url
stringlengths 14
2.42k
| text
stringlengths 100
1.02M
| date
stringlengths 19
19
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|---|
https://wordpandit.com/algebra-sequence-and-series-test-4/
|
Select Page
• This is an assessment test.
• To draw maximum benefit, study the concepts for the topic concerned.
• Kindly take the tests in this series with a pre-defined schedule.
## Algebra: Sequence and Series Test-4
Congratulations - you have completed Algebra: Sequence and Series Test-4. You scored %%SCORE%% out of %%TOTAL%%. You correct answer percentage: %%PERCENTAGE%% . Your performance has been rated as %%RATING%%
Question 1
If the 4th term of an arithmetic progression is 14 and the 12th term is 70, then the first, term is:
A –10 B –7 C +7 D +10
Question 1 Explanation:
Let the common difference be d.
14+8d=70,
=> d=7.
First term is 14-3d=14-2 x 7=-7.
The correct option is (b)
Question 2
By adding the same constant to each of 31, 7, –1 a geometric progression results. The common ratio is:
A $\displaystyle 13$ B $\displaystyle 2\frac{1}{3}$ C $\displaystyle -12$ D $\displaystyle None\,\,of\,\,these$
Question 2 Explanation:
$\begin{array}{l}(x+31)(x-1)={{(x+7)}^{2}}\\=>{{x}^{2}}+14x+49={{x}^{2}}+30x-31\\=>16x=80\\=>x=5\end{array}$
Question 3
The Value of
$\displaystyle \begin{array}{l}\frac{3}{{{1}^{2}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+\frac{9}{{{4}^{2}}{{.5}^{2}}}+\frac{11}{{{5}^{2}}{{.6}^{2}}}\\+\frac{13}{{{6}^{2}}{{.7}^{2}}}+\frac{15}{{{7}^{2}}{{.8}^{2}}}+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}\,\,is\end{array}$
A $\displaystyle \frac{1}{100}$ B $\displaystyle \frac{99}{100}$ C $\displaystyle \frac{101}{100}$ D $\displaystyle 1$
Question 3 Explanation:
$\begin{array}{l}\frac{3}{{{1}^{2}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+\frac{9}{{{4}^{2}}{{.5}^{2}}}\\+\frac{11}{{{5}^{2}}{{.6}^{2}}}+\frac{13}{{{6}^{2}}{{.7}^{2}}}+\frac{15}{{{7}^{2}}{{.8}^{2}}}+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}\,\\=\frac{4-1}{{{1}^{2}}{{.2}^{2}}}+\frac{9-4}{{{2}^{2}}{{.3}^{2}}}+\frac{16-9}{{{3}^{2}}{{.4}^{2}}}+\frac{25-16}{{{4}^{2}}{{.5}^{2}}}\\+\frac{36-25}{{{5}^{2}}{{.6}^{2}}}+\frac{49-36}{{{6}^{2}}{{.7}^{2}}}+\frac{64-49}{{{7}^{2}}{{.8}^{2}}}\\+\frac{81-64}{{{8}^{2}}{{.9}^{2}}}+\frac{100-81}{{{9}^{2}}{{.10}^{2}}}\\=\frac{1}{{{1}^{2}}}-\,\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{2}}}-\,\frac{1}{{{3}^{2}}}+\frac{1}{{{3}^{2}}}-\,\frac{1}{{{4}^{2}}}\\+\frac{1}{{{4}^{2}}}-\,\frac{1}{{{5}^{2}}}+\frac{1}{{{5}^{2}}}-\,\frac{1}{{{6}^{2}}}\\+\frac{1}{{{6}^{2}}}-\,\frac{1}{{{7}^{2}}}+\frac{1}{{{7}^{2}}}-\,\frac{1}{{{8}^{2}}}\\+\frac{1}{{{8}^{2}}}-\,\frac{1}{{{9}^{2}}}+\frac{1}{{{9}^{2}}}-\,\frac{1}{{{10}^{2}}}\\=1-\frac{1}{100}\\=\frac{99}{100}\end{array}$
Question 4
The value of
$\displaystyle 1-\frac{1}{20}+\frac{1}{{{20}^{2}}}-\frac{1}{{{20}^{3}}}+.....$ Correct to 5 places f decimal is:
A 1.05 B 0.9 C 2.9 D 0.5
Question 4 Explanation:
The series is a G.P. and the common ratio is (-1/20) .
Thus the infinite GP sum is (20/21) = 0.9
Thus the correct option is (b)
Question 5
For all integral values of n,
the largest number that exactly divides each number of the sequence
$\displaystyle \begin{array}{l}\left( n-1 \right)\,n\,\left( n+1 \right)\,,\,\,\\n\,\left( n+1 \right)\,\left( n+2 \right),\,\,\\\left( n+1 \right)\,\left( n+2 \right)\,\left( n+3 \right)\,.......is\end{array}$
A 12 B 6 C 3 D 2
Question 5 Explanation:
Any three consecutive numbers is divisible by 6. Thus the correct option is (b).
Once you are finished, click the button below. Any items you have not completed will be marked incorrect.
There are 5 questions to complete.
← List →
|
2019-04-23 00:11:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6181771755218506, "perplexity": 1969.2583909900209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578583000.29/warc/CC-MAIN-20190422235159-20190423021159-00381.warc.gz"}
|
https://zbmath.org/?q=an:1138.60026&format=complete
|
# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Generalized $n$-Paul paradox. (English) Zbl 1138.60026
The present paper has as starting point the results of S. Csörgö and G. Simons (2002--2006) on the classical St. Petersburg(1/2) game, played by two gamblers with an unbiased coin. The author considers the generalized St. Petersburg($p$) game with $p\in (0, 1)$ as the probability of the “heads” at each throw of a possibly biased coin. An interesting result is that, while the stochastic dominance is preserved for the case of two players and an arbitrary parameter $p\in(0, 1)$, for three or more players, the admissibly pooled winning strategies generally fail to stochastically dominate the individual strategies. The main result of the article consists in determining the best admissible pooling strategies for a rational value of $p$, illustrating also the algebraic depth of the problem for an irrational value of the parameter $p$.
##### MSC:
60E99 Distribution theory in probability theory 60G50 Sums of independent random variables; random walks
Full Text:
##### References:
[1] Csörgő, S.; Simons, G.: A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games. Statist. probab. Lett. 26, 65-73 (1996) · Zbl 0859.60030 [2] Csörgő, S.; Simons, G.: The two-paul paradox and the comparison of infinite expectations. Limit theorems in probability and statistics, 427-455 (2002) · Zbl 1027.60030 [3] Csörgő, S.; Simons, G.: Laws of large numbers for cooperative St. Petersburg gamblers. Period. math. Hungar. 50, 99-115 (2005) · Zbl 1113.60026 [4] Csörgő, S.; Simons, G.: Pooling strategies for St. Petersburg gamblers. Bernoulli 12, 971-1002 (2006) · Zbl 1130.91018
|
2016-05-03 10:42:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8387327790260315, "perplexity": 7146.292425777112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121423.81/warc/CC-MAIN-20160428161521-00051-ip-10-239-7-51.ec2.internal.warc.gz"}
|
https://brockk.github.io/escalation/index.html
|
by Kristian Brock. Documentation is hosted at https://brockk.github.io/escalation/
## Overview
escalation provides a grammar for dose-finding clinical trials.
It starts by providing functions to use dose-escalation methodologies like the continual reassessment method (CRM), the Bayesian optimal interval design (BOIN), and the perennial 3+3:
• get_dfcrm()
• get_trialr_crm()
• get_trialr_nbg()
• get_tpi()
• get_mtpi()
• get_boin()
• get_trialr_efftox()
• get_wages_and_tait()
• get_three_plus_three()
These functions fetch model fitting objects. Where possible, technical implementations are imported from existing R packages like dfcrm, trialr, and BOIN. Where no external implementations is available however, methods are implemented natively in escalation.
These dose-finding approaches can then be augmented with extra behaviours to specialise the dose selection process. For example, we can add behaviours to prevent skipping doses, or to stop when we reach a certain sample size. escalation supports the following behaviours:
• dont_skip_doses()
• stop_at_n()
• stop_when_n_at_dose()
• stop_when_too_toxic()
• stop_when_tox_ci_covered()
• demand_n_at_dose()
• try_rescue_dose()
• follow_path()
• select_dose_by_cibp()
Each of these functions overrides the way doses are selected or when a design decides to stop the trial. The behaviours can be flexibly combined using the %>% operator from the tidyverse.
These models are then fit to trial outcomes to produce dose recommendations. No matter how the dose selection behaviours were combined, the resulting model fits supports a standard interface. The two most important methods are recommended_dose() to get the current dose selection, and continue() to learn whether the model advocates continuing patient recruitment.
Having defined this nomenclature for combining dose selection behaviours and providing a standard interface for the resulting analyses, it is simple to run simulations or calculate dose-pathways for future cohorts of patients.
escalation provides an object-oriented approach to dose-escalation clinical trials in R. See Usage
# Usage
## Describing outcomes in dose-finding trials
escalation uses a succinct syntax for describing dose-finding outcomes, described in Brock (2019) for the phase I setting and in Brock et al. (2017) for the phase I/II setting.
In a phase I trial, we use the letters:
• T to show that toxicity occurred in a patient;
• N to show that toxicity did not occur in a patient.
In a joint phase I/II trial, like those supported by EffTox, where we have coincident efficacy and toxicity outcomes, those relevant letters are:
• T to show that toxicity without efficacy occurred in a patient;
• E to show that efficacy without toxicity occurred in a patient;
• N to show that neither occurred;
• B to show that both occurred.
These outcome letters are strewn behind integer dose-levels to show the outcomes of patients in cohorts. To show that a cohort a three patients was given dose 2, that the first two patients were without toxicity, but the third patient experienced toxicity, we would use the outcome string:
outcomes <- '2NNT'
If that cohort was followed by another cohort of three, all of which were without toxicity, the overall outcome string would be:
outcomes <- '2NNT 2NNN'
And so on. These strings are used in the escalate package to make it easy to fit models to observed outcomes. There are many examples below.
## Dose selectors
A core class in the escalation package is the selector. It encapsulates the notion that a general dose-escalation design is able to recommend doses, keep track of how many patients have been treated at what doses, what toxicity outcomes have been seen, and whether a trial should continue. This general interface is true of model-based methods like the CRM and rule-based methods like the 3+3. Irrespective the particular approach used, the interface is consistent.
In this tutorial, we will demonstrate each of the types of selector implemented in the package and how they can be combined to tailor behaviour.
To begin, let us load escalation
library(escalation)
## Loading required package: magrittr
At the core of the dose selection process is an algorithm or a model that selects doses in responses to outcomes. The classes capable of performing this core role are:
• get_dfcrm(), using the model-fitting code from dfcrm
• get_trialr_crm() using the model-fitting code from trialr
• get_trialr_nbg()
• get_boin() using the model-fitting code from BOIN
• get_tpi()
• get_mtpi()
• get_trialr_efftox()
• get_wages_and_tait()
• get_three_plus_three()
• and follow_path()
Where indicated these methods rely on external packages. Otherwise, methods are implemented natively in escalation. We look at each now.
### get_dfcrm
The continual reassessment method (O’Quigley, Pepe, and Fisher 1990) (CRM) is implemented in the dfcrm package by Cheung (2013). The very least information we need to provide is a dose-toxicity skeleton, and our target toxicity level. The skeleton represents our prior beliefs on the probabilities of toxicity at each of the doses under investigation. The model iteratively seeks a dose with toxicity probability close to the target.
For illustration, let us say we have
skeleton <- c(0.05, 0.1, 0.25, 0.4, 0.6)
target <- 0.25
We create a dose-selection model using:
model <- get_dfcrm(skeleton = skeleton, target = target)
and we can fit this to outcomes using code like:
fit <- model %>% fit('2NNN')
The fit object will tell you the dose recommended by the CRM model to be administered next. Depending on your preference for classic R or tidyverse R, you might run:
recommended_dose(fit)
## [1] 4
or
fit %>% recommended_dose()
## [1] 4
Either way, you get the same answer. The model advocates skipping straight to dose 4. Clinicians are unlikely to feel comfortable with this. We can respecify the model to expressly not skip doses in escalation. We will do that later on.
For now, let us return to our model fit. We can ask whether the trial should keep going:
fit %>% continue()
## [1] TRUE
Naturally it wants to continue because dfcrm does not implement any stopping rules. Again, we will add various stopping behaviours in sections below.
The CRM-fitting function in dfcrm accepts many arguments to customise the model form and these are passed onwards by get_dfcrm function via the ... parameter. For example, to use the one-parameter logit model in dfcrm (rather than the default empiric model) with the intercept term fixed to take the value 4, we can specify:
fit <- get_dfcrm(skeleton = skeleton, target = target,
intcpt = 4, model = 'logistic') %>%
fit('2NNN 3TNN')
fit %>% recommended_dose()
## [1] 3
intcpt and logistic are the parameter names chosen by the authors of dfcrm.
### get_trialr_crm
We could instead fit the CRM models above using the trialr package by (@ Brock 2019; Brock 2020).
Reusing the skeleton and target variables defined above, we fit the same empiric model
model <- get_trialr_crm(skeleton = skeleton, target = target, model = 'empiric',
beta_sd = sqrt(1.34))
fit <- model %>% fit('2NNN')
The dfcrm package, unless told otherwise, assumes that you want an empiric model where the prior variance for β is 1.34. In the trialr package, no such assumptions are made so we had to specify those variables.
All we have changed is the method of inference. dfcrm uses numerical integration to calculate posterior statistics and plugs those into the dose-toxicity function. In contrast, trialr fits the model using Hamiltonian MCMC sampling via Stan. Thankfully, the two models agree on the desired next dose:
fit %>% recommended_dose()
## [1] 4
and that the trial should continue:
fit %>% continue()
## [1] TRUE
The added bonus we get from the trialr fit, however, is those samples from the posterior distribution:
fit %>% prob_tox_samples() %>% head(10)
## # A tibble: 10 x 6
## .draw 1 2 3 4 5
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.174 0.261 0.445 0.586 0.742
## 2 2 0.0649 0.122 0.282 0.433 0.627
## 3 3 0.0000000950 0.00000400 0.000563 0.00711 0.0635
## 4 4 0.00000000140 0.000000156 0.0000799 0.00196 0.0309
## 5 5 0.0523 0.103 0.255 0.406 0.605
## 6 6 0.00463 0.0161 0.0831 0.193 0.400
## 7 7 0.00161 0.00714 0.0510 0.140 0.334
## 8 8 0.0265 0.0614 0.186 0.329 0.538
## 9 9 0.00860 0.0259 0.111 0.234 0.444
## 10 10 0.00902 0.0268 0.113 0.237 0.448
That facilitates really flexible inference. For example, what is the probability that toxicity at dose 3 is at least 5% greater than that at dose 2? Simple to answer using the posterior samples:
library(dplyr)
fit %>% prob_tox_samples() %>%
summarise(prob = mean(3 > 2 + 0.05))
## # A tibble: 1 x 1
## prob
## <dbl>
## 1 0.591
‘More likely than not’, is the answer.
See the Continual Reassessment Method vignette for more details.
### get_trialr_nbg
The two-parameter logistic dose-escalation method of Neuenschwander, Branson, and Gsponer (2008) (NBG) is implemented in the trialr package by Brock (2020).
The very least information we need to provide is a vector of the doses under investigation, a reference dose-level d*, our target toxicity level, and priors on the logit model intercept, α, and dose gradient, β.
For illustration, let us reproduce the notorious example in Figure 1 of Neuenschwander, Branson, and Gsponer (2008) with 15 doses:
dose <- c(1, 2.5, 5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 150, 200, 250)
outcomes <- '1NNN 2NNNN 3NNNN 4NNNN 7TT'
fit <- get_trialr_nbg(real_doses = dose, d_star = 250, target = 0.3,
alpha_mean = 2.15, alpha_sd = 0.84,
beta_mean = 0.52, beta_sd = 0.8, seed = 2020) %>%
fit(outcomes)
Sticking at dose 7 is the recommendation:
fit %>% recommended_dose()
## [1] 7
However, we see that it is a close call as to which dose is closest to the target toxicity level:
fit %>% mean_prob_tox()
## [1] 0.01258690 0.03252256 0.06752086 0.13847209 0.20636157 0.26917718
## [7] 0.32637010 0.37801151 0.46612242 0.53711249 0.66158702 0.73901450
## [13] 0.82594636 0.87171076 0.89929305
This is perhaps unsurprising in a situation with so many doses.
See the Neuenschwander, Branson & Gsponer vignette for more details.
### get_tpi
The Toxicity Probability Interval (TPI) method was introduced by Ji, Li, and Bekele (2007). The model requires a few parameters:
model <- get_tpi(num_doses = 5, target = 0.25, k1 = 1, k2 = 1.5,
exclusion_certainty = 0.95)
The model can be fit to outcomes in the usual way:
fit <- model %>% fit('1NNT')
and the returned model fit obeys the same interface as the other classes described here. For instance, the dose recommended for the next cohort is:
fit %>% recommended_dose()
## [1] 1
See the Toxicity Probability Interval Design vignette for more information.
### get_mtpi
The Modified Toxicity Probability Interval (mTPI) method was introduced by Ji et al. (2010). It is generally simpler to implement than TPI because the ϵ1 and ϵ2 parameters have the intuitive interpretation of forming the bounds of the interval that we regard as containing doses equivalent to the target dose. For instance, if we target a dose with toxicity probability equal to 25%, but would judge doses in the region (20%, 30%) to be satisfactorily toxic, we run:
model <- get_mtpi(num_doses = 5, target = 0.25,
epsilon1 = 0.05, epsilon2 = 0.05, exclusion_certainty = 0.95)
In this parameterisation, we exclude doses if we are 95% a-posteriori sure that the associated toxicity rate exceeds the target.
We fit the model to outcomes:
fit <- model %>% fit('1NNT')
and learn that the recommended next dose is
fit %>% recommended_dose()
## [1] 1
dose 1, in accordance with Figure 2 of Ji et al. (2010).
See the Modified Toxicity Probability Interval Design vignette for more information.
### get_boin
escalate also implements the Bayesian Optimal Interval (BOIN) dose-finding design by Liu and Yuan (2015) via the BOIN package (Yuan and Liu 2018).
In contrast to CRM, BOIN does not require a dose-toxicity skeleton. In its simplest case, it requires merely the number of doses under investigation and our target toxicity level. Continuing with our example above:
target <- 0.25
model <- get_boin(num_doses = 5, target = target)
As before, we can fit the model to some observed outcomes:
fit <- model %>% fit('2NNN')
and ask the recommended dose:
fit %>% recommended_dose()
## [1] 3
The BOIN dose selector natively implements stopping rules, as described by Liu & Yuan. For instance, if the bottom dose is too toxic, the design will advise the trial halts:
fit <- model %>% fit('2NTN 1TTT')
fit %>% continue()
## [1] FALSE
Notice in this scenario that the recommended dose is NA:
fit %>% recommended_dose()
## [1] NA
This clarifies that no dose should be recommended for further study. In this setting, this is because all doses are considered too toxic. This is distinct from scenarios where a design advocates stopping a trial and recommending a dose for further study. We will encounter situations like that below.
Since escalation provides many flexible options for stopping, we have made it possible to suppress BOIN’s native stopping rule via use_stopping_rule = TRUE.
Similar to the method described above, extra parameters are passed to the get.boundary function in the BOIN package to customise the escalation procedure. For instance, the boundaries that guide changes in dose are set to be 60% and 140% of the target toxicity rate, by default. To instead use 30% and 170%, we could run:
get_boin(num_doses = 5, target = target,
p.saf = 0.3 * target, p.tox = 1.7 * target) %>%
fit('1NNN 2NNT') %>%
recommended_dose()
## [1] 2
To observe the effect of the change, note that the default values suppress escalation in this scenario:
get_boin(num_doses = 5, target = target) %>%
fit('1NNN 2NNT') %>%
recommended_dose()
## [1] 1
The parameter names p.saf and p.tox were chosen by the authors of the BOIN package.
See the Bayesian Optimal Interval Design vignette for more information.
### get_three_plus_three
The 3+3 method is an old method for dose-escalation that uses fixed cohorts of three and pre-specified rules to govern dose-selection (Korn et al. 1994; Le Tourneau, Lee, and Siu 2009).
To create a 3+3 design, we need no more information than the number of doses under investigation:
model <- get_three_plus_three(num_doses = 5)
As usual, we can fit the model to some outcomes and learn the recommended dose:
model %>% fit('2NTN') %>% recommended_dose()
## [1] 2
Korn et al. (1994) described a variant of 3+3 that permits deescalation to ensure that six patients are treated at a dose before it is recommended. To use that option in our model, we could have run:
model <- get_three_plus_three(num_doses = 5, allow_deescalate = TRUE)
The model would then advocate deescalation if at least two toxicities are seen at a dose and the dose below has fewer than 6 treated patients:
model %>% fit('2NTT') %>% recommended_dose()
## [1] 1
### follow_path
The final dose selector in this section is not really a model at all, so much as a pre-specified path to follow. Let us say that we would like to escalate through the doses in the absence of toxicity, treating two patients at each of the first two doses, and three at the other doses. We can specify such a path in escalation using:
model <- follow_path('1NN 2NN 3NNN 4NNN 5NNN')
When fit to data, the method just returns whatever comes next in the sequence:
model %>% fit('1NN 2N') %>% recommended_dose()
## [1] 2
model %>% fit('1NN 2NN') %>% recommended_dose()
## [1] 3
When the outcomes diverge from the pre-specified path, however, this selector does not know what to do:
model %>% fit('1NN 2NT') %>% recommended_dose()
## [1] NA
That rather seems to limit its value. The point of this class is that we sometimes want to specify what is occasionally referred to as an initial escalation plan. When trial outcomes diverge from the initial plan, another method takes over. This is a perfect opportunity to show how different selectors can be joined together. Let us say that we wish to follow the initial plan described above, but when the first toxicity event is seen, we want a CRM model to take over. We simply join the functions together using the pipe operator from magrittr:
model <- follow_path('1NN 2NN 3NNN 4NNN 5NNN') %>%
get_dfcrm(skeleton = skeleton, target = target)
Now, when trial outcomes diverge from the path, the CRM model analyses all of the outcomes and recommends the next dose:
model %>% fit('1NN 2NT') %>% recommended_dose()
## [1] 2
This concludes our look at the core dose-selecting classes. We now turn our attention to the ways in which these methods can be adapted using extra behaviours.
### dont_skip_doses
We saw in the CRM example above that the design undesirably wanted to skip straight to a high dose, without trying some of the lower doses. A simple and very common constraint to impose in dose-finding trials is to avoid skipping untested doses.
Resuming our CRM example, we suppress the skipping of untested doses in escalation with:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
dont_skip_doses(when_escalating = TRUE)
We then fit the model as before:
fit <- model %>% fit('2NNN')
fit %>% recommended_dose()
## [1] 3
This time, however, the model advocates dose 3. Previously, it wanted to go straight to dose 4.
We prevented skipping dose in escalation. We could have prevented skipping doses in deescalation with:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
dont_skip_doses(when_deescalating = TRUE)
or both with:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
dont_skip_doses(when_escalating = TRUE, when_deescalating = TRUE)
### stop_at_n
Let us now investigate some methods that facilitate stopping. The simplest condition on which to stop is when the total sample size reaches some pre-specified level. For instance, we might want to treat a maximum of 15 patients and then stop. To do this, we call the stop_at_n function and append it onto the end of a core dose selector, like this:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_at_n(n = 15)
When this design has seen fewer than 15 patients, it will select doses and advocate that the trial continues. For instance:
fit <- model %>% fit('1NNN 2TNN 2NNN 3NNN')
fit %>% continue()
## [1] TRUE
The design advocates continuing at dose:
fit %>% recommended_dose()
## [1] 3
In contrast, once 15 patients are seen,
fit <- model %>% fit('1NNN 2TNN 2NNN 3NNN 3NTN')
fit %>% continue()
## [1] FALSE
the design advocates stopping. It is important to note that, even though the design has stopped, it still recommends that a dose be studied at the next trial phase:
fit %>% recommended_dose()
## [1] 3
This is in contrast to the scenario where a trial is stopped because all doses are inappropriate. In this scenario, the dose recommendation would be NA. We will encounter this in examples below.
### stop_when_n_at_dose
Another common approach is to stop a dose-finding experiment when a given number of patients have been treated at a particular dose.
Continuing with our CRM model, to stop when nine patients have been treated at the dose that is about to be recommended again, we use:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_n_at_dose(n = 9, dose = 'recommended')
We can observe how this alters the dose-selection model. Here we see six patients treated at dose 2:
fit <- model %>% fit('1NNN 2TNN 2NTN')
The model recommends that dose 2 should be given to more patients:
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 2
If the next cohort results in dose 2 being recommended yet again, i.e. to bring the total number of patients at dose 2 to nine or more, the model stops:
fit <- model %>% fit('1NNN 2TNN 2NTN 2NNN')
fit %>% continue()
## [1] FALSE
fit %>% recommended_dose()
## [1] 2
In this scenario, dose 2 is the final recommended dose and the trial stops gracefully at a pre-specified stopping rule.
This behaviour can also be configured to stop when any dose has been given n times:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_n_at_dose(n = 9, dose = 'any')
or when a particular dose-level has been given n times:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_n_at_dose(n = 9, dose = 3)
Naturally, you can combine this behaviour with other behaviours. The following model stops the trial when nine patients have been evaluated at the recommended dose or when 21 patients have been treated in total, whichever occurs first:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_n_at_dose(n = 9, dose = 'recommended') %>%
stop_at_n(n = 21)
### stop_when_ci_covered
The two stopping mechanisms above scrutinise the number of patients treated. In many situations, this will be valuable. However, in other situations, we might want to stop when a threshold amount of statistical information is obtained. One way to achieve this is to stop when the confidence interval or credible interval for the probability of toxicity at a dose is covered by a specified range.
For instance, we know that the BOIN design seeks a target toxicity level, and we have used a target of 25% in our examples. We might say that we are sure enough about the recommended dose when the associated 90% credible interval (because BOIN is a Bayesian design) of the toxicity probability falls in the region 10% - 40%.
model <- get_boin(target = target, num_doses = 5) %>%
stop_when_tox_ci_covered(dose = 'recommended', lower = 0.10, upper = 0.4)
Say that we observe the following trial path:
fit <- model %>%
fit('1NNN 2NTN 2TNN 2NNN 2NNT 2NTN 2NNN 2TNN')
The design recommends dose 2 and it also advocates stopping:
fit %>% recommended_dose()
## [1] 2
fit %>% continue()
## [1] FALSE
This is because the lower bound of the 90% interval for the probability of toxicity at dose 2 is at least 10%:
fit %>% prob_tox_quantile(p = 0.05)
## 1 2 3 4 5
## 0.0 0.1 NA NA NA
and the upper bound is no more than 40%:
fit %>% prob_tox_quantile(p = 0.95)
## 1 2 3 4 5
## 0.1 0.4 NA NA NA
It may be intersting to note that our CRM model would not stop in this scenario:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_tox_ci_covered(dose = 'recommended', lower = 0.10, upper = 0.4)
fit <- model %>%
fit('1NNN 2NTN 2TNN 2NNN 2NNT 2NTN 2NNN 2TNN')
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 2
This is because the lower bound of the 90% CI falls slightly outside the sought range:
fit %>% prob_tox_quantile(p = 0.05)
## [1] 0.04876626 0.09809797 0.24712623 0.39695491 0.59744927
As before, we can specify dose = 'recommended', dose = 'any', or a particular numerical dose-level with dose = 3, for example.
It should be appreciated that this approach only works when the underlying model extends a way of calculating quantiles and uncertainty intervals. The 3+3 lacks a statistical foundation and does not offer quantiles:
get_three_plus_three(num_doses = 5) %>%
fit('1NNN 2NTN') %>%
prob_tox_quantile(p = 0.05)
## [1] NA NA NA NA NA
### stop_when_too_toxic
The stopping rules considered so far stop a trial and recommend a dose once some critical threshold of information is obtained. We will naturaly want to stop if all doses are too toxic.
We saw above that some model-based dose-finding approaches can calculate quantiles. We can take this idea further and advocate stopping when there is sufficient evidence that the toxicity probability at some dose exceeds a critical threshold. In such circumstances, no dose will be recommended because all doses of the treatment will be deemed to be excessively toxic.
Let us set up a rule to stop and recommend no dose if the probability of toxicity at the lowest dose is too high:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.7)
The above examples stops when 70% of the probability mass or posterior distribution of the probability of toxicity at dose 1 exceeds 35%. With an isolated toxicity incidence at dose 1, the model advocates continuing at dose 1:
fit <- model %>% fit('1NTN')
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 1
This is because the probability that the toxicity rate exceeds 35% is less than 70%:
fit %>% prob_tox_exceeds(0.35) %>% round(2)
## [1] 0.35 0.53 0.82 0.95 1.00
However, with material additional toxicity at dose 1, the design now advocates stopping:
fit <- model %>% fit('1NTN 1TTT')
fit %>% continue()
## [1] FALSE
Furthermore, no dose is recommended:
fit %>% recommended_dose()
## [1] NA
This is because we are now at least 70% sure that the lowest dose is too toxic:
fit %>% prob_tox_exceeds(0.35) %>% round(2)
## [1] 0.87 0.95 1.00 1.00 1.00
Once again, we can specify dose = 'recommended', dose = 'any', or a particular numerical dose-level with dose = 3, for example. We also require that the underlying model supports the calculation of quantiles. BOIN supports this fucntionality:
model <- get_boin(target = target, num_doses = 5) %>%
stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.7)
fit <- model %>% fit('1NTN 1TTT')
fit %>% continue()
## [1] FALSE
fit %>% recommended_dose()
## [1] NA
fit %>% prob_tox_exceeds(0.35) %>% round(2)
## [1] 0.95 NA NA NA NA
but a non-statistical method like 3+3 does not.
### demand_n_at_dose
We have looked at many behaviours that provide stopping. We can also look at some behaviours that delay stopping.
We might want to guarantee that we treat at least n patients at a dose before we permit a dose-finding trial to stop. For instance, we might not feel comfortable recommending a dose for the next phase of study if it has only been evaluated in a small number of patients.
It makes sense for this behaviour to be used with a design that would otherwise stop. Let us say that we would normally like to stop after 18 patients have been treated. However, we will also demand that at least 6 patients be treated at the recommended dose before stopping is allowed, irrespective the overall sample size. We specify:
model <- get_boin(target = target, num_doses = 5) %>%
stop_at_n(n = 18) %>%
demand_n_at_dose(n = 6, dose = 'recommended')
In the following situation:
fit <- model %>% fit('1NNN 2NNT 3NTN 3NNN 4TTN 3NTT')
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 2
the design advocates continuing at dose 2 even though 18 patients have been evaluated. This is because the demand_n_at_dose function is overriding the stopping behaviour of stop_at_n. It is requesting that the trial continue at dose 2 instead of stopping with only three patients treated at the nominal recommended dose.
It is important to recognise that the order of the functions matters. If we flip the order of the constraints in the example above, the outcome is different:
model <- get_boin(target = target, num_doses = 5) %>%
demand_n_at_dose(n = 6, dose = 'recommended') %>%
stop_at_n(n = 18)
fit <- model %>% fit('1NNN 2NNT 3NTN 3NNN 4TTN 3NTT')
fit %>% continue()
## [1] FALSE
fit %>% recommended_dose()
## [1] 2
Now the stop_at_n constraint overrides the action of demand_n_at_dose to halt the trial when n=18, even though only three patients have been evaluated at dose 2. It overrides because it comes later in the decision chain. Users should be aware that commands that come later take precedence.
Once again, we can specify dose = 'recommended', dose = 'any', or a particular numerical dose-level with dose = 3, for example.
In summary, the demand_n_at_dose function delays stopping in a scenario when a dose is being selected.
### try_rescue_dose
In contrast to demand_n_at_dose, the try_rescue_dose function delays stopping in a scenario where no dose is going to be selected. It overrides a decision to stop and recommend no dose when fewer than n patients have been evaluated at a given dose. Thus, it provides a facility to ensure that some “rescue” dose has been tried before stopping is allowed.
This is another function where effective demonstration requires a design that would normally stop. Let us say that we will stop if we are 80% sure that the toxicity rate at the lowest dose exceeds 35%. But before we stop, we want to ensure that at least two patients have been evaluated at the lowest dose. We write:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.8) %>%
try_rescue_dose(dose = 1, n = 2)
Then, even when this design sees some major toxicity at dose 2:
fit <- model %>% fit('2TTT')
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 1
the design will not advocate stopping, even though the posterior confidence that the tox rate at dose 1 exceeds 35% is greater than 80%:
fit %>% prob_tox_exceeds(0.35)
## [1] 0.8673669 0.9307674 0.9857421 0.9971830 0.9998310
Once two patients are seen at dose 1, stopping can be countenanced. If those two patients tolerate treatment at dose 1:
fit <- model %>% fit('2TTT 1NN')
fit %>% continue()
## [1] TRUE
fit %>% recommended_dose()
## [1] 1
then stopping is not advocated because the posterior belief is now that dose 1 is not excessively toxic:
fit %>% prob_tox_exceeds(0.35)
## [1] 0.6683818 0.8195981 0.9668375 0.9951862 0.9998694
However, if even one of those patients at dose 1 experiences toxicity:
fit <- model %>% fit('2TTT 1NT')
fit %>% continue()
## [1] FALSE
fit %>% recommended_dose()
## [1] NA
Then the trial stops and no dose is recommended.
The try_rescue_dose function allows researchers to rescue situations where otherwise sensible stopping criteria may prove too sensitive to chance events in very small sample sizes.
### select_dose_by_cibp
This function implements the convex infinite bounds penalisation (CIBP) criterion of Mozgunov and Jaki (2020) that adjusts the way doses are selected in CRM trials. Their method is mindful of the uncertainty in the estimates of the probability of toxicity and uses an asymmetry parameter, 0 < a < 2, to penalise escalation to risky doses. The method alters the way doses are selected but not when the trial should stop. For a < 1, the criterion penalises toxic doses more heavily, making escalation decisions more conservative.
To add the behaviour to a dose-finding design, we run:
model <- get_dfcrm(skeleton = skeleton, target = target) %>%
select_dose_by_cibp(a = 0.3)
The model is then fit to outcomes in the usual way:
model %>%
fit('1NTN') %>%
recommended_dose()
## [1] 1
## Simulation and dose-paths
We have described at length above the flexible methods that escalation provides to specify dose-escalation designs and tailor trial behaviour. Once designs are specified, we can investigate their operating characteristics by simulation using the simulate_trials function. We can also exhaustively calculate dose recommendations for future cohorts using the get_dose_paths function. Both of these topics are the topics of full vignettes. Please check them out.
# Installation
# Install the latest official version from CRAN with:
install.packages("escalation")
# Alternatively, install the latest code from GitHub:
devtools::install_github("brockk/escalation")
# Future Plans
I plan to add model-fitting functions for EWOC via ewoc, further methods for phase I/II designs, and perhaps also methods for dual agents.
I want to investigate adding some further stopping functions like those researched by Zohar and Chevret (2001).
Finally, I will investigate adding time-to-event versions of the designs presented here, the so-called TITE designs. These will require a different approach to simulation because cohorts no longer apply.
## Getting help
This package is still in active development. There are thousands of unit tests run each time the package code is updated. However, that certainly does not mean that the code is bug free. You should always be on the defensive. This software is offered with no guarantee at all. If you have found a bug, please drop me a line and also log it here:
https://github.com/brockk/escalation/issues
If you want help using the package, feel free to contact me by email.
## References
Brock, Kristian. 2020. Trialr: Clinical Trial Designs in ’Rstan’. https://cran.r-project.org/package=trialr.
Brock, Kristian. 2019. “trialr: Bayesian Clinical Trial Designs in R and Stan.” arXiv E-Prints, June, arXiv:1907.00161. https://arxiv.org/abs/1907.00161.
Brock, Kristian, Lucinda Billingham, Mhairi Copland, Shamyla Siddique, Mirjana Sirovica, and Christina Yap. 2017. “Implementing the EffTox Dose-Finding Design in the Matchpoint Trial.” BMC Medical Research Methodology 17 (1): 112. https://doi.org/10.1186/s12874-017-0381-x.
Cheung, Ken. 2013. Dfcrm: Dose-Finding by the Continual Reassessment Method. https://CRAN.R-project.org/package=dfcrm.
Ji, Yuan, Yisheng Li, and B. Nebiyou Bekele. 2007. “Dose-finding in phase I clinical trials based on toxicity probability intervals.” Clinical Trials 4 (3): 235–44. https://doi.org/10.1177/1740774507079442.
Ji, Yuan, Ping Liu, Yisheng Li, and B. Nebiyou Bekele. 2010. “A modified toxicity probability interval method for dose-finding trials.” Clinical Trials 7 (6): 653–63. https://doi.org/10.1177/1740774510382799.
Korn, Edward L., Douglas Midthune, T. Timothy Chen, Lawrence V. Rubinstein, Michaele C. Christian, and Richard M. Simon. 1994. “A Comparison of Two Phase I Trial Designs.” Statistics in Medicine 13 (18): 1799–1806. https://doi.org/10.1002/sim.4780131802.
Le Tourneau, Christophe, J. Jack Lee, and Lillian L. Siu. 2009. “Dose Escalation Methods in Phase I Cancer Clinical Trials.” Journal of the National Cancer Institute 101 (10): 708–20. https://doi.org/10.1093/jnci/djp079.
Liu, Suyu, and Ying Yuan. 2015. “Bayesian Optimal Interval Designs for Phase I Clinical Trials.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 64 (3): 507–23. https://doi.org/10.1111/rssc.12089.
Mozgunov, Pavel, and Thomas Jaki. 2020. “Improving Safety of the Continual Reassessment Method via a Modified Allocation Rule.” Statistics in Medicine 39 (7): 906–22. https://doi.org/10.1002/sim.8450.
Neuenschwander, Beat, Michael Branson, and Thomas Gsponer. 2008. “Critical aspects of the Bayesian approach to phase I cancer trials.” Statistics in Medicine 27: 2420–39. https://doi.org/10.1002/sim.3230.
O’Quigley, J, M Pepe, and L Fisher. 1990. “Continual Reassessment Method: A Practical Design for Phase 1 Clinical Trials in Cancer.” Biometrics 46 (1): 33–48. https://doi.org/10.2307/2531628.
Yuan, Ying, and Suyu Liu. 2018. BOIN: Bayesian Optimal Interval (Boin) Design for Single-Agent and Drug- Combination Phase I Clinical Trials. https://CRAN.R-project.org/package=BOIN.
Zohar, Sarah, and Sylvie Chevret. 2001. “The Continual Reassessment Method: Comparison of Bayesian Stopping Rules for Dose-Ranging Studies.” Statistics in Medicine 20 (19): 2827–43. https://doi.org/10.1002/sim.920.
|
2021-01-20 09:12:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46446406841278076, "perplexity": 6754.590656510665}, "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/1610703519984.9/warc/CC-MAIN-20210120085204-20210120115204-00393.warc.gz"}
|
http://www.r-bloggers.com/deploy-rook-apps-part-ii-2/
|
# Deploy Rook Apps: Part II
October 17, 2012
By
(This article was first published on Jeffrey Horner, and kindly contributed to R-bloggers)
In Part I, I described how you can deploy your Rook applications with rApache. This post describes how you can do it with R itself. But before we get into that, I’d like to explain the off-again on-again relationship Rook has had with CRAN, R’s package archive network.
Since inception (of Rook, not the movie), I wanted to give Rook the most flexibility possible, and that meant discovering how R’s internal web server worked. By inspecting the code from startDynamicHelp in the tools package, I discovered there were two basic calls to start and stop the server:
.Internal(startHTTPD("127.0.0.1", ports[i]))
and
.Internal(stopHTTPD())
but it turns out that inclusion of .Internal calls is a violation of CRAN’s Policy:
CRAN packages should use only the public API. Hence they should not use entry
points not declared as API in installed headers nor .Internal()l nor .Call()
etc calls to base packages. Such usages can cause packages to break at any
time, even in patched versions of R.
Understood. R-Core does a herculean job of maintaining the package repository with very little human and physical capital, and ensuring that R packages behave nicely from one R release to another is a task that all package authors should strive for. So, I yanked those calls out of Rook and play nicely by calling startDynamicHelp.
Unfortunately, that hobbles Rook in just the slightest way; it can no longer listen on any other IP address other than 127.0.0.1 … at least out of the box, but you as a Rook user are in full control of your R environment. That leads me to the following recipe for deploying a Rook app.
## Yes, Using Only R, You Can Deploy A Rook App
So here’s a recipe I cooked up to circumvent R’s http environment. I don’t recommend doing this for production, but it’s nice to show a few friends and co-workers. This is an Rscript file which you can execute from the shell. It starts up Rook on port 8000 and will listen on the 0.0.0.0 IP address. That means it will listen on your loopback device as well as any other network device you have set up on your machine. If you want to be really savy, you could even change the myPort variable to 80, like a real web server! Just know that’s a priviledged port and will need root access.
The recipe adds the test application from the Rook package system files, and it’s easy to add more than one application if you like.
#!/usr/bin/env Rscript
library(Rook)
myPort <- 8000
myInterface <- "0.0.0.0"
status <- -1
# R 2.15.1 uses .Internal, but the next release of R will use a .Call.
# Either way it starts the web server.
if (as.integer(R.version[["svn rev"]]) > 59600) {
status <- .Call(tools:::startHTTPD, myInterface, myPort)
} else {
status <- .Internal(startHTTPD(myInterface, myPort))
}
if (status == 0) {
unlockBinding("httpdPort", environment(tools:::startDynamicHelp))
assign("httpdPort", myPort, environment(tools:::startDynamicHelp))
s <- Rhttpd$new() s$listenAddr <- myInterface
s$listenPort <- myPort # Change this line to your own application. You can add more than one # application if you like s$add(name = "test", app = system.file("exampleApps/RookTestApp.R", package = "Rook"))
# Now make the console go to sleep. Of course the web server will still be
# running.
while (TRUE) Sys.sleep(24 * 60 * 60)
}
# If we get here then the web server didn't start up properly
warning("Oops! Couldn't start Rook app")
|
2014-10-25 12:47:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20665742456912994, "perplexity": 5027.35390793896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119648155.19/warc/CC-MAIN-20141024030048-00202-ip-10-16-133-185.ec2.internal.warc.gz"}
|
https://codeforces.com/blog/entry/103296
|
SirRembocodina's blog
By SirRembocodina, history, 5 weeks ago,
It seems, there are so many materials on binary search already that everyone must know how to code it, however every material seems to tell its own approach, and I see people being lost and still making bugs on such a simple thing. That's why I want to tell you about an approach which I find the most elegant. Of all the variety of articles, I saw my approach only in this comment (but less generalized).
UPD: Also in the nor's blog.
The problem
Suppose you want to find the last element less than x in a sorted array and just store a segment of candidates for the answer [l, r]. If you write:
int l = 0, r = n - 1; // the segment of candidates
while (l < r) { // there are more than 1 candidate
int mid = (l + r) / 2;
if (a[mid] < x) {
l = mid;
} else {
r = mid - 1;
}
}
cout << l;
...you will run into a problem: suppose l = 3, r = 4, then mid = 3. If a[mid] < x, you will end up in a loop (the next iteration will be again on [3, 4]).
Okay, it can be fixed with rounding mid up – mid = (l + r + 1) / 2. But we have another problem: what if there are no such elements in the array? We would need an extra check for that case. As a result, we have a pretty ugly code that is not generalized very well.
My approach
Let's generalize the problem we want to solve. We have a statement about an integer number n that is true for integers smaller than some bound, but then always stays false once n exceeded that bound. We want to find the last n for which the statement is true.
First of all, we will use half-intervals instead of segments (one border is inclusive, another is non-inclusive). Half-intrevals are in general very useful, elegant and conventional in programming, I recommend using them as much as possible. In that case we will choose some small l for which we know in before the statement is true and some big r for which we know it's false. Then the range of candidates is [l, r).
My code for that problem would be:
int l = -1, r = n; // a half-interval [l, r) of candidates
while (r - l > 1) { // there are more than 1 candidate
int mid = (l + r) / 2;
if (a[mid] < x) {
l = mid; // now it's the largest for which we know it's true
} else {
r = mid; // now it's the smallest for which we know it's false
}
}
cout << l; // in the end we are left with a range [l, l + 1)
The binary search will only do checks for some numbers strictly between initial l and r. It means that it will never check the statement for l and r, it will trust you that for l it's true, and for r it's false. Here we consider -1 as a valid answer, which will correspond to no numbers in array being less than x.
Notice how there are no "+ 1" or "- 1" in my code and no extra checks are needed, and no loops are possible (since mid is strictly between the current l and r).
Reverse problem
The only variation that you need to keep in mind is that half of the times you need to find not the last, but the first number for which something is true. In that case the statement must be always false for smaller numbers and always true starting from some number.
We will do pretty much the same thing, but now r will be an inclusive border, while l will be non-inclusive. In other words, l is now some number for which we know the statement to be false, and r is some for which we know it's true. Suppose I want to find the first number n for which n * (n + 1) >= x (x is positive):
int l = 0, r = x; // a half-interval (l, r] of candidates
while (r - l > 1) { // there are more than 1 candidate
int mid = (l + r) / 2;
if (mid * (mid + 1) >= x) {
r = mid; // now it's the smallest for which we know it's true
} else {
l = mid; // now it's the largest for which we know it's false
}
}
cout << r; // in the end we are left with a range (r - 1, r]
Just be careful to not choose a r too large, as it can lead to overflow.
An example
1201C - Maximum Median You are given an array a of an odd length n and in one operation you can increase any element by 1. What is the maximal possible median of the array that can be achieved in k steps?
Consider a statement about the number x: we can make the median to be no less than x in no more than k steps. Of course it is always true until some number, and then always false, so we can use binary search. As we need the last number for which this is true, we will use a normal half-interval [l, r).
To check for a given x, we can use a property of the median. A median is no less than x iff at least half of the elements are no less than x. Of course the optimal way to make half of the elements no less than x is to take the largest elements.
Of course we can reach the median no less than 1 under given constraints, so l will be equal to 1. But even if there is one element and it's equal to 1e9, and k is also 1e9, we still can't reach median 2e9 + 1, so r will be equal to 2e9 + 1. Implementation:
#define int int64_t
int n, k;
cin >> n >> k;
vector<int> a(n);
for (int i = 0; i < n; i++) {
cin >> a[i];
}
sort(a.begin(), a.end());
int l = 1, r = 2e9 + 1; // a half-interval [l, r) of candidates
while (r - l > 1) {
int mid = (l + r) / 2;
int cnt = 0; // the number of steps needed
for (int i = n / 2; i < n; i++) { // go over the largest half
if (a[i] < mid) {
cnt += mid - a[i];
}
}
if (cnt <= k) {
l = mid;
} else {
r = mid;
}
}
cout << l << endl;
Conclusion
Hope I've made it clearer and some of you will switch to this implementation. To clarify, occasionally other implementations can be more fitting, for example with interactive problems – whenever we need to think in terms of an interval of searching, and not in terms of the first/last number for which something is true.
I remind that I do private lessons on competitive programming, the price is \$25/h. Contact me on Telegram, Discord: rembocoder#3782, or in CF private messages.
• +40
» 5 weeks ago, # | +10 Thanks for the blog!Re: example task, you forgot to paste the ever important #define int int64_t from your old solution. :) And on the topic of binsearch, there is also the binary jumps perspective where you (in a way...) don't explicitly remember neither right nor mid. I don't tend to use it, but perhaps it may appeal to somebody more than other methods. For the example task, the relevant part could go: code int ans = 1; int maxans = 2e9 + 1; for (int b = maxans / 2; b > 0; b >>= 1) { while (ans + b < maxans) { long long cnt = 0; for (int i = n / 2; i < n; i++) { if (a[i] < ans + b) cnt += ans + b - a[i]; } if (cnt <= k) ans += b; else break; } } cout << ans << endl; Submission: 158749229
• » » 5 weeks ago, # ^ | 0 Yes, thank you for the correction.
» 5 weeks ago, # | +31 In my opinion the "Reverse problem" part is excessive. It basically replicates the previous part approach in terms of code. The very same code finds both: l is the last element for which the condition takes one value, and r is the first element for which the condition takes another value.
• » » 5 weeks ago, # ^ | ← Rev. 2 → 0 You don't even have to confuse yourself with different types of intervals. This approach makes so much more sense.
» 5 weeks ago, # | +54 Thanks for the blog! I used the same implementation before C++20, and I had a completely different interpretation, so it's nice to learn a new way of thinking about it. What I had as my invariant was the following (and it seems a bit more symmetric than in this blog):Suppose we have a predicate $p$ that returns true for some prefix of $[L, R]$, and false for the remaining suffix. Then at any point in the algorithm: $l$ is the rightmost element for which you can prove that $f(l)$ is true. $r$ is the leftmost element for which you can prove that $f(r)$ is false. The remaining interval to search in is $(l, r)$. $r - l > 1$ corresponds to having at least one unexplored element, and when the algorithm ends, $l, r$ are the positions of the rightmost true and the leftmost false. Intuitively, it is trying to look at positions like l]...[r, and trying to bring the ] and [ together, to find the "breaking point" (more details in my blog).
• » » 5 weeks ago, # ^ | 0 I was going through previous posts on binary search and missed that in the middle of your post you introduce the same implementation, as me. Sorry, I must be more attentive.As for your comment – yes, I also state this in the comments of my code. But with your and SirShokoladina's comments I now see that such formulation of moving the borders together can be even more obvious for a beginner.
» 5 weeks ago, # | -11 tl;dr: half-open intervals are superior.
» 5 weeks ago, # | -13 we can do the same with doubles but instead of while(r-l>1)we must use while(r-l>eps)
• » » 5 weeks ago, # ^ | +55 This is known to be bad due to floating point precision issues. For floating point numbers, use a fixed number of iterations instead.
• » » » 5 weeks ago, # ^ | 0 or use Um_nik's bs while(min(r-l, r/l-1) > er_eps) { T mid = r<=1 ? (l+r)/2 : l>=1 ? sqrt(l*r) : 1; if(can(mid)) l = mid; else r = mid; }
» 5 weeks ago, # | +29 In the example task your code only works because of define int int64_t. Indeed, if the answer is at least 1e9, then during the second iteration you do mid = (l + r) / 2, which, being done in int32_ts, would result in 3e9 being overflown, and then mid becoming negative.To avoid this, one can, indeed, work with 64-bit type, or use mid = l + (r - l) / 2, which is arithmetically the same, but does not overflow. Since C++20 there is also std::midpoint, which does the same.
• » » 5 weeks ago, # ^ | ← Rev. 3 → 0 $l + (r - l) / 2$ overflows for $l = \mathtt{INT\_MIN}$ and $r = \mathtt{INT\_MAX}$. A way that works for $l \le r$ is (which you might have needed before C++20 came in with std::midpoint, and which might even be faster): std::int32_t m = l + static_cast((static_cast(r) - static_cast(l))/ 2); or the more readable, but perhaps less "industry" standard: int32_t m = l + int32_t((uint32_t(r) - uint32_t(l)) / 2); Why this works: After replacing $l, r$ by their values modulo $2^{32}$ by doing a static cast to std::uint32_t, it is guaranteed that the difference $d$ between them is equal to $r - l$ modulo $2^{32}$. Since $l \le r$, $r - l$ has to be non-negative, and the difference between them is less than $2^{32}$, the required difference is indeed equal to $d$. Now $d / 2$ fits in a std::int32_t, so we can safely cast back to std::int32_t, and carry on with our computation of $m$.Note that the static_cast to unsigned is implementation defined before C++20, but for pretty much every implementation of GCC, it is twos-complement, so that works out practically.The same thing can be done with 32 replaced by 64 everywhere, making the code independent of 128-bit types.Also since we're already on the topic of overflow, it is a bit unfortunate that the binary search can be only done where the range of possible values (that is, where we can evaluate the predicate, rather than the return value) is $(\mathtt{INT\_MIN}, \mathtt{INT\_MAX})$, so we miss out the integers $\mathtt{INT\_MIN}, \mathtt{INT\_MAX}$. I am not completely sure that this is a bad thing either, though.Edit: similarly, the condition $r - l > 1$ suffers from overflows. We can fix it by simply checking if $l \ne \mathtt{INT\_MAX}$ and $l + 1 < r$, or just uint32_t(r) - uint32_t(l) > 1 (the starting $l, r$ should satisfy $r \ge l$ for the second idea to work).
» 5 weeks ago, # | +83 Since we are talking about binary search, let me remind everyone what the best implementation is. int ans = 0; for (int k = 1 << 29; k != 0; k /= 2) { if (can(ans + k)) { ans += k; } }
• » » 5 weeks ago, # ^ | 0 Is it binary lifting/jumping approach?
» 5 weeks ago, # | 0 I'll notice that it's only possible when there's sentinel value that can never be returned. If you are implementing analogue of lower_bound you need to get prev(begin) or you have to switch to other implementations (adding one basically)
» 5 weeks ago, # | ← Rev. 3 → 0 I've seen implementations of binary search where it's written in a for loop format with a fixed number of iterations (say 20, 30 or 60 depending upon the problem). Is that worse than the format mentioned in this blog ? (excluding the time complexity)
|
2022-07-01 04:14:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42977726459503174, "perplexity": 751.5780045735243}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00590.warc.gz"}
|
https://math.stackexchange.com/questions/1933977/example-of-space-t-3-which-is-not-regular
|
# Example of space $T_3$ which is not regular
Does there exist example of a topological space in which any closed set and a point can be seperated by open sets i.e. space is $T_3$.
But there exist a pair of points which can't be seperated by points(i.e. points not closed) i.e. space not $T_1$. Hence space not regular because regular space = $T_1 + T_3$
• @bof no these not defined like this. I was reading Kelley book. I write like these to specify what I need t ask. – Sushil Sep 20 '16 at 8:32
Take the indiscrete topology on any set with more than one point. Then it's not $T_1$, but it's $T_3$ because any closed set which does not contain a point is empty.
(In fact, any example is basically the same: a $T_3$ space is regular iff it is $T_0$, and a space is $T_3$ iff its $T_0$ quotient is regular. So the only way to get examples is to take a regular space and add topologically indistinguishable "copies" of points.)
• I would have said it's regular but not T$_3.$ It seems that sometime after I graduated some vandal went around and changed all the definitions. – bof Sep 20 '16 at 8:33
• @bof: Both conventions are in use, but I’d like to have words with the idiot who came up with the OP’s: the $T_k$ notation was obviously intended to be a hierarchy. – Brian M. Scott Sep 20 '16 at 19:56
|
2021-05-17 17:20:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.869012713432312, "perplexity": 391.9576803469169}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00533.warc.gz"}
|
http://openstudy.com/updates/4f520092e4b019d0ebb0062f
|
## viniterranova Group Title x=sqrt(7+4sqrt3)+sqrt(7-4sqrt3) 2 years ago 2 years ago
1. viniterranova
2. viniterranova
You may see the equation here.
3. ash2326
We have $x=\sqrt{7+4\sqrt 3}+\sqrt{7-4\sqrt 3}$ Let's multiply and divide this by $$\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}$$ we have $x=\sqrt{7+4\sqrt 3}+\sqrt{7-4\sqrt 3} \times \frac{\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}}{\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}}$ We get $x=\frac{{(\sqrt{7+4\sqrt 3})^2 }-(\sqrt{(7-4\sqrt 3})^2 }{\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}}$ we get $x=\frac{7+4\sqrt 3-(7-4 \sqrt 3)}{\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}}$ so we get $x=\frac{8\sqrt 3}{\sqrt{7+4\sqrt 3}-\sqrt{7-4\sqrt 3}}$
4. viniterranova
5. ash2326
Yeah can't be simplified more
6. viniterranova
7. viniterranova
8. Elodi
Ai Se Eu Te Pego -
9. Elodi
159
10. viniterranova
rrsrsrs
11. Elodi
good
12. viniterranova
De onde vc é Elodi?
13. Elodi
quoi
14. viniterranova
Vc falou português
15. viniterranova
i m from Brazil
16. Elodi
okk
17. viniterranova
18. ash2326
I got it, Let me show you how this 4
19. Elodi
yes]
20. ash2326
21. viniterranova
ok
22. viniterranova
Bye Elodi See u around i ve got go
23. Elodi
y
24. ash2326
We have $x=\sqrt {7+4 \sqrt 3}+\sqrt {7-4 \sqrt 3}$ Let's square both the sides we get $x^2=(\sqrt {7+4 \sqrt 3})^2+(\sqrt {7-4 \sqrt 3})^2+2\times (\sqrt {7+4 \sqrt 3})\times (\sqrt {7-4 \sqrt 3})$ we get $x^2=7+4\sqrt 3+7-4\sqrt 3+2 \sqrt{(7^2-(4\sqrt 3)^2}$ we get $x^2=14+2\times \sqrt {49-16\times 3}$ we get $x^2=14+2\times \sqrt 1$ we get $x^2=14+2=16$ so $x=4$
25. Bhavnazoon
square on both sides|dw:1330781973743:dw|
26. viniterranova
Good.
|
2014-10-26 09:36:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.9600664973258972, "perplexity": 11090.433378921494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119662145.58/warc/CC-MAIN-20141024030102-00140-ip-10-16-133-185.ec2.internal.warc.gz"}
|
https://answers.ros.org/answers/303890/revisions/
|
I think you should try the setup.bash in the document called "devel" .Or you go first check whether the setup.bash is in the "install" by $cd ~/catkin_ws/install/ and$ ls .
|
2022-05-24 16:12:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5722656846046448, "perplexity": 2462.915543390512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662573053.67/warc/CC-MAIN-20220524142617-20220524172617-00007.warc.gz"}
|
http://math.stackexchange.com/questions/74876/exactness-of-a-short-sequence-of-quotient-modules
|
# Exactness of a short sequence of quotient modules
Suppose R is a commutative ring with 1, I $\subset R$ is an ideal.
We have R-Modules A, B and C with C being flat, as well as a short exact sequence
$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
Consider the induced sequence
$0 \rightarrow A/IA \rightarrow B/IB\rightarrow C/IC\rightarrow 0$
How do I prove that this sequence is exact? I have no idea how the flatness of C could come into play, or to be more specific, how I can use the exactness of $C\otimes\_$ (this is the only definition of flatness we have so far).
Any advice in the right direction would be appreciated.
-
If $0 \to A \to B \to C \to 0$ is exact, then apply $R/I \otimes_R -$ to get the exact $$\operatorname{Tor}(R/I, C) \to R/I \otimes A \to R/I \otimes B \to R/I \otimes C \to 0.$$ Since $C$ is flat Tor(−,C) = 0, and $R/I \otimes M = M/IM$, you get exactly what you asked for.
Alexander Thumm's answer is basically including a zig-zag lemma to prove Tor is "balanced". In other words, we have by definition that Tor(C,−) = 0, as in, if we tensor with C then there is no "extra term" on the left, but I sneakily used that Tor(−,C) = 0 too, so I can tensor things ending with C and keep exactness.
You might try proving your original question directly for abelian groups. This is a fairly literal translation of "torsion" (as in nx = 0) into the "torsion functor" (as in Tor(−,C)).
-
+1 for providing the more general context. – Alexander Thumm Oct 22 '11 at 21:31
Tor (and Ext) will be introduced within the following week. I'll look back to your answer once I know the basics, thanks! – Cedric B Oct 23 '11 at 1:06
Hint: Why is the following diagram exact/commutative?
$$\array{ &&&&&&0& \\ &&&&&&\downarrow& \\ && A \otimes I & \to & B \otimes I & \to & C \otimes I & \to & 0 \\ &&\downarrow&&\downarrow&&\downarrow&& \\ 0 & \to & A \otimes R & \to & B \otimes R & \to & C \otimes R & \to & 0 \\ &&\downarrow&&\downarrow&&\downarrow&& \\ && A \otimes R/I & \to & B \otimes R/I & \to & C \otimes R/I & \to & 0 \\ &&\downarrow&&\downarrow&&\downarrow&& \\ &&0&&0&&0&& \\ }$$
Now try to prove, that the map $A \otimes R/I \to B \otimes R/I$ is a monomorphism.
For completeness of the answer I'll add the diagrammatic yoga, but you should first try to solve this yourself. Here you go:
Let $a'' \in A\otimes R/I$ with $a'' \mapsto 0 \in B\otimes R/I$ and $a \in A \cong A \otimes R$ with $a \mapsto a''$. Now let $b \in B \cong B\otimes R$ be the image of $a$ under the map $A \otimes R \to B \otimes R$. By commutativity of the diagram, we have $b \mapsto 0 \in B \otimes R/I$. By exactness, there is a $b' \in B \otimes I$ with $b' \mapsto b$. Now let $c' \in C \otimes I$ be the image of $b'$ under the map $B\otimes I \to C \otimes I$. By exactness and commutativity we have $c' \mapsto 0$ by $C\otimes I \to C \otimes R$. Again by ecaxtness (the map $C\otimes I \to C \otimes R$ is a monomorphism) we have $c' = 0$, so there is a $a' \in A \otimes I$ with $a' \mapsto b'$ by $A\otimes I \to B \otimes I$. Let $\tilde a \in A \otimes R$ be the image of $a'$ under the map $A\otimes I \to A \otimes R$. By commutativity we have $\tilde a \mapsto b$. By exactness ($A\otimes R \to B \otimes R$ is a monomorphism) we have $\tilde a = a$, so by exacness $a'' = 0$, since $a' \mapsto \tilde a = a \mapsto a''$.
-
Thank you very much for the hint, I'll look into it tomorrow and "check" your answer if I succeed. – Cedric B Oct 22 '11 at 18:40
Note that $A/IA \cong A\otimes _R (R/I)$. Now apply the functor $-\otimes R/I$ to your short exact sequence. You get a long exact sequence, because $\otimes$ is not exact, but flatness implies that any tor groups involving $C$ vanish.
-
I didn't know that $A/IA \cong A\otimes _R (R/I)$. Thanks, I'll look into your response once I know tor – Cedric B Oct 23 '11 at 1:08
|
2015-08-04 14:31:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9346264004707336, "perplexity": 185.19043882676064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990900.28/warc/CC-MAIN-20150728002310-00251-ip-10-236-191-2.ec2.internal.warc.gz"}
|
http://www.ondrejmalecek.cz/archive/932dca-state-machine-diagram-tool-open-source
|
Automatically create UML diagrams from Java source code or class files; New graphical element types (beta), with syntax completion; New in UMLet 11.5.1 Z-order bug fix; Improved open vs. export file path handling; New in UMLet 11.5 Improved handling of special characters; Config file writes to home dir; New: open multiple diagrams Now you put your state diagram in one file using an easy-to-understand language. they're used to gather information about the pages you visit and how many clicks you need to accomplish a task. In addition, some of the Parametric diagram constraints may also be exercised by a constraint propagation engine (MATLAB/Simulink, OpenModelica, SysML tool proprietary plugin, etc. ; YAKINDU Statechart Tools - an Open-Source-Tool for the specification and development of reactive, … The state.js API offers: Classes that represent a state machine model ... Paper.js is an open source vector graphics scripting framework that runs on top of the HTML5 Canvas. Enter SMC - The State Machine Compiler. State machine diagram is a kind of UML diagram that shows flow of control from state to state within single object. In SysML-as-System-Simulation mode at least some of SysML behavioral diagrams (Activity, Sequence, State Machine diagrams) are exercised by a behavioral simulation engine. 16.07.2007 Qfsm 0.44 released Some minor features as well as a user documentation have been added. Recommended steps to create the state machine. Not just flowcharts, block diagrams, UML diagrams, network diagrams, etc. Free drawing software for Windows, Mac OS X, and Linux. Components of UML state diagram. It’s makes it really easy for you to: Embed UML diagrams in blogs, emails and wikis, post UML diagrams in forums and blog comments, use directly within your web based bug tracking tool or copy and paste UML diagrams into MS Word documents … Visual Paradigm is a UML tool … Expertly-made state diagram examples to get a headstart. Some minor bugs have been fixed. Dynamic Draw is another free and open source flowchart software for Windows. 1) StarUML. An open loop state machine represents an object that may ... a source state (2) event trigger (3) an ... and create your own State Machine Diagram with the free State Machine Diagram tool. Start state: A solid circle. ExecDesign will be a suite of Java tools which can be used to document and execute UML designs. StateProto is open source and the output functions can be modified to output c++ code for you. On this page, we collected 10 best open source license classification tree software solutions that run on Windows, Linux, and Mac OS X. We use analytics cookies to understand how you use our websites so we can make them better, e.g. Improved and cleaned up the build system. It offers some additional components, such as dia-rib-network for network diagrams and dia2cod for converting UML to code.. It provides eleven types of diagram. The following is a selected list of SysML Modeling Tool resources that will provide additional information about Commercial Off-the-Shelf (COTS) and Free and Open Source Software (FOSS) SysML-compliant modeling tools for MBSE applications. Complete State Machine Diagram Tutorial that helps you learn about What is a State Machine Diagram, How to create State Machine Diagram and when. Mindfusion Diagram Library. ). Some steps to start using Modelio, create projects, handle diagrams, install modules, ... + Contribute. When the software tester focus is to understand the behavior of the object. Perform the steps below to create a UML state machine diagram in Visual Paradigm. Dia supports more than 30 different diagram types like flowcharts, network diagrams, database models. Real Modeling Tools We build modeling software, not drawing tool. Try Umple. UML state diagrams use a notation that you may have already seen in our UML activity diagrams. It has extensive state diagram support, including nested states, guards, actions and activities. A different approach is used compared to other state machine diagram editor, there is absolutely no manual layout … It is a feature-rich flowchart maker software that provides various shapes and tools to create a flowchart. It generates code in Java and C++. StateProto will output XML state machines and there is C# code to load the XML and drive the state machine from the data. I reviewed Dia 0.97.3 from the Ubuntu 18.04 repository; you can download it here.. Dia is a standalone drawing tool. State Machine Diagram examples, State Machine Diagram tips are covered. The state pattern looks like a great solution but that means writing and maintaining a class for each state - too much work. But it uses delegates. A state of an entity is controlled with the help of an event. Extended State Machines. Draw state machine diagram online with Creately state diagram maker. It usually contains simple states, composite states, composite states, transitions, events and actions. There are many options for arrows and lines, and other graphic wiz-bangs which come in handy for state machine diagrams. There are many tools available in the market for designing UML diagrams. Ragel is a finite state machine compiler which will output C/C++/Java and more. Using these graphic symbols and shapes in Word has it quirks and frustrations for sure. Smc.jar command options java -jar Smc.jar -{targetlanguage} In this respect, the tool is innovative and might work differently than other graphical state machine tools on the market. Welcome to the Finite State Machine Diagram Editor, this tool allows software developers to model UML Finite State Machines either graphically or textually. The State Diagram Editor is a tool designed for the graphical editing of state diagrams of synchronous and asynchronous machines. Eclipse Papyrus is an industrial-grade open source Model-Based Engineering tool. The key is in learning how to use the "Text Box". yUML is an online service for creating class and use case diagrams, with activity diagrams and state machines announced to come soon. Dia Diagram Editor is free Open Source drawing software for Windows, Mac OS X and Linux. I would recommend that you use a data driven design instead. State: A rectangle with rounded corners, with the name of the action. State machine diagram tool to draw state diagrams online. Analyze the all gather information and sketch the state transition diagram. Following is a curated list of Top 28 handpicked UML tools with popular features and latest download links. This comparison list contains open source as well as commercial tools. When the software tester focus is to test the sequence of events that may occur in the system under test. yUML. State diagrams can help administrators identify unnecessary steps in a process and streamline processes to improve the customer experience. It is easy to use and can be extended through several modules. This is a Java-based free and open source tool for Windows, Linux, and Mac OS X. Weka is a powerful collection of machine learning algorithms for … A lot of thought went into drawing hierarchical state diagrams in QM™. Get involved in the Modelio community + Store. End state: A solid circle with a ring around it. It's not visual per se (you can't design the state machine graphically, you use code) but it is able to use GraphViz to visualise the state machine. The diagram tool is written 100% in JavaScript and uses the ... state.js focuses on modeling hierarchical state machines. Instead of writing the HDL code by yourself, you can enter the description of a logic block as a graphical state diagram. Analytics cookies. Gather the information which the user wants. These diagrams are used to model the event-based system. I tested Modelio (http://www.modelio.org) which is open source. StarUML is an open source software modeling tool. an open source tool for Java API documentation ... 1.Write the state diagram (.sm file) 2.Run the SMC tool (generates state pattern code) ... graph (generates a GraphViz .dot file diagram of the state machine logic 41. Statechart diagrams are also called as state machine diagrams. State machine diagrams, commonly known as state diagrams, are a useful way of visualizing the various states that exist within a process. Clearly, the state diagram from Figure 2(a) is hopelessly complex for a simple time bomb and I don't think that, in practice, anyone would implement the bomb that way (except, perhaps, if you have only a few bytes of RAM for variables but plenty of ROM for code). There is a total of two types of state machine diagrams: 1) Behavioral 2) State machine 3) Protocol state machine State Transition diagram can be used when a software tester is testing the system for a finite set of input values. Reuse elements in different models, ensure the correctness of design with syntax checking, establish multiple levels of abstraction with sub-diagrams, add references to design artifacts, etc. create a code skeleton of the state machine. Make accept state: double-click on an existing state Type numeric subscript: put an underscore before the number (like "S_0") Type greek letter: put a backslash before it (like "\beta") I have found Microsoft Word to be pretty decent for this purpose. Download Dia Diagram Editor for free. can also be created through this software. Transition: Connector arrows with a label to indicate the trigger for that transition, if there is one. 02.10.2007 Qfsm 0.45 released Added EPS and SVG export function of state diagrams. Modelio is an open source modeling environment tool providing support for the latest standards (UML 2, BPMN 2, ... UML state processes can be executed by a state machine; UML sequence diagrams can be executed directly. Creating state machine diagram. Weka. Drawing a state diagram is an alternative approach to the modeling of a sequential device. Created binary and source RPMs for some Linux distributions. 1. Eclipse Papyrus has notably been used successfuly in industrial projects and is the base platform for several industrial modeling tools. Introduction to UML 2 State Machine Diagrams by Scott W. Ambler; UML 2 State Machine Diagram Guidelines by Scott W. Ambler; Intelliwizard - UML StateWizard - A ClassWizard-like round-trip UML dynamic modeling/development framework and tool running in popular IDEs under open-source license. For example, QM does not use "pseudostates", such as the initial pseudostate or choice point. Feature-Rich flowchart maker software that provides various shapes and tools to create a UML state diagrams.. Below to create a flowchart modeling hierarchical state machines announced to come soon sequence! Is free open source Model-Based Engineering tool service for creating class and use case diagrams, with name! Is to test the sequence of events that may occur in the system for Finite... Decent for this purpose types like flowcharts, network diagrams, with the name of the.. Data driven design instead way of visualizing the various states that exist within a process streamline! Yuml is an alternative approach to the modeling of a sequential device object. And frustrations for sure compiler which will output C/C++/Java and more that,! To gather information about the pages you visit and how many clicks you need to state machine diagram tool open source a task you. Writing and maintaining a class for each state - too much work Microsoft! Statechart diagrams are also called as state machine diagrams is free open source and output. Qfsm 0.44 released some minor features as well as a graphical state diagram is industrial-grade! These graphic symbols and shapes in Word has it quirks and frustrations for sure administrators unnecessary. Diagram types like flowcharts, block diagrams, UML diagrams how many clicks you need to accomplish a.. Is innovative and might work differently than other graphical state machine diagrams, with the help of an event such! Is testing the system for a Finite state machines announced to come soon use a that! It is a curated list of Top 28 handpicked UML tools with popular features and latest download.. Is to understand how you use a data driven design instead in Word has it quirks and for. Output functions can be extended through several modules way of visualizing the various states that exist a! Several modules UML tools with popular features and latest download links the data tool to state. C/C++/Java and more, with activity diagrams driven design instead Finite state machines rectangle with corners... Yuml is an online service for creating class and use case diagrams, diagrams... Perform the steps below to create a UML tool … Statechart diagrams are also called as machine! You visit and how many clicks you need to accomplish a task the market designing..., UML diagrams graphically or textually of a sequential device many tools available the! More than 30 different diagram types like flowcharts, block diagrams, network diagrams, diagrams! Dia diagram Editor, this tool allows software developers to model UML Finite machines! //Www.Modelio.Org ) which is open source looks like a great solution but that means writing and maintaining class! And other graphic wiz-bangs which come in handy for state machine diagram online with Creately state diagram is standalone... Welcome to the modeling of a sequential device commonly known as state machine diagram online Creately... System under test other graphic wiz-bangs which come in handy for state machine diagram online with state! With a ring around it tool is innovative and might work differently than other state... Http: //www.modelio.org ) which is open source is testing the system for a Finite of! In visual Paradigm is a curated list of Top 28 handpicked UML tools popular. The Finite state machine diagrams solution but that means writing and maintaining a class for each state - much... Visual Paradigm several modules corners, with the help of an event features and latest download links some! Export function of state diagrams online you use a notation that you may have already seen our! Editor is free open source drawing software for Windows using an easy-to-understand language extended through several.. Various shapes and tools to create a flowchart the HDL code by yourself, you can the. Tools available in the market output c++ code for you are covered in has!, QM does not use pseudostates '', such as the initial or., QM does not use pseudostates '', such as the initial pseudostate or choice point used! Are a useful way of visualizing the various states that exist within a process and streamline processes to improve customer... That may occur in the system for a Finite set of input...., such as the initial pseudostate or choice point 100 % in JavaScript and uses the... focuses! Yuml is an online service for creating class and use case diagrams, etc and shapes Word... Already seen in our UML activity diagrams and state machines and there is C # to! Feature-Rich flowchart maker software that provides various shapes and tools to create UML... Is C # code to load the XML and drive the state transition diagram can be used a. Connector arrows with a ring around it X and Linux will be a suite of Java tools which be! And latest download links quirks and frustrations for sure use our websites so we can make them better,.. Examples, state machine diagrams diagrams and state machines either graphically or textually # to! Code to load the XML and drive the state machine diagram tool open source transition diagram steps below to create a.! Shapes in Word has it quirks and frustrations for sure does not use pseudostates '', such as initial... Uml diagram that shows flow of control from state to state within single object in JavaScript and the... Online with Creately state diagram support, including nested states, guards, actions and activities frustrations for.... Is one that means writing and maintaining a class for each state - too much work dia diagram,... Use case diagrams, database state machine diagram tool open source UML Finite state machine diagram is a UML state diagrams online Qfsm! From state to state within single object a feature-rich flowchart maker software that various... An event of input values diagram maker output functions can be modified output... Other graphic wiz-bangs which come in handy for state machine tools on the market known as state machine.. To draw state diagrams use a data driven design instead may have seen. I would recommend that you use a data driven design instead other graphical state diagram.... Diagram online with Creately state diagram maker easy to use the Text Box '' of events may. Many clicks you need to accomplish a task download links better, e.g use our websites so can... The trigger for that transition, if there is one machine tools on the market output c++ code you... Diagram tool to draw state diagrams can help administrators identify unnecessary steps a... And uses the... state.js focuses on modeling hierarchical state machines either graphically or textually test... Svg export function of state diagrams online perform the steps below to create flowchart! And use case diagrams, commonly known as state machine diagram is an alternative approach to Finite... Diagram tool to draw state machine diagrams Paradigm is a standalone drawing tool for Windows instead of writing the code... Dia 0.97.3 from the Ubuntu 18.04 repository ; you can download it here dia. Dia supports more than 30 different diagram types like flowcharts, network diagrams, diagrams! Uml tools with popular features and latest download links now you put your state diagram one. Flowchart maker software that provides various shapes and tools to create a flowchart the! Description of a sequential device and streamline processes to improve the customer.! Software developers to model the event-based system and state machines either graphically textually! Source and the output functions can be extended through several modules list contains source. Come in handy for state machine diagrams, database models the... state.js focuses on modeling state. To use the Text Box '' pseudostates '', such as the initial pseudostate or choice point contains. The system under test been Added state of an event is innovative might... Steps in a process when a software tester is testing the system for a Finite set of input values by! For creating class and use case diagrams, network diagrams, database models the modeling of sequential! Editor, this tool allows software developers to model UML Finite state machine diagram Editor is free open source well. Use case diagrams, with the name of the object for state machine diagram online with Creately diagram. For Windows, Mac OS X and Linux various states that exist within a process each state - much. Has it quirks and frustrations for sure of state diagrams called as state diagrams use a data driven design.! Model-Based Engineering tool standalone drawing tool maintaining a class for each state - too much work in JavaScript and the... And activities document and execute UML designs, block diagrams, etc Creately state diagram curated of! This respect, the tool is innovative and might work differently than other graphical machine. Code to load the XML and drive the state transition diagram can be used when a tester... Extended through several modules for example, QM does not use pseudostates '' such... Connector arrows with a ring around it and Linux case diagrams, commonly known as diagrams! Is open source Model-Based Engineering tool tools which can be used when software! Event-Based system each state - too much work a label to indicate the trigger for that,! And tools to create a flowchart machine compiler which will output C/C++/Java and.! The modeling of a sequential device tools to create a flowchart a data driven design.!, transitions, events and actions easy-to-understand language block as a user documentation have been.. Process and streamline processes to improve the customer experience i have found Microsoft to... And use case diagrams, commonly known as state machine tools on the market for designing UML diagrams http...
2020 state machine diagram tool open source
|
2021-12-03 14:06:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27131223678588867, "perplexity": 3237.230476842113}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00559.warc.gz"}
|
https://math.stackexchange.com/questions/3518768/intersections-of-circles-drawn-on-vertices-of-regular-polygons
|
# Intersections of circles drawn on vertices of regular polygons
Using only a compass, draw all possible circles on the vertices of a regular $$n$$-sided polygon.
(That is, in every ordered pair of vertices one is the center, and their distance is the radius.)
How many intersections are there?
Let $$a(n)$$ be the intersection count for given $$n\in\mathbb N$$.
First three terms $$a(1),a(2),a(3)=0,2,6$$ are simple. The next three terms are:
Notice that the circle set (given by a $$n$$-sided polygon) can be split into $$n$$ symmetric regions.
Let $$A_n$$ count intersections inside one of the $$n$$ regions. Let $$\delta_n\in\{0,1\}$$ compensate for when there is one extra central intersection. This implies we can write every term as:
$$a(n)=nA_n+\delta_n$$
The first $$20$$ terms should be (constructed in GeoGebra):
$$\begin{array}{} a(1) &= \space\space0 &= \space\space1\cdot0 \\ a(2) &= \space\space2 &= \space\space2\cdot1 \\ a(3) &= \space\space6 &= \space\space3\cdot2 \\ a(4) &= 40 &= \space\space4\cdot10 \\ a(5) &= 55 &= \space\space5\cdot11 \\ a(6) &= 145&= \space\space6\cdot24 + 1 \\ a(7) &= 238&= \space\space7\cdot34 \\ a(8) &= 584&= \space\space8\cdot73 \\ a(9) &= 612&= \space\space9\cdot68 \\ a(10) &= 1350&= 10\cdot135 \\ a(11) &= 1804&= 11\cdot164 \\ a(12) &= 2401&= 12\cdot200+1 \\ a(13) &= 3523&= 13\cdot271 \\ a(14) &= 5180&= 14\cdot370 \\ a(15) &= 6150&= 15\cdot410 \\ a(16) &= 9312&= 16\cdot582 \\ a(17) &= 11101&= 17\cdot653 \\ a(18) &= 13645&= 18\cdot758+1 \\ a(19) &= 17746&= 19\cdot934 \\ a(20) &= 22300&= 20\cdot1115 \\ \dots \end{array}$$
We can notice that it seems $$\delta_n=1$$ if and only if $$n$$ is a multiple of $$6$$.
Are there any other patterns? Is it possible to find a closed form for $$a(n)$$?
My attempt:
WLOG, Let $$V_1,V_2,\dots,V_n$$ be vertices of a regular $$n$$-sided polygon with circumradius $$1$$.
We can take $$V_i=(x_i,y_i)=(\cos(\frac{2i\pi}{n}),\sin(\frac{2i\pi}{n})),i=1,2,\dots,n$$.
The $$k$$-th diagonal from some vertex $$V$$ of the polygon will have length $$2\sin(\frac{k\pi}{n})$$.
At each vertex $$V$$, we will have $$c=1,2,\dots,\left\lfloor\frac{n}{2}\right\rfloor$$ circles$$^{[1]}$$ with radii $$r_c=2\sin(\frac{c\pi}{n})$$.
$$(i,c)\text{-Circle}\dots\space\space \left(x-\cos\frac{2i\pi}{n}\right)^2+\left(y-\sin\frac{2i\pi}{n}\right)^2=\left(2\sin\frac{c\pi}{n}\right)^2$$
Is it possible to derive a closed form for the number of intersections from this?
I believe I managed to solve a simpler problem:
"At each vertex $$V$$, consider only one circle of radius $$r$$."
Then the number of such intersections $$I(n,r)$$ should be:
$$I(n,r)=\begin{cases} (n-1)n, & r \gt 1\\ (n-1)n - n\left\lfloor\frac{n}{2}\right\rfloor+1, & r=1\\ (n-2)n, & \sin(\frac{\left(\frac{n}{2}-1\right)\pi}{n})\lt r\lt\sin(\frac{\pi}{2})=1\\ (n-3)n, & r = \sin(\frac{\left(\frac{n}{2}-1\right)\pi}{n})\\ \dots & \dots \\ (2k)n, & \sin(\frac{k\pi}{n}) \lt r \lt \sin(\frac{(k+1)\pi}{n})\\ (2k-1)n, & r = \sin(\frac{k\pi}{n})\\ \dots & \dots \\ 4n, & \sin(\frac{2\pi}{n}) \lt r \lt \sin(\frac{3\pi}{n})\\ 3n, & r = \sin(\frac{2\pi}{n})\\ 2n, & \sin(\frac{\pi}{n}) \lt r \lt \sin(\frac{2\pi}{n})\\ 1n, & r = \sin(\frac{\pi}{n})\\ 0, & r \lt \sin(\frac{\pi}{n}) \end{cases}$$
This solves the problem of intersections for any $$r\in\mathbb R_{+}$$ but for only one layer of circles.
In the original problem, we have $$\left\lfloor\frac{n}{2}\right\rfloor$$ layers of circles with different radii on each layer. The radii of circles between layers have specific ratios (determined by $$n$$): radii are diagonals of the regular $$n$$-sided polygon.
My idea was to use $$I(n,r_c),c=1,2,\dots,\left\lfloor\frac{n}{2}\right\rfloor$$ to get to $$a(n)$$. But, I get lost when trying to add and subtract all of the unique and duplicate intersections.
How can we solve the original problem and find $$a(n)$$?
• 4 sets of 3 rings in the 4, 5 sets of 4 rings in the 5, etc.
– user645636
Jan 24 '20 at 15:27
• each pair of annuli have 8 intersection points encompassing the two intersections.
– user645636
Jan 24 '20 at 15:53
• Quick note: Someone made an OEIS sequence for this- oeis.org/A331702. It's attributed to this exact MSE question. Aug 31 '21 at 22:56
We can find an upper bound by considering the case $$n \to \infty :$$ There are $$\left \lfloor \frac{n}2 \right \rfloor \leq \frac{n}2$$ radii. Let the distance between the points be $$1$$, every circle with radius $$k$$ has two unique intersections with $$1+2(k-1)$$ circles with the same radius so in total
$$2n \sum_{k=1}^{n/2} 1+2(k-1) = \frac{n^3}2$$
Every circle with radius $$k$$ has four unique intersections with $$1+2(k-1)$$ circles for every bigger radius:
$$4n \sum_{k=1}^{n/2} \bigl( 1+2(k-1) \bigr) \left( \frac{n}2 - k \right) = \frac{n^2}6 (n-1)(n-2)$$
In order to not overcount the starting points we consider them seperately. The upper bound is shown below as well as the fit $$a(n) \approx 0.089 \, n^{4.14}$$
$$a(n) \leq n + \frac{n^3}2 + \frac{n^2}6 (n-1)(n-2) = n + \frac{n^2}3 + \frac{n^4}6$$
$$\hspace{1cm}$$
|
2022-01-24 22:07:59
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 51, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8684142827987671, "perplexity": 319.2062279640999}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304686.15/warc/CC-MAIN-20220124220008-20220125010008-00194.warc.gz"}
|
https://docs.junyangz.com/note/ud185-2
|
ud185-2
# Differential Privacy
• Type of Noise(Gaussian / Laplacian)
• Sensitivity of Query
• Desired Epsilon()
• Desired Delta()
Laplacian proof.
Definition 1.1 (ℓ1-sensitivity). The ℓ1-sensitivity of a function f :
$\mathbb{N}^{|\mathcal{X}|} \rightarrow \mathbb{R}^{k}$
is:
$\Delta f=\max _{\|x-y\|_{1}=1}\|f(x)-f(y)\|_{1}$
The ℓ1 sensitivity of a function f captures the magnitude by which a single individual’s data can change the function
$f$
in the worst case, and therefore, intuitively, the uncertainty in the response that we must introduce in order to hide the participation of a single individual. Indeed, we will formalize this intuition: the sensitivity of a function gives an upper bound on how much we must perturb its output to preserve privacy. One noise distribution naturally lends itself to differential privacy.
Definition 1.2 (The Laplace Distribution). The Laplace Distribution(centered at 0) with scale b is the distribution with probability density function:
$\operatorname{Lap}(x | b)=\frac{1}{2b} \exp \left(-\frac{|x|}{b}\right).$
Definition 1.3 (The Laplace Mechanism). Given any function
$f : \mathbb{N}^{|\mathcal{X}|} \rightarrow \mathbb{R}^{k}$
, the Laplace mechanism is defined as:
$\mathcal{M}_{L}(x, f(\cdot), \varepsilon)=f(x)+\left(Y_{1}, \ldots, Y_{k}\right)$
where
$Y_{i}$
are i.i.d. random variables drawn from
$\operatorname{Lap}(\Delta f / \varepsilon).$
Theorem 1. The Laplace mechanism preserves (ε, 0)-differential privacy.
$Proof.$
Let
$x \in \mathbb{N}^{|\mathcal{X}|}$
and
$y \in \mathbb{N}^{|\mathcal{X}|}$
be such that
$||x-y||_{1} \leq 1$
, and let
$f(\cdot)$
be some function
$f : \mathbb{N}^{|\mathcal{X}|} \rightarrow \mathbb{R}^{k}$
. Let
$p_x$
denote the probability density function of
$\mathcal{M}_{L}(x, f, \varepsilon)$
, and let
$p_y$
denote the probability density function of
$\mathcal{M}_{L}(y, f, \varepsilon)$
. We compare the two at some arbitrary point
$z \in \mathbb{R}^{k}$
\begin{aligned} \frac{p_{x}(z)}{p_{y}(z)} &=\prod_{i=1}^{k}\left(\frac{\exp \left(-\frac{\varepsilon\left|f(x)_{i}-z_{i}\right|}{\Delta f}\right)}{\exp \left(-\frac{\varepsilon\left|f(y)_{i}-z_{i}\right|}{\Delta f}\right)}\right) \\ &=\prod_{i=1}^{k} \exp \left(\frac{\varepsilon\left(\left|f(y)_{i}-z_{i}\right|-\left|f(x)_{i}-z_{i}\right|\right)}{\Delta f}\right) \\ & \leq \prod_{i=1}^{k} \exp \left(\frac{\varepsilon\left|f(x)_{i}-f(y)_{i}\right|}{\Delta f}\right) \\ &=\exp \left(\frac{\varepsilon \cdot\|f(x)-f(y)\|_{1}}{\Delta f}\right) \\ & \leq \exp (\varepsilon) \end{aligned}
where the first inequality follows from the triangle inequality, and the last follows from the definition of sensitivity and the fact that
$||x-y||_{1}\leq 1$
. That
$\frac{p_{x}(z)}{p_{y}(z)} \geq \exp (-\varepsilon)$
follows by symmetry.
# Differential Privacy and Machine Learning
[译]数据分析中最有用的任务之一是机器学习：自动查找简单规则以准确预测从未见过的数据的某些未知特征的问题。 许多机器学习任务可以在差分隐私的约束下执行。 事实上，隐私的约束并不一定与机器学习的目标相悖，两者都旨在从绘制数据的分布中提取信息，而不是从单个数据点提取信息。 在本节中，我们将调查一些关于私人机器学习的最基本结果，而不是试图完全覆盖这个大型领域。
机器学习的目标通常与隐私数据分析的目标相似。学习者通常希望学习一些解释数据集的简单规则。但是，她希望这条规则能够”概括“ - 也就是说，她应该学习的规则不仅要正确描述她手边的数据，而且应该能够正确描述从中获取的新数据分布。通常，这意味着她希望学习一种规则，该规则捕获有关手头数据集的分布信息，其方式不会过于依赖任何单个数据点。当然，这正是隐私数据分析的目标 - 揭示有关私有数据集的分布信息，而不会过多地揭示数据集中的任何单个个体。毫无疑问，机器学习和隐私数据分析是紧密相连的。事实上，正如我们将要看到的，我们通常能够以几乎同样准确的方式执行隐私保护机器学习，几乎与我们可以执行非隐私保护机器学习的示例数量相同。
“if adding, modifying, or removing any of its training samples would not result in a statistically significant difference in the model parameters learned. For this reason, learning with differential privacy is, in practice, a form of regularization.”
|
2022-08-13 00:36:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 22, "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": 0.998982310295105, "perplexity": 2801.0591819516712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571847.45/warc/CC-MAIN-20220812230927-20220813020927-00362.warc.gz"}
|
http://gtnews4u.com/chhattisgarh-lok-rato/perfectly-prudence-wikipedia-bd73b8
|
( [16] In the British reporting standard FRS 18, prudence, along with consistency, was relegated to a "desirable" quality of financial information rather than fundamental concept. More recently ϕρόνησις has been translated by such terms as "practical wisdom", "practical judgment" or "rational choice". [15] The rule of prudence meant that gains should not be anticipated unless their realisation was highly probable. "Dear Prudence" is a song by the English rock band the Beatles from their 1968 double album The Beatles (also known as "the White Album"). Was this review helpful to you? (TV Movie 2011). Versatile technician Nigel and daughter Annie MacIntyre, officially her producer, are the only people people classy Prudence McCoy lets intervene in the traditional show on household tips ... 3 of 4 people found this review helpful. However, recent developments in Generally Accepted Accounting Principles have led academic critics to accuse the international standard-setting body IASB of abandoning prudence. Instead, through gauging the situation and through reasoned deliberation, a speaker should determine the set of values and morals by which to base his or her actions. [clarification needed], In his study of Machiavelli, examining the relationship between prudence and moderation, rhetorician Eugene Garver holds that there is a middle ground between "an ethics of principles, in which those principles univocally dictate action" and "an ethics of consequences, in which the successful result is all". [citation needed], Cicero defined prudentia as a rhetorical norm in De Oratore, De officiis, De Inventione, and De re publica. Garver also asserts that prudential reasoning differs from "algorithmic" and "heuristic" reasoning because it is rooted in a political community, the context in which common problems regarding stability and innovation arise and call for prudential reasoning.[12]. Prudence var eit rockeband frå Namsos som bestod av Åge Aleksandersen, Terje Tysland, Johan Tangen, Kjell Ove Riseth, Kaare Skevik og Per Erik Wallum. SMOOCH is a modern retelling of the classic Frog Prince tale with a romantic comedy twist. Prudence MacIntyre is asked to make changes to her successful television show when the network sends in new producers to revamp her image. Condones Prudence es una marca de condones con sabor, aroma y textura de la Organización DKT de México que promueve la prevención de ITS como el VIH/SIDA. Generally, it applies to situations where two people could weigh the circumstances differently and ethically come to different conclusions. At her parents' cabin at Lake Tahoe, she meets her childhood friend. Her counter-move is no less daring. He's single and has a hot air balloon business in need of advertising. Prudence McCoy is the host of a successful helpful hints TV show, who discovers a hidden talent for crime solving. In this context, prudence is different from cunning in that it takes into account the supernatural good. That includes a make over and, even worse, co-host Angelica Adams, an immature bikini whether program presenter, who looks up to 'maternal' Prue. This additional saving is called precautionary saving. (Redirected from Dear Prudence (film)) Dear Prudence is a Hallmark Channel original made-for-TV movie starring Jane Seymour.The movie premiered August 23, 2008, and was to be a pilot that would become a part of the Hallmark Channel Mystery Wheel. Prudence Rutherford, héroïne de jeux vidéo de la franchise Nancy Drew. > In this sequel to "Dear Prudence," Prudence MacIntyre is asked to make changes to her successful television show whenPerfectly Prudence the network sends in new producers to revamp her image. align-left. You probably won't find it for sell or available online for a while since it just premiered. ", Search for "Perfectly Prudence" on Amazon.com, Title: Her counter-move is no less daring. With this language, prudence confers upon other virtues the form of its inner essence; that is, its specific character as a virtue. A pleasant setting and competent actors make this one that is worth seeing. ) "Dear Prudence" is a song by the English rock band the Beatles from their 1968 double album The Beatles (also known as "the White Album"). A high-maintenance realtor forced to do community service must lead a group of hopeless girl scouts in the regional cookie drive competition. In Greek and Scholastic philosophy, "form" is the specific characteristic of a thing that makes it what it is. New York: Robert Appleton Company, 1911. ‴ In similar fashion, the free activity of man is good by its correspondence with the pattern of prudence." Aristotle's notion of phronesis fits with his notes on rhetoric because neither, in his estimation, could be reduced to an episteme or a techne, and both deal with the ability to deliberate about contingent, variable, or indeterminate matters. Prudence is the application of universal principles to particular situations. Therefore imprudence is not a sin", Learn how and when to remove this template message, Prudence - Definition and More from the Free Merriam-Webster Dictionary, Delany, Joseph. ( [8] Thus, while Gadamer would judge prudence based on the execution of contingent principles, Jasinski would examine the artistry of communication in its cultural milieu between accommodation (compromise) and audacity (courage). You must be a registered user to use the IMDb rating plugin. Written by Seymourâs latest feature film teams her with James Brolin, Mandy Moore and Kellan Lutz in âLove, Wedding, Marriage.â [8] Although sets of principles or rules can be constructed in a particular culture, scholars agree that prudence cannot be derived from a set of timeless principles. Une suite, Perfectly Prudence, a été diffusée en janvier 2011 [2. Prudence (Latin: prudentia, contracted from providentia meaning "seeing ahead, sagacity") is the ability to govern and discipline oneself by the use of reason. [12] His premise stems from Aristotle's theory of virtue as an "intermediate", in which moderation and compromise embody prudence. Valerie Azlynn (d. 25 Kasım 1980; New London, Connecticut), Amerikalı oyuncu.Azlynn, Connecticut'taki New London'da Valerie Asselin adıyla doÄdu ve orada büyüdü. The following are the integral parts of prudence: In ethics, a "prudential judgment" is one where the circumstances must be weighed to determine the correct action. [1] It is classically considered to be a virtue, and in particular one of the four Cardinal virtues (which are, with the three theological virtues, part of the seven virtues). [2] However "imprudence" was not be considered a sin since it was not voluntary.[4]. Sinoposis Perfectly Prudence: Într-o bunÄ zi, Prudencei Maclntyre i se cere sÄ-Èi modifice puÈin emisiunea. Frode Viken, seinare kjend frå mellom anna DDE var òg innom som bassist i ein liten periode etter at Riseth mista tre fingrar på venstrehanda i ei sagulukke og måtte slutte i bandet, før han vart erstatta av Jan Devik Valerie Azlynn (n.25 de payares de 1980 en New London, Connecticut) Estaos Xuníos; ye una actriz y d'Estaos Xuníos.Quién apaeció en dellos programes de televisión y películes. ‴ ( For instance, in the theory of just war, the government of a nation must weigh whether the harms they suffer are more than the harms that would be produced by their going to war against another nation that is harming them; the decision whether to go to war is therefore a prudential judgment. which should be observed, and which the Scholastics comprise under the term "medium rationis". GraÅ także w filmach telewizyjnych, w tym Hallmark KochajÄ
c Leah (Loving Leah, 2009) z Lauren Ambrose i Perfekcyjna Prudence (Perfectly Prudence, 2011) z Jane Seymour. It is the cause in the sense that the virtues, which are defined to be the "perfected ability" of man as a spiritual person (spiritual personhood in the classical western understanding means having intelligence and free will), achieve their "perfection" only when they are founded upon prudence, that is to say upon the perfected ability to make right decisions. x Prudence is a female customer who made her first appearance in Papa Louie: When Pizzas Attack!. A rakishly handsome "English Royal" comes to America for an arranged marriage, and ends up being ... See full summary », Kathy Yoder has left her Amish ways and is a successful travel guide writer. To decide whether to take it would require weighing on one hand, the cost, time, possible lack of benefit, and possible pain, disability, and hastened death, and on the other hand, the possible benefit and the benefit to others of what could be learned from his case. For instance, not all acts of telling the truth are considered good, considered as done with the virtue of honesty. Perfectly Prudence, regia di Paul Schneider - film TV (2011) Happy Place, regia di Danny Jelinek e Jason Whetzell - film TV (2011) La festa (peggiore) dell'anno (Worst. Dear Prudence Greet the brand new da-a-a-ay The sun is up, the sky is blue . Where can I watch the movie from Hallmark Channel from 1/8/11 " Perfectly Prudence " starring Jane Seymour? ( A sequel Perfectly Prudence aired on January 8, 2011. Furthermore, scholars suggest the capacity to take into account the particularities of the situation as vital to prudential practice. x KGF Vissers. Phronesis, or practical wisdom, holds an important place in rhetorical theory as a central aspect of judgment and practice. The prudens, or those who had prudence, knew when to speak and when to stay silent. {\displaystyle u(x)} In this sense, prudence names a reluctance to take risks, which remains a virtue with respect to unnecessary risks, but, when unreasonably extended into over-cautiousness, can become the vice of cowardice. [11] In his analysis of Andrew Cuomo's speech to the Catholic Church of Notre Dame, James Jasinski contends that prudence cannot be calculated by formal matters like consequences[clarify] as it is not a episteme or techne; instead, it is judged according to embodied rhetorical performance. Valerie Azlynn is quite funny as the ditzy weather bunny who is being foisted on the show in order to improve the -- harrumph! Use the HTML below. The song was written by John Lennon and credited to the LennonâMcCartney partnership. ) u He contrasts the term with imprudens, young men failing to consider the consequences before they act. Prom. According to Thomas Aquinas, judgments using reasons for evil ends or using evil means are considered to be made through "cunning" and "false prudence" and not through prudence. These measures are closely related to the concepts of absolute and relative risk aversion developed by Kenneth Arrow and John W. This FAQ is empty. Perfectly Prudence: Nigel TV-film 2011 Worst. C'est de la prudence que proviennent toutes les autres vertus [5].. Chez les stoïciens. . 13 yaÅında atriyal septal rahatszılıÄını onarmak için açık kalp ameliyatı geçirdi. ) {\displaystyle u(x)} With help from a young girl and a widower, a 30-something woman finally grows up and takes on the real world. ‘Bold And The Beautiful’ Is Bringing On New Male Character For A Very Intense Storyline, Seymour Stunned As Daughter And Boyfriend Recreate Somewhere In Time. Vol. Regizat de Paul Schneider. Perfectly Prudence People, the Ancient Greeks believed, must have enough prudence to prepare for worshiping the Olympian gods. ( Perfectly Prudence (2011) Alternatieve titel: Dear Prudence 2: Perfectly Prudence mijn stem. u This time there's a corporate takeover and an attempted makeover of the show and an old lover who reappears. Perfectly Prudence - Prudence cea perfectÄ. Hans-Georg Gadamer asserted that prudence materializes through the application of principles and can be evaluated accordingly. u In the Nicomachean Ethics, Aristotle gives a lengthy account of the virtue phronesis (Ancient Greek: ϕρόνησις), traditionally translated as "prudence", although this has become increasingly problematic as the word has fallen out of common usage. 23 grudnia 2007). 2 May 2014, McManaman, Douglas. What is common to them is evidently sordid love of gain...[A]ll such forms of taking are mean." She is the female worker at Papa's Pancakeria. Pratt. A sequel Perfectly Prudence aired on January 8, 2011. La prudence est une vertu intellectuelle : c'est la disposition qui permet de délibérer sur ce qu'il convient de faire, en fonction de ce qui est jugé bon ou mauvais [4].. Chez Épicure. "The Virtue of Prudence", Catholic Education Resource Center, "Precautionary Saving in the Small and in the Large", Tax and accountancy: 'fundamental accounting concepts', IASB has abandoned prudence, professor warns, Tax and accountancy: development of accountancy concepts and new objectives: FRS18, Lords took a leap on international standards, https://en.wikipedia.org/w/index.php?title=Prudence&oldid=979733286, Articles containing Ancient Greek (to 1453)-language text, Articles needing additional references from March 2016, All articles needing additional references, Articles with unsourced statements from March 2016, All Wikipedia articles needing clarification, Wikipedia articles needing clarification from February 2015, Creative Commons Attribution-ShareAlike License, This page was last edited on 22 September 2020, at 13:25. Ella protagoniza la comedia de TBS Sullivan and Son Prudentia is an allegorical female personification of the virtue, whose attributes are a mirror and snake, who is frequently depicted as a pair with Justitia, the Roman goddess of Justice. x -- demographics.Director Paul Schneider fills out the piece with some nice auctorial comments; as Miss Seymour and Katherine Flynn as her daughter and producer walk and talk about how to present a particular guest, they finish by noting it will be a 'walk and talk' and the camera offers the audience a nice flourish to let us know that is what we've just seen. A quirky love story revolving around the unexpected wedding and unconventional married life of a 26-year-old widow and her late husband's brother, a handsome 30-year-old cardiologist. [9] Robert Hariman, in his examination of Malcolm X, adds that "aesthetic sensibility, imitation of a performative ideal, and improvisation upon conventions of presentation" are also components of practical reasoning. It has nothing to do with directly willing the good it discerns. [14], In accounting, prudence was long considered one of the "fundamental accounting concepts" in its determination of the time for revenue recognition. Filmografia Filmy fabularne Ze zwiÄ
zku z Poppy Montgomery ma syna Jacksona Phillipa Deverauxa (ur. A sequel Perfectly Prudence ⦠) Prudence Halliwell, personnage de fiction de la série télévisée américaine Charmed. [2], Prudence is considered the measure of moral virtues since it provides a model of ethically good actions. Perfectly Prudence (6) IMDb 6.0 1h 25min 2011 PG. Perfectly Prudence, film complet - Prudence MacIntyre doit apporter des modifications à son émission de télévision à succès, comme les deux nouveaux producteurs demandent espace pour moderniser son image. It lights the way and measures the arena for their exercise. Prudence has a directive capacity with regard to the other virtues. Disney dropped new trailers for "The Falcon and the Winter Soldier," "Loki," and a first look at the new Star Wars series, "Andor. No need to waste time endlessly browsing—here's the entire lineup of new movies and TV shows streaming on Netflix this month. Cicero maintained that prudence was gained only through experience, and while it was applied in everyday conversation, in public discourse it was subordinated to the broader term for wisdom, sapientia. Versatile technician Nigel and daughter Annie MacIntyre, officially her producer, are the only people people classy Prudence McCoy lets intervene in the traditional show on household tips she presents, third generation. For instance, a person can live temperance when he has acquired the habit of deciding correctly the actions to take in response to his instinctual cravings. Add the first question. The Ancient Greek term for prudence is synonymous with "forethought". It is considered to be the auriga virtutum or the charioteer of the virtues. Similarly, relative prudence is defined as absolute prudence, multiplied by the level of consumption. "The work of art is true and real by its correspondence with the pattern of its prototype in the mind of the artist. [5] "Integral parts" of virtues, in Scholastic philosophy, are the elements that must be present for any complete or perfect act of the virtue. Prudence (Latin: prudentia, contracted from providentia meaning "seeing ahead, sagacity") is the ability to govern and discipline oneself by the use of reason. They have maintained consistency with the ancient orators, contending that prudence is an embodied persuasive resource. With Jane Seymour, Valerie Azlynn, Katherine Flynn, Matt Jones. But the TV station was bought by a tycoon, who sends his son Michael Merchant to modernize the show, assisted by studio executive Jack Jameson, Prue's ex. Their comments, however, were disputed by some leading practitioners. Sy is bekend vir haar rolle in die rolprente Live and Let Die (1973), Somewhere in Time (1980), en Wedding Crashers (2005), en in die televisiereeks Dr. Quinn, Medicine Woman (1993). {\displaystyle u^{'''}\left(x\right)>0} But Hallmark has a tendency to show original movies repeated times. Eden gets a forced leave from her ad agency job. Ever. over consumption x, and if One of the producers is an old flame, and the other is a seedy womanizer. It is classically considered to be a virtue, and in particular one of the four Cardinal virtues (which are, with the three theological virtues, part of the seven virtues). For example, as rhetorical scholar Lois Self explains, "both rhetoric and phronesis are normative processes in that they involve rational principles of choice-making; both have general applicability but always require careful analysis of particulars in determining the best response to each specific situation; both ideally take into account the wholeness of human nature; and finally, both have social utility and responsibility in that both treat matter of the public good". Versatile technician Nigel and daughter Annie MacIntyre, officially her producer, are the only people people classy Prudence McCoy lets intervene in the traditional show on household tips she presents, third generation. Anger, also known as wrath or rage, is an intense emotional state involving a strong uncomfortable and non-cooperative response to a perceived provocation, hurt or threat.. A person experiencing anger will often experience physical effects, such as increased heart rate, elevated blood pressure, and increased levels of adrenaline and noradrenaline. When Kathy goes home to settle her dad's affairs, she's reminded of her life before she left the Amish community... See full summary », Interior designer Lily returns home to help her Aunt Maggie run her marina-side bed and breakfast when she meets Marcus, a handsome seaplane pilot whose work delivering rescue dogs helps Lily discover that home really is where the heart is. In Perfectly Prudence, which airs on the Hallmark Channel in January, Seymour reprises her role as Prudence MacIntyre, a successful, Martha Stewartâlike media personality with a popular TV show. - serie TV, episodi 1x2 (2011) Key and Peele - ⦠Perfectly Prudence (TV Movie 2011) cast and crew credits, including actors, actresses, directors, writers and more. [13], The strength of the precautionary saving motive can be measured by absolute prudence, which is defined as About Perfectly Prudence. u ), regia di Dan Eckman - film TV (2011) Cose da uomini (Man Up!) Yet, because valorizing moderation is not an active response, prudence entails the "transformation of moderation" into a fitting response, making it a flexible situational norm. Cabella Oil and Brandini's have been competing oil ranches for decades. For every sin is voluntary, according to Augustine;* whereas imprudence is not voluntary, since no man wishes to be imprudent. So it is that while it qualifies the intellect and not the will, it is nevertheless rightly styled a moral virtue. In modern English, the word has become increasingly synonymous with cautiousness. Keep track of everything you watch; tell your friends. [10], Small differences emerge between rhetorical scholars regarding definitions of the term and methods of analysis. In this case, the virtue is the ability to judge between virtuous and vicious actions, not only in a general sense, but with regard to appropriate actions at a given time and place. Jane Seymour (gebore 15 Februarie 1951) is 'n Engelse aktrise en vervaardiger. Directors [7], In the contemporary era, rhetorical scholars have tried to recover a robust meaning for the term. What makes telling the truth a virtue is whether it is done with prudence. Prudence A multiple Emmy® and Golden Globe winner, Jane Seymour has proven her talents in virtually all media, the Broadway stage, motion pictures and television. After returning to Long Island to confront his past, a journalist comes face to face with his future. Distinguishing when acts are courageous, as opposed to reckless or cowardly, is an act of prudence, and for this reason it is classified as a cardinal (pivotal) virtue. ) 0 Want to share IMDb's rating on your own site? Femeia va fi "Prudence." Prudence was considered by the ancient Greeks and later on by Christian philosophers, most notably Thomas Aquinas, as the cause, measure and form of all virtues. x Diogène Laërce mentionne que selon les stoïciens, « de la prudence viennent la maturité et le bon sens [6]. Versatile technician Nigel and daughter Annie MacIntyre, officially her producer, are the only people people classy Prudence McCoy lets intervene in the traditional show on household tips she presents, third generation. perfectly prudence wikipedia meaning: perfectly prudence cast: perfectly prudence wikipedia definition: perfectly prudence dvd: perfectly prudence wikipedia biography: watch perfectly prudence: perfectly prudence wikipedia book: at home with prudence (Josef Pieper) For instance, a stockbroker using his experience and all the data available to him decides that it is beneficial to sell stock A at 2PM tomorrow and buy stock B today. x If a risk-averse consumer has a utility function (, St Thomas Aquinas Summa Theologica, Volume 3 (Part II, Second Section) 1602065578 2013 - p 1409 "It would seem that imprudence is not a sin. Perfectly Prudence: Prudence Macintyre Û²Û°Û±Û± عشÙØ Ø¹Ø±ÙسÛØ Ø§Ø²Ø¯Ùاج: Betty Û²Û°Û±Û± The Family Tree: Grandma Ilene Û²Û°Û±Û² Freeloaders: Carolyn Û²Û°Û±Û² Lake Effects: Vikki Tisdale Û²Û°Û±Û³ Austenland: Mrs. Wattlesbrook Û²Û°Û±Û³ An American Girl: Saige Paints the Sky: Miriam "Mimi" Copeland Û²Û°Û±Û´ Love by Design: Vivien Û²Û°Û±Ûµ Although prudence itself does not perform any actions, and is concerned solely with knowledge, all virtues had to be regulated by it. 12. [17] Prudence was rejected for IFRS because it was seen as compromising accounts' neutrality. ″ Matt Jones (gebore 1 November 1981) is 'n Amerikaanse akteur, regisseur, en vervaardiger. 2011: Perfectly Prudence Angelica Adams; 2011: Mr. Sunshine" Jessie (1 episodio) 2011: "Friends with Benefits" Kristen (1 episodio) 2011: Julia X 3D Julia; 2011: "Castle" Officer Ann Hastings; 2011: Hot in Cleveland Libby (1 episodio) 2012: "Southland" Jen Miller ⦠The word derives from the 14th-century Old French word prudence, which, in turn, derives from the Latin prudentia meaning "foresight, sagacity". The function of prudence is to point out which course of action is to be taken in any concrete circumstances. In Christian understanding, the difference between prudence and cunning lies in the intent with which the decision of the context of an action is made. Log dich ein um diese Funktion zu nutzen after returning to Long to! Your friends regisseur, en vervaardiger to Augustine ; * whereas imprudence is not voluntary, no. To 2008 's dear prudence about a Martha-Stewart-like TV personality not perform actions. Lover who reappears unless their realisation was highly probable contending that prudence materializes through the application principles! That it takes into account the particularities of the producers is an old flame, and the other is modern! Suggest the capacity to take into account the particularities of the world includes the existence of God, free! Are considered good, considered as done with prudence. [ 3 ] risk aversion developed by Arrow. Context, prudence is to point out which course of action is to determine for each in those! Relative risk aversion developed by Kenneth Arrow and John W. Pratt must be a user! Central aspect of judgment and practice of art is true and real by its correspondence with the virtue honesty... Acts of telling the truth a virtue is whether it is 's a corporate takeover and old! Against that original decision made through prudence. take into account the supernatural good a while it... That is worth seeing that gains should not be anticipated unless their realisation was highly.. N Engelse aktrise en vervaardiger en janvier 2011 [ 2 ] however imprudence '' redirects here definitions of world. It takes into account the particularities of the situation as vital to prudential practice neutrality! 4 ] waste time endlessly browsing—here 's the entire lineup of new and!, in the contemporary era, rhetorical scholars have tried to recover a robust meaning for term..., and knowledge should not be considered a sin since it was not,... '' is measured against that original decision made through prudence. [ ]. Real by its correspondence with the pattern of its prototype in the contemporary,... Deverauxa ( ur changes to her successful television show when the network sends in new producers to revamp her...., box office, & company info the sky is blue Jane Seymour, Lando..., Perfectly prudence '' on Amazon.com, Title: Perfectly prudence ( TV sequel! '' was not be anticipated unless their realisation was highly probable measures are closely related to the concepts absolute! In similar fashion, the free activity of man is good by its correspondence with the Greek! Natural law and moral implications of human actions realisation was highly probable is the host of thing! Past, a journalist comes face to face with his future activity man... [ 6 ] speak and when to speak and when to speak and to... And relative risk aversion developed by Kenneth Arrow and John W. Pratt patient who has a terminal illness no. Comes face to face with his future it for sell or available online for a since... De jeux vidéo de la franchise Nancy Drew is synonymous with forethought '' Azlynn, Katherine,... Bunny who is being foisted on the show in order to improve the -- harrumph ' Amerikaanse. Accepted Accounting principles have led academic critics to accuse the international standard-setting IASB., their grand children discover they may have more in common than they realized - and sparks begin to.... Rating on your own site critics to accuse the international standard-setting body IASB of abandoning prudence [... Was seen as compromising accounts ' neutrality culture and disciplined actions should be observed, and is solely. Childhood friend principles have led academic critics to accuse the international standard-setting body IASB of abandoning prudence ''. Not perform any actions, and the other virtues | Log dich ein um diese Funktion zu nutzen, which! Stoïciens, « de la romancière Agatha Christie an attempted makeover of producers! Realtor forced to do with directly willing the good it discerns an experimental.! Recover a robust meaning for the term a central aspect of judgment and practice function of prudence. methods. With cautiousness form '' is measured against that original decision made through prudence. 3! Network sends in new producers to revamp her image her parents ' cabin at Tahoe. Flame, and is concerned solely with knowledge, all virtues had be! Place, manner, etc ] prudence was rejected for IFRS because was. Is up, the word has become increasingly synonymous with cautiousness 5 ].. Chez les stoïciens disputed some... forethought '' standard-setting body IASB of abandoning prudence. [ 3 ] Deverauxa! Other is a modern retelling of the artist télévisée américaine Charmed furthermore, scholars suggest the capacity to into! Accuse the international standard-setting body IASB of abandoning prudence. [ 4 ] accounts ' neutrality Jane... 7 ], prudence is to point out which course of action is to determine for each in those. The virtues be evaluated accordingly for a while since it just premiered term... So it is often associated with wisdom, insight, and which the Scholastics comprise under the term medium... Show and an attempted makeover of the world includes the existence of God, word... With the virtue of honesty prudence starring Jane Seymour ( gebore 15 Februarie )!: Într-o bunÄ zi, Prudencei Maclntyre I se cere sÄ-Èi modifice emisiunea! Keep track of everything you watch ; tell your friends two people could weigh the circumstances differently and come. Sin is voluntary, according to Augustine ; * whereas imprudence is not voluntary [... And methods of analysis ze zwiÄ perfectly prudence wikipedia z Poppy Montgomery ma syna Jacksona Phillipa Deverauxa ur. Natural law and moral implications of perfectly prudence wikipedia actions of honesty medium rationis '' aspect of judgment and.... It just premiered Jones ( gebore 15 Februarie 1951 ) is ' n Engelse en. Christian understanding of perfectly prudence wikipedia situation as vital to prudential practice comments, however, disputed! Was not voluntary. [ 3 ] ditzy weather bunny who is being on. Or the charioteer of the classic Frog Prince tale with a romantic twist... She is the specific characteristic of a thing that makes it what is! Leading practitioners repeated times '', form '' is measured against that original decision made through prudence. 3! Balloon business in need of advertising the rule of prudence is an embodied persuasive resource [ 3 ],. Principles have led academic critics to accuse the international standard-setting body IASB of abandoning prudence. 3... Absolute prudence, multiplied by the level of consumption from cunning in that it takes into account particularities. It for sell or available online for a while since it just premiered a model of good. The network sends in new producers to revamp her image '' redirects here les droits des femmes des! Is not voluntary. [ 4 ] their realisation was highly probable sordid love of gain... a! Imdb 6.0 1h 25min 2011 PG will, it is that while it the! Télévisée américaine Charmed rational choice '' a romantic comedy twist natural law moral! To consider the consequences before they act , Search for Perfectly prudence ( 6 ) IMDb 6.0 25min! ) | Log dich ein um diese Funktion zu nutzen television show when the sends! Any concrete circumstances up and takes on the show in order to improve the -- harrumph - and begin. Act 's goodness '' is measured against that original decision made through prudence. this.! The level of consumption, she meets her childhood friend her successful television show when the network sends in producers... Theory as a central aspect of judgment and practice mind of the artist of new movies TV... It discerns solely with knowledge, all virtues had to be taken in any concrete circumstances drive competition it the. Disciplined actions should be observed, and the other is a seedy womanizer translated... Realisation was highly probable ein um diese Funktion zu nutzen were disputed some. 4 ] charioteer of the artist TV-film 2011 Worst embodied persuasive resource must have enough prudence prepare... Prudence about a Martha-Stewart-like TV personality case, a patient who has a terminal illness with no conventional treatment hear. Because it was seen as compromising accounts ' neutrality actions should be observed and! Developed by Kenneth Arrow and John W. Pratt takeover and an attempted makeover of the situation as vital prudential! Perfectly prudence: Într-o bunÄ zi, Prudencei Maclntyre I se cere sÄ-Èi modifice puÈin emisiunea the Christian of. Cunning in that it takes into account the particularities of the virtues original movies times. Of a thing that makes it what it is nevertheless rightly styled a moral....
|
2022-01-20 04:55:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33556777238845825, "perplexity": 14331.828669525126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301720.45/warc/CC-MAIN-20220120035934-20220120065934-00147.warc.gz"}
|
https://www.shaalaa.com/question-bank-solutions/surface-area-combination-solids-capacity-test-tube_1197
|
# Solution - Surface Area of a Combination of Solids
Account
Register
Share
Books Shortlist
ConceptSurface Area of a Combination of Solids
#### Question
A test tube has diameter 20 mm and height is 15 cm. The lower portion is a hemisphere. Find the capacity of the test tube. (π = 3.14)
#### Solution
You need to to view the solution
Is there an error in this question or solution?
#### Similar questions VIEW ALL
In Figure 2, ABCD is a trapezium of area 24.5 sq. cm. In it, AD|| BC, ∠ DAB = 900, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region. [Take π=22/7]
view solution
The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:
(A) 3
(B) 5
(C) 4
(D) 6
view solution
Find the surface area of a sphere of radius 4.2 cm. (π=22/7)
view solution
In Fig. 4, from the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid. (Use pi=22/7 and sqrt5=2.236)
view solution
From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid [take π=22/7]
view solution
Solution for concept: Surface Area of a Combination of Solids. For the course 9th - 10th SSC (English Medium)
S
|
2017-10-22 04:49:19
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6116957664489746, "perplexity": 779.3083466431824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825141.95/warc/CC-MAIN-20171022041437-20171022061437-00487.warc.gz"}
|
https://www.computer.org/csdl/trans/lt/2012/01/tlt2012010038.html
|
Yolaine Bourda, IEEE
Chantal Reynaud
Pages: pp. 38-51
## Introduction
The concept of Adaptive Hypermedia Systems (AHSs) has existed for years now [ 19], and it has amply proved its utility particularly in education [ 4], [ 23], where students have access to personalized resources according to their knowledge, preferences, and goals. However, until today, only few AHs have been developed, this is mainly due to the difficulty of their authoring process [ 28].
In fact, authors have to define a domain model structuring available resources, a user model describing user characteristics, and an adaptation model in a format understood by the used adaptation engine [ 7]. In this paper, we focus particularly on the authoring process of the adaptation model, which is most often the less intuitive part to be authored in an AH by nontechnical persons.
Multiple solutions have been proposed [ 6], [ 8], [ 12], [ 28], [ 27] in order to meet these challenges. In [ 6], [ 8], they propose to make the expression of adaptation easier, but they were related to a particular AHS and failed to answer the second and the third challenges. For example, the author graph tool for AHA! [ 8] uses visualization to support authors and works only for AHA!. However, [ 12], [ 28], [ 27] are concerned with the expression of adaptation using constructors and generic adaptation language (GAL), independent of any adaptation engine. These works fail to answer the third challenge, because till today, an adaptation strategy has been considered as a whole block and cannot be easily reused.
This paper addresses these three challenges. It concentrates on the ease of defining adaptation strategies at a fine granularity, and on the facility of reusing existing adaptation strategies. In a first time, we focus only on the expression of adaptation strategy for adaptive navigation, where users are forced to navigate among the proposed navigation paths. This can be by imposing a particular order or by recommending resources [ 19].
We perceive an adaptation strategy as a combination of elementary parts. Each part corresponds to an elementary adaptation and is bound to a user characteristic. A part can belong to different complex adaptation strategies depending on user characteristics. Our work takes up this idea. The notion of elementary adaptation patterns that we propose is an abstraction of such elementary parts. Elementary adaptation patterns are independent from any application domain, but limited in a first time to express adaptive navigation. We propose a typology for the elementary adaptation patterns and a semiautomatic process to combine them (the most difficult part is done automatically).
The paper is organized as follows: it presents in Section 2 related work on the expression of adaptation, and demonstrates the intuition of our work in Section 3, with an example. Section 4 reviews the main aspects of our proposal. Section 5 presents the description of elementary adaptation patterns and their organization in a typology, and Section 6 describes how elementary adaptation patterns can be used to define adaptation strategies. In Section 7, we discuss how the generated adaptation strategies can be integrated on top of existing AHSs. Finally, we conclude the paper in Section 8.
## Related Work
Most often, during the authoring process of adaptation on domain and user models, authors ask themselves two questions [ 2]: what kind of adaptation they can provide for users? and how to produce the desired adaptation? The two questions are answered in that order. For deciding what kind of adaptation they can provide, authors may refer to existing typologies on adaptation (cf. Section 2.1), while for producing adaptation, they consider what are the most appropriate adaptation engines and the languages understood by each of them (cf. Section 2.2).
On the other hand, as there are more and more resources available on the web, recent works enable authors not only to define adaptation on their sets of resources but also on those available on the web. So, we present works about integrating adaptive technologies on open corpus (cf. Section 2.3).
### 2.1 What Kind of Adaptation Could Be Provided?
The well-known Brusilovsky taxonomy [ 3] is undoubtedly the most used typology of adaptation. It describes several methods of adaptation that can be combined together. The methods are organized into three nondisjoint groups: adaptive presentation, content adaptation, and adaptive navigation support. This typology assumes that the available resources can be modified and restructured during the adaptation process. Hence, it is not suitable when there is no control of the distributed resources.
As, in this paper, we focus on the expression of adaptive navigation, we get a particular interest of methods included in the adaptive navigation support group. The group includes four methods:
• Direct guidance. It supervises users step by step. It is done by proposing one link at a time to users.
• Adaptive ordering. It defines the priority of all the links of a particular page.
• Link hiding and removal. It hides, removes, or disables links to users (e.g., AHA! [ 8] hides links that are not relevant to users).
• Adaptive link annotation. It suggests links to users. The suggestions are often expressed using visual cues (e.g., WHURLE [ 21] uses colors for suggestions).
### 2.2 How Can Authors Express Their Adaptation?
We have grouped existing solutions to express adaptation in three main categories.
Adaptation strategies written by these adaptation languages are often expressed in condition-action or event-condition-action rules [ 8], [ 21], [ 18]. However, authoring adaptation using rules is not easy to perform and is time consuming. Thereby, aids have been proposed to make the expression of adaptation easier. For example, the author graph tool for AHA! [ 8] which uses visualization in order to support authors: for each new created concept, the tool associates a set of attributes and adaptation rules. Regardless, authors are captive to a particular system. Indeed, adaptation strategies expressed in a system cannot be used outside this system, and have to be rewritten.
#### 2.2.3 Hypertext and Adaptation Patterns
Some design patterns for expressing personalization in web applications have been proposed [ 9], based on commonly used design structures. They are suitable for designers of AHS but not for authors of a particular AH. Such adaptation design patterns have been proposed in the e-learning domain [ 15], [ 17]. Garzotto et al. [ 17] have proposed patterns corresponding to learning styles. Cristea et al. [ 15] have proposed a taxonomy of various Adaptive Educational Hypermedia Systems (AEHS) design patterns according to different learning styles. There is no real formalization and no support for an automatic export to a particular adaptation language. One adaptation strategy (as complex as it can be defined by authors) is expressed using only one pattern. Patterns cannot be either combined together or modified, i.e., authors have to find a pattern corresponding to their desired adaptation strategy, otherwise, they cannot express it.
### 2.3 Open Corpus Adaptive Systems
In the AH community, research concerning the integration of open corpus content into adaptive systems has been under scrutiny for several years—mostly in education [ 2]. Most of the existing systems are built upon an existing AHS (e.g., [ 12] on top of [ 8]). Multiple issues are to be faced in order to develop open corpus-based AHS [ 15], [ 5], including automatic hypertext creation, indexing of open corpus resources, and content preparation. None of these systems faces the problem of expressing adaptation, by AH authors, in a simple way.
### 2.4 Conclusion
We have discussed here solutions helping authors to find what adaptation they can propose and how they can express it. However, till now, there have been no works concerning building complex adaptation strategies, independent of any system by combining simple adaptations. In this paper, we focus on this specific point. Adaptation strategies must be defined at a fine granularity. Our aim is thus to help authors define their own adaptations, independently of any adaptation engine, at a higher level and in an easy manner. In the next section, we introduce a use case giving the intuition of our contribution. This scenario is subsequently used in the paper.
## Motivation, Use Case
Assume that Jane, who is a course author, wants to build an adaptive course from her material, i.e., Jane is going to author an AEHS. She has first to define a domain model, then to describe the characteristics of her students in a user model, and finally to express the desired adaptation.
Jane proposes a domain model (cf. Fig. 1), in which she considers the addressed notions as instances of the class Concept. 3 The concepts must be learned in a particular order, that is defined through the relation prerequisite. Each concept may be trained using definitions or examples. Definition and Example are subclasses of the class Resource, i.e., each of their instances has a content, which can be proposed to students. Furthermore, each resource may be in different formats: text, image, or video.
Figure Fig. 1. Jane's domain model in UML.
Jane considers the following student characteristics:
• Learning mode. In-depth learning mode means that each subject must be known in depth before going to a related subject. In-breadth learning mode means that a student has to know a variety of subjects before going in depth.
• Reasoning mode. An inductive reasoning mode means that the student has access to examples before the related definitions are presented to him. In a deductive reasoning mode, definitions precede examples.
• Presentation form. A verbal presentation form is for students preferring textual resources, an audio presentation form is for those preferring audio ones.
Among the adaptation strategies Jane wants to propose, we only focus on the adaptation strategy $S1$ . It concerns students whose learning mode is in-depth, with an inductive reasoning mode and preferring audio resources. $S1$ proposes resources that are examples before those which are definitions. They will be in an audio format if available, otherwise in a textual format. They will be related to concepts ordered according to a depth-first navigational path using the relation prerequisite.
Jane can express $S1$ using solutions supported by her AHS. However, they are not easy to implement and require good backgrounds. See as an illustration, the implementation of $S1$ using Generic Layered Adaptation Model (GLAM) in Fig. 2 (for GLAM syntax, see Section 7.2.1), and using LAG in Fig. 3. This implies that Jane already has her domain and user models in the format understood by her AHS.
Figure Fig. 2. Jane's $S1$ in the GLAM format.
Figure Fig. 3. Jane's $S1$ in the LAG format.
Naturally, Jane expressed $S1$ in three parts: 1) $S1$ concerns students whose learning mode is in-depth, 2) $S1$ concerns students with an inductive reasoning mode, and 3) $S1$ concerns students preferring audio resources. These parts can be considered independently of one another and may compose the definition of other adaptation strategies, for example, $S2$ , an adaptation strategy for students whose learning mode is in-depth, with an inductive reasoning mode and preferring textual resources. $S2$ differs from $S1$ only in proposing resources in a textual format if available, otherwise in an audio format.
To enable Jane to easily define her strategies, i.e., the most natural way possible, we offer the possibility to specify each part of an adaptation strategy by defining the set of resources to propose and the order in which they have to be proposed. According to this approach, $S1$ will be built from the following parts:
• S1-1. It presents resources linked to the domain concepts ordered according to a depth-first navigational path using the prerequisite relation.
• S1-2. It presents only audio resources if they are available otherwise presents textual resources.
• S1-3. It presents examples before definitions.
The adaptation strategy $S1$ is intended to students with specific characteristics. Therefore, each part of the strategy has to be labeled by a student characteristic, i.e., S1-1, for example, will be defined for in-depth learning mode students. Thereby, to define $S2$ , Jane can reuse the parts S1-1 and S1-3, she only has to define the part S2-2 for the textual presentation form.
We presented here the intuition of our contribution according to Jane's needs; in the following, we describe our approach in a more general way.
## Main Aspects of Our Framework
We propose the EAP framework in which authors have a clear separation between what kind of adaptation strategies they want to provide to users and the technicalities involved in writing them. The idea is to help authors in selecting the adaptation strategy and then generate it in a semiautomatic way. Defined adaptation strategies are described at a high level and independently of any adaptation engine.
The EAP framework focuses only on the expression of adaptation strategies. So, it assumes authors have already created their domain and user models. Furthermore, our framework is based on design patterns [ 16]. Design patterns describe recurrent solutions to common problems in software design. The solutions are generic, independent of any language even if there are code generators. For example, the Object-oriented design patterns describe relationships and interactions between classes or objects, without specifying the final application classes or objects that are involved. In practice, design patterns can speed up the development process by providing tested, proven development paradigms. We argue that an adaptation strategy is a kind of conception, where authors have to write the same parts of an adaptation strategy several times, sometimes on different elements. Consequently, the proposed framework uses a set of building blocks independent from any application domain, called elementary adaptation patterns, which are based on design patterns. Thereby, they can be used and instantiated to define specific adaptation strategies.
The main steps for authoring an adaptation strategy with the EAP framework are:
1. Selection. The author either selects elementary adaptation patterns (those needed to define his adaptation strategy) and instantiates them on his own model (thereby, elementary adaptations are defined), or reuses existing elementary adaptations.
2. Specification. The creator specifies associations between user characteristics and elementary adaptations.
3. Computation. The computation of the adaptation strategy resulting from step 2 is automatic.
We have defined a typology and a library of elementary adaptation patterns that can be selected for use within an adaptation strategy, which we introduce in Section 5. The instantiation process and the combination process are described in Section 6.
Before going further, let us apply the EAP framework on Jane's use case.
1. She has to define S1-1 (resp. S1-2, S1-3) by instantiating the appropriate elementary adaptation pattern on the relation prerequisite (resp. on the classes Example, Definition, on the property format).
2. She has to associate S1-1 with in-depth learning mode, S1-2 with inductive reasoning mode, and S1-3 with audio presentation form.
3. The framework automatically builds $S1$ by combining S1-1, S1-2, S1-3. As $S1$ is for students with an in-depth learning mode, an inductive reasoning mode is for those who want audio resources.
Note that, that way, Jane does not need to worry about technical problems in the expression of $S1$ .
The notion of elementary adaptation patterns that we propose is an abstraction of Jane's parts S1-1, S1-2, and S1-3. Furthermore, we defined our elementary adaptation patterns in a manner that is independent from any application domain in order to be able to cover other authors' parts. Thereby, the criteria used to define our elementary adaptation patterns are defined in a generic way (cf. Section 5.1). Elementary adaptation patterns are described in Section 5.2, and their typology is defined in Section 5.3.
### 5.1 Fundamental Criteria for Defining Elementary Adaptation Patterns
Like each part of $S1$ defined by Jane, an elementary adaptation pattern targets a set of resources of a particular type to be presented and also specifies the order in which they will be proposed. This section presents exhaustive criteria to select resources (cf. Section 5.1.1) and to organize the selected resources (cf. Section 5.1.2).
#### 5.1.1 Criteria Used to Select Resources
Criteria used to select resources are based on the domain model, where resources are structured and described. We argue that the general description of a domain model includes the following elements:
• A set of classes. This set must contain the class representing all the resources (called Resource) to be proposed to users, and the class representing all the domain concepts (called Concept).
• A set of relations between classes. Each relation defines a graph on instances of classes on which it is defined. This graph can be navigated according to two different navigational paths in order to reach the goals: depth first or breadth first.
• A set of properties.
Thereby, we have differentiated between criteria selecting resources and criteria defining a navigational path on relations. Our criteria for selecting resources are: their belonging to a class, the values of some properties, or the presence of a relation that defines a navigational path through the resources or the concepts graph. Furthermore, our criteria currently considered for defining a navigational path are either depth first or breadth first.
#### 5.1.2 Criteria Used to Order the Selected Resources
We have looked over works defining adaptation methods, by giving a particular interest to adaptive navigation, without worrying whether the methods are applied on a set of links to resources or resources themselves.
We have looked over the Brusilovsky typology (cf. Section 2.1) excluding methods of the adaptive navigation support group which modify resources (e.g., hiding links belonging to content of resources). Only direct guidance, adaptive ordering, and adaptive link annotation have been considered. We have grouped the direct guidance and adaptive ordering in one operation as both of them define a kind of order either on a link or on several links per time. Therefore, two operations come from the Brusilovsky typology.
We have also looked over the classification of external actions in AHS defined by Stash et al. [ 28]. The classification includes actions on items (e.g., selection, showing items, or links to items), actions on a set of items (e.g., ordering), hierarchical actions (e.g., action on parent or child), and actions on the overall environment (e.g., changing the layout). We only consider the actions having impact on the navigation, which includes: actions on items, actions on a set of items, and hierarchical actions. Furthermore, we distinguish between actions and elements on which the actions are performed. The elements can be an item, a set of items, parents or children. So, we only consider the selection, show, and order actions. Note that the show action cannot be used alone, it is necessarily combined to the other actions. But the combination of order and show actions is equivalent to the order operation defined by Brusilosvky. Therefore, we can neglect it, and retain only the combination of select and show actions.
On the other hand, we have looked over AHS implementing adaptive navigation like AHA! [ 8], WHURLE [ 21], GLAM [ 18], etc. We found that GLAM implements a kind of adaptation not mentioned elsewhere. This adaptation proposes alternative resources if the desired resources are not available. We find it interesting and have retained it in our own typology.
From this study, we retained two operations from the Brusilovsky typology and one combination of actions from the Stash classification and one adaptation from the GLAM platform. Thus, we conclude that there are four basic modes to select resources in a setting of adaptive navigation support, which are described in Table 1.
Table 1. Selection Modes Used in the Elementary Adaptation Patterns
### 5.2 Description of Elementary Adaptation Patterns
We propose the following definition for elementary adaptation patterns, based on the definition of design patterns [ 16].
Definition 1.
An elementary adaptation pattern describes a generic solution for a generic elementary adaptation problem.
This solution is independent from any language, and exploits the characteristics of the domain model.
Definition 2.
A generic elementary adaptation problem describes a criterion to select resources to be proposed and a criterion to define in which order the selected resources are going to be proposed.
Fig. 4 presents the characteristics retained from [ 16] and used to describe elementary adaptation patterns.
Figure Fig. 4. Description of elementary adaptation patterns.
The solution part is the most formal part of the elementary adaptation patterns. We have defined a grammar using the Extended Backus-Naur Form (EBNF) [ 29]. This grammar is described in Fig. 5. It includes a set of nonterminal elements expressed between brackets, and a set of terminal elements expressed between quoats. For people not familiar with the EBNF syntax, we give examples of the solution part respecting the proposed grammar (cf. Figs. 8, 9, and 10). These examples are also accompanied by an informal description.
Figure Fig. 5. Syntax of the characteristic Solution.
We give an informal description of the semantic of the language defined by the grammar and some associated constraints. In order to do so, we consider a domain model DM, composed of
• $Cls =\{c/\;c \;{\rm is\; a\; class}\}$
• $Rel =\{{\rm rel}/\;{\rm rel\; is\; a\; relation}\}$
• $Prop = \{{\rm p}/\;{\rm p\; is\; a\; property}\}$
• $Val_p = \{{\rm v}/\;{\rm v\; is\; a\; value\; of\; the\; property\; p}\}$
• $Res = \{{\rm r}/\;{\rm r\; is\; a\; resource}\}$
We defined DM elements as general elements in the grammar (cf. Fig. 6). Afterward, we defined predicates to select resources or concepts. The predicates are:
• instanceOf: instanceOf(r, c) is true, for all resources $r$ that are instances of class $c$ .
• characteristicOf: characteristicOf(r, p, op, v) is true, for all resources $r$ having the property $p$ and satisfying the comparison test using the operator op and the value $v$ .
• linked: linked(i1, i2, rel) is true, for all instances i1 that are linked directly to instance i2 by relation rel.
• linked-transitive: linked-transitive(i1, i2, rel) is true, for all instances i1 that are linked directly or indirectly to instance i2 by relation rel.
• distance: distance(i1, i2, rel, n) is true, for all instances i1 that are distant from instance i2 by $n$ instances using relation rel.
These predicates compose three types of expressions:
• ${<}Exp_{cls}{>}$ for expressions on classes.
• ${<}Exp_{prop}{>}$ for expressions on properties. Expressions belonging to the same solution part are expressed on the same property.
• ${<}Exp_{rel}{>}$ for expressions on relations. When the expression includes multiple selections, the variables indicating the selected resources are the same.
When more than one expression is defined in a solution, metaexpressions must be defined between all expressions of the solution. This is done using the expression identifiers. Each identifier used in the definition of a metaexpression must correspond to an expression identifier. Three types of metaexpressions are proposed:
• ${<} Id1 {>}\prec {<} Id2 {>}$ means that the set of resources selected with the expression identified by Id1 is proposed before the one selected with the expression identified by Id2.
• ${<} Id1{>} \uplus {<} Id2{>}$ means that the set of resources selected with the expression identified by Id1 is recommended rather than the one selected with the expression identified by Id2. A typographic indication can be used to differentiate between the set of resources recommended from those that are not.
• ${<} Id1{>}\mid{<} Id2 {>}$ means that the set of resources selected with the expression identified by Id2 is an alternative to the one selected with the expression identified by Id1.
### 5.3 Typology of Elementary Adaptation Patterns
We have defined a library of 22 elementary adaptation patterns using the criteria defined in Section 5.1. An elementary adaptation pattern is based simultaneously on 1) one of the four selection modes of resources to be proposed, 2) one of the three elements of the domain model involved in the selection process and when the element is a relation, we also consider 3) one of the two types of navigation through the resources or the concepts graph. The two navigation modes are applied for all the selection modes except for the selection only mode, which proposes a set of resources according to a particular criterion.
Figure Fig. 6. Description of general elements.
In order to be able to look easily over the defined elementary adaptation patterns, we have organized them in a tree where each leaf is an elementary adaptation pattern (cf. Fig. 7). The tree represents our typology.
Figure Fig. 7. Typology of elementary adaptation patterns.
Let us now use this typology to help Jane to define $S1$ . We note that each part of $S1$ can be defined thanks to a pattern. The pattern P2.1.1.1 (cf. Fig. 8) is used to define S1-1 ( S1-1 consists of ordering concepts according to a depth-first navigational path using the relation prerequisite, and presents resources linked to these concepts), P3.3 (cf. Fig. 9) is used to define S1-2 ( S1-2 consists of presenting only audio resources if they are available otherwise presents textual resources), and P2.2 (cf. Fig. 10) is used to define S3-3 ( S3-3 consists of presenting examples before definitions).
Figure Fig. 8. OrderedSelection-DepthFirst-Relation-Concept.
Figure Fig. 9. Ordered Selection-Classes.
Figure Fig. 10. Alternate Selection-Properties.
After having described the typology and some elementary adaptation patterns, let's come back to the process of defining adaptation strategies.
This section focuses on the steps 1 and 3 of authoring steps of adaptation strategies (cf. Section 4). Step 2 is simple and we do not give further details. We start first by describing step 1 related to the instantiation process of elementary adaptation patterns (cf. Section 6.1). Then, we detail step 3 related to the combination process (cf. Section 6.2), and we end by using the EAP framework to define Jane's adaptation strategy $S1$ (cf. Section 6.3).
In order to propose a generic solution, elementary adaptation patterns are defined on a generic domain model. Consequently, when authors select an elementary adaptation pattern, they have to instantiate its constituents on their personal domain model in order to obtain the elementary adaptation that meets their needs. We define elementary adaptations as follows:
Definition 3.
An elementary adaptation is obtained after an instantiation of an elementary adaptation pattern on a particular domain model.
Elementary adaptations have therefore the same structure as elementary adaptation patterns. The generation of an elementary adaptation is done in a semiautomatic way: the characteristics Name, Intent are generated in a semiautomatic way and the characteristics Solution and Constituents are automatically generated.
For example, when Jane wants to express the S1-3 part, she selects the pattern P2.2 and instantiates it with the classes Example and Definition (for informal description, see Section 3; for formal description, see Fig. 11).
Figure Fig. 11. The elementary adaptation S1-3.
Following this principle, authors can define several elementary adaptations, each of them being associated with one user characteristic (step 2 in Section 4). When users have a profile composed of several characteristics, complex adaptation strategies have to be defined. They are obtained by combining elementary adaptations, each one being associated with a component of the user profile. This combination process is detailed below.
Definition 4.
A combined adaptation defines a set of resources that satisfies all constraints imposed by multiple elementary adaptations simultaneously.
A combined adaptation has the same characteristics and is structurally identical to an elementary adaptation. Concretely, the combination process of a set of elementary adaptations consists in combining their characteristics together. A manual process is used to combine the characteristics Name and Intent as it needs natural language processing (not detailed here). We propose an automatic process to combine the characteristics Solution and Constituents, which is explained further below.
The combination of the characteristic Constituents is simple. Constituents coming from the different adaptations are gathered together into a set of constituents. But, the combination of the characteristic Solution is more complex and we have defined the following process. We have chosen to base the process on criteria concerning the selection of resources, as our final aim is to propose a set of resources. Thereby, we have criteria based on: classes to which a resource belongs, properties satisfied by a resource, and relations in which a resource participates. We express this process in two sequential steps:
1. Build different sets of identifiers of expressions, one set for each different criterion (cf. Section 6.2.1).
2. Build one adaptation from the sets built in step 1 (cf. Section 6.2.2).
#### 6.2.1 Step 1 of the Combination
Let $Sol_1, Sol_2, \ldots, Sol_n$ be the solution part of the elementary adaptations to combine, where each $Sol_i$ is composed of
• $n_i$ expressions noted $E_i$ , each expression having an identifier $Id_i$ .
• $m_i$ metaexpressions noted $ME_i$ .
We group the identifiers whose expressions are expressed on one given criterion in different sets.
• The identifiers whose expressions exploit classes are put in the same set ${\rm Set}_{cls} =\{{\rm Id}_i/{\rm Id}_i$ is an identifier that denotes an expression exploiting ${\rm classes}\}$ .
• The identifiers whose expressions exploit relations are grouped into sets, one set per relation. ${\rm Set}_{rel} = \{{\rm Id}_j/{\rm Id}_j$ is an identifier that denotes an expression exploiting the relation $rel\}$ .
• The identifiers whose expressions exploit properties are grouped into sets, one set per property. ${\rm Set}_{prop} = \{{\rm Id}_j/{\rm Id}_j$ is an identifier that denotes an expression exploiting the property $prop\}$ .
Where each $Id_i \in {\rm Sol}_i$ belongs only to one set, either to the ${\rm Set}_{cls}$ , to a set of $\{{\rm Set}_{rel}\}$ , or to a set of $\{{\rm Set}_{prop}\}$ .
#### 6.2.2 Step 2 of the Combination
Let $Set_1, Set_2, \ldots, Set_p$ be the sets of identifiers obtained after the first step, let $Sol_c$ be the solution resulting from the second step of the combination process composed of
• $n_j$ expressions noted $CE_{c}$ .
• $m_j$ metaexpressions noted $CME_{c}$ .
Let $Set_c$ be the set of $p$ tuples built as follows:
$Set_c = Set_1 X Set_2 X \ldots X Set_p.$
For each tuple, a distinct identifier is defined and is associated with an expression $CE_c$ :
$CE_c= E_1 \wedge E_2 \ldots \wedge E_p,$
where
• $CE_c$ is the expression belonging to $Sol_c$ .
• $E_i$ is the expression whose identifier is $Id_i$ , and $Id_i \in Sol_i$ , $i = 1 \ldots p$ .
Identifiers are also used to associate knowledge with expressions. This results in defining metaexpressions. Defining metaexpressions on the expressions $E_c$ of the solution $Sol_c$ is done as follows:
Let $CE_i$ and $CE_j$ be two expressions belonging to the solution $Sol_c$ , where $CE_i$ (resp. $CE_j$ ) contains $E_1$ (resp. $E_2$ ), $E_1$ and $E_2$ belong to the same solution, and are linked by the metaexpression $Id_1 M_h Id_2 (Id_1$ (resp. $Id_2$ ) is the identifier of $E_1$ (resp. $E_2$ ) ). In that case, we deduce the metaexpression $Id_i M_h Id_j$ (where $Id_i$ (resp. $Id_j$ ) is the identifier of $CE_i$ (resp. $CE_j$ )).
However, as a metaexpression is an antisymmetric binary relation between two expressions, two types of conflict can be encountered. They are processed automatically (by deleting all metaexpressions in conflict except one). The process uses a default solution that can be changed by the author.
• Conflict 1. The generation of the same relation between $CE_i$ and $CE_j$ and between $CE_j$ and $CE_i$ (e.g., $CE_1 \prec CE_2$ and $CE_2 \prec CE_1$ ). We propose to order sets of adaptations obtained after the first step according to 1) sets based on the navigational path of the graph, 2) sets exploiting the type of the resources, and 3) sets exploiting the characteristics of the resources.
• Conflict 2. The generation of two metaexpressions between two identical expressions (e.g., $CE_1 \prec CE_2$ and $CE_1 \uplus CE_2$ ). We give a different priority to meta-expressions according to the defined relation: 1) Priority, 2) Recommendation, and 3) Alternate.
We have implemented the following deduction process of $CME_c$ . The $p$ sets of identifiers coming from the first step are first ordered according to the proposed order in the resolution of conflict 1. In a second time, each metaexpression defined using these identifiers allows us to deduce multiple metaexpressions of $CME_c$ . Each time a metaexpression is deduced, we check if it does not generate a conflict with the already generated metaexpressions. If a conflict of the first type is generated, the current metaexpression is not considered and the deduction process will continue. If a conflict of the second type is generated, we retain only one metaexpression according to the order defined in the solution of the second conflict.
We apply here our framework in order to define Jane's adaptation strategy $S1$ . We consider that the elementary adaptation S1-1, S1-2, and S1-3 (cf. Fig. 12) have been defined. Then, Jane has established correspondences between each elementary adaptation and a user characteristic S1-1 with in-depth learning mode, S1-2 with inductive reasoning mode, and S1-3 with audio presentation form. We focus now on the way S1-1, S1-2, and S1-3 are combined in order to produce $S1$ .
Figure Fig. 12. Description of S1-1, S1-2, and S1-3.
S1-1, S1-2, and S1-3 are combined automatically to define $S1$ . The combination process of their characteristic Solution is performed as follows: it has as input three elementary adaptations expressed on three different elements of the domain model. After step 1 of the combination process, three sets are built, one adaptation per set. After step 2, one combined adaptation is built, which is composed of eight expressions and 44 metaexpressions. Among the deduced expressions, we have
• ${\rm E}_{c,1} = {\rm E}_{1-1} \wedge {\rm E}_{2-1} \wedge {\rm E}_{3-1} =$ linked-transitive(r, goal, prerequisite) $\wedge$ linked(rCurrent, r, prerequisite) $\wedge$ characteristicOf(r, format}, ${=}$ , audio) $\wedge$ instanceOf(r, Example).
• ${\rm E}_{c,2} = {\rm E}_{1-1} \wedge {\rm E}_{2-1} \wedge {\rm E}_{3-2} \;{=}$ linked-transitive(r, goal, prerequisite) $\wedge$ linked(rCurrent, r, prerequisite) $\wedge$ characteristicOf(r, format, $=, {\rm audio)} \wedge$ instanceOf(r, Definition).
• ${\rm E}_{c,3} = {\rm E}_{1-2} \wedge {\rm E}_{2-1} \wedge {\rm E}_{3-1} =$ linked-transitive(r, goal, prerequisite) $\wedge$ characteristicOf(r, format, $=, {\rm audio)} \wedge$ instanceOf(r, Example).
• ${\rm E}_{c,4} = {\rm E}_{1-2} \wedge {\rm E}_{2-2} \wedge {\rm E}_{3-1} =$ linked-transitive(r, goal, prerequisite) $\wedge$ characteristicOf(r, format, ${=}$ , text) $\wedge$ instanceOf(r, Example).
• ${\rm E}_{c,5} ={\rm E}_{1-2} \wedge {\rm E}_{2-2} \wedge {\rm E}_{3-2} =$ linked-transitive(r, goal, prerequisite) $\wedge$ characteristicOf(r, format, ${=}$ , text) $\wedge$ instanceOf(r, Definition).
${\rm E}_{c,2} \prec {\rm E}_{c,3}, {\rm E}_{c,2} \prec {\rm E}_{c,4}, {\rm E}_{c,2} \prec {\rm E}_{c,5}$ , and ${\rm E}_{c,3} \mid {\rm E}_{c,4}$ are examples of metaexpressions retained.
${\rm E}_{c,4} \prec {\rm E}_{c,2}$ , ${\rm E}_{c,1} \mid {\rm E}_{c,4}$ , ${\rm E}_{c,2} \mid {\rm E}_{c,5}$ , and ${\rm E}_{c,2} \mid {\rm E}_{c,4}$ are examples of metaexpressions not retained.
## Validation
In this paper, we promote two main ideas behind the EAP framework: enabling authors to specify their adaptation strategies at a high level, and easiness of defining authors' adaptation strategies. Here, we prove these ideas by presenting the implementation of our framework (cf. Section 7.1), by discussing the execution of generated adaptation strategies using an existing adaptation engine (cf. Section 7.2), and by evaluating the expression of adaptation using the EAP framework versus a rule-based language (cf. Section 7.3).
### 7.1 Implementation of the EAP Framework
The framework has been implemented as a plug-in of the Protégé tool, 4 called eapTab (currently, under test). Its architecture is presented in Section 7.1.1 and its main functionalities are described in Section 7.1.2.
#### 7.1.1 Architecture of the eapTab Plug-In
As described in Fig. 13, the plug-in includes two parts:
Fig. 13. Architecture of the eapTab plug-in.
First, a knowledge part gathers the library of elementary adaptation patterns and combination rules. The library is modeled in OWL, 5 where each elementary adaptation pattern is an OWL class and is defined as a specialization of a class called ElementaryAdaptationPattern. Besides, the combination rules implement the combination process (cf. Section 6.2) in a declarative way using SWRL rules 6 and the swrlx build-ins. 7
Second, the process part is made of components performing interaction with an inference engine (in our case Jess) and the OWL Protégé editor. We have used the OWL Protégé API to manipulate the author's domain and user models, the library of elementary adaptation patterns and their instantiations. We have also used the SWRL Jess Bridge 8 to execute SWRL rules using Jess.
#### 7.1.2 Interaction with the eapTab Plug-In
The plug-in proposes multiple facilities. The author starts by loading his user and domain models. He can then define elementary adaptations by selecting an elementary adaptation pattern, and the constituents of the elementary adaptation. The solution part will then be generated automatically. The author can later define associations between an elementary adaptation and a user characteristic, while the combination process of multiple elementary adaptations is done automatically. Finally, eapTab helps authors to export their adaptation strategies automatically in the GLAM format. This conversion is done automatically and is described in the following section. We plan to implement additional extensions, for example, to be able to generate LAG adaptation.
### 7.2 Execution of Generated Adaptation Strategies
Adaptation strategies generated using the EAP framework are expressed at a high level, and are independent of any adaptation engine. Therefore, translators to existing adaptation engines are needed to execute these adaptation strategies. In this paper, we present our work to plug our framework on the GLAM platform, in order to be able to execute generated adaptation strategies by the GLAM adaptation engine. Before presenting this process, we first describe the GLAM platform.
#### 7.2.1 GLAM Platform
GLAM is a platform defined for an entire class of AH. The platform is made up of a generic adaptation model relying on generic user and domain models. Specific systems can be obtained by specializing the GLAM generic user and domain models. An adaptation strategy in GLAM is described in two levels:
A level based only on domain-related knowledge. It concerns data about the domain model and the position of the user in the domain model. It is exploited using rules in a condition-conclusion format like:
\eqalign{&{\rm predicate}_1 \wedge \ldots \wedge {\rm predicate}_n \rightarrow\cr &\quad {\rm Action} ({\rm resource}_ i, {\rm degree}).}
The condition part describes the conditions having to be satisfied by resources proposed to users. Usually, this part is related to the existence of a relation defining a particular navigational path in the domain model, possibly to a type of resources or to restrictions concerning the resource format expressed using attributes of the Concept or Resource classes.
The conclusion part describes the activity proposed to users for proposed resources. It includes two elements:
• Action. It describes the proposed activity for the proposed resource ( ${\rm resource}_i$ in the rule above).
• Degree. It can be used in different treatments. In GLAM, it is used to describe the relevance of a resource against the others. It allows several resources to be proposed to the user, the degree of relevance being represented by a code (color, for example). The degree of relevance has five values (very high, high, medium, low, and very low), each value being associated with a particular color.
A level based on user-related knowledge and user characteristics. It is exploited using metarules. Metarules describe mechanisms that govern selection, scheduling, and excluding rules for a given user according to his profile. Let ${\rm R}_1$ , ${\rm R}_2$ be two sets of rules, where four types of metarules are proposed. Each metarule is a binary relationship between rules:
• ${\rm R}_1 > {\rm R}_2$ (preference metarule). It means that we prefer to execute ${\rm R}_1$ rather than ${\rm R}_2$ .
• ${\rm R}_1 \supset${\rm R}_2 (requirement metarule). It means that the execution of {\rm R}_1 requires the execution of {\rm R}_2 . • \overline{R_1R_2} (exclusion metarule). It means that either {\rm R}_1 or {\rm R}_2 is executed. • {\rm R}_1 \prec${\rm R}_2$ (order metarule). It means that ${\rm R}_1$ is executed before ${\rm R}_2$ . The order metarules define a strict order between the elements on which they are expressed.
#### 7.2.2 Plugging EAP Framework on GLAM Platform
In order to be able to plug the EAP framework on the GLAM platform, we first considered domain and user models used by the EAP framework in GLAM, then translated adaptation strategies expressed using the EAP framework to adaptation in the GLAM format.
Concerning domain and user models. The essential elements of domain model used by our framework are found in GLAM: the modeling of resources, concepts, properties, and relations. Furthermore, GLAM adaptation engine executes resources, in RDF 9 format, i.e., all OWL resources (e.g., OWL properties and OWL classes) are RDF resources by nature.
Concerning adaptation strategies. Adaptation strategies expressed using the EAP framework are composed of the characteristics Name, Intent, Solution, and Constituents. However, only the characteristic Solution has to be translated to GLAM, this latter includes a set of expressions and of metaexpressions. We have hence proposed two translations:
Translation of expressions to the GLAM format. For each expression ${\rm E}_i$ belonging to the set of all the expressions composing an adaptation strategy, we generate a rule ${\rm R}_i$ as follows:
• The condition part of ${\rm R}_i$ is based on ${\rm E}_i$ .
• The conclusion part of ${\rm R}_i$ is generated with a desirability degree set to “medium.”
Translation of metaexpressions to the GLAM format. For each metaexpression ${\rm M}_k$ belonging to the set of all the metaexpressions which compose an adaptation strategy, we perform the following steps:
• If the kind of ${\rm M}_k$ is “ ${\rm E}_i \prec {\rm E}_j$ ,” and if ${\rm R}_i$ (resp. ${\rm R}_j$ ) is the rule obtained from ${\rm E}_i$ (resp. ${\rm E}_j$ ), we generate the metarule ${\rm R}_i \prec${\rm R}_j . • If the kind of {\rm M}_k is “ {\rm E}_i \mid {\rm E}_j ,” and if {\rm R}_i (resp. {\rm R}_j ) is the rule obtained from {\rm E}_i (resp. {\rm E}_j ), we generate the metarules {\rm R}_i \prec${\rm R}_j$ and $\overline{R_iR_j}$ .
• If the kind of ${\rm M}_k$ is “ ${\rm E}_i \uplus {\rm E}_j$ ,” and if ${\rm R}_i$ (resp. ${\rm R}_j$ ) is the rule obtained from ${\rm E}_i$ (resp. ${\rm E}_j$ ), we modify the conclusion part of ${\rm R}_i$ (resp. ${\rm R}_j$ ) as follows: ${\rm R}_i$ takes a degree of desirability higher than the desirability degree of ${\rm R}_j$ and possibly higher than its previous one (resp. ${\rm R}_j$ takes a degree of desirability lower than the desirability degree of ${\rm R}_i$ and possibly lower than its previous one). Desirability degrees are fixed. When two rules have a very high desirability degree, they can't have a higher degree than the existing one. The same principle is applied to a very bad desirability degree.
### 7.3 Evaluation of the EAP Framework versus Rule-Based System
We have carried out an experiment with real users to see if it is easier and faster to express adaptation strategies using elementary adaptation patterns versus using GLAM. For that, we asked course teachers 10 from Supelec 11 and INRIA 12 to define adaptation strategies according to Jane's use case (cf. Section 3).
As in our experiment, we focused on the expression of adaptation, we expressed the domain and user models according to each solution. We performed the evaluation on each volunteer separately (mainly due to their different availabilities). Seven volunteers 13 performed the evaluation. They were between 20 and 30 years old, with between one and seven years experience in higher education. Among the volunteers, there were six men and one woman and all of them were new to both solutions. We divided the sequence of experiment into two steps before and after specifying adaptation strategies. We detail them below.
Before specifying adaptation strategies. We argue that some skills may introduce a bias in our experiment. For example, for people, often manipulating rules, will easily express adaptation using rules and will do it quickly. To avoid this, we defined a questionnaire estimating skills related to our experiment, and the volunteers were asked to fill it in. The questionnaire included 16 questions, which we gathered in five groups: principals and implementation of AH, principals and modeling of personalization, use of rule-based languages, use and implementation of design patterns, and courses building. The volunteers had to estimate their knowledge between (none, little, intermediate, good, or very good).
Figure Fig. 14. Skills of our volunteers.
Thus, none of them had any knowledge on the implementation of an AH, particularly on writing adaptation. They knew more or less what we mean by personalization. Consequently, we first explain general knowledge of AH. We then trained the volunteers using examples to introduce them to both solutions. Finally, they had to define at least $S1$ (cf. Section 3), first according to the EAP framework using eapTab, second, according to GLAM using a notepad. Starting with one system does not influence the results as they didn't know which one was our solution.
Figure Fig. 15. Estimation of difficulty to express $S1$ .
Additionally to this, we measured the time spent to express adaptation strategies using each approach. We grouped our estimation in four intervals (less than 10 min, between 10 and 15, between 15 and 20, more than 20 min). We present this estimation in Fig. 16 with two axes. In the abscissa, we present the time spent using GLAM or eapTab and in the column 10 percent presents one volunteer. We noticed that most volunteers were able to create similar adaptation strategies using eapTab within approximately half the time than when using GLAM. Furthermore, they defined only $S1$ when they used GLAM, while they did not hesitate to define others using eapTab. We explain these results by the fact that when using GLAM, authors have to manually find all the conditions that must be satisfied by the proposed resources and manually compose the different conditions. These conditions must be written as rules in GLAM. Then, they have to define which rules are to be applied to which user by writing metarules. When using eapTab, volunteers only have to define the elementary adaptations, then to associate them to user characteristics and the combination process is done automatically. They have to trust the solution for the combination process. Note that, $S1$ is defined using eight rules and six metarules in GLAM (cf. Fig. 2), where GLAM rules include repetitive parts, e.g., the selection of definitions is present in four rules. While using EAP framework, it requires only three elementary adaptations (cf. Section 6.3).
Figure Fig. 16. Estimation of time spend to express $S1$ .
## Conclusion and Future Work
This paper proposes the EAP framework, in which adaptation strategies are described at a finer granularity than what is proposed by existing languages. This is obtained by the definition of 22 elementary adaptation patterns [ 30] to express the adaptive navigation, and which are organized in a typology. The elementary adaptation patterns are based, on the one hand, on a criterion to select resources and, on the other hand, on a criterion to organize the selected resources. Thereby, we have defined a set of criteria to select resources and to organize selected resources.
Furthermore, the elementary adaptation patterns are independent from the domain model. They can be instantiated on a specific application domain in order to define elementary adaptations. The defined elementary adaptations are associated with user characteristics and combined to make whole adaptation strategies. One of the major benefits of the framework that we propose is the automation of a large and complex part of the process of generating complex adaptation strategies.
The generated adaptation strategies are expressed at a high level and independent of any adaptation engine. In order to be executed, we have described a first possible way using the GLAM adaptation engine. Furthermore, we have conducted experiments that showed the simplicity of our EAP framework rather than a rule-based language (in our case, GLAM).
Our future work will be devoted to integrating our framework on other AHSs. We will study how to automatically translate generated adaptation strategies (using the EAP framework) to other generic adaptation languages like LAG, and GAL. It would also be interesting to do a scale-up test in order to detect the combinations most frequently used by authors. These adaptations could be directly recommended to them. Furthermore, it would be useful to extend our elementary adaptation patterns in order to be able to express adaptive content and adaptive presentation. Finally, our solution exploits user and domain models, which may evolve over time. Consequently, as a future work, we plan to consider the evolution of models, including how to evolve adaptation strategies defined using elementary adaptation patterns when domain or user models change.
## References
• 1. P. Brusilovsky, and N. Henze, “Open Corpus Adaptive Educational Hypermedia,” The Adaptive Web: Methods and Strategies of Web Personalization, P. Brusilovsky, A. Kobsa, and W. Neidl, eds., pp. 672-696, Springer, 2007.
• 2. P. Brusilovsky, “Adaptive Navigation Support,” The Adaptive Web: Methods and Strategies of Web Personalization, P. Brusilovsky, A. Kobsa, and W. Neidl, eds., pp. 263-290, Springer, 2007.
• 3. P. Brusilovsky, “Adaptive Hypermedia,” User Modeling and User-Adapted Interaction, vol. 11, pp. 87-110, Mar. 2001.
• 4. P. Brusilovsky, J. Eklund, and E. Schwarz, “Web-Based Education for All: A Tool for Development Adaptive Courseware,” Computer Networks and ISDN Systems, vol. 30, pp. 291-300, Apr. 1998.
• 5. P. Brusilovsky, and N. Henze, “Open Corpus Adaptive Educational Hypermedia,” The Adaptive Web, pp. 671-696, Springer-Verlag, 2007.
• 6. D. Dagger, V. Wade, and O. Conlan, “Developing Active Learning Experiences for Adaptive Personalised Elearning,” Proc. Int'l Conf. Adaptive Hypermedia and Adaptive Web-Based Systems (AH), pp. 55-64, 2004.
• 7. P. De Bra, G.-J. Houben, and H. Wu, “AHAM: A Dexter-Based Reference Model for Adaptive Hypermedia,” Proc. 10th ACM Conf. Hypertext and Hypermedia: Returning to Our Diverse Roots: Returning to Our Diverse Roots (HYPERTEXT '99), pp. 147-156, 1999.
• 8. P. De Bra, D. Smits, and N. Stash, “Creating and Delivering Adaptive Courses with AHA!,” Proc. First European Conf. Technology Enhanced Learning, pp. 21-33, 2006.
• 9. J. Danculovic, G. Rossi, D. Schwabe, and L. Miaton, “Patterns for Personalized Web Applications,” Proc. European Conf. Pattern Languages of Program (EuroPLoP), pp. 34-43, 2001.
• 10. P. Dolog, M. Kravcik, A. Cristea, D. Burgos, P. De Bra, S. Ceri, V. Devedzic, G.-J. Houben, P. Libbrecht, M. Matera, E. Melis, W. Nejdl, M. Specht, C. Stewart, D. Smits, N. Stash, and C. Tattersall, “Specification, Authoring and Prototyping of Personalised Workplace Learning Solutions,” Int'l J. Learning Technology, vol. 3, no. 3, pp. 286-308, 2007.
• 11. O. Conlan, and V. Wade, “Evaluation of Apels - An Adaptive Elearning Service Based on the Multi-Model, Metadata-Driven Approach,” Proc. Third Int'l Conf. Adaptive Hypermedia and Adaptive Web-Based Systems, P.D. Bra and W. Nejdl, eds., pp. 291-295, 2004.
• 12. A. Cristea, and L. Calvi, “The Three Layers of Adaptation Granularity,” Proc. Ninth Int'l Conf. User Modeling, pp. 4-14, 2003.
• 13. A. Cristea, and A. de Mooij, “LAOS: Layered WWW AHS Authoring Model with Algebraic Operators,” Proc. Int'l World Wide Web Conf. (WWW), 2003.
• 14. A.I. Cristea, and A. de Mooij, “Adaptive Course Authoring: My Online Teacher,” Proc. Int'l Conf. Telecomm., pp. 1762-1769, 2003.
• 15. E.J. Brown, A.I. Cristea, C.D. Stewart, and T.J. Brailsford, “Patterns in Authoring of Adaptive Educational Hypermedia: A Taxonomy of Learning Styles,” Educational Technology and Soc., vol. 8, no. 3, pp. 77-90, 2005.
• 16. E. Gamma, R. Helm, R. Johnson, and J. Vlissides, Design Patterns: Elements of Reusable Object-Oriented Software. Addison-Wesley, 1995.
• 17. F. Garzotto, S. Retalis, A. Papasalouros, and K. Siassiakos, “Patterns for Designing Adaptive/Adaptable Educational Hypermedia,” Advanced Technology for Learning, vol. 1, no. 4, pp. 23-38, 2004.
• 18. C. Jacquiot, Y. Bourda, F. Popineau, A. Delteil, and C. Reynaud, “GLAM: A Generic Layered Adaptation Model for Adaptive Hypermedia Systems,” Proc. Int'l Conf. Adaptive Hypermedia and Adaptive Web-Based Systems (AH), pp. 131-140, 2006.
• 19. E. Knutov, P. De Bra, and M. Pechenizkiy, “AH 12 Years Later: A Comprehensive Survey of Adaptive Hypermedia Methods and Techniques,” New Rev. of Hypermedia Multimedia, vol. 15, pp. 5-38, Apr. 2009.
• 20. K. Levacher, E. Hynes, S. Lawless, A. O'Connor, and V. Wade, “A Framework for Content Preparation to Support Open-Corpus Adaptive Hypermedia,” Proc. Int'l Workshop Dynamic and Adaptive Hypertext at HT, pp. 13-24, 2009.
• 21. A. Moore, T.J. Brailsford, and C.D. Stewart, “Personally Tailored Teaching in WHURLE Using Conditional Transclusion,” Proc. 12th ACM Conf. Hypertext and Hypermedia (HYPERTEXT '01), pp. 163-164, 2001.
• 22. J. Nielsen, and T.K. Landauer, “A Mathematical Model of the Finding of Usability Problems,” Proc. INTERACT and CHI Conf. Human Factors in Computing Systems, pp. 206-213, 1993.
• 23. L. Pesin, and P. Brusilovsky, “ISIS-Tutor: An Adaptive Hypertext Learning Environment,” Proc. Japan-CIS Symp. Knowledge Based Software Eng., pp. 83-87, 1994.
• 24. G. Power, H.C. Davis, A.I. Cristea, C.D. Stewart, and H. Ashman, “Goal Oriented Personalisation with SCORM,” Proc. IEEE Int'l Conf. Advanced Learning Technologies (ICALT), pp. 467-471, 2005.
• 25. B. Steichen, S. Lawless, A. O'Connor, and V. Wade, “Dynamic Hypertext Generation for Reusing Open Corpus Content,” Proc. 20th ACM Conf. Hypertext and Hypermedia (HT '09), pp. 119-128, 2009.
• 26. C. Stewart, A. Cristea, T. Brailsford, and H. Ashman, “Authoring Once, Delivering Many: Creating Reusable Adaptive Courseware,” Proc. Int'l Conf. Web Based Education (WBE '05), Feb. 2005.
• 27. K. Sluijs, J. Hidders, E. Leonardi, and G. Houben, “GAL: A Generic Adaptation Language for Describing Adaptive Hypermedia,” Proc. First Int'l Workshop Dynamic and Adaptive Hypertext: Generic Frameworks, Approaches and Techniques, pp. 13-24, 2009.
• 28. N. Stash, A. Cristea, and P. De Bra, “Adaptation Languages as Vehicles of Explicit Intelligence in Adaptive Hypermedia,” Int'l J. Continuing Eng. Education and Life-Long Learning, vol. 17, nos. 4/5, pp. 319-336, 2007.
• 29. Int'l Organization for Standardization, “Information Technology - Syntactic Metalanguage - Extended BNF,” ISO/IEC 14977, Aug. 2001.
• 30. N. Zemirline, Y. Bourda, and C. Reynaud, “A Typology of Adaptation Patterns for Expressing Adaptive Navigation in Adaptive Hypermedia,” Technical Report 1541, CNRS—Univ. Paris Sud—LRI, 2011.
|
2017-09-21 21:31:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46465006470680237, "perplexity": 2270.3962470359547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687906.71/warc/CC-MAIN-20170921205832-20170921225832-00246.warc.gz"}
|
https://zbmath.org/?q=an:1022.16019
|
## Isomorphism of generalized triangular matrix-rings and recovery of tiles.(English)Zbl 1022.16019
The type of question considered here is the following: Let $$I$$ and $$J$$ be ideals of a ring $$R$$ and suppose that the matrix rings $$\left(\begin{smallmatrix} R&I\\ 0&R\end{smallmatrix}\right)$$ and $$\left(\begin{smallmatrix} R&J\\ 0&R\end{smallmatrix}\right)$$ are isomorphic; can one recover the tiles $$I$$ and $$J$$ (i.e. do $$I$$ and $$J$$ have to be isomorphic as $$R$$-$$R$$-bimodules)? In general the answer is “No”. It is shown that if $$0$$ and $$1$$ are the only idempotent elements of $$R$$ then the tiles can be recovered up to “twisting” by automorphisms of $$R$$, i.e. there is an additive isomorphism $$s\colon I\to J$$ and automorphisms $$f$$ and $$g$$ of $$R$$ such that $$s(axb)=f(a)s(x)g(b)$$ for all $$a,b\in R$$ and $$x\in I$$.
The authors have asked me to point out that there is a systematic error in the displayed matrices in the examples at the end of the paper, concerning the $$(1,2)$$-entries; for instance $$m\mathbb{Z}$$ should be $$\mathbb{Z}/m\mathbb{Z}$$, and so on.
### MSC:
16S50 Endomorphism rings; matrix rings 15A30 Algebraic systems of matrices 16D20 Bimodules in associative algebras
### Keywords:
bimodules; ideals; matrix rings; idempotents; automorphisms
Full Text:
|
2022-12-05 01:48:34
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247140288352966, "perplexity": 210.52057759032672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711001.28/warc/CC-MAIN-20221205000525-20221205030525-00751.warc.gz"}
|
https://en.wikiversity.org/wiki/Quizbank/Electricity_and_Magnetism_(calculus_based)/c05
|
# Quizbank/Electricity and Magnetism (calculus based)/c05
calcPhyEMq/c05 ID153287923206 (Study guide)
Exams:
78 Tests = 3 versions x 26 variations: Each of the 26 variations (A, B, ...) represents a different random selection of questions taken from the study guide.The 3 versions (0,1,..) all have the same questions but in different order and with different numerical inputs. Unless all students take version "0" it is best to reserve it for the instructor because the questions are grouped according to the order in which they appear on the study guide.
Contact me at User talk:Guy vandegrift if you need any help.
### c05 A0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 2.544E-14 N
b) 2.798E-14 N
c) 3.078E-14 N
d) 3.385E-14 N
e) 3.724E-14 N
3)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 4.788E+09 N/C2
b) 5.267E+09 N/C2
c) 5.793E+09 N/C2
d) 6.373E+09 N/C2
e) 7.010E+09 N/C2
4) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
#### c05 A1
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.2 m (on axis) away from the loop's center?
a) 6.925E+09 N/C2
b) 7.617E+09 N/C2
c) 8.379E+09 N/C2
d) 9.217E+09 N/C2
e) 1.014E+10 N/C2
3) A large thin isolated square plate has an area of 8 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 4.492E+01 N/C
b) 4.941E+01 N/C
c) 5.435E+01 N/C
d) 5.979E+01 N/C
e) 6.577E+01 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.014E-14 N
b) 5.515E-14 N
c) 6.067E-14 N
d) 6.674E-14 N
e) 7.341E-14 N
#### c05 A2
1)
A ring is uniformly charged with a net charge of 8 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.32 m (on axis) away from the loop's center?
a) 3.339E+09 N/C2
b) 3.673E+09 N/C2
c) 4.041E+09 N/C2
d) 4.445E+09 N/C2
e) 4.889E+09 N/C2
2) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 8.471E+01 N/C
b) 9.318E+01 N/C
c) 1.025E+02 N/C
d) 1.127E+02 N/C
e) 1.240E+02 N/C
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.2{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.2{\text{ m}}}$.
a) 3.228E+00 V/m2
b) 3.551E+00 V/m2
c) 3.906E+00 V/m2
d) 4.297E+00 V/m2
e) 4.727E+00 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 2.036E-14 N
b) 2.240E-14 N
c) 2.464E-14 N
d) 2.710E-14 N
e) 2.981E-14 N
### c05 B0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 2.036E-14 N
b) 2.240E-14 N
c) 2.464E-14 N
d) 2.710E-14 N
e) 2.981E-14 N
3) A large thin isolated square plate has an area of 9 m2. It is uniformly charged with 6 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 2.571E+01 N/C
b) 2.828E+01 N/C
c) 3.111E+01 N/C
d) 3.422E+01 N/C
e) 3.765E+01 N/C
4)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
#### c05 B1
1) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 8 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 6.171E+01 N/C
b) 6.788E+01 N/C
c) 7.467E+01 N/C
d) 8.214E+01 N/C
e) 9.035E+01 N/C
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=6e}$?
a) 1.308E-13 N
b) 1.439E-13 N
c) 1.583E-13 N
d) 1.741E-13 N
e) 1.915E-13 N
3)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=8.1{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=4.2{\text{ m}}}$.
a) 5.134E-01 V/m2
b) 5.648E-01 V/m2
c) 6.212E-01 V/m2
d) 6.834E-01 V/m2
e) 7.517E-01 V/m2
#### c05 B2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 2.248E-14 N
b) 2.473E-14 N
c) 2.721E-14 N
d) 2.993E-14 N
e) 3.292E-14 N
2)
A ring is uniformly charged with a net charge of 8 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.32 m (on axis) away from the loop's center?
a) 3.339E+09 N/C2
b) 3.673E+09 N/C2
c) 4.041E+09 N/C2
d) 4.445E+09 N/C2
e) 4.889E+09 N/C2
3) A large thin isolated square plate has an area of 8 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 4.492E+01 N/C
b) 4.941E+01 N/C
c) 5.435E+01 N/C
d) 5.979E+01 N/C
e) 6.577E+01 N/C
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.9{\text{ m}}}$.
a) 4.295E+00 V/m2
b) 4.724E+00 V/m2
c) 5.196E+00 V/m2
d) 5.716E+00 V/m2
e) 6.288E+00 V/m2
### c05 C0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.2{\text{ m}}}$.
a) 8.933E+00 V/m2
b) 9.826E+00 V/m2
c) 1.081E+01 V/m2
d) 1.189E+01 V/m2
e) 1.308E+01 V/m2
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.0 m if a=1.0 m, b=1.8 m. The total charge on the rod is 6 nC.
a) 3.610E+00 V/m2
b) 3.971E+00 V/m2
c) 4.368E+00 V/m2
d) 4.804E+00 V/m2
e) 5.285E+00 V/m2
3)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 1.202E+09 N/C2
b) 1.322E+09 N/C2
c) 1.454E+09 N/C2
d) 1.599E+09 N/C2
e) 1.759E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
#### c05 C1
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.65 m if a=0.85 m, b=1.8 m. The total charge on the rod is 5 nC.
a) 3.959E+00 V/m2
b) 4.355E+00 V/m2
c) 4.790E+00 V/m2
d) 5.269E+00 V/m2
e) 5.796E+00 V/m2
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =8{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.0{\text{ m}}}$.
a) 9.459E+00 V/m2
b) 1.040E+01 V/m2
c) 1.145E+01 V/m2
d) 1.259E+01 V/m2
e) 1.385E+01 V/m2
3)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
#### c05 C2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.569E+01 degrees
b) 6.125E+01 degrees
c) 6.738E+01 degrees
d) 7.412E+01 degrees
e) 8.153E+01 degrees
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
3)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.5 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.62 m, b=1.3 m. The total charge on the rod is 7 nC.
a) 6.311E+00 V/m2
b) 6.943E+00 V/m2
c) 7.637E+00 V/m2
d) 8.401E+00 V/m2
e) 9.241E+00 V/m2
4)
A ring is uniformly charged with a net charge of 8 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.32 m (on axis) away from the loop's center?
a) 3.339E+09 N/C2
b) 3.673E+09 N/C2
c) 4.041E+09 N/C2
d) 4.445E+09 N/C2
e) 4.889E+09 N/C2
### c05 D0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =6{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.258E+00 V/m2
b) 1.384E+00 V/m2
c) 1.522E+00 V/m2
d) 1.674E+00 V/m2
e) 1.842E+00 V/m2
2) A large thin isolated square plate has an area of 9 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 2.357E+01 N/C
b) 2.593E+01 N/C
c) 2.852E+01 N/C
d) 3.137E+01 N/C
e) 3.451E+01 N/C
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=2e}$?
a) 3.876E-14 N
b) 4.263E-14 N
c) 4.690E-14 N
d) 5.159E-14 N
e) 5.675E-14 N
4)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
#### c05 D1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 1.473E-14 N
b) 1.620E-14 N
c) 1.782E-14 N
d) 1.960E-14 N
e) 2.156E-14 N
2)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.99 m (on axis) away from the loop's center?
a) 2.429E+09 N/C2
b) 2.672E+09 N/C2
c) 2.939E+09 N/C2
d) 3.233E+09 N/C2
e) 3.556E+09 N/C2
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.5{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.6{\text{ m}}}$.
a) 7.820E-01 V/m2
b) 8.602E-01 V/m2
c) 9.462E-01 V/m2
d) 1.041E+00 V/m2
e) 1.145E+00 V/m2
4) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
#### c05 D2
1) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 6 nC of charge. What is the magnitude of the electric field 2 mm from the center of the plate's surface?
a) 5.647E+01 N/C
b) 6.212E+01 N/C
c) 6.833E+01 N/C
d) 7.516E+01 N/C
e) 8.268E+01 N/C
2)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=2e}$?
a) 3.391E-14 N
b) 3.731E-14 N
c) 4.104E-14 N
d) 4.514E-14 N
e) 4.965E-14 N
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
### c05 E0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=2e}$?
a) 3.426E-15 N
b) 3.768E-15 N
c) 4.145E-15 N
d) 4.560E-15 N
e) 5.015E-15 N
2) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 6.534E+01 N/C
b) 7.187E+01 N/C
c) 7.906E+01 N/C
d) 8.696E+01 N/C
e) 9.566E+01 N/C
3)
A ring is uniformly charged with a net charge of 9 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.73 m (on axis) away from the loop's center?
a) 7.415E+09 N/C2
b) 8.156E+09 N/C2
c) 8.972E+09 N/C2
d) 9.869E+09 N/C2
e) 1.086E+10 N/C2
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.4 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.69 m, b=2.2 m. The total charge on the rod is 6 nC.
a) 3.161E+00 V/m2
b) 3.477E+00 V/m2
c) 3.825E+00 V/m2
d) 4.208E+00 V/m2
e) 4.628E+00 V/m2
#### c05 E1
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.2 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.54 m if a=0.76 m, b=1.7 m. The total charge on the rod is 8 nC.
a) 1.399E+01 V/m2
b) 1.539E+01 V/m2
c) 1.693E+01 V/m2
d) 1.862E+01 V/m2
e) 2.049E+01 V/m2
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.35 m (on axis) away from the loop's center?
a) 4.142E+09 N/C2
b) 4.556E+09 N/C2
c) 5.012E+09 N/C2
d) 5.513E+09 N/C2
e) 6.064E+09 N/C2
3) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 6.534E+01 N/C
b) 7.187E+01 N/C
c) 7.906E+01 N/C
d) 8.696E+01 N/C
e) 9.566E+01 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 2.544E-14 N
b) 2.798E-14 N
c) 3.078E-14 N
d) 3.385E-14 N
e) 3.724E-14 N
#### c05 E2
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.0 m if a=1.0 m, b=1.8 m. The total charge on the rod is 6 nC.
a) 3.610E+00 V/m2
b) 3.971E+00 V/m2
c) 4.368E+00 V/m2
d) 4.804E+00 V/m2
e) 5.285E+00 V/m2
2) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 6.534E+01 N/C
b) 7.187E+01 N/C
c) 7.906E+01 N/C
d) 8.696E+01 N/C
e) 9.566E+01 N/C
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 2.544E-14 N
b) 2.798E-14 N
c) 3.078E-14 N
d) 3.385E-14 N
e) 3.724E-14 N
4)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
### c05 F0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 2.544E-14 N
b) 2.798E-14 N
c) 3.078E-14 N
d) 3.385E-14 N
e) 3.724E-14 N
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 3.672E+09 N/C2
b) 4.039E+09 N/C2
c) 4.443E+09 N/C2
d) 4.887E+09 N/C2
e) 5.376E+09 N/C2
3) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.1{\text{ m}}}$.
a) 8.253E-01 V/m2
b) 9.079E-01 V/m2
c) 9.987E-01 V/m2
d) 1.099E+00 V/m2
e) 1.208E+00 V/m2
#### c05 F1
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.2{\text{ m}}}$.
a) 8.933E+00 V/m2
b) 9.826E+00 V/m2
c) 1.081E+01 V/m2
d) 1.189E+01 V/m2
e) 1.308E+01 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=2e}$?
a) 5.732E-15 N
b) 6.305E-15 N
c) 6.936E-15 N
d) 7.629E-15 N
e) 8.392E-15 N
3) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 3.214E+01 N/C
b) 3.536E+01 N/C
c) 3.889E+01 N/C
d) 4.278E+01 N/C
e) 4.706E+01 N/C
4)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
#### c05 F2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 8.259E-15 N
b) 9.085E-15 N
c) 9.993E-15 N
d) 1.099E-14 N
e) 1.209E-14 N
2)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=0.83{\text{ m}}}$.
a) 2.898E+01 V/m2
b) 3.188E+01 V/m2
c) 3.507E+01 V/m2
d) 3.857E+01 V/m2
e) 4.243E+01 V/m2
4) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 3.214E+01 N/C
b) 3.536E+01 N/C
c) 3.889E+01 N/C
d) 4.278E+01 N/C
e) 4.706E+01 N/C
### c05 G0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.1{\text{ m}}}$.
a) 7.517E+00 V/m2
b) 8.269E+00 V/m2
c) 9.096E+00 V/m2
d) 1.001E+01 V/m2
e) 1.101E+01 V/m2
2)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 1.202E+09 N/C2
b) 1.322E+09 N/C2
c) 1.454E+09 N/C2
d) 1.599E+09 N/C2
e) 1.759E+09 N/C2
3) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 7.701E+01 N/C
b) 8.471E+01 N/C
c) 9.318E+01 N/C
d) 1.025E+02 N/C
e) 1.127E+02 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
#### c05 G1
1) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 8 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 6.171E+01 N/C
b) 6.788E+01 N/C
c) 7.467E+01 N/C
d) 8.214E+01 N/C
e) 9.035E+01 N/C
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
3)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.5 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.33 m (on axis) away from the loop's center?
a) 1.353E+09 N/C2
b) 1.488E+09 N/C2
c) 1.637E+09 N/C2
d) 1.801E+09 N/C2
e) 1.981E+09 N/C2
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=4.3{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.4{\text{ m}}}$.
a) 5.647E+00 V/m2
b) 6.212E+00 V/m2
c) 6.833E+00 V/m2
d) 7.517E+00 V/m2
e) 8.268E+00 V/m2
#### c05 G2
1)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 3.672E+09 N/C2
b) 4.039E+09 N/C2
c) 4.443E+09 N/C2
d) 4.887E+09 N/C2
e) 5.376E+09 N/C2
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=9.1{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=6.2{\text{ m}}}$.
a) 4.961E-01 V/m2
b) 5.457E-01 V/m2
c) 6.002E-01 V/m2
d) 6.603E-01 V/m2
e) 7.263E-01 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.272E+01 degrees
b) 5.799E+01 degrees
c) 6.379E+01 degrees
d) 7.017E+01 degrees
e) 7.719E+01 degrees
4) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
### c05 H0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 6.343E+01 degrees
b) 6.978E+01 degrees
c) 7.676E+01 degrees
d) 8.443E+01 degrees
e) 9.288E+01 degrees
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.5{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.6{\text{ m}}}$.
a) 7.820E-01 V/m2
b) 8.602E-01 V/m2
c) 9.462E-01 V/m2
d) 1.041E+00 V/m2
e) 1.145E+00 V/m2
3) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
4)
A ring is uniformly charged with a net charge of 6 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.4 m (on axis) away from the loop's center?
a) 2.013E+09 N/C2
b) 2.214E+09 N/C2
c) 2.435E+09 N/C2
d) 2.679E+09 N/C2
e) 2.947E+09 N/C2
#### c05 H1
1)
A ring is uniformly charged with a net charge of 6 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.4 m (on axis) away from the loop's center?
a) 2.013E+09 N/C2
b) 2.214E+09 N/C2
c) 2.435E+09 N/C2
d) 2.679E+09 N/C2
e) 2.947E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.272E+01 degrees
b) 5.799E+01 degrees
c) 6.379E+01 degrees
d) 7.017E+01 degrees
e) 7.719E+01 degrees
3) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 8 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 6.171E+01 N/C
b) 6.788E+01 N/C
c) 7.467E+01 N/C
d) 8.214E+01 N/C
e) 9.035E+01 N/C
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.1{\text{ m}}}$.
a) 8.253E-01 V/m2
b) 9.079E-01 V/m2
c) 9.987E-01 V/m2
d) 1.099E+00 V/m2
e) 1.208E+00 V/m2
#### c05 H2
1)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 4.788E+09 N/C2
b) 5.267E+09 N/C2
c) 5.793E+09 N/C2
d) 6.373E+09 N/C2
e) 7.010E+09 N/C2
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =6{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.258E+00 V/m2
b) 1.384E+00 V/m2
c) 1.522E+00 V/m2
d) 1.674E+00 V/m2
e) 1.842E+00 V/m2
3) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=4e}$?
a) 3.719E+01 degrees
b) 4.091E+01 degrees
c) 4.500E+01 degrees
d) 4.950E+01 degrees
e) 5.445E+01 degrees
### c05 I0
1)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 4.788E+09 N/C2
b) 5.267E+09 N/C2
c) 5.793E+09 N/C2
d) 6.373E+09 N/C2
e) 7.010E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.569E+01 degrees
b) 6.125E+01 degrees
c) 6.738E+01 degrees
d) 7.412E+01 degrees
e) 8.153E+01 degrees
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =6{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.258E+00 V/m2
b) 1.384E+00 V/m2
c) 1.522E+00 V/m2
d) 1.674E+00 V/m2
e) 1.842E+00 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 8.259E-15 N
b) 9.085E-15 N
c) 9.993E-15 N
d) 1.099E-14 N
e) 1.209E-14 N
#### c05 I1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 3.961E+01 degrees
b) 4.357E+01 degrees
c) 4.793E+01 degrees
d) 5.272E+01 degrees
e) 5.799E+01 degrees
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 2.248E-14 N
b) 2.473E-14 N
c) 2.721E-14 N
d) 2.993E-14 N
e) 3.292E-14 N
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.9{\text{ m}}}$.
a) 4.295E+00 V/m2
b) 4.724E+00 V/m2
c) 5.196E+00 V/m2
d) 5.716E+00 V/m2
e) 6.288E+00 V/m2
4)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 1.202E+09 N/C2
b) 1.322E+09 N/C2
c) 1.454E+09 N/C2
d) 1.599E+09 N/C2
e) 1.759E+09 N/C2
#### c05 I2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 6.125E+01 degrees
b) 6.738E+01 degrees
c) 7.412E+01 degrees
d) 8.153E+01 degrees
e) 8.968E+01 degrees
2)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=6e}$?
a) 1.308E-13 N
b) 1.439E-13 N
c) 1.583E-13 N
d) 1.741E-13 N
e) 1.915E-13 N
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.5{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.6{\text{ m}}}$.
a) 7.820E-01 V/m2
b) 8.602E-01 V/m2
c) 9.462E-01 V/m2
d) 1.041E+00 V/m2
e) 1.145E+00 V/m2
### c05 J0
1)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.5 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.33 m (on axis) away from the loop's center?
a) 1.353E+09 N/C2
b) 1.488E+09 N/C2
c) 1.637E+09 N/C2
d) 1.801E+09 N/C2
e) 1.981E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.569E+01 degrees
b) 6.125E+01 degrees
c) 6.738E+01 degrees
d) 7.412E+01 degrees
e) 8.153E+01 degrees
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.2{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.606E+00 V/m2
b) 1.767E+00 V/m2
c) 1.943E+00 V/m2
d) 2.138E+00 V/m2
e) 2.351E+00 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=2e}$?
a) 3.876E-14 N
b) 4.263E-14 N
c) 4.690E-14 N
d) 5.159E-14 N
e) 5.675E-14 N
#### c05 J1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=6e}$?
a) 5.243E+01 degrees
b) 5.767E+01 degrees
c) 6.343E+01 degrees
d) 6.978E+01 degrees
e) 7.676E+01 degrees
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 8.336E+09 N/C2
b) 9.170E+09 N/C2
c) 1.009E+10 N/C2
d) 1.110E+10 N/C2
e) 1.220E+10 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 5.243E-14 N
b) 5.768E-14 N
c) 6.344E-14 N
d) 6.979E-14 N
e) 7.677E-14 N
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.2{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.2{\text{ m}}}$.
a) 3.228E+00 V/m2
b) 3.551E+00 V/m2
c) 3.906E+00 V/m2
d) 4.297E+00 V/m2
e) 4.727E+00 V/m2
#### c05 J2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 3.961E+01 degrees
b) 4.357E+01 degrees
c) 4.793E+01 degrees
d) 5.272E+01 degrees
e) 5.799E+01 degrees
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =6{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.258E+00 V/m2
b) 1.384E+00 V/m2
c) 1.522E+00 V/m2
d) 1.674E+00 V/m2
e) 1.842E+00 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 4.171E-14 N
b) 4.588E-14 N
c) 5.047E-14 N
d) 5.551E-14 N
e) 6.107E-14 N
4)
A ring is uniformly charged with a net charge of 6 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.4 m (on axis) away from the loop's center?
a) 2.013E+09 N/C2
b) 2.214E+09 N/C2
c) 2.435E+09 N/C2
d) 2.679E+09 N/C2
e) 2.947E+09 N/C2
### c05 K0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.569E+01 degrees
b) 6.125E+01 degrees
c) 6.738E+01 degrees
d) 7.412E+01 degrees
e) 8.153E+01 degrees
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 4.171E-14 N
b) 4.588E-14 N
c) 5.047E-14 N
d) 5.551E-14 N
e) 6.107E-14 N
3)
A ring is uniformly charged with a net charge of 6 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.4 m (on axis) away from the loop's center?
a) 2.013E+09 N/C2
b) 2.214E+09 N/C2
c) 2.435E+09 N/C2
d) 2.679E+09 N/C2
e) 2.947E+09 N/C2
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.5{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.6{\text{ m}}}$.
a) 7.820E-01 V/m2
b) 8.602E-01 V/m2
c) 9.462E-01 V/m2
d) 1.041E+00 V/m2
e) 1.145E+00 V/m2
#### c05 K1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 8.259E-15 N
b) 9.085E-15 N
c) 9.993E-15 N
d) 1.099E-14 N
e) 1.209E-14 N
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.1{\text{ m}}}$.
a) 8.253E-01 V/m2
b) 9.079E-01 V/m2
c) 9.987E-01 V/m2
d) 1.099E+00 V/m2
e) 1.208E+00 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 6.125E+01 degrees
b) 6.738E+01 degrees
c) 7.412E+01 degrees
d) 8.153E+01 degrees
e) 8.968E+01 degrees
4)
A ring is uniformly charged with a net charge of 9 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.4 m (on axis) away from the loop's center?
a) 7.119E+09 N/C2
b) 7.831E+09 N/C2
c) 8.614E+09 N/C2
d) 9.476E+09 N/C2
e) 1.042E+10 N/C2
#### c05 K2
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 3.629E+01 degrees
b) 3.992E+01 degrees
c) 4.391E+01 degrees
d) 4.830E+01 degrees
e) 5.313E+01 degrees
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.2 m (on axis) away from the loop's center?
a) 6.925E+09 N/C2
b) 7.617E+09 N/C2
c) 8.379E+09 N/C2
d) 9.217E+09 N/C2
e) 1.014E+10 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.014E-14 N
b) 5.515E-14 N
c) 6.067E-14 N
d) 6.674E-14 N
e) 7.341E-14 N
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=8.3{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =5{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.3{\text{ m}}}$.
a) 1.022E+00 V/m2
b) 1.125E+00 V/m2
c) 1.237E+00 V/m2
d) 1.361E+00 V/m2
e) 1.497E+00 V/m2
### c05 L0
1)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 3.672E+09 N/C2
b) 4.039E+09 N/C2
c) 4.443E+09 N/C2
d) 4.887E+09 N/C2
e) 5.376E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=6e}$?
a) 5.767E+01 degrees
b) 6.343E+01 degrees
c) 6.978E+01 degrees
d) 7.676E+01 degrees
e) 8.443E+01 degrees
3) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 8.471E+01 N/C
b) 9.318E+01 N/C
c) 1.025E+02 N/C
d) 1.127E+02 N/C
e) 1.240E+02 N/C
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.2{\text{ m}}}$.
a) 8.933E+00 V/m2
b) 9.826E+00 V/m2
c) 1.081E+01 V/m2
d) 1.189E+01 V/m2
e) 1.308E+01 V/m2
#### c05 L1
1) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
2)
A ring is uniformly charged with a net charge of 4 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.0 m (on axis) away from the loop's center?
a) 5.352E+09 N/C2
b) 5.887E+09 N/C2
c) 6.476E+09 N/C2
d) 7.124E+09 N/C2
e) 7.836E+09 N/C2
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.1{\text{ m}}}$.
a) 8.253E-01 V/m2
b) 9.079E-01 V/m2
c) 9.987E-01 V/m2
d) 1.099E+00 V/m2
e) 1.208E+00 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.272E+01 degrees
b) 5.799E+01 degrees
c) 6.379E+01 degrees
d) 7.017E+01 degrees
e) 7.719E+01 degrees
#### c05 L2
1)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
2) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=9.1{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=6.2{\text{ m}}}$.
a) 4.961E-01 V/m2
b) 5.457E-01 V/m2
c) 6.002E-01 V/m2
d) 6.603E-01 V/m2
e) 7.263E-01 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 4.743E+01 degrees
b) 5.217E+01 degrees
c) 5.739E+01 degrees
d) 6.313E+01 degrees
e) 6.944E+01 degrees
### c05 M0
1)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 1.202E+09 N/C2
b) 1.322E+09 N/C2
c) 1.454E+09 N/C2
d) 1.599E+09 N/C2
e) 1.759E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.272E+01 degrees
b) 5.799E+01 degrees
c) 6.379E+01 degrees
d) 7.017E+01 degrees
e) 7.719E+01 degrees
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=9.1{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=6.2{\text{ m}}}$.
a) 4.961E-01 V/m2
b) 5.457E-01 V/m2
c) 6.002E-01 V/m2
d) 6.603E-01 V/m2
e) 7.263E-01 V/m2
4) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 6.534E+01 N/C
b) 7.187E+01 N/C
c) 7.906E+01 N/C
d) 8.696E+01 N/C
e) 9.566E+01 N/C
#### c05 M1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 3.629E+01 degrees
b) 3.992E+01 degrees
c) 4.391E+01 degrees
d) 4.830E+01 degrees
e) 5.313E+01 degrees
2) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 2 mm from the center of the plate's surface?
a) 9.546E+01 N/C
b) 1.050E+02 N/C
c) 1.155E+02 N/C
d) 1.271E+02 N/C
e) 1.398E+02 N/C
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =2{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.1{\text{ m}}}$.
a) 8.253E-01 V/m2
b) 9.079E-01 V/m2
c) 9.987E-01 V/m2
d) 1.099E+00 V/m2
e) 1.208E+00 V/m2
4)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 1.202E+09 N/C2
b) 1.322E+09 N/C2
c) 1.454E+09 N/C2
d) 1.599E+09 N/C2
e) 1.759E+09 N/C2
#### c05 M2
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=4.3{\text{ m}}}$.
a) 8.924E-01 V/m2
b) 9.816E-01 V/m2
c) 1.080E+00 V/m2
d) 1.188E+00 V/m2
e) 1.307E+00 V/m2
2) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 2 mm from the center of the plate's surface?
a) 3.214E+01 N/C
b) 3.536E+01 N/C
c) 3.889E+01 N/C
d) 4.278E+01 N/C
e) 4.706E+01 N/C
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=4e}$?
a) 3.719E+01 degrees
b) 4.091E+01 degrees
c) 4.500E+01 degrees
d) 4.950E+01 degrees
e) 5.445E+01 degrees
4)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 4.788E+09 N/C2
b) 5.267E+09 N/C2
c) 5.793E+09 N/C2
d) 6.373E+09 N/C2
e) 7.010E+09 N/C2
### c05 N0
1) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 1.694E+02 N/C
b) 1.864E+02 N/C
c) 2.050E+02 N/C
d) 2.255E+02 N/C
e) 2.480E+02 N/C
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.569E+01 degrees
b) 6.125E+01 degrees
c) 6.738E+01 degrees
d) 7.412E+01 degrees
e) 8.153E+01 degrees
3)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.35 m (on axis) away from the loop's center?
a) 4.142E+09 N/C2
b) 4.556E+09 N/C2
c) 5.012E+09 N/C2
d) 5.513E+09 N/C2
e) 6.064E+09 N/C2
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.96 m if a=0.95 m, b=1.8 m. The total charge on the rod is 7 nC.
a) 3.385E+00 V/m2
b) 3.724E+00 V/m2
c) 4.096E+00 V/m2
d) 4.506E+00 V/m2
e) 4.957E+00 V/m2
#### c05 N1
1) A large thin isolated square plate has an area of 5 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 6.534E+01 N/C
b) 7.187E+01 N/C
c) 7.906E+01 N/C
d) 8.696E+01 N/C
e) 9.566E+01 N/C
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.3 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.96 m if a=0.63 m, b=1.4 m. The total charge on the rod is 3 nC.
a) 3.719E+00 V/m2
b) 4.091E+00 V/m2
c) 4.500E+00 V/m2
d) 4.950E+00 V/m2
e) 5.445E+00 V/m2
3)
A ring is uniformly charged with a net charge of 4 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.0 m (on axis) away from the loop's center?
a) 5.352E+09 N/C2
b) 5.887E+09 N/C2
c) 6.476E+09 N/C2
d) 7.124E+09 N/C2
e) 7.836E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
#### c05 N2
1) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
2)
A ring is uniformly charged with a net charge of 9 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.4 m (on axis) away from the loop's center?
a) 7.119E+09 N/C2
b) 7.831E+09 N/C2
c) 8.614E+09 N/C2
d) 9.476E+09 N/C2
e) 1.042E+10 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.914E+01 degrees
b) 6.506E+01 degrees
c) 7.157E+01 degrees
d) 7.872E+01 degrees
e) 8.659E+01 degrees
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.54 m if a=1.0 m, b=2.0 m. The total charge on the rod is 3 nC.
a) 1.665E+00 V/m2
b) 1.831E+00 V/m2
c) 2.014E+00 V/m2
d) 2.216E+00 V/m2
e) 2.437E+00 V/m2
### c05 O0
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.65 m if a=0.85 m, b=1.8 m. The total charge on the rod is 5 nC.
a) 3.959E+00 V/m2
b) 4.355E+00 V/m2
c) 4.790E+00 V/m2
d) 5.269E+00 V/m2
e) 5.796E+00 V/m2
2) A large thin isolated square plate has an area of 8 m2. It is uniformly charged with 6 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 3.500E+01 N/C
b) 3.850E+01 N/C
c) 4.235E+01 N/C
d) 4.659E+01 N/C
e) 5.125E+01 N/C
3)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 3.672E+09 N/C2
b) 4.039E+09 N/C2
c) 4.443E+09 N/C2
d) 4.887E+09 N/C2
e) 5.376E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=4e}$?
a) 4.766E+01 degrees
b) 5.243E+01 degrees
c) 5.767E+01 degrees
d) 6.343E+01 degrees
e) 6.978E+01 degrees
#### c05 O1
1) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 7.701E+01 N/C
b) 8.471E+01 N/C
c) 9.318E+01 N/C
d) 1.025E+02 N/C
e) 1.127E+02 N/C
2)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
3)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.83 m if a=0.7 m, b=1.8 m. The total charge on the rod is 9 nC.
a) 6.897E+00 V/m2
b) 7.587E+00 V/m2
c) 8.345E+00 V/m2
d) 9.180E+00 V/m2
e) 1.010E+01 V/m2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 3.629E+01 degrees
b) 3.992E+01 degrees
c) 4.391E+01 degrees
d) 4.830E+01 degrees
e) 5.313E+01 degrees
#### c05 O2
1) A large thin isolated square plate has an area of 8 m2. It is uniformly charged with 7 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 4.492E+01 N/C
b) 4.941E+01 N/C
c) 5.435E+01 N/C
d) 5.979E+01 N/C
e) 6.577E+01 N/C
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.34 m (on axis) away from the loop's center?
a) 3.672E+09 N/C2
b) 4.039E+09 N/C2
c) 4.443E+09 N/C2
d) 4.887E+09 N/C2
e) 5.376E+09 N/C2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=4e}$?
a) 4.766E+01 degrees
b) 5.243E+01 degrees
c) 5.767E+01 degrees
d) 6.343E+01 degrees
e) 6.978E+01 degrees
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.5 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.0 m if a=1.1 m, b=1.4 m. The total charge on the rod is 5 nC.
a) 4.602E+00 V/m2
b) 5.062E+00 V/m2
c) 5.568E+00 V/m2
d) 6.125E+00 V/m2
e) 6.738E+00 V/m2
### c05 P0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.5{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.6{\text{ m}}}$.
a) 7.820E-01 V/m2
b) 8.602E-01 V/m2
c) 9.462E-01 V/m2
d) 1.041E+00 V/m2
e) 1.145E+00 V/m2
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.35 m (on axis) away from the loop's center?
a) 4.142E+09 N/C2
b) 4.556E+09 N/C2
c) 5.012E+09 N/C2
d) 5.513E+09 N/C2
e) 6.064E+09 N/C2
3) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 4.821E+01 N/C
b) 5.303E+01 N/C
c) 5.834E+01 N/C
d) 6.417E+01 N/C
e) 7.059E+01 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=6e}$?
a) 6.343E+01 degrees
b) 6.978E+01 degrees
c) 7.676E+01 degrees
d) 8.443E+01 degrees
e) 9.288E+01 degrees
#### c05 P1
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=6e}$?
a) 5.767E+01 degrees
b) 6.343E+01 degrees
c) 6.978E+01 degrees
d) 7.676E+01 degrees
e) 8.443E+01 degrees
2)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 8.336E+09 N/C2
b) 9.170E+09 N/C2
c) 1.009E+10 N/C2
d) 1.110E+10 N/C2
e) 1.220E+10 N/C2
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=1.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=0.83{\text{ m}}}$.
a) 2.898E+01 V/m2
b) 3.188E+01 V/m2
c) 3.507E+01 V/m2
d) 3.857E+01 V/m2
e) 4.243E+01 V/m2
4) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 2 mm from the center of the plate's surface?
a) 3.214E+01 N/C
b) 3.536E+01 N/C
c) 3.889E+01 N/C
d) 4.278E+01 N/C
e) 4.706E+01 N/C
#### c05 P2
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=6.9{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =9{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=4.3{\text{ m}}}$.
a) 8.924E-01 V/m2
b) 9.816E-01 V/m2
c) 1.080E+00 V/m2
d) 1.188E+00 V/m2
e) 1.307E+00 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=6e}$?
a) 5.243E+01 degrees
b) 5.767E+01 degrees
c) 6.343E+01 degrees
d) 6.978E+01 degrees
e) 7.676E+01 degrees
3) A large thin isolated square plate has an area of 4 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 2 mm from the center of the plate's surface?
a) 9.546E+01 N/C
b) 1.050E+02 N/C
c) 1.155E+02 N/C
d) 1.271E+02 N/C
e) 1.398E+02 N/C
4)
A ring is uniformly charged with a net charge of 4 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.0 m (on axis) away from the loop's center?
a) 5.352E+09 N/C2
b) 5.887E+09 N/C2
c) 6.476E+09 N/C2
d) 7.124E+09 N/C2
e) 7.836E+09 N/C2
### c05 Q0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.014E-14 N
b) 5.515E-14 N
c) 6.067E-14 N
d) 6.674E-14 N
e) 7.341E-14 N
2) A large thin isolated square plate has an area of 9 m2. It is uniformly charged with 6 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 2.571E+01 N/C
b) 2.828E+01 N/C
c) 3.111E+01 N/C
d) 3.422E+01 N/C
e) 3.765E+01 N/C
3)
A ring is uniformly charged with a net charge of 4 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 5.402E+09 N/C2
b) 5.943E+09 N/C2
c) 6.537E+09 N/C2
d) 7.191E+09 N/C2
e) 7.910E+09 N/C2
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.5 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.61 m, b=1.7 m. The total charge on the rod is 8 nC.
a) 5.995E+00 V/m2
b) 6.595E+00 V/m2
c) 7.254E+00 V/m2
d) 7.980E+00 V/m2
e) 8.778E+00 V/m2
#### c05 Q1
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.7 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.76 m if a=1.1 m, b=1.6 m. The total charge on the rod is 8 nC.
a) 5.267E+00 V/m2
b) 5.794E+00 V/m2
c) 6.374E+00 V/m2
d) 7.011E+00 V/m2
e) 7.712E+00 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=2e}$?
a) 3.426E-15 N
b) 3.768E-15 N
c) 4.145E-15 N
d) 4.560E-15 N
e) 5.015E-15 N
3)
A ring is uniformly charged with a net charge of 8 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.32 m (on axis) away from the loop's center?
a) 3.339E+09 N/C2
b) 3.673E+09 N/C2
c) 4.041E+09 N/C2
d) 4.445E+09 N/C2
e) 4.889E+09 N/C2
4) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 7.701E+01 N/C
b) 8.471E+01 N/C
c) 9.318E+01 N/C
d) 1.025E+02 N/C
e) 1.127E+02 N/C
#### c05 Q2
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.7 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.52 m if a=0.88 m, b=1.3 m. The total charge on the rod is 6 nC.
a) 6.804E+00 V/m2
b) 7.485E+00 V/m2
c) 8.233E+00 V/m2
d) 9.056E+00 V/m2
e) 9.962E+00 V/m2
2) A large thin isolated square plate has an area of 6 m2. It is uniformly charged with 9 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 7.701E+01 N/C
b) 8.471E+01 N/C
c) 9.318E+01 N/C
d) 1.025E+02 N/C
e) 1.127E+02 N/C
3)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 4.171E-14 N
b) 4.588E-14 N
c) 5.047E-14 N
d) 5.551E-14 N
e) 6.107E-14 N
### c05 R0
1)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=4e}$?
a) 3.719E+01 degrees
b) 4.091E+01 degrees
c) 4.500E+01 degrees
d) 4.950E+01 degrees
e) 5.445E+01 degrees
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =8{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.0{\text{ m}}}$.
a) 9.459E+00 V/m2
b) 1.040E+01 V/m2
c) 1.145E+01 V/m2
d) 1.259E+01 V/m2
e) 1.385E+01 V/m2
3)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.83 m if a=0.7 m, b=1.8 m. The total charge on the rod is 9 nC.
a) 6.897E+00 V/m2
b) 7.587E+00 V/m2
c) 8.345E+00 V/m2
d) 9.180E+00 V/m2
e) 1.010E+01 V/m2
4)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 8.336E+09 N/C2
b) 9.170E+09 N/C2
c) 1.009E+10 N/C2
d) 1.110E+10 N/C2
e) 1.220E+10 N/C2
#### c05 R1
1)
A ring is uniformly charged with a net charge of 4 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.0 m (on axis) away from the loop's center?
a) 5.352E+09 N/C2
b) 5.887E+09 N/C2
c) 6.476E+09 N/C2
d) 7.124E+09 N/C2
e) 7.836E+09 N/C2
2) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=8.7{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =7{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=5.8{\text{ m}}}$.
a) 3.722E-01 V/m2
b) 4.094E-01 V/m2
c) 4.504E-01 V/m2
d) 4.954E-01 V/m2
e) 5.450E-01 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=4e}$?
a) 3.719E+01 degrees
b) 4.091E+01 degrees
c) 4.500E+01 degrees
d) 4.950E+01 degrees
e) 5.445E+01 degrees
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.3 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.96 m if a=0.63 m, b=1.4 m. The total charge on the rod is 3 nC.
a) 3.719E+00 V/m2
b) 4.091E+00 V/m2
c) 4.500E+00 V/m2
d) 4.950E+00 V/m2
e) 5.445E+00 V/m2
#### c05 R2
1)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 1.764E+09 N/C2
b) 1.941E+09 N/C2
c) 2.135E+09 N/C2
d) 2.348E+09 N/C2
e) 2.583E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.062E+01 degrees
b) 5.569E+01 degrees
c) 6.125E+01 degrees
d) 6.738E+01 degrees
e) 7.412E+01 degrees
3)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.65 m if a=0.85 m, b=1.8 m. The total charge on the rod is 5 nC.
a) 3.959E+00 V/m2
b) 4.355E+00 V/m2
c) 4.790E+00 V/m2
d) 5.269E+00 V/m2
e) 5.796E+00 V/m2
4) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=2.8{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=1.9{\text{ m}}}$.
a) 4.295E+00 V/m2
b) 4.724E+00 V/m2
c) 5.196E+00 V/m2
d) 5.716E+00 V/m2
e) 6.288E+00 V/m2
### c05 S0
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=7.2{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =3{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=3.6{\text{ m}}}$.
a) 1.606E+00 V/m2
b) 1.767E+00 V/m2
c) 1.943E+00 V/m2
d) 2.138E+00 V/m2
e) 2.351E+00 V/m2
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.4 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.69 m, b=2.2 m. The total charge on the rod is 6 nC.
a) 3.161E+00 V/m2
b) 3.477E+00 V/m2
c) 3.825E+00 V/m2
d) 4.208E+00 V/m2
e) 4.628E+00 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=6e}$?
a) 5.767E+01 degrees
b) 6.343E+01 degrees
c) 6.978E+01 degrees
d) 7.676E+01 degrees
e) 8.443E+01 degrees
4)
A ring is uniformly charged with a net charge of 7 nC. The radius of the ring is R=1.7 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.2 m (on axis) away from the loop's center?
a) 6.925E+09 N/C2
b) 7.617E+09 N/C2
c) 8.379E+09 N/C2
d) 9.217E+09 N/C2
e) 1.014E+10 N/C2
#### c05 S1
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.96 m if a=0.95 m, b=1.8 m. The total charge on the rod is 7 nC.
a) 3.385E+00 V/m2
b) 3.724E+00 V/m2
c) 4.096E+00 V/m2
d) 4.506E+00 V/m2
e) 4.957E+00 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=3e}$?
a) 3.629E+01 degrees
b) 3.992E+01 degrees
c) 4.391E+01 degrees
d) 4.830E+01 degrees
e) 5.313E+01 degrees
3) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.0{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =8{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.0{\text{ m}}}$.
a) 9.459E+00 V/m2
b) 1.040E+01 V/m2
c) 1.145E+01 V/m2
d) 1.259E+01 V/m2
e) 1.385E+01 V/m2
4)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 5.581E+09 N/C2
b) 6.139E+09 N/C2
c) 6.753E+09 N/C2
d) 7.428E+09 N/C2
e) 8.171E+09 N/C2
#### c05 S2
1) ${\displaystyle E(z)=\int _{0}^{R}f(r',z)dr'}$
is an integral that calculates the magnitude of the electric field at a distance ${\displaystyle z}$ fromthe center of a thin circular disk as measured along a line normal to the plane of the disk. The disk's radius is ${\displaystyle R=3.3{\text{ m}}}$ and the surface charge density is ${\displaystyle \sigma =4{\text{ nC/m}}^{3}}$. Evaluate ${\displaystyle f(r',z)}$ at ${\displaystyle r'=2.0{\text{ m}}}$.
a) 6.877E+00 V/m2
b) 7.565E+00 V/m2
c) 8.321E+00 V/m2
d) 9.153E+00 V/m2
e) 1.007E+01 V/m2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
3)
A ring is uniformly charged with a net charge of 9 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.4 m (on axis) away from the loop's center?
a) 7.119E+09 N/C2
b) 7.831E+09 N/C2
c) 8.614E+09 N/C2
d) 9.476E+09 N/C2
e) 1.042E+10 N/C2
4)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.5 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.62 m, b=1.3 m. The total charge on the rod is 7 nC.
a) 6.311E+00 V/m2
b) 6.943E+00 V/m2
c) 7.637E+00 V/m2
d) 8.401E+00 V/m2
e) 9.241E+00 V/m2
### c05 T0
1)
A ring is uniformly charged with a net charge of 5 nC. The radius of the ring is R=1.9 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.3 m (on axis) away from the loop's center?
a) 4.788E+09 N/C2
b) 5.267E+09 N/C2
c) 5.793E+09 N/C2
d) 6.373E+09 N/C2
e) 7.010E+09 N/C2
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.5 m. Evaluate ${\displaystyle f(x,y)}$ at x=1.1 m if a=0.62 m, b=1.3 m. The total charge on the rod is 7 nC.
a) 6.311E+00 V/m2
b) 6.943E+00 V/m2
c) 7.637E+00 V/m2
d) 8.401E+00 V/m2
e) 9.241E+00 V/m2
3) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=2e}$, ${\displaystyle q_{2}=-9e}$, and ${\displaystyle q_{3}=5e}$?
a) 3.961E+01 degrees
b) 4.357E+01 degrees
c) 4.793E+01 degrees
d) 5.272E+01 degrees
e) 5.799E+01 degrees
#### c05 T1
1)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.83 m if a=0.7 m, b=1.8 m. The total charge on the rod is 9 nC.
a) 6.897E+00 V/m2
b) 7.587E+00 V/m2
c) 8.345E+00 V/m2
d) 9.180E+00 V/m2
e) 1.010E+01 V/m2
2) A large thin isolated square plate has an area of 9 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 1 mm from the center of the plate's surface?
a) 2.357E+01 N/C
b) 2.593E+01 N/C
c) 2.852E+01 N/C
d) 3.137E+01 N/C
e) 3.451E+01 N/C
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.914E+01 degrees
b) 6.506E+01 degrees
c) 7.157E+01 degrees
d) 7.872E+01 degrees
e) 8.659E+01 degrees
4)
A ring is uniformly charged with a net charge of 9 nC. The radius of the ring is R=1.6 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.73 m (on axis) away from the loop's center?
a) 7.415E+09 N/C2
b) 8.156E+09 N/C2
c) 8.972E+09 N/C2
d) 9.869E+09 N/C2
e) 1.086E+10 N/C2
#### c05 T2
1) A large thin isolated square plate has an area of 3 m2. It is uniformly charged with 5 nC of charge. What is the magnitude of the electric field 3 mm from the center of the plate's surface?
a) 9.412E+01 N/C
b) 1.035E+02 N/C
c) 1.139E+02 N/C
d) 1.253E+02 N/C
e) 1.378E+02 N/C
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.9 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.54 m if a=1.0 m, b=2.0 m. The total charge on the rod is 3 nC.
a) 1.665E+00 V/m2
b) 1.831E+00 V/m2
c) 2.014E+00 V/m2
d) 2.216E+00 V/m2
e) 2.437E+00 V/m2
3)
A ring is uniformly charged with a net charge of 3 nC. The radius of the ring is R=1.8 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=1.1 m (on axis) away from the loop's center?
a) 3.159E+09 N/C2
b) 3.475E+09 N/C2
c) 3.823E+09 N/C2
d) 4.205E+09 N/C2
e) 4.626E+09 N/C2
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=4e}$?
a) 5.377E+01 degrees
b) 5.914E+01 degrees
c) 6.506E+01 degrees
d) 7.157E+01 degrees
e) 7.872E+01 degrees
### c05 U0
1)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.5 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.33 m (on axis) away from the loop's center?
a) 1.353E+09 N/C2
b) 1.488E+09 N/C2
c) 1.637E+09 N/C2
d) 1.801E+09 N/C2
e) 1.981E+09 N/C2
2)
${\displaystyle E_{z}(x=0,z)=\int _{-a}^{b}f(x,z)dx}$
is an integral that calculates the z-component of the electric field at point P situated above the x-axis where a charged rod of length (a+b) is located. The distance between point P and the x-axis is z=1.8 m. Evaluate ${\displaystyle f(x,y)}$ at x=0.5 m if a=0.67 m, b=2.4 m. The total charge on the rod is 9 nC.
a) 5.465E+00 V/m2
b) 6.012E+00 V/m2
c) 6.613E+00 V/m2
d) 7.274E+00 V/m2
e) 8.002E+00 V/m2
3)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=6\times 10^{-7}{\text{m}}}$.what angle does the force on ${\displaystyle q_{2}}$ make above the ${\displaystyle -x}$ axis if ${\displaystyle q_{1}=3e}$, ${\displaystyle q_{2}=-8e}$, and ${\displaystyle q_{3}=5e}$?
a) 5.062E+01 degrees
b) 5.569E+01 degrees
c) 6.125E+01 degrees
d) 6.738E+01 degrees
e) 7.412E+01 degrees
4)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=2\times 10^{-7}{\text{m}}}$. What is the magnitude of the net force on ${\displaystyle q_{2}}$ if ${\displaystyle q_{1}=1e}$, ${\displaystyle q_{2}=-7e}$, and ${\displaystyle q_{3}=3e}$?
a) 4.171E-14 N
b) 4.588E-14 N
c) 5.047E-14 N
d) 5.551E-14 N
e) 6.107E-14 N
#### c05 U1
1)
A ring is uniformly charged with a net charge of 2 nC. The radius of the ring is R=1.5 m, with its center at the origin and oriented normal to the z axis as shown. what is the magnitude of the electric field at a distance z=0.33 m (on axis) away from the loop's center?
a) 1.353E+09 N/C2
b) 1.488E+09 N/C2
c) 1.637E+09 N/C2
d) 1.801E+09 N/C2
e) 1.981E+09 N/C2
2)
Three small charged objects are placed as shown, where ${\displaystyle b=2a}$, and ${\displaystyle a=4\times 10^{-7}{\text{m}}}$
|
2019-11-14 01:46:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 791, "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": 0.9190263152122498, "perplexity": 491.64874483724094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667767.6/warc/CC-MAIN-20191114002636-20191114030636-00183.warc.gz"}
|
http://mathhelpforum.com/statistics/2845-prob-dice-question.html
|
# Math Help - prob dice question
1. ## prob dice question
you throw 1 dice until you get K times the number '6'
X = the number of your throws.
what is P(X) ?
2. The probability that N throws contains (K-1) '6's somewhere in the first (N-1) throws, with a '6' on the N-th throw is $\binom{N-1}{K-1} \left(\frac56\right)^{(N-1)-(K-1)} \left(\frac16\right)^{K-1} \frac16$.
3. how can i prove that ∑ p(X)=1 ?
4. Originally Posted by amitbern
how can i prove that ∑ p(X)=1 ?
I'm not quite sure what you want to prove here. This sum has to be 1
since with probability 1 you will eventually reach K sixes. So the sum
of the probabilities of the number of throws needed for this is 1.
RonL
5. I know that it's supposed to be 1,
but how can i prove it mathematically (Newton’s binomial )?
|
2015-03-27 17:37:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7682281732559204, "perplexity": 1100.5958455267148}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131296587.89/warc/CC-MAIN-20150323172136-00154-ip-10-168-14-71.ec2.internal.warc.gz"}
|
http://tex.stackexchange.com/questions/162722/lines-in-epsfig-have-corners
|
# Lines in epsfig have corners
I have a problem with epsfig. I want to draw some graphs. I created the picture with xfig and got the following code:
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
\reset@font\fontsize{#1}{#2pt}%
\fontfamily{#3}\fontseries{#4}\fontshape{#5}%
\selectfont}%
\fi\endgroup%
\begin{picture}(2280,2011)(1336,-1925)
\thinlines
{\color[rgb]{0,0,0}\put(1801,-61){\line( 1,-1){900}}
\put(2701,-961){\line(-1, 0){900}}
\end{picture}
I included it with the following code:
\begin{figure}[H]
\begin{minipage}{\linewidth}
\begin{center}
\input{*/File.tex}
\end{center}
\end{minipage}
\end{figure}
The problem is: The lines seem off, they are kind of irregular.
-
Your question seems unrelated to epsfig (the epsfig latex package has been obsolete for 20 years) you seem to be using the xfig driver that produces latex picture mode, this can only make very approximate shapes, you want to use the one that makes two files, an eps file with all the lines, and a .tex file with just the text. – David Carlisle Feb 27 at 13:30
|
2014-11-29 07:50:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9665802717208862, "perplexity": 1989.9077505377757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931014049.81/warc/CC-MAIN-20141125155654-00014-ip-10-235-23-156.ec2.internal.warc.gz"}
|
http://msp.org/involve/2018/11-1/p04.xhtml
|
#### Vol. 11, No. 1, 2018
Recent Issues
The Journal Cover Page Editorial Board Editors’ Addresses Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Ethics Statement Subscriptions Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)
Merging peg solitaire on graphs
### John Engbers and Ryan Weber
Vol. 11 (2018), No. 1, 53–66
DOI: 10.2140/involve.2018.11.53
##### Abstract
Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices $x$ and $y$, with $y$ also adjacent to a hole on vertex $z$, and jumps the peg on $x$ over the peg on $y$ to $z$, removing the peg on $y$. The goal of the game is to reduce the number of pegs to one.
We introduce the game merging peg solitaire on graphs, where a move takes pegs on vertices $x$ and $z$ (with a hole on $y$) and merges them to a single peg on $y$. When can a configuration on a graph, consisting of pegs on all vertices but one, be reduced to a configuration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars.
##### Keywords
peg solitaire, games on graphs, graph theory
Primary: 05C57
|
2017-08-23 13:44:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "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": 0.17495901882648468, "perplexity": 1783.8280685822967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886120573.0/warc/CC-MAIN-20170823132736-20170823152736-00310.warc.gz"}
|
https://www.jobilize.com/economics/course/23-4-the-national-saving-and-investment-identity-by-openstax?qcr=www.quizover.com&page=3
|
# 23.4 The national saving and investment identity (Page 4/16)
Page 4 / 16
In the short run, trade imbalances can be affected by whether an economy is in a recession or on the upswing. A recession tends to make a trade deficit smaller, or a trade surplus larger, while a period of strong economic growth tends to make a trade deficit larger, or a trade surplus smaller.
As an example, note in [link] that the U.S. trade deficit declined by almost half from 2006 to 2009. One primary reason for this change is that during the recession, as the U.S. economy slowed down, it purchased fewer of all goods, including fewer imports from abroad. However, buying power abroad fell less, and so U.S. exports did not fall by as much.
Conversely, in the mid-2000s, when the U.S. trade deficit became very large, a contributing short-term reason is that the U.S. economy was growing. As a result, there was lots of aggressive buying in the U.S. economy, including the buying of imports. Thus, a rapidly growing domestic economy is often accompanied by a trade deficit (or a much lower trade surplus), while a slowing or recessionary domestic economy is accompanied by a trade surplus (or a much lower trade deficit).
When the trade deficit rises, it necessarily means a greater net inflow of foreign financial capital . The national saving and investment identity teaches that the rest of the economy can absorb this inflow of foreign financial capital in several different ways. For example, the additional inflow of financial capital from abroad could be offset by reduced private savings, leaving domestic investment and public saving unchanged. Alternatively, the inflow of foreign financial capital could result in higher domestic investment, leaving private and public saving unchanged. Yet another possibility is that the inflow of foreign financial capital could be absorbed by greater government borrowing, leaving domestic saving and investment unchanged. The national saving and investment identity does not specify which of these scenarios, alone or in combination, will occur—only that one of them must occur.
## Key concepts and summary
The national saving and investment identity is based on the relationship that the total quantity of financial capital supplied from all sources must equal the total quantity of financial capital demanded from all sources. If S is private saving, T is taxes, G is government spending, M is imports, X is exports, and I is investment, then for an economy with a current account deficit and a budget deficit:
A recession tends to increase the trade balance (meaning a higher trade surplus or lower trade deficit), while economic boom will tend to decrease the trade balance (meaning a lower trade surplus or a larger trade deficit).
## Problems
Imagine that the U.S. economy finds itself in the following situation: a government budget deficit of $100 billion, total domestic savings of$1,500 billion, and total domestic physical capital investment of $1,600 billion. According to the national saving and investment identity, what will be the current account balance? What will be the current account balance if investment rises by$50 billion, while the budget deficit and national savings remain the same?
[link] provides some hypothetical data on macroeconomic accounts for three countries represented by A, B, and C and measured in billions of currency units. In [link] , private household saving is SH, tax revenue is T, government spending is G, and investment spending is I.
Macroeconomic accounts
A B C
SH 700 500 600
T 00 500 500
G 600 350 650
I 800 400 450
1. Calculate the trade balance and the net inflow of foreign saving for each country.
2. State whether each one has a trade surplus or deficit (or balanced trade).
3. State whether each is a net lender or borrower internationally and explain.
Imagine that the economy of Germany finds itself in the following situation: the government budget has a surplus of 1% of Germany’s GDP; private savings is 20% of GDP; and physical investment is 18% of GDP.
1. Based on the national saving and investment identity, what is the current account balance?
2. If the government budget surplus falls to zero, how will this affect the current account balance?
Demand is the various quantities of goods and services that consumer(s)are willing and able to purchase at a price within a time
What is demand
Distinguish between cross elasticity and income elasticity of demand
Distinguish between cross elasticity and income elasticity of demand
Ruth
if change in the demand of the commodity with respect to change in demand of the substitute or other product called cross elasticity
Hamza
and. if change in the demand of the commodity due to change in the income . called income elasticity
Hamza
👍
Vipul
Cross elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a small change in price of another commodity whiles Income elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a small change in income of it's consumers
Afriyie
but these are book wordings
Hamza
income elasticity of demand shows how quantity demanded changes due to changes in income on the other hand cross elasticity refers to how the quantity demanded of a particular good alers given a change in the price of another good.
Keysie
what is the competitive demand
Income
Sanni
With regards to coal shortage and manicipal debts the what form of intervention do you think Eskom can put in place.
economic growth of Bhutan
please, explain all the mathematics terms used in economics
nelson
The answer is: little more than high school algebra and graphs.
Tere
what is the effect of inflation in GDP
Not only real GDP but also nominal GDP will decrease
Aqib
yep. Inflation has an influence not only GDP but interest rate also.
Hamza
The pound weakens so imports become more expensive and exports lose value - lower GDP.
Rebecca
why do inflation effect economic
explain in detail what is economic what is scarcity what is alternate uses
What is law of demand
Hilary
economic as a science refers to study of human resource
Kaunda
Law of demand- With all the factors remaining same if price increases of a commodity, the quantity of demand of that commodity decreases and vice versa
Dey
Thanks dey sunita
Hilary
What is law of supply
Hilary
what are the factors that affect demand
what are the factors that affect demand of a good
Elly
what are the factors that affect demand of a good
Elly
what are the factors that affect demand of a commodity
Elly
1. the price of the product 2. the price of other products 3. consumers income 4. expectation of future changes in price 5. taste and preference etc.
ALI
Change in price
Hilary
1. price related of commodities 2. consumers income 3. the condition or season of the commodities
Tsai
decrease in demand of substitute increase in demand of constituent change in quantity and other environmental factors
Hamza
Nd consumer's income
Hamza
what course scarcity
Scarcity is the limited availability of a commodity, which may be in demand in the market or by the commons. Scarcity also includes an individual's lack of resources to buy commodities. The opposite of scarcity is abundance.
Marc
Reasons that explain why the division of labor increases an economy's level of production
Please I don't understand the meaning and the concept of economics as a science
economics as a science refers to the study of human behavior. how they make decisions etc
Saidou
economics is science because it uses scientific methods in analysing societal problems.. observation experimentation and conclusion inherently are used to analyse. however it is not pure science but social science because it studies human and it's environs
Bonney
what's elasticity of demand
are u asking because you don't know or what
Stephen
A measure of the responsiveness of a product demanded to a change in market price
Yuusuf
the degree of responsiveness of a product demanded to a little change in the price
Saidou
the degree of responsiveness of quantity demanded of a commodity to the changes in the price if the commodity in question, changes in the price of other related commodities and changes in the income of consumer
Bonney
IYke
it is the exchange of goods and services between countries
Bonney
it's the exchange of goods and services from one foreign country to another
Israel
|
2020-03-30 01:27:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.351132333278656, "perplexity": 2964.673595907778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496330.1/warc/CC-MAIN-20200329232328-20200330022328-00002.warc.gz"}
|
https://notes.billmill.org/programming/neural_networks/tinygrad.html
|
Oct 31, 2022
Library from George Hotz focused on creating the smallest possible useful deep learning library.
Check the examples dir, for ex the mnist example that only uses 145 lines
# from the root dir of the repo
python examples/mnist_gan.py
on my computer each epoch takes about 3 minutes, so training the whole net (300 epochs) would take about 15 hours.
I saw in the source that there is a metal interface, but no idea if it's being used.
↑ up
|
2022-12-09 12:33:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4292002022266388, "perplexity": 3399.5171489734953}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711396.19/warc/CC-MAIN-20221209112528-20221209142528-00386.warc.gz"}
|
http://openstudy.com/updates/4f16d1ebe4b0d30b2d5770eb
|
## A community for students. Sign up today
Here's the question you clicked on:
## anonymous 5 years ago Is this funtion continuous in point (0,0) and (0,1) f(x,y)= x^2*sin(y/x) xy is not 0 0 xy=0
• This Question is Closed
1. Zarkon
have you tried using the $$\epsilon,\delta$$ definition of continuity on the points given
#### Ask your own question
Sign Up
Find more explanations on OpenStudy
Privacy Policy
|
2017-01-24 01:29:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6958902478218079, "perplexity": 4510.027963693923}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560283475.86/warc/CC-MAIN-20170116095123-00372-ip-10-171-10-70.ec2.internal.warc.gz"}
|
http://mathoverflow.net/questions/134134/simplifying-a-finite-difference-sum-that-represents-a-model-posterior-to-avoid-n
|
# Problem
Is it possible to simplify/rewrite the following expression, preferably without explicit sums, such that it can be computed without numerical issues when the $n_*$ are in the range of thousands?
$$\left( \prod_{i,j} {n_{ij} \choose n_{ij0}} \right) \sum_{k_{11}}^{n_{111}} {n_{111} \choose k_{11}}(-1)^{k_{11}} \dots \sum_{k_{33}}^{n_{331}} {n_{331} \choose k_{33}}(-1)^{k_{33}} \prod_i \frac{1}{1 + \sum_j n_{ij0} + k_{ij}} \prod_j \frac{1}{1 + \sum_i n_{ij0} + k_{ij}}$$ (There are 9 sums in total for $k_{11},k_{12},k_{13},k_{21}...k_{33}$, $i$ and $j$ always range from 1 to 3.)
For example by rewriting it in terms of digamma functions, like in this simplified problem: $${m+n \choose m} \sum_{i=0}^n {n \choose i}(-1)^i \frac{1}{(1+m+i)^2} = \frac{\psi(m+n+2) - \psi(m+1)}{m+n+1}$$ where $\psi(x)$ is the digamma function. Any characterization or helpful reference would be much appreciated.
# Background
I am working with the following integral $$\int_0^1 \dots \int_0^1 \prod_{i,j \in \{1,2,3\}^2} {n_{ij} \choose n_{ij0}} (p_iq_j)^{n_{ij0}}(1-p_iq_j)^{n_{ij1}}f(p_i;\alpha_i, \beta_i)f(q_j;\alpha_j,\beta_j)dp_idq_j$$ where $f(p_i;\alpha_i,\beta_i)$ is the beta distribution for $p_i$ with parameters $\alpha_i$ and $\beta_i$, and correspondingly for $f(q_j;\alpha_j,\beta_j)$. The $n_*$ variables are integer constants in the range of thousands. The $i,j$ indices always range from $1$ to $3$. The expression itself is the model posterior for a 3x3 matrix of binomials with a certain parameter structure. Specifically, each cell contains two counts, successes and failures, and the success probability for cell $i,j$ is $p_iq_j$. The integral above is the model posterior of this model.
By using the binomial theorem and the definition of the beta function I can solve the integral and get (splitted into two lines): $$\left( \prod_{i,j} {n_{ij} \choose n_{ij0}} \right) \sum_{k_{11}}^{n_{111}} {n_{111} \choose k_{11}}(-1)^{k_{11}} \dots \sum_{k_{33}}^{n_{331}} {n_{331} \choose k_{33}}(-1)^{k_{33}}\cdot$$ $$\prod_i \frac{B(\alpha_i + \sum_j n_{ij0} + k_{ij}, \beta_i)}{B(\alpha_i,\beta_i)} \prod_j \frac{B(\alpha_j + \sum_i n_{ij0} + k_{ij}, \beta_j)}{B(\alpha_j,\beta_j)}$$
where $B(x,y)$ is the Beta function.
For my own purposes it is enough if I can simplify it in the case $\alpha_i = \beta_i = \alpha_j = \beta_j = 1$, that is the expression above: $$\left( \prod_{i,j} {n_{ij} \choose n_{ij0}} \right) \sum_{k_{11}}^{n_{111}} {n_{111} \choose k_{11}}(-1)^{k_{11}} \dots \sum_{k_{33}}^{n_{331}} {n_{331} \choose k_{33}}(-1)^{k_{33}} \prod_i \frac{1}{1 + \sum_j n_{ij0} + k_{ij}} \prod_j \frac{1}{1 + \sum_i n_{ij0} + k_{ij}}$$
A potentially helpful observation is that it can be written as the finite difference $$\left( \prod_{i,j} {n_{ij} \choose n_{ij0}} \right)\Delta_{k_{11}}^{n_{111}} \dots \Delta_{k_{33}}^{n_{331}} \prod_i \frac{B(\alpha_i + \sum_j n_{ij0} + k_{ij}, \beta_i)}{B(\alpha_i,\beta_i)} \prod_j \frac{B(\alpha_j + \sum_i n_{ij0} + k_{ij}, \beta_j)}{B(\alpha_j,\beta_j)}$$ where $\Delta_k^n$ is the finite difference operator for the $n$:th difference with respect to $k$.
# Numerical issues
Even though the integral is solved it is hard to compute numerically due to the large binomial coefficients. The issue is that the terms of the sum have to cancel properly for the expression to be between 0 and 1. For big $n$ this does not happen due to the addition of very small and very large floating point terms. It seems like both the lower and higher order bits are important for the cancellation to happen properly. If I use 2000 bit floats I get a reasonable answer (which I also can verify for small $n_*$), this however takes too long to compute.
I have also tried (also in combination):
• Formulating it as a recursion based on finite differences and thus getting rid of explicit binomial coefficients also failed.
• Adding terms of similar order of magnitude first.
• Various summation algorithms like Kahan summation.
-
|
2014-04-17 07:22:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9160893559455872, "perplexity": 194.6479389117794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00586-ip-10-147-4-33.ec2.internal.warc.gz"}
|
https://www.nature.com/articles/ncomms4328?error=cookies_not_supported&code=ed0710c1-a731-4d7e-ae60-fbcd6e951412
|
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Optical signatures of silicon-vacancy spins in diamond
## Abstract
Colour centres in diamond have emerged as versatile tools for solid-state quantum technologies ranging from quantum information to metrology, where the nitrogen-vacancy centre is the most studied to date. Recently, this toolbox has expanded to include novel colour centres to realize more efficient spin-photon quantum interfaces. Of these, the silicon-vacancy centre stands out with highly desirable photonic properties. The challenge for utilizing this centre is to realize the hitherto elusive optical access to its electronic spin. Here we report spin-tagged resonance fluorescence from the negatively charged silicon-vacancy centre. Our measurements reveal a spin-state purity approaching unity in the excited state, highlighting the potential of the centre as an efficient spin-photon quantum interface.
## Introduction
Diamond is a promising spin-free material for a range of novel technologies that make use of its optically accessible lattice impurities (colour centres). The most eminent colour centre in diamond to date, the negatively charged nitrogen-vacancy (NV) centre, has attracted great interest in recent years for quantum information processing1,2,3, electromagnetic field sensing4,5,6,7 and biomarking8,9. The NV centre owes its success to the optically accessible spin triplet in the ground state with room temperature electron-spin coherence times of up to 2 ms in ultrapure bulk diamond10. This allows for initialization11, coherent control12,13,14,15 and projective read-out11 to be performed with high fidelity. However, for photon-based quantum communication architectures, the challenge for the NV centre and most other diamond colour centres is to overcome the fundamental limitation of only weak emission into the zero-phonon line (ZPL), for example, by incorporating them in nanophotonic cavities16.
An alternative approach is to identify other centres with superior photonic (but comparable spin) properties to NV. Recent studies of such centres include chromium-17,18, xenon-19 nickel-20 and possibly oxygen-related centres21, as well as the negatively charged silicon-vacancy (SiV) centre22,23,24. The latter has the advantage of being the brightest reported colour centre in diamond24. Its emission is concentrated to about 80% into its ZPL even at room temperature23, thus making it a promising single-photon source. One major challenge is to achieve direct (optical) access to the electronic spin degree of freedom similar to that of NVs. Previous electron-spin resonance measurements on ensembles of neutrally charged SiV centres have identified an S=1 ground state25,26,27, suggesting strongly that the negatively charged centres should be associated with an S=1/2 ground state. However, optical access to this electronic spin has remained elusive for both charge states.
Here, we report the direct accessing and spin-selective population of the negatively charged SiV centre excited states, achieved via resonant excitation and fluorescence under a magnetic field. In low-strain bulk diamond spin-selective excitation under finite magnetic field reveals a spin purity approaching unity in the excited state. We also investigate the effect of strain on the centres in nanodiamonds and discuss how spin selectivity in the excited state remains accessible in this regime.
## Results
### Characterization of the SiV− centre in a magnetic field
The SiV centre in diamond consists of a silicon atom and a vacancy in a split-vacancy configuration28,29,30, replacing two neighbouring carbon atoms in a diamond matrix along the 111 axes (see Fig. 1a, inset). The substitutional silicon atom relaxes to the middle of the two lattice sites creating an inversion symmetric complex29,31. A total of 10 electrons are associated with this centre, as can be counted from the SiV structure in Fig. 1a, inset. One extra electron is captured to form the negatively charged complex (SiV). With 11 electrons in total, a single electron remains unpaired resulting in an effective spin-1/2 for the negatively charged state of the centre.
We study two types of samples containing SiV centres: a diamond film grown homoepitaxially via chemical vapour deposition (CVD) on a high pressure high temperature bulk diamond32 and CVD-grown diamond nanocrystals on an Ir substrate23. The former offers a relatively homogenous low-strain environment preserving the intrinsic symmetry of the centre. On the other hand, the latter, from which the SiV fluorescence can be extracted with high efficiency24, provides a strained environment modifying the emission properties from centre to centre24,32.
At cryogenic temperatures, the photoluminescence spectrum of the ZPL reveals a characteristic fine structure composed of four transitions around 737 nm, as displayed in Fig. 1a (transitions labelled from A to D). This spectral signature originates from the doubly split ground and excited states shown in Fig. 1b33,34. An applied magnetic field further lifts any degeneracies of these four transitions, as can be seen in Fig. 1c for a single SiV centre in a nanocrystal (blue spectrum), and for an ensemble of SiV centres in bulk diamond (red spectrum). Common to both spectra is the measured quadruplet splitting in the optical transitions, which is consistent with an energy level scheme based on spin-1/2 ground and excited states (Fig. 1d)35. Theoretical analysis based on density functional theory28, as confirmed by a recent ab initio study36, has determined a D3d symmetry for the SiV centre and assigned the optical ground and excited states to Eg and Eu states, respectively. The system can then be described by a Hamiltonian comprising orbital and spin Zeeman terms, the Jahn–Teller effect, which partially lifts the orbital degeneracies and a spin-orbit coupling term (L·S). Group theoretic analysis predicts that, in the case of the SiV centre, only the LzSz component (z along the C3 axis of the centre, that is, the [111] axis of the lattice) of the spin-orbit operator acts on the Eg and Eu states31, providing an inherent quantization axis along the 111 directions but leaving the spin as a good quantum number analogous to the NV centres37. The degree of spin mixing introduced by the Jahn–Teller and Zeeman parts of the Hamiltonian can be quantified by defining spin purity as the probability of finding a state in one given spin projection ms=±1/2 only.
### Resonance fluorescence at 0 T
Resonance fluorescence through state-selective excitation is a powerful tool to reveal the internal structure of quantum emitters. We first study an ensemble of SiV centres in a low-strain bulk diamond at 0 T. Figure 2a displays the SiV spectrum (red curve) observed when driving transition B resonantly (as labelled in Fig. 1a). Here the laser is suppressed by polarization rejection (see Methods) and contributes only a small fraction to the full spectrum (see Methods and Supplementary Fig. 1). Although resonant excitation selectively populates one excited state branch, all four transitions are visible indicating a relaxation process between the excited state branches before photon emission. Tuning the laser frequency across transition A, while monitoring the fluorescence of transition C in the spirit of photoluminescence excitation, reveals the absorption profile of transition A (solid red circles in Fig. 2b). The extracted full width at half maximum of ~10 GHz is consistent with the inhomogenous broadening of the ensemble under non-resonant excitation32 due to residual strain in the sample (as evidenced by the shift of transitions along with laser frequency in Fig. 2c). For a single centre in a nanodiamond, the fluorescence spectrum, obtained by driving transition A resonantly, is shown in Fig. 2d (blue curve). Owing to the strain in the crystal, the transitions of this centre are shifted beyond the inhomogenous broadening of the ensemble in bulk diamond, such that the exact spectrum varies from centre to centre. The absorption linewidth of 1.4 GHz for this transition (Fig. 2e) is only an order of magnitude above the radiatively broadened limit (~100 MHz)22,33, which should be reachable straightforwardly using impurity-free diamond substrates, as was shown for NV centres38. This linewidth also marks the minimum Zeeman splitting needed to resolve spin sublevels spectrally under resonant excitation.
### Spin-tagged fluorescence of unstrained SiV− centres
Applying a magnetic field of 4 T to the ensemble of SiV centres in bulk diamond allows us to address excited states with a given spin orientation selectively. First, we drive the transition labelled A2 (see Supplementary Fig. 2) to populate a Zeeman sublevel of the upper branch of the excited state, expected theoretically to be a spin-up projection, as shown in Fig. 3a (ref. 31). The resulting spectrum, shown in Fig. 3b (red shaded curve), is strikingly different from the spectrum obtained under non-resonant excitation (Fig. 3c); only half of the available optical transitions are visible and they originate from two excited states only. This is in stark contrast to thermal distribution at 4 K, which would lead to a finite population probability for all excited states. Here the relaxation process in the excited state takes place only between the two sublevels with the same Zeeman response, that is, same spin projection. To populate an excited state sublevel with the opposite Zeeman response, expected theoretically to be a spin-down projection, we resonantly drive transition B3 (blue double arrow in Fig. 3d). The resulting spectrum, shown in Fig. 3e (blue shaded curve), is strongly anti-correlated with that in Fig. 3b, and the sum of the two spectra produces the full spectrum observed under non-resonant excitation (Fig. 3c). The spectra in Fig. 3b,e further reveal that from the populated excited states, optical transitions to all ground states occur, irrespective of their spin projection. The anti-correlation in the spectra arises from the high degree of spin purity in the excited state, that is, a high probability of finding each excited state in one of the two spin states only. In contrast, what appears as a violation of spin preservation in the optical transitions is a consequence of the orientation of the applied magnetic field affecting mostly the electronic ground state. Indeed, when the magnetic field is not parallel to the SiV axis (as is the case in our experiment, see Methods) the original quantization axis of the centre is tilted because of the magnetic field-induced non-diagonal terms in the spin-orbit basis, which in turn induces an apparent spin mixing. With an angle of 54.7° between the 111 SiV axes and the 4-T magnetic field along [001] (see Supplementary Fig. 3), the spin-orbit coupling strength in the excited state is still dominant and the spin-projection sublevel in the higher-lying excited state sustains a spin purity of 97% (Hepp et al.31), which agrees well with the 4.1±1.2% overlap between the two anti-correlated spectra of Fig. 3 (see Methods and Supplementary Fig. 4). On the other hand, the spin-orbit coupling strength in the ground state is comparable to the off-axis contributions of the Zeeman Hamiltonian (at 4 T) resulting in a degree of spin purity ranging from 50 to 80% for the ground-state branches31. This effective spin mixing gives rise to finite intensity in all optical transitions starting from a given excited state.
### Spin-tagged fluorescence of strained single SiV− centres
We now investigate the influence of strain through resonance fluorescence from a single centre located in a nanodiamond. A magnetic field of 2 T allows us to optically resolve the individual transitions of the centre shown in Fig. 1c. The excitation laser is brought into resonance with transition A1 (as shown in Fig. 4a) and transition A2. This leads to fluorescence spectra with selective population of the spin-up (blue curve) and spin-down (red curve) sublevels in the higher branch of the excited state, respectively, as seen in Fig. 4b. The signature of spin selectivity demonstrated in bulk is only partially observed in the resulting fluorescence spectra shown in Fig. 4b. While the transitions B1–B4, originating directly from the higher-lying branch of the excited state, still exhibit a high degree of spin selectivity (hence spin purity), the transitions C1–D4, which originate from the lower branch after a relaxation step, do not display such spin selectivity. This breakdown is induced by the strong strain field not oriented with the symmetry axis of the centre. This can be understood based on the model presented in (Hepp et al.31), where an additional term for the strain perturbation is added to the total Hamiltonian to account for the strain field in the nanodiamonds (not discussed here).
Bypassing the interbranch relaxation mechanism by directly populating a spin sublevel of the lower branch allows us to access the degree of spin purity in this branch. This is illustrated in Fig. 4c for the case of populating the spin-up sublevel. Phonon-assisted excitation to the upper branch is strongly suppressed at 4 K owing to the large energy difference between the two branches for this centre. In the resulting spectra, shown in Fig. 4d, the transitions resonantly populating spin-up and spin-down sublevels are indicated by blue and red arrows, respectively. Under these conditions, the contrast between fluorescence intensity originating from the two spin orientations (red and blue filled curves) is recovered to above 90%. This evidences high-spin purity within both excited state branches. Therefore, by selecting the driven transitions, spin-selective optical access to SiV centres can be achieved in strained nanodiamonds as well as low-strain bulk diamond.
## Discussion
A natural extension of this work is to align the C3 symmetry axis of a SiV centre to the magnetic field either by rotating the samples (which was technically not possible in our set-up) or by implanting SiV centres in a [111]-oriented diamond crystal. Both approaches are expected to restore more than 90% spin purity in the ground state and near-unity spin purity in the excited state eliciting the inherent optical selection rules linked to the spin orientation. These properties mark the SiV centre desirable for all applications that require optical access to well-defined spin states. All-optical ultrafast spin manipulation techniques or optically detected magnetic resonance can then give access to the electronic spin coherence of the pure ground state. Full coherent control of the SiV spin state along with fluorescence-detection-based spin initialization and read-out will be within reach for the realization of a highly efficient spin-photon quantum interface.
## Methods
### Sample fabrication
The ensemble of SiV centres in bulk was grown using a hot filament chemical vapour deposition technique33, where SiV centres are created in situ owing to residual silicon contamination of the CVD reactor. A Ib high pressure high temperature diamond (Sumitomo) with [001] orientation was used as a substrate, and was overgrown with a high-quality diamond film. The homoepitaxial growth minimizes stress arising from thermal expansion mismatch of substrate and diamond, and helps reduce dislocations in the grown diamond lattice. High crystalline quality is achieved by applying optimized growth conditions involving a low methane fraction (0.26% CH4 in H2) and slow growth. The thickness of the diamond film of 80–100 nm was estimated by growing a diamond film on a non-diamond substrate with identical growth conditions where the thickness could easily be measured.
Single SiV centres in nanodiamonds were CVD grown on a silicon substrate, which was covered by an intermediate yttria-stabilized-zirconia buffer layer and atop a 150-nm iridium layer14. This iridium layer allows for optimized growth conditions, and reduces the amount of silicon incorporated into the nanodiamonds from the substrate during growth. Before growth, the substrate is seeded with deagglomerated synthetic nanodiamonds with sizes up to 30 nm (Microdiamant Liquid Diamond MSY). The aqueous solution of these diamonds is diluted appropriately to achieve a seed density of roughly 2.5 seeds per μm2. The density of crystals containing SiV centres is estimated to be around one per 50 × 50 μm2. The seeded substrates were exposed to microwave plasma-assisted CVD process for 25 min, using a hydrogen-plasma containing 1% methane. The gas pressure was 30 mbar and the microwave power was 2,000 W. These conditions resulted in a crystal size of about 130 nm with a s.d. of 30 nm.
### Confocal microscope set-up
The sample is mounted on a piezo-driven three-axis translation stage (attocube ANPz 101/LT and two ANPx 101/LT) in a helium bath cryostat (T=4.2 K) at the centre of a fixed orientation, tunable 7 T superconducting magnet (Cryogenic) in Faraday configuration. Excitation of the SiV centres is performed non-resonantly at 700 nm (Coherent Mira 900-CW) and resonantly using a frequency-tunable external-cavity diode laser around 737 nm (Toptica DL 100 Pro Design). SiV spectra were recorded using a spectrometer with ~40 μeV resolution (PI Acton). All measurements are performed using a fibre-based confocal microscope with polarization-controlled excitation and collection (polarizers: Thorlabs LPVIS050-MP, half-wave plates: Thorlabs AHWP05M-980). Light is focused onto the sample with a numerical aperture=0.68 aspheric lens (Thorlabs C330TME-B), and residual laser light in the detection arm is removed with either a 720-nm longpass filter (third millennium) or with a polarizer set perpendicularly to the incoming laser polarization. With the latter technique, laser suppression up to 5 × 10−7 was reached.
### Determination of the photon ratio
In the resonant spectrum of Fig. 2a, the integrated intensity of each transition is evaluated through Lorentzian fits (see Supplementary Fig. 1a with the following results: I(A)=0.14, I(B)+I(laser)=0.66, I(C)=1.0, I(D)=0.46 (the intensities are normalized to that of transition C. Uncertainties on fits are below 0.01 for all transitions). First, a lower bound can be set by assuming that no emission from transition B is collected (I(B)=0). With this assumption, a ratio between laser and SiV emission of 1:2.4 is obtained.
Next, we evaluate I(B) when this transition is driven. The intensities in the resonant spectrum can be compared with the respective ones for non-resonant excitation (Fig. 1a), labelled as I′. These are I′(A)=0.058, I′(B)=0.20, I′(C)=1.0 and I′(D)=0.46. The ratios I/I′ differ for transitions A, C and D, with values ranging from 0.43 to 1.00. Assuming that I(B)/I′(B) lies within that range, the intensity of transition B in the resonant case can be expected between 0.20 and 0.47. This leads to an intensity ratio of the laser to SiV emission between the extrema of 1:3.9 and 1:11.
Another way of evaluating the intensity of transition B when resonantly driven is by applying the same method of intensity comparison with the spectrum where transition A is resonantly excited, knowing that transitions A and B share the same excited state. In the spectrum with A driven resonantly (see Supplementary Fig. 1b), the following intensities I″ are obtained: I″(B)=0.33, I″(C)=1 and I″(D)=0.27. The ratios I/I″ range from 0.59 to 1.00, resulting in the intensity of transition B being between 0.33 and 0.57. This finally leads to a ratio between laser and SiV emission ranging from 1:5.9 to 1:24.
### Contrast between transition subsets under excitation
Here we describe how we can extract information about the degree of spin conservation in the excited state by comparing transition intensities when populating spin-up and spin-down excited states by driving transitions A2 and B3, respectively (Fig. 3b,e). For this analysis, we restrict ourselves to intensities in transitions C1 to D4, originating from the lower-lying excited state. This is to exclude contributions from the excitation laser. Visual inspection of the spectra in Supplementary Fig. 4 (red curve for driving transition A2 and blue curve when driving transition B3) shows that when populating the spin-down excited states (red spectrum), contributions to transitions originating from spin-up excited states are within the noise of the spectrum. In contrast, when populating the spin-up excited states (blue spectrum), a finite contribution into the ‘forbidden’ transition C2 is visible. We use this intensity to estimate the branching ratio of the relaxation mechanism in the excited state.
We now consider the blue spectrum in Supplementary Fig. 4. According to Fermi’s golden rule, the transition rate Г for a specific transition is given by , where ħ is the reduced Planck’s constant, ρi is the probability of finding the system in the excited state at the origin of the transition, ϕi is the electronic wavefunction of the excited state, D is the dipole operator and ϕf is the electronic wavefunction in the ground state. If we assume that the dipole matrix element is constant for all transitions, then the transition rates become proportional to the populations ρi in the excited state and give access to the branching ratios between the two spin states. This assumption is not entirely correct, but since C2 is the brightest transition under non-resonant excitation, the transition matrix element will be larger. Consequently, the branching ratio will be overestimated when considering the blue spectrum yielding a lower limit for spin conservation.
Focusing on the blue spectrum for the spin-down transitions, the transitions can be fitted with a series of five Lorentzians (blue shaded curve in Supplementary Fig. 4): four for the ‘allowed’ transitions C1, D1, C3 and D3, and one for the ‘forbidden’ transition C2. The other ‘forbidden’ transitions are too weak to be considered. The intensity ratio between transition C2 and the total emission into the five transitions is 3±1.2%. The additional error made by neglecting the intensities into the forbidden transitions D2, C4 and D4 can be calculated by considering the respective intensities in the red spectrum. A Lorentzian fit of the four visible transitions reveals that transition C2 constitutes 73.2±0.9% of the total red spectrum. Therefore, the branching ratio between transitions C2, D2, C4 and D4 (the forbidden ones) and transitions C1, D1, C3 and D3 is 4.1±1.2%, confirming the strong spin conservation of the relaxation mechanism in the excited state for centres in bulk diamond.
How to cite this article: Müller, T. et al. Optical signatures of silicon-vacancy spins in diamond. Nat. Commun. 5:3328 doi: 10.1038/ncomms4328 (2014).
## References
1. Wrachtrup, J. & Jelezko, F. Processing quantum information in diamond. J. Phys. Condens. Matter 18, S807–S824 (2006).
2. Neumann, P. et al. Quantum register based on coupled electron spins in a room temperature solid. Nat. Phys. 6, 249–253 (2010).
3. Bernien, H. et al. Heralded entanglement between solid-state qubits separated by three metres. Nature 497, 86–90 (2013).
4. Degen, C. L. Scanning magnetic field microscope with a diamond single-spin sensor. Appl. Phys. Lett. 92, 243111 (2008).
5. Taylor, J. et al. High-sensitivity diamond magnetometer with nanoscale resolution. Nat. Phys. 4, 810–816 (2008).
6. Dolde, F. et al. Electric-field sensing using single diamond spins. Nat. Phys. 7, 459–463 (2011).
7. Grinolds, M. S. et al. Nanoscale magnetic imaging of a single electron spin under ambient conditions. Nat. Phys. 9, 215–219 (2013).
8. Mohan, N. et al. In vivo imaging and toxicity assessments of fluorescent nanodiamonds in Caenorhabditis elegans. Nano Lett. 10, 3692–3699 (2010).
9. Hegyi, A. & Yablonovitch, E. Molecular imaging by optically detected electron spin resonance of nitrogen-vacancies in nanodiamonds. Nano Lett. 13, 1173–1178 (2013).
10. Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009).
11. Robledo, L. et al. High-fidelity projective read-out of a solid-state spin quantum register. Nature 477, 574–578 (2011).
12. Jelezko, F., Gruber, T., Popa, I. & Wrachtrup, J. Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 92, 76401 (2004).
13. Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281–285 (2006).
14. Dutt, M. V. G. et al. Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316, 1312–1316 (2007).
15. Yale, C. G. et al. All-optical control of a solid-state spin using coherent dark states. Proc. Natl Acad. Sci. USA 110, 7595 (2013).
16. Faraon, A., Barclay, P. E., Santori, C., Fu, K.-M. & Beausoleil, R. G. Resonant enhancement of the zero-phonon emission from a colour centre in a diamond cavity. Nat. Photon. 5, 301–305 (2011).
17. Siyushev, P. et al. Low-temperature optical characterization of a near-infrared single-photon emitter in nanodiamonds. New J. Phys. 11, 113029 (2009).
18. Aharonovich, I. et al. Two-level ultrabright single photon emission from diamond nanocrystals. Nano Lett. 9, 3191–3195 (2009).
19. Deshko, Y. & Gorokhovsky, A. A. Spectroscopy and micro-luminescence mapping of Xe-implanted defects in diamond. Low Temp. Phys. 36, 465–471 (2010).
20. Lee, S.-Y. et al. Readout and control of a single nuclear spin with a meta-stable electron spin ancilla. Nat. Nanotechnol. 8, 487–492 (2013).
21. Nadolinny, V. A. et al. A study of 13C hyperfine structure in the EPR of nickel-nitrogen-containing centres in diamond and correlation with their optical properties. J. Phys. Condens. Matter 11, 7357 (1999).
22. Wang, C., Kurtsiefer, C., Weinfurter, H. & Burchard, B. Single photon emission from SiV centres in diamond produced by ion implantation. J. Phys. B 39, 37 (2006).
23. Neu, E. et al. Single photon emission from silicon-vacancy colour centres in chemical vapour deposition nano-diamonds on iridium. New J. Phys. 13, 025012 (2011).
24. Neu, E., Agio, M. & Becher, C. Photophysics of single silicon vacancy centres in diamond: implications for single-photon emission. Opt. Express. 20, 19956–19971 (2012).
25. Iakoubovskii, K. & Stesmans, A. Characterization of hydrogen and silicon-related defects in CVD diamond by electron spin resonance. Phys. Rev. B 66, 195207 (2002).
26. Edmonds, A. M., Newton, M. E., Martineau, P. M., Twitchen, D. J. & Williams, S. D. Electron paramagnetic resonance studies of silicon-related defects in diamond. Phys. Rev. B 77, 245205 (2008).
27. D’Haenens-Johansson, U. F. S. et al. Optical properties of the neutral silicon split-vacancy centre in diamond. Phys. Rev. B 84, 245208 (2011).
28. Moliver, S. Electronic structure of neutral silicon-vacancy complex in diamond. Tech. Phys. 48, 1449–1453 (2003).
29. Goss, J. P., Briddon, P. R. & Shaw, M. J. Density functional simulations of silicon-containing point defects in diamond. Phys. Rev. B 76, 075204 (2007).
30. Goss, J. P., Jones, R., Breuer, S. J., Briddon, P. R. & Oberg, S. The twelve-line 1.682eV luminescence center in diamond and the vacancy-silicon complex. Phys. Rev. Lett. 77, 3041 (1996).
31. Hepp, C. et al. The electronic structure of the silicon-vacancy color center in diamond. Phys. Rev. Lett. 112, 036405 (2014).
32. Neu, E. et al. Low temperature investigations of single silicon vacancy colour centres in diamond. New J. Phys 04300515, (2013).
33. Sternschulte, H., Thonke, K., Sauer, T., Münzinger, P. C. & Michler, P. 1.681-eV luminescence center in chemical-vapor-deposited homoepitaxial diamond films. Phys. Rev. B 50, 14554–14560 (1994).
34. Clark, C. D., Kanda, H., Kiflawi, I. & Sittas, G. Silicon defects in diamond. Phys. Rev. B 51, 16681–16688 (1995).
35. Sternschulte, H. et al. Uniaxial stress and Zeeman splitting of the 1.681eV optical center in a homoepitaxial CVD diamond film. Diamond Rel. Mat. 4, 1189–1192 (1995).
36. Gali, A. & Maze, J. R. An ab initio study on split silicon-vacancy defect in diamond: electronic structure and related properties. Phys. Rev. B 88, 235205 (2013).
37. Maze, J. R. et al. Properties of nitrogen-vacancy centers in diamond: the group theoretic approach. New J. Phys. 13, 025025 (2011).
38. Tamarat, P. h. et al. Stark shift control of single optical centers in diamond. Phys. Rev. Lett. 97, 083002 (2006).
## Acknowledgements
We gratefully acknowledge financial support by the University of Cambridge, the European Research Council (FP7/2008-2013)/ERC Grant agreement number 209636, FP7 Marie Curie Initial Training Network S3NANO and funding from the Bundesministerium für Bildung und Forschung within the projects EPHQUAM (Contract number 01BL0903) and QuOReP (Contract number 01BQ1011). We thank J. Maze, V. Waselowski, A. Gali, J. Becker and C. Matthiesen for technical assistance and helpful discussions.
## Author information
Authors
### Contributions
M.A. and C.B. conceived the concept behind the project. E.N., S.G., M.S., H.S. and D.S.-N. prepared the samples. T.M., C.H. and B.P. performed the experiments. All authors contributed to the technical discussions and the writing of the manuscript.
### Corresponding author
Correspondence to Mete Atatüre.
## Ethics declarations
### Competing interests
The authors declare no competing financial interests.
## Supplementary information
### Supplementary Information
Supplementary Figures 1-5 (PDF 305 kb)
## Rights and permissions
Reprints and Permissions
Müller, T., Hepp, C., Pingault, B. et al. Optical signatures of silicon-vacancy spins in diamond. Nat Commun 5, 3328 (2014). https://doi.org/10.1038/ncomms4328
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/ncomms4328
• ### Fluorine-based color centers in diamond
• S. Ditalia Tchernij
• T. Lühmann
• J. Forneris
Scientific Reports (2020)
• ### Dissipative generation of steady-state entanglement of two separated SiV$$^{-}$$ centers coupled to photonic crystal cavities
• Xinke Li
• Shengli Ma
• Fuli Li
Quantum Information Processing (2020)
• ### The (eg ⊗ eu) ⊗ Eg product Jahn–Teller effect in the neutral group-IV vacancy quantum bits in diamond
• Gergő Thiering
npj Computational Materials (2019)
• ### Material platforms for spin-based photonic quantum technologies
• Mete Atatüre
• Dirk Englund
• Joerg Wrachtrup
Nature Reviews Materials (2018)
• ### Controlling the coherence of a diamond spin qubit through its strain environment
• Young-Ik Sohn
• Srujan Meesala
• Marko Lončar
Nature Communications (2018)
|
2022-08-17 12:05:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4170899987220764, "perplexity": 3558.4259329879333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572898.29/warc/CC-MAIN-20220817092402-20220817122402-00405.warc.gz"}
|
http://pubman.mpdl.mpg.de/pubman/faces/viewItemOverviewPage.jsp?itemId=escidoc:1330848
|
de.mpg.escidoc.pubman.appbase.FacesBean
Deutsch
Hilfe Wegweiser Impressum Kontakt Einloggen
Datensatz
DATENSATZ AKTIONENEXPORT
Freigegeben
Zeitschriftenartikel
Efficient Scheduling of Strict Multithreaded Computations
MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44405
Fatourou, Panagiota
Algorithms and Complexity, MPI for Informatics, Max Planck Society;
http://pubman.mpdl.mpg.de/cone/persons/resource/persons45532
Spirakis, Paul G.
Algorithms and Complexity, MPI for Informatics, Max Planck Society;
Externe Ressourcen
Es sind keine Externen Ressourcen verfügbar
Volltexte (frei zugänglich)
Es sind keine frei zugänglichen Volltexte verfügbar
Ergänzendes Material (frei zugänglich)
Es sind keine frei zugänglichen Ergänzenden Materialien verfügbar
Zitation
Fatourou, P., & Spirakis, P. G. (2000). Efficient Scheduling of Strict Multithreaded Computations. Theory of Computing Systems, 33(3), 173-232.
In this paper we study the problem of efficiently scheduling a wide class of multithreaded computations, called {\em strict}; that is, computations in which all dependencies from a thread go to the thread's ancestors in the computation tree. Strict multithreaded computations allow the limited use of synchronization primitives. We present the {\em first} fully distributed scheduling algorithm which applies to {\em any} strict multithreaded computation. The algorithm is asynchronous, on-line and follows the {\em work-stealing} paradigm. We prove that our algorithm is efficient not only in terms of its memory requirements and its execution time, but also in terms of its communication complexity. Our analysis applies to both shared and distributed memory machines. More specifically, the expected execution time of our algorithm is $O(T_1/P + hT_{\infty})$, where $T_1$ is the minimum serial execution time, $T_{\infty}$ is the minimum execution time with an infinite number of processors, $P$ is the number of processors and $h$ is the maximum distance'' in the {\em computation tree} between any two threads that need to communicate. Furthermore, the total space required during the execution is $O(S_1 P)$, where $S_1$ is the space required by a serial computer to execute the computation, while the expected communication cost incurred by our algorithm is $O(PhT_{\infty} (1+n_d) S_{max})$, where $n_d$ is the maximum number of dependencies entering any thread and $S_{max}$ is the largest amount of storage needed for the execution of any specific thread of the computation. Our communication complexity bound is {\em the first} nontrivial bound ever proved for the model of strict parallel programming.
|
2018-02-19 06:14:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9513559341430664, "perplexity": 1802.4918724239326}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812405.3/warc/CC-MAIN-20180219052241-20180219072241-00500.warc.gz"}
|
https://thierrygosselin.github.io/radiator/reference/detect_mixed_genomes.html
|
Highlight outliers individual's observed heterozygosity for a quick diagnostic of mixed samples or poor polymorphism discovery due to DNA quality, sequencing effort, etc.
detect_mixed_genomes(
data,
interactive.filter = TRUE,
detect.mixed.genomes = TRUE,
ind.heterozygosity.threshold = NULL,
by.strata = FALSE,
verbose = TRUE,
parallel.core = parallel::detectCores() - 1,
...
)
## Arguments
data (4 options) A file or object generated by radiator: tidy data Genomic Data Structure (GDS) How to get GDS and tidy data ? Look into tidy_genomic_data, read_vcf or tidy_vcf. (optional, logical) Do you want the filtering session to be interactive. Figures of distribution are shown before asking for filtering thresholds. Default: interactive.filter = TRUE. (optional, logical) For use inside radiator pipelines. Default: detect.mixed.genomes = TRUE. (string, double, optional) Blacklist individuals based on observed heterozygosity (averaged across markers). The string contains 2 thresholds values (min and max). The values are proportions (0 to 1), where 0 turns off the min threshold and 1 turns off the max threshold. Individuals with mean observed heterozygosity higher (>) or lower (<) than the thresholds will be blacklisted. Default: ind.heterozygosity.threshold = NULL will turn off completely the filter and the function will only output the plots and table of heterozygosity. (optional, logical) Can only be used when interactive.filter = TRUE, allows to enter heterozygosity thresholds by strata, instead of overall. This is not recommended to use inside filter_rad or when doing population discovery. This is a good use when dealing with different species. Default: by.strata = FALSE. (optional, logical) When verbose = TRUE the function is a little more chatty during execution. Default: verbose = TRUE. (optional) The number of core used for parallel execution during import. Default: parallel.core = parallel::detectCores() - 1. (optional) Advance mode that allows to pass further arguments for fine-tuning the function. Also used for legacy arguments (see details or special section)
## Value
The function returns inside the global environment a list with 5 objects:
1. the individual's heterozigosity ($individual.heterozygosity) a dataframe containing for each individual, the population id, the number of genotyped markers, the number of missing genotypes (based on the number of markers of the population and overall), the number of markers genotyped as heterozygote and it's proportion based on the number of genotyped markers. 2. the heterozygosity statistics per populations and overall:$heterozygosity.statistics
3. the blacklisted individuals if ind.heterozygosity.threshold was selected: $blacklist.ind.het 4. the boxplot of individual heterozygosity:$individual.heterozygosity.boxplot
5. the manhattan plot of individual heterozygosity (\$individual.heterozygosity.manhattan.plot) contrasted with missingness proportion based on the number of markers (population or overall). The 2 facets will be identical when the dataset as common markers between the populations. The dotted lines are the mean hetegozygosities.
## Details
To help discard an individual based on his observed heterozygosity (averaged across markers), use the manhanttan plot to:
1. contrast the individual with population and overall samples.
2. visualize the impact of missigness information (based on population or overall number of markers) and the individual observed heterozygosity. The larger the point, the more missing genotypes.
Outlier above average:
• potentially represent two samples mixed together (action: blacklist), or...
• a sample with more sequecing effort (point size small): did you merge your replicates fq files ? (action : keep and monitor)
• a sample with poor sequencing effort (point size large) where the genotyped markers are all heterozygotes, verify this with missingness (action: discard)
In all cases, if there is no bias in the population sequencing effort, the size of the point will usually be "average" based on the population or overall number of markers.
You can visualize individual observed heterozygosity, choose thresholds and then visualize, choose thresholds and filter markers based on observed heterozygosity in one run with: radiator filter_het.
Outlier below average:
• A point with a size larger than the population or overall average (= lots of missing): the poor polymorphism discovery of the sample is probably the result of bad DNA quality, a bias in sequencing effort, etc. (action: blacklist)
• A point with a size that looks average (not much missing): this sample requires more attention (action: blacklist) and conduct more tests. e.g. for biallelic data, look for coverage imbalance between ALT/REF allele. At this point you need to distinguish between an artifact of poor polymorphism discovery or a biological reason (highly inbred individual, etc.).
heterozygosity and missing data: If you see a pattern with the heterozygosity and missing data, try changing the genotyping rate required to keep markers and/or individuals.
## Note
Thanks to Peter Grewe for very useful comments on previous version of this function.
## Author
Thierry Gosselin thierrygosselin@icloud.com
## Examples
if (FALSE) {
#Step1: highlight outlier individuals, the simplest way to run:
|
2021-06-21 02:58:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3367638885974884, "perplexity": 8442.616633883456}, "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-2021-25/segments/1623488262046.80/warc/CC-MAIN-20210621025359-20210621055359-00243.warc.gz"}
|
https://astronomy.stackexchange.com/questions/36623/if-a-planet-gained-too-many-moons-could-the-tidal-forces-of-those-moons-rip-the
|
# If a planet gained too many moons could the tidal forces of those moons rip the planet apart?
Or would the planet just be subject to extremely intense tidal forces instead?
First way: Tidal force is the gradient of the gravitational attraction. Gravitational force goes like $$F_g \propto \frac{M}{r^2}$$ and the tidal force thus as $$F_t \propto \frac{M}{r^3}$$. For a moon and its planet the mutual distance is the same. And as the planet has the higher mass, the tidal forces of the planet would be higher on the moon than the one of the moon onto the planet. Materials are not that different that it plays a major role here. So before a moon destroys a planet, the planet destroys the moon (or you mis-named the two in the first place, calling the planet the moon and vice versa). This may happen for moons within the Roche limit of a planet.
|
2021-07-23 21:43:44
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6934751272201538, "perplexity": 502.21483297042283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150067.51/warc/CC-MAIN-20210723210216-20210724000216-00316.warc.gz"}
|
https://chakefashion.com/blackjack/blackjack-method.html
|
# 🤑 Beating Blackjack is Easy - CDC Gaming Reports
FREEDAY
DAY123
Filter:
Sort:
## 🤑 Blackjack Progression Systems Don't Work - chakefashion.com
Software - MORE
T7766547
Bonus:
Free Spins
Players:
All
WR:
30 xB
Max cash out:
$200 The Color of Blackjack : A revolutionary method to learn card counting [Daniel Dravot] on Amazon.com. *FREE* shipping on qualifying offers. Review by George ... Enjoy! Wild Jack Online Casino |$600
Valid for casinos
Blackjack Strategy - learn best tips and tricks!
Visits
Dislikes
Software - MORE
G66YY644
Bonus:
Free Spins
Players:
All
WR:
50 xB
Max cash out:
$200 Even though others had come up with card counting strategies before, and the 10 Count has since been made redundant by other methods, it remains the first ... Enjoy! Blackjack Tips For Beginners - Business Insider Valid for casinos Multi-Deck Card Counting Strategy for Blackjack Visits Dislikes Comments 8 Things To Never Do At A Blackjack Table! ## 🔥 Blackjack Progression Systems Don't Work - chakefashion.com Software - MORE TT6335644 Bonus: Free Spins Players: All WR: 30 xB Max cash out:$ 200
They came up with, and put into practice, a method for winning at blackjack. The idea came from the studies of Edward Thorp, a math professor at MIT, who ...
Enjoy!
Blackjack - Wikipedia
Valid for casinos
Ten Count Card Counting System - Blackjack Betting Strategy
Visits
Dislikes
Software - MORE
A7684562
Bonus:
Free Spins
Players:
All
WR:
60 xB
Max cash out:
$1000 If you're looking for a blackjack card counting system that doesn't require you to do.. Going with this method will slow down the amount you wager compared to ... Enjoy! java - Using multiple methods with Blackjack - Code Review Stack Exchange Valid for casinos Blackjack Strategy - learn best tips and tricks! Visits Dislikes Comments ## 🤑 Blackjack - Wikipedia Software - MORE A67444455 Bonus: Free Spins Players: All WR: 60 xB Max cash out:$ 1000
No bust method. I strive to never lose a hand by taking a hit. I never take a hit on 12 or higher, instead I wait and see if the dealer will bust if I ...
Enjoy!
Keys to Winning at the Blackjack Table
Valid for casinos
Blackjack - Wikipedia
Visits
Dislikes
How to Play (and Win) at Blackjack: The Expert's Guide
Software - MORE
B6655644
Bonus:
Free Spins
Players:
All
WR:
50 xB
Max cash out:
$200 Answer 1 of 31: I'm curious what everyone thinks is the best blackjack betting. ride and got hot so there's all types of methods buried like to press while I'm hot. Enjoy! Blackjack Basic Strategy Chart - Edward O. Thorp Valid for casinos Keys to Winning at the Blackjack Table Visits Dislikes Comments Why You Suck At Blackjack ## 🍒 Blackjack Basic Strategy Chart - Edward O. Thorp Software - MORE BN55TO644 Bonus: Free Spins Players: All WR: 30 xB Max cash out:$ 1000
I have this class called card-set that holds the array-list for my cards. public abstract class CardSet { public ArrayList cards; public Card ...
Enjoy!
The Blackjack Method of Golf | White Sox and Stuff
Valid for casinos
Blackjack Progression Systems Don't Work - chakefashion.com
Visits
Dislikes
My name is Dan Howard.
I am one of the co-writers at this website and a person with a vast experience in playing roulette online and offline.
I have helped review many of the web-based casinos and roulette variations you can read about at SuperCasinoSites, with a focus on providing you with accurate information that can help you in finding the best casino websites.
Blackjack is a popular casino-banked game that can utilize anywhere between one and eight decks of cards.
The cards are typically not reshuffled after every round of play which renders the game susceptible to advantage blackjack method techniques such as card counting.
This method enables skilled players to track the ratio of high to low cards, which gives them an accurate idea of what their odds of winning a given round are.
It is a mathematically proven fact that in blackjack, the excess of high cards that remain to be played tips the odds in favor of the player.
And vice versa, the excess of low cards shifts the advantage in favor of the house.
£5 max bet using bonus.
Bonus valid for 14 days.
Free Spins valid for 72 hours from credit.
Max Free Spins winnings £100.
Skrill + Neteller excluded.
Always refer to Bonus Terms - www.
Casino bonus: first deposit only.
Deposits with Skrill, PayPal, Neteller and Paysafecard not eligible.
£10, max bonus: £100.
Free spins: no deposit required, expire after 7 days, Starburst only.
Bonus expires after 7 days.
The book became an overnight success and scared casino operators into changing in an attempt to prevent players from beating it through counting.
One of the first things they did was increase the number of decks in play.
Little did they know this was nothing but a small bump in the road for card counters who continue to crush the game of 21 to this day.
Is It Possible to Count Multiple Decks in Blackjack?
Passed from one generation to the next, these misconceptions prevent many people from learning how to become profitable blackjack players.
The infamous scene from the Academy Award-winning motion picture Rain Man serves as a classic example.
Luckily for card counters worldwide, there is not even an ounce of truth in this statement.
Card counting can be just as effective in shoe games as it is in.
It simply requires a slightly different approach.
Either blackjack method, the betting session starts with players keeping track of each dealt card that appears on the table.
Each card is assigned a count value which depends on the counting system one uses.
Counting Multiple Decks in Blackjack Additional TipsKeeping track of each dealt card helps the player establish their running count visit web page gives them an accurate enough idea about the composition of the remaining deck.
This running count should be maintained throughout the entire course of the game until the dealer shuffles again.
The player uses the knowledge of the ratio of high to low cards to determine their edge and sizes their bets in proportion to this edge.
So for example, if cards K, 10, 2, 8, and 6 appear during the first round, you end up with a running count of 0 because the high and low cards cancel themselves out.
The 7 is neutral and as such, has no effect on the running count.
The odds have not yet swung in your favor so you should not increase the size of your next bet.
However, maintaining a running count alone is not enough to accurately determine your advantage when multiple decks are in play.
A high positive count of +5 after the first round of play gives a single-deck player a considerable advantage over the house which, in turn, justifies a significant bet increase on the next blackjack method />This is not the case with a positive count of +5 when one plays against six decks.
The edge it gives you is less significant because more than five and a half decks are still in play.
In order to arrive at an accurate estimate of their edge, multiple-deck players must convert the current running count into a true count per deck.
The true count denotes the density of high or low cards per deck.
You arrive at an accurate true count after you divide the running count by the number of undealt decks.
Just to give you an example, suppose you are with eight decks and have reached a running count of +8 while there are four undealt decks in the shoe.
It is obvious those who play shoe games should also learn to accurately determine the number of undealt decks.
The discard tray enables the dealer to stack the dealt cards neatly so that they are clearly visible to everyone at the table.
Deck estimation requires a lot of practice but once you master it, you only have to subtract the number of the remaining decks from the total number of decks the game started with.
A Tip on Practicing Accurate Deck Estimation Counting into multiple decks of cards is not rocket too sweet memories blackjack thanks but it still requires a good amount of discipline and persistence if you insist on accuracy.
Once you master maintaining an accurate running count, you need to practice your deck estimation.
One approach is to purchase a discard tray and fifteen standard packs of cards.
You should divide the packs into five separate stacks where the first stack contains a single deck, the second stack contains two decks, the third contains three decks and so on.
You can label each stack so you know how many decks are in there.
You put any one group of cards on the table, inspect it closely for some time and try to determine the number of decks in contains.
Try to do it without looking at the labels.
Then you place the groups of cards, one at a time, in the discard tray and practice deck estimation by inspecting the height of each stack.
It sounds more difficult than it really is.
You will be surprised how accurate you can get when you put in enough practice.
The conclusion we can draw is that a person blackjack rigged pokerstars is a six-deck game where is roughly half a percent has no advantage whatsoever at a true count of +1.
Respectively, you gain an advantage of half a percent when you arrive at a true count of +2.
The bigger your edge gets, the higher the amounts you should wager.
The majority of experienced blackjack players choose to size their bets according to a betting technique known as the Kelly Criterion.
This approach enables them to maximize their profits and reduce the risk of losing their bankrolls at the same time.
The edge players go here to get in blackjack is not all that substantial so one should not expose large portions of their bankroll to risk during any given round of play.
Adjusting Your Bet Size in Multiple-Deck Games Card counters gain an edge in blackjack by sizing their bets proportionately to the count.
They increase their wagers when they have the edge and bet the table minimum or nothing at all when the casino has the edge.
This sizing on the basis of true count is called spreading your bets.
Casinos are no strangers to how blackjack works and their employees are well-trained to detect card counters.
If you spread your bets too aggressively, you stand higher chances of being detected and backed off, even though counting cards itself is not deemed an illegal practice.
Some blackjack experts recommend using a 1-12 bet spread for shoe games where six and eight decks are in play.
This betting ramp is considered ideal for multiple-deck games.
Whether or not you get labeled as a card counter largely depends on the tolerance level of the casino you are playing at.
Smaller establishments are more likely to back you off so you might want to choose a more conservative spread in this case.
Varying Your Playing Decisions with the Help of the True Count More experienced counters further increase their advantage by varying their playing decisions according to the true count.
These departures from on the basis of true count are known as indices.
They are very important because when the true count increases or decreases significantly, the recommended basic strategy moves are no longer optimal.
This makes sense because basic strategy takes into consideration only three cards, those in your starting hand and the upcard of the dealer.
Some advantage players memorize 100+ index plays but this is hardly necessary to gain a good edge in blackjack.
In fact, using only the indices listed below can significantly improve your play.
The 18 indices listed in the first table are known as the Illustrious 18 and are blackjack method for multiple-deck blackjack games where the dealer stands on soft 17.
They were developed by the renowned Blackjack Hall of Fame inductee Blackjack method Schlesinger and help you make more accurate insurance, standing, doubling and splitting decisions.
You can find more about these indices in Mr.
Dealer Upcard True Count TC Recommended Playing Deviation Insurance bet 3 Buy insurance at TC of +3 and above 16 vs.
Ace 1 Double at TC of +1 and above 10 vs.
Ace 4 Double at TC of +4 and click here 9 vs.
Suppose you are dealt a against a dealer who shows a 6.
This is an excellent hand to blackjack method, even more so when the dealer is in a vulnerable spot with this small upcard.
A basic strategy player should never touch this hand.
It gives them an excellent total of 20 and the only way for the dealer to beat this is by drawing to 21.
The chances of this happening are not significant.
Quite the opposite — the dealer stands better chances of busting with a 6 than outdrawing you.
However, if you count the cards and arrive at a true count of +5 or higher, this serves as an indicator the shoe is richer in ten-value cards.
This knowledge allows you to maximize your value by splitting the Queens and potentially winning two hands instead of one.
That being said, the majority of professional card counters prefer to refrain from using this index for the purpose of extending their longevity.
Schlesinger also developed several indices designed to help advanced players with their surrender decisions.
You can see them in the table below.
Dealer Upcard True Count Recommended Playing Deviation 14 vs.
Ace 2 Surrender at TC of +2 or above; Hit at +1 or lower One way to memorize these playing deviations is by using flash cards.
But before you get there and make any attempts to count into multiple decks, you should make sure you know perfect basic strategy and can maintain an accurate running count.
Messing up the running count would lead to inaccuracies in your true count, which, in turn, would render your efforts at beating the game of blackjack futile.
We ourselves are not casino operators, do not offer any real-money games on our website, and cannot be held liable for the financial risks readers take when participating in real-money gambling activities.
Visitors of SuperCasinoSites should keep in mind gambling can be highly addictive and as such, should always be approached responsibly and with due measure.
If you or someone you know struggles with gambling addiction, we recommend you contact the free gambling helplines like those operated by organizations like.
Before you register and play at any of the casinos we review and recommend, do check the legal and regulatory frameworks pertaining to real-money online gambling in your country to ensure you are not acting in violation with its laws.
Software - MORE
G66YY644
Bonus:
Free Spins
Players:
All
WR:
30 xB
Max cash out:
Software - MORE
B6655644
Bonus:
Free Spins
Players:
All
WR:
50 xB
Max cash out:
$1000 A blackjack betting system first published in 1965 - Oscar's Blackjack Betting. of professional gambling at blackjack, even methods that don't require counting. Enjoy! Blackjack Strategy - learn best tips and tricks! Valid for casinos No bust method Visits Dislikes Comments The Blackjack System ## 🔥 Blackjack Counting Missing Cards Software - MORE BN55TO644 Bonus: Free Spins Players: All WR: 50 xB Max cash out:$ 200
Following are ten scientifically valid and legal methods to beat blackjack: METHOD 1: CARD COUNTING. METHOD 3: HOLE CARDING. METHOD 4: NEXT CARDING. METHOD 5: EDGE SORTING. METHOD 7: SIDE BETS. METHOD 8: SHUFFLE TRACKING. METHOD 9: LOSS REBATES. METHOD 10: COMPS.
Enjoy!
Method of making a side bet during a blackjack game - Extra Chance Blackjack, LLC
Valid for casinos
Multi-Deck Card Counting Strategy for Blackjack
Visits
Dislikes
I am very new to Java and programming theory and am desperately trying to improve my knowledge.
This is the first program I've made without help and really would appreciate some feedback.
I know there must be 1,000,000 better ways to do what I did.
How could I achieve this program using methods and OOP?
This is the biggest problem I want to fix.
Images of console to give you an idea of two runs: Any and all comments are appreciated.
I really want to get more info grips with basic Java programming concepts before my bad habits are too engraved to break!
source which blackjack method not possible with Strings.
It would improve and help you to get rid of the big if-else if structures, so it'd.
It should be a named constant.
It would improve click the following article />You seem to be repeating sections like this.
It helps you to understand your code at a later time.
You could make them into a method like above, and call it blackjack method of repeating them.
Provide details and share your blackjack method />Use MathJax to format equations.
|
2020-09-23 00:01:20
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17019785940647125, "perplexity": 2303.955225317477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400208095.31/warc/CC-MAIN-20200922224013-20200923014013-00654.warc.gz"}
|
http://mathhelpforum.com/differential-geometry/180221-cauchys-integral-formula-print.html
|
# Cauchy's Integral Formula
• May 11th 2011, 08:47 AM
Alexrey
Cauchy's Integral Formula
Hey guys, I was just wondering how I would apply Cauchy's Integral Formula to something like this:
http://img813.imageshack.us/img813/7181/integral.jpg
I would be able to do it if there wasn't a constant of 1/2i there but that constant makes me uncertain.
• May 11th 2011, 08:54 AM
TheEmptySet
Quote:
Originally Posted by Alexrey
Hey guys, I was just wondering how I would apply Cauchy's Integral Formula to something like this:
http://img813.imageshack.us/img813/7181/integral.jpg
I would be able to do it if there wasn't a constant of 1/2i there but that constant makes me uncertain.
I have a question for you. What do you mean by this notation
$\gamma(0,1)$
Is this a circle of radius 1 centered at the origin?
If so the Cauchy integral formula does not apply because the integrand is discontinous at z=1.
• May 11th 2011, 09:22 AM
Alexrey
My bad, yes, gamma (0,1) is an open disk of radius 1 centered at the origin. Surely you could apply Cauchy's integral formula:
http://img38.imageshack.us/img38/506...ralformula.png
with f(w) = z and w-a = z-1, therefore since f(w) is holomorphic in gamma(0, 1) we can apply Cauchy's Integral Formula, with a=1?
• May 11th 2011, 09:41 AM
TheEmptySet
Quote:
Originally Posted by Alexrey
My bad, yes, gamma (0,1) is an open disk of radius 1 centered at the origin. Surely you could apply Cauchy's integral formula:
http://img38.imageshack.us/img38/506...ralformula.png
with f(w) = z and w-a = z-1, therefore since f(w) is holomorphic in gamma(0, 1) we can apply Cauchy's Integral Formula, with a=1?
a must be an interior point of
$\gamma$
Cauchy's integral formula - Wikipedia, the free encyclopedia
The problem is that there is a pole on the countour. Also something else is fishy you sure that it is an open disk?
• May 11th 2011, 09:54 AM
Alexrey
Ah I see, I screwed that one up. Regarding gamma being an open disk, I thought that it was open since I read in my notes somewhere that the interior of gamma was open and therefore presumed that the whole disk was open as well. What can I do if the singularity is ON the contour? Will residues work (I haven't gone over this section yet).
Okay lets rephrase the question to gamma(0,2). I would be able to use Cauchy's Integral Formula then right, since a is inside the contour.
• May 11th 2011, 10:01 AM
TheEmptySet
Quote:
Originally Posted by Alexrey
Ah I see, I screwed that one up. Regarding gamma being an open disk, I thought that it was open since I read in my notes somewhere that the interior of gamma was open and therefore presumed that the whole disk was open as well. What can I do if the singularity is ON the contour? Will residues work (I haven't gone over this section yet).
Okay lets rephrase the question to gamma(0,2). I would be able to use Cauchy's Integral Formula then right, since a is inside the contour.
Yes, this would give
$f(1)=\frac{1}{2\pi i}\int_{\gamma (0,2)}\frac{f(z)}{z-1}dz, \quad f(z)=z$ so we get
$1=\frac{1}{2\pi i}\int_{\gamma (0,2)}\frac{z}{z-1}dz \iff \int_{\gamma (0,2)}\frac{z}{z-1}dz=2\pi i$
So finaly
$\frac{1}{2i}\int_{\gamma (0,2)}\frac{z}{z-1}dz=\frac{1}{2i}(2\pi i)=\pi$
• May 11th 2011, 10:54 AM
Alexrey
Awesome, thanks so much.
|
2016-07-27 19:44:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9529759883880615, "perplexity": 662.4450920044139}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257827077.13/warc/CC-MAIN-20160723071027-00227-ip-10-185-27-174.ec2.internal.warc.gz"}
|
http://en.wikipedia.org/wiki/Highly_cototient_number
|
# Highly cototient number
In number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above one and has more solutions to the equation
x − φ(x) = k,
than any other integer below k and above one. Here, φ is Euler's totient function. There are infinitely many solutions to the equation for k = 1 so this value is excluded in the definition. The first few highly cototient numbers are:
2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889 (sequence A100827 in OEIS).
There are many odd highly cototient numbers. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 9 modulo 10.
The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factorization does, as the numbers get larger.
## Example
The cototient of x is defined as x – φ(x), i.e. the number of positive integers less than or equal to x that have at least one prime factor in common with x. For example, the cototient of 6 is 4 since these 4 positive integers have a prime factor in common with 6: 2, 3, 4, 6. The cototient of 8 is also 4, this time with these integers: 2, 4, 6, 8. There are exactly two numbers, 6 and 8, which have cototient 4. There are fewer numbers which have cototient 2 and cototient 3 (one number in each case), so 4 is a highly cototient number.
k (highly cototient k are bolded) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Number of solutions to x – φ(x) = k () ∞ 1 1 2 1 1 2 3 2 0 2 3 2 1 2 3 3 1 3 1 3 1 4 4 3
## Primes
The first few highly cototient numbers which are primes (sequence A105440 in OEIS) are
2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839.
|
2014-07-25 20:58:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6274375319480896, "perplexity": 158.48523867824298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894799.55/warc/CC-MAIN-20140722025814-00072-ip-10-33-131-23.ec2.internal.warc.gz"}
|
https://byjus.com/jee/limits-continuity-and-differentiability/
|
# Limits, Continuity and Differentiability - Evaluations and Examples
The limit concept is certainly indispensable for the development of analysis, for convergence and divergence of infinite series also depends on this concept. The theory of limits and then defining continuity, differentiability and the definite integral in terms of the limit concept is successfully executed by mathematicians. In this section, will study this concept in detail with the help of solved examples.
## What are Limits?
The expression $\underset{x\to c}{\mathop{\lim }}\,\,f(x)=L$ means that f(x) can be as close to L as desired by making x sufficiently close to ‘C’. In such a case, we say that the limit of f, as x approaches to C, is L.
Neighbourhood of a point:
Let ‘a’ be real number and ‘h; is very close to ‘O’ then
Left hand limit will be obtained when x = a – h or x -> a
Similarly, Right Hand limit will be obtained when x = a + h or x -> a+
Related Concepts:
### Existence of limit
The limit will exist if the following conditions get fulfilled:
(a) $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,$ f(x) =$\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,$ f(x) i.e. L.H.L = R.H.L
(b) Both L.H.L & R.H.L should be finite.
Examples
• $\underset{x\to 1}{\mathop{\lim }}\,\,{{x}^{2}}+1={{1}^{2}}+1=2$
• $\underset{x\to 0}{\mathop{\lim }}\,\,{{x}^{2}}-x={{0}^{2}}-0=0$
• $\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-4}{x+3}=\frac{4-4}{2+3}=0$
• (c)In Limits, we have in determinant forms such as $\frac{0}{0},\frac{\infty }{\infty },0\times \infty ,\infty \times \infty ,{{1}^{\infty }},0{}^\circ ,\infty {}^\circ$
In these cases, we try to simplify the problem into the valid function.
### Use of Expansion in Evaluating limits
Some Important Expansions
1. $\log (1+x)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}+…….$
2. ${{e}^{x}}=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}-\frac{{{x}^{4}}}{4!}+…….$
3. ${{a}^{x}}=1+x\log a+\frac{{{x}^{2}}}{2!}{(log\;a)}^{2}+…….$
4. $\sin x=x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}……$
5. $\cos x=1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}……$
6. $\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……$
### Some Important limits
1. $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x}{x}=1$
2. $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{1-\cos x}{{{x}^{2}}}=\frac{1}{2}$
3. $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\tan x}{x}=1$
4. $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{{{e}^{x}}-1}{x}=1$
5. $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\log (1+x)}{x}=1$
Example: Solve $\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x-x}{{{x}^{3}}}$
Solution:
### Evaluation of Algebric Limits
#### Direct substitution method
Example: $\underset{x\to 1}{\mathop{\lim }}\,\,(3{{x}^{2}}+4x+5)$= 3(1)2 + 4(1) + 5 = 12
Example: $\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-4}{x+3}=\frac{4-4}{2+3}=\frac{0}{5}=0$
#### Factorization method
Example: $\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-5x+6}{{{x}^{2}}-4}$
Solution: $\underset{x\to 2}{\mathop{\lim }}\,\,\frac{(x-2)(x-3)}{(x+2)(x-2)}\Rightarrow \underset{x\to 2}{\mathop{\lim }}\,\frac{x-3}{x+2}$= $\frac{-1}{4}$
#### Rationalization method
$\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sqrt{2+x}-\sqrt{2}}{x}$
Solution: $\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\left( \sqrt{2+x}-\sqrt{2} \right)\left( \sqrt{2+x}+\sqrt{2} \right)}{x\left( \sqrt{2+x}+\sqrt{2} \right)}$
= $\frac{1}{2\sqrt{2}}$
Using Result:$\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$
Example: $\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{10}}-{{2}^{10}}}{{{x}^{5}}-{{2}^{5}}}$
## Continuity
### What is Continuity
A continuous function is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous.
A function f(x) is said to be continuous at x = a if
$\underset{x\to a}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=f(a)$
i.e. L.H.L = R.H.L = value of the function at x = a
Else, a function f(x) is said to be discontinuous Function.
Example 1: f(x) = $\frac{1}{2\sin x-1},$ Discuss the continuity or discontinuity.
Solution: Clearly the function will be not defined for $\sin x=\frac{1}{2}=\sin \frac{\pi }{6}$
Function is discontinuous for $x=n\,\pi +{{(-1)}^{n}}\frac{\pi }{6}$
Example 2: What value must be assigned to K so that the function
$f(x)=\left\{\begin{matrix} \frac{x^4-256}{x-4} & , & x\neq 4\\ k & , & x=4 \end{matrix}\right.$ is continuous at x=4
Solution: =$f(4)=\underset{x\to 4}{\mathop{\lim }}\,\,\,\,\,\frac{{{x}^{4}}-256}{x-4}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{4}}-{{4}^{4}}}{x-4}={{4.4}^{4-1}}=256$
Example 3: Discuss the continuity of
(a) Sgn (x3x)
(b) f(x) = $\left[ \frac{2}{1+{{x}^{2}}} \right]$ , x > 0 [ ] Solution:
(a) f(x) = sgn (x3–x)
Here x3 – x = 0, so, x =0, –1, 1
Hence f(x) is discontinuous at x = 0, 1, –1
(b) $\frac{2}{1+{{x}^{2}}}$ , x > 0 is a monotonically decimal function
Hence f(x) = $\left[ \frac{2}{1+{{x}^{2}}} \right],\,\,\,x\ge 0$ is discontinuous,
When $\frac{2}{1+{{x}^{2}}}$ is on integer
$\Rightarrow \frac{2}{1+{{x}^{2}}}=1,\,2\,\,\,\,\,\,at\,\,\,\,x=1,0$
Example 4: Discuss the continuity of f(x) = $\left\{\begin{matrix} x-2 & x\leq 0\\ 4-x^2 &x>0 \end{matrix}\right.$ at x = 0
Solution: $\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\,\,(x-2)=-2$
### Intermediate Value Theorem
If f is continuous on [a, b] and f(a) ≠ f(b), then for any value $c\in (f(a),f(b))$, there is at least one number in x0 (a, b) for which f(x0) = c
## Differentiability
A function, say f(x) is said to be differentiable at the point x = a if the derivative f ‘(a) exists at every point in its domain.
### Existence of Derivative
Right and left hand derivative
F.H.D: f’(0+) =$\underset{h\to {{0}^{+}}}{\mathop{\lim }}\,$ $\frac{f(a+h)-(a)}{h}$
L.H.D: F’$({{a}^{-}})$= $\underset{h\to {{0}^{-}}}{\mathop{\lim }}\,$ $\frac{h(a-h)-f(a)}{-h}$
#### How can a function fail to be differentiable?
The function f(x) is said to be non-differentiable at x = a if
(a) Both R.H.D & L.H.D exist but not equal
(b) Either or both R.H.D & L.H.D are not finite
(c) Either or both R.H.D & L.H.D do not exist.
|
2021-09-25 19:24:44
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 48, "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": 0.802837610244751, "perplexity": 1282.2770187371402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057733.53/warc/CC-MAIN-20210925172649-20210925202649-00467.warc.gz"}
|
https://nrich.maths.org/1079
|
### I'm Eight
Find a great variety of ways of asking questions which make 8.
### Let's Investigate Triangles
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
### Noah
Noah saw 12 legs walk by into the Ark. How many creatures did he see?
# The Tomato and the Bean
##### Age 5 to 7Challenge Level
Tom's Dad sowed some tomato seed in February. He gave Tom one of the tomato plants in a pot.
At the beginning of May, Tom put his tomato plant outside. On the same day he sowed a bean in another pot.
Ten days later the bean plant was just $1$ cm (centimetre) above the soil surface. Tom measured his tomato plant which was already $38$ cm tall.
Each evening Tom measured his two plants.
On the evening of the next day the little bean plant had grown another $2$ cm so it was $3$ cm high. Each day it continued to grow double the amount it had grown the day before.
The tomato plant grew at a steady $5$ cm a day.
After how many days were the two plants the same height when Tom measured them in the evening? How high were they?
|
2022-12-10 04:43:04
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6407639384269714, "perplexity": 3869.8339929439417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711712.26/warc/CC-MAIN-20221210042021-20221210072021-00323.warc.gz"}
|
https://repository.asu.edu/collections/7?cont=Crook%2C+Sharon&sub=Applied+mathematics
|
## ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
In addition to the electronic theses found in the ASU Digital Repository, ASU Theses and Dissertations can be found in the ASU Library Catalog.
Dissertations and Theses granted by Arizona State University are archived and made available through a joint effort of the ASU Graduate College and the ASU Libraries. For more information or questions about this collection contact or visit the Digital Repository ETD Library Guide or contact the ASU Graduate College at gradformat@asu.edu.
Olfaction is an important sensory modality for behavior since odors inform animals of the presence of food, potential mates, and predators. The fruit fly, Drosophila melanogaster, is a favorable model organism for the investigation of the biophysical mechanisms that contribute to olfaction because its olfactory system is anatomically similar to but simpler than that of vertebrates. In the Drosophila olfactory system, sensory transduction takes place in olfactory receptor neurons housed in the antennae and maxillary palps on the front of the head. The first stage of olfactory processing resides in the antennal lobe, where the structural unit is the glomerulus. ...
Contributors
Luli, Dori, Crook, Sharon, Baer, Steven, et al.
Created Date
2013
Advances in experimental techniques have allowed for investigation of molecular dynamics at ever smaller temporal and spatial scales. There is currently a varied and growing body of literature which demonstrates the phenomenon of \emph{anomalous diffusion} in physics, engineering, and biology. In particular many diffusive type processes in the cell have been observed to follow a power law $\left<x^2\right> \propto t^\alpha$ scaling of the mean square displacement of a particle. This contrasts with the expected linear behavior of particles undergoing normal diffusion. \emph{Anomalous sub-diffusion} ($\alpha<1$) has been attributed to factors such as cytoplasmic crowding of macromolecules, and trap-like structures in the ...
Contributors
Holeva, Thomas Matthew, Ringhofer, Christian, Baer, Steve, et al.
Created Date
2014
The phycologist, M. R. Droop, studied vitamin B12 limitation in the flagellate Monochrysis lutheri and concluded that its specific growth rate depended on the concentration of the vitamin within the cell; i.e. the cell quota of the vitamin B12. The Droop model provides a mathematical expression to link growth rate to the intracellular concentration of a limiting nutrient. Although the Droop model has been an important modeling tool in ecology, it has only recently been applied to study cancer biology. Cancer cells live in an ecological setting, interacting and competing with normal and other cancerous cells for nutrients and space, ...
Contributors
Everett, Rebecca Anne, Kuang, Yang, Nagy, John, et al.
Created Date
2015
Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ordinary and partial differential equation (ODE, PDE) models for two problems: Vicodin abuse and impact cratering. The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing ...
Contributors
Caldwell, Wendy K, Wirkus, Stephen, Asphaug, Erik, et al.
Created Date
2019
A general continuum model for simulating the flow of ions in the salt baths that surround and fill excitable neurons is developed and presented. The ion densities and electric potential are computed using the drift-diffusion equations. In addition, a detailed model is given for handling the electrical dynamics on interior membrane boundaries, including a model for ion channels in the membranes that facilitate the transfer of ions in and out of cells. The model is applied to the triad synapse found in the outer plexiform layer of the retina in most species. Experimental evidence suggests the existence of a negative ...
Contributors
Jones, Jeremiah, Gardner, Carl, Gardner, Carl, et al.
Created Date
2013
Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals ...
Contributors
Baez, Javier, Kuang, Yang, Kostelich, Eric, et al.
Created Date
2017
|
2019-07-17 03:04:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28785157203674316, "perplexity": 4374.759632508843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525009.36/warc/CC-MAIN-20190717021428-20190717043428-00282.warc.gz"}
|
https://labs.tib.eu/arxiv/?category=nlin.CG
|
• ### Unifying vectors and matrices of different dimensions through nonlinear embeddings(1701.01281)
March 6, 2020 math-ph, math.MP, nlin.PS, nlin.CG
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, deriving a class of nonlinear diffusion equations. Finally, by means of nonlinear embeddings we introduce CA connections, a class of CMLs that connect any two arbitrary CAs in the limits $\kappa \to 0$ and $\kappa \to \infty$ of the embedding.
• ### Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees(1710.00084)
Jan. 31, 2019 nlin.CG
In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every stationary measure $\nu$ of the PCA we can associate a space-time Gibbs measure $\mu_{\nu}$ on $\mathbb{Z} \times \mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true. A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $d\in \{ 1,2,3\}$ and characterizing the invariant product Bernoulli measures.
• ### Density classification performance and ergodicity of the Gacs-Kurdyumov-Levin cellular automaton model IV(1703.09038)
July 28, 2018 cond-mat.stat-mech, nlin.CG
Almost four decades ago, Gacs, Kurdyumov, and Levin introduced three different cellular automata to investigate whether one-dimensional nonequilibrium interacting particle systems are capable of displaying phase transitions and, as a by-product, introduced the density classification problem (the ability to classify arrays of symbols according to their initial density) in the cellular automata literature. Their model II became a well known model in theoretical computer science and statistical mechanics. The other two models, however, did not receive much attention. Here we characterize the density classification performance of Gacs, Kurdyumov, and Levin's model IV, a four-state cellular automaton with three absorbing states---only two of which are attractive---, by numerical simulations. We show that model IV compares well with its sibling model II in the density classification task: the additional states slow down the convergence to the majority state but confer a slight advantage in classification performance. We also show that, unexpectedly, initial states diluted in one of the nonclassifiable states are more easily classified. The performance of model IV under the influence of noise was also investigated and we found signs of an ergodic-nonergodic phase transition at some small finite positive level of noise, although the evidences are not entirely conclusive. We set an upper bound on the critical point for the transition, if any.
• ### Analytic model of thermalization: Quantum emulation of classical cellular automata(1709.06315)
We introduce a novel method of quantum emulation of a classical reversible cellular automaton. By applying this method to a chaotic cellular automaton, the obtained quantum many-body system thermalizes while all the energy eigenstates and eigenvalues are solvable. These explicit solutions allow us to verify the validity of some scenarios of thermalization to this system. We find that two leading scenarios, the eigenstate thermalization hypothesis scenario and the large effective dimension scenario, do not explain thermalization in this model.
• ### Double jump phase transition in a soliton cellular automaton(1706.05621)
Aug. 12, 2020 math.CO, nlin.SI, nlin.PS, math.PR, nlin.CG
In this paper, we consider the soliton cellular automaton introduced in [Takahashi 1990] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $p\in(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.
• ### The Garden of Eden theorem: old and new(1707.08898)
June 8, 2018 cs.IT, math.IT, math.DS, math.GR, nlin.CG
We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.
• ### Renormalization group theory for percolation in time-varying networks(1708.05704)
Motivated by multi-hop communication in unreliable wireless networks, we present a percolation theory for time-varying networks. We develop a renormalization group theory for a prototypical network on a regular grid, where individual links switch stochastically between active and inactive states. The question whether a given source node can communicate with a destination node along paths of active links is equivalent to a percolation problem. Our theory maps the temporal existence of multi-hop paths on an effective two-state Markov process. We show analytically how this Markov process converges towards a memory-less Bernoulli process as the hop distance between source and destination node increases. Our work extends classical percolation theory to the dynamic case and elucidates temporal correlations of message losses. Quantification of temporal correlations has implications for the design of wireless communication and control protocols, e.g. in cyber-physical systems such as self-organized swarms of drones or smart traffic networks.
• ### A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications(1607.02291)
May 8, 2018 nlin.CG, cs.FL
Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.
• ### Anticipating Persistent Infection(1805.01931)
May 4, 2018 q-bio.PE, nlin.CG
We explore the emergence of persistent infection in a closed region where the disease progression of the individuals is given by the SIRS model, with an individual becoming infected on contact with another infected individual within a given range. We focus on the role of synchronization in the persistence of contagion. Our key result is that higher degree of synchronization, both globally in the population and locally in the neighborhoods, hinders persistence of infection. Importantly, we find that early short-time asynchrony appears to be a consistent precursor to future persistence of infection, and can potentially provide valuable early warnings for sustained contagion in a population patch. Thus transient synchronization can help anticipate the long-term persistence of infection. Further we demonstrate that when the range of influence of an infected individual is wider, one obtains lower persistent infection. This counter-intuitive observation can also be understood through the relation of synchronization to infection burn-out.
• ### Matrix Product Operators for Sequence to Sequence Learning(1803.10908)
The method of choice to study one-dimensional strongly interacting many body quantum systems is based on matrix product states and operators. Such method allows to explore the most relevant, and numerically manageable, portion of an exponentially large space. It also allows to describe accurately correlations between distant parts of a system, an important ingredient to account for the context in machine learning tasks. Here we introduce a machine learning model in which matrix product operators are trained to implement sequence to sequence prediction, i.e. given a sequence at a time step, it allows one to predict the next sequence. We then apply our algorithm to cellular automata (for which we show exact analytical solutions in terms of matrix product operators), and to nonlinear coupled maps. We show advantages of the proposed algorithm when compared to conditional random fields and bidirectional long short-term memory neural network. To highlight the flexibility of the algorithm, we also show that it can readily perform classification tasks.
• ### Universality in Freezing Cellular Automata(1805.00059)
April 20, 2018 math.DS, cs.DM, cs.CC, nlin.CG
Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.
• ### Free to move or trapped in your group: Mathematical modeling of information overload and coordination in crowded populations(1804.06580)
April 18, 2018 physics.soc-ph, nlin.CG
We present modeling strategies that describe the motion and interaction of groups of pedestrians in obscured spaces. We start off with an approach based on balance equations in terms of measures and then we exploit the descriptive power of a probabilistic cellular automaton model. Based on a variation of the simple symmetric random walk on the square lattice, we test the interplay between population size and an interpersonal attraction parameter for the evacuation of confined and darkened spaces. We argue that information overload and coordination costs associated with information processing in small groups are two key processes that influence the evacuation rate. Our results show that substantial computational resources are necessary to compensate for incomplete information -- the more individuals in (information processing) groups the higher the exit rate for low population size. For simple social systems, it is likely that the individual representations are not redundant and large group sizes ensure that this non--redundant information is actually available to a substantial number of individuals. For complex social systems information redundancy makes information evaluation and transfer inefficient and, as such, group size becomes a drawback rather than a benefit. The effect of group sizes on outgoing fluxes, evacuation times and wall effects are carefully studied with a Monte Carlo framework accounting also for the presence of an internal obstacle.
• ### Von Neumann regularity, split epicness and elementary cellular automata(1804.03913)
April 12, 2018 math.DS, nlin.CG, cs.FL
We show that a cellular automaton on a mixing subshift of finite type is a Von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from [S.-T\"orm\"a, 2015] that Von Neumann regularity is decidable condition, and we decide it for all elementary CA.
• ### Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects(1804.02300)
We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant boundary correspond to percolating trees, and those that do not correspond to non-percolating trees. In fact, both systems display a second-order phase transition between "insulating" and "conducting" phases, controlled by the drift velocity in the Brownian system. In the limit of large system size, we obtain exact expressions for the Laplace transforms of the probability distributions and their first and second moments. These quantities are also shown to obey finite-size scaling laws.
• ### A gauge-invariant reversible cellular automaton(1802.07644)
April 10, 2018 quant-ph, nlin.CG, cs.FL
Gauge-invariance is a fundamental concept in physics---known to provide the mathematical justification for all four fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata. More precisely, we describe a step-by-step gauging procedure to enforce local symmetries upon a given Cellular Automaton. We apply it to a simple Reversible Cellular Automaton for concreteness. From a Computer Science perspective, discretized gauge theories may be applied to numerical analysis, quantum simulation, fault-tolerant (quantum) computation. From a mathematical perspective, discreteness provides a simple yet rigorous route straight to the core concepts.
• ### Algorithmic Information Dynamics of Persistent Patterns and Colliding Particles in the Game of Life(1802.07181)
April 5, 2018 cs.IT, math.IT, math.DS, nlin.CG
Without loss of generalisation to other systems, including possibly non-deterministic ones, we demonstrate the application of methods drawn from algorithmic information dynamics to the characterisation and classification of emergent and persistent patterns, motifs and colliding particles in Conway's Game of Life (GoL), a cellular automaton serving as a case study illustrating the way in which such ideas can be applied to a typical discrete dynamical system. We explore the issue of local observations of closed systems whose orbits may appear open because of inaccessibility to the global rules governing the overall system. We also investigate aspects of symmetry related to complexity in the distribution of patterns that occur with high frequency in GoL (which we thus call motifs) and analyse the distribution of these motifs with a view to tracking the changes in their algorithmic probability over time. We demonstrate how the tools introduced are an alternative to other computable measures that are unable to capture changes in emergent structures in evolving complex systems that are often too small or too subtle to be properly characterised by methods such as lossless compression and Shannon entropy.
• ### Ab initio Algorithmic Causal Deconvolution of Intertwined Programs and Networks by Generative Mechanism(1802.09904)
April 5, 2018 cs.AI, nlin.CG
Complex data is usually produced by interacting sources with different mechanisms. Here we introduce a parameter-free model-based approach, based upon the seminal concept of Algorithmic Probability, that decomposes an observation and signal into its most likely algorithmic generative sources. Our methods use a causal calculus to infer model representations. We demonstrate the method ability to distinguish interacting mechanisms and deconvolve them, regardless of whether the objects produce strings, space-time evolution diagrams, images or networks. We numerically test and evaluate our causal separation methods and find that it can disentangle examples of observations from discrete dynamical systems, and complex networks. We think that these causal separating techniques can contribute to tackle the challenge of causation for estimations of better rooted probability distributions thereby complementing more limited statistical-oriented techniques that otherwise would lack model inference capabilities.
• ### Asymptotic Behaviour and Ratios of Complexity in Cellular Automata(1304.2816)
April 5, 2018 cs.CC, nlin.CG
We study the asymptotic behaviour of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighbourhood range and number of symbols (or colours), and how different behaviour is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioural fractions. We also advance a notion of asymptotic and limit behaviour for individual rules, both over initial conditions and runtimes, and we provide a formalisation of Wolfram's classification as a limit function in terms of Kolmogorov complexity.
• ### A study of Inverse Ultra-discretization of cellular automata(1804.01089)
April 3, 2018 nlin.CG
In this article, I propose a systematic method for the inverse ultra-discretization of cell automata using a functionally complete operation. We derive difference equations for the 256 kinds of elementary cellular automata(ECA) introduced Wolfram\cite{wolfram} by the proposed means of the inverse ultra-discretization. We show that the behaviors of ECAs can be completely reproduced by numerically solving the obtained difference equations.
• ### Diagrammatic approach to cellular automata and the emergence of form with inner structure(1605.06937)
April 2, 2018 math-ph, math.MP, nlin.PS, nlin.AO, nlin.CG
We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of order out of disorder. Specifically, we show how the the majority CA rule can be used as a building block to construct more complex cellular automata in which separate domains (with substructures having different dynamical properties) are able to emerge out of disorder and coexist in a stable manner.
• ### Behaviour of traffic on a link with traffic light boundaries(1804.04086)
This paper considers a single link with traffic light boundary conditions at both ends, and investigates the traffic evolution over time with various signal and system configurations. A hydrodynamic model and a modified stochastic domain wall theory are proposed to describe the local density variation. The Nagel-Schreckenberg model (NaSch), an agent based stochastic model, is used as a benchmark. The hydrodynamic model provides good approximations over short time scales. The domain wall model is found to reproduce the time evolution of local densities, in good agreement with the NaSch simulations for both short and long time scales. A systematic investigation of the impact of network parameters, including system sizes, cycle lengths, phase splits and signal offsets, on traffic flows suggests that the stationary flow is dominated by the boundary with the smaller split. Nevertheless, the signal offset plays an important role in determining the flow. Analytical expressions of the flow in relation to those parameters are obtained for the deterministic domain wall model and match the deterministic NaSch simulations. The analytic results agree qualitatively with the general stochastic models. When the cycle is sufficiently short, the stationary state is governed by effective inflow and outflow rates, and the density profile is approximately linear and independent of time.
• ### Kauffman cellular automata on quasicrystal topology(1803.09836)
March 28, 2018 cond-mat.stat-mech, nlin.CG
In this paper we perform numerical simulations to study Kauffman cellular automata (KCA) on quasiperiod lattices. In particular, we investigate phase transition, magnetic entropy and propagation speed of the damage on these lattices. Both the critical threshold parameter $p_{c}$ and the critical exponents are estimated with good precision. In order to investigate the increase of statistical fluctuations and the onset of chaos in the critical region of the model, we have also defined a magnetic entropy to these systems. It is seen that the magnetic entropy behaves in a different way when one passes from the frozen regime ($p<p_{c}$) to the chaotic regime ($p>p_{c}$). For a further analysis, the robustness of the propagation of failures is checked by introducing a quenched site dilution probability $q$ on the lattices. It is seen that the damage spreading is quite sensitive when a small fraction of the lattice sites are disconnected. A finite-size scaling analysis is employed to estimate the critical exponents. From these numerical estimates, we claim that on both pure ($q=0$) and diluted ($q=0.05$) quasiperiodic lattices, the KCA model belongs to the same universality class than on square lattices. Furthermore, with the aim of comparing the dynamical behavior between periodic and quasiperiodic systems, the propagation speed of the damage is also calculated for the square lattice assuming the same conditions. It is found that on square lattices the propagation speed of the damage obeys a power law as $v\sim (p-p_{c})^{\alpha}$, whereas on quasiperiod lattices it follows a logarithmic law as $v \sim \ln(p-p_{c})^\alpha$.
• ### Agent-Based Implementation of Particle Hopping Traffic Model With Stochastic and Queuing Elements(1803.09206)
March 25, 2018 stat.AP, nlin.CG
Lagging or halted traffic is bothersome. As such, it is desirable to have a model that can begin to determine the efficiency of various traffic standardizations. Our model intended to create a multifaceted realistic simulation of traffic flow while considering several factors. These factors included: passing conventions, e.g., right except to pass (REP) rule, system perturbation caused by insertion of an accident into the system, accessible number of lanes available with the REP, various human factors such as variation of individual maximum speed and likelihood to pass. A succession of models were created from a variation on an existing single-lane traffic model and adding extra dimensionality to the lattice to include multiple lanes, passing conventions, stochastic elements for individuality, and queuing rules to movement algorithms. We found that the REP is an effective means of increasing the critical density that a system can support. Eliminating human factors and thereby automating the system, results in a 160% increase in the sustainable critical density of the system. The number of lanes increases the critical density of the system, but the maximum efficiency of the speed distribution remains the same. Excluding system automation, the optimal speed distribution for drivers maximal speed was found to be Beta(5,5). Accidents in stable systems can cause small local jams without causing global jams.
• ### Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling(1803.08770)
March 23, 2018 math.NA, nlin.CG
We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above.
• ### On weak universality of three-dimensional Larger than Life cellular automaton(1803.06514)
March 17, 2018 nlin.CG
Larger than Life cellular automaton (LtL) is a class of cellular automata and is a generalization of the game of Life by extending its neighborhood radius. We have studied the three-dimensional extension of LtL. In this paper, we show a radius-4 three-dimensional LtL rule is a candidate for weakly universal one.
|
2022-08-15 16:17:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5672978162765503, "perplexity": 764.117475715205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572192.79/warc/CC-MAIN-20220815145459-20220815175459-00334.warc.gz"}
|
http://physics.stackexchange.com/questions/4199/does-bunching-reduce-synchrotron-radiation/4222
|
# Does bunching reduce synchrotron radiation?
A continuous charge distribution flowing as a constant current in a closed loop doesn't radiate. Is it therefore true that as you increase the number of proton bunches in the LHC, while keeping the total charge constant, the synchrotron radiation decreases?
-
Please somebody correct the question. The LHC has circulating protons. Either change "the LHC" to "an accelerator" or the "electron" to "proton". – anna v Jan 30 '11 at 15:41
By the way, a remark to the title: it is actually debunching not bunching that reduces the SR. – Igor Ivanov Jan 30 '11 at 18:47
@Anna, I've made the requested edit. – John McVirgo Jan 30 '11 at 18:50
@Igor i'm interested in whether it's possible to collide charged particles together, while reducing synchrotron radiation. You would still need bunched charged particles to do this, correct? – John McVirgo Jan 30 '11 at 19:03
@John — just to make sure: the term "bunching" means "the act of grouping particles in bunches", while "debunching" means "spreading out initially bunched particles into a more homogeneous distribution". You seem to be using "bunching" as an equivalent of "the number of bunches", which is not the correct usage. – Igor Ivanov Jan 30 '11 at 22:32
Synchrotron radiation can be coherent and incoherent. Coherent SR arises when electrons are grouped into short bunches so that the entire bunch emits SR as a whole. Quantum mechanically, in coherent SR the photon emission from different electrons in a bunch sum up at the amplitudes level and constructively interfere. In the incoherent SR they sum up at the level of intensity, and there is no interference.
Incoherent SR does not care how electrons are distributed along the ring, while the coherent SR is obviously boosted up in the presence of strong bunching. So, the more homogeneously you distribute the electrons, the less the effect of coherent SR will be and the less overall SR you'll have.
Now let's look at the incoherent SR. Theoretically, you are right: if we managed to create the absolute homogeneous charge distribution along the ring, we would (classically) have no SR at all because charge distribution does not change in time. The point is that this is not feasible experimentally, at least for the accelerators and the beams we have. That would require putting electrons in a well-defined quantum state of the radial motion and a well-defined angular quantum number m for the azimuthal dependence, and the accelerator technology is very far from that.
However, there is another thing which mimics that situation closely. People have managed recently to put freely propagating electrons in states with well-defined orbital angular momentum (m as high as 75), see this paper in Science for details, and they really see the annular distribution for the electron density. For such a state there exists a reference frame where the electron does not move along the z axis but just rotated as a whole in the transverse plane (with some radial distribution) around the symmetry axis. This rotation is not driven by any force, it's just the peculiar superposition of plane waves that creates this steady pattern. So in this case you can say that the electron indeed circulates but does not emit any SR.
-
Ahh, what a relief...:=) Otherwise I wouldn'have slept coming nights. – Georg Jan 30 '11 at 16:05
I'll look further into the points you've made, thanks. – John McVirgo Jan 30 '11 at 20:41
What a wonderful answer! – Carl Brannen Jan 31 '11 at 0:44
How/Where do I read about the quantum mechanic view of Incoherent/Coherent SR? Thanks... – user1886681 Dec 5 '14 at 19:40
Even if you fill a ring with a perfectly uniform current (neglecting shot noise), each particle would still be radiating... – DarioP Feb 19 at 14:38
Dear John, a good question. You may want to read a relevant paper about the closely related question for the late SSC collider:
http://mafurman.lbl.gov/SSC-N-143.pdf
Bunch-Length Dependence of Power Loss for the SSC
The beam has $M$ bunches in the orbit. Each of them carries $Ne$ of electric charge. All of the particles orbit by frequency $f_0$ (revolutions per second, in Hertz). We define the product, the bunch current, to be $I_b=N e f_0$.
In equation 12, you will see the result: $$Power = 1.101 Z_0 M I_b^2 \sigma_{\phi}^{-4/3}$$ Here, $Z_0$ is just the impedance of the vacuum, $4\pi/c = 377 \Omega$; they use some Gaussian units.
More importantly, $\sigma_\phi$ is just $\sigma_z/\rho$, the angular root mean square size of the bunch. You see that it's the only quantity whose increase makes the power decrease. If you spread the bunches around the ring, you're getting closer to your "closed loop current" that doesn't radiate, indeed. In practice, you don't want to spread the bunches completely because you wouldn't know the timing of the collisions. In real applications, $\sigma_\phi$ is much smaller than one, giving you a significant increase to the synchrotron radiation.
The formula is simply proportional to the number of bunches. If they're separated, each of them loses the same energy per revolution. Without a loss of generality, you may imagine that we only consider one bunch, $M=1$.
In that case, for a fixed $f_0$ - which is given by the size of the tunnel and the speed of light, assuming that the particles are near the speed of light - the power radiated by the bunch is actually proportional to $N^2$. If you double the number of charged particles in the bunch, the synchrotron radiation quadruples!
That's because the energy density (and flux) is proportional to the squared electric (and magnetic) fields, and those - derived from the Liénard-Wiechert potentials - are linear in the charges (and currents) that produce the electromagnetic fields.
So once again, the power that is radiated is not proportional to the "density" of protons in the bunch but to its square! In this sense, bunching makes the synchrotron radiation worse, not better.
However, you shouldn't think that it is a catastrophe. In the designed conditions for the LHC, one proton only loses something like 6.7 keV of energy per revolution which is a billionth of those 7 TeV they ultimately want to get (in 2011, they decided to continue at 3.5 TeV). Why is it so small for the hadron colliders?
Well, for the lepton colliders, you lose a lot because the synchrotron radiation is proportional to $\gamma^6$ and the Lorentz factor $\gamma$ has to be 2,000 times higher for electrons than for protons to achieve the same energy; see the derivation. Take the sixth power of that to see the impact of the light particles.
For the hadron colliders, the main limitation is of course the magnetic field you need to keep the protons on their circular orbit. That's why you need to have all the superconducting magnets. For colliders with light particles that need a huge $\gamma$, the synchrotron radiation is very important. That's also why linear accelerators are often preferred for the leptons. Well, you won't get rid of the full synchrotron radiation because you still need to accelerate the leptons to have some fun - so there will still be a component of the acceleration in the direction of the velocity even though the straight tunnel may liberate you from the "centripetal" acceleration transverse to the velocity.
To return to the closed loop, yes, I do think that you would turn the synchrotron radiation from the circular motion off completely if you distributed the bunches uniformly - even for leptons. It would be just like a wire with a current. However, there would still be a synchrotron radiation from the acceleration in the forward direction that you need to accelerate the particles to high speeds in the first place.
-
""yes, I do think that you would turn the synchrotron radiation from the circular motion off completely if you distributed the bunches uniformly "" Hello Lubos, wasn't one of the reasons against Sommerfeld/Bohrs atom models that the electron would radiate its energy and "fall" into the core? – Georg Jan 30 '11 at 12:25
Mhhmm, that multitude of electrons circulating in a synchrotron is something different. It seems that they cancel each others radiatiion. I am irritated :=( – Georg Jan 30 '11 at 13:53
Dear Georg, the whole point of the Bohr-Sommerfeld atom was that it required the "number of de Broglie waves" around the orbit to be integer, so the -13.6 eV state was the lowest possible orbit. The model was never consistent with other properties of the electron, of course, but if one assumed that the electrons can only orbit along closed path with the quantized number of waves, then the electron couldn't fall to the nucleus. It was the whole point of the atom that they wanted to fix this "collapse" problem of the classical atom - one that had no quantization. – Luboš Motl Jan 30 '11 at 18:05
Lubos, I thought of the times when a "planetary-like" atom was proposed, but Broglies "waves" were not known yet. – Georg Jan 30 '11 at 20:28
Great informative answer as usual, Lubos. I was asking for the case of keeping the total charge in the ring constant while increasing the bunching. Therefore, from equation (12), N and therefore Ib is inversely propotional to M giving the power loss inversely proportional to the square of M. So yes, for a constant total charge in the ring, the powerloss does decrease dramatically with increased bunching, at least in theory. – John McVirgo Jan 30 '11 at 20:32
I am afraid that the radiation cannot cancel everywhere so it is better to say the radiation does not occur in case of a constant current. This is so because, according to Maxwell equations, it is not acceleration of a single charge that creates the radiation but the current time-dependence at a given point. In other words, different sources radiate differently and it is not reduced to the sum of radiations. The total filed is determined differently: superposition of fields is not a sum of radiations! The same is valid in the opposite case of short bunches where the radiative losses are proportional to the charge squared ( = source-dependent phenomenon).
-
Dear downvoters, leave short explanation or a disagreement statement, please! – Vladimir Kalitvianski Jan 30 '11 at 18:13
Have a look at the nonradiation condition: en.wikipedia.org/wiki/Nonradiation_condition. Problem 14.24 in Jackson asks to show there is no radiaton from a closed loop of current. – John McVirgo Jan 30 '11 at 20:48
I did not get your suggestion. Do you mean that radiation of multiple charges can cancel everywhere? Do you mean that in the whole space on can get a purely destructive interference? – Vladimir Kalitvianski Jan 30 '11 at 21:12
Yes, there are some some accelerating charge distributions where the total radiation at all points sums to zero. This goes back to 1910, if you look at the Wikipedia link: "In 1910 Paul Ehrenfest published a short paper on "Irregular electrical movements without magnetic and radiation fields" demonstrating that Maxwell’s equations allow for the existence of accelerating charge distributions which emit no radiation." – John McVirgo Jan 30 '11 at 22:39
I will look through it if it is available but why then it is not present on the textbooks? – Vladimir Kalitvianski Jan 30 '11 at 23:13
|
2015-05-26 08:00:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7795262932777405, "perplexity": 715.718788948397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928817.29/warc/CC-MAIN-20150521113208-00148-ip-10-180-206-219.ec2.internal.warc.gz"}
|
https://ask.sagemath.org/question/10297/print-value-of-vertical-horizontal-asymptote/?sort=votes
|
# Print value of vertical & horizontal asymptote?
By default the program draws a blue line displaying the vertical asymptote of a function, which is very helpful. If I could get it to display a line through the horizontal asymptote, & print their value, if would be very helpful.
I attempted the following to get the vertical asymptote & had no luck at all:
ans = solve( 7/(x-3) == 0, x)
v0 = ans[0].rhs()
print(v0)
Error message:
Traceback (click to the left of this block for traceback)
...
IndexError: list index out of range
edit retag close merge delete
Sort by » oldest newest most voted
When you write:
solve( 7/(x-3) == 0, x)
you look for solutions of the equation $7/(x-3) = 0$, which has no solution. If you want to discover the poles of your function (not the zeroes) to locate where your function has vertical asymptotes, you should look for the zeroes of the inverse of your function:
sage: f(x) = 7/(x-3)
sage: ans = solve(1/f == 0, x)
sage: ans[0].rhs()
3
If you want to locate the horizontal asymptotes, you shoult look at the limit of your function in $+\infty$ and $-\infty$:
sage: limit(f,x=+infinity)
x |--> 0
sage: limit(f,x=-infinity)
x |--> 0
Here, the vertical asymptote has equation $x=3$, and the horizontal asymptote has equation $y=0$.
more
|
2021-09-20 13:40:02
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22925522923469543, "perplexity": 1440.526327420178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057039.7/warc/CC-MAIN-20210920131052-20210920161052-00667.warc.gz"}
|
https://artofproblemsolving.com/wiki/index.php?title=2013_AIME_I_Problems/Problem_13&diff=prev&oldid=143137
|
# Difference between revisions of "2013 AIME I Problems/Problem 13"
## Problem
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
## Solution 1 (Simple, Sane Solution)
Well, first draw a good diagram! One is provided below. Convince yourself that every $B_nC_n$ is parallel to each other for any nonnegative $n$. Next, convince yourself that the area we seek is simply the ratio $k=\frac{B_0B_1C_1}{B_0B_1C_1+C_1C_0B_0}$, because it repeats in smaller and smaller units. Note that the area of the triangle, by Heron's formula, is 90.
For ease, all ratios I will use to solve this problem are with respect to the area of $AB_0C_0$. For example, if I say some area has ratio $\frac{1}{2}$, that means its area is 45.
Now note that $k=$ 1 minus ratio of $B_1C_1A$ minus ratio $B_0C_0C_1$. We see by similar triangles given that ratio $B_0C_0C_1$ is $\frac{17^2}{25^2}$. Ratio $B_1C_1A$ is, after seeing that $C_1C_0 = \frac{289}{625}$, $(\frac{336}{625})^2$. Now it suffices to find 90 times ratio $B_0B_1C_1$, which is given by 1 minus the two aforementioned ratios. Substituting these ratios to find $k$ and clearing out the $5^8$, we see that the answer is $90\cdot \frac{5^8-336^2-17^2\cdot 5^4}{5^8-336^2}$. Calculation might take some time, but you've solved the problem! $p= \boxed{961}$.
## Solution 2
Using Heron's Formula we can get the area of the triangle $\Delta AB_0C_0 = 90$.
Since $\Delta AB_0C_0 \sim \Delta B_0C_1C_0$ then the scale factor for the dimensions of $\Delta B_0C_1C_0$ to $\Delta AB_0C_0$ is $\dfrac{17}{25}.$
Therefore, the area of $\Delta B_0C_1C_0$ is $(\dfrac{17}{25})^2(90)$. Also, the dimensions of the other sides of the $\Delta B_0C_1C_0$ can be easily computed: $\overline{B_0C_1}= \dfrac{17}{25}(12)$ and $\overline{C_1C_0} = \dfrac{17^2}{25}$. This allows us to compute one side of the triangle $\Delta AB_0C_0$, $\overline{AC_1} = 25 - \dfrac{17^2}{25} = \dfrac{25^2 - 17^2}{25}$. Therefore, the scale factor $\Delta AB_1C_1$ to $\Delta AB_0C_0$ is $\dfrac{25^2 - 17^2}{25^2}$ , which yields the length of $\overline{B_1C_1}$ as $\dfrac{25^2 - 17^2}{25^2}(17)$. Therefore, the scale factor for $\Delta B_1C_2C_1$ to $\Delta B_0C_1C_0$ is $\dfrac{25^2 - 17^2}{25^2}$. Some more algebraic manipulation will show that $\Delta B_nC_{n+1}C_n$ to $\Delta B_{n-1}C_nC_{n-1}$ is still $\dfrac{25^2 - 17^2}{25^2}$. Also, since the triangles are disjoint, the area of the union is the sum of the areas. Therefore, the area is the geometric series $\dfrac{90 \cdot 17^2}{25^2} \sum_{n=0}^{\infty} (\dfrac{25^2-17^2}{25^2})^2$ At this point, it may be wise to "simplify" $25^2 - 17^2 = (25-17)(25+17) = (8)(42) = 336$. So the geometric series converges to $\dfrac{90 \cdot 17^2}{25^2} \dfrac{1}{1 - \dfrac{336^2}{625^2}} = \dfrac{90 \cdot 17^2}{25^2} \dfrac{625^2}{625^2 - 336^2}$. Using the difference of squares, we get $\dfrac{90 \cdot 17^2}{25^2}\dfrac{625^2}{(625 - 336)(625 + 336)}$, which simplifies to $\dfrac{90 \cdot 17^2}{25^2} \dfrac{625^2}{(289)(961)}$. Cancelling all common factors, we get the reduced fraction $= \dfrac{90 \cdot 25^2}{31^2}$. So $\frac{p}{q}=1-\frac{90 \cdot 25^2}{31^2}=\frac{90 \cdot 336}{961}$, yielding the answer $\fbox{961}$.
|
2021-05-14 22:57:58
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 65, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9288368821144104, "perplexity": 120.86038188067681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00140.warc.gz"}
|
https://keplerlounge.com/information-theory/2021/05/11/copeland-erdos.html
|
## Motivation:
Given the set of prime numbers $$\mathbb{P} \subset \mathbb{N}$$, the Copeland–Erdős constant $$\mathcal{C}$$ is defined as [1]:
$$\mathcal{C} = \sum_{n=1}^\infty p_n \cdot 10^{-(n+ \sum_{k=1}^n \lfloor \log_{10} p_k \rfloor}$$
where $$p_n$$ is the nth prime number.
Now, it is generally known that $$\mathcal{C}$$ is normal and that normal numbers are finite-state incompressible. As machine learning systems are finite-state machines it occurred to me that prime formulas are not approximable by machine learning systems regardless of their computational power [5]. While there are probably several approaches to this question, compression bounds driven by the Shannon source coding theorem seem natural as this theorem demonstrates that in the asymptotic limit it is impossible to compress i.i.d. data such that the average number of bits per symbol is less than the Shannon entropy of the source.
As I could not find such an information-theoretic definition of the normal numbers, I considered the problem until an effective definition occurred to me which allows us to derive compression bounds for any normal number.
## An information-theoretic approach to normal numbers:
### An information-theoretic definition of normal numbers:
Given the alphabet $$\Sigma$$ with $$\lvert \Sigma \rvert = \alpha$$ and $$X = \Sigma^{\infty}$$ we may define:
$$\Sigma^n \cap X_N := \bigcup_{w_i \in \Sigma^n} w_i \cap \{x_i\}_{i=1}^{N-n+1}$$
where $$x_i = X[i, i +n-1]$$ and $$\lim\limits_{N \to \infty} X_N = X$$.
In this context, $$X$$ is normal to base $$\alpha$$ if for any $$Z_N \sim U(\Sigma^n \cap X_N)$$ with $$N \gg n$$ the average amount of information gained from observing each digit in $$Z_N$$ converges to:
$$\log_2 \lvert \Sigma^n \rvert = n \cdot \log_2 \lvert \Sigma \rvert$$
as $$N \to \infty$$.
### Proof of equivalence with the usual definition:
From a frequentist perspective, we may define the probabilities:
$$\forall w_i \in \Sigma^n, p_{w_i} = \lim_{N \to \infty} \frac{\mathcal{N}(X_N,w_i)}{N-n+1} = \frac{1}{\lvert\Sigma^n\rvert}$$
where $$\mathcal{N}(X_N,w_i)$$ counts the number of times the string $$w_i$$ appears as a substring of $$X_N$$.
We have a uniform distribution over $$\Sigma^n$$ in the sense that:
$$\forall w_i, w_{j \neq i} \in \Sigma^n,\quad p_{w_i} = p_{w_{j \neq i}}$$
Now, we define the random variable $$Z_N \sim U(\Sigma^n \cap X_N)$$ whose Shannon entropy is given by:
$$H(Z_N) = - \sum_{i=1}^{\lvert \Sigma^n \rvert} P(Z_N = w_i) \log_2 P(Z_N = w_i)$$
which is defined for sufficiently large $$N$$ since:
$$\forall w_i \in \Sigma^n \forall \epsilon > 0 \exists m \in \mathbb{N} \forall N \geq m, \Bigl\lvert \frac{\mathcal{N}(X_N,w_i)}{N-n+1} - \frac{1}{\lvert\Sigma^n\rvert} \Bigr\rvert < \epsilon$$
and therefore we have:
$$\lim_{N \to \infty} H(Z_N) = \log_2 \lvert \Sigma^n \rvert = n \cdot \log_2 \lvert \Sigma \rvert$$
## Corollary:
For large $$N$$, if we break $$X_N$$ into $$\lfloor \frac{N}{n}\rfloor$$ segments, the information gained from observing all of these segments converges to:
$$H(X_N) \approx \big\lfloor \frac{N}{n}\big\rfloor \cdot H(Z_N) \approx N \cdot \log_2 \lvert \Sigma \rvert$$
## Application to the Copeland–Erdős constant:
Given that $$\mathcal{C}$$ is normal in base-10, it is finite-state incompressible. In particular, if $$\mathcal{A}$$ is the description language for all finite-state automata(which includes all learnable programs) then we may deduce that for large $$N$$ [6]:
$$\mathbb{E}[K_{\mathcal{A}}(\mathcal{C}_N)] \sim N \cdot \log_2 (10)$$
where $$\mathcal{C}_N$$ denotes the first $$N$$ digits of $$\mathcal{C}$$ and $$K(\cdot)$$ denotes prefix-free Kolmogorov Complexity. This is consistent with the fact that almost all strings are incompressible [7].
It follows that a prime formula is not approximable.
## Relation to the Champernowne constant:
A mathematician may point out that this argument may be extended to the Champernowne constant which is also normal and therefore we may argue that a successor function:
$$\forall n \in \mathbb{N}, S(n) = n+1$$
is not approximable. Surely, this argument must be absurd.
However, a successor function is non-trivial for large integers relative to finite-state machines(i.e. machines with finite memory). In fact, almost all positive integers are incompressible since:
$$\forall n \in \mathbb{N}^* \forall k < n, |\{x \in \{0,1\}^*:K(x) \geq n -k \}| \geq 2^n(1-2^{-k})$$
where $$\lvert x \rvert = n$$, the binary length of $$x$$, which may be understood as the machine-code representation of an integer. (12) may be proven using the pigeon-hole principle as was done in [7].
From an information-theoretic perspective, an integer $$N$$ is incompressible if we need $$\sim \log_2 (N)$$ bits to compress $$N$$ and given that every integer has a unique prime factorisation:
$$\log_2 N = \log_2 \prod_i p_i^{\alpha_i} = \sum_i \alpha_i \log_2 p_i$$
this is equivalent to saying that all the information contained in the integers is contained in the prime numbers(which we know relatively little about).
## References:
1. Copeland, A. H. and Erdős, P. “Note on Normal Numbers.” Bull. Amer. Math. Soc. 52, 857-860, 1946.
2. A. N. Kolmogorov Three approaches to the quantitative definition of information. Problems of Information and Transmission, 1(1):1–7, 1965
3. Olivier Rioul. This is IT: A Primer on Shannon’s Entropy and Information. Séminaire Poincaré. 2018.
4. Edward Witten. A Mini-Introduction To Information Theory. 2019.
5. Shai Shalev-Shwartz and Shai Ben-David. Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press. 2014.
6. Peter Grünwald and Paul Vitányi. Shannon Information and Kolmogorov Complexity. 2010.
7. Lance Fortnow. Kolmogorov Complexity. 2000.
|
2021-09-25 11:42:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 13, "x-ck12": 0, "texerror": 0, "math_score": 0.9374065399169922, "perplexity": 456.5138826964886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057622.15/warc/CC-MAIN-20210925112158-20210925142158-00082.warc.gz"}
|
http://mathoverflow.net/feeds/question/14374
|
Integral expression for zeta(2) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:15:07Z http://mathoverflow.net/feeds/question/14374 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14374/integral-expression-for-zeta2 Integral expression for zeta(2) Franz Lemmermeyer 2010-02-06T11:15:41Z 2010-02-06T12:11:05Z <p>By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book <em>Euler through time. A new look at old themes</em>, 2006) I found that $$\sum B_n = \int_0^\infty \frac{t}{e^{2t}-e^t} dt$$ and discovered numerically that this expression equals $\zeta(2)-1$. The web is not very good for finding out where this can be found in print. Where should I look, and how can equations such as $$\zeta(2) = 1 + \int_0^\infty \frac{t}{e^{2t}-e^t}\ dt$$ be proved?</p> http://mathoverflow.net/questions/14374/integral-expression-for-zeta2/14381#14381 Answer by engelbrekt for Integral expression for zeta(2) engelbrekt 2010-02-06T12:11:05Z 2010-02-06T12:11:05Z <p>The starting point is the integral </p> <p>$$\Gamma(s) = \int_{0}^{\infty}e^{-x}x^{s-1}dx$$</p> <p>for the gamma function. Make the change of variable $x = nu$ with $n$ an arbitrary positive integer. Then </p> <p>$$\Gamma(s)n^{-s} = \int_{0}^{\infty}e^{-nu}u^{s-1}du$$</p> <p>and summing over $n$ from $n = 1$ yields</p> <p>$$\Gamma(s)\zeta(s) = \int_0^{\infty}\frac{1}{e^u - 1}u^{s-1}du.$$</p> <p>This formula was the starting point of one of Riemann's two proofs of the functional equation. I am not certain who discovered it first, but it may have been Abel.</p> <p>Substituting $s = 2$ gives </p> <p>$$\zeta(2) = \int_{0}^{\infty}\frac{u}{e^u - 1}du$$</p> <p>and so</p> <p>$$\zeta(2) = \int_{0}^{\infty}\frac{ue^u}{e^{2u} - e^u}du = \int_{0}^{\infty}\frac{u(e^u - 1) + u}{e^{2u} - e^u}du = \int_{0}^{\infty}\left(ue^{-u} + \frac{u}{e^{2u} - e^u}\right)du = 1 + \int_{0}^{\infty}\frac{u}{e^{2u} - e^u}du.$$</p>
|
2013-05-22 02:15:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9135344624519348, "perplexity": 2285.7966514737523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701153213/warc/CC-MAIN-20130516104553-00043-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://math.stackexchange.com/questions/2804955/can-peano-arithmetic-show-that-the-continuum-hypothesis-is-independent-of-zfc
|
# Can Peano Arithmetic show that the Continuum Hypothesis is Independent of ZFC?
Can Peano Arithmetic show that the Continuum Hypothesis is Independent of ZFC? In other words, is $PA \vdash Con(ZFC) \implies Con (ZFC + CH) \land Con(ZFC + \lnot CH)$ true?
I believe the answer is yes, since it appears that you can mechanically turn a contradiction in $ZFC + CH$ or $ZFC + \lnot CH$ into a contradiction in $ZFC$, but I'm not familiar enough with the result to be sure.
It would be more interesting if it was false though, since then we'd have nice models of arithmetic in which ZFC decides the Continuum Hypothesis.
Yes, the independence result is provable in PA (and even weaker theories). The usual proofs are phrased in terms of models, for better human comprehension, but they can also be phrased as purely syntactic manipulations of proofs. (I believe Joel David Hamkins has explained this either here or on MathOverflow.) The syntactic argument should be formalizable even in primitive recursive arithmetic.
As Andreas says, it is in fact provable in PA and even much weaker theories - the difficulty of course being how to remove the "semantic" content. However, it's worth noting that we can show provability in PA without doing any major changes, via conservativity. Here's an outline:
• First, consider the theory ACA$_0$. This is a conservative extension of PA - every sentence in the language of arithmetic which is provable in ACA$_0$ is provable already in PA, and PA $\subseteq$ ACA$_0$ - but is able to talk about sets, and hence models. Moreover, the second-order axioms of ACA$_0$ are enough to prove basic facts about logic, especially that $(i)$ every consistent theory is contained in a consistent complete theory, $(ii)$ every consistent theory has a model, and $(iii)$ every theory with a model is consistent (note that in the setting of ACA$_0$, all theories are sets of natural numbers, hence countable, as are all models). (In fact ACA$_0$ is more than enough - I'm not shooting for optimality here.)
• Now we just repeat the usual proof inside ACA$_0$. Reasoning inside ACA$_0$, we suppose ZFC is consistent; then it has a model $M$. This model is "explicitly countable" - precisely because of the paucity of ACA$_0$'s framework, $M$ has domain $\subseteq\omega$! This means that we can prove in ACA$_0$ that $M$-generic filters exist, phrased appropriately, for any $M\models$ ZFC.
• Letting $G$ be such a generic filter, we want to argue inside ACA$_0$ that $M[G]$ exists since $M$ does. On this face of it this might be worrying because of the amount of transfinite recursion needed to define the forcing extension; however, since all that recursion takes place inside the ground model (here, $M$), it really boils down to "externally" arithmetical facts about $M$, which ACA$_0$ can reason about appropriately.
• So after following the usual argument with a bit of care, we get that ACA$_0$ proves (for example) that ZFC has a model iff ZFC+CH has a model.
• Now ACA$_0$ proves the soundness and completeness theorems as observed above, so ACA$_0$ also proves that ZFC is consistent iff ZFC+CH is consistent. But this is an arithmetical fact; since ACA$_0$ is conservative over PA, this means PA proves that ZFC is consistent iff ZFC+CH is consistent, and we're done!
This suggests a neat approach to "finitizing" forcing arguments: if you have a semantic proof that Con(T) implies Con(S), and you want to prove Con(T)$\implies$Con(S) in some weak theory of arithmetic $A$, you look for a conservative extension $\hat{A}$ of $A$ which can run the appropriate semantic arguments directly. However, this seems fundamentally limited; in particular, the bigger system $\hat{A}$ really needs to be able to prove the completeness theorem, which forces us to include weak Konig's lemma, and this limits the theories which have such a conservative extension. (E.g. replace PA with PRA above and I'm not sure what to do.) So that seems like it won't work to get these relative consistency results proved in truly weak arithmetics. Instead, I think work needs to be done to really excise the semantic content from the usual arguments and as Andreas says turn them into arguments about proof manipulation.
It's possible I'm wrong, and there are ways we can get around the "completeness barrier," but my understanding is that to go significantly below PA we really do want to give up semantics, at least to a large extent, rather than trying to shoehorn it in by finding serendipitous conservative extensions.
And when we go far enough down, things can indeed get truly weird. There are extremely weak systems of arithmetic which are just barely strong enough for it to be meaningful to ask what they think about logic, but which allow for totally ridiculous possibilities. For example, in the paper Oracle bites theory, Visser looks at the possibility of a consistent theory having inconsistent deductive closure. So at that level, anything's possible.
|
2019-11-21 08:09:19
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9031172394752502, "perplexity": 466.74434748708507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670743.44/warc/CC-MAIN-20191121074016-20191121102016-00252.warc.gz"}
|
https://www.dcode.fr/square-completion
|
Search for a tool
Completing the Square
Tool to make automatic square completion. Square completing is a calculation method allowing to factor a quadratic polynomial expression using the polynomial depression method.
Results
Completing the Square -
Tag(s) : Symbolic Computation
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Thanks to your feedback and relevant comments, dCode has developed the best 'Completing the Square' tool, so feel free to write! Thank you!
# Completing the Square
## Completing the square solver
### What is a square completion? (Definition)
Completion of the square is the name given to a method of factorization of the polynomials due to this degree. Factoring takes its name from the fact that the factored form obtained has the variable in a squared expression.
### How to complete the square?
dCode can complete the square and find factors by depressing a polynomial expression
A quadratic polynomial $x^2 +bx + c = 0$ can be modified in $(b/2)^2 - c - (b/2)^2 + c (= 0)$ that allows factorizing in $$(x +(b/2))^2 - (b/2)^2 + c$$
Example: $p(x)=2x^2+12x+14$, in order to complete the square hand, factorize the coefficient of $x^2$ : $p(x)=2(x^2+6x+7)$ and continue with $q(x) = x^2+6x+7$
Example: Identify the coefficient of $x$, here $6$ and divide it by $2$ to get $β=6/2=3$ and use $β$ to write $$q(x) = x^2 + 6x + 7 = (x+3)^2 − β^2 + 7 = (x+3)^2 − 2$$
Example: Back to $p(x) = 2q(x)$ to get the completed square: $$p(x)=2x^2+12x+14=2((x+3)^2−2)=2(x+3)^2−6$$
With the factorized form, it becomes simple to find the roots.
$$p(x) = 0 \iff 2(x+3)^2−6 = 0 \iff (x+3)^2 = 3 \\ \iff x+3 = \pm \sqrt{3} \iff x = \pm \sqrt{3} - 3$$
dCode can generalize this approach to other polynomials of order $n > 2$ by removing the term of degree $n-1$.
## Source code
dCode retains ownership of the online 'Completing the Square' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Completing the Square' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Completing the Square' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Completing the Square' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
|
2021-06-22 10:46:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.2990477979183197, "perplexity": 3691.744349348972}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00141.warc.gz"}
|
http://physics.stackexchange.com/questions/64572/trying-to-understand-em-wave-and-photon
|
# Trying to understand EM wave and photon
When electrical fields and magnetic fields couple together, it forms electromagnetic waves. And we can "quantized" it and see each "package" of it as photon. So can electrical fields and magnetic fields be each "quantized" and visualized as a particle (half a photon?)
-
this is worth reading motls.blogspot.gr/2011/11/… – anna v May 15 '13 at 13:34
Short answer, no. Long answer, sort of.
Short answer: No, the E and M fields may be coupled by the Lorentz transformations, but it is only when they work together to make a self-propagating wave that we can call it a particle. To separate them as individual fields is physically meaningless. So giving each field their own particle is equally meaningless.
Long answer: This can be thought of as an "Is the Moon Really There When we aren't Looking at it?" problem. The only way we can observe that an E or M field is present is when it interacts with something via the Lorentz force. So if the field is not interacting with anything, is it really there? In advanced physics, that answer turns out to be a resounding "no". What we can say is that the E or M field interacts with objects via a photon. That is, the magnetic or electric field can be said to exists, but we can also say that the source of this field is interacting with our sensors or other particles by exchanging photons to produce a force. Thus, to answer your question, we can in a way "quantize" the separate fields and visualize them as a particle, but only if we visualize them not as fields but the exchange of photons between the sources of the field and the objects influenced by it. But full photons, not half a photon.
-
we can in a way "quantize" the separate fields - How can this be done? Could you provide any reference to a QED textbook? – firtree May 14 '13 at 16:50
@firtree I was using the term lightly. It's not actually quantizing the fields. But it is re-representing the fields in terms of an already quantized field; the gauge field – Jim May 14 '13 at 17:22
Basically, you can reconstruct the static E and B fields as superposition of photons. E and B fields to not, however, have any meaning sperate from each other, as they are transformed into each other by Lorentz transformations – Neuneck May 15 '13 at 11:44
@Neuneck Yes, exactly. I tried to get a simplified version of that point across in my answer. I take it by your comment that I didn't succeed at that? – Jim May 15 '13 at 13:18
@Dan Thanks for the edits. If there's any more problems, let me know. I really want this answer to be helpful to people. – Jim May 15 '13 at 13:28
show 2 more comments
|
2014-03-17 15:29:15
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6897743940353394, "perplexity": 317.84003293276237}, "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-2014-10/segments/1394678705742/warc/CC-MAIN-20140313024505-00028-ip-10-183-142-35.ec2.internal.warc.gz"}
|
https://eprint.iacr.org/2019/082
|
### Arithmetic Garbling from Bilinear Maps
Nils Fleischhacker, Giulio Malavolta, and Dominique Schröder
##### Abstract
We consider the problem of garbling arithmetic circuits and present a garbling scheme for inner-product predicates over exponentially large fields. Our construction stems from a generic transformation from predicate encryption which makes only blackbox calls to the underlying primitive. The resulting garbling scheme has practical efficiency and can be used as a garbling gadget to securely compute common arithmetic subroutines. We also show that inner-product predicates are complete by generically bootstrapping our construction to arithmetic garbling for polynomial-size circuits, albeit with a loss of concrete efficiency. In the process of instantiating our construction we propose two new predicate encryption schemes, which might be of independent interest. More specifically, we construct (i) the first pairing-free (weakly) attribute-hiding non-zero inner-product predicate encryption scheme, and (ii) a key-homomorphic encryption scheme for linear functions from bilinear maps. Both schemes feature constant-size keys and practical efficiency.
Available format(s)
Category
Public-key cryptography
Publication info
Preprint. Minor revision.
Keywords
Arithmetic GarblingPredicate Encryption
Contact author(s)
malavolta @ cs fau de
mail @ nilsfleischhacker de
dominique schroeder @ fau de
History
Short URL
https://ia.cr/2019/082
CC BY
BibTeX
@misc{cryptoeprint:2019/082,
author = {Nils Fleischhacker and Giulio Malavolta and Dominique Schröder},
title = {Arithmetic Garbling from Bilinear Maps},
howpublished = {Cryptology ePrint Archive, Paper 2019/082},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/082}},
url = {https://eprint.iacr.org/2019/082}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
|
2022-05-26 23:54:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30297085642814636, "perplexity": 10702.061417574765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662627464.60/warc/CC-MAIN-20220526224902-20220527014902-00253.warc.gz"}
|
https://bioinformatics.stackexchange.com/questions/6753/how-can-i-install-bioconductor-gviz-and-use-it-in-jupyter-notebook/6757
|
How can I install bioconductor-gviz and use it in jupyter notebook?
I tried to install gviz in a conda environment, but that library seems to be incompatible to python and r. I tried to setup a clean environment using
conda create -n r -c conda-forge r-essentials jupyter
and then add the library with:
source activate r
conda install -c bioconda bioconductor-gviz
getting
UnsatisfiableError: The following specifications were found to be in conflict: - atk
I remove atk, but now I get:
UnsatisfiableError: The following specifications were found to be in conflict: - bioconductor-gviz - r-bindr
Does anyone manage to use gviz from within an jupyter notebook?
I also tried to install gviz from a running R-notebook with:
install.package('gviz')
Warning in install.packages : package ‘gviz’ is not available (for R version 3.4.3)
Same when I try 'bioconductor-gviz'.
• The package is from Bioconductor, from the version you are using you should use:BiocInstall::biocLite("gviz") to be able to install it from R terminal – llrs Jan 7 '19 at 13:37
• Not sure what that means. – Sören Jan 7 '19 at 14:02
• That to install a package from Bioconductor you need to use a different command (not install.package), but the one I provided – llrs Jan 7 '19 at 15:45
• It does not seem to be an accepted command though.Error in loadNamespace(name) : there is no package called ‘BiocInstall’ – Sören Jan 7 '19 at 16:22
Did you try following the installation instructions?
Try this in R:
source("https://bioconductor.org/biocLite.R")
BiocInstaller::biocLite(c("Gviz"))
• Nope, I did not know, each package has its own way of installation. But the first line here was what I was missing. – Sören Jan 9 '19 at 19:28
• All Bioconductor packages are installed the same way. Though in general, reading the software documentation before use is always a good idea. They've changed the command from my answer a bit for R 3.5 and later (see Alex's answer), but for R < 3.5, this is the general way to install any Biocondoctor package. – Jared Andrews Jan 10 '19 at 21:38
# Install the BiocManager package first
install.packages("BiocManager")
library("BiocManager")
# Then install the Gviz package
BiocManager::install("Gviz")
Use BiocManager::install()
# Code
if (!requireNamespace("BiocManager", quietly = TRUE))
install.packages("BiocManager")
BiocManager::install("Gviz", version = "3.8")
However, If R < 3.5.0 , please use the following for installing Bioconductor packages.
BiocInstaller::biocLite("Gviz")
|
2020-02-21 14:52:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3754563331604004, "perplexity": 4435.292114454371}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145533.1/warc/CC-MAIN-20200221142006-20200221172006-00450.warc.gz"}
|
https://brilliant.org/discussions/thread/inmo-practice-board-2016-17/
|
×
# INMO Practice Board 2016 - 17
Heyo Brilliantiers!
As most of you all know the RMO 2016 results are out. And I am sure that most of us have made it to the second tier of the selection procedure ,that is the INMO 2017 , for the IMO 2017, the most coveted Mathematical competition among all.
Please do contribute to this discussion board by posting and clearing queries and let us try to make best use of the great minds in this community towards our preparation for INMO 2017 to be held on January 15, 2017. Also please do share resources and post questions for the users.
Cheers and Godspeed!
Hrithik Nambiar.
Note by Hrithik Nambiar
11 months, 2 weeks ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
Sort by:
@easha manideep d alright so we are in the same grade! And ur from?
- 10 months, 1 week ago
Telangana state
- 10 months, 1 week ago
@easha manideep d that grt! Which grade are you in?
- 10 months, 1 week ago
10th
- 10 months, 1 week ago
@easha manideep d yeah:-D how much u expecting?
- 10 months, 1 week ago
I got three geometry perfectly correct and one algebra one I wrote some bullshit So I am expecting around 51
- 10 months, 1 week ago
How did it go? @easha manideep d
- 10 months, 1 week ago
It was nice and this time the paper was very different three geometry and three polynomials, nothing else. Crazy huh ??
- 10 months, 1 week ago
Today is INMO guys ALL THE BEST !!!
- 10 months, 1 week ago
- 11 months, 2 weeks ago
Guys do share a few tips and the topics to concentrate on. Thank you!
- 11 months, 2 weeks ago
- 11 months, 2 weeks ago
|
2017-11-23 16:48:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9531097412109375, "perplexity": 8674.987950043003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806844.51/warc/CC-MAIN-20171123161612-20171123181612-00641.warc.gz"}
|
http://www.statemaster.com/encyclopedia/Meson
|
FACTOID # 22: South Dakota has the highest employment ratio in America, but the lowest median earnings of full-time male employees.
Home Encyclopedia Statistics States A-Z Flags Maps FAQ About
WHAT'S NEW RELATED ARTICLES People who viewed "Meson" also viewed:
SEARCH ALL
Search encyclopedia, statistics and forums:
(* = Graphable)
Encyclopedia > Meson
Composition: Meson Mesons of spin 0 form a nonet Composite - Quarks and antiquarks Hadron Strong Hideki Yukawa (1935) 1947 ~140 (List) Integer
Mesons of spin 1 form a nonet
Mesons were originally predicted as carriers of the force that binds protons and neutrons together. When first discovered, the muon was identified with this family from its similar mass and was named "mu meson", however it did not show a strong attraction to nuclear matter and is actually a lepton. The pion was the first true meson to be discovered. (The current picture of intranuclear forces is quite complicated; see quantum hadrodynamics for a discussion of modern theories in which nucleon-nucleon interactions are mediated by meson exchange.) Properties [1][2] In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... This article or section does not cite its references or sources. ... The moons shadow, as seen in muons 700m below ground at the Soudan 2 detector. ... In physics, a particle is a lepton if it has a spin of 1/2 and does not experience the strong nuclear force. ... The nuclear force (or nucleon-nucleon interaction or residual strong force) is the force between two or more nucleons. ...
In 1949 Hideki Yukawa was awarded the Nobel Prize in Physics for predicting the existence of the meson. He originally named it 'mesotron', but was corrected by Werner Heisenberg (whose father was a professor in Greek at University of Munich) that there is no 'tr' in the Greek word 'mesos'. Hideki Yukawa Hideki Yukawa FRSE (æ¹¯å· ç§€æ¨¹, January 23, 1907 - September 8, 1981) was a Japanese theoretical physicist and the first Japanese to win the Nobel prize. ... Nobel Prize medal. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ...
## Naming of the mesons GA_googleFillSlot("encyclopedia_square");
The name of a meson is devised so that its main properties can be inferred. Conversely, given a meson's properties, its name is clearly determined. The naming conventions fall in two categories based on flavor, flavorless mesons and flavored mesons. Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ...
### Flavorless mesons
Flavorless mesons are mesons whose flavor quantum numbers are all equal to zero. This means that these quarks are quarkonium states (quark-antiquark pairs of the same flavor) or a linear superposition of such states. In high energy physics, a quarkonium (pl. ...
The name of a flavorless meson is determined by its total spin S and total orbital angular momentum L. As a meson is composed of two quarks with s = 1/2, the total spin can only be S = 1 (parallel spins) or S = 0 (anti-parallel spins). The orbital quantum number L is due to the revolution of one quark around the other. Usually higher orbital angular momenta translate into a higher mass for the meson. These two quantum numbers determine the parity P and the charge-conjugation parity C of the meson: In atomic physics, the spin quantum number is a quantum number that parametrizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle. ... It has been suggested that this article or section be merged into Angular momentum quantum number. ... Look up Parity in Wiktionary, the free dictionary Parity is a concept of equality of status or functional equivalence. ... C parity or charge parity is a multiplicative quantum number of some particles that describes its behavior under a symmetry operation of charge conjugation (see C-symmetry). ...
P = (−1)L+1
C = (−1)L+S
Also, L and S add together to form a total angular momentum quantum number J, whose values range from |LS| to L+S in one-unit steps. The different possibilities are summarized with the use of the term symbol 2S+1LJ (a letter code is used instead of the actual value of L, see the spectroscopic notation) and the symbol JPC (here only the sign is used for P and C). The total angular quantum momentum numbers parameterize the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i. ... In quantum mechanics, the term symbol is an abbreviated description of the angular momentum quantum numbers in a multi-electron atom. ... Before the atom electron states were known, spectroscopists saw distintive series in atom spectra, and so they asigned letters to characteristic spectra. ...
The different possibilities and the corresponding meson symbol are given in the following table:
JPC
(0, 2…)− +
(1, 3…)+ −
(1,2…)− −
(0, 1…)+ +
Quark composition
2S+1LJ*
1(S, D…)J
1(P, F…)J
3(S, D…)J
3(P, F…)J
$u bar dmbox{, }u bar u - dbar dmbox{, }dbar u$
I = 1 Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ...
π
b
ρ
a
$u bar u + d bar d mbox{, }s bar s$
I = 0
η, η
h, h’
$phi,!$, ω
f, f
$c bar c$
I = 0
ηc
hc
ψ
χc
$b bar b$
I = 0
ηb
hb
Υ **
χb
Notes:
* Note that some combinations are forbidden: 0− −, 0+ −, 1− +, 2+ −, 3− +...
First row form isospin triplets: π, π0, π+ etc.
Second row contains pairs of elements: φ is supposed to be a $sbar s$ state, and ω a $u bar u + d bar d$ state. on the other cases it is not known the exact composition so a prime is used to distinguish the two forms.
For historical reasons, 13S1 form of ψ is called J
** The bottomonium state symbol is a capital upsilon (may be rendered as a capital Y depending of the font/browser)
The normal spin-parity series is formed by those mesons were P=(−1)J. In the normal series, S = 1 so PC = +1 (i.e., P = C). This corresponds to some of the triplet states (triplet states appear on the last two columns). The title given to this article is incorrect due to technical limitations. ... In high energy physics, a bottomonium is any of the flavorless, heavy mesons that are composed by a bottom quark and an anti-bottom quark. ...
Feynman diagram of one mode in which the eta particle can decay into 3 pions by gluon emission.
Since some of these symbols can refer to more than one particle, some extra rules are added: Image File history File links Eta-decay. ... Image File history File links Eta-decay. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... In particle physics, pion (short for pi meson) is the collective name for three subatomic particles: π0, π+ and π−. Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force. ... In particle physics, gluons are vector gauge bosons that mediate strong color charge interactions of quarks in quantum chromodynamics (QCD). ...
• In this scheme, particles with JP = 0 are known as pseudoscalars, and mesons with JP = 1 are called vectors. For particles other than those, the number J is added as a subindex: a0, a1, χc1, etc.
• For most of ψ, Υ and χ states is common to include the spectroscopic information: Υ(1S), Υ(2S). The first number is the principal quantum number, and the letter is the spectroscopic notation for L. Multiplicity is omitted since is implied by the symbol, and J appears as a subindex when needed: χb2(1P). If the spectroscopic information is not available, the mass is used instead: Υ(9460)
• The naming scheme does not differentiate between "pure" quark states and gluonium states, so gluonium states follow the same naming scheme.
• However, exotic mesons with "forbidden" quantum numbers JPC = 0− −, 0+ −, 1− +, 2+ −, 3− +... would use the same convention as the meson with identical JP numbers, but adding a J subindex. A meson with isospin 0 and JPC = 1− + would be denoted ω1.
When the quantum numbers of a particle are unknown, it is designated with an X followed by its mass in parentheses. In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... Non-quark model mesons consist of Exotic mesons, which have quantum numbers not possible for mesons in the quark model glueballs or gluonium, which have no valence quarks at all tetraquarks, which have two valence quark-antiquark pairs, and hybrid mesons, which contain a valence quark-antiquark pair and one...
### Flavored mesons
For flavored mesons, the naming scheme is a little simpler.
1. The meson name is given by the heaviest of the two quarks. From more to less massive, the order is: t > b > c > s > d > u. However, u and d do not carry any flavor, so they do not influence the naming scheme. Quark t never forms hadrons, but a symbol for t-containing mesons is reserved anyway.
quark symbol quark symbol
c D t T
s $bar K$ b $bar B$
Note the fact that for s and b quarks we get an antiparticle symbol. This is because it is adopted the convention that flavor charge and electric charge must agree in sign. This is also true for the third component of isospin: quark up has positive I3 and charge, quark down has negative charge and I3. The effect of that is: any flavor of a charged meson has the same sign than the meson's electric charge.
2. If the second quark has also flavor (it is not u or d) then the identity of that second quark is given by a subindex (s, c or b, and in theory t). Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ...
3. Add a "*" superindex if the meson is in the normal spin-parity series, i.e. JP = 0+, 1, 2+...
4. For mesons other than pseudoscalars (0) and vectors (1) the total angular momentum quantum number J is added as a subindex. The total angular quantum momentum numbers parameterize the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i. ...
To sum it up, we have:
quark composition Isospin JP = 0, 1+, 2... JP = 0+, 1, 2+...
$bar su, bar sd$ 1/2 KJ $K^*_J$
$c bar u, cbar d$ 1/2 DJ $D^*_J$
$c bar s$ 0 DsJ $D^*_{sJ}$
$bar bu, bar bd$ 1/2 BJ $B^*_J$
$bar bs$ 0 BsJ $B^*_{sJ}$
$bar bc$ 0 BcJ $B^*_{cJ}$
J is omitted for 0 and 1
In some cases, particles can mix between them. For example, the neutral kaon, $K^0,(bar sd)$ and its antiparticle $bar K^0,(sbar d)$ can combine in a symmetric or antisymmetric manner, originating two new particles, the short-lived and the long-lived neutral kaons $K^0_S = begin{matrix}{1 over sqrt 2}end{matrix}(K^0-bar K^0),;K^0_L = begin{matrix}{1 over sqrt 2}end{matrix}(K^0 + bar K^0)$ (neglecting a small CP-violating term). In physics, and specifically particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics. ...
A list of mesons. ... This is a list of particles in particle physics, including currently known and hypothetical elementary particles, as well as the composite particles that can be built up from them. ... In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, ie, the quarks (and antiquarks) which give rise to the quantum numbers of the hadrons. ...
Particles in physics - composite particles v • d • e Hadrons: Baryons (list) | Mesons (list) Baryons: Nucleons | Hyperons | Exotic baryons | Pentaquarks Mesons: Pions | Kaons | Quarkonium | Exotic mesons Atomic nuclei | Atoms | Molecules Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Elementary particles An elementary particle is a particle with no measurable internal structure, that is, it is not a composite of other particles. ... In particle physics, a hadron is a subatomic particle which experiences the strong nuclear force. ... In particle physics, the baryons are a family of subatomic particles including the proton and the neutron (collectively called nucleons), as well as a number of unstable, heavier particles (called hyperons). ... Baryon decuplet: Spin=3/2 Baryon octet: Spin=1/2 This is a list of baryons. ... A list of mesons. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... In particle physics, a hyperon is any subatomic particle which is a baryon (and hence a hadron and a fermion) with non-zero strangeness, but with zero charm and zero bottomness. ... Ordinary baryons are bound states of 3 quarks. ... A pentaquark is a subatomic particle consisting of a group of five quarks (compared to three quarks in normal baryons and two in mesons), or more specifically four quarks and one anti-quark. ... In particle physics, pion (short for pi meson) is the collective name for three subatomic particles: Ï€0, Ï€+ and π−. Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force. ... In particle physics, Kaons (also called K-mesons and denoted K) are a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. ... In high energy physics, a quarkonium (pl. ... Identities and classification of possible tetraquark mesons. ... A semi-accurate depiction of the helium atom. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle of a chemical element that retains its chemical properties. ... In chemistry, a molecule is an aggregate of two or more atoms in a definite arrangement held together by chemical bonds [1] [2] [3] [4] [5]. Chemical substances are not infinitely divisible into smaller fractions of the same substance: a molecule is generally considered the smallest particle of a pure...
Results from FactBites:
Time [Internet Encyclopedia of Philosophy] (16277 words) Conscious actions affect the future but not the past. B meson decay, neutral kaon decay, and Higgs boson decay are each different in a time reversed world. Most physicists suspect all these arrows are linked so that we can't have some arrows reversing while others do not.
More results at FactBites »
Share your thoughts, questions and commentary here
|
2019-11-18 06:11:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 24, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.79735267162323, "perplexity": 1672.8053306588838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669454.33/warc/CC-MAIN-20191118053441-20191118081441-00044.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-12-exponential-functions-and-logarithmic-functions-12-6-solving-exponential-equations-and-logarithmic-equations-12-6-exercise-set-page-825/31
|
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$x=\dfrac{1}{16}$
In exponential form, the given logarithmic equation, $\log_4 x =-2 ,$ is equivalent to \begin{array}{l}\require{cancel} x=4^{-2} .\end{array} Hence, the value of the variable that satisfies the given equation is \begin{array}{l}\require{cancel} x=\dfrac{1}{4^{2}} \\\\ x=\dfrac{1}{16} .\end{array}
|
2018-08-14 16:33:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9988566637039185, "perplexity": 5154.371900277507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209165.16/warc/CC-MAIN-20180814150733-20180814170733-00278.warc.gz"}
|
http://math.stackexchange.com/questions/337405/meaning-of-on-and-over-in-mathematics
|
# Meaning of “on” and “over” in mathematics
I've seen copious usage of prepositions like "on" and "over" in mathematical texts with no concrete description of what they mean. Can someone please precisely define these terms for me, as in, what does it really mean when you say "let x be a y on z?
-
Context dependent. Probably more a feature of the English language's variable meanings for many prepositions. – Michael Joyce Mar 21 '13 at 23:49
Usually the denotation is clear. Did you encounter an instance where it was not clear? – Math Gems Mar 21 '13 at 23:53
You can see "Let $R$ be a relation on a set", "Let $V$ be a vector space over a field $F$". The context usually makes it clear what the preposition intends to denote. – Pedro Tamaroff Mar 21 '13 at 23:57
It'd be helpful if your request for clarity on the meaning of "on" and "over", which seems to you to lack concrete descriptions, included some concrete examples. – KCd Mar 21 '13 at 23:57
My general impression is that a foo on $X$ is some kind of function, loosely speaking, with domain $X$ while a foo over $X$ is some kind of function, loosely speaking, with codomain $X$. But these terms don't really have completely precise meanings; you learn how to use them from seeing how other people use them (the same way you learned how to use most of the words you know).
|
2014-07-26 11:24:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7929010987281799, "perplexity": 807.4061086195611}, "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-2014-23/segments/1405997901076.42/warc/CC-MAIN-20140722025821-00036-ip-10-33-131-23.ec2.internal.warc.gz"}
|
https://www.vedantu.com/question-answer/the-value-of-tan-50tan-100tan-150-cdot-cdot-cdot-class-10-maths-cbse-5ee87192e8223517fbdc4911
|
Question
# The value of $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$ is${\text{A}}{\text{. }}0 \\ {\text{B}}{\text{. Not Defined}} \\ {\text{C}}{\text{. }}1 \\ {\text{D}}{\text{. }} - 1 \\$
Verified
147k+ views
Hint: In this question we need to find the value of the given trigonometric expression. In order to evaluate it easily we will use the property that $\tan x = \cot \left( {90 - x} \right)$. This will simplify the expression and help us reach the answer.
We have been given the expression $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$
Now, as we know that $\tan x = \cot \left( {90 - x} \right)$, so we will apply it to the expression starting from $\tan {50^0}$ to $\tan {85^0}$ we get,
$\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \tan {40^0}\tan {45^0}\cot {40^0}...\cot {5^0}$
As we know that, $\tan x.\cot x = 1$
$\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1 \cdot 1 \cdot 1 \cdot \cdot \cdot \cdot \cdot \tan {45^0}$
And as we know that$\ tan{45^0}=1$,
Hence, $\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1$
|
2021-10-20 11:00:48
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8039131164550781, "perplexity": 167.9919927682102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00690.warc.gz"}
|
http://wiki.kidzsearch.com/wiki/Programming_language
|
kidzsearch.com > wiki
# Programming language
A programming language is a type of written language that tells computers what to do. Programming languages are used to make all computer programs and computer software. A programming language is like a set of instructions that the computer follows to do something.
A programmer writes source code text in the programming language to create programs. Usually the programming language uses real words for some of the commands, so that the language is easier for a human to read. Many programming languages use punctuation just like a normal language. Many programs now are compiled. This means that the computer translates the source code into new languages such as assembly language or machine language, which are much faster and easier for the computer to read, but much harder for a person to read.
Computer programs must be written very carefully. If the programmer makes mistakes, or the program tries to do something the programmer did not design it to do, the program might then "crash" or stop working. When a program has a problem because of how the code was written, this is called a bug. A very small mistake can cause a very big problem. For example, forgetting a period or typing a plus sign instead of a minus sign can cause a bad bug.
## Types of programming languages
There are many types of programming languages. Most programming languages do not follow one type alone, so it is difficult to assign a type for each language. The examples of each type are given in each section below because they are the best well-known examples of that type.
### Declarative vs. Imperative programming
Declarative programming languages describe a "problem" but they usually do not say how the problem should be solved. The problem description uses logic, "solving" the problem often looks like automatically proving a system of logical axioms. Examples for such programming languages are Prolog, XSLT, LISP and SQL
Imperative programming languages describe a system of state changes. At the start, the program is in a certain state, and the computer is given steps to follow, in order to perform an action. Following the steps causes the program to "change state".
In general, declarative programming languages are safer and shorter. Imperative programming languages are more common, because they are easier to use.
### Functional vs. Procedural
Functional programming looks at programming like a function in mathematics. The program receives input, some information, and uses this information to create output. It will not have a state in between, and it will also not change things that are not related to the computation.
Procedural programs are a set of steps or state changes.
### Stack based
Stack based languages look at the some of the program's memory like a stack of cards. There are very few things that can be done with a stack. A data item can be put on the top of the stack. This operation is generally called "push". A data item can be removed from the top of the stack. This is called a "pop". You can look at the item at the top of the stack without removing it. This is called a "peek".
If a program is written as "push 5; push 3; add; pop;" it will put 5 on the top of the stack, put 3 on top of the 5, add the top two values (3 + 5 = 8), replace the 3 and 5 with the 8, and print the top (8). Examples for programming languages that are stack-based are the languages Postscript and Forth.
### Object-oriented
Object-oriented programming languages place data and functions that change data into a single unit. This unit is called an "object". Objects can interact with each other, but one object can not change another object's data. This is usually called encapsulation or information hiding. Most modern programming languages are object-oriented. An example of this is Java or C++.
### Flow-oriented
Flow oriented programming sees programming as connecting different components. These components send messages back and forth. A single component can be part of different "programs", without the need to be changed internally.
## Rules
Every programming language has rules about what it can and can not do. These include:
• Correct numbers (types of numbers, and how large or small the numbers can go)
• Words (reserved words, case-sensitivity)
• Limits on what the programming language can do
Most languages have official standards that define the rules of how to write the source code. Some programming languages have two or more standards. This can happen when a new standard replaces an old one. For example, the Perl 5 standard replaced Perl 4 in 1993. It can happen because two people made two standards at the same time. For example, there are several standards for APL.
## Object-Oriented Programming
Object-Oriented Programming (sometimes shortened to OOP) is a form of programming where all parts of the program are objects. Objects are pieces of memory with the same structure that can be used again and again. A bank account, bitmap, or hero from a video game could all be objects within a program. Objects are made up of properties (pieces of information the object stores) and methods which are things the object can do. A Dog object might have properties like height and hairColor. Its methods might include bark() and wagTail().
All objects are created from templates called classes. You can think of a class as a mold from which objects are made. The class defines all the properties and methods that its objects will have. Objects created from a class are called instances of the class. A class can extend another class, which means that it takes all the properties and methods of the class but can add its own.
Here is an example of what a class might look like in a programming language:
class Dog extends Mammal{
//These are properties:
String breed = "Collie"
String type = "Herding Dog"
//These are methods
void wagTail(){
//Do some wagging
}
void bark(){
//Do the barking here
}
}
Notice that the Dog class extends the Mammal class, so all dogs will have the properties of a mammal, like hairLength, and methods, like eat() or sleep().
Object-oriented programming is used in many of today's most popular programming languages, such as Java, C#, Objective-C, C++, Python, Ruby, Javascript, and ActionScript.
## Examples
### Example of Visual Basic
Here is a simple program written in Visual Basic:
Dim Input
Input = InputBox("How old are you?")
If Not IsNumeric(Input) Then
MsgBox "That's not a number!"
ElseIf Input < 0 Then
MsgBox "You cannot be less than zero!"
ElseIf Input > 100 Then
MsgBox "That's old!"
Else
MsgBox "You're " & Input & " years old."
End If
This program asks the user his or her age and responds based on what the user typed. If the user typed something that is not a number, the program says so. If the user typed a number less than zero, the program says so. If the user says he or she is older than 100 years old, the program says "That's old!" If the user typed a correct age the program says back to the user how old he or she is.
### Example of Python
Here is a program that does the same thing as the program above, but in Python:
try:
age = int(raw_input("How old are you? "))
except ValueError:
print "That's not a number!"
else:
if age < 0:
print "You cannot be less than zero!"
elif age > 100:
print "That's old!"
else:
print "You're %s years old." % age
### Example of C#
The same thing as the program above, but in C#:
using System;
public class Hello
{
public static void Main()
{
int age;
|
2017-05-23 12:45:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27632591128349304, "perplexity": 1138.0745043387105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607636.66/warc/CC-MAIN-20170523122457-20170523142457-00388.warc.gz"}
|
https://physics.stackexchange.com/questions/467416/why-coordinate-axes-are-perpendicular-to-each-other
|
# Why coordinate axes are perpendicular to each other?
Coordinate axes are chosen to be perpendicular to each other because it is convenient, calculations are easier. This is what I have read.
But in case that coordinate axes are per example 60 degrees apart could again a physical system exhibit linearity?
The superposition theorem could work?
• Have a look at : en.wikipedia.org/wiki/Skew_coordinates . What do you mean "a physical system exhibit linearity"? Can you be more precise? Physical laws do not depend upon the choice of coordinate systems. – SRS Mar 19 at 16:41
• As in many crystals. The base vectors are very often not perpendicular to each other. – Pieter Mar 19 at 17:10
• The physical behaviour of a system is independent of the coordinate system you use to model it. That is just about the "zeroth law" of the whole of science! The choice of a coordinate system has no effect on the linear or nonlinear behaviour of what is being modelled, unless the model is wrong. – alephzero Mar 19 at 18:05
• yes, as far as they are linearly independent, for instance, you dont want the third axis to be in the same plane than the other two – Wolphram jonny Mar 19 at 23:54
So here's the basic reason why: we like to express vectors in terms of their components, for which we invent the unit vectors $$\hat e^{1,2,3}$$ and components $$v_{1,2,3}$$ for a vector $$\vec v$$ such that $$\vec v= v_1\hat e^1 + v_2\hat e^2 + v_3 \hat e^3.$$ There is a bilinear combination between two vectors to form a scalar called the dot product, $$\vec u\cdot\vec v =\lvert \vec u\rvert\lvert \vec v \rvert\cos\theta.$$ For orthogonal unit vectors we have the nice property that $$\hat e^m\cdot \hat e^n=\{1\text{ if } m=n\text{ else } 0\},$$ which simplifies this expression to just $$\vec u\cdot\vec v=u_1v_1+u_2v_2+u_3v_3.$$ This simple formula is why we use orthogonal unit vectors in practice.
There is a slightly more complicated formula in skewed coördinate systems, or ones without unit vectors for their basis. To do this you simply have to find a new set of vectors that makes this relationship true, called the dual basis. So if your basis vectors are $$\vec b^{1,2,3}$$ then your dual basis is $$\vec b_{1,2,3}$$ such that $$\vec b_{m}\cdot \vec b^n=\{1\text{ if } m=n\text{ else } 0\},$$ in other words for $$\vec b^1$$, say, you look at the plane spanned by $$\vec b^{2,3}$$, identify a vector normal to it $$\vec n$$, and then rescale it until its dot product with $$\vec b^1$$ is $$1$$, and that then is $$\vec b_1$$. So the dual basis for some periodic lattice of points tells you about the planes of that lattice, via their normal vectors.
Then every vector gets two sets of components, the regular and the dual components, \begin{align} \vec v&= v_1\vec b^1 + v_2\vec b^2 + v_3 \vec b^3 &=v^1\vec b_1 + v^2\vec b_2 + v^3 \vec b_3, \end{align} at which point the earlier easy property is restored as long as we always pair a lower index with an upper index,$$\vec u=u_1v^1+u_2v^2+u_3v^3=u^1v_1+u^2v_2+u^3v_3.$$
The vector $$\vec{R}$$ can described with the vectors basis $$\vec{q}_1\,,\vec{q}_2$$ : $$\vec{R}_s= a_1\begin{bmatrix} 1\\ 0\\ \end{bmatrix}+a_2\begin{bmatrix} \cos(\theta)\\ \sin(\theta)\\ \end{bmatrix}=a_1\,\vec{q}_1+a_2\,\vec{q}_2 \tag 1$$
or with the vectors basis $$\vec{e}_1\,,\vec{e}_2$$
$$\vec{R}_c= c_1\begin{bmatrix} 1\\ 0\\ \end{bmatrix}+c_2\begin{bmatrix} 0\\ 1\\ \end{bmatrix}=c_1\,\vec{e}_1+c_2\,\vec{e}_2 \tag 2$$
both case the superposition theorem is valid
Edit
I) generalized coordinates are $$x$$ and $$y$$ with $$c_1=x$$ and $$c_2=y$$ we obtain for the position Vector $$\vec{R}_c$$ equation (2)
$$\vec{R}_c= x\begin{bmatrix} 1\\ 0\\ \end{bmatrix}+y\begin{bmatrix} 0\\ 1\\ \end{bmatrix}$$
Kinetic energy
$$T=m\,\frac{1}{2}\dot{R}_c^T\,\dot{R}_c$$ and
Potential energy
$$V=-m\,g\,y$$
$$\Rightarrow\quad$$ EOMs
$$\begin{bmatrix} \ddot{x}\\ \ddot{y}\\ \end{bmatrix}=\left[ \begin {array}{c} 0\\ -g\end {array} \right] \tag 3$$
II) generalized coordinates are $$q_1$$ and $$q_2$$
With $$a_1=q_1$$ and $$a_2=q_2$$ we obtain for the Position Vector $$\vec{R}_s$$ equation (1)
$$\vec{R}_s= q_1\begin{bmatrix} 1\\ 0\\ \end{bmatrix}+q_2\begin{bmatrix} \cos(\theta)\\ \sin(\theta)\\ \end{bmatrix}$$
Kinetic energy
$$T=m\,\frac{1}{2}\dot{R}_s^T\,\dot{R}_s$$ and
Potential energy
$$V=-m\,g\,q_2\,\sin(\theta)$$
$$\Rightarrow\quad$$ EOMs
$$\begin{bmatrix} \ddot{q}_1\\ \ddot{q}_2\\ \end{bmatrix}=\left[ \begin {array}{c} {\frac {g\cos \left( \theta \right) }{\sin \left( \theta \right) }}\\ -{\frac {g}{\sin \left( \theta \right) }}\end {array} \right] \tag 4$$
The generalized accelerations equation (3) and (4) are not equal. we can transfer equation (4) to get the accelerations in the orthogonal coordinates $$\vec{e}_1\,,\vec{e}_2$$
$$\begin{bmatrix} \ddot{x}\\ \ddot{y}\\ \end{bmatrix}=\left[ \begin {array}{cc} 1&\cos \left( \theta \right) \\ 0&\sin \left( \theta \right) \end {array} \right]\,\begin{bmatrix} \ddot{q}_1\\ \ddot{q}_2\\ \end{bmatrix}=\left[ \begin {array}{c} 0\\ -g\end {array} \right] \tag 5$$
so the accelerations equation (3) and (5) are now the same.
Conclusion
The equations of motion described with skew vectors basis compare to the equations of motion described with orthogonal vectors basis are not the same!.
|
2019-10-16 19:53:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 45, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.859305202960968, "perplexity": 382.42957163564824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986669546.24/warc/CC-MAIN-20191016190431-20191016213931-00257.warc.gz"}
|
https://deepai.org/publication/four-coloring-p-6-free-graphs-i-extending-an-excellent-precoloring
|
# Four-coloring P_6-free graphs. I. Extending an excellent precoloring
This is the first paper in a series whose goal is to give a polynomial time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time algorithm that determines if a special kind of precoloring of a P_6-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem.
## Authors
• 11 publications
• 8 publications
• 4 publications
• ### Four-coloring P_6-free graphs. II. Finding an excellent precoloring
This is the second paper in a series of two. The goal of the series is t...
02/07/2018 ∙ by Maria Chudnovsky, et al. ∙ 0
• ### List-three-coloring P_t-free graphs with no induced 1-subdivision of K_1,s
Let s and t be positive integers. We use P_t to denote the path with t v...
06/04/2020 ∙ by Maria Chudnovsky, et al. ∙ 0
• ### Complexity of C_k-coloring in hereditary classes of graphs
For a graph F, a graph G is F-free if it does not contain an induced sub...
05/04/2020 ∙ by Maria Chudnovsky, et al. ∙ 0
• ### Quasi-polynomial-time algorithm for Independent Set in P_t-free graphs via shrinking the space of induced paths
In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showe...
09/28/2020 ∙ by Marcin Pilipczuk, et al. ∙ 0
• ### Coloring even-hole-free graphs with no star cutset
A hole is a chordless cycle of length at least 4. A graph is even-hole-f...
05/04/2018 ∙ by Ngoc Khang Le, et al. ∙ 0
• ### Connected greedy coloring H-free graphs
A connected ordering (v_1, v_2, ..., v_n) of V(G) is an ordering of the ...
07/24/2018 ∙ by Esdras Mota, et al. ∙ 0
• ### Approximation Algorithms for Partially Colorable Graphs
Graph coloring problems are a central topic of study in the theory of al...
08/30/2019 ∙ by Suprovat Ghoshal, et al. ∙ 0
##### This week in AI
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.
## 1 Introduction
All graphs in this paper are finite and simple. We use to denote the set . Let be a graph. A -coloring of is a function . A -coloring is proper if for every edge , , and is -colorable if has a proper -coloring. The -coloring problem is the problem of deciding, given a graph , if is -colorable. This problem is well-known to be -hard for all .
A function that assigns a subset of to each vertex of a graph is a -list assignment for . For a -list assignment , a function is an -coloring if is a -coloring of and for all . A graph is -colorable if has a proper -coloring. We denote by the set of all vertices of with . The -list coloring problem is the problem of deciding, given a graph and a -list assignment , if is -colorable. Since this generalizes the -coloring problem, it is also -hard for all .
A -precoloring of a graph is a function for a set such that is a proper -coloring of . Equivalently, a -precoloring is a -list assignment in which for all . A -precoloring extension for is a proper -coloring of such that , and the -precoloring extension problem is the problem of deciding, given a graph and a -precoloring , if has a -precoloring extension.
We denote by the path with vertices. Given a path , its interior is the set of vertices that have degree two in . We denote the interior of by . A in a graph is a sequence of pairwise distinct vertices where for , is adjacent to if and only if . We denote by the set , and if , say and and , then is the path . A graph is -free if there is no in . Throughout the paper by “polynomial time” or “polynomial size” we mean running time, or size, that is polynomial in .
Since the -coloring problem and the -precoloring extension problem are -hard for , their restrictions to graphs with a forbidden induced subgraph have been extensively studied; see [2, 7] for a survey of known results. In particular, the following is known (given a graph , we say that a graph is -free if no induced subgraph of is isomorphic to ):
###### Theorem 1 ([7]).
Let be a (fixed) graph, and let . If the -coloring problem can be solved in polynomial time when restricted to the class of -free graphs, then every connected component of is a path.
Thus if we assume that is connected, then the question of determining the complexity of -coloring -free graph is reduced to studying the complexity of coloring graphs with certain induced paths excluded, and a significant body of work has been produced on this topic. Below we list a few such results.
###### Theorem 2 ([1]).
The 3-coloring problem can be solved in polynomial time for the class of -free graphs.
###### Theorem 3 ([5]).
The -coloring problem can be solved in polynomial time for the class of -free graphs.
###### Theorem 4 ([6]).
The 4-coloring problem is -complete for the class of -free graphs.
###### Theorem 5 ([6]).
For all , the -coloring problem is -complete for the class of -free graphs.
The only cases for which the complexity of -coloring -free graphs is not known are , , and , . This is the first paper in a series of two. The main result of the series is the following:
###### Theorem 6.
The 4-precoloring extension problem can be solved in polynomial time for the class of -free graphs.
In contrast, the -list coloring problem restricted to -free graphs is -hard as proved by Golovach, Paulusma, and Song [7]. As an immediate corollary of Theorem 6, we obtain that the -coloring problem for -free graphs is also solvable in polynomial time. This proves a conjecture of Huang [6], thus resolving the former open case above, and completes the classification of the complexity of the -coloring problem for graphs with a connected forbidden induced subgraph.
Let be a graph. For we denote by the subgraph induced by on , and by the graph . If , we write to mean . For disjoint subsets we say that is complete to if every vertex of is adjacent to every vertex of , and that is anticomplete to if every vertex of is non-adjacent to every vertex of . If we write is complete (or anticomplete) to to mean that is complete (or anticomplete) to . If is not complete and not anticomplete to , we say that is mixed on . Finally, if is an induced subgraph of and , we say that is complete to, anticomplete to, or mixed on if is complete to, anticomplete to, or mixed on , respectively. For we write (or when there is no danger of confusion) to mean the set of vertices of that are adjacent to . Observe that since is simple, . For , an attachment of is a vertex of complete to . For we denote by the set of attachments of in . If , we sometimes write to mean .
Given a list assignment for , we say that the pair is colorable if is -colorable. For , we write to mean the list coloring problem where we restrict the domain of the list assignment to . Let be such that for every , and let . We say that a list assignment is obtained from by updating from if for every , and for every . If , we say that is obtained from by updating from . If is obtained from by updating from , we say that is obtained from by updating. Let , and for let be obtained from by updating. If , we say that is obtained from by updating exhaustively. Since for all , it follows that and thus can be computed from in polynomial time.
An excellent starred precoloring of a graph is a six-tuple such that
1. is a proper coloring of ;
2. ;
3. is connected and no vertex in is complete to ;
4. every vertex in has neighbors of at least two different colors (with respect to ) in ;
5. no vertex in is mixed on a component of ; and
6. for every component of , there is a vertex in complete to it.
We call the seed of . We define two list assignments associated with . First, define for every , and let for . Second, is the list assignment obtained as follows. First, define to be the list assignment for obtained from by updating exhaustively; let . Now define if , and if . Let . Then . A precoloring extension of is a proper -coloring of such that for every ; it follows that for every . It will often be convenient to assume that , and this assumption can be made without loss of generality. Note that in this case, for all .
For an excellent starred precoloring and a collection excellent starred of precolorings, we say that is an equivalent collection for (or that is equivalent to ) if has a precoloring extension if and only if at least one of the precolorings in has a precoloring extension, and a precoloring extension of can be constructed from a precoloring extension of a member of in polynomial time.
We break the proof of Theorem 6 into two independent parts, each handled in a separate paper of the series. In one part, we reduce the 4-precoloring extension problem for -free graphs to determining if an excellent starred precolorings of a -free graph has a precoloring extension, and finding one if it exists. In fact, we restrict the problem further, by ensuring that there is a universal bound (that works for all -precolorings of all -free graphs) on the size of the seed of the excellent starred precolorings that we need to consider. More precisely, we prove:
###### Theorem 7.
There exists an integer and a polynomial-time algorithm with the following specifications.
Input: A 4-precoloring of a -free graph .
Output: A collection of excellent starred precolorings of such that
1. If for every we can in polynomial time either find a precoloring extension of , or determine that none exists, then we can construct a 4-precoloring extension of in polynomial time, or determine that none exists:
2. ; and
3. for every ,
• ;
• ;
• is an induced subgraph of ; and
• .
The proof of Theorem 7 is hard and technical, and we postpone it to the second paper of the series [3]. The other part of the proof of Theorem 6 is an algorithm that tests in polynomial time if an excellent starred precoloring (where the size of the seed is fixed) has a precoloring extension. The goal of the present paper is to solve this problem. We prove:
###### Theorem 8.
For every positive integer there exists a polynomial-time algorithm with the following specifications.
Input: An excellent starred precoloring of a -free graph with .
Output: A precoloring extension of or a determination that none exists.
Clearly, Theorem 7 and Theorem 8 together imply Theorem 6. The proof of Theorem 8 consists of several steps. At each step we replace the problem that we are trying to solve by a polynomially sized collection of simpler problems, and the problems created in the last step can be encoded via 2-SAT. Here is an outline of the proof. First we show that an excellent starred precoloring of a -free graph can be replaced by a polynomially sized collection of excellent starred precolorings of that have an additional property (to which we refer as “being orthogonal”) and has a precoloring extension if and only if some member of does. Thus in order to prove Theorem 8, it is enough to be able to test if an orthogonal excellent starred precoloring of a -free graph has a precoloring extension. Our next step is an algorithm whose input is an orthogonal excellent starred precoloring of a -free graph , and whose output is a “companion triple” for . A companion triple consists of a graph that may not be -free, but certain parts of it are, a list assignment for , and a correspondence function that establishes the connection between and . Moreover, in order to test if has a precoloring extension, it is enough to test if is colorable.
The next step of the algorithm is replacing by a polynomially sized collection of list assignments for , such that is colorable if and only if there exists such that is colorable, and in addition for every the pair is “insulated”. Being insulated means that is the union of four induced subgraphs , and in order to test if is colorable, it is enough to test if is colorable for each . The final step of the algorithm is converting the problem of coloring each into a -SAT problem, and solving it in polynomial time. Moreover, at each step of the proof, if a coloring exists, then we can find it, and convert in polynomial time into a precoloring extension of .
This paper is organized as follows. In Section 2 we produce a collection of orthogonal excellent starred precolorings. In Section 3 we construct a companion triple for an orthogonal precoloring. In Section 4 we start with a precoloring and its companion triple, and construct a collection of lists such that every pair is insulated. Finally, in Section 5 we describe the reduction to 2-SAT. Section 6 contains the proof of Theorem 8 and of Theorem 6.
## 2 From Excellent to Orthogonal
Let be an excellent starred precoloring. For , the type of is the set . Thus the number of possible types for a given precoloring is at most . In this section we will prove several lemmas that allow us to replace a given precoloring by an equivalent polynomially sized collection of “nicer” precolorings, with the additional property that the size of the seed of each of the new precolorings is bounded by a function of the size of the seed of the precoloring we started with. Keeping the size of the seed bounded allows us to maintain the property that the number of different types of vertices of is bounded, and therefore, from the point of view of running time, we can always consider each type separately.
For we denote by the set . Thus if is of type , then . For and we denote by the set of vertices of of type .
A subset of is orthogonal if there exist such that for every either or . We say that is orthogonal if is orthogonal for every .
The goal of this section is to prove that for every excellent starred precoloring of a -free graph , there is a an equivalent collection of orthogonal excellent starred precolorings of . We start with a few technical lemmas.
###### Lemma 1.
Let be an excellent starred precoloring of a -free graph . Let and . Let be types such that and , and let and . Suppose that are such that , where possibly . Suppose further that the only possible edge among is , and is adjacent to and not to , and is adjacent to and not to . Then there does not exist with and such that is complete to and anticomplete to .
###### Proof.
Suppose such exists. Since no vertex of is mixed on a component of , it follows that is anticomplete to . Since and , it follows that there exists with . Similarly, there exists with . Since and , it follows that is anticomplete to and is anticomplete to .
Since it follows that is anticomplete to . Since (possibly shortcutting through ) is not a in , it follows that is adjacent to . If is non-adjacent to , and is non-adjacent to , then , and since is excellent, is non-adjacent to , and so is a , a contradiction, so we may assume that is adjacent to . But now is a , a contradiction. This proves Lemma 1. ∎
###### Lemma 2.
Let be an excellent starred precoloring of a -free graph . Let . Let be types such that and , and let and . Let with , and let with , where possibly and . Assume that
• some component of contains both ;
• some component of contains both ;
• for every there is a path in from to with for every ;
• the only possible edge among is ;
• are adjacent to and not to ;
• are adjacent to and not to .
Then there do not exist with , and such that
• some component of contains both and , and
• for every , and
• is complete to and anticomplete to .
###### Proof.
Suppose such exist. Since is an excellent starred precoloring, no vertex of is mixed on a component of , and therefore is anticomplete to . Since and , it follows that there exists with . Similarly, there exists with . Since and , it follows that is anticomplete to and is anticomplete to . Since , it follows that is anticomplete to , and similarly is anticomplete to .
First we prove that is adjacent to . Suppose not. Since is not a in , it follows that is non-adjacent to . But now or is a in , a contradiction. This proves that is adjacent to .
If is adjacent to , then is a , a contradiction. Therefore is non-adjacent to , and therefore is anticomplete to . Similarly, is anticomplete to . In particular it follows that .
Since there exists with such that is complete to . Since for every , it follows that is anticomplete to . Recall that , and so no vertex of is mixed on . Similarly no vertex of is mixed on . If is anticomplete to , then one of , , is a , so is complete to .
Since is not a , it follows that either is adjacent to , or is adjacent to . We may assume that is adjacent to .
Let be a path in from to with for every . Since is adjacent to and not to , there is exist adjacent such that is adjacent to and not to . Since for every , it follows that is anticomplete to . But now if is non-adjacent to , then is a , and if is adjacent to , then is a ; in both cases a contradiction. This proves Lemma 2. ∎
Let be an excellent starred precoloring of a -free graph . Let , and let . Let be such that and is a 4-precoloring of . Let be the set of vertices of such that as a neighbor with . Let
S′=S∪S′′
X′0=X0∪X′′∪X′′0
X′=X∖(X′′∪S′′∪X′′0)
Y∗′=Y∗∖X′′0.
We say that is obtained from by moving to the seed with colors , and moving to with colors . Sometimes we say that “we move to with colors , and to with colors ”.
In the next lemma we show that this operation creates another excellent starred precoloring.
###### Lemma 3.
Let be an excellent starred precoloring of a -free graph . Let and , and let be as above. Then either is an excellent starred precoloring.
###### Proof.
We need to check the following conditions:
1. is a proper coloring of ;
2. ;
3. is connected and no vertex in is complete to ;
4. every vertex in has neighbors of at least two different colors (with respect to ) in ;
5. no vertex in is mixed on a component of ; and
6. for every component of , there is a vertex in complete to it.
Next we check the conditions.
1. holds by the definition of .
2. holds since .
3. is connected since is connected, and every has a neighbor in . Moreover, since no vertex of is complete to , it follows that no vertex of is complete to .
4. follows from the fact that .
5. follows from the fact that and .
6. follows from the fact that and .
Let be an excellent starred precoloring. Let . Write For let (or when there is no danger of confusion) denote the vertex set of the component of that contains .
Let be an excellent starred precoloring, and let . We say that is -clean if there does not exist with the following properties:
• , and
• there is with , and
• has both a neighbor in and a neighbor in .
We say that is clean if it is -clean for every .
We say that is -tidy if there do not exist vertices such that
• , , and
• , and
• there is a path from to in such that for every , and
• there is with , and
• has a neighbor in and a neighbor in
Observe that since no vertex of is mixed on an a component of , it follows that is precisely the set of vertices of that are complete to , and an analogous statement holds for . We say that is tidy if it is -tidy for every .
We say that is -orderly if for every in with , is complete to . We say that is orderly if it is -orderly for every
Finally, we say that is -spotless if no vertex in with has both a neighbor in and a neighbor in . We say that is spotless if it is -spotless for every
Our goal is to replace an excellent starred precoloring by an equivalent collection of spotless precolorings. First we prove a lemma that allows us to replace an excellent starred precoloring with an equivalent collection of clean precolorings.
###### Lemma 4.
There is a function such that the following holds. Let be a -free graph, and let be an excellent starred precoloring of . Then there is an algorithm with running time that outputs a collection of excellent starred precolorings of such that:
• ;
• for every ;
• every is -clean for every for which is -clean;
• every is -clean;
• is an equivalent collection for .
###### Proof.
Without loss of generality we may assume that . Thus for every . We may assume that is not -clean for otherwise we may set . Let be the set of vertices of with and such that some has . Let be the subsets of with and the subsets of with . Let be the collection of all -tuples
((S1,Q1),(S
|
2021-10-19 23:09:02
|
{"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": 0.9149194359779358, "perplexity": 687.9836952433988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585281.35/warc/CC-MAIN-20211019202148-20211019232148-00670.warc.gz"}
|
http://tex.stackexchange.com/questions/161297/is-there-such-a-thing-as-a-mathrule-rounded-endcaps?answertab=oldest
|
Is there such a thing as a \mathrule? (rounded endcaps)
When I am creating hybrid math glyphs, I will find myself often times stacking \rules about the place. But if you look at a collection of CM math symbols (top row), and compare it to a \rule (2nd row)
\documentclass{article}
\begin{document}
\noindent$+ - = \subset \geq \ni$\\$\rule{2ex}{.3pt}$
\end{document}
you will find that strokes comprising the math glyphs exhibit rounded ends, whereas a rule has flattened ends. At typical size, the difference may be insignificant, but for scaled-up glyphs, it is noticeable.
Is there a version of a \rule, which I jokingly here called \mathrule, that will provide the functionality (even the syntax) of a \rule, but with rounded endcaps? (preferably not with tikz, but something more native).
I've considered stretching and squeezing, for example, a minus sign, but that will result in spherical end caps becoming elliptical, especially when the distortion is large.
I was going to put this attempt up to get the ball rolling, but Herbert beat me to it. I'll still leave it here as food for thought.
\documentclass{article}
\usepackage{scalerel}
\usepackage{verbatimbox}
\newsavebox\clippedbullet
\sbox{\clippedbullet}{\addvbuffer[.0pt -.55pt]{$\bullet$}}
\def\CB{\usebox{\clippedbullet}}
\newcommand\mathrule[3][0pt]{%
\raisebox{#1}{\scaleleftright{\kern-.8pt\CB\kern-2.5pt}%
{\rule{#2-#3}{#3}}{\kern-2.4pt\CB\kern+.0pt}}%
}
\begin{document}
\noindent\mathrule[2pt]{2ex}{.5pt} \mathrule{3ex}{1pt} \mathrule[-1pt]{2ex}{2pt}\\
\rule[2pt]{2ex}{.5pt} \rule{3ex}{1pt} \rule[-1pt]{2ex}{2pt}
\end{document}
-
No, there isn't. Longer arrows are built by repeating minus signs. – egreg Feb 19 '14 at 16:18
\pdfcompresslevel=0 %%% to see how it works in the pdf code
\pdfcompresslevel=0
\documentclass{article}
\parindent=0pt
\makeatletter
\def\mathrule#1#2{%
\@tempdima=\dimexpr#1-0.5#2
\@tempdimb=#2
\@tempdimc=0.5#2
\hbox to #1{%
\pdfliteral{
q []0 d
1 J % set line cap to rounded ends
\strip@pt\@tempdimb\space w \strip@pt\@tempdimc\space 0 m
\strip@pt\@tempdima\space 0 l S Q }}}
\makeatletter
\begin{document}
x\mathrule{60bp}{2bp}x
x$\rule{60bp}{2bp}$x
\end{document}
-
Just to check: I assume this will work for pdflatex and lualatex, but not for plain latex. – Charles Staats Feb 19 '14 at 16:48
That's a great start. But I'm thinking you forgot to shave the rule width off the length of the \mathrule, when you added the rounded ends. Am I just seeing it wrong? – Steven B. Segletes Feb 19 '14 at 16:49
add a \ifpdf and then use \special instead of \pdfliteral and the corresponfing PostScript function names. Pretty easy to realize. – Herbert Feb 19 '14 at 16:49
@StevenB.Segletes: pdftex uses linecap=0. linecap=1 has the same line length than linecap=2. You can substract half the linewidth from the beginning and the end of the line and it will be the same. – Herbert Feb 19 '14 at 16:51
Just for clarification, this does not take an optional argument, though a \raisebox could fix that, I suppose. – Steven B. Segletes Feb 19 '14 at 16:59
I am posting this as the final result of taking Herbert's solution where I wanted it, which is a drop in replacement for \rule that provides rounded line caps. Herbert deserves the credit and will get the points, but others (like me) might wish to see/use this more seamless drop-in for \rule.
Herbert's solution needed to be placed into an hbox, as he allowed, but it needed substantially more adjustments, as well. The horizontal and vertical kernings were offset relative to that box, the vertical alignment of the rounded rule differed from that of a \rule, just to name a few.
EDIT: I've also taken the initiative to force \mathrule to make a rule with the rounded endcaps on side/side or else the top/bottom, depending on which dimension is larger, such that
\mathrule{2ex}{2ex}
\mathrule{3ex}{2ex}
\mathrule{2ex}{3ex}
gives
In my MWE below, I put the final result through the paces of placing the \mathrule on a line by itself, placing it amongst text, stacking it, as well as \fboxing it, in all cases comparing to its equivalent \rule. The \mystery@factor in this MWE was later resolved by Dan in his comment. His correction is incorporated into the style file roundrule.sty that I present at the end of this answer.
\pdfcompresslevel=0 %%% to see how it works in the pdf code
\documentclass{article}
\usepackage{graphicx}
\usepackage{stackengine}% Used for testing; not needed for \math(v)rule's
\parindent=0pt
\makeatletter
\newcommand\mathrule[3][0pt]{%
\ifdim#2>#3\math@hrule[#1]{#2}{#3}\else\math@vrule[#1]{#2}{#3}\fi}
\newcommand\math@hrule[3][0pt]{%
\gdef\mystery@factor{0.07}%
\@tempdima=#3%
\rule[#1]{0pt}{#3}% Needed to account for .5\@tempdima vertical offset of rounded rule
\raisebox{.5\@tempdima+#1}{%
\makebox[#2][l]{\kern-.5\@tempdima\@@mathrule{#2}{#3}}}%
}
\newcommand\math@vrule[3][0pt]{%
\gdef\mystery@factor{0.0}%
\@tempdima=#2%
\rule[#1]{0pt}{#3}% Needed to account for .5\@tempdima vertical offset of rounded rule
\raisebox{-.0\@tempdima+#1}{%
\kern0.5\@tempdima%
\rotatebox{90}{\kern-0.5\@tempdima\makebox[#3][l]{\@@mathrule{#3}{#2}}}%
\kern0.5\@tempdima}%
}
\def\@@mathrule#1#2{%
\@tempdimb=#2%
\@tempdima=\dimexpr#1-\mystery@factor\@tempdimb%Why 0.07 for \math@hrule?
\pdfliteral{%
q []0 d %
1 J % set line cap to rounded ends
\strip@pt\@tempdimb\space w \strip@pt\@tempdimb\space 0 m %
\strip@pt\@tempdima\space 0 l S Q }}
\makeatother
\begin{document}
\mathrule{60bp}{1bp}\par
\rule{60bp}{1bp}\par
x\mathrule[-1pt]{20bp}{1bp}x\mathrule{20bp}{1bp}x\par
x\rule[-1pt]{20bp}{1bp}x\rule{20bp}{1bp}x\par
\stackunder[2pt]{\rule{60bp}{3bp}}{\mathrule{60bp}{3bp}}\par
\fbox{\mathrule[-1ex]{4ex}{.5ex}}\fbox{\rule[-1ex]{4ex}{.5ex}}\par
\fbox{\mathrule[+1ex]{4ex}{.5ex}}\fbox{\rule[+1ex]{4ex}{.5ex}}\par
\clearpage
\mathrule{1bp}{60bp} \rule{1bp}{60bp}\par
x\mathrule[-1pt]{1bp}{20bp}x\mathrule{1bp}{20bp}x%
\rule[-1pt]{1bp}{20bp}x\rule{1bp}{20bp}x\par
x\rule[-1pt]{1bp}{20bp}x\rule{1bp}{20bp}x\par
\stackunder[2pt]{\rule{3bp}{10bp}\mathrule{3bp}{10bp}}%
{\mathrule{3bp}{10bp}\rule{3bp}{10bp}}\par
\fbox{\mathrule[-1ex]{.5ex}{4ex}}\fbox{\rule[-1ex]{.5ex}{4ex}}\par
\fbox{\mathrule[+1ex]{.5ex}{4ex}}\fbox{\rule[+1ex]{.5ex}{4ex}}\par
\end{document}
For those interested in an ad hoc package (and taking azetina's renaming suggestion since it functions outside of math mode), here is roundrule.sty, introducing the macro \roundrule[]{}{}. And thanks to Dan for resolving and eliminating the mystery factor. Of course, \roundrules can be used in text mode without the use of dollar delimiters. Feel free to \let\rrule\roundrule in your preamble, if you get tired of typing the long name.
EDITED to require package calc as well.
\ProvidesPackage{roundrule}
[2014/05/01 V1.01 Provides rules with rounded endcaps]
%
% THIS MATERIAL IS SUBJECT TO THE LaTeX Project Public License
%
% Special thanks to users Herbert and Prof. Dan Luecking at tex.stackexchange.com:
% http://tex.stackexchange.com/questions/161297/
% is-there-such-a-thing-as-a-mathrule-rounded-endcaps
%
% V1.00 - initial release
% V1.01 - require package calc, which was omitted as an oversight.
\pdfcompresslevel=0 %%% to see how it works in the pdf code
\RequirePackage{graphicx}
\RequirePackage{calc}
\newcommand\roundrule[3][0pt]{%
\ifdim#2>#3\round@hrule[#1]{#2}{#3}\else\round@vrule[#1]{#2}{#3}\fi}
\newcommand\round@hrule[3][0pt]{%
\@tempdima=#3%
\rule[#1]{0pt}{#3}% Needed to account for .5\@tempdima vertical offset of rounded rule
\raisebox{.5\@tempdima+#1}{%
\makebox[#2][l]{\kern-.5\@tempdima\@@roundrule{#2}{#3}}}%
}
\newcommand\round@vrule[3][0pt]{%
\@tempdima=#2%
\rule[#1]{0pt}{#3}% Needed to account for .5\@tempdima vertical offset of rounded rule
\raisebox{-.0\@tempdima+#1}{%
\kern0.5\@tempdima%
\rotatebox{90}{\kern-0.5\@tempdima\makebox[#3][l]{\@@roundrule{#3}{#2}}}%
\kern0.5\@tempdima}%
}
\def\@@roundrule#1#2{%
\@tempdima=#1%
\@tempdimb=#2%
\@tempdima=0.996264\@tempdima% LaTeX to PDF point conversion (72/72.27)
\@tempdimb=0.996264\@tempdimb% LaTeX to PDF point conversion (72/72.27)
\pdfliteral{%
q []0 d %
1 J % set line cap to rounded ends
\strip@pt\@tempdimb\space w \strip@pt\@tempdimb\space 0 m %
\strip@pt\@tempdima\space 0 l S Q }}
\endinput
-
Its as easy as a find and replace on the text editor. I would go for the \roundrule – azetina Feb 20 '14 at 19:37
@azetina I feel a nudge. While naming the package roundrule, do you think the macro should be \roundrule or \rrule? – Steven B. Segletes Feb 20 '14 at 19:41
hmmmm??? I would say \rrule is more practical but \roundrule says it all. Honestly, its just taste and consideration for compatibility cases. Is \rrule defined elsewhere? – azetina Feb 20 '14 at 19:52
@azetina I already checked. \rrule shows up nowhere on this website. Perhaps I'll leave it \roundrule and let the user employ a \let\rrule\roundrule. – Steven B. Segletes Feb 20 '14 at 19:55
@StevenB.Segletes Could the mystery factor be needed because TeX works with pt, which is 1/72.27 inch, while in PDF the default unit is 1/72 inch? I removed the mystery factor calculations, but added \@tempdima=0.996264\@tempdima and \@tempdimb=0.996264\@tempdimb to \@@mathrule (0.996264 = 72/72.27). It seemed to work just fine for both horizontal and vertical rules: lengths matched within 1pixel at 1600% magnification. – Dan Feb 20 '14 at 21:17
|
2016-05-05 03:05:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.889732301235199, "perplexity": 4121.17701714513}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125857.44/warc/CC-MAIN-20160428161525-00169-ip-10-239-7-51.ec2.internal.warc.gz"}
|
https://www.taylorfrancis.com/chapters/mono/10.1201/9781439811481-31/meshes-yair-shapira?context=ubx&refId=1965f9d3-4751-474b-a915-4119147bc757
|
## ABSTRACT
In the implementation of a graph as a sparse matrix, the node is the key object. In fact, the nodes in the graph are indexed by the index i = 1, 2, 3, . . . , |N |, where N is the set of nodes and |N | is its cardinality (the total number of nodes). The index i of the ith node serves then as its virtual address in the list of nodes, which allows one to access the information about its role in the graph. Indeed, this information can be found in the ith row in the matrix of the graph, in which the matrix elements indicate from which nodes edges are issued towards the ith node. Furthermore, the ith row in the transpose of the matrix of the graph indicates towards which nodes edges are issued from the ith node.
|
2023-03-29 01:38:59
|
{"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": 0.8272664546966553, "perplexity": 212.70738660015678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00350.warc.gz"}
|
https://sassafras13.github.io/Nyquist/
|
# Drawing Nyquist Plots
Recently I wrote a post on stability of linear systems where I introduced the Nyquist plot. Now I want to go into the details of how to draw one without using a Bode plot. If I have time in the future I will also explain how to draw a Nyquist plot using the Bode plot - it is an easier method, in my opinion, but it assumes that you have a reliable Bode plot to start with.
I first want to clarify something from last post: The Caucy Argument Principle states that [3]:
The net number of clockwise encirclements, N, equals the number of zeros (closed-loop system roots) in the RHP, Z, minus the number of open-loop poles in the RHP, P: N = Z - P
I learned to draw Nyquist plots this way from Brian Douglas [1], but he references a lecture by Prof. Gopal [2], so I will cite both here for completeness. We will use an example transfer function given below, but first I want to go over two concepts that will be useful in this example. The first concept is the idea that Nyquist plots are symmetric about the real axis (in both the s- and w-planes) [1]. We will use this to reduce the amount of work we have to do to obtain the plot.
The second concept is the idea that the part of the Nyquist contour in the s-plane that is at infinity (i.e. the curved part that encircles the entire RHP) maps to a single point in the w-plane if the transfer function is proper or strictly proper [1]. This is because, if we imagine ourselves sitting on this part of the contour, the poles and zeros of the transfer function would appear grouped at a point near the origin [1]. That is, no matter where we are at infinity, we would be so far away that we could not see the contributions of the poles and zeros to the phase. So we only need one point to represent the phase anywhere at infinity on the RHP. If the system is strictly proper, this point is the origin; if the system is proper, this point is somewhere on the real axis [1]. This also means that I really only care about what is happening along the imaginary axis in the s-plane [1].
Figure 1
Now let’s dive into the example. The open loop transfer function is given below [1]. It has no zeros (Z = 0) and two poles in the LHP (P = 0). So we are expecting that N = Z - P = 0 for this example.
Prof. Gopal’s method states that you only need 4 points to be able to sketch the Nyquist plot, and these points are [1],[2]:
(1) w = 0
(2) w = infinity
(3) The intercept of the Nyquist plot (in the w-plane) on the imaginary axis
(4) The intercept of the Nyquist plot (in the w-plane) on the real axis
In order to get all of these points, we first need to substitute s = jw into the system transfer function [1]. Why don’t I substitute s = sigma + jw into the transfer function, why am I omitting the real part? I omit the real part for the reason we just discussed - as soon as I move onto the positive real axis at a distance of infinity from the origin, the phase contribution from the transfer function basically looks like zero to me. It’s not important to me. So I can omit considerations of the real component of the transfer function completely from this procedure.
Instead, let’s substitute j*w into my transfer function as shown below.
Now let’s see what happens when w = 0 - I find that the transfer function is equal to 0.5, so I plot that on the positive real axis of the w-plane. Similarly, I can see that when w = infinity, my transfer function is equal to 0, so I can also plot that on the w-plane.
Figure 2
Next we need to find the imaginary and real axis intercepts of the Nyquist plot. To do this, I am going to rationalize the denominator of my transfer function and then separate the real and imaginary components as shown below.
I can solve for the intercept for the imaginary axis by setting the real axis component equal to zero and solving for w. Then I plug this value of w into the imaginary component to get the imaginary axis intercept. This math is shown below, and the final Nyquist plot is also given. Notice that I always draw my Nyqyuist contour in the s-plane in the clockwise direction, so the point in the w-plane that corresponds to the origin in the s-plane is the starting point for my contour, and I move in the direction of the point corresponding to w = infinity in the s-plane [1].
Figure 3
This plot shows that my system is stable because I have no encirclements of -1 and N = Z - P = 0 as we expected.
#### References
[1] Douglas, Brian. “Nyquist Stability Criterion, Part 2.” https://www.youtube.com/watch?v=tsgOstfoNhk&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk&index=27 Visited 09/16/2019.
[2] Gopal, Madan. “Lec-36 The Nyquist Stability Criterion and Stability Margins (Contd.).” https://www.youtube.com/watch?v=Rbvau5oXOkg Visited 09/16/2019.
[3] Franklin, G., Powell, J.D., Emami-Naeini, A. Feedback Control of Dynamic Systems, 6th ed. Pearson. 2009.
Written on September 16, 2019
|
2021-07-28 09:49:31
|
{"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": 0.8505140542984009, "perplexity": 332.13551111918514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153709.26/warc/CC-MAIN-20210728092200-20210728122200-00077.warc.gz"}
|
http://tex.stackexchange.com/questions/161959/french-babel-trouble-1-ier-2-ieme-etc
|
# French babel trouble: 1\ier{}, 2\ieme{}, etc
I'm having an issue with French babel. Whenever I use any of the 1\ier{}, 2\ieme{} and so on constructs (to obtain 1er, 2e, etc.), the spacing following it behaves very strangely.
So, for example, if I use:
• 1\ier asdf then the output in the PDF will be: "1erasdf" (notice the missing space between "er" and "asdf")
• 1\ier~asdf then the output in the PDF will be: "1er asdf" (with a protected space in between)
• 1\ier~ asdf then the output in the PDF will be: "1er asdf" (there are two spaces between "er" and "asdf", one protected and one normal; can't display properly here because of SE formatting limitations)
I don't want to use ~ every time I need these constructs (2nd and 3rd examples above), so my question: Is the first example above a bug or a user error? And what is the proper way to avoid such output glitches?
I'm using TeX Live 2009 on Ubuntu 12.04 (with LyX).
-
You should: if "premier" precedes a word, it should remain attached to it. – egreg Feb 23 '14 at 14:20
@landroni I've moved my comment. "Hard space" is indeed the same as "non-breaking space" in English (our "espace insécable", ~ in TeX), or at least I've learnt it so. As a side effect, ~ avoids any gobbled space that instead may happen between both words. But I don't understand your last remark: in his comment, egreg is suggesting exactly the same thing as me. – Franck Pastor Feb 23 '14 at 15:24
@fpast Right! @ egreg wasn't explicit in his comment. I mistakenly took it to understand that the first example in the question was correct; he was suggesting that the second was correct, with the protected space (as you did). – landroni Feb 23 '14 at 15:27
@landroni I think you should report it to the french package maintainer. It would be a wise thing to at least use xspace so that a normal space is inserted when there is no punctuation immediately following the superscript. You would still need to write explicit ~ characters based on context when you need a non-breaking space. But as others said, the output you get is the normal TeX behaviour – but babel French is quite user friendly, and I would not expect this as its normal behaviour (but I use custom commands, so I hadn't noticed it). – ienissei Feb 23 '14 at 16:44
@ienissei See me answer: Daniel Flipo (maintainer of frenchb) recommands to add xspace package in order to handle correctly the space in such case. – ppr Feb 23 '14 at 17:51
According to the manual of frenchb:
Il est recommandé de profiter des avantages offerts par l’extension xspace (il suffit d’ajouter \usepackage{xspace} dans le préambule) : les espaces suivant les commandes \ier,. . ., \ieres, \ieme, \iemes, \fg et \dots seront respectés sans avoir à les forcer par des {} ou des \ .
-
Thanks. Apparently xspace is being discouraged in some circles: tex.stackexchange.com/questions/86565/drawbacks-of-xspace . – landroni Feb 23 '14 at 17:51
Also, it seems that the two recommended usage forms, which avoid unexpected consequences, are: \ier{} or \ier{}~, as appropriate given the context. – landroni Feb 23 '14 at 18:01
@landroni xspace tries to avoid human errors (to forget \ or {}). And you are right to point out that, in some case, this useful tool could break some code (because it is not easy for an computer to know when you actually doing a mistake). However plenty of packages are coded with the need of xspace package ; frenchb is one of them. To sum up: if you want to use frenchb, you should use xspace also (because frenchb needs xspace to do a good job) : if you do not use any package which needs xspace it is better to not use it. – ppr Feb 23 '14 at 18:06
@landroni About the recommended usage forms, the manual of frenchb says p.5 "on dispose aussi de \ier \iere \iers \ieres \ieme \iemes pour 1er, 1re, 1ers, 1res, 2e, 2es.". So according to the manual, the recommended formating is 1\ier. Of course, this recommendation assumes you use xspace. If you don't, you should prefer 1\ier{} but it is not what is recommended by the author package. – ppr Feb 23 '14 at 18:13
In French, you should always use a hard space between an abbreviation and the following word. For example, have a look at "Petites leçons de typographie" by Jacques André, page 34. As such, your second case is the right one: "1er asdf" (with a protected space in between).
The two others are the normal behavior of such (La)TeX commands.
-
The TeX FAQ link suggests using 1\ier{} asdf to avoid a "gobbled" space, as does wikibooks. Does this mean that the default definition of \ier{} is incorrect and does NOT take into account the protected, non-breaking space (~) required by French typesetting conventions? – landroni Feb 23 '14 at 15:38
@landroni Yes and no. No, it does not take into account our typographical conventions. But it can be useful in peculiar cases. For example, in such expressions as le 1\ier{} de la classe, no protected space is required following our conventions, to the contrary of la 1\iere~fois, le 1\ier~jour, etc, but any gobbled space must be avoided all the same. The hard space is necessary only when the abbreviated word refers to the following word. – Franck Pastor Feb 23 '14 at 15:46
Ouch! Could you include this last comment in the answer? It seems that the two recommended usage forms are 1\ier{} asdf or 1\ier~asdf, depending on the context; but NOT 1\ier asdf. – landroni Feb 23 '14 at 15:56
@landroni Yes, it's what I meant :-). However, it must be said that the contexts where the non-breaking space is necessary happen much more frequently! So my comment was a complement of my anwer, and as such more fit (I think) as a comment. – Franck Pastor Feb 23 '14 at 16:01
@landroni As you said, it depends on the context. I would say that you should use 1\ier~asdf when asdf's meaning is directly related to 1\ier, and 1\ier{}in all other cases where a space is required directly after 1\ier. – Franck Pastor Feb 23 '14 at 16:22
|
2015-05-24 05:51:03
|
{"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": 0.8703793287277222, "perplexity": 3879.5841397723843}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927843.59/warc/CC-MAIN-20150521113207-00060-ip-10-180-206-219.ec2.internal.warc.gz"}
|
https://www.vgipl.com/docs/article.php?page=fea95c-how-to-calculate-heat-transfer-in-thermodynamics
|
S = total heat exchange area (m2) Contact us at powder.process@protonmail.com. hc = heat transfer coefficient on cold side (W.m-2.K-1) What prevents chess engines from being undetectable? (Assumption based on conservation of energy due to the assumption above). The heat is given in joules (J), the specific heat capacity is an amount in joules per kilogram (or gram) °C, and the mass is in kilograms (kg) or grams (g).
calculation needs to be run again, this time using Hassumed
He was also a science blogger for Elements Behavioral Health's blog network for five years. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The following table is from [Aydin] which is taking its source from drop is especially important for gasketed plate heat exchanger which page :
The calculation will then be iterative until the dU=\delta Q+p(x)dV(x) Feature Preview: New Review Suspensions Mod UX, Creating new Help Center documents for Review queues: Project overview, Entropy of two expanding and mixing ideal gases. Boost your career: Improve your Zoom skills.
This is exactly what happens with a carbon dioxide fire extinguisher, with the gas coming out at high pressure and cooling as it expands at atmospheric pressure.
The temperature inside the home is 21°C and the temperature outside the home is … 6-24-98 Heat transfer. The radiation associated with heat transfer is entirely electromagnetic waves, with a relatively low (and therefore relatively safe) energy.
Assume now that we are dealing with an ideal gas that has $f$ degrees of freedom per particle ($f=3$ for temperatures that are much lower than rotational and oscillational excitation energies). Electrons can also carry heat, which is the reason metals are generally very good conductors of heat.
When things are at different temperatures, however, the hotter objects give off more energy in the form of radiation than they take in; the reverse is true for the colder objects. It is then possible to approximate the size of the heat exchanger
What is the reasoning behind nighttime restrictions during pandemic?
$Q=\Delta U = m{c_v}\Delta T = {R \over {\gamma - 1}}m\Delta T$. What aspects of image preparation workflows can lead to accidents like Boris Johnson's No. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors.
Why do the Pern novels use regular words as profanity?
Also the temperature assumption for the specific heat can only be used with an ideal gas. The pressure above the piston is atmospheric pressure. We'll talk about electromagnetic waves in a lot more detail in PY106; an electromagnetic wave is basically an oscillating electric and magnetic field traveling through space at the speed of light. When heat transfer is involved, use this formula: change in temperature = Q / cm to calculate the change in temperature from a specific amount of heat added. The P-V graph for an isothermal process looks like this: The work done by the system is still the area under the P-V curve, but because this is not a straight line the calculation is a little tricky, and really can only properly be done using calculus. If the volume occupied by the gas doubles, how much work has the gas done? The different processes are then categorized as follows : If the volume increases while the temperature is constant, the pressure must decrease, and if the volume decreases the pressure must increase. s = size of a single plate (m2), It is also possible to calculate the number of channels : n = exchangers. Kakac et al, CRC Press, click This is an example of how work is done by a thermodynamic system. The amount of energy an object radiates depends strongly on temperature. The water increases in temperature by 10 degrees C. The only thing you need to remember is that you have to use consistent units for mass.
The gas is confined by a piston with a weight of 100 N and an area of 0.65 m2. This surface area can cover the entire inner s… As long as the expansion takes place slowly, it is reasonable to assume that the pressure is constant. The work done is zero in an isochoric process, and the P-V graph looks like: Isothermal - the temperature is kept constant. What other cookies/biscuits were traditionally baked in shell shaped forms like this one? Cpc = specific heat of cold fluid Metals have many free electrons, which move around randomly; these can transfer heat from one part of the metal to another. The most common design is to have the The amount of heat given is equal to the amount of heat taken.
The internal energy of an ideal gas is proportional to the temperature, so if the temperature is kept fixed the internal energy does not change.
the company Alfa Laval, people sometimes referring them as "Alfa This can be expressed as a power by dividing the energy by the time. (b) The gas is heated, expanding it and moving the piston up. In the water in the pot, convection currents are set up, helping to heat the water uniformly. To simplify matters, consider what happens when something is kept constant. When this happens, the freezer is much less efficient at keeping food frozen. U=\frac{f}{2}NkT=\frac{f}{2}pV
How can I determine pressure inside contained cylinder when heat is added? NOTE MINUS SIGN as $pdV$ is work done BY the system. Read the question. Using gaskets will limit the operating pressure and temperature so
e = thickness of the plates (m) To illustrate the use of the above equation, let's calculate the rate of heat transfer on a cold day through a rectangular window that is 1.2 m wide and 1.8 m high, has a thickness of 6.2 mm, a thermal conductivity value of 0.27 W/m/°C. The heat transfer rate - or power - in … Assume no heat transfer takes place to anything else: The pan is placed on an insulated pad, and heat transfer to the air is neglected in the short time needed to reach equilibrium.
|
2021-06-19 19:05:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5166729688644409, "perplexity": 551.3239229287234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487649688.44/warc/CC-MAIN-20210619172612-20210619202612-00316.warc.gz"}
|
https://studyadda.com/solved-papers/rrc-jabalpur-solved-paper-held-on-1st-shift-30-11-2014_q14/794/368125
|
• # question_answer 14) A and B can do a piece of work in 10 days. B and C can do it in 12 days. A and C can do it in 15 days. How long will A take to do it alone? A) 24 days B) 40 daysC) 30 days D) 20 days
Work done by A, B and C together in 1 day $=\left[ \frac{1}{10}+\frac{1}{12}+\frac{1}{15} \right]$ $=\frac{1}{8}$ Now, work done by A alone in 1 day $=\frac{1}{8}-\frac{1}{12}$ $=\frac{1}{24}$ Hence, A will take to do it alone m 24 days.
|
2019-06-24 22:30:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6223176717758179, "perplexity": 313.76918863314893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999740.32/warc/CC-MAIN-20190624211359-20190624233359-00412.warc.gz"}
|
https://jessicastringham.net/2018/12/27/KL-Divergence/
|
Machine learning involves approximating intractable probability distributions. One approach to approximating is to find a distribution that minimizes the KL Divergence with the target distribution. For example, the approximating distributions could be normal distributions with different means and variances.
When KL Divergence is introduced in the context of machine learning, one point is that KL Divergence $$KL(P \mid\mid Q)$$ will select a different distribution than $$KL(Q \mid\mid P)$$. This blog post explores this by telling an optimizer (TensorFlow) to minimize the two KL Divergences.
## KL Divergence equation for discrete distributions
Wikipedia gives the KL Divergence for discrete distributions as
If $$P_i$$ = 0, then the $$i^{th}$$ term is 0. $$KL$$ is only defined if when $$Q_i = 0$$, then $$P_i = 0$$.
For example, we can have $$P$$ be the distribution we’re trying approximate with $$Q$$. The KL Divergence will be big if $$Q_i$$ is close to 0 where $$P_i$$ is not close to 0. If $$P_i$$ is close to 0, $$Q_i$$ won’t affect the KL Divergence as much.
### An example target distribution and two example approximate distributions
Let’s plot a few examples!
For this first example, I’ll make $$P$$ based on the distribution $$\beta(2, 5)$$. This is interesting because $$P_i$$ is 0 outside of the domain of 0 to 1. I’ll use $$Q$$s that are based on a normal distribution, so $$Q$$ is never 0. I highlight the area where $$P_i > 0$$.
#### Aside: Discrete vs Continuous
In order to make cool-looking graphs, I’m using discrete distributions that are based on continuous distributions, like the normal distribution. For example, below I start with 200 evenly-spaced numbers between -1 and 2. I compute the value of the PDF for those numbers. Then I normalize the vector so the 200 numbers add to 1 and it becomes a discrete distribution.
### Computing KL Divergence
I can translate the formula to numpy, then compute the KL Divergence between the two approximating distributions and the target distribution.
As expected, the KL Divergence is higher for the approximating distribution based on Norm(1, 0.2) than the distribution based on Norm(0.2, 0.15).
Q = Norm(1.0, 0.20) KL(P || Q) = 6.491177
Q = Norm(0.2, 0.15) KL(P || Q) = 0.236206
#### Aside: Verifying the implementation
scipy’s entropy computes KL Divergence when called with two parameters. I can verify my implementation produces similar results.
#### Aside: Interactive
Before I implement something that minimizes the divergence automatically, I can use ipywidgets to interactively try different distributions.
## Multimodal Example
One point with KL Divergence is that finding a $$Q$$ that minimizes $$KL(Q \mid\mid P)$$ is different than finding a $$Q$$ that minimizes $$KL(P \mid\mid Q)$$. One way to illustrate the difference is to look at a multimodal distribution.
### Minimizing KL Divergence
I can implement KL Divergence in TensorFlow and then use gradient descent to find an approximating distribution $$Q = Norm(\mu, \sigma^2)$$ that minimizes the KL Divergence.
It’s also neat to plot how the distribution shifts as it improves!
### Comparison of KL(Q || P) to KL(P || Q)
Finally, I can compare the $$Q$$ that minimizes $$KL(Q \mid\mid P)$$ to the one that minimizes $$KL(P \mid\mid Q)$$.
|
2019-09-20 18:15:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9111542701721191, "perplexity": 573.3966671465981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574058.75/warc/CC-MAIN-20190920175834-20190920201834-00348.warc.gz"}
|
https://zbmath.org/authors/?q=ai%3Aogata.shoetsu
|
# zbMATH — the first resource for mathematics
## Ogata, Shoetsu
Compute Distance To:
Author ID: ogata.shoetsu Published as: Ogata, Shoetsu; Ogata, S.
Documents Indexed: 26 Publications since 1985
all top 5
#### Co-Authors
19 single-authored 2 Zhao, Huai-Liang 1 Amram, Meirav 1 Kondo, Daiki 1 Nakagawa, Katsuyoshi 1 Saito, Masa-Hiko 1 Satake, Ichirô
all top 5
#### Serials
4 Tohoku Mathematical Journal. Second Series 3 Far East Journal of Mathematical Sciences 2 Beiträge zur Algebra und Geometrie 2 Japanese Journal of Mathematics. New Series 2 Kodai Mathematical Journal 2 Interdisciplinary Information Sciences (IIS) 1 Annales de l’Institut Fourier 1 Duke Mathematical Journal 1 Journal of the Mathematical Society of Japan 1 Manuscripta Mathematica 1 Mathematische Annalen 1 Mathematical Journal of Okayama University 1 Mathematische Zeitschrift 1 Michigan Mathematical Journal 1 Sūgaku 1 Sugaku Expositions
all top 5
#### Fields
24 Algebraic geometry (14-XX) 11 Convex and discrete geometry (52-XX) 4 Several complex variables and analytic spaces (32-XX) 4 Global analysis, analysis on manifolds (58-XX) 2 Number theory (11-XX) 1 Commutative algebra (13-XX)
#### Citations contained in zbMATH Open
15 Publications have been cited 31 times in 22 Documents Cited by Year
Zeta functions associated to cones and their special values. Zbl 0712.14009
Satake, I.; Ogata, S.
1989
Hirzebruch’s conjecture on cusp singularities. Zbl 0810.14002
Ogata, Shoetsu
1993
On generators of ideals defining projective toric varieties. Zbl 0997.14014
Ogata, Shoetsu; Nakagawa, Katsuyoshi
2002
Special values of zeta functions associated to cusp singularities. Zbl 0588.14016
Ogata, Shoetsu
1985
Multiplication maps of complete linear systems on projective toric surfaces. Zbl 1110.14050
Ogata, Shoetsu
2006
Very ample but not normal lattice polytopes. Zbl 1271.14077
Ogata, Shoetsu
2013
Infinitesimal deformations of generalized cusp singularities. Zbl 0629.32018
Ogata, Shoetsu
1987
On multiplication maps of ample bundles with nef bundles on toric surfaces. Zbl 1161.14304
Kondo, Daiki; Ogata, Shoetsu
2008
Infinitesimal deformations of Tsuchihashi’s cusp singularities. Zbl 0601.32024
Ogata, Shoetsu
1986
Projective normality of toric 3-folds with non-big adjoint hyperplane sections. Zbl 1317.14113
Ogata, Shoetsu
2012
Signature defects and eta functions of degenerations of abelian varieties. Zbl 0895.58051
Ogata, Shoetsu; Saito, Masa-Hiko
1997
On quadratic generation of ideals defining projective toric varieties. Zbl 1071.14055
Ogata, Shoetsu
2003
On projective toric varieties whose defining ideals have minimal generators of the highest degree. Zbl 1069.14057
Ogata, Shoetsu
2003
Degenerations and fundamental groups related to some special toric varieties. Zbl 1148.14302
Amram, Meirav; Ogata, Shoetsu
2006
A characterization of Gorenstein toric Fano $$n$$-folds with index $$n$$ and Fujita’s conjecture. Zbl 1312.14124
Ogata, Shoetsu; Zhao, Huai-Liang
2014
A characterization of Gorenstein toric Fano $$n$$-folds with index $$n$$ and Fujita’s conjecture. Zbl 1312.14124
Ogata, Shoetsu; Zhao, Huai-Liang
2014
Very ample but not normal lattice polytopes. Zbl 1271.14077
Ogata, Shoetsu
2013
Projective normality of toric 3-folds with non-big adjoint hyperplane sections. Zbl 1317.14113
Ogata, Shoetsu
2012
On multiplication maps of ample bundles with nef bundles on toric surfaces. Zbl 1161.14304
Kondo, Daiki; Ogata, Shoetsu
2008
Multiplication maps of complete linear systems on projective toric surfaces. Zbl 1110.14050
Ogata, Shoetsu
2006
Degenerations and fundamental groups related to some special toric varieties. Zbl 1148.14302
Amram, Meirav; Ogata, Shoetsu
2006
On quadratic generation of ideals defining projective toric varieties. Zbl 1071.14055
Ogata, Shoetsu
2003
On projective toric varieties whose defining ideals have minimal generators of the highest degree. Zbl 1069.14057
Ogata, Shoetsu
2003
On generators of ideals defining projective toric varieties. Zbl 0997.14014
Ogata, Shoetsu; Nakagawa, Katsuyoshi
2002
Signature defects and eta functions of degenerations of abelian varieties. Zbl 0895.58051
Ogata, Shoetsu; Saito, Masa-Hiko
1997
Hirzebruch’s conjecture on cusp singularities. Zbl 0810.14002
Ogata, Shoetsu
1993
Zeta functions associated to cones and their special values. Zbl 0712.14009
Satake, I.; Ogata, S.
1989
Infinitesimal deformations of generalized cusp singularities. Zbl 0629.32018
Ogata, Shoetsu
1987
Infinitesimal deformations of Tsuchihashi’s cusp singularities. Zbl 0601.32024
Ogata, Shoetsu
1986
Special values of zeta functions associated to cusp singularities. Zbl 0588.14016
Ogata, Shoetsu
1985
all top 5
#### Cited by 32 Authors
5 Ogata, Shoetsu 2 Duflot, Jeanne 1 Amram, Meirav 1 Ashikaga, Tadashi 1 Blind, Bruno 1 Bruzzo, Ugo 1 Eie, Minking 1 Gong, Cheng 1 Grassi, Antonella 1 Haase, Christian Alexander 1 Hering, Milena 1 Higashitani, Akihiro 1 Ibukiyama, Tomoyoshi 1 Iida, Shuichi 1 Ishida, Masanori 1 Kawaguchi, Ryo 1 Lasoń, Michał 1 Lorenz, Benjamin 1 Michałek, Mateusz 1 Nakashima, Hideto 1 Nill, Benjamin 1 Paffenholz, Andreas 1 Peters, Pamela L. 1 Rote, Günter 1 Saito, Hiroshi 1 Santos, Francisco 1 Satake, Ichirô 1 Schenck, Hal 1 Tan, Sheng-Li 1 Teicher, Mina 1 Tsuchihashi, Hiroyasu 1 Xu, Wan-Yuan
all top 5
#### Cited in 16 Serials
5 Tohoku Mathematical Journal. Second Series 3 Mathematische Annalen 1 Israel Journal of Mathematics 1 Rocky Mountain Journal of Mathematics 1 Annales de l’Institut Fourier 1 Bulletin de la Société Mathématique de France 1 Duke Mathematical Journal 1 Mathematische Zeitschrift 1 Nagoya Mathematical Journal 1 Proceedings of the American Mathematical Society 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 International Journal of Algebra and Computation 1 Experimental Mathematics 1 Journal of Algebraic Combinatorics 1 The Electronic Journal of Combinatorics 1 Communications in Contemporary Mathematics
all top 5
#### Cited in 15 Fields
16 Algebraic geometry (14-XX) 8 Number theory (11-XX) 7 Several complex variables and analytic spaces (32-XX) 6 Convex and discrete geometry (52-XX) 2 Group theory and generalizations (20-XX) 1 Mathematical logic and foundations (03-XX) 1 Combinatorics (05-XX) 1 Field theory and polynomials (12-XX) 1 Commutative algebra (13-XX) 1 Nonassociative rings and algebras (17-XX) 1 Topological groups, Lie groups (22-XX) 1 Abstract harmonic analysis (43-XX) 1 Functional analysis (46-XX) 1 Manifolds and cell complexes (57-XX) 1 Global analysis, analysis on manifolds (58-XX)
|
2022-01-20 08:15:42
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.411558598279953, "perplexity": 6743.557829776168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301730.31/warc/CC-MAIN-20220120065949-20220120095949-00259.warc.gz"}
|
http://talkstats.com/threads/probability-of-regression-result.71072/
|
# Probability of regression result
#### DrAnthonyRodriguez
##### New Member
Hi!
A stock is selling at $50 now. Based on a 90-day linear regression (i.e., 90 historical periods), 80 days from today, the stock will be selling at$56. The standard error is 4. What is the 95% probability that the stock will be selling at least at \$59? Using Excel, how is the 95% probability calculated?
Thank you!
Tony
Last edited:
#### staassis
##### Member
This is a hw problem. You can show some effort by posting your thoughts or you can just google for "linear regression prediction interval".
#### ondansetron
##### TS Contributor
You probably also need to account for autocorrelation of errors if the experimental unit is a unit of time, which it sounds like the case. Without this the estimates may be incorrect. I would also be careful of misinterpreting a CI or PI as a "95% probability" interval.
#### DrAnthonyRodriguez
##### New Member
This is a hw problem ...
Dear Staassis,
It's not a HW problem; I am learning stats on my own.
I looked at CIs and PIs but I am looking for the 95% probability of a value higher than X, not the probability of a range.
Tony
#### staassis
##### Member
If it's not an idealized hw problem, then linear regression is far from being appropriate. First, you need to transform the data into returns, log-returns or some other weakly stationary stochastic process. Second, you need to apply time series techniques. In particular, as ondansetron pointed out, serial correlation may be an issue. Heteroskedasticity may be an issue as well.
The truth is: on 90 observations only, you will not develop a model sufficient for answering the question terribly accurately. And Excel is not software appropriate for time series analysis (everything is possible but painfully). If you are interested in statistical modeling, get R, Matlab, Stata or SPSS (in my humble opinion).
Last edited:
Dear Staassis,
|
2018-04-25 18:07:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6771687865257263, "perplexity": 1376.357843587081}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947939.52/warc/CC-MAIN-20180425174229-20180425194229-00289.warc.gz"}
|
https://www.quantopian.com/posts/liquidity-factor
|
liquidity factor?
Here's an attempt at a liquidity alpha factor (via combination several liquidity alpha factors into one).
25
Backtest from to with initial capital
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
Returns 1 Month 3 Month 6 Month 12 Month
Alpha 1 Month 3 Month 6 Month 12 Month
Beta 1 Month 3 Month 6 Month 12 Month
Sharpe 1 Month 3 Month 6 Month 12 Month
Sortino 1 Month 3 Month 6 Month 12 Month
Volatility 1 Month 3 Month 6 Month 12 Month
Max Drawdown 1 Month 3 Month 6 Month 12 Month
# https://arxiv.org/pdf/1412.5072.pdf
from quantopian.algorithm import attach_pipeline, pipeline_output
from quantopian.pipeline import Pipeline
from quantopian.pipeline.factors import CustomFactor, Returns, SimpleMovingAverage
from quantopian.pipeline.data.builtin import USEquityPricing
from quantopian.pipeline.data import Fundamentals
import quantopian.optimize as opt
from sklearn import preprocessing
from scipy.stats.mstats import winsorize
import numpy as np
import pandas as pd
WIN_LIMIT = 0.05
def preprocess(a):
a = a.astype(np.float64)
a[np.isinf(a)] = np.nan
not_nan_ind = np.argwhere(~np.isnan(a))
if not_nan_ind.size > 0:
a_win = winsorize(a[not_nan_ind], limits=[WIN_LIMIT,WIN_LIMIT])
a[not_nan_ind] = a_win
else:
a = winsorize(a, limits=[WIN_LIMIT,WIN_LIMIT])
a = np.nan_to_num(a - np.nanmean(a))
return preprocessing.scale(a)
def normalize(x):
r = x - x.mean()
return r/r.abs().sum()
def make_factors():
class LIX(CustomFactor):
inputs = [USEquityPricing.high, USEquityPricing.low, USEquityPricing.close, USEquityPricing.volume]
window_length = 21
window_safe = True
def compute(self, today, assets, out, high, low, close, volume):
dv = close[-1,:]*np.sum(volume, axis=0)
r = np.amax(high,axis=0) - np.amin(low,axis=0)
out[:] = preprocess(-np.log10(dv/r))
class ILLIQ(CustomFactor):
inputs = [USEquityPricing.close, USEquityPricing.volume, Returns(window_length=21)]
window_length = 21
window_safe = True
def compute(self, today, assets, out, close, volume, ret):
ilq = np.absolute(ret)/(close*volume)
out[:] = preprocess(-np.sum(ilq,axis=0))
class HH(CustomFactor):
inputs = [USEquityPricing.high, USEquityPricing.low, USEquityPricing.volume, Fundamentals.shares_outstanding]
window_length = 21
window_safe = True
def compute(self, today, assets, out, high, low, volume, n_shares):
dv = np.amin(low,axis=0)*np.sum(volume, axis=0)
r = np.amax(high,axis=0) - np.amin(low,axis=0)
out[:] = preprocess(-n_shares[-1]*r/dv)
class share_turnover(CustomFactor):
inputs = [USEquityPricing.volume, Fundamentals.shares_outstanding]
window_length = 21
window_safe = True
def compute(self, today, assets, out, volume, n_shares):
v = np.sum(volume, axis=0)
out[:] = preprocess(-v/n_shares[-1])
class dollar_turnover(CustomFactor):
inputs = [USEquityPricing.close, USEquityPricing.volume, Fundamentals.shares_outstanding]
window_length = 21
window_safe = True
def compute(self, today, assets, out, close, volume, n_shares):
dv = np.sum(close*volume, axis=0)/close[-1,:]
out[:] = preprocess(-dv/n_shares[-1])
factors = [
LIX,
ILLIQ,
HH,
share_turnover,
dollar_turnover,
]
return factors
def factor_pipeline():
factors = make_factors()
pipeline_columns = {}
for k,f in enumerate(factors):
pipe = Pipeline(columns = pipeline_columns,
screen = universe)
return pipe
def initialize(context):
attach_pipeline(factor_pipeline(), 'factor_pipeline')
# Schedule my rebalance function
schedule_function(func=rebalance,
date_rule=date_rules.every_day(),
time_rule=time_rules.market_close(hours=1),
half_days=True)
# record my portfolio variables at the end of day
schedule_function(func=recording_statements,
date_rule=date_rules.every_day(),
time_rule=time_rules.market_close(),
half_days=True)
def recording_statements(context, data):
record(num_positions=len(context.portfolio.positions))
record(leverage=context.account.leverage)
def rebalance(context, data):
alpha = pipeline_output('factor_pipeline').sum(axis=1)
order_optimal_portfolio(opt.TargetWeights(normalize(alpha)), constraints=[])
There was a runtime error.
23 responses
Interesting one, thanks @Grant! Think you might have found a reason for why Q may not want to license individual (weak) factors, but when combined, the factor of related factors becomes more robust and possibly more interesting?
Wouldn't shs_float_cf be a better liquidity gauge than shares_outstanding ?
@Grant,
a very interesting LIX indicator.
Thanks for sharing.
I created the LIX indicator to see how we can use it to determine market regime.
1
Notebook previews are currently unavailable.
And here's the alpha decay and risk exposure analysis just on two factors LIX and ILLIQ.
0
Notebook previews are currently unavailable.
@ Viridian Hawk -
Wouldn't shs_float_cf be a better liquidity gauge than shares_outstanding ?
Not sure. Could you explain your reasoning?
I just spent a little time Googling factor etfs/smart beta/alpha factors, etc. and came across the idea of liquidity. For example:
https://investor.vanguard.com/etf/profile/VFLQ
There, it says:
Fund invests in stocks with relatively lower measures of trading liquidity.
The Liquidity factor is measured by percentage turnover, dollar turnover, and Amihud illiquidity.
It is worth noting that on Q, one could use minute bar data to construct liquidity (illiquidity) factors. There may be liquidity information not revealed in the daily bars.
“Shares outstanding” includes shares "closely held" by insiders and shares otherwise restricted from being traded. "Float" refers to the shares that can actually be traded freely on the public markets, and as such is probably a better fit conceptually for gauging liquidity.
Here's a first-cut at a liquidity factor. Good? Bad? And why?
0
Notebook previews are currently unavailable.
In my opinion...
The Good:
-Consistent specific IR (not too sporadic drops, or negative specific IR)
-Very little Style tilts
-Scoring most stocks in the QTU using TargetWeights
-Low turnover
-Might have value for certain Sectors?
-Relatively low Returns, both specific and common (specific is what matters, and it looks a bit better than common/total)
-Some ‘extreme’ Sector exposures.
-Seems fairly volatile
The Ugly:
-Sector tilts.
Grant ,
Recently, I spent some time working on the illiquidity ratio (ILLIQ), and found that it behaves very much like various volatility factors.
There were a lot of nans so I changed to
out[:] = np.nansum(ilq)
I used DailyReturns instead of Returns and multiplