meant to be ❤️❤️
1
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Only files 8MB or smaller of the following types are supported: JPEG, PNG, GIF
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2021-01-24 07:02:07
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{"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": 0.8439115881919861, "perplexity": 11145.121525391549}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703547333.68/warc/CC-MAIN-20210124044618-20210124074618-00247.warc.gz"}
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https://tex.stackexchange.com/questions/395011/more-and-less-space-in-cases
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# More and Less Space in Cases [duplicate]
Why do I have this space (red arrow)? Why I do not have more space (orange arrow)?
I use cases and doublespacing.
Even I use \everydisplay\expandafter{\the\everydisplay\setstretch{1}}, because of doublespacing, as suggested in Matrix within equation, the problem does not solve.
I have tried array, dcases, drcases, but it does not work.
Thank you!
## marked as duplicate by Mico, Troy, Mensch, Schweinebacke, diabonasOct 7 '17 at 17:27
• @Mico I have not issued any command \renewcommand\baselinestretch{2}. I use the harvard-thesis.cls and I use doublespacing. – Vassilis Chasiotis Oct 7 '17 at 15:26
• Ah, the piece of information related to the use of the setspace package and the \doublespacing directive is absolutely crucial. – Mico Oct 7 '17 at 15:31
• You should study the query Matrix within equation and, in particular, the answer provided by Herbert, viz., issuing the instruction \everydisplay\expandafter{\the\everydisplay\setstretch{1}} in the preamble. – Mico Oct 7 '17 at 15:46
You can have some symmetry using \bigstrut but you get a better result with some caution loading empheq and using the eponymous environment:
\documentclass[12pt, a4paper]{article}
\usepackage{setspace}
\usepackage{empheq, bigstrut}
\begin{document}
\doublespacing
With \texttt{ dcases: }
$\begin{cases} x + y = q^2 \\ \biggl[\dfrac{N}{q^2}\biggr]_{\bigstrut[b]} x + \biggl(\biggl[\dfrac{N}{q^2}\biggr] + 1\biggr)y = N \end{cases}$%
With \verb|\empheqlbrace:|
\begin{empheq}[left=\empheqlbrace]{align*}
& x + y = q^2\\
& \biggl[\dfrac{N}{q^2}\biggr]x + \biggl(\biggl[\dfrac{N}{q^2}\biggr]_{\bigstrut[b]} + 1\biggr)y = N
\end{empheq}
\end{document}
• Note that the OP has revealed a crucial piece of information in the meantime, viz., that he/she uses the setspace package and a \doublespacing directive. Do you want to adjust your answer accordingly? – Mico Oct 7 '17 at 15:33
• @Bernand It works perfect with \empheqlbrace. But I need to use \\[-2.6ex] instead of \\[-2ex] to be symmetric. Maybe because of \doublespacing as @Mico said. – Vassilis Chasiotis Oct 7 '17 at 15:43
• Probably. The exact value has to be found by trial and error. Normally, there would be no correction to do, but here there's a discrepancy between the two rows. – Bernard Oct 7 '17 at 15:45
• @VassilisChasiotis: I've found a better solution, using bigstrut. Please see my updated answer. – Bernard Oct 7 '17 at 16:07
• @Bernard It is much better! I can also use align,align* with numbering easier. Thank you! – Vassilis Chasiotis Oct 7 '17 at 16:13
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2019-08-21 03:20:17
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https://mailman.ntg.nl/pipermail/ntg-context/2015/082124.html
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# [NTG-context] question about \xmlconcat
Pablo Rodriguez oinos at gmx.es
Sun May 10 18:36:16 CEST 2015
On 05/10/2015 05:43 PM, Hans Hagen wrote:
> On 5/10/2015 5:36 PM, Pablo Rodriguez wrote:
>> [...]
>> What I think I understand now is that it is impossible (it doesn’t make
>> sense) to concatenate elements (pardon my inaccuracy) inside one of
>> these elements.
>
> indeed, in that case you need to set a mode after the first and act upon
> that state
>
> (btw, don't use sequential - stream based - flushing for everything,
> sometimes it makes more sense to flush explicitly)
Many thanks for the explanation and your advice, Hans.
I suspect that the explicit flush requires to have a clearer element
tree than the one I have in mind (I’m still eating some leafs :-)).
Many thanks for your help,
Pablo
--
http://www.ousia.tk
More information about the ntg-context mailing list
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2022-05-19 12:29:22
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http://physics.stackexchange.com/questions/51653/looking-backwards-in-time-at-yourself
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# Looking backwards in time at yourself
If a person on Earth today is looking at a star, say, 10 billion light years away, is it possible that some of the atoms he is looking at will eventually go on to make him?
-
If the atoms travel at half the speed of light, wouldn't you have to live another 10 billion years for them to reach you? – RedGrittyBrick Jan 19 '13 at 22:42
If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Jan 20 '13 at 6:26
As pointed out, the atoms of that star would have had to travel faster than light to reach earth in time to form you. But if you really want to look back in time at yourself, a mirror will do the trick. And the farther away you stand from it, the further back in time you will see yourself. :) – Wouter Jan 20 '13 at 10:21
Thanks team! Wouldn't it be incredible to 'see' the reflected light rays from Earth's history today?? We could point our cameras at a big mirror somewhere in the universe and 'see' the Library of Alexandria, the dinosaurs etc etc!! – wideangle123 Jan 20 '13 at 10:56
Hi T-Fox. I don't provide contradiction to your quote: "Nothing travels faster than light". Actually, objects with zero mass, can travel at $c$. Whilst on the other hand, objects having negative mass, can travel faster than light. By saying mass, I really meant rest mass. The problem is, we haven't observed any superluminal objects yet and we won't. Maybe that's the reason I didn't argue with your quote ;-) – Waffle's Crazy Peanut Jan 20 '13 at 5:09
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2015-05-29 00:21:01
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http://www.chegg.com/homework-help/questions-and-answers/line-charge-uniform-density-lextends-z-l-2-z-l-2-along-z-axis-apply-coulomb-slaw-obtain-ex-q683519
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A line of charge with uniform density ρlextends between z=-L/2 and z=L/2 along the z-axis. Apply Coulomb'slaw to obtain an expression for the electric field at any pointP(r,φ,0) on the x-y plane. Show that your result reduces tothe expression given by E = D / ε0 = rDr / ε0 = r ρl /2πε0r as the length L is extended toinfinity. (r is properly defined as the radial distance vector fromthe line charge to the observation point i.e. r is perpendicular toline of charge)
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2015-08-03 19:24:57
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https://physics.stackexchange.com/questions/294105/do-photons-have-absolute-motion
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# Do photons have absolute motion?
Absolute rest is not possible as the concept of motion is relative. But can we assert that photons have absolute motion as the observers in all the frames of reference would agree to the same value of speed of light.
Do we consider frames of references moving at speed of light, if we do than above assertion won't be correct.
• By tagging this with general-relativity you open a whole 'nother can of worms. There are perfectly good ways of looking at things in GR when the speed of light at places distant from you is not constant. – dmckee Nov 22 '16 at 5:52
• We cannot consider frames of references moving at speed of light because Lorentz transofmation formulas become undefined at $v=c$. We can only take $v<c$ and try to obtain some limiting results at $v\rightarrow c$, but they should be interpreted with caution. – Alexey Sokolik Nov 22 '16 at 9:25
• – ACuriousMind Nov 22 '16 at 14:01
• Also related: physics.stackexchange.com/q/16018/2451 – Qmechanic Nov 23 '16 at 8:14
In a certain sense you are right: there is no inertial frame of reference in which a photon propagating in vacuum is locally at rest.
When measured locally and in an inertial frame, the speed of light is $c$ independently of the inertial frame we choose: this is one of the postulates of special relativity.
Notice anyway the keyword locally: on larger scales, where spacetime cannot be considered flat, we have to use the formalism of general relativity, and things may be very different: see for example this question & answer and also this one.
Also, it is important that we choose an inertial frame: if the frame of reference is not inertial, the speed of light may be different from c.
The similar behaviors found in the presence of gravitational fields and of non-inertial frames is no coincidence: the equivalence principle actually tells us that the forces experienced in a gravitational field are the same as those expereinced in a non-inertial frame of reference.
All photons move on individual vectors through inertial space at exactly C. From this undisputed experimental fact, we can conclude that all photons move within the same inertial reference frame. This assumption leads to the conclusion that each photon is a material body with an intrinsic mass (m=p/c) and dimension (=h/mc) within this frame of absolute photon rest. Any motion of a photon’s source, relative to rest, has no effect on the photon’s velocity, but transforms its intrinsic mass and wavelength in proportion to the absolute motion of the source. This same transformation occurs between the photon’s intrinsic momentum at c and its observed momentum of . The Doppler effect allows the accurate measurement of any difference in velocity along a vector between source and observer, but it prevents any measurement of the three intrinsic inertial vectors of source, observer, and photon. To verify the existence of this elusive preferred reference frame, and to measure its position, the observer must look to the results of the Michelson-Morley experiment, the Uhuru pulsar observations and the Pound-Rebka measurements.
I would just comment but I don't have enough reputation (sorry ;))
• Welcome on Physics SE and thank you for the contribution :) You might want to see this help page for typesetting formulas :) – Sanya Nov 23 '16 at 10:20
• There are two problems with your first statement: it is difficult, if not impossible, to make any sense of the notion of an inertial frame moving at $c$. Secondly, even if we were talking about a group of objects moving at the same speed $v<c$ relative to some observer, one cannot infer that they share the same inertial frame from their all moving at the same speed $v$. The directions of the velocity vectors can be different, in which case so are the inertial frames. – WetSavannaAnimal Nov 23 '16 at 10:56
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2019-10-21 13:34:50
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https://math.stackexchange.com/questions/3278919/how-to-graph-this-derivative
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How to graph this derivative?
Problem:
The volume of a cylinder equals 𝑉 cubic inches, where 𝑉 is a constant. Find the proportions of the cylinder that minimize the total surface area.
I know how to get the answer to this problem. What I have trouble with is visualizing what the graph of $$\frac{dS}{dr}$$ is if $$S(r)$$ is the total surface area as a function of the radius. The equation for $$\frac{dS}{dr}$$ is $$\frac{dS}{dr}=\frac{4\pi r^3-2V}{r^2}$$ and since $$V=\pi r^2h$$, shouldn't the total surface area be a function of both the radius and the height, so basically $$S(r,h)$$.
• You can use a $3$ dimensional surface with axes $S,r,h$. Why not ask the same question for $V$ itself which depends on both $r$ and $h$? – Peter Foreman Jun 30 '19 at 16:16
• What you want to graph $S(r,h)$ or dS/dr ? – Ajay Mishra Jun 30 '19 at 16:29
• @AjayMishra $\frac{dS}{dr}$, but how do I graph this if I don't know what $V$ is? – user532874 Jun 30 '19 at 16:39
• @user532874 see my answer. – Ajay Mishra Jun 30 '19 at 16:45
Here, Green axis : $$h$$, Red axis: $$r$$
Since, $$\cfrac{dS}{dr} = \cfrac{4 \pi r^3 - 2V}{r^2} = \cfrac{4 \pi r^3 - 2 \pi r^2 h}{r^2} = 2 \pi ( 2r - h)$$ If I define $$z = \cfrac{dS}{dr}$$ ,then $$z = 2 \pi (2r-h)$$ is just equation of a plane.
• How can $\frac{dS}{dr}$ have just $dr$ on the bottom when both $r$ and $h$ are inputs? – user532874 Jun 30 '19 at 16:49
• Oh, it is just $\cfrac{ \partial S}{ \partial r}$ . Are you familiar with partial derivatives? – Ajay Mishra Jun 30 '19 at 16:53
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2020-06-03 22:10:13
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https://crypto.stackexchange.com/questions/19163/salsa20-is-invertible-useless-in-ctr-mode/19165
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# salsa20 is invertible, useless in CTR mode?
I'm reading the salsa20 spec (sorry for the newbie question, i'm really a newbie here). Every paragraph ends with : "this operation is invertible", I suppose the whole salsa20 algorithm is.
1. If I use salsa20 in bloc cipher CTR mode, this property does not seems mandatory to me, am I right?
2. As such, any cryptographically-robust hash function (one-way only) could be used for CTR encryption?
3. Isn't it easier to design a robust hash than a robust cipher?
• How exactly do you use salsa20 in block cipher CTR mode? Salsa20 is not a block cipher, it is a stream cipher. – mikeazo Sep 16 '14 at 13:20
• @mikeazo, Salsa20 runs the "Salsa20 hash function" in "CTR mode". The terminology there is a bit confusing, especially as "hash function" doesn't mean cryptographic hash, so it's no wonder if the question isn't exactly correctly stated. – otus Sep 16 '14 at 13:29
• yup, sorry for the confusion. One said : What is well understood can be clearly expressed :) It's not well understood yet. Both answer were perfect, I picked one. Thank you very much – yota Sep 16 '14 at 13:49
• @mikeazo More of an aside: While it’s indeed a stream cipher, there are some pages that add to the confusion by hinting at the fact that “Salsa20/r resembles the operation of a block cipher in counter mode.” (source: ecrypt.eu.org, emphasis mine). – e-sushi Sep 16 '14 at 14:32
## 2 Answers
Every paragraph ends with : "this operation is invertible", I suppose the whole salsa20 algorithm is.
The Salsa20 quarterround, and thus rowround, columnround and doubleround are invertible. However, the whole Salsa20 core is not because the initial state is added to the state after iterating the rounds (cf. page 6 in the spec).
1. If I use salsa20 in bloc cipher CTR mode, this property does not seems mandatory to me, am I right?
If the whole cipher core was invertible, then a known plaintext attack would reveal the key. So not being invertible is a requirement, somewhere along the line.
1. As such, any cryptographically-robust hash function (one-way only) could be used for CTR encryption?
This is correct. See, for example this question.
1. Isn't it easier to design a robust hash than a robust cipher?
Maybe, but block ciphers are usually much faster than a hash function in CTR mode. Hash functions are optimized to consume data, not output it. Stream ciphers, OTOH, are occasionally even faster, like you can see if you compare Salsa20 and AES in software implementations.
## How does Salsa20 work?
1. The basic building block of salsa20 is a fixed 512 bit permutation. This is similar to a block cipher with a fixed and publicly know key (or a zero bit key if you prefer). Since it has no key input, you can't use it with block cipher modes of operation.
2. The next step in Salsa20 is a feed-forward by adding the input into the output, turning it into an unkeyed hash function/one-way-function.
If $P(x)$ is a permutation and thus reversible, $P(x)\oplus x$ is one way.
This is the standard way for turning a reversible function into an irreversible one, and is often used to build compression functions.
3. The input of this 512 bit function gets divided into several parts: A 256 bit key, an 8 byte stream identifier (IV) an 8 byte counter and 16 bytes of fixed padding. Note that the padding is necessary to remove certain symmetries from the raw hash function.
This turns the unkeyed hash into a keyed one, formally a PRF (pseudo random function)
4. Finally to produce a long output Salsa20 calls the core function with a different counter for each 512 bit block. This is very similar to the CTR mode used with block ciphers.
## To address your questions:
2) As such, any cryptographically-robust hash function (one-way only) could be used for CTR encryption?
A generalized counter mode can be used with any PRF, which is the formalization of a keyed hash.
Building a PRF from an unkeyed hash is easy in practice and secure for most hashes we use, such as SHA-2 or SHA-3. Theoretically it's tricky and probably needs the random oracle assumption.
1) If I use Salsa20 in block cipher CTR mode, this property does not seems mandatory to me, am I right?
If you interpret Salsa20 as a 512 bit block cipher it's clearly not secure, since it has a zero bit key and undesirable symmetries on top of that. So you can't use it in CTR mode as it's defined for block ciphers.
On the other hand, the Salsa20 function as described in Step 3 above is believed to be a PRF, and Step 4 is a kind of counter mode, so in a way you're always using Salsa20 in CTR mode.
3) Isn't it easier to design a robust hash than a robust cipher?
Pretty much all hashes we use are built from a permutation, keyed (aka block cipher) or unkeyed. The key idea is that the irreversible operation is where collisions happen, so we want only a single one of those and it should be as far as possible from what the attacker controls.
So I don't think it's easier to build a hash than a permutation or block cipher. In some ways it's harder, since we often request collision resistance from hashes, whereas ciphers are generally used with random keys the attacker has no control over.
• I'm not sure to understand how block cipher CTR is (or not) alike salsa20 stream cipher. Looking at the spec, and at en.wikipedia.org/wiki/… I can not spot the differences, neither why you can't use "salsa20 in CTR mode as it's defined for block ciphers". Isn't the salsa20 initial state precisely made of the key, and a CTR index (+ nonce) ? – yota Sep 18 '14 at 11:45
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2020-01-29 20:27:39
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http://bioconductor.riken.jp/packages/devel/bioc/vignettes/RVS/inst/doc/RVS.html
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# 1 Introduction
Rare Variant Sharing (RVS) implements tests of association and linkage between rare genetic variant genotypes and a dichotomous phenotype, e.g. a disease status, in family samples (Sherman et al. (2018)). The tests are based on probabilities of rare variant sharing by relatives under the null hypothesis of absence of linkage and association between the rare variants and the phenotype and apply to single variants or multiple variants in a region (e.g. gene-based test).
# 2 Setting up Pedigree Data
For this example experiment we will consider four family types. A pair of first cousins, a pair of second cousins, a triple of first cousins, and a triple of second cousins. RVS comes with several example pedigrees and these four types can be found in the samplePedigrees list.
data(samplePedigrees)
# store the pedigrees
fam_type_A <- samplePedigrees$firstCousinPair fam_type_B <- samplePedigrees$secondCousinPair
fam_type_C <- samplePedigrees$firstCousinTriple fam_type_D <- samplePedigrees$secondCousinTriple
# re-label the family ids for this example
fam_type_A$famid <- rep('SF_A', length(fam_type_A$id))
fam_type_B$famid <- rep('SF_B', length(fam_type_B$id))
fam_type_C$famid <- rep('SF_C', length(fam_type_C$id))
fam_type_D$famid <- rep('SF_D', length(fam_type_D$id))
## 2.2 Plotting a Pedigree
In order to see the pedigree structure we can use the plot function provided by the kinship2 package. In this family we have three second cousins that have been sequenced.
plot(fam_type_D)
# 3 Calculating Sharing Probabilities
## 3.1 Sharing Probability for One Family, One Variant
The simplest use of the RVS package is to compute the probability that all sequenced subjects in a pedigree share a rare variant, given that it is seen it at least one of the subjects. For more information about this calculation see the documentation for the function RVsharing and the Appendix.
In this case we compute the probability for the family of three second cousins. Note that in the case of a single family and variant, the sharing probability can be interpreted as a p-value.
p <- RVsharing(fam_type_D)
## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342
## 3.2 P-Value for Multiple Families, One Variant
In the case of a single variant seen across multiple families, we can compute the individual sharing probabilities with RVsharing, but the sharing probabilities can no longer be interpreted as a p-value for the sharing pattern of the variant across the families. The function multipleFamilyPValue can be used to compute the p-value which is defined as the sum of all sharing probabilities across families at most as large as the sharing probability observed.
# compute the sharing probabilities for all families
fams <- list(fam_type_A, fam_type_B, fam_type_C, fam_type_D)
sharing_probs <- suppressMessages(RVsharing(fams))
signif(sharing_probs, 3)
## SF_A SF_B SF_C SF_D
## 0.06670 0.01590 0.01180 0.00134
# compute p-value for this sharing pattern
sharing_pattern <- c(TRUE, TRUE, FALSE, FALSE)
names(sharing_pattern) <- names(sharing_probs)
multipleFamilyPValue(sharing_probs, sharing_pattern)
## [1] 0.002125543
The sharing_pattern vector indicates whether or not the variant is observed in all sequenced subjects.
## 3.3 P-Value for Multiple Families, Multiple Variants
The function multipleVariantPValue generalizes multipleFamilyPValue across multiple variants. This function takes a SnpMatrix instead of a specific sharing pattern. The behavior of this function could be achieved by converting every column of a SnpMatrix into a sharing pattern across families and applying multipleFamilyPValue across the columns.
The first step is reading in the genotype data. See the Data Input vignette in the snpStats package for examples using different file types. Here we use a pedigree file in the LINKAGE format. See here for an example of reading genotypes data from a Variant Call Format (VCF) file.
pedfile <- system.file("extdata/sample.ped.gz", package="RVS")
sample <- snpStats::read.pedfile(pedfile, snps=paste('variant', LETTERS[1:20], sep='_'))
In this data set we have 3 copies of each family type. The sharing probabilities for this set of families are:
A_fams <- lapply(1:3, function(i) samplePedigrees$firstCousinPair) B_fams <- lapply(1:3, function(i) samplePedigrees$secondCousinPair)
C_fams <- lapply(1:3, function(i) samplePedigrees$firstCousinTriple) D_fams <- lapply(1:3, function(i) samplePedigrees$secondCousinTriple)
fams <- c(A_fams, B_fams, C_fams, D_fams)
famids <- unique(sample$fam$pedigree)
for (i in 1:12)
{
fams[[i]]$famid <- rep(famids[i], length(fams[[i]]$id))
}
sharingProbs <- suppressMessages(RVsharing(fams))
signif(sharingProbs, 3)
## SF_A1 SF_A2 SF_A3 SF_B1 SF_B2 SF_B3 SF_C1 SF_C2 SF_C3 SF_D1
## 0.06670 0.06670 0.06670 0.01590 0.01590 0.01590 0.01180 0.01180 0.01180 0.00134
## SF_D2 SF_D3
## 0.00134 0.00134
When we call the function on the genotypes from a snpMatrix as follows, it converts them into a sharing pattern assuming the rare variant is the allele with the lowest frequency in the family sample:
result <- multipleVariantPValue(sample$genotypes, sample$fam, sharingProbs)
## Ignoring 0 variants not present in any subject
ppvals <- result$potential_pvalues ppvals_sub <- ppvals[names(pvals)] # subset potential p-values plot(-log10(ppvals[order(ppvals)]), ylab="-log10 p-value", col="blue", type="l", xaxt="n", xlab="variants", ylim=c(0,8)) xlabel <- sapply(names(ppvals)[order(ppvals)], function(str) substr(str, nchar(str), nchar(str))) axis(1, at=1:length(ppvals), labels=xlabel) points(-log10(pvals[order(ppvals_sub)]), pch=20, cex=1.3) bcut <- 0.05/(1:20) lines(1:20,-log10(bcut),col="red",type="b",pch=20) ## 3.5 Minor Allele Frequency Sensitivity Analysis When the minor allele frequency (MAF) is known in the population, then an exact sharing probability can be calculated using the alleleFreq parameter. Here we analyze the sensitivity of our p-values to the population MAF, using the 3 most significant variants. Note that variants which don’t reach their potential p-values, indicating some families only have partial sharing, are much more sensitive to the MAF. # calculate p-values for each MAF freq <- seq(0,0.05,0.005) variants <- names(sort(result$pvalues))[1:3]
pvals <- matrix(nrow=length(freq), ncol=length(variants))
pvals[1,] = sort(result$pvalues)[1:3] for (i in 2:length(freq)) { sharingProbs <- suppressMessages(RVsharing(fams, alleleFreq=freq[i])) pvals[i,] <- multipleVariantPValue(sample$genotypes[,variants], sample$fam, sharingProbs)$pvalues
}
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
## Ignoring 0 variants not present in any subject
colnames(pvals) <- variants
# plot p-values as a function of MAF
plot(NULL, xlim=c(min(freq),max(freq)), ylim=c(0,max(pvals)), type='l',
xlab="minor allele frequency", ylab="p-value",
main="sensitivity of p-value to allele frequency in three variants")
lines(freq, pvals[,1], col="black")
lines(freq, pvals[,2], col="red")
lines(freq, pvals[,3], col="blue")
legend(min(freq), max(pvals), legend=colnames(pvals), col=c("black", "red", "blue"), lty=1)
# 4 Joint analysis of multiple variants
The power of single variant analyses is limited due to the small number of families where a rare variant is seen (often a single family). Even if no individual variant has a significant p-value, it is possible for multiple variants considered across multiple families to exhibit an unusual amount of sharing. The procedure to test multiple rare variants depends whether the variants are far apart or close together in a short genomic region, spanning a single gene for instance.
## 4.1 Enrichment Test
The enrichmentPValue function can compute a single p-value for all variants seen in all families assuming the variants are independent. This assumption is reasonable when variants are sufficiently far apart to be unlinked, such as rare deletions scattered over the whole genome as analyzed by Fu et al. (2017). The computation is implemented using a binary tree algorithm described by Fu et al. (2017). When calculating this p-value, note that a very small p-value may result in a very long computation time. Because of this, we can pass a minimum p-value threshold, where the greater of this threshold and the actual p-value will be returned.
enrichmentPValue(sample$genotypes, sample$fam, sharingProbs, 0.001)
## Ignoring 0 variants not present in any subject
## [1] 0.001
## 4.2 Gene-based analysis
Joint analysis of rare variants within a gene (typically single nucleotide variants and short indels, possibly filtered based on functional annotations) is another approach to increase statistical power. Here the assumption of independence of rare variants does not hold when variants are seen in the same family, and the solution described by Bureau et al. (2018) and implemented in the RVgene function is to keep the variant with the sharing pattern having the lowest probability (usually the variant shared by the largest number of affected relatives in the family). The gene-based analysis with the RVgene function is illustrated in section 4.2 along with another new feature: the partial sharing test.
# 5 Partial sharing test
Phenocopies, diagnosis error and intra-familial genetic heterogeneity in complex disorders result in disease susceptibility variants being shared by a subset of affected subjects. In order to detect such causal variants, a partial sharing test was defined by Bureau et al. (2018) where the p-value is the probability of sharing events as or more extreme as the observed event. A more extreme sharing event is defined as having lower probability and involving more carriers of the variant.
## 5.1 Precomputing Sharing Probabilities and Number of Carriers for all Possible Carrier Subsets
In order to perfore the partial sharing test, the RVgene function requires the lists pattern.prob.list of vectors of sharing probabilities and N.list of number of carriers for all possible affected carrier subsets in each family in the sample being analyzed. The arguments of the RVsharing function allowing the computation of sharing probabilities by a subset of affected subjects are described here. The elements of both of these lists must have the same names as the pedigree objects in the ped.listfams argument. When all affected subjecs in a family are final descendants, the sharing probabilities and number of carriers for all subsets can be generated automatically. Here is an exanple with three second cousins:
carriers = c(15,16,17)
carrier.sets = list()
for (i in length(carriers):1)
carrier.sets = c(carrier.sets, combn(carriers,i,simplify=FALSE))
fam15157.pattern.prob = sapply(carrier.sets,function (vec) RVsharing(samplePedigrees$secondCousinTriple,carriers=vec)) ## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342 ## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396 ## Probability subjects 15 17 among 15 17 16 share a rare variant: 0.009396 ## Probability subjects 16 17 among 16 17 15 share a rare variant: 0.009396 ## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235 ## Probability subjects 16 among 16 15 17 share a rare variant: 0.3235 ## Probability subjects 17 among 17 15 16 share a rare variant: 0.3235 fam15157.N = sapply(carrier.sets,length) When the splitPed option is TRUE, the generation of all carrier subsets is performed within the RVsharing function, which then returns the vector of sharing probabilities for all subsets. So the following code is equivalent to the sapply of RVsharing above: fam15157.pattern.prob = RVsharing(samplePedigrees$secondCousinTriple,splitPed=TRUE)
## Founder 1 subped size 17
## Founder 4 subped size 5
## Founder 6 subped size 5
## Founder 8 subped size 5
## Founder 10 subped size 3
## Founder 12 subped size 3
## Founder 14 subped size 3
## Probability every subset of subjects among 15 16 17 share a rare variant:
## [1] 0.001342 0.009396 0.009396 0.009396 0.323500 0.323500 0.323500
While this code applies to any configuration of affected final descendants, symmetries in the relationships of these third cousins results in equal sharing probabilities for multiple subsets. Subsets with the same probabilities are equivalent, and the optional argument nequiv.list can be used to indicate the number of equivalent subset for each sharing probability. While shorter vectors in pattern.prob.list and N.list result in more efficient computation, identification of the equivalent subsets is not easily automated, and will usually require custom code for each pedigree in a sample. With three second cousins we can use:
fam15157.pattern.prob = c(RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16,17)), RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15))) ## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342 ## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396 ## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235 fam15157.N = 3:1 fam15157.nequiv = c(1,3,3) It is then easy to check that the distribution sums to 1: sum(fam15157.pattern.prob*fam15157.nequiv) ## [1] 1 When some affected subjects are not final descendants, some subsets are incompatible with a variant being IBD among carriers. Assume individual 3, the grand-father of subject 15 in family 15157, is also affected and his genotype is available. fam15157 <- samplePedigrees$secondCousinTriple
fam15157$affected[3] = 1 plot(fam15157) Then the carrier subsets (15,16,17), (15,16) and (15,17) involving subject 15 but not 3 are incompatible with sharing IBD and must be removed from the list of subsets. The code then becomes: carriers = c(3,15,16,17) carrier.sets = list() for (i in length(carriers):1) carrier.sets = c(carrier.sets, combn(carriers,i,simplify=FALSE)) carrier.sets ## [[1]] ## [1] 3 15 16 17 ## ## [[2]] ## [1] 3 15 16 ## ## [[3]] ## [1] 3 15 17 ## ## [[4]] ## [1] 3 16 17 ## ## [[5]] ## [1] 15 16 17 ## ## [[6]] ## [1] 3 15 ## ## [[7]] ## [1] 3 16 ## ## [[8]] ## [1] 3 17 ## ## [[9]] ## [1] 15 16 ## ## [[10]] ## [1] 15 17 ## ## [[11]] ## [1] 16 17 ## ## [[12]] ## [1] 3 ## ## [[13]] ## [1] 15 ## ## [[14]] ## [1] 16 ## ## [[15]] ## [1] 17 carrier.sets = carrier.sets[-c(5,9,10)] fam15157.pattern.prob = sapply(carrier.sets,function (vec) RVsharing(fam15157,carriers=vec,useAffected=TRUE)) ## Probability subjects 3 15 16 17 among 3 15 16 17 share a rare variant: 0.001121 ## Probability subjects 3 15 16 among 3 15 16 17 share a rare variant: 0.007848 ## Probability subjects 3 15 17 among 3 15 16 17 share a rare variant: 0.007848 ## Probability subjects 3 16 17 among 3 15 16 17 share a rare variant: 0.003363 ## Probability subjects 3 15 among 3 15 16 17 share a rare variant: 0.05493 ## Probability subjects 3 16 among 3 15 16 17 share a rare variant: 0.02354 ## Probability subjects 3 17 among 3 15 16 17 share a rare variant: 0.02354 ## Probability subjects 16 17 among 3 15 16 17 share a rare variant: 0.004484 ## Probability subjects 3 among 3 15 16 17 share a rare variant: 0.1648 ## Probability subjects 15 among 3 15 16 17 share a rare variant: 0.2152 ## Probability subjects 16 among 3 15 16 17 share a rare variant: 0.2466 ## Probability subjects 17 among 3 15 16 17 share a rare variant: 0.2466 fam15157.N = sapply(carrier.sets,length) Notice the use of the option useAffected=TRUE with affected subjects who are not final descendants. Again, one can check that the distribution sums to 1: sum(fam15157.pattern.prob) ## [1] 1 Precomputed sharing probabilities and numbers of carriers can be used directly to obtain p-values of observed sharing events, by summing the probability of all events as or more extreme as the one observed (both in terms of sharing probability and number of carriers), i.e. this is a one-sided exact test. For instance, if subjects 3, 16 and 17 share a rare variant, the probability of that event is pobs = RVsharing(fam15157,carriers=c(3,16,17),useAffected=TRUE) ## Probability subjects 3 16 17 among 3 15 16 17 share a rare variant: 0.003363 The p-value of that sharing event is then: sum(fam15157.pattern.prob[fam15157.pattern.prob<=pobs & fam15157.N >= 3]) ## [1] 0.004484305 The RVgene function enables these computations with more than one family harboring the same or different variants. Once the vectors of sharing probabilities and number of carriers have been computed for all families in the sample, they need to be stored in lists. We return to the original second cousin triple family and add a first and second cousin triple family. Then we create the lists of pattern probabilities, number of equivalent subsets and number of carriers in the subsets. fam15157.pattern.prob = c(RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16,17)),
RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15,16)), RVsharing(samplePedigrees$secondCousinTriple,carriers=c(15)))
## Probability subjects 15 16 17 among 15 16 17 share a rare variant: 0.001342
## Probability subjects 15 16 among 15 16 17 share a rare variant: 0.009396
## Probability subjects 15 among 15 16 17 share a rare variant: 0.3235
fam15157.N = 3:1
fam15157.nequiv = c(1,3,3)
fam28003.pattern.prob = c(RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36,104,110)), RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36,104)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(104,110)), RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(36)),
RVsharing(samplePedigrees$firstAndSecondCousinsTriple,carriers=c(104))) ## Probability subjects 36 104 110 among 36 104 110 share a rare variant: 0.00277 ## Probability subjects 36 104 among 36 104 110 share a rare variant: 0.00831 ## Probability subjects 104 110 among 104 110 36 share a rare variant: 0.04155 ## Probability subjects 36 among 36 104 110 share a rare variant: 0.3352 ## Probability subjects 104 among 104 36 110 share a rare variant: 0.3019 fam28003.N = c(3,2,2,1,1) fam28003.nequiv = c(1,2,1,1,2) ex.pattern.prob.list = list("15157"=fam15157.pattern.prob,"28003"=fam28003.pattern.prob) ex.nequiv.list = list("15157"=fam15157.nequiv,"28003"=fam28003.nequiv) ex.N.list = list("15157"=fam15157.N,"28003"=fam28003.N) ## 5.2 Example of Analysis of the Rare Variants in the Genomic Sequence of a Gene We now turn to the analysis of variants observed in the simulated genomic sequence of the gene PEAR1 in a sample of related affected subjects. The processing of the sequence data results in Variant Call Format (VCF) files, which can be read into R with the function readVcf from the variantAnnotation package. Two VCF objects obtained with readVcf from VCF files of sequence data for the second cousin triple and first and second cousin triple families are contained in the famVCF data. These VCF files are converted to snpMatrix objects using the genotypeToSnpMatrix function. data(famVCF) fam15157.snp = VariantAnnotation::genotypeToSnpMatrix(fam15157.vcf) ## non-single nucleotide variations are set to NA ## Warning in .local(x, ...): non-diploid variants are set to NA fam28003.snp = VariantAnnotation::genotypeToSnpMatrix(fam28003.vcf) ## non-single nucleotide variations are set to NA ## Warning in .local(x, ...): non-diploid variants are set to NA RVgene requires lists of the snpMatrix and pedigree objects for these two families. The names given to the elements of these lists are not used by RVgene and are thus arbitrary. Family IDs are extracted from the famid element of the pedigree objects. Please note that currently RVgene does not accept a pedigreeList, but only a plain list of pedigree objects. ex.SnpMatrix.list = list(fam15157=fam15157.snp$genotypes,fam28003=fam28003.snp$genotypes) ex.ped.obj = list(fam15157=samplePedigrees$secondCousinTriple,fam28003=samplePedigrees$firstAndSecondCousinsTriple) In the sequence segment, one can specify which variants are rare and possibly satisfy other filtering criteria (e.g. coding variants) using the sites argument. Here, we will focus on two sites: 92 where the three second cousins of family 15157 share the rare allele and 119 where the two first cousins of family 28003 share the rare allele but not their second cousin. sites = c(92,119) ex.SnpMatrix.list[["fam15157"]][,sites[1]]@.Data ## rs187756653 ## 1 00 ## 2 00 ## 3 00 ## 4 00 ## 5 00 ## 6 00 ## 7 00 ## 8 00 ## 9 00 ## 10 00 ## 11 00 ## 12 00 ## 13 00 ## 14 00 ## 15 02 ## 16 02 ## 17 02 ex.SnpMatrix.list[["fam28003"]][,sites[2]]@.Data ## rs199628333 ## 3 00 ## 4 00 ## 6 00 ## 7 00 ## 11 00 ## 12 00 ## 13 00 ## 15 00 ## 27 00 ## 28 00 ## 36 01 ## 103 00 ## 104 02 ## 109 00 ## 110 02 Finally, the call to RVgene returns the P-value of the exact rare variant sharing test allowing for sharing by a subset of affected subjects (p), the P-value of the exact rare variant sharing test requiring sharing by all affected subjects (pall) and the minimum achievable p-value if all affected subjects were carriers of a rare variant (potentialp). RVgene(ex.SnpMatrix.list,ex.ped.obj,sites,pattern.prob.list=ex.pattern.prob.list,nequiv.list=ex.nequiv.list,N.list=ex.N.list,type="count") ##$p
## [1] 0.000159884
##
## $pall ## [1] 0.001342282 ## ##$potentialp
## [1] 3.718232e-06
# 6 Appendix
## 6.1 Rare Variant Sharing Probability Assuming One Founder Introduces the Variant
In this case, we assume the variant is rare enough so that the probability of more than one founder introducing it to the pedigree is negligible. This is the default scenario for RVsharing.
We define the following random variables:
*$$C_i$$: Number of copies of the RV received by subject $$i$$,
*$$F_j$$: Indicator variable that founder $$j$$ introduced one copy of the RV into the pedigree,
For a set of $$n$$ subjects descendants of $$n_f$$ founders we want to compute the probability $\begin{eqnarray*} P[\mbox{RV shared}] &=& P[C_1 = \dots = C_n = 1 | C_1 + \dots + C_n \geq 1] \nonumber \\[0.5em] &=& \frac{P[C_1 = \dots = C_n = 1 ]}{P[C_1 + \dots + C_n \geq 1]} \nonumber \\[0.5em] &=& \frac{\sum_{j=1}^{n_f} P[C_1 = \dots = C_n = 1 | F_j] P[F_j]} {\sum_{j=1}^{n_f} P[C_1 + \dots + C_n \geq 1 | F_j]P[F_j]}, \label{sharingp} \end{eqnarray*}$ where the expression on the third line results from our assumption of a single copy of that RV among all alleles present in the $$n_f$$ founders. The probabilities $$P[F_j] = {1 \over n_f}$$ cancel from the numerator and denominator.
## 6.2 Rare Variant Sharing Probabilities for a Subset of Affected Pedigree Members
By default, RVsharing will compute the probability that all of the final descendants share the variant given that it is seen in at least one of them. Final descendants are defined as subjects of the pedigree with no children. This event can be customized with the carriers and useAffected arguments.
If the argument carriers is provided, then the probability of all carriers having the variant given it is seen in at least one subject in the union of the final descendants and the carriers will be computed.
If the argument useAffected is TRUE and the pedigree has a slot for affected, then the probability of all carriers having the variant given it is seen in at least one affected will be computed.
These two arguments can be used individually or in combination, the only restriction is that carriers must be a subset of affected.
ped <- samplePedigrees$firstCousinTriple ped$affected[9] <- 0
plot(ped)
p <- RVsharing(ped)
## Probability subjects 9 10 11 among 9 10 11 share a rare variant: 0.01176
p <- RVsharing(ped, useAffected=TRUE)
## Probability subjects 10 11 among 10 11 share a rare variant: 0.06667
p <- RVsharing(ped, carriers=c(7,9,10))
## Probability subjects 7 9 10 among 7 9 10 11 share a rare variant: 0.01064
p <- RVsharing(ped, carriers=c(10,11), useAffected=TRUE)
## Probability subjects 10 11 among 10 11 share a rare variant: 0.06667
## 6.3 Using Monte Carlo Simulation
RVsharing also allows for estimating sharing probabilities through Monte Carlo simulation. The primary use of this feature is for calculating sharing probabilities under non standard assumptions about the founders. However, this feature is available for the standard assumptions as well. To run a monte carlo simulation, specify all parameters as normal and additionally provide the nSim parameter specifying how many simulations should be run.
p <- RVsharing(samplePedigrees$firstCousinPair, alleleFreq=0.01) ## Probability subjects 7 8 among 7 8 share a rare variant: 0.07631 p <- RVsharing(samplePedigrees$firstCousinPair, alleleFreq=0.01, nSim=1e5)
## Probability subjects 7 8 among 7 8 share a rare variant: 0.07707
This method allows for more complex relationships among the founders to be given. RVsharing allows for a complete distribution among the founders to be passed in as the parameter founderDist. This function should accept a single argument, N, and should return a vector of length N with values in {0,1,2} representing the number of copies of the variant each founder has.
# assumption that 1 founder introduces variant
fDist <- function(N) sample(c(rep(0,N-1), 1))
p <- RVsharing(samplePedigrees$firstCousinPair, nSim=1e5, founderDist=fDist) ## Probability subjects 7 8 among 7 8 share a rare variant: 0.06461 p <- RVsharing(samplePedigrees$firstCousinPair)
## Probability subjects 7 8 among 7 8 share a rare variant: 0.06667
# References
Bureau, Alexandre, Ferdouse Begum, Margaret A Taub, Jacqueline Hetmanski, Margaret M Parker, Hasan Albacha-Hejazi, Alan F Scott, et al. 2018. “Inferring Disease Risk Genes from Sequencing Data in Multiplex Pedigrees Through Sharing of Rare Variants.” Http://Biorxiv.org/Cgi/Content/Short/285874v1.
Bureau, Alexandre, Samuel G Younkin, Margaret M Parker, Joan E Bailey-Wilson, Mary L Marazita, Jeffrey C Murray, Elisabeth Mangold, Hasan Albacha-Hejazi, Terri H Beaty, and Ingo Ruczinski. 2014. “Inferring Rare Disease Risk Variants Based on Exact Probabilities of Sharing by Multiple Affected Relatives.” Bioinformatics (Oxford, England) 30 (15): 2189–96. https://doi.org/10.1093/bioinformatics/btu198.
Fu, Jack, Terri H Beaty, Alan F Scott, Jacqueline Hetmanski, Margaret M Parker, Joan E Bailey Wilson, Mary L Marazita, et al. 2017. “Whole Exome Association of Rare Deletions in Multiplex Oral Cleft Families.” Genetic Epidemiology 41 (1): 61–69. https://doi.org/10.1002/gepi.22010.
Sherman, Thomas, Jack Fu, Robert B Scharpf, Alexandre Bureau, and Ingo Ruczinski. 2018. “Detection of rare disease variants in extended pedigrees using RVS,” November. https://doi.org/10.1093/bioinformatics/bty976.
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2022-11-29 02:21:04
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https://raweb.inria.fr/rapportsactivite/RA2019/cqfd/uid27.html
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PDF e-Pub
## Section: New Results
### Hamilton-Jacobi-Bellman Inequality for the Average Control of Piecewise Deterministic Markov Processes
Abstract : The main goal of this work is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity $\lambda$ and on the transition measure $Q$ of the process.
Authors : O.L.V. Costa; F. Dufour (CQFD)
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2020-07-11 17:18:14
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http://math.blogoverflow.com/2014/07/24/two-points-determine-a-line-three-a-quadratic-what-has-that-got-to-do-with-cds/
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# Two points determine a line, three a quadratic — what has that got to do with CDs?
July 24, 2014 by . 4 comments
In this post I describe how simple facts about polynomials are applied in correcting errors, for example scratches on compact disks. The same technique is used in many other places, e.g. in the 2-dimensional QuickResponse bar codes.
## Elements
The two facts from algebra that we need are:
Theorem 1. A polynomial of degree $$n$$ has at most $$n$$ zeros.
Theorem 2. If $$(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)$$ are $$n$$ points on the $$xy$$-plane such that $$x_i\neq x_j$$ whenever $$i\neq j$$, then there is a unique polynomial $$f(x)$$ of degree $$<n$$ such that $$f(x_i)=y_i$$ for all $$i$$.
So, if $$n=2$$, we want the polynomial $$f(x)$$ to be linear. In that case, the graph $$y=f(x)$$ will be the line passing through the points $$(x_1,y_1)$$ and $$(x_2,y_2)$$. Similarly, when $$n=3$$ we want the polynomial to be (at most) quadratic, and we want its graph to pass through the given three points. Finding the coefficients of such a quadratic is not too arduous an exercise in linear systems of equations. For general $$n$$ there is a known formula for the polynomial $$f(x)$$ called Lagrange’s interpolation polynomial . The uniqueness of such a polynomial follows from Theorem 1. If $$f_1(x)$$ and $$f_2(x)$$ were two different polynomials of degree $$<n$$ passing through all these $$n$$ points, then their difference $$f_1(x)-f_2(x)$$ is also of degree $$<n$$ and vanishes at all the points $$x_i, i=1,2,\ldots,n$$, which is impossible by Theorem 1.
## Extending a message using a polynomial
The applications I discuss are about communication. We have two parties, a transmitter and a receiver. The transmitter wants to send a message to the receiver. We assume that they have in advance agreed upon a method of coding the messages to sequences of numbers $$y_1,y_2,\ldots,y_k$$ for some natural number $$k$$. The simplest way of communicating would be for the transmitter to simply write this list of numbers to a channel that the receiver can later read. The channel could be something like a note that you pass to a classmate or it could be compact disk, where the transmitter just writes the numbers. It could be something fancier like a radio frequency band, or an optical fiber, but we ignore the physical nature of the channel here.
If everything goes well, the receiver can read the message correctly. But, they may misread one or more of the numbers. Or, one of the numbers may get smeared and become illegible while the note is exchanging hands. What to do? The basic idea is to add some redundancy to the message. We seek to extend the message from $$k$$ numbers to $$n$$ numbers, $$n>k$$, in such a way that the receiver can recover from a few such mishaps. A useful way of doing this is to:
• Agree in advance on a sequence of $$n$$ values for $$x$$-coordinates $$x_1,\ldots,x_n$$.
• View the message $$y_1,y_2,\ldots,y_k$$ as a sequence of $$k$$ points $$(x_1,y_1), (x_2,y_2),\ldots, (x_k,y_k).$$
• Find the unique polynomial $$f(x)$$ of degree $$<k$$ passing through all these $$k$$ points.
• Extend the message to $$y_1,y_2,\ldots,y_n$$ in such a way that $$y_i=f(x_i)$$ for all $$i=1,2,\ldots,n$$. Note that the actual ‘payload’ message consists of the $$k$$ first numbers, the remaining $$r=n-k$$ are the redundant numbers.
## Toy Example #1 – recovery from an illegible number
In the Figures below I always use $$x_1=1,x_2=2,\ldots,x_n=n$$ for simplicity. Let us consider as an example the case $$k=2, n=3$$, so our polynomials will always have degree $$\le1$$, and their graphs are thus lines. Let us further select the message to be $$(y_1,y_2)=(3,4)$$. In this case the polynomial $$f(x)=x+2$$ as $$f(1)=y_1$$ and $$f(2)=y_2$$. As $$f(3)=5$$ the extended message is then $$(y_1,y_2,y_3)=(3,4,5)$$. In the figure below the payload part of the message is formed by the $$y$$-coordinates of the black dots and the $$y$$-coordinates of the red dot(s) are there for redundancy. I also include the entire graph of $$f(x)$$ to make what follows clearer.
How does this minimum amount of redundancy help? Consider the scenario when the second numbers was, indeed, smeared and the receiver could only read the sequence as $$(3,?,5)$$. Now, our scheme allows the receiver to deduce that the missing number must be equal to $$4$$. This is because they know that the points $$(1,3), (2,?)$$ and $$(3,5)$$ must all be on the same line. With the knowledge that two points determine a line, the value of $$y_2=4$$. What the receiver sees is shown in the Figure below.
Observe that it is immaterial here which of the three numbers was garbled. The receiver knows two points on the line, and can thus recover the polynomial $$f(x)$$ and hence also the message.
## Toy example #2 – spotting an error, and correcting it
However, adding a single redundant number does not protect us from other kinds of errors. Suppose that one of the numbers is misread, and the receiver reads $$(y_1,y_2,y_3)=(3,2,5)$$ instead of the correct sequence above. Now they cannot cope. The receiver can notice that the three points are not collinear, so they know that something went wrong. But, they have no way of knowing which of the three numbers is incorrect. Any pair of points gives a line all right, but now the received version of the message matches equally well with three distinct lines — all passing through two out of three points. The picture is as follows.
How can we cope with such errors? The solution is to add more redundancy. Let’s try $$k=2, n=4, r=4-2=2$$ with two redundant numbers. Let’s again use the message $$(y_1,y_2)=(3,4)$$. This time we add a fourth point $$(x_4,y_4)=(4,f(4))=(4,6)$$, so the extended message (payload + redundancy) is now $$(3,4,5,6)$$. The picture is as follows.
If, again, one of the numbers is read incorrectly, then this time we can tell which one is incorrect, because the other three points are collinear.
Theorem 3. Given four points $$P_1,P_2,P_3,P_4$$ on the plane there can be at most a single line containing at least three of those points.
Proof. Assume contrariwise that there are two lines, $$L_1$$ and $$L_2$$, both passing through at least three of the points. Because at most one of the four points is not on $$L_1$$, and similarly at most one of the points is not on $$L_2$$, we see that the lines $$L_1$$ and $$L_2$$ share at least $$4-1-1=2$$ of the points. But there is a unique line through those two points, so $$L_1=L_2$$. Q.E.D.
If in our toy example the receiver again makes a mistake with the second number and reads $$(y_1,y_2,y_3,y_4)=(3,2,5,6)$$, it will be easy for them to tell from the following picture, which point is the odd one out. Even without the plot of $$y=f(x)$$ that I added to make things easier for you.
## Coping with many problems in a single message
Ok, let us generalize. Assume that the payload consists of $$k$$ numbers, and that we tag $$r=n-k$$ redundant numbers to it so the entire transmitted list has $$n$$ numbers. What kind of problems does this scheme protect us against? Above we encountered two kinds of problems. An erasure happens when the receiver cannot make out the written number. Observe that the receiver still knows the $$x$$-coordinate of that point (those were agreed in advance) — only the $$y$$-coordinate is missing. Another type of problem was an error, an incorrectly read number. From the toy examples we already got the feeling that, of these two, an error is more difficult to cope with. We needed two redundant numbers to correct a single error, but a single redundant number was enough to cope with an erasure.
Assume that we want to cater for the possibility of $$t$$ errors and $$e$$ erasures in a single transmission. How many redundant numbers do we need to cope with that? Altogether, the message gives us $$n=k+r$$ points. The erasures mean that we only have $$n-e$$ points remaining, and must manage with them. The goal is that these should somehow determine a unique polynomial of degree $$<k$$. The solution is provided by the following generalization of Theorem 3.
Theorem 4. Given $$k+2t$$ points with distinct $$x$$-coordinates on the plane there can be at most a single polynomial of degree $$<k$$ whose graph passes through all but at most $$t$$ of the points.
Proof. I am feeling a bit naughty and leaving this as an exercise. Use Theorem 2 in the same way that I used the fact “two points determine a unique line” in the proof of Theorem 3.
Corollary. When we append $$r$$ redundant numbers to a payload of $$k$$ according to this scheme we can cope with a combination of $$e$$ erasures and $$t$$ errors, if $$r\ge 2t+e.$$
So we see that adding redundancy makes the scheme more robust against the prescribed types of problems created by the channel. Of course, the added redundancy comes with a cost. A redundant number consumes the same amount of channel resources as a payload number (space on a CD or that piece of paper). There the user of this kind of a system needs to consider the probabilities of both errors and erasures, and then settle on a certain quality of service such that the inequality of Corollary is satisfied with a high enough probability.
## Snakes in the paradise – need more algebra
There is one serious concern that I need to mention and address. Above I have loosely used the concept of a number without specifying what I meant by that. The figures undoubtedly left you with the impression that the numbers are usual real numbers. Unfortunately that won’t do. The basic reason is that to write down a real number with full precision takes infinitely many bits. But the messages on e.g. CDs consist of bytes. Programmers are used to viewing the value of a byte as an integer $$b$$ in the range $$0\le b\le 255$$. Even if we restrict the payload numbers to this range we cannot guarantee that the redundant numbers to be tagged would also oblige. Consider the following example of $$k=7, r=4$$ and the payload $$(y_1,\ldots,y_7)=(3, 7, 0, 1, 9, 6, 8)$$. Here’s what the Lagrange interpolation polynomial looks like with these inputs.
Where are the red dots indicating the redundant numbers? Well, you can guess from the plot that the polynomial $$f(x)$$ (it is of degree six) will increase very rapidly after the last payload. Indeed, this sextic polynomial gives the values $$f(8)=164, f(9)=903, f(10)=3129$$ and $$f(11)=8464$$ as the redundant numbers. So we face a serious overflow problem. Here’s a rescaled version of the plot.
Underflow is also possible as the redundant numbers may very well be negative. One might also worry that even though the payload consists of integers the redundant numbers might not. After all, Lagrange’s interpolation formula introduces divisions. Actually, as in the above overflow example, this never happens. But this is an artefact of the way we set up things. I leave this as an exercise for the interested reader.
Exercise. Let $$f(x)\in\mathbb{Q}[x]$$ be the Lagrange interpolation polynomial (of degree $$<k$$) passing through the points $$(1,y_1), (2,y_2), \ldots, (k,y_k)$$, where all the numbers $$y_i, i=1,2,\ldots,k$$ are integers. Show that $$f(n)$$ is then an integer for all $$n\in\mathbb{Z}$$.
But, we absolutely don’t want to use more than a single byte to present the redundant numbers. Or, more precisely, we insist that a redundant number should use the same amount of memory as a number in the payload part (a byte, a word,…). People familiar with elementary number theory may already feel the urge to reinterpret byte as a residue class modulo $$256$$, i.e. as an element of the residue class ring $$R=\mathbb{Z}_{256}$$. Unfortunately the algebra breaks down if we do that. To wit, already Theorem 1 fails dramatically. As an example, consider the linear polynomial $$f(x)=16x$$. Even though it is linear, it has no less than sixteen distinct zeros in the ring $$R$$, namely the residue classes of the form $$\overline{16m}$$ for $$m=0,1,\ldots,15$$. When you learn a bit more about the structures in abstract algebra you realize that the problem is that the ring $$R$$ is not a field. But there is a field with 256 elements that fits the bill exactly! Its construction requires familiarity with polynomial rings and their quotient rings, and I don’t want to reproduce scores of pages from a first textbook on abstract algebra. To a reader who knows what is going on I will tell that
$$\mathbb{F}_{256}\cong\mathbb{F}_2[x]/\langle x^8+x^4+x^3+x^2+1\rangle$$
is a popular way of describing this field. A consequence of this is that the total length (payload + redundancy) of the messages in bytes cannot exceed $$256$$. This is because at that point we run out of $$x$$-coordinates!
Coding theory is a topic straddling mathematics and communication engineering. It deals (among other things) in the design of schemes, like the one outlined here, aiming at coping with errors produced into messages by the channel. Two American coding theorists, Irving S. Reed and Gustave Solomon were the first to figure out this way of using the algebra of finite fields. One of their inventions was to choose the sequence of $$x$$-coordinates in a clever way that makes the calculation of the redundant numbers very easy (they can skip the time consuming use of Lagrange’s interpolation formula entirely). The resulting codes are called Reed–Solomon codes (or just RS-codes) in their honor. The Wikipedia-article on RS-codes describes the algebra very differently, but in the end it is equivalent. That description reflects the more efficient calculation of the redundant part. I chose to present the codes in this way because it makes it easier to understand why these codes can cope with the prescribed types of errors.
## Practical issues
I will close my contribution with a few remarks about certain practical aspects. The receiver faces the task of figuring out the correct polynomial. In the toy example of four points we could just take any pair of points (there are only six pairs), and check whether the line determined by them passes through a third point. But if we use a RS-code capable of correcting five errors with parameters $$k=240, r=10$$, then the receiver following an analogous approach would need to generate $$\binom{250}{10}=219005316087032475$$ Lagrange interpolation polynomials of degree $$239$$, and check if one gives a good enough match. Not efficient at all! Thankfully, much more efficient algorithms have been developed by several other coding theorists.
The other remark I want to make has to do with compact disks. There, Reed-Solomon codes are used in the following way that produces good protection against typical scratching of the disk surface. The data on a CD is split into blocks like the one shown below.
Here each dot represents a single byte (actually, there is another error correcting code involved in reading a single byte from the disk that one is working at the level of individual bits and does not concern us here). The rows and columns in such a matrix are all encoded using an appropriate RS-code. The black dots are payload bytes. The encoding is done as follows. First, we add to each column four redundant bytes (green). Then we also add four redundant bytes to all the rows — in the payload part (red) as well as in the redundant part (blue). The bytes of a single row are close to each other on the CD, so the CD-player can read them quickly. It attempts to correct eventual errors. Usually it succeeds, but if that row is in a scratchy area, it fails, and then marks the entire row as unreliable data (i.e. an erasure). But different rows are more spaced out on the CD surface, so there is a high hope that only few rows from a given a matrix are lost in this fashion. If at most four of the rows are marked as erasures the column by column RS-code can recover the lost data. For the CD-player to be able to retrieve such matrices from the disk fast the entire matrix is on the same track (at the same distance from the disk center). Thus, a circular scratch may wipe out enough of the matrix, and the scheme will fail.
## A happy thought: algebraic tools triumph in solving problems arising in communication.
• G.T.R says:
Thank for this informative and well-written post: I didn’t know polynomial interpolation could be used to communicate data.
• Kartik says:
When I was reading this, this thought came in my mind: “Where is the upvote button on this one?”
• leo says:
Nice!
• Please fix the latex stuff. It is using the format of WordPress like dollar latex dollar instead of the mathjax format dollar dollar.
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2017-01-18 22:04:11
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http://ncatlab.org/nlab/show/homomorphism
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# Contents
## Idea
In a restrictive sense, a homomorphism is a function between (the underlying sets) of two algebras that preserves the algebraic structure.
More generally, a homomorphism is a function between structured sets that preserves whatever structure there is around. Even more generally, ‘homomorphism’ is just a synonym for ‘morphism’ in any category, the structured sets being generalised to arbitrary objects.
Note: The word “homomorphism” has also traditionally been used for what we call a (weak) 2-functor between bicategories.
## Definitions
Traditionally, a homomorphism between two magmas $A$ and $B$ is a function
$\phi\colon A \to B$
of the underlying sets that respects the binary operation in that for all $a_1, a_2$ in $A$ we have
$\phi(a_1 \cdot a_2) = \phi(a_1) \cdot \phi(a_2) \,.$
This definition gives us the correct notion of magma homomorphism, semigroup homomorphsim, and group homomorphism, but it is actually a bit of a coincidence that it works for groups. It does not give the correct definition of monoid homomorphism, since it doesn't properly treat the identity elements. (However, the correct notion of monoid isomorphism can still be constructed from this inadequate definition of homomorphism.)
A rng homomorphism is a function between rngs that is a homomorphism for both the additive group and the multiplicative semigroup. (For rings with identity, this is again inadequate.)
### Identity-preserving (monoids, rings)
A homomorphism between two monoids $A$ and $B$ is a semigroup homomorphism
$\phi\colon A \to B$
of the underlying semigroups that preserves identity elements in that we have
$\phi(1_A) = 1_B \,.$
This definition of monoid homomorphism is a special case (via delooping) of the definition of functor.
It is a theorem that a semigroup homomorphism between groups must be a monoid homomorphism (and additionally must preserve inverse elements, which is also necessary to be the correct definition of group homomorphism.)
A ring homomorphism is a function between rings that is a homomorphism for both the additive group and the multiplicative monoid. Traditional ring theory sometimes actually uses rng homomorphisms even when the rngs in question are assumed to have identity elements, so be careful when reading old books.
### General
More generally, a homomorphism between sets equipped with any algebraic structure is a map preserving this structure. This can be made precise using Lawvere theories, monads, etc.
Generalizing further, we may simply treat ‘homomorphism’ as a synonym for ‘morphism’ in any category, although there is a strong tendency to use ‘homomorphism’ in the case of ‘algebraic’ categories: for example, nobody seems to speak of a homomorphism between topological spaces (continuous maps), or between manifolds. Here we put ‘algebraic’ in scare quotes to indicate the field of algebra; even morphisms in algebraic categories from other fields (such as the category of compacta) are not usually called homomorphisms.
Revised on November 28, 2014 08:58:35 by Urs Schreiber (82.224.164.72)
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2015-01-27 22:58:58
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-4-quadratic-functions-and-factoring-4-3-solve-x-squared-bx-c-0-4-3-exercises-skill-practice-page-256/58
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## Algebra 2 (1st Edition)
$$\pm8$$
Here, we see that the factors must multiply to get $7$ and add to get $b$. The only factors of 7 are $-1\ and\ -7$ and $1\ and\ 7$. Thus, the only values of b that work are: $$-1-7=-8 \\ AND \\ 1+7=8$$
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2021-03-03 03:13:16
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https://www.springerprofessional.de/set-operads-in-combinatorics-and-computer-science/2285472
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scroll identifier for mobile
main-content
## Über dieses Buch
This monograph has two main objectives. The first one is to give a self-contained exposition of the relevant facts about set operads, in the context of combinatorial species and its operations. This approach has various advantages: one of them is that the definition of combinatorial operations on species, product, sum, substitution and derivative, are simple and natural. They were designed as the set theoretical counterparts of the homonym operations on exponential generating functions, giving an immediate insight on the combinatorial meaning of them. The second objective is more ambitious. Before formulating it, authors present a brief historic account on the sources of decomposition theory. For more than forty years decompositions of discrete structures have been studied in different branches of discrete mathematics: combinatorial optimization, network and graph theory, switching design or boolean functions, simple multi-person games and clutters, etc.
## Inhaltsverzeichnis
### 1. Introduction
Abstract
In this chapter the main ideas involved in the notion of a set operad are explained by studying the example of parenthesized injective words, and a historical account on set operads is provided. Finally, we state the main objectives of this monograph.
Miguel A. Méndez
### 2. Preliminaries on Species and Set Operads
Abstract
The formal definition of a species and its exponential generating function are given. Based on the intuitive definition of the operation of substitution of species we give an informal definition of a set operad.
Miguel A. Méndez
### 3. Operations on Species and Operads
Abstract
In this chapter operations on species (sum, product, Hadamard product, derivative, and substitution) are introduced by mimicking the analogous operations on exponential generating functions. The operations of substitution and product define two important monoidal categories on species. In this context, operads are defined as monoids in the monoidal category of species with respect to the substitution. Species with a monoidal structure with respect to the operation of product are simply called monoids. As a consequence of the chain rule for species, it is proved that the derivative sends operads into monoids. A construction of families of partially ordered sets from cancellative operads (respectively, cancellative monoids) is obtained. The substitutional (respectively, multiplicative) inverse of the exponential generating function of a cancellative operad (respectively, cancellative monoid) is proved to be the M\"obius generating function of the respective associated family of posets.
Miguel A. Méndez
### 4. Decomposition Theory
Abstract
In this chapter the main results on decomposition theory are revisited using the tools of species and set operads. We begin by giving the definition of module domain operads, and by introducing the sub-classes of partitive and weakly partitive operads. If an operad is either partitive or weakly partitive, each of its structures has a unique factorization into prime factors. We then introduce the amalgam operation, based upon which we can construct operads where a unique factorization is still valid, but whose amalgam factors are not weakly partitive, thus widening the spectrum of possibilities of unique factorizable structures.
Miguel A. Méndez
Abstract
A rigid finite structure is one without automorphisms other than the identity. Structures constructed over totally ordered sets are rigid. Thus, the natural context for the definition of rigid operads is the species on linear orders or L-species. In the context of $$\mathcal{L}$$-species there are two different substitution operations; ordinal and shuffle. These operations lead to two kinds of operads, respectively non-symmetric and shuffle.
Miguel A. Méndez
### 6. Posets from Cancellative Operads and Koszul Duality
Abstract
A species $${\mathcal{G}}$$ is said to be homogeneous concentrated in m, if $${\mathcal{G}}[k]=\emptyset$$ for $$k\neq m$$. Based on previous work of Fresse [Fre04], B. Vallette [Val07] proved that a quadratic cancellative operad generated by a homogeneous species is Koszul if and only if the maximal intervals of the associated posets P Q are Cohen–Macaulay. In this chapter, we give an account of Vallete’s results and generalize his criterion for Koszulness. We show that the homogeneity assumption is not necessary. The result is still valid for any quadratic cancellative operad.
Miguel A. Méndez
### Backmatter
Weitere Informationen
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2019-03-22 12:15:43
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https://socratic.org/questions/what-is-the-slope-of-the-line-passing-through-0-0-2-7
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# What is the slope of the line passing through (0,0); (-2,7)?
Apr 28, 2018
$- \frac{7}{2}$
#### Explanation:
Slope ($m$) through two points $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right)$: $\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
$\frac{7 - 0}{- 2 - 0}$
$\frac{7}{- 2}$
$- \frac{7}{2}$
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2022-10-02 06:45:25
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https://access.openupresources.org/curricula/our6-8math-v1/7/students/7/my_reflections.html
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# Unit 7: My Reflections
## Lesson 1: Relationships of Angles
• I can find unknown angle measures by reasoning about adjacent angles with known measures.
• I can recognize when an angle measures $90^\circ$, $180^\circ$, or $360^\circ$.
• I can recognize when adjacent angles are complementary or supplementary.
• I can find unknown angle measures by reasoning about complementary or supplementary angles.
• I can determine if angles that are not adjacent are complementary or supplementary.
• I can explain what vertical angles are in my own words.
## Lesson 4: Solving for Unknown Angles
• I can reason through multiple steps to find unknown angle measures.
• I can recognize when an equation represents a relationship between angle measures.
## Lesson 5: Using Equations to Solve for Unknown Angles
• I can write an equation to represent a relationship between angle measures and solve the equation to find unknown angle measures.
## Lesson 6: Building Polygons (Part 1)
• I can show that the 4 side lengths that form a quadrilateral can be rearranged to form different quadrilaterals.
• I can show that the 3 side lengths that form a triangle cannot be rearranged to form a different triangle.
## Lesson 7: Building Polygons (Part 2)
• I can show whether or not 3 side lengths will make a triangle.
• I can reason about a figure with an unknown angle.
## Lesson 8: Triangles with 3 Common Measures
• I understand that changing which sides and angles are next to each other can make different triangles.
## Lesson 9: Drawing Triangles (Part 1)
• Given two angle measures and one side length, I can draw different triangles with these measurements or show that these measurements determine one unique triangle or no triangle.
## Lesson 10: Drawing Triangles (Part 2)
• Given two side lengths and one angle measure, I can draw different triangles with these measurements or show that these measurements determine one unique triangle or no triangle.
## Lesson 11: Slicing Solids
• I can picture different cross sections of prisms and pyramids.
• I can explain that when a three dimensional figure is sliced it creates a face that is two dimensional.
## Lesson 12: Volume of Right Prisms
• I can explain why the volume of a prism can be found by multiplying the area of the base and the height of the prism.
## Lesson 13: Decomposing Bases for Area
• I can calculate the the volume of a prism with a complicated base by decomposing the base into quadrilaterals or triangles.
## Lesson 14: Surface Area of Right Prisms
• I can picture the net of a prism to help me calculate its surface area.
• I can find and use shortcuts when calculating the surface area of a prism.
## Lesson 15: Distinguishing Volume and Surface Area
• I can decide whether I need to find the surface area or volume when solving a problem about a real-world situation.
## Lesson 16: Applying Volume and Surface Area
• I can solve problems involving the volume and surface area of children’s play structures.
## Lesson 17: Building Prisms
• I can build a triangular prism from scratch.
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2021-04-23 09:17:07
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https://www.plant-ecology.com/EN/Y2017/V41/I7/749
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Chin J Plant Ecol ›› 2017, Vol. 41 ›› Issue (7): 749-760.
• Research Articles •
### Spatiotemporal variation and scale effect of canopy leaf area index of larch plantation on a slope of the semi-humid Liupan Mountains, Ningxia, China
Ze-Bin LIU1, Yan-Hui WANG1,*(), Yu LIU2, Ao TIAN1, Ya-Rui WANG1, Hai-Jun ZUO1
1. 1Institute of Forest Ecology, Environment and Protection, Chinese Academy of Forestry, Key Laboratory of Forestry Ecology and Environment of State Forestry Administration, Beijing 100091, China
and
2College of Soil and Water Conservation, Beijing Forestry University, Beijing 100083, China
• Received:2016-09-13 Accepted:2017-02-28 Online:2017-07-10 Published:2017-08-21
• Contact: Yan-Hui WANG
• About author:KANG Jing-yao(1991-), E-mail: kangjingyao_nj@163.com
Abstract:
Aims Leaf area index (LAI) is an important canopy structure parameter characterizing ecological and hydrological processes, such as forest growth, canopy interception and transpiration. Forest LAI is limited by both light and soil water availability, thus may vary with slope position and seasonality. This study is aimed at the spatiotemporal variation of LAI and its relationship with environmental variables. Methods A 34-years-old Larix gmelinii var. principis-rupprechtii planted forest situated on a typical slope located in a small watershed of Xiangshuihe within Liupan Mountains was selected for LAI observations. Sixteen plots along a 30 m wide transect along the slope was surveyed from May to October of 2015 to measure the monthly canopy LAI. Important findings It showed there was a remarkable difference of LAI among slope positions. The LAI in May decreased toward downslope direction with a scale effect of -0.02/100 m. Whereas for the period from June to August, LAI showed a nonlinear variation along slope positions: increasing from to top slope downward, reaching its maximum at the middle slope, and then decreasing to the slope foot. The scale effect of LAI was +0.15/100, +0.16/100, and +0.18/100 m in the slope range (downward positive) of 0-244.2 m, but -0.09/100, -0.08/100, and -0.07/100 m in the slope range of 244.2-425.1 m for June, July and August, respectively. The LAI increased toward downslope in September and October, with a slope scale effect of +0.03/100 m and +0.09/100 m, respectively. The seasonal variation of LAI-slope relationship showed a shift from the light and temperature control in the early growing season, to the soil water resources control in the mid growing season, and then to an integrated control of many factors in the late growing season. In the early growing season when soil moisture and nutrients were abundant, terrain shading limited the leaf growth in middle and downslope. From early to the mid growing season, the soil moisture on the slope was quickly depleted due to fast evapotranspiration and poor moisture retention of the coarse soil. On the other hand, average solar height increased, and allowed direct light radiation to penetrate to the middle then downslope. The result is that the leaf growth in the middle slope was the strongest in the mid growing season. During the late growing season, the temperature decreased fast in the mountain top to incur earlier leaf fall than the mountain foot. Thus the LAI exhibited the increasing trend toward the downslope.
http://jtp.cnki.net/bilingual/detail/html/ZWSB201707005
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2023-02-02 13:47:07
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http://math.stackexchange.com/questions/234433/generalizing-the-catalan-numbers?answertab=votes
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# Generalizing the Catalan Numbers
Preliminaries
There are many equivalent definitions of the Catalan Numbers. I'll use this one:
A Dyck Word is a string consisting on $n$ X's and $n$ Y's such that no initial segment of the string has more Y's than X's. For example, XXYYXY is a Dyck Word, but XXYYYX is not, because of the leading substring XXYYY. The $n^{th}$ Catalan number $C_n$ is equal to the number of Dyck Words of length $2n$.
There are a number of well-known formulas for computing the Catalan numbers. One is $C_n = \frac{1}{n+1} {2n \choose n}$.
My Question
I want to generalize the Catalan Numbers as follows: Suppose a Dyck Word is allowable if no leading substring has $k$ or more Y's than X's. Then $D^k_n$ gives the number of (modified) Dyck Words of length $2n$. So $C_n = D^1_n$.
What is a formula for $D^k_n$?
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Your generalized Catalan numbers have a combinatorial interpretation. Just as the Dyck words encode Dyck paths, your generalized Catalan numbers $D_n^k$ is the number of Dyck-like paths which lie at most $k-1$ steps below the $x$-axis.
Therefore $D_n^2$ is the number of paths from $(0,\ 0)$ to $(2n,\ 0)$ which lie above $y=-1$. From the reflection principle for lattice paths, if we have a path which at some point intercepts $y=-k$ then the number of such paths is equal to the number of paths which end at $(2n,\ -2k)$. Therefore the number of such paths is $\binom{2n}{n-k}$.
The number of paths which lie above $y=-1$ can then be seen as composed of the following groups $$\{\text{# paths above y=0}\} + \{\text{# paths intercepting y=-1}\} - \{\text{# paths intercepting y=-2}\}$$ which gives $$D_n^2 = D_n^1 + \binom{2n}{n-1} - \binom{2n}{n-2}$$ This is a shifted Catalan number as helopticor has noticed. For $D_n^3$, we similarly have $$D_n^3 = D_n^1 + D_n^2 + \binom{2n}{n-2} - \binom{2n}{n-3} = D_n^1 + \binom{2n}{n-1} - \binom{2n}{n-3}$$ You can verify that continuing in this manner, the full generalization is $$D_n^k = \binom{2n}{n} - \binom{2n}{n-k}$$
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We can modify the standard recursion satisfied by the Catalan numbers.
First, consider the case $k=2$. A $k=2$ word is either already a Dyck word (which can happen in $D^1_n$ ways), or there's a first place where there are more Ys than Xs. In the latter case, the word looks like $w_1YXw_2$ where $w_1$ is a Dyck word (since we're referring to the first place where Ys overtake Xs) and $w_2$ is any $k=2$ word of the appropriate size (since the Xs and Ys have been evened out before w_2). Thus, $D_{n}^{2}=D_{n}^{1}+\sum_{i=0}^{n-1}D_{i}^{1}D_{n-1-i}^{2}$. Notice that if we define $D_{-1}^{2}=1$, this is really the recursion for the Catalan numbers, with a shift, so that $D_{n}^{2}=D_{n+1}^{1}$, but that fact won't generalize, so we press on.
For $k>2$, either the word is actually a $k-1$ word, or there's a first place where the word gets maximum initial Ys. I had wanted to say "so it begins with a $k-1$ word that is not a $k-2$ word, followed by YX" to get the recursion $D_{n}^{k}=D_{n}^{k-1}+\sum_{i=0}^{n-1}(D_{i}^{k-1}-D_{i}^{k-2})D_{n-1-i}^{k}$ for $k\ge2$ if we follow the convention that $D_{n}^{0}=0$". But that's flawed, because a $k-1$ word ends with enough Xs to cancel out any initial Ys. (Incidentally, for $k=3$ anyway this potentially incorrect sequence seems to be a Catalan transform of the fibonacci numbers as listed on OEIS.)
Edit: Upon thinking things over some more, I found an uglier recursion for $k=3$ manually. As before, either the word is a $k=2$ word (so a count of $D_{n}^{2}$), or it has an initial string with two un-Xed Ys in a row. The first time this happens, it may be a YYXX, so we have $w_1$YYXX$w_2$ where $w_1$ is a $k=2$ word and $w_2$ is a $k=3$ word. However, it may be that the first time this happens we have YYXYXX, or YYXYXYXX, etc. If $i$ is the number of intervening "XY"s, then we can sum over $i$ to get all of the possibilities: $D_{n}^{3}=D_{n}^{2}+\sum_{i=0}^{n-2}\sum_{j=0}^{n-2-i}D_{j}^{2}D_{n-2-i-j}^{3}$. This appears to be the second differences of the Catalan numbers.
The problem is that for $k$ beyond three this gets very hairy very fast, so I'd hope there's a better approach that can get something that works for all k. My bet is that maybe some recursion can be gotten by adding another parameter to allow partial modified Dyck words to be counted, as that's what needs to be done inside of YYYX...X for the $k=4$ case. I don't have any more insight right now.
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2015-07-01 21:43:22
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https://wikimili.com/en/Aspect_ratio_(image)
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Aspect ratio (image)
Last updated
Common image aspect ratios
1:1
Square. Used in some social networks, and in few devices.
1.2:1 (6:5)
Fox Movietone aspect ratio
1.25:1 (5:4)
Early television & large-format computer monitors
1.3:1 (4:3)
Traditional television & computer monitor standard
1.375:1 (11:8)
Academy standard film aspect ratio
1.43:1
IMAX motion picture film format
1.5:1 (3:2)
Classic 35 mm still photographic film
1.56:1 (14:9)
Used to create an acceptable picture on both 4:3 and 16:9 televisions
1.6:1 (16:10)
A common computer screen ratio
1.6180:1 (${\displaystyle \varphi }$:1)
The golden ratio
1.6:1 (5:3)
A common European widescreen standard; Paramount format; [1] native Super 16 mm film
1.7:1 (16:9)
HD video standard; US & UK digital broadcast TV standard
1.9:1
DCI standard for 4K & 2K; Digital IMAX
2.2:1
Standard 70mm film
2.35:1, 2.39:1 or 2.4:1
A current widescreen cinema standard
2.414:1 (δS:1)
The silver ratio
3.5:1 or 3.6:1 (32:9 or 18:5)
Super Ultrawide, Ultra-WideScreen 3.6
4:1
Used only in Napoléon (1927)
The aspect ratio of an image describes the proportional relationship between its width and its height. It is commonly expressed as two numbers separated by a colon, as in 16:9. For an x:y aspect ratio, no matter how big or small the image is, if the width is divided into x units of equal length and the height is measured using this same length unit, the height will be measured to be y units.
Contents
For example, in a group of images that all have an aspect ratio of 16:9, one image might be 16 inches wide and 9 inches high, another 16 centimeters wide and 9 centimeters high, and a third might be 8 yards wide and 4.5 yards high. Thus, aspect ratio concerns the relationship of the width to the height, not an image's actual size.
16:9 (1.77:1 = 42:32) is an aspect ratio with a width of 16 units and height of 9.
Some common examples
The most common aspect ratios used today in the presentation of films in cinemas are 1.85:1 and 2.39:1. [2] Two common videographic aspect ratios are 4:3 (1.3:1), [lower-alpha 1] the universal video format of the 20th century, and 16:9 (1.7:1), universal for high-definition television and European digital television. Other cinema and video aspect ratios exist, but are used infrequently.
Film, also called movie or motion picture, is a visual art used to simulate experiences that communicate ideas, stories, perceptions, feelings, beauty or atmosphere by the means of recorded or programmed moving images along with other sensory stimulations. The word "cinema", short for cinematography, is often used to refer to filmmaking and the film industry, and to the art form that is the result of it.
Videography refers to the process of capturing moving images on electronic media and even streaming media. The term includes methods of video production and post-production. It could be considered the video equivalent of cinematography. The advent of digital video recording in the late 20th century blurred the distinction between videography and cinematography, as in both methods the intermittent mechanism became the same. Nowadays, any video work outside commercial motion picture production could be called videography. A videographer is a person who works in the field of videography and/or video production. News broadcasting relies heavily on live television where videographers engage in electronic news gathering (ENG) of local news stories.
High-definition television (HDTV) is a television system providing an image resolution that is of substantially higher resolution than that of standard-definition television. This can be either analog or digital. HDTV is the current standard video format used in most broadcasts: terrestrial broadcast television, cable television, satellite television, Blu-rays, and streaming video.
In still camera photography, the most common aspect ratios are 4:3, 3:2, and more recently found in consumer cameras, 16:9. [3] Other aspect ratios, such as 5:3, 5:4, and 1:1 (square format), are used in photography as well, particularly in medium format and large format.
Medium format has traditionally referred to a film format in still photography and the related cameras and equipment that use film. Nowadays, the term applies to film and digital cameras that record images on media larger than 24 mm × 36 mm (full-frame), but smaller than 4 in × 5 in.
Large format refers to any imaging format of 4×5 inches (102×127 mm) or larger. Large format is larger than "medium format", the 6×6 cm or 6×9 cm size of Hasselblad, Mamiya, Rollei, Kowa, and Pentax cameras, and much larger than the 24×36 mm (0.95×1.42 inch) frame of 35 mm format.
With television, DVD and Blu-ray Disc, converting formats of unequal ratios is achieved by enlarging the original image to fill the receiving format's display area and cutting off any excess picture information (zooming and cropping), by adding horizontal mattes (letterboxing) or vertical mattes (pillarboxing) to retain the original format's aspect ratio, by stretching (hence distorting) the image to fill the receiving format's ratio, or by scaling by different factors in both directions, possibly scaling by a different factor in the center and at the edges (as in Wide Zoom mode).
DVD is a digital optical disc storage format invented and developed in 1995. The medium can store any kind of digital data and is widely used for software and other computer files as well as video programs watched using DVD players. DVDs offer higher storage capacity than compact discs while having the same dimensions.
Digital zoom is a method of decreasing the apparent angle of view of a digital photographic or video image. Digital zoom is accomplished by cropping an image down to a centered area with the same aspect ratio as the original, and usually also interpolating the result back up to the pixel dimensions of the original. It is accomplished electronically, with no adjustment of the camera's optics, and no optical resolution is gained in the process.
The pillarbox effect occurs in widescreen video displays when black bars are placed on the sides of the image. It becomes necessary when film or video that was not originally designed for widescreen is shown on a widescreen display, or a narrower widescreen image is displayed within a wider aspect ratio, such as a 16:9 image in a 2.39:1 frame. The original material is shrunk and placed in the middle of the widescreen frame.
Practical limitations
In motion picture formats, the physical size of the film area between the sprocket perforations determines the image's size. The universal standard (established by William Dickson and Thomas Edison in 1892) is a frame that is four perforations high. The film itself is 35 mm wide (1.38 in), but the area between the perforations is 24.89 mm × 18.67 mm (0.980 in × 0.735 in), leaving the de facto ratio of 4:3, or 1.3:1. [4]
A sprocket or sprocket-wheel is a profiled wheel with teeth, or cogs, that mesh with a chain, track or other perforated or indented material. The name 'sprocket' applies generally to any wheel upon which radial projections engage a chain passing over it. It is distinguished from a gear in that sprockets are never meshed together directly, and differs from a pulley in that sprockets have teeth and pulleys are smooth.
William Kennedy-Laurie Dickson was a Scottish inventor who devised an early motion picture camera under the employment of Thomas Edison.
Thomas Alva Edison was an American inventor and businessman who has been described as America's greatest inventor. He developed many devices in fields such as electric power generation, mass communication, sound recording, and motion pictures. These inventions, which include the phonograph, the motion picture camera, and the long-lasting, practical electric light bulb, have had a widespread impact on the modern industrialized world. He was one of the first inventors to apply the principles of organized science and teamwork to the process of invention, working with many researchers and employees. He established the first industrial research laboratory.
With a space designated for the standard optical soundtrack, and the frame size reduced to maintain an image that is wider than tall, this resulted in the Academy aperture of 22 mm × 16 mm (0.866 in × 0.630 in) or 1.375:1 aspect ratio.
Sound-on-film is a class of sound film processes where the sound accompanying a picture is recorded onto photographic film, usually, but not always, the same strip of film carrying the picture. Sound-on-film processes can either record an analog sound track or digital sound track, and may record the signal either optically or magnetically. Earlier technologies were sound-on-disc, meaning the film's soundtrack would be on a separate phonograph record.
The Academy ratio of 1.375:1 is an aspect ratio of a frame of 35mm film when used with 4-perf pulldown. It was standardized by the Academy of Motion Picture Arts and Sciences as the standard film aspect ratio in 1932, although similar-sized ratios were used as early as 1928.
Cinema terminology
The motion picture industry convention assigns a value of 1.0 to the image's height; an anamorphic frame (since 1970, 2.39:1) is often incorrectly described (rounded) as 2.40:1 or 2.40 ("two-four-oh"). After 1952, a number of aspect ratios were experimented with for anamorphic productions, including 2.66:1 and 2.55:1. [5] A SMPTE specification for anamorphic projection from 1957 (PH22.106-1957) finally standardized the aperture to 2.35:1. [5] An update in 1970 (PH22.106-1971) changed the aspect ratio to 2.39:1 in order to make splices less noticeable. [5] This aspect ratio of 2.39:1 was confirmed by the most recent revision from August 1993 (SMPTE 195-1993). [5]
In American cinemas, the common projection ratios are 1.85:1 and 2.39:1. Some European countries have 1.6:1 as the wide screen standard. The "Academy ratio" of 1.375:1 was used for all cinema films in the sound era until 1953 (with the release of George Stevens' Shane in 1.6:1). During that time, television, which had a similar aspect ratio of 1.3:1, became a perceived threat to movie studios. Hollywood responded by creating a large number of wide-screen formats: CinemaScope (up to 2.6:1), Todd-AO (2.20:1), and VistaVision (initially 1.50:1, now 1.6:1 to 2.00:1) to name just a few. The "flat" 1.85:1 aspect ratio was introduced in May 1953, and became one of the most common cinema projection standards in the U.S. and elsewhere.
The goal of these various lenses and aspect ratios was to capture as much of the frame as possible, onto as large an area of the film as possible, in order to fully utilize the film being used. Some of the aspect ratios were chosen to utilize smaller film sizes in order to save film costs while other aspect ratios were chosen to use larger film sizes in order to produce a wider higher resolution image. In either case the image was squeezed horizontally to fit the film's frame size and avoid any unused film area. [6]
Movie camera systems
Development of various film camera systems must ultimately cater to the placement of the frame in relation to the lateral constraints of the perforations and the optical soundtrack area. One clever wide screen alternative, VistaVision, used standard 35 mm film running sideways through the camera gate, so that the sprocket holes were above and below frame, allowing a larger horizontal negative size per frame as only the vertical size was now restricted by the perforations. There were even a limited number of projectors constructed to also run the print-film horizontally. Generally, however, the 1.50:1 ratio of the initial VistaVision image was optically converted to a vertical print (on standard four-perforation 35 mm film) to show with the standard projectors available at theaters, and was then masked in the projector to the US standard of 1.85:1. The format was briefly revived by Lucasfilm in the late 1970s for special effects work that required larger negative size (due to image degradation from the optical printing steps necessary to make multi-layer composites). It went into obsolescence largely due to better cameras, lenses, and film stocks available to standard four-perforation formats, in addition to increased lab costs of making prints in comparison to more standard vertical processes. (The horizontal process was also adapted to 70 mm film by IMAX, which was first shown at the Osaka '70 Worlds Fair.)
Super 16 mm film was frequently used for television production due to its lower cost, lack of need for soundtrack space on the film itself (as it is not projected but rather transferred to video), and aspect ratio similar to 16:9 (the native ratio of Super 16 mm is 15:9). It also can be blown up to 35 mm for theatrical release and therefore is sometimes used for feature films.
Current video standards
1:1 (Square)
Square displays are rarely used in devices [7] [8] and monitors. [9] Nonetheless, video consumption on social apps has grown rapidly and led to the emergence of new video formats more suited to mobile devices that can be held in horizontal and vertical orientations. In that sense, square video was popularized by mobile apps such as Instagram and has since been supported by other major social platforms including Facebook and Twitter. It can fill nearly twice as much screen space compared to 16:9 format (when the device is held differently while viewing from how video was recorded).
4:3 standard
4:3 (1.3:1) (generally read as "Four-Three", "Four-by-Three", or "Four-to-Three") for standard television has been in use since the invention of moving picture cameras and many computer monitors used to employ the same aspect ratio. 4:3 was the aspect ratio used for 35 mm films in the silent era. It is also very close to the 1.375:1 Academy ratio, defined by the Academy of Motion Picture Arts and Sciences as a standard after the advent of optical sound-on-film. By having TV match this aspect ratio, movies originally photographed on 35 mm film could be satisfactorily viewed on TV in the early days of the medium (i.e. the 1940s and the 1950s).
With the adoption of high definition television, the majority of modern televisions are now produced with 16:9 displays instead. Apple's iPad series of tablets, however, continue to use 4:3 displays (despite other Apple products typically using widescreen aspect ratios) to better suit use as an e-reader. [10]
16:9 standard
16:9 (1.7:1) (generally named as "Sixteen-by-Nine", "Sixteen-Nine", and "Sixteen-to-Nine") is the international standard format of HDTV, non-HD digital television and analog widescreen television PALplus. Japan's Hi-Vision originally started with a 5:3 (= 15:9) ratio but converted when the international standards group introduced a wider ratio of 5⅓ to 3 (= 16:9). Many digital video cameras have the capability to record in 16:9 (= 42:32), and 16:9 is the only widescreen aspect ratio natively supported by the DVD standard. DVD producers can also choose to show even wider ratios such as 1.85:1 and 2.39:1 [2] within the 16:9 DVD frame by hard matting or adding black bars within the image itself. However, it was used often in British TVs in the United Kingdom in the 1990s.
1.85:1
When cinema attendance dropped, Hollywood created widescreen aspect ratios in order to differentiate the film industry from TV, with one of the most common being the 1.85:1 ratio. [11]
2:1
The 2:1 aspect ratio was first used in the 1950s for the RKO Superscope format. [12] [13]
Since 1998, cinematographer Vittorio Storaro has advocated for a format named "Univisium" that uses a 2:1 format. [14] It is designed to be a compromise between the cinema 2.39:1 aspect ratio and the HD-TV broadcast 16:9 ratio. Univisium has gained little traction in the theatrical film market, but has recently been used by Netflix and Amazon Video for productions such as House of Cards and Transparent , respectively. This aspect ratio is standard on the acquisition formats mandated by these content platforms and is not necessarily a creative choice. [15]
Moreover, some mobile devices, such as the LG G6, LG V30, Huawei Mate 10 Pro, Google Pixel 2 XL, OnePlus 5T and Sony Xperia XZ3, are embracing the 2:1 format (advertised as 18:9), as well as the Samsung Galaxy S8, Samsung Galaxy Note 8, Samsung Galaxy S9 and Samsung Galaxy Note 9 with a slightly similar 18.5:9 format. [16] [17] The Apple iPhone X also has a similar screen ratio of 19.5:9 (2.16:1).
2.35:1 and 2.39:1
Anamorphic format is the cinematography technique of shooting a widescreen picture on standard 35 mm film or other visual recording media with a non-widescreen native aspect ratio. When projected, image have an approximated 2.35:1 or 2.39:1 (often rounded to 2.4:1) aspect ratio. "21:9 aspect ratio" is actually 64:27 (= 43:33), or approximately 2.37:1, and is a near both cinematic movie aspect ratios.
Mobile devices are now starting to use the 21:9 format, such as the Sony Xperia 1.
Vertical video
Another trend arising from the massive use of smartphones is Vertical video (9:16), that is intended for viewing in portrait mode. It was popularized by Snapchat and is also now being adopted by Twitter and Facebook. Instagram stories are based on this aspect ratio as well.
Obtaining height, width, and area of the screen
Often, screen specifications are given by their diagonal length. The following formulae can be used to find the height (h), width (w) and area (A), where r stands for ratio, written as a fraction, and d for diagonal length.
${\displaystyle h={\frac {d}{\sqrt {r^{2}+1}}}\qquad w={\frac {r\times d}{\sqrt {{r^{2}}+1}}}\qquad A={\frac {r\times d^{2}}{{r^{2}}+1}}={\frac {d^{2}}{r+{\frac {1}{r}}}}}$
Distinctions
This article primarily addresses the aspect ratio of images as displayed, which is more formally referred to as the display aspect ratio (DAR). In digital images, there is a distinction with the storage aspect ratio (SAR), which is the ratio of pixel dimensions. If an image is displayed with square pixels, then these ratios agree; if not, then non-square, "rectangular" pixels are used, and these ratios disagree. The aspect ratio of the pixels themselves is known as the pixel aspect ratio (PAR) – for square pixels this is 1:1 – and these are related by the identity:
SAR × PAR = DAR.
Rearranging (solving for PAR) yields:
PAR = DAR/SAR.
For example, a 640 × 480 VGA image has a SAR of 640/480 = 4:3, and if displayed on a 4:3 display (DAR = 4:3), has square pixels, hence a PAR of 1:1. By contrast, a 720 × 576 D-1 PAL image has a SAR of 720/576 = 5:4, but is displayed on a 4:3 display (DAR = 4:3), so by this formula it would have a PAR of (4:3)/(5:4) = 16:15.
However, because standard definition digital video was originally based on digitally sampling analog television, the 720 horizontal pixels actually capture a slightly wider image to avoid loss of the original analog picture. In actual images, these extra pixels are often partly or entirely black, as only the center 704 horizontal pixels carry actual 4:3 or 16:9 image. Hence, the actual pixel aspect ratio for PAL video is a little different from that given by the formula, specifically 12:11 for PAL and 10:11 for NTSC. For consistency, the same effective pixel aspect ratios are used even for standard definition digital video originated in digital form rather than converted from analog. For more details refer to the main article.
In analog images such as film there is no notion of pixel, nor notion of SAR or PAR, and "aspect ratio" refers unambiguously to DAR. Actual displays do not generally have non-square pixels, though digital sensors might; they are rather a mathematical abstraction used in resampling images to convert between resolutions.
Non-square pixels arise often in early digital TV standards, related to digitalization of analog TV signals – whose horizontal and vertical resolutions differ and are thus best described by non-square pixels – and also in some digital videocameras and computer display modes, such as Color Graphics Adapter (CGA). Today they arise particularly in transcoding between resolutions with different SARs.
DAR is also known as image aspect ratio and picture aspect ratio, though the latter can be confused with pixel aspect ratio.
Visual comparisons
Comparing two different aspect ratios poses some subtleties – when comparing two aspect ratios, one may compare images with equal height, equal width, equal diagonal, or equal area. More amorphous questions include whether particular subject matter has a natural aspect ratio (panoramas being wide, full-length images of people being tall), or whether a particular ratio is more or less aesthetically pleasing, for example the golden ratio (~1.618).
Televisions and other displays typically list their size by their diagonal. Given the same diagonal, a 4:3 screen has more area compared to 16:9. For CRT-based technology, an aspect ratio that is closer to square is cheaper to manufacture. The same is true for projectors, and other optical devices such as cameras, camcorders, etc. For LCD and plasma displays, however, the cost is more related to the area. Producing wider and shorter screens can yield the same advertised diagonal, but with less area.
The following compares crops of an image at 4:3 and 16:9 ratios, with different dimensions set equal. Note that either image (or both) can be cropped; one aspect doesn't necessarily show more detail than the other.
• Images using the same diagonal size:
4:3 (1.3:1) 16:9 (1.7:1)
• Images using the same area / same number of pixels:
4:3 (1.3:1) 16:9 (1.7:1)
• Images using the same height / same vertical size:
4:3 (1.3:1) 16:9 (1.7:1)
• Images using the same width/ same horizontal size:
4:3 (1.3:1) 16:9 (1.7:1)
Previous and currently used aspect ratios
See list of common resolutions for a listing of computer resolutions and aspect ratios.
See list of film formats for a full listing of film formats, including their aspect ratios.
• 1.19:1 (19:16): Sometimes referred to as the Movietone ratio, this ratio was used briefly during the transitional period when the film industry was converting to sound, from 1926 to 1932 approx. It is produced by superimposing an optical soundtrack over a full-gate 1.3 aperture in printing, resulting in an almost square image. Films shot in this ratio are often projected or transferred to video incorrectly using a 1.37 mask or squashed to 1.37. Examples of films shot in the Movietone ratio include Sunrise, M, Hallelujah! and The Lighthouse. [18] [19]
• 1.25:1 (5:4): Once-popular aspect for larger format computer monitors, especially in the guise of mass-produced 17" and 19" LCD panels or 19" and 21" CRTs, using 1280×1024 (SXGA) or similar resolutions. Notably one of the few popular display aspect ratios narrower than 4:3, and one popularised by business (CAD, DTP) rather than entertainment use, as it is well-suited to full-page layout editing. Historically, 5:4 was also the original aspect ratio of early 405-line television broadcasts, which progressed to a wider 4:3 as the idea of broadcasting cinema films gained traction.
• 1.3:1 (4:3): 35 mm original silent film ratio, today commonly known in TV and video as 4:3. Also standard ratio for MPEG-2 video compression. This format is still used in many personal video cameras today and has influenced the selection or design of other aspect ratios. It is the standard Super 35mm ratio.
• 1.37:1: 16 mm and 35 mm standard ratio.
• 1.375:1 (11:8): 35 mm full-screen sound film image, nearly universal in movies between 1932 and 1953. Officially adopted as the Academy ratio in 1932 by AMPAS. Rarely used in theatrical context nowadays, but occasionally used for other context.
• 1.43:1: IMAX format. IMAX productions use 70 mm wide film (the same as used for 70 mm feature films), but the film runs through the camera and projector horizontally. This allows for a physically larger area for each image.
• 1.5:1 (3:2): The aspect ratio of 35 mm film used for still photography when 8 perforations are exposed. Also the native aspect ratio of VistaVision, for which the film runs horizontally. Used on the Chrome OS-based Chromebook Pixel Notebook PC, the Game Boy Advance portable game console, the Surface Pro 3 laplet and Surface Studio.
• 1.5:1 (14:9): Widescreen aspect ratio sometimes used in shooting commercials etc. as a compromise format between 4:3 and 16:9. When converted to a 16:9 frame, there is slight pillarboxing, while conversion to 4:3 creates slight letterboxing. All widescreen content on ABC Family's SD feed until January 2016 were presented in this ratio.
• 1.6:1 (16:10 = 8:5): Widescreen computer monitor ratio (for instance 1920×1200 resolution).
• 1.6:1 (5:3): 35 mm widescreen ratio, originally invented by Paramount Pictures, now a standard among several European countries.[ which? ] It is also the native Super 16 mm frame ratio. Sometimes this ratio is rounded up to 1.6:1. From the late 1980s to the early 2000s, Walt Disney Feature Animation's CAPS program animated their features in the 1.6:1 ratio (a compromise between the 1.85:1 theatrical ratio and the 1.3:1 ratio used for home video), this format is also used on the Nintendo 3DS's top screen as well.
• 1.75:1 (7:4): Early 35 mm widescreen ratio, primarily used by MGM and Warner Bros. between 1953 and 1955, and since abandoned, though Disney has cropped some of its post-50's Full Screen films to this ratio for DVD, including The Jungle Book .
• 1.7:1 (16:9 = 42:32): Video widescreen standard, used in high-definition television, one of three ratios specified for MPEG-2 video compression. Also used increasingly in personal video cameras. Sometimes this ratio is rounded up to 1.78:1.
• 1.85:1 (37:20): 35 mm US and UK widescreen standard for theatrical film. Introduced by Universal Pictures in May, 1953. Projects approximately 3 perforations ("perfs") of image space per 4 perf frame; films can be shot in 3-perf to save cost of film stock. Also the ratio of Ultra 16 mm.
• 1.896:1 (256:135): DCI / SMPTE digital cinema basic resolution container aspect ratio. [20]
• 2:1 : Recently popularized by the Red Digital Cinema Camera Company. Original SuperScope ratio, also used in Univisium. Used as a flat ratio for some American studios in the 1950s and abandoned in the 1960s. Also used in recent mobile phones such as the LG G6, Google Pixel 2 XL, HTC U11+, Xiaomi MIX 2S and Huawei Mate 10 Pro, while the Samsung Galaxy S8, Note 8, and S9 use the similar 18.5:9 ratio.
• 2.2:1 (11:5): 70 mm standard. Originally developed for Todd-AO in the 1950s. Specified in MPEG-2 as 2.21:1, but hardly used.
• 2.35:1 (~47:20): 35 mm anamorphic prior to 1970, used by CinemaScope ("'Scope") and early Panavision. The anamorphic standard has subtly changed so that modern anamorphic productions are actually 2.39, [2] but often referred to as 2.35 anyway, due to old convention. (Note that anamorphic refers to the compression of the image on film to maximize an area slightly taller than standard 4-perf Academy aperture, but presents the widest of aspect ratios.) All Indian Bollywood films released after 1972 are shot in this standard for theatrical exhibition.
• 2.37:1 (64:27 = 43:33): TVs were produced with this aspect ratio between 2009 and 2012 [21] and marketed as "21:9 cinema displays". But this aspect ratio is still seen on higher end monitors, and are sometimes called UltraWide monitors.
• 2.39:1 (~43:18): 35 mm anamorphic from 1970 onwards. Aspect ratio of current anamorphic widescreen theatrical viewings. Often commercially branded as Panavision format or 'Scope'.
• 2.4:1 (12:5): Rounded notation of 2.39:1, also as 2.40:1. Blu-ray Disc film releases may use only 800 instead of 803 or 804 lines of the 1920×1080 resolution, resulting in an even 2.4:1 aspect ratio.
• 2.55:1 (~23:9): Original aspect ratio of CinemaScope before optical sound was added to the film in 1954. This was also the aspect ratio of CinemaScope 55.
• 2.59:1 (~13:5): Cinerama at full height (three specially captured 35 mm images projected side-by-side into one composite widescreen image).
• 2.6:1 (8:3): Full frame output from Super 16 mm negative when an anamorphic lens system has been used. Effectively, an image that is of the ratio 24:9 is squashed onto the native 15:9 aspect ratio of a Super 16 mm negative.
• 2.76:1 (~11:4): Ultra Panavision 70/MGM Camera 65 (65 mm with 1.25× anamorphic squeeze). Used only on a handful of films between 1957 and 1966 and two films in the 2010s, for some sequences of How the West Was Won (1962) with a slight crop when converted to three strip Cinerama, and films such as It's a Mad, Mad, Mad, Mad World (1963) and Ben-Hur (1959). Quentin Tarantino used it for The Hateful Eight (2015), Gareth Edwards for Rogue One (2016), Kirill Serebrennikov for Leto (2018).
• 3.5:1 (32:9): In 2017, Samsung and Phillips announced 'Super UltraWide displays', with aspect ratio of 32:9.
• 3.6:1 (18:5): In 2016, IMAX announced the release of films in 'Ultra-WideScreen 3.6' format, [22] with an aspect ratio of 36:10. [23] Ultra-WideScreen 3.6 video format didn't spread, as cinemas in an even wider ScreenX 270° format were released. [24]
• 4:1: Rare use of Polyvision, three 35 mm 1.3:1 images projected side by side. First used in 1927 on Abel Gance's Napoléon .
• 12:1: Circle-Vision 360° developed by the Walt Disney Company in 1955 for use in Disneyland. Uses nine 4:3 35 mm projectors to show an image that completely surrounds the viewer. Used in subsequent Disney theme parks and other past applications.
Aspect ratio releases
Original aspect ratio (OAR)
Original Aspect Ratio (OAR) is a home cinema term for the aspect ratio or dimensions in which a film or visual production was produced – as envisioned by the people involved in the creation of the work. As an example, the film Gladiator was released to theaters in the 2.39:1 aspect ratio. It was filmed in Super 35 and, in addition to being presented in cinemas and television in the Original Aspect Ratio of 2.39:1, it was also broadcast without the matte, altering the aspect ratio to the television standard of 1.3:1. Because of the varied ways in which films are shot, IAR (Intended Aspect Ratio) is a more appropriate term, but is rarely used.
Modified aspect ratio (MAR)
Modified Aspect Ratio is a home cinema term for the aspect ratio or dimensions in which a film was modified to fit a specific type of screen, as opposed to original aspect ratio. Modified aspect ratios are usually either 1.3:1 (historically), or (with the advent of widescreen television sets) 1.7:1 aspect ratio. 1.3:1 is the modified aspect ratio used historically in VHS format. A modified aspect ratio transfer is achieved by means of pan and scan or open matte, the latter meaning removing the cinematic matte from a 1.85:1 film to open up the full 1.3:1 frame. Another name for it is "prescaled" aspect ratio".
Problems in film and television
Multiple aspect ratios create additional burdens on directors and the public, and confusion among TV broadcasters. It is common for a widescreen film to be presented in an altered format (cropped, letterboxed or expanded beyond the original aspect ratio). It is also not uncommon for windowboxing to occur (when letterbox and pillarbox happen simultaneously). For instance, a 16:9 broadcast could embed a 4:3 commercial within the 16:9 image area. A viewer watching on a standard 4:3 (non-widescreen) television would see a 4:3 image of the commercial with 2 sets of black stripes, vertical and horizontal (windowboxing or the postage stamp effect). A similar scenario may also occur for a widescreen set owner when viewing 16:9 material embedded in a 4:3 frame, and then watching that in 16:9. Active Format Description is a mechanism used in digital broadcasting to avoid this problem. It is also common that a 4:3 image is stretched horizontally to fit a 16:9 screen to avoid pillar boxing but distorts the image so subjects appear short and fat.
Both PAL and NTSC have provision for some data pulses contained within the video signal used to signal the aspect ratio (See ITU-R BT.1119-1 – Widescreen signaling for broadcasting). These pulses are detected by television sets that have widescreen displays and cause the television to automatically switch to 16:9 display mode. When 4:3 material is included (such as the aforementioned commercial), the television switches to a 4:3 display mode to correctly display the material. Where a video signal is transmitted via a European SCART connection, one of the status lines is used to signal 16:9 material as well.
Still photography
Common aspect ratios in still photography include:
• 1:1
• 5:4 (1.25:1)
• 4:3 (1.3:1)
• 3:2 (1.5:1)
• 5:3 (1.6:1)
• 16:9 (1.7:1)
• 3:1
Many digital still cameras offer user options for selecting multiple image aspect ratios. Some achieve this through the use of multi-aspect sensors (notably Panasonic), while others simply crop their native image format to have the output match the desired image aspect ratio.
1:1
Is the classic Kodak image, and is available as a choice in some digital still cameras, and hearkens back to the days of film cameras when the square image was popular with photographers using twin lens reflex cameras. These medium format cameras used 120 film rolled onto spools. The 6 × 6 cm image size was the classic 1:1 format in the recent past. 120 film can still be found and used today. Many Polaroid instant films were designed as square formats. Furthermore, up until August 2015, photo-sharing site Instagram only allowed users to upload images in 1:1 format. In 2017, Fujifilm added the 1:1 Instax Square format to their lineup of instant film cameras.
5:4
Common in large and medium format photography, and still in common use for prints from digital cameras in the 8"×10" size.
4:3
Is used by most digital point-and-shoot cameras, Four Thirds system, Micro Four Thirds system cameras and medium format 645 cameras. The 4:3 digital format popularity was developed to match the then prevailing digital displays of the time, 4:3 computer monitors.
The next several formats have their roots in classic film photography image sizes, both the classic 35 mm film camera, and the multiple format Advanced Photo System (APS) film camera. The APS camera was capable of selecting any of three image formats, APS-H ("High Definition" mode), APS-C ("Classic" mode) and APS-P ("Panoramic" mode).
3:2
is used by classic 35 mm film cameras using a 24 mm × 36 mm image size, and their digital derivatives represented by DSLRs. Typical DSLRs come in two flavors, the so-called professional "full frame" (24 mm × 36 mm) sensors and variations of smaller, so called "APS-C" sensors. The term "APS" is derived from another film format known as the APS and the "-C" refers to "Classic" mode, which exposed images over a smaller area (25.1 mm × 16.7 mm) but retaining the same "classic" 3:2 proportions as full frame 35 mm film cameras.
When discussing DSLR's and their non-SLR derivatives, the term APS-C has become an almost generic term. The two major camera manufacturers Canon and Nikon each developed and established sensor standards for their own versions of APS-C sized and proportioned sensors. Canon actually developed two standards, APS-C and a slightly larger area APS-H (not to be confused with the APS-H film format), while Nikon developed its own APS-C standard, which it calls DX. Regardless of the different flavors of sensors, and their varying sizes, they are close enough to the original APS-C image size, and maintain the classic 3:2 image proportions that these sensors are generally known as an "APS-C" sized sensor.
The reason for DSLR's image sensors being the flatter 3:2 versus the taller point-and-shoot 4:3 is that DSLRs were designed to match the legacy 35 mm SLR film, whereas the majority of digital cameras were designed to match the predominant computer displays of the time, with VGA, SVGA, XGA and UXGA all being 4:3. Widescreen computer monitors did not become popular until the advent of HDTV, which uses a 16:9 image aspect ratio.
16:9
16:9 is another format that has its roots in the APS film camera.[ citation needed ] Known as APS-H (30.2 mm × 16.7 mm), with the "-H" denoting "High Definition", the 16:9 format is also the standard image aspect ratio for HDTV. 16:9 is gaining popularity as a format in all classes of consumer still cameras which also shoot High Definition (HD) video. When still cameras have an HD video capability, some can also record stills in the 16:9 format, ideal for display on HD televisions and widescreen computer displays.
3:1
is another format that can find its roots in the APS film camera. Known as APS-P (30.2 × 9.5 mm), with the -P" denoting "Panorama", the 3:1 format was used for panorama photography. The APS-P panorama standard is the least adhered to any APS standard, and panoramic implementation varies with by manufacturer on different cameras, with the only commonality being that the image is much longer than it is tall, in the classic "panorama" style.
Common print sizes in the U.S. (in inches) include 4×6 (1.5), 5×7 (1.4), 4×5 and 8×10 (1.25), and 11×14 (1.27); large format cameras typically use one of these aspect ratios. Medium-format cameras typically have format designated by nominal sizes in centimeters (6×6, 6×7, 6×9, 6×4.5), but these numbers should not be interpreted as exact in computing aspect ratios. For example, the usable height of 120-format roll film is 56mm, so a width of 70mm (as in 6×7) yields an aspect ratio of 4:5 — ideal for enlarging to make an 8×10" portrait. Print sizes are usually defined by their portrait dimensions (tall) while equipment aspect ratios are defined by their landscape dimensions (wide, flipped sideways). A good example of this a 4×6 print (6 inch wide by 4 inch tall landscape) perfectly matches the 3:2 aspect ratio of a DSLR/35 mm, since 6/2=3 and 4/2=2.
For analog projection of photographic slides, projector and screen use a 1:1 aspect ratio, supporting horizontal and vertical orientation equally well. In contrast, digital projection technology typically supports vertically oriented images only at a fraction of the resolution of landscape-oriented images. For example, projecting a digital still image having a 3:2 aspect ratio on a 16:9 projector employs 84.3% of available resolution in horizontal orientation, but only 37.5% in vertical orientation.
Citations
1. "BBC Academy - Academy - Beyond HD". web.archive.org. 27 June 2017.
2. The 2.39:1 ratio is commonly labeled 2.40:1, e.g., in the American Society of Cinematographers' American Cinematographer Manual (Many widescreen films before the 1970 SMPTE revision used 2.35:1).
3. "Panasonic Introduces 2 New Cameras". India: Tech Tree. Archived from the original on 2009-01-23.Cite journal requires |journal= (help)
4. Burum, Stephen (2004). American Cinematographer Manual (9th ed.). ASC Press. ISBN 0-935578-24-2.
5. "ALEXA Anamorphic De-squeeze". Arri. 2011-07-07. Retrieved 2014-06-21.
6. "Anamorphic Now" (PDF). Film and Digital Times (53): 24–31. April 2013. Retrieved 2014-06-21.
7. "BlackBerry Passport – Full phone specifications". www.gsmarena.com. Retrieved 2018-11-29.
8. "Sony SmartWatch 3 SWR50 – Full phone specifications". www.gsmarena.com. Retrieved 2019-01-24.
9. "Eizo's 27-inch 3K display is perfectly square - Geek.com". Geek.com. 2014-11-20. Retrieved 2018-11-29.
10. McElhearn, Kirk (2010-02-01). "Why Apple chose the iPad's screen format". Macworld. Retrieved 2019-07-24.
11. Berger, John L. (2019). "Aspect Ratios and Camera Formats". www.widescreen.org. Retrieved 2018-10-30.
12. "Widescreen Museum – CinemaScope Derivatives – Superscope 1". www.widescreenmuseum.com. Retrieved 2018-11-02.
13. "The Aspect Ratio of 2.00 : 1 is Everywhere | VashiVisuals". vashivisuals.com. Retrieved 2018-11-02.
14. Eelvee (4 March 2007). ". . : : VITTORIO STORARO : : . .: What is UNIVISIUM?".
15. O'Falt, Chris (2017-04-04). "What Amazon and Netflix's Demand for 4K Means for Documentaries". IndieWire. Retrieved 2018-05-10.
16. Petrov, Daniel. "So, what is this 2:1 Univisium display ratio on the LG G6 and likely the S8?". Phone Arena.
17. The official Honor website displays that it has an 18:9 ratio (visit the gaming tab).
18. Scott Eyman, The Speed of Sound: Hollywood and the Talkie Revolution, 1926–1930, New York, Simon & Schuster (1997), p. 222.
19. "'The Lighthouse,' 'The Witch' and the Horror of Robert Eggers". The Hollywood Reporter. Retrieved 2019-11-10.
20. Arne Nowak (October 2010). "Digital Cinema Technologies from the Archive's Perspective" (PDF). p. 4. Retrieved May 16, 2016.
21. Goddard, Louis. Philips discontinuing super-wide Cinema 21:9 TVs due to lack of demand. The Verge. 2012-08-28. Retrieved 2013-03-18.
22. "Voyage of Time: The IMAX® Experience in Ultra-Widescreen". IMAX.com. Dec 7, 2016. Retrieved April 27, 2018.
23. Kristopher Tapley (Dec 5, 2016). "'Ultra Widescreen' Version of Terrence Malick's 'Voyage of Time' Set for Release". variety.com. Retrieved April 27, 2018.
24. Aftab, Kaleem. "Introducing Screen X, Cinema in 270 Degrees | Filmmaker Magazine". Filmmaker Magazine. Retrieved 2018-10-12.
Related Research Articles
The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side – the ratio of width to height, when the rectangle is oriented as a "landscape".
Letterboxing is the practice of transferring film shot in a widescreen aspect ratio to standard-width video formats while preserving the film's original aspect ratio. The resulting videographic image has mattes above and below it; these mattes are part of the image. LBX or LTBX are the identifying abbreviations for films and images so formatted.
Pan and scan is a method of adjusting widescreen film images so that they can be shown in fullscreen proportions of a standard definition 4:3 aspect ratio television screen, often cropping off the sides of the original widescreen image to focus on the composition's most important aspects.
Standard-definition television is a television system which uses a resolution that is not considered to be either high or enhanced definition. SDTV and high-definition television (HDTV) are the two categories of display formats for digital television (DTV) transmissions. "Standard" refers to the fact that it was the prevailing specification for broadcast television in the mid- to late-20th century.
Widescreen images are images that are displayed within a set of aspect ratios used in film, television and computer screens. In film, a widescreen film is any film image with a width-to-height aspect ratio greater than the standard 1.37:1 Academy aspect ratio provided by 35 mm film.
35 mm film is the film gauge most commonly used for motion pictures and chemical still photography. The name of the gauge refers to the width of the photographic film, which consists of strips 34.98 ± 0.03 millimetres (1.3772 ± 0.0012 in) wide. The standard negative pulldown for movies is four perforations per frame along both edges, which results in 16 frames per foot of film. For still photography, the standard frame has eight perforations on each side.
VistaVision is a higher resolution, widescreen variant of the 35 mm motion picture film format which was created by engineers at Paramount Pictures in 1954.
16 mm film is a historically popular and economical gauge of film. 16 mm refers to the width of the film; other common film gauges include 8 and 35 mm. It is generally used for non-theatrical film-making, or for low-budget motion pictures. It also existed as a popular amateur or home movie-making format for several decades, alongside 8 mm film and later Super 8 film. Eastman Kodak released the first 16 mm "outfit" in 1923, consisting of a camera, projector, tripod, screen and splicer, for \$335. RCA-Victor introduced a 16 mm sound movie projector in 1932, and developed an optical sound-on-film 16 mm camera, released in 1935.
CinemaScope is an anamorphic lens series used, from 1953 to 1967, and less often later, for shooting widescreen movies that, crucially, could be screened in theatres using existing equipment, albeit with a lens adapter. Its creation in 1953 by Spyros P. Skouras, the president of 20th Century Fox, marked the beginning of the modern anamorphic format in both principal photography and movie projection.
Anamorphic widescreen is a process by which a comparatively wide widescreen image is horizontally compressed to fit into a storage medium with a narrower aspect ratio, reducing the horizontal resolution of the image while keeping its full original vertical resolution. Compatible play-back equipment can then expand the horizontal dimension to show the original widescreen image. This is typically used to allow one to store widescreen images on a medium that was originally intended for a narrower ratio, while using as much of the frame – and therefore recording as much detail – as possible.
Negative pulldown is the manner in which an image is exposed on a film stock, described in the number of film perforations spanned by an individual frame. It can also describe the orientation of the image on the negative, whether it is captured horizontally or vertically. Changing the number of exposed perforations allows a cinematographer to change both the aspect ratio of the image and the size of the area on the film stock that the image occupies.
Techniscope or 2-perf is a 35 mm motion picture camera film format introduced by Technicolor Italia in 1960. The Techniscope format uses a two film-perforation negative pulldown per frame, instead of the standard four-perforation frame usually exposed in 35 mm film photography. Techniscope's 2.33:1 aspect ratio is easily cropped to the 2.39:1 widescreen ratio, because it uses half the amount of 35 mm film stock and standard spherical lenses. Thus, Techniscope release prints are made by anamorphosizing and enlarging each frame by a factor of two.
Univisium is a proposed universal film format created by cinematographer Vittorio Storaro, ASC, AIC and his son, Fabrizio, to unify all future theatrical and television movies into one respective aspect ratio of 2:1. The proposed format also includes new standards for projection to maximize the efficiencies of the Univisium format.
Note: For a quick understanding of numbers like 1/2.3, skip to table of sensor formats and sizes. For a simplified discussion of image sensors see image sensor.
Anamorphic format is the cinematography technique of shooting a widescreen picture on standard 35 mm film or other visual recording media with a non-widescreen native aspect ratio. It also refers to the projection format in which a distorted image is "stretched" by an anamorphic projection lens to recreate the original aspect ratio on the viewing screen. The word anamorphic and its derivatives stem from the Greek words meaning "formed again". In the late 1990s and 2000s, anamorphic lost popularity in comparison to "flat" formats such as Super 35 with the advent of digital intermediates; however in the years since digital cinema cameras and projectors have become commonplace, anamorphic has experienced a considerable resurgence of popularity, due in large part to the higher base ISO sensitivity of digital sensors, which facilitates shooting at smaller apertures.
The technology of television has evolved since its early days using a mechanical system invented by Paul Gottlieb Nipkow in 1884. Every television system works on the scanning principle first implemented in the rotating disk scanner of Nipkow. This turns a two-dimensional image into a time series of signals that represent the brightness and color of each resolvable element of the picture. By repeating a two-dimensional image quickly enough, the impression of motion can be transmitted as well. For the receiving apparatus to reconstruct the image, synchronization information is included in the signal to allow proper placement of each line within the image and to identify when a complete image has been transmitted and a new image is to follow.
21:9 is a consumer electronics (CE) marketing term to describe the ultra-widescreen aspect ratio of 64:27, designed to show films recorded in CinemaScope and equivalent modern anamorphic formats. The main benefit of this screen aspect ratio, compared to the more common 16:9, is the absence of the black bars at the top and bottom of the screen when viewing content in this format, and a constant display height when displaying other content with a lesser aspect ratio.
The graphics display resolution is the width and height dimension of an electronic visual display device, such as a computer monitor, in pixels. Certain combinations of width and height are standardized and typically given a name and an initialism that is descriptive of its dimensions. A higher display resolution in a display of the same size means that displayed photo or video content appears sharper, and pixel art appears smaller.
Ultra-wide or Ultrawide formats refers to photos, videos, and displays, with aspect ratios significantly wider than 2:1. There were multiple moves in history, towards a wider display aspect ratio, including one by Disney. Some moves were successful, while others saw limited success.
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2019-11-15 09:54:35
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http://www.math.sci.hokudai.ac.jp/?p=27929
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## 偏微分方程式セミナー(2017/11/10): Some study on the blowup problem of scale-invariant damping wave equation with Strauss type exponent, Tu Ziheng 氏 (北海道大学)
2017年 11月 10日 16時 30分 ～ 2017年 11月 10日 17時 30分
Tu Ziheng 氏 (北海道大学)
In this talk, I would like to share some of my recent study on this problem. By introducing the modified Bessel function of second kind $K_\nu$, the blowup range is extended from $p_F(n)\leq p < p_s(n+2\mu)$ to $p_F(n)\leq p < p_s(n+\mu)$ for the sub-Strauss exponent. For the critical Strauss exponent, applying the plane wave fourmula and utilizing the limiting behavior of $K_\nu$, the desired lifespan estimate is obtained.
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2018-02-20 15:26:26
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https://hal.inria.fr/inria-00457603
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# On the Price of Anarchy and the Optimal Routing of Parallel non-Observable Queues
1 MESCAL - Middleware efficiently scalable
Inria Grenoble - Rhône-Alpes, LIG - Laboratoire d'Informatique de Grenoble
Abstract : We consider a network of parallel, non-observable queues and analyze the price of anarchy'', an index measuring the worst-case performance loss of a decentralized system with respect to its centralized counterpart. Our analysis is undertaken from the new point of view where the router has the memory of previous dispatching choices, which significantly complicates the nature of the problem. In the limiting regime where the demands proportionally grow with the network capacity, we provide a tight lower bound on the socially-optimal response time and a tight upper bound on the price of anarchy by means of convex programming. Then, we exploit this result to show, by simulation, that the billiard routing scheme yields a response time which is remarkably close to our lower bound, implying that billiards minimize response time. To study the added-value of non-Bernoulli routers, we introduce the price of forgetting'' and prove that it is bounded from above by two, which is tight in heavy-traffic. Finally, other structural properties are derived numerically for the price of forgetting. These claim that the benefit of having memory in the router is independent of the network size and heterogeneity, while monotonically depending on the network load only. These properties yield simple product-forms well-approximating the socially-optimal response time.
Keywords :
Type de document :
Rapport
[Research Report] 2010
Domaine :
Littérature citée [2 références]
https://hal.inria.fr/inria-00457603
Contributeur : Jonatha Anselmi <>
Soumis le : jeudi 18 février 2010 - 10:53:03
Dernière modification le : jeudi 11 janvier 2018 - 06:21:39
Document(s) archivé(s) le : jeudi 18 octobre 2012 - 15:20:32
### Fichier
AnselmiGaujal.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : inria-00457603, version 1
### Citation
Jonatha Anselmi, Bruno Gaujal. On the Price of Anarchy and the Optimal Routing of Parallel non-Observable Queues. [Research Report] 2010. 〈inria-00457603〉
### Métriques
Consultations de la notice
## 332
Téléchargements de fichiers
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2018-01-22 14:44:56
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https://darkgenesis.zenithmoon.com/xna-to-silverxnapart-2-getting-our-xna-project-running-in-silverlight/
|
In Part 1 I went over some of the things you’ll need to do to get your current XNA project ready for use in the Silverlight XNA integration which I’ve dubbed as SilverXNA for this tutorial series.
Today I’m going to walk you through this by migrating the Platformer Sample from the AppHub Educational content into a new SilverXNA project.
If you want to read more about what happens under the covers between the Silverlight and XNA frameworks you can read one of my previous posts here, mostly technical stuff and a Silverlight primer fro XNA devs.
I’m endeavouring to keep this tutorial open to all levels of dev’s, so if some of the instructions are a bit basic for you, just skim read them as needed (just pay attention )
Full source for the completed project can be found here on codeplex
The main focus of this chapter is to just get us running and the problems I’ve faced in getting this to just run, nothing fancy just simple baby steps to show off the impact of the changes were going to make later.
Now if you are converting your own project along side me with this tutorial, make sure you have read through Part one and got a heads up of the main impacts to your project.
Follow along with the series here:
Also Channel 9 are running a similar video series here if you prefer videos!
## Lets get the new project started
So keeping it simple we’ll create a new SilverXNA project, thankfully in the latest Beta 2 phase of the tools they have fixed one of my pet peeves where the Silverlight version of the “Windows Phone Rich Graphics Application” and the XNA “Windows Phone Silverlight and XNA application” which was that the two projects were completely different and in fact one of them didn’t work out of the box. (you can read more about this here, in fact the Rich Graphics app used to be the XNA project and the Silverlight one was called a “Windows Phone 3D graphics application” ), so that’s sorted now but if you compare the two projects there are subtle differences, why they are not the same project for both with the same name is beyond anyone’s guess.
So now it doesn’t matter which one you pick as they are both effectively the same so just pick the one nearest to you (I used the XNA version when prepping for this tutorial series and am now using the Silverlight one for the sample project )
Once you have got it setup you should see the new SilverXNA solution with it’s three projects, A Silverlight C# project(if you chose C# that is, if you are running VB then it’ll obviously be VB, but this tutorial is written for C# so you’ll just have to follow along and convert in your head, the same tricks will work through), an XNA game library (the bridge between Silverlight and XNA) and the XNA Content Project.
If you run this now you’ll get the two new starter screens:
Main Page (Start) Game Page
Nothing spectacular but it does give us a chance to see a good old clean Cornflower Blue page again .
## Brining in the Rain
So with our new project setup, first thing we need to do is bring in our XNA game project with a twist. make sure you have downloaded the Platformer sample from the AppHub first and have it unpacked somewhere.
First remove the “Content” project as we are going to be using the one from our XNA game project, next right click on the Solution and select “Add –> New Project” then select from the XNA branch of the New Project wizard the “Windows Phone Game Library (4.0)” project, name it something appropriate as this is were we are going to copy the Platformer code to (I used PlatformerGameLibrary).
Next Right-Click on the new PlatformerGameLibrary project and select “Add –> Existing Item” which will pop up the File Browse wizard and navigate to the folder where the Platformer sample game code is located, this is key especially if you want to maintain a multi-platform project where we want to share code.
Now select all the “.CS” files with the exception of the “Program.cs” and “Platformergame.cs”. We need to manage in the code from the game.cs file so it fit’s properly with the new SilverXNA project plus we don’t want it interfering with the build, as for the Program.CS file Windows Phone doesn’t even use the
If you just want a separate SilverXNA project in which case just copy the code directly into your new library or even the existing library that came with the solution and skip the previous step.
Next Right click the Solution and select “Add –> Existing Project” and browse to the location of the Platformer sample Content folder, then select the content project there. You should now end up with the following:
So our project is now made up of: The Content project from our existing XNA solution unchanged (although make sure it’s a phone variation of the Content project with assets resized and compressed appropriately using the same asset names as it’s XBOX / PC counterpart if you have one) The unmodified code (we’ll soon change that) from our original XNA game project The Silverlight page project for our solution, we should always endeavour to only put Silverlight specific functionality or presentation in here to keep consistency with other platforms The default XNA Game library that came with our solution. You could quite happily remove this at this point but I have kept it in for now (just in case ), if you are just building a SilverXNA project and not worried about consistency across platforms then feel free to just use this for all your game logic and XNA drawing code.
## And now comes the Breaking
At this point we have pure XNA code in a Silverlight project, so not only will it not run it won’t even compile….
Now if you are doing your own project you should have already done the prerequisites to your project from the instructions in Part 1 but I’m going to re-iterate through them here for the Platformer sample.
References
First off tidy up the references and add the ones we require for each project:
• Reference the Content Project from the SilverXNA (main project link library) project
• Reference the PlatformerGameLibrary project from the Main Silverlight Project
• Add a reference to “Microsoft.Phone.Sensors” to the PlatformerGameLibrary project (as we are using the accelerometer)
• Add a reference to “Microsoft.Phone” to the PlatformerGameLibrary project (as we are using the some native API’s)
Change the scope of the base objects in the PlatformerGameLibrary
Edit the following files and simply make the classes and enumerations within them “Public” so they will be exposed outside the game library
• AnimationPlayer.cs
• Circle.cs
• Enemy.cs
• Gem.cs
• Level.cs
• Player.cs
• Tile.cs
For example, change the following:
From /// <summary>
/// Facing direction along the X axis.
/// </summary>
enum FaceDirection
{
Left = -1,
Right = 1,
}
/// <summary>
/// A monster who is impeding the progress of our fearless adventurer.
/// </summary>
class Enemy
{
public Level Level
{
get { return level; }
}
Level level; To /// <summary>
/// Facing direction along the X axis.
/// </summary>
public enum FaceDirection
{
Left = -1,
Right = 1,
}
/// <summary>
/// A monster who is impeding the progress of our fearless adventurer.
/// </summary>
public class Enemy
{
public Level Level
{
get { return level; }
}
Level level;
Replace GameTime references to just use the TimeSpan ElapsedGameTime variable in the function parameters
In the Update and Draw functions of the above classes replace “GameTime gametime” with “TimeSpan elapsedGameTime” where applicable
From public void Update(GameTime gameTime)
{ To public void Update(TimeSpan elapsedGameTime)
{
Fix code that originally use the GameTime variable
Again in the above classes update and draw functions, remove references that use the gameTime variable as it was passed to the function so that it now uses the elapsedGameTime variable
From float elapsed = (float)gameTime.ElapsedGameTime.TotalSeconds; To float elapsed = (float)elapsedGameTime.TotalSeconds;
Final tidy ups
Clean up the remaining broken references in Player.cs and Level.cs by replacing any mentions of “gameTime” with “elapsedGameTime”. There are a few dotted around including some internal functions such as “DoJump” and “gem.Update(gameTime)”
There’s always one exception
The only reference I was unable to fix was in “Gem.cs” where the project actualy makes use of the “TotalGameTime” property of the original GameTime class. As it’s only one reference I decided to overlook this and just replaced it with “elapsedGameTime” just to keep things simple and it doesn’t overly affect the end result. If it were really important or if it was my own project I may have looked to refactor this a bit better or as stated before passed both elapsed and total time to the function that needed it.
At this point your project should compile with no errors, granted if you run it you will just have an empty blue screen but we know the game code compiles fine.
## Final cut
So with our project in a state where it will work with SilverXNA its time to get this show on the road, now we just need to copy all the relevant bits from the original PlatformerGame..cs and get them working in our new project.
Constructor and Variables
First we need to sort out all the primary variables our game uses and the initialisation logic, so open up “GamePage.XAML.cs” and add the following variables to the top of the GamePage class just below the content manager, Timer and spritebatch variables (fixing any broken references as you go):
// Global content.<br>
private SpriteFont hudFont;
<p>private Texture2D winOverlay;<br>
private Texture2D loseOverlay;<br>
private Texture2D diedOverlay;</p>
<p>// Meta-level game state.<br>
private int levelIndex = -1;<br>
private Level level;<br>
private bool wasContinuePressed;</p>
<p>// When the time remaining is less than the warning time, it blinks on the hud<br>
private static readonly TimeSpan WarningTime = TimeSpan.FromSeconds(30);</p>
<p>// We store our input states so that we only poll once per frame, <br>
// then we use the same input state wherever needed<br>
private GamePadState gamePadState;<br>
private KeyboardState keyboardState;<br>
private TouchCollection touchState;<br>
private AccelerometerState accelerometerState;</p>
<p>// The number of levels in the Levels directory of our content. We assume that<br>
// levels in our content are 0-based and that all numbers under this constant<br>
// have a level file present. This allows us to not need to check for the file<br>
// or handle exceptions, both of which can add unnecessary time to level loading.<br>
private const int numberOfLevels = 3;
And add the following to the end of the GamePage constructor:
Accelerometer.Initialize();
A few notes about the above before we continue, I’ve kept in the references to the Keyboard and Gamepad states, mainly to keep uniformity down the line in the main game project, you can remove them if you with but at present they are doing no harm as they are effectively not being used anyway, its all just about cross platform support.
As for the Accelerometer, I’ve had to tweak it slightly, as we only want the accelerometer on when the GamePage is running and stop it when it’s not but the current AccelerometerHelper is not designed that way, so to avoid the game crashing when you run it twice I have replaced the error exception when it is started twice with a return statement, fix this if you wish or just check out the debugger when it breaks. (edit Accelerometer.cs and replace the entire line of “Throw InvalidOperationException” with a simple “Return” statement)
Navigation and Content Loading
Now unlike native XNA where the Content Loading process is handled by the Main Game framework in SilverXNA we have to do it manually, now this can be both a blessing and a pain just because of the many ways and events available to us when a page is navigated to and from. the Navigated to event is fired just after the page has been constructed in memory but before it is presented to the screen so it is a handy place to load content that needs to be drawn to the screen.
With all XNA games knowing what content to load and when is very important so only load what you immediately need and delay loading anything else till later (like in the onLoaded event which is when the page has finished presenting to the screen or better yet offload it to another thread when the page has finished loading else you may lock up screen drawing if it takes a long time)
In this case we don’t have that much to load so we can do it all at once, just add the following in the “onNaviagatedTo” function just before the “Timer.Start()” call:
LoadContent();
and then add the respective function just after the “OnNavigatedTo” function:
/// <summary><br>
/// LoadContent will be called once per game and is the place to load<br>
/// all of your content.<br>
/// </summary><br>
void LoadContent()<br>
{<br>
// Load fonts<br>
hudFont = contentManager.Load<spritefont>("Fonts/Hud");
<p>// Load overlay textures<br>
winOverlay = contentManager.Load<texture2d>("Overlays/you_win");<br>
loseOverlay = contentManager.Load<texture2d>("Overlays/you_lose");<br>
diedOverlay = contentManager.Load<texture2d>("Overlays/you_died");</p>
<p>//Known issue that you get exceptions if you use Media PLayer while connected to your PC<br>
//See http://social.msdn.microsoft.com/Forums/en/windowsphone7series/thread/c8a243d2-d360-46b1-96bd-62b1ef268c66<br>
//Which means its impossible to test this from VS.<br>
//So we have to catch the exception and throw it away<br>
try<br>
{<br>
MediaPlayer.IsRepeating = true;<br>
MediaPlayer.Play(contentManager.Load<song>("Sounds/Music"));<br>
}<br>
catch { }</p>
<p>LoadNextLevel();<br>
}
In this I’ve fixed up for you the Content Manager references to use the SilverXNA one over the use of the Game class “Content” references, in SilverXNA the Content Manager is exposed as a service unlike XNA where is it just provided out of the box.
Don’t forget to fix the references to the MediaPlayer, ignore the missing “LoadNextLevel” function for now as we will get to that later.
Pages should really unload their content if the assets are not going to be used any more so keep this in mind and handle the unloading of any unneeded content in the “onNavigatedFrom” function. As we are constantly using the same assets in this game we need not bother.
Game Update function
Next up is the update function (if for no other reason that Update is called before Draw), thankfully in the project template has already wired up the Timer and associated events for Update and Draw so just add the following to the onUpdate function in the GamePage.XAML.cs file:
// Handle polling for our input and handling high-level input<br>
HandleInput();
<p>// update our level, passing down the GameTime along with all of our input states<br>
level.Update(e.ElapsedTime, keyboardState, gamePadState, touchState,<br>
accelerometerState, this.Orientation.ToXNAOrientation());
Not much to talk about here just basic game logic stuff, again ignore the red squiggles as we will come back to it later. Note the reference to “e.ElapsedTime”, this comes from the “GameTimerEventArgs” we talked about recently which is the replacement for the XNA GameTime class, here is where you would need to also grab the TotalGameTime if needed.
Game Draw Function
As with the Update function, the Draw function is just admin stuff really and I’ve tidied up references to the Spritebatch as they were slightly different, so just add the following under the comment in the “onDraw” function:
spriteBatch.Begin();
<p>level.Draw(e.ElapsedTime, spriteBatch);</p>
<p>DrawHud();</p>
<p>spriteBatch.End();
You know the drill broken bits to be fixed shortly
The last bits, the supporting actors
And finally (well almost) here’s the rest of the supporting code and functions from the original Platformer Game sample, I’ve tweaked and prodded where needed just to line up use of GameTime and such but I’ve not had to change much, granted a lot of this could be handled from within the game library, for others however this actually helps us because it’s stuff I’ going to rip out and replace with Silverlight:
SO just paste the following after the “onDraw” function:
private void DrawHud()<br>
{<br>
Microsoft.Xna.Framework.Rectangle titleSafeArea = <br>
SharedGraphicsDeviceManager.Current.GraphicsDevice.Viewport.TitleSafeArea;<br>
Vector2 hudLocation = new Vector2(titleSafeArea.X, titleSafeArea.Y);<br>
Vector2 center = new Vector2(titleSafeArea.X + titleSafeArea.Width / 2.0f,<br>
titleSafeArea.Y + titleSafeArea.Height / 2.0f);
<p>// Draw time remaining. Uses modulo division to cause blinking when the<br>
// player is running out of time.<br>
string timeString = "TIME: " + level.TimeRemaining.Minutes.ToString("00") + <br>
":" + level.TimeRemaining.Seconds.ToString("00");<br>
Color timeColor;<br>
if (level.TimeRemaining &gt; WarningTime ||<br>
level.ReachedExit ||<br>
(int)level.TimeRemaining.TotalSeconds % 2 == 0)<br>
{<br>
timeColor = Color.Yellow;<br>
}<br>
else<br>
{<br>
timeColor = Color.Red;<br>
}<br>
DrawShadowedString(hudFont, timeString, hudLocation, timeColor);</p>
<p>// Draw score<br>
float timeHeight = hudFont.MeasureString(timeString).Y;<br>
DrawShadowedString(hudFont, "SCORE: " + level.Score.ToString(), hudLocation + new Vector2(0.0f, timeHeight * 1.2f),<br>
Color.Yellow);</p>
<p>// Determine the status overlay message to show.<br>
Texture2D status = null;<br>
if (level.TimeRemaining == TimeSpan.Zero)<br>
{<br>
if (level.ReachedExit)<br>
{<br>
status = winOverlay;<br>
}<br>
else<br>
{<br>
status = loseOverlay;<br>
}<br>
}<br>
else if (!level.Player.IsAlive)<br>
{<br>
status = diedOverlay;<br>
}</p>
<p>if (status != null)<br>
{<br>
// Draw status message.<br>
Vector2 statusSize = new Vector2(status.Width, status.Height);<br>
spriteBatch.Draw(status, center - statusSize / 2, Color.White);<br>
}<br>
}</p>
<p>private void DrawShadowedString(SpriteFont font, string value, Vector2 position, Color color)<br>
{<br>
spriteBatch.DrawString(font, value, position + new Vector2(1.0f, 1.0f), Color.Black);<br>
spriteBatch.DrawString(font, value, position, color);<br>
}</p>
<p>private void HandleInput()<br>
{<br>
touchState = TouchPanel.GetState();<br>
accelerometerState = Accelerometer.GetState();</p>
<p>bool continuePressed = touchState.AnyTouch();</p>
<p>// Perform the appropriate action to advance the game and<br>
// to get the player back to playing.<br>
if (!wasContinuePressed &amp;&amp; continuePressed)<br>
{<br>
if (!level.Player.IsAlive)<br>
{<br>
level.StartNewLife();<br>
}<br>
else if (level.TimeRemaining == TimeSpan.Zero)<br>
{<br>
if (level.ReachedExit)<br>
LoadNextLevel();<br>
else<br>
ReloadCurrentLevel();<br>
}<br>
}</p>
<p>wasContinuePressed = continuePressed;<br>
}</p>
<p>private void LoadNextLevel()<br>
{<br>
// move to the next level<br>
levelIndex = (levelIndex + 1) % numberOfLevels;</p>
<p>// Unloads the content for the current level before loading the next one.<br>
if (level != null)<br>
level.Dispose();</p>
<p>// Load the level.<br>
string levelPath = string.Format("Content/Levels/{0}.txt", levelIndex);<br>
using (Stream fileStream = TitleContainer.OpenStream(levelPath))<br>
level = new Level((Application.Current as App).Services, fileStream, levelIndex);<br>
}</p>
<p>private void ReloadCurrentLevel()<br>
{<br>
--levelIndex;<br>
LoadNextLevel();<br>
}
There are a few comments I should make about this block however, most was just the simple fixes that I’ve stated many times before here, other took a little head scratching namely where the Graphics Device was involved.
Because SilverXNA uses a “SharedGraphicsDeviceManager” over the traditional XNA “GraphicsDevice” you need to be a bit more specific in what you are referring to so note that I have replaced the original line for accessing the “Tile Safe Area” from:
Rectangle titleSafeArea = GraphicsDevice.Viewport.TitleSafeArea;
To:
Microsoft.Xna.Framework.Rectangle titleSafeArea =<br>
SharedGraphicsDeviceManager.Current.GraphicsDevice.Viewport.TitleSafeArea;
So it’s just a point of reference to note the difference but something definitely to keep in mind, there are also TWO rectangle classes, so in instances like this you have to be very specific, cannot rule out one or the other because both might be needed.
I did also remove all of the code for keyboard and gamepad updates as they were no longer needed in the “HandleInput” function.
The other thing of note is the section at the end for loading the next level, I had to change the reference to he Game “Services” repository (part of the Game class again) to the reference held in the App.XAML.CS class, like many things with the reduced / modified framework you just need to know where to look for stuff and if it is held in the Application class for the Silverlight application then you just need to refer to it in this manner thus:
(Application.Current as App).Services
Lastly don’t forget your references, those Streams are not going to find themselves.
Last Man Standing
You know when I said there is only one exception, well I wasn’t being completely honest there, you will still have one broken link if you compile the above for an extension method I had to create to handle the Orientation enumeration differences between XNA and Silverlight (and I did warn you in Part 1 )
Just add the following to the very end of the “GamePage.XAML.cs” file after the GamePage class, it’s a simple enough Extension method that will return an XNA orientation from a defined Silverlight Orientation, this can then be fed to whatever XNA code in your project is handling drawing / input for orientation, as stated before this is to make it easer on multi-platform XNA projects so you don’t need a load of #IF statements to balance everything:
public static class OrientationExtensions<br>
{<br>
public static DisplayOrientation ToXNAOrientation(this PageOrientation input)<br>
{<br>
switch (input)<br>
{<br>
case PageOrientation.Landscape:<br>
case PageOrientation.LandscapeLeft:<br>
return DisplayOrientation.LandscapeLeft;<br>
case PageOrientation.LandscapeRight:<br>
return DisplayOrientation.LandscapeRight;<br>
default:<br>
return DisplayOrientation.Portrait;<br>
}<br>
}<br>
}
I could have put this into it’s own class as your supposed to but it was the end of a very long day and it made ore sense to keep all the changes and additional stuff to a minimum.
## Mind the water jump at the end of the course
Now you would be forgiven if you thought we were done and sure enough the game will compile and run with only one minor little bugg’et, as shown below:
Now one reason for this is simple, by default XNA will start games in Landscape, in Silverlight the default is Portrait, simples . So we just need to tell the GamePage that we would like it in Landscape Pretty please (note that every page needs to be set to Landscape or Portrait unlike XNA where it is set once on start-up and only changes if you tell it to or if the user rotates the device if supported)
So for the first time here, edit the GamePage.XAML and change the following line from:
SupportedOrientations="Portrait" Orientation="Portrait"
To
SupportedOrientations="Landscape" Orientation="Landscape"
Not so bad for your first bit of XAML editing, you can do this from code of course but with Silverlight it’s better to set the default in XAML and only override if needed.
A key thing to note though is that unlike XNA where we only have one screen and have to use game states to manage different setup’s of the game world whereas in Silverlight we can have many pages all of which can be either completely Silverlight, completely XNA or a mixture of both which gives us greater flexibility and avoid complex screen management scenarios.
The best example I have seen of this was the car setup for a racing game, each different view that you had to select tracks, setup the car, select add-on’s and customisations was a separate page each with it’s own logic or input strategy before launching the main racing part of the game, if done in XNA you had to manage many states and write complex routines to handle all the variations or start making compromises to avoid breaking points.
## End of the line
That’s a wrap people, print it and go and get more coffee.
As stated before the full source for the completed project can be found here on codeplex
So now we have our game running in SilverXNA, the series will now return to our regular broadcasting schedule and focus on the little things, the big advantages of using SilverXNA and the simplicity it brings for our games.
Light’s out please
|
2021-10-28 17:21:52
|
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|
https://math.stackexchange.com/questions/2107249/expectation-stopped-brownian-motion-with-drift
|
# Expectation stopped Brownian motion with drift
Let $\{X_t:t\geq 0\}$ be a Brownian motion with drift $\mu>0$ and define a stopping time $\tau$ by $$\tau=\inf\{t\geq 0:X_t=a\}.$$ Now I want to show that $$\mathbb{E}(e^{-\lambda\tau})=e^{(\mu-\sqrt{\mu^2+2\lambda})a}$$ for $\lambda>0$. Now as a hint I know that I need to use the martingale $M_t=e^{\alpha X_t-\alpha\mu t-\frac{1}{2}\alpha^2t}$. Obviously I need to use Doobs optional stopping theorem but I do not know how. Anyone has a suggestion?
1. Check that for fixed $\alpha>0$ the process $$M_t := \exp \left(\alpha X_t-\alpha \mu t- \frac{1}{2} \alpha^2 t \right)$$ is a martingale (with respect to the canonical filtration of the Brownian motion).
2. By the optional stopping theorem, $$\mathbb{E}(M_{\tau \wedge t}) = \mathbb{E}(M_0) = 1, \qquad t \geq 0.$$
3. Show that $|M_{t \wedge \tau}| \leq e^{\alpha a}$. Deduce from the dominated convergence theorem that $$\mathbb{E}(M_{\tau}) = 1.$$
4. Since $(X_t)_{t \geq 0}$ has continuous sample paths, we have $X_{\tau}=a$. Hence, $$M_{\tau} = e^{\alpha a} \exp \left( - \left[ \mu \alpha + \frac{1}{2} \alpha^2 \right] \tau \right).$$
5. It follows from step 3 and 4 that $$\mathbb{E} \exp\left( - \left[ \mu \alpha + \frac{1}{2} \alpha^2 \right] \tau \right) = e^{-\alpha a}.$$ Setting $\lambda := \mu \alpha + \frac{1}{2} \alpha^2$ proves the assertion.
• Thanks for the help, it helped me a lot! I have a follow-up question though: I want to prove that with this drift $a>0$, that $\tau<\infty$ with probability one by taking the limit $\lambda \to 0$, but I can't figure out how $a>0$ plays a big role in proving this using equation from step 5? – higuys Jan 28 '17 at 15:26
• @higuys I suppose you mean with drift $\mu>0$ (and not drift $a>0$)....? (Just as a side remark: If you find the answer useful, you can upvote it by clicking on the up arrow next to it.) – saz Jan 28 '17 at 15:34
• @higuys I used in step 3 that $\tau<\infty$ almost surely. Note that if $\tau(\omega)=\infty$ for some $\omega$, then $$M_{t \wedge \tau}(\omega) = M_{t}(\omega) \xrightarrow[]{t \to \infty} 0,$$ this follows from the fact that $X_t(\omega) < a$ for all $t$ (as $\tau(\omega)=\infty$) and $\mu>0$. Hence, by the dominated convergence theorem, $$\mathbb{E}(M_{\tau} 1_{\{\tau<\infty\}}+ 0 \cdot 1_{\{\tau=\infty\}})=1.$$ Therefore, we get $$\mathbb{E}(e^{-\lambda \tau} 1_{\{\tau<\infty\}}) = e^{(\mu-\sqrt{\mu^2+2\lambda})a}.$$ Now you can let $\lambda \to 0$ to conclude $\mathbb{P}(\tau<\infty)=1$. – saz Jan 28 '17 at 17:55
• @DaneelOlivaw 1.What do you mean by "couldn't we just state that [...]"? How do you justify that $\mathbb{E}(M_{t \wedge \tau}) = \mathbb{E}(M_{0 \wedge \tau})$? The optional sampling theorem shows that $(M_{t \wedge \tau})_t$ is a martingale and so the assertion follows; I fail to see how to prove this without the optional sampling theorem. 2. Typically you have to assume continuity of the sample paths. What exactly do you mean by "$X_{\tau}$ is not random variable"? $X_{\tau}=a$ is a random variable for sure (well, it's constant, but nevertheless it's a random variable). – saz Jun 30 '17 at 10:48
• @DaneelOlivaw Well, yes, there are several names for this result (optional sampling/optional stopping/...). In my answer I use the fact that $(M_t)_t$ martingale implies that $(M_{t \wedge \tau})_t$ is a martingale, and that's it. – saz Jun 30 '17 at 11:24
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2019-05-22 02:38:44
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https://mathematica.stackexchange.com/questions/40267/user-defined-usage-message-template-removing-repeated-text?noredirect=1
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# User-defined usage message template; removing repeated text
In Mathematica 9.0.1, I want to write a user-defined template for a function that I wrote: myfunc
Remove[myfunc]
myfunc::usage = "myfunc[x,y,z] is a function.
myfunc[x,y,z,w] is another function."
with the result:
As you can see, within the description myfunc is (in my opinion) unnecessarily repeated. In fact, templates for built-in functions do not behave that way. For example, look at the template for the Integrate function:
The function name Integrate is not repeated in the description. How do I tell Mathematica not to repeat the name of the function inside the description?
The built-in function templates displayed by the Input Assistant when Shift+Ctrl+K-ing the symbol name are NOT drawn from either
• the package file that holds the symbol. Check and edit e.g. CalendarDaysPlus in $InstallationDirectory\AddOns\Packages\Calendar\Calendar.m. It has one simple usage though the Input Assistant shows two. • the usage file: $InstallationDirectory\SystemFiles\Kernel\TextResources\English\Usage.m. It does contain the multiple usage messages for e.g. CalendarDaysPlus or Integrate, but editing them will only change ?f and Message[f::usage] prints, but not the Input Assistant templates.
The usage messages in Usage.m are probably programmatically created from the same source during initialization that is used to format the templates. This source is in v9 (before, it was FunctionInformation.m):
file = ToFileName[{$InstallationDirectory, "SystemFiles", "Kernel", "TextResources", CurrentValue["Language"]}, "FunctionInformation2.m"] The file is read when the FrontEnd initializes (at line 376 in GetFEKernelInit.tr): ToFileName[{$InstallationDirectory, "SystemFiles", "FrontEnd", "TextResources"},
"GetFEKernelInit.tr"]
While "FunctionInformation2.m" is compressed, it is easy to create a wrapper that makes it readable. Here I only print the information on System symbols (and omit package symbols that form the first part of data). Warning: the list is huge, 2871 symbols are printed.
format[list_List] := OpenerView[{list[[1]], Column@{
"ArgumentsPattern" -> list[[2]],
"Help" -> If[list[[3]] =!= None, Panel@Column[
MapThread[Row[{##}, " "] &, {DisplayForm /@ list[[3]],
DisplayForm /@ list[[4]]}], Alignment -> Left]],
"Options" -> If[Length@list > 4, list[[5]], ""],
"LocalVariables" -> If[Length@list > 5, list[[6]], ""]
}}];
data = Uncompress@Get@file;
format /@ Last@Last@data // Column
The structure is: first symbols without options, then symbols with options, then symbols with options and available "LocalVariables" specification. What you are looking for is the list[[4]] part, that lists the textual part of each line of help (while list[[3]] lists the formula part for each line).
Unfortunately, I don't see any easy ways to add custom syntax information/template to "FunctionInformation2.m" in the way you would like it to be. It's a pity WRI does not allow users to manipulate the Input Assistant in a more usable fashion.
The short answer is: there isn't currently a way to do so.
As @istvan has correctly guessed, both the usage messages and the templates come from a common source, which is in fact the source notebooks used to generate the documentation. For those familiar with the DocumentationTools included with Wolfram Workbench, these are very similar. These source notebooks have a fair amount of markup that the build systems--for usages, templates, and documentation center--can use to precisely separate out which pieces go where, and produce these nicely formatted templates.
Of course, we still have the fall-back system of automatically generating templates from usage messages. It doesn't allow for the same level of control. This isn't because we don't care, but because its a very hard problem. Without the extra tagging, it's not obvious which part of the usage message to put in the template and which to be in the description. In fact, the InputForm of a formatted is string sufficiently complicated that just reliably picking it apart and splitting it into different pieces without causes pink boxes to appear is not trivial. For that reason, it is unlikely for the automatic usage message -> template system to be upgraded to support the nice templates of V9 and later.
Note: since version 11, the FunctionInformation2.m file is in fact no longer used for templates of system functions. We are working on making the new system user extendible, but as with all features under development, we cannot promise when it might be released.
• Many thanks for your clarifying answer. To be honest, my opinion is that constructing a template out of a ::usage string was a design flaw. The nontriviality of it should have been the clue to avoid it in the first place. If it isn't too late already, please get rid of the system in the future versions, and provide a separate function CreateFrontEndUsageTemplate[...]` which can support the extra tagging a simple string cannot. – QuantumDot Jul 7 '17 at 14:17
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2021-07-29 14:48:28
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https://repository.uantwerpen.be/link/irua/93848
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Title Evaluation of manganese-bodies removal in historical stained glass windows via $SR-\mu-XANES/XRF$ and $SR-\mu-CT$ Author Cagno, Simone Nuyts, Gert Bugani, Simone De Vis, Kristel Schalm, Olivier Caen, Joost Helfen, Lukas Cotte, Marine Reischig, Peter Janssens, Koen Faculty/Department Faculty of Sciences. Chemistry Publication type article Publication 2011 London , 2011 Subject Chemistry Source (journal) Journal of analytical atomic spectrometry. - London Volume/pages 26(2011) :12 , p. 2442-2451 ISSN 0267-9477 ISI 000297030400009 Carrier E Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract The speed and effectiveness of a conservation treatment used for stained glass windows have been investigated. Dark-coloured Mn-rich stains can be found in the alteration layer of ancient glass artefacts and cause the surface to turn brown/black: this phenomenon is known as Mn-browning or Mn-staining. While in glass manganese is present in the +II or +III oxidation states, in the Mn-rich bodies, manganese is in a higher oxidation state (+IV). In restoration practice, mildly reducing solutions are employed to eliminate the dark colour and restore the clear appearance of the glass. In this paper the effectiveness and side effects of the use of hydroxylamine hydrochloride for this purpose are assessed. Archaeological fragments of stained glass windows, dated to the 14th century and originating from Sidney Sussex College, Cambridge (UK), were examined by means of synchrotron radiation (SR) based microscopic X-ray Absorption Near-Edge Spectroscopy (μ-XANES) and microscopic X-Ray Fluorescence (μ-XRF) and with high resolution computed absorption tomography (μ-CT) before, during and after the treatment. The monitoring of the glass fragments during the treatment allows us to better understand the manner in which the process unfolds and its kinetics. The results obtained reveal that the hydroxylamine hydrochloride treatment is effective, but also that it has a number of unwanted side effects. These findings are useful for optimizing the time and other modalities of the Mn-reducing treatment as well as minimizing its unwanted results. E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000297030400009&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000297030400009&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000297030400009&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle
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2017-02-26 14:41:21
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https://stats.stackexchange.com/questions/478363/assumptions-of-ols-and-linear-mixed-models
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# Assumptions of OLS and linear mixed models
I've only taken a few statistics courses, and so I apologize if any of my questions are rudimentary, however, I'm wondering if someone could explain/direct me to resources regarding the correct process of testing model assumptions, model fitting, and also the consequences behind not meeting model assumptions such as normality, homoscedasticity, etc?
My impression is that it's important to meet model assumptions because otherwise, the mathematics of the models will not work. However, this link, caught my eye in that it seems to explain that you can still estimate parameters, and though it's not ideal for hypothesis testing (is this assessment of p-value?), you can get around this with bootstrapping.
Is this true in both OLS and in linear mixed effects modeling where you have to account for random effects? Additionally, someone suggested to me that it's not practical in real life settings to meet all the assumptions because data is rarely perfect (ie: not always normally distributed, has a lot of variance, etc). My understanding is if the data isn't meeting assumptions, that suggests it's the wrong model and using the model just leads to inaccurate results. Is this true, or in real life settings of analysis, are model assumptions rarely ever met? This doesn't seem true to me, though in my own experience, I have been having difficulty meeting many assumptions or rectifying them with transformations, which makes it hard for me even proceed in my analyses.
Any thoughts/advice on this to clear things up would be very appreciated.
It is worth remembering that the reason assumptions are made is usually so that statistical tests may be carried out and that the estimators have certain desirable properties (like unbiasedness and consistency). Many "assumptions" are better considered as "conditions" that are needed in order to make certain inferences.
Common assumptions are:
• that the model matrix is of full rank (ie. no perfect collinearity). This is necessary for the estimates to even exist.
• that the relationship between the linear predictor and the outcome is linear. This is necessary for the estimates to be unbiased.
• that the samples are independent. This is necessary for the estimates to be consistent and for them to have nice distributional properties. Mixed effects models are often used when this assumption is invalidated due to repeated measures/clustering/nesting.
• that the residuals are homoskedastic. This is necessary to make valid inferences.
Note that mild departures from these assumptions are to be expected.
This is not intended to be an exhaustive answer. People have written textbooks on these matters, and they are discussed at length in answers to other questions on this site.
• thank you @Robert Long i guess i'm wondering how to get a good sense of "mild" departures? Does this come from experience? – molecularrunner Jul 23 at 19:15
• Indeed. Experience with these matters allows for judgements to be made. Generally it is folly to expect assumptions for hold perfectly. – Robert Long Jul 23 at 20:02
• What I would suggest is to use some publicly available data, fit some models and try to assess how well the assumptions are met. If you use R then there are lots of datasets built in and many others included in packages. Start with linear models (lm()) and look at the output of plot(model) which is really all you need to get going – Robert Long Jul 23 at 20:12
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2020-09-25 06:25:10
|
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https://nips.cc/Conferences/2010/ScheduleMultitrack?event=1993
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Timezone: »
Workshop
Optimization for Machine Learning
Suvrit Sra · Sebastian Nowozin · Stephen Wright
Fri Dec 10 07:30 AM -- 06:30 PM (PST) @ Westin: Emerald A
Our workshop focuses on optimization theory and practice that is relevant to machine learning. This proposal builds on precedent established by two of our previously well-received NIPS workshops:
(@NIPS08) http://opt2008.kyb.tuebingen.mpg.de/
(@NIPS
09) http://opt.kyb.tuebingen.mpg.de/
Both these workshops had packed (often overpacked) attendance almost throughout the day. This enthusiastic reception reflects the strong interest, relevance, and importance enjoyed by optimization in the greater ML community.
One could ask why does optimization attract such continued interest? The answer is simple but telling: optimization lies at the heart of almost every ML algorithm. For some algorithms textbook methods suffice, but the majority require tailoring algorithmic tools from optimization, which in turn depends on a deeper understanding of the ML requirements. In fact, ML applications and researchers are driving some of the most cutting-edge developments in optimization today. The intimate relation of optimization with ML is the key motivation for our workshop, which aims to foster discussion, discovery, and dissemination of the state-of-the-art in optimization, especially in the context of ML.
The workshop should realize its aims by:
* Providing a platform for increasing the interaction between researchers from optimization, operations research, statistics, scientific computing, and machine learning;
* Identifying key problems and challenges that lie at the intersection of optimization and ML;
* Narrowing the gap between optimization and ML, to help reduce rediscovery, and thereby accelerating new advances.
ADDITIONAL BACKGROUND AND MOTIVATION
Previous talks at the OPT workshops have covered frameworks for convex programs (D. Bertsekas), the intersection of ML and optimization, especially in the area of SVM training (S. Wright), large-scale learning via stochastic gradient methods and its tradeoffs (L. Bottou, N. Srebro), exploitation of structured sparsity in optimization (Vandenberghe), and randomized methods for extremely large-scale convex optimization (A. Nemirovski). Several important realizations were brought to the fore by these talks, and many of the dominant ideas will appear in our book (to be published by MIT Press) on Optimization for Machine learning.
Given the above background it is easy to acknowledge that optimization is indispensable to machine learning. But what more can we say beyond this obvious realization?
The ML community's interest in optimization continues to grow. Invited tutorials on optimization will be presented this year at ICML (N. Srebro) and NIPS (S. Wright). The traditional point of contact'' between ML and optimization - SVM - continues to be a driver of research on a number of fronts. Much interest has focused recently on stochastic gradient methods, which can be used in an online setting and in settings where data sets are extremely large and high accuracy is not required. Regularized logistic regression is another area that has produced a recent flurry of activity at the intersection of the two communities. Many aspects of stochastic gradient remain to be explored, for example, different algorithmic variants, customizing to the data set structure, convergence analysis, sampling techniques, software, choice of regularization and tradeoff parameters, parallelism. There also needs to be a better understanding of the limitations of these methods, and what can be done to accelerate them or to detect when to switch to alternative strategies. In the logistic regression setting, use of approximate second-order information has been shown to improve convergence, but many algorithmic issues remain. Detection of combined effects predictors (which lead to a huge increase in the number of variables), use of group regularizers, and dealing with the need to handle very large data sets in real time all present challenges. <br> <br>To avoid becoming lopsided, in our workshop we will also admit thenot particularly large scale' setting, where one has time to wield substantial computational resources. In this setting, high-accuracy solutions and deep understanding of the lessons contained in the data are needed. Examples valuable to MLers may be exploration of genetic and environmental data to identify risk factors for disease; or problems dealing with setups where the amount of observed data is not huge, but the mathematical models are complex.
#### Author Information
##### Suvrit Sra (MIT)
Suvrit Sra is a faculty member within the EECS department at MIT, where he is also a core faculty member of IDSS, LIDS, MIT-ML Group, as well as the statistics and data science center. His research spans topics in optimization, matrix theory, differential geometry, and probability theory, which he connects with machine learning --- a key focus of his research is on the theme "Optimization for Machine Learning” (http://opt-ml.org)
##### Stephen Wright (UW-Madison)
Steve Wright is a Professor of Computer Sciences at the University of Wisconsin-Madison. His research interests lie in computational optimization and its applications to science and engineering. Prior to joining UW-Madison in 2001, Wright was a Senior Computer Scientist (1997-2001) and Computer Scientist (1990-1997) at Argonne National Laboratory, and Professor of Computer Science at the University of Chicago (2000-2001). He is the past Chair of the Mathematical Optimization Society (formerly the Mathematical Programming Society), the leading professional society in optimization, and a member of the Board of the Society for Industrial and Applied Mathematics (SIAM). Wright is the author or co-author of four widely used books in numerical optimization, including "Primal Dual Interior-Point Methods" (SIAM, 1997) and "Numerical Optimization" (with J. Nocedal, Second Edition, Springer, 2006). He has also authored over 85 refereed journal papers on optimization theory, algorithms, software, and applications. He is coauthor of widely used interior-point software for linear and quadratic optimization. His recent research includes algorithms, applications, and theory for sparse optimization (including applications in compressed sensing and machine learning).
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2021-09-21 07:35:20
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https://answers.ros.org/answers/390234/revisions/
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Thank you for sharing, looks really cool. I haven’t tried this hat but reviewing the buildhat source code for motors management for example https://github.com/RaspberryPiFoundation/python-build-hat/blob/main/buildhat/motors.py
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2022-05-20 21:47:14
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https://math.stackexchange.com/questions/2493771/lhospitals-rule-for-lim-n-rightarrow-infty-an-bn-frac1n
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# L'Hospital's rule for $\lim_{n\rightarrow \infty} (a^n + b^n)^\frac{1}{n}$.
I have recently come across a problem concerning the limit of a sequence given as, $$\lim_{n\rightarrow \infty} (a^n + b^n)^\frac{1}{n}$$
Where $0<a<b$. Solving the limit is not a concern. But applying L'Hospital's rule to this limit is one to me.
On rearranging we get, $b \lim_{n\rightarrow \infty} ((\frac{a}{b}) ^n + 1)^\frac{1}{n}$. The solver had then applied the L'Hospital's rule to evaluate the limit. Here, as $n\rightarrow \infty$, the limit approaches the form $1^0$. I have tried to think about this limit approaching an indeterminate form. It does not. From my knowledge of Cauchy's Mean Value theorem, I say that we cannot apply the rule to determinate forms. But it seems to me that we can because it worked for the solver. I think I am not clear on this. Can somebody help?
• You're right, L'Hôpital's rule doesn't apply here. And the limit is $b$. So what answer does the solver give? – Matthew Leingang Oct 28 '17 at 15:18
• Note that if $n$ is an integer variable then the L'Hospital's Rule does not apply. Rather one can use Cesaro-Stolz under similar circumstances. For the current limit, neither Cesaro-Stolz nor L'Hospital's Rule is applicable. The answer is arrived at easily as $b$ – Paramanand Singh Oct 28 '17 at 15:25
• To be more specific the situation is like using L'Hospital's Rule for limit $\lim_{x\to \infty} \dfrac{1}{x}$. – Paramanand Singh Oct 28 '17 at 15:27
• @ParamanandSingh If L'Hôpital's rule gives an answer when $n$ is considered a continuous variable, the limit when $n$ is an integer variable is the same. (It doesn't work in the other direction, for sure.) – Matthew Leingang Oct 28 '17 at 15:30
• @MatthewLeingang: agree but when the problem is of sequences I prefer to use the theorems on sequences. Or if one wants to apply L'Hospital's Rule or any other method applicable to continuous variable then it is better to be explicit and mention this. – Paramanand Singh Oct 28 '17 at 15:34
Why not $$y= (a^n + b^n)^\frac{1}{n}\implies \log(y)=\frac{\log(a^n + b^n)}{n}$$ and then $$\frac{a^n \log (a)+b^n \log (b)}{a^n+b^n}$$ Now, divide everything by $b^n$ since $b>a$ to get the limit of $\log(y)$ and then the limit of $y$.
• Ah! I now realize where I went wrong. I totally overlooked the fact that taking the natural log on both sides has nothing to do with L'Hospital's rule. It was a silly mistake. Thank you. – R004 Oct 28 '17 at 16:30
Note that $\log(1+x)\le x$ for $x>-1$. Therefore we have for $a<b$
$$1\le \left(1+\left(\frac ab\right)^n\right)^{1/n}\le e^{\frac1n \left(\frac ab\right)^n}\tag 1$$
whence applying the squeeze theorem to $(1)$ yields the coveted limit
$$\lim_{n\to \infty}\left(1+\left(\frac ab\right)^n\right)^{1/n}=1$$
And we are done!
One can take a different view by an expansion which is $$(1 + x)^{1/n} = 1 + \frac{1}{n} \, \frac{x}{1!} + \frac{1}{n} \, \left( \frac{1}{n} - 1\right) \, \frac{x^{2}}{2!} + \mathcal{O}\left(\frac{1}{n^{3}}\right)$$ for which, $0 < a <b$, $$\left(1 + \left(\frac{a}{b}\right)^{n} \right)^{1/n} = 1 + \frac{1}{n} \, \left(\frac{a}{b}\right)^{n} + \frac{1}{2 \, n} \, \left( \frac{1}{n} - 1\right) \, \left(\frac{a}{b}\right)^{2n} + \mathcal{O}\left(\frac{1}{n^{3}}\right)$$ and \begin{align} \lim_{n \to \infty} (a^n + b^n)^{1/n} &= b \, \lim_{n \to \infty} \left(1 + \left(\frac{a}{b}\right)^{n} \right)^{1/n} = 1. \end{align}
If $a=b$ then $$2^{1/n} = 1 + \frac{1}{n} + \frac{1}{2 \, n} \, \left( \frac{1}{n} - 1\right) + \mathcal{O}\left(\frac{1}{n^{3}}\right)$$ which leads to $$\lim_{n \to \infty} 2^{1/n} = 1.$$
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2021-01-23 13:51:03
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https://doc.visionappster.com/latest/cameras/configurator/parameters/
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# Camera driver parameters
Each camera has a set of configuration parameters that affect things like image format, acquisition and transmission. In Camera Configurator, click a camera in the list on the left side of the view and its parameter list appears in the Parameters view.
The parameter name is shown in the first column and the current value of the parameter in the second column. Parameters that cannot be edited are shown in a cursive font. Changing a parameter value may affect also other parameter values, their editability or even their meanings. In that case, the affected parameters are shown after the parameter that is affecting them, with their names slightly indented. To change the value of a parameter, click the parameter value. The changed value is set in the camera immediately and you can see the effect by taking an image. Note that some of the parameters may be editable only when the camera is not capturing images.
## Driver parameter sets
To create a configuration that will be set in a camera when an app using it is started, a parameter set needs to be saved. After configuring the camera parameters to the desired values, you can save the configuration to a parameter set by clicking the Save as button in the Parameters view. It saves the current values of all the camera parameters and loading the parameter set will set them all exactly to the same values. Note that if you have set up the localhost or any other remote connection in the Builder (see Remote connections menu) the save dialog has a checkbox for each remote connection. At least one must be checked and the parameter set is saved with the given name in all the checked ones.
If an application has an Image Source that has the autoConfig parameter set to true, it will use the Driver parameter set that is associated with the camera, as explained in Device Configuration. After having saved a new parameter set, there will be a warning shown that the currently viewed parameter set (MyParameterSet2 in the picture) is not associated with the camera.
To change the associated parameter set, edit the Associated parameters setting shown first in the camera parameter list.
One parameter set can be associated with many cameras that share the same settings. Note that the associated parameter set is not automatically loaded in Camera Configurator when a camera is selected or the view is switched. Instead, the parameter set needs to be explicitly loaded.
An existing parameter set can be loaded with the Load button. The parameter list will then show the values that were set in the camera based on the loaded parameter set.
The Delete button shows a list of all the saved parameter sets and lets you delete the ones that are not needed any more. The deleted parameter sets are also dissociated from all the cameras.
The same parameter set control buttons are found also in Calibration and Positioning views, which both have their own parameter sets.
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2021-01-17 17:50:30
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https://blog.akendo.eu/post/2021-03-15-habit-score/
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# Habit Score
For long, I’ve used the Habit Tracker[0] to keep track of my daily routines. A while ago, I’ve decided to stop using my Smartphone to reduce distracting. Instead, the task warrior has become my tool to keep track of daily routines. This comes with some downsides. I do not have that neat graphs anymore.
To compensated, I’ve taken some time to figure out how the App is rendering the graph. After some search, I think I manage to find the formula that is used to compute the score value of a given habit. When we know this value, we can be used to generate an own graph.
$$s = 0.5^{\frac{\sqrt(f)}{13}}$$
Wait, what? $$0.5^{\frac{\sqrt{f}}{13}}$$ ? That’s an odd function. I have to admit, it took some time until I understood what the idea is. This is a graceful descending graph. The function will hit zero eventually. Important is the follow quote[2] here:
* Given the frequency of the habit, the previous score, and the value of
* the current checkmark, computes the current score for the habit.
*
* The frequency of the habit is the number of repetitions divided by the
* length of the interval. For example, a habit that should be repeated 3
* times in 8 days has frequency 3.0 / 8.0 = 0.375.
*/
A little bit below we can see the computation that is made:
val multiplier = Math.pow(0.5, sqrt(frequency) / 13.0)
var score = previousScore * multiplier
score += checkmarkValue * (1 - multiplier)
return score
I wasn’t able to find any specify documentation within the project why this formula is the way it is. Habits require a certain amount of reiteration before it becomes a true habit. It might be that one of the paper that state this has made up this formula and the developer has adapted it. It could be also possible, that it was completely invented by him? I’m not certain.
How I understood the code is as follows: You compute for a fixed sized interval the number of repetitions. How often should you do this giving habit per week? In the graph on top of the page, we see the different interval lengths that available. There are fixed for weeks, months or years, when I remembered correctly. In the example the interval is 8 days what feels arbitrary, but it is just an example. Anyway, we then have the frequency.
At this point I’m a bit uncertain in how the app counts the actual number of repetitions. It could be that the frequency is a fixed value and the app computed via the value for a checkmark. Or it, changes frequency by calculate the actual repetitions one has done per interval into. It makes more sense to me that it is mainly done via the checkmark value because it is boolean.
One thing that left me to wonder is the 13[3]? Why 13? I guess to represent the period something take until it becomes really a habit. When you remove it, it takes five days. I wrote the code in python to imitate it, but left the 13 as a factor away.
def score(frequency,previousScore, Value):
multipler = math.pow(0.5, math.sqrt(frequency))
score = previousScore * multipler
score += Value * (1 - multipler)
return score
Now we loop through and see:
It is day 0
Score is 0.5
It is day 1
Score is 0.75
It is day 2
Score is 0.875
It is day 3
Score is 0.9375
It is day 4
Score is 0.96875
It is day 5
Score is 0.984375
The formula converging against 1, but will never reach 1. This make sense because it represents percentage. Besides, when it has reached 0.99 it does not matter anyway, you would count it as 100%. We may keep playing with the code until we see the period in which a habit becomes habitual. This is fun because usually, we do not tinker with mathematical formula this way.
It also raises some concern here. Think about it, some people that use this app taking this as an indicated without understanding why it is this way. For them, it is just like: Hey here is a graph, and it looks great. Let this becomes a metric for this routine now. Yacs…
Browsing through the code base and I must say, Kotlin feels odd. However, this project is especially useful to learn from because it covers different platforms. I think that the code is organized as follows: A core part written in java to allow cross-platform support and then platform specify code. For instance, iOS or Android. It also has some web and server related resources. This make sense to keep the code arranged this way. I did just not expect this.
Any way, that it for now.
So far, akendo
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2021-04-14 20:48:23
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https://www.jobilize.com/physics2/course/14-4-rl-circuits-inductance-by-openstax?qcr=www.quizover.com&page=2
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# 14.4 Rl circuits (Page 3/4)
Page 3 / 4
Check Your Understanding Verify that RC and L/R have the dimensions of time.
Check Your Understanding (a) If the current in the circuit of in [link] (b) increases to $90\text{%}$ of its final value after 5.0 s, what is the inductive time constant? (b) If $R=20\phantom{\rule{0.2em}{0ex}}\text{Ω}$ , what is the value of the self-inductance? (c) If the $20\text{-}\text{Ω}$ resistor is replaced with a $100\text{-}\text{Ω}$ resister, what is the time taken for the current to reach $90\text{%}$ of its final value?
a. 2.2 s; b. 43 H; c. 1.0 s
Check Your Understanding For the circuit of in [link] (b), show that when steady state is reached, the difference in the total energies produced by the battery and dissipated in the resistor is equal to the energy stored in the magnetic field of the coil.
## Summary
• When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
$I\left(t\right)=\frac{\text{ε}}{R}\left(1-{e}^{\text{−}Rt\text{/}L}\right)=\frac{\text{ε}}{R}\left(1-{e}^{\text{−}t\text{/}{\tau }_{L}}\right)$ (turning on),
where the initial current is ${I}_{0}=\epsilon \text{/}R.$
• The characteristic time constant $\tau$ is ${\tau }_{L}=L\text{/}R,$ where L is the inductance and R is the resistance.
• In the first time constant $\tau ,$ the current rises from zero to $0.632{I}_{0},$ and to 0.632 of the remainder in every subsequent time interval $\tau .$
• When the inductor is shorted through a resistor, current decreases as
$I\left(t\right)=\frac{\epsilon }{R}{e}^{\text{−}t\text{/}{\tau }_{L}}$ (turning off).
Current falls to $0.368{I}_{0}$ in the first time interval $\tau$ , and to 0.368 of the remainder toward zero in each subsequent time $\tau .$
## Conceptual questions
Use Lenz’s law to explain why the initial current in the RL circuit of [link] (b) is zero.
As current flows through the inductor, there is a back current by Lenz’s law that is created to keep the net current at zero amps, the initial current.
When the current in the RL circuit of [link] (b) reaches its final value $\text{ε}\text{/}R,$ what is the voltage across the inductor? Across the resistor?
Does the time required for the current in an RL circuit to reach any fraction of its steady-state value depend on the emf of the battery?
no
An inductor is connected across the terminals of a battery. Does the current that eventually flows through the inductor depend on the internal resistance of the battery? Does the time required for the current to reach its final value depend on this resistance?
At what time is the voltage across the inductor of the RL circuit of [link] (b) a maximum?
At $t=0$ , or when the switch is first thrown.
In the simple RL circuit of [link] (b), can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current?
If the emf of the battery of [link] (b) is reduced by a factor of 2, by how much does the steady-state energy stored in the magnetic field of the inductor change?
1/4
A steady current flows through a circuit with a large inductive time constant. When a switch in the circuit is opened, a large spark occurs across the terminals of the switch. Explain.
Describe how the currents through ${R}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{R}_{2}$ shown below vary with time after switch S is closed.
Initially, ${I}_{R1}=\frac{\epsilon }{{R}_{1}}$ and ${I}_{R2}=0$ , and after a long time has passed, ${I}_{R1}=\frac{\epsilon }{{R}_{1}}$ and ${I}_{R2}=\frac{\epsilon }{{R}_{2}}$ .
Discuss possible practical applications of RL circuits.
## Problems
In [link] , $\epsilon =12\phantom{\rule{0.2em}{0ex}}\text{V}$ , $L=20\phantom{\rule{0.2em}{0ex}}\text{mH}$ , and $R=5.0\phantom{\rule{0.2em}{0ex}}\text{Ω}$ . Determine (a) the time constant of the circuit, (b) the initial current through the resistor, (c) the final current through the resistor, (d) the current through the resistor when $t=2{\tau }_{L},$ and (e) the voltages across the inductor and the resistor when $t=2{\tau }_{L}.$
For the circuit shown below, $\epsilon =20\phantom{\rule{0.2em}{0ex}}\text{V}$ , $L=4.0\phantom{\rule{0.2em}{0ex}}\text{mH,}$ and $R=5.0\phantom{\rule{0.2em}{0ex}}\text{Ω}$ . After steady state is reached with ${\text{S}}_{1}$ closed and ${\text{S}}_{2}$ open, ${\text{S}}_{2}$ is closed and immediately thereafter $\left(\text{at}\phantom{\rule{0.2em}{0ex}}t=0\right)$ ${\text{S}}_{1}$ is opened. Determine (a) the current through L at $t=0$ , (b) the current through L at $t=4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{s}$ , and (c) the voltages across L and R at $t=4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{s}$ .
a. 4.0 A; b. 2.4 A; c. on R : $V=12\phantom{\rule{0.2em}{0ex}}\text{V}$ ; on L : $V=7.9\phantom{\rule{0.2em}{0ex}}\text{V}$
The current in the RL circuit shown here increases to $40\text{%}$ of its steady-state value in 2.0 s. What is the time constant of the circuit?
How long after switch ${\text{S}}_{1}$ is thrown does it take the current in the circuit shown to reach half its maximum value? Express your answer in terms of the time constant of the circuit.
$0.69\tau$
Examine the circuit shown below in part (a). Determine dI/dt at the instant after the switch is thrown in the circuit of (a), thereby producing the circuit of (b). Show that if I were to continue to increase at this initial rate, it would reach its maximum $\epsilon \text{/}R$ in one time constant.
The current in the RL circuit shown below reaches half its maximum value in 1.75 ms after the switch ${\text{S}}_{1}$ is thrown. Determine (a) the time constant of the circuit and (b) the resistance of the circuit if $L=250\phantom{\rule{0.2em}{0ex}}\text{mH}$ .
a. 2.52 ms; b. $99.2\phantom{\rule{0.2em}{0ex}}\text{Ω}$
Consider the circuit shown below. Find ${I}_{1},{I}_{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{3}$ when (a) the switch S is first closed, (b) after the currents have reached steady-state values, and (c) at the instant the switch is reopened (after being closed for a long time).
For the circuit shown below, $\epsilon =50\phantom{\rule{0.2em}{0ex}}\text{V}$ , ${R}_{1}=10\phantom{\rule{0.2em}{0ex}}\text{Ω,}$ and $L=2.0\phantom{\rule{0.2em}{0ex}}\text{mH}$ . Find the values of ${I}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{2}$ (a) immediately after switch S is closed, (b) a long time after S is closed, (c) immediately after S is reopened, and (d) a long time after S is reopened.
a. ${I}_{1}={I}_{2}=1.7\phantom{\rule{0.2em}{0ex}}A$ ; b. ${I}_{1}=2.73\phantom{\rule{0.2em}{0ex}}A,{I}_{2}=1.36\phantom{\rule{0.2em}{0ex}}A$ ; c. ${I}_{1}=0,{I}_{2}=0.54\phantom{\rule{0.2em}{0ex}}A$ ; d. ${I}_{1}={I}_{2}=0$
For the circuit shown below, find the current through the inductor $2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.2em}{0ex}}\text{s}$ after the switch is reopened.
Show that for the circuit shown below, the initial energy stored in the inductor, $L{I}^{2}\left(0\right)\text{/}2$ , is equal to the total energy eventually dissipated in the resistor, ${\int }_{0}^{\infty }{I}^{2}\left(t\right)Rdt$ .
proof
what is unit
is electric field directly proportional to the squared of a distance
The bullet 2.00cm long is fired at 420/s and passes straight through a 10.0 cm thick board existing at 280 m/s.What is the average acceleration of the bullet through the board?
FAUSTINA
an unstretched spring is 12cm long .A load of 5N stretched it to 15cm .how long will it be under a load of 15N?
Morapeli
hi
Africa
hi
Benjamin
Benjamin how are u are u a freshman in the university
Africa
like 100 level
Africa
yes sir
Benjamin
l need a physics tutor
Benjamin
I think the best tutor is God and organic tutor in YouTube that guy is good
Africa
me too 100level
Africa
wow nice
Benjamin
from Nigeria and u
Africa
I am from Nigeria and u wow nice that something I use to always say
Africa
am from Ghana
Benjamin
ok
Africa
studying what
Africa
Compare the electric flux through the surface of a cube of side length a that has a charge q at its center to the flux through a spherical surface of radius a with a charge q at its center.
please I want to know how to solve increase in length
Ujah
Why a charged capacitor has potential difference but not emf
what is the dimension symbol of temperature?
what is the dimension symbol of temperature?
Keren
what's the meaning of enthalpy in terms of latent heat, internal energy, phase change
how to convert Kelvin to centigrade
what is the s, p, d, f in this table
Sangeetha
s, p, d, f in this table
Sangeetha
what kind of table this
Vengata
what are waves
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities
Discuss how would orient a planar surface of area A in a uniform electric field of magnitude E0 to obtain (a) the maximum flux and (b) the minimum flux through the area.
I'm just doing the first 3 with this message. but thankyou for the time your obviously intending to support us with. viva la accumulation
Marcel
Find the net capacitance of the combination of series and parallel capacitors shown belo
what is ohm?
calculate ideal gas pressure of 0.300mol,v=2L T=40°c
what is principle of superposition
what are questions that are likely to come out during exam
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2021-05-08 22:58:38
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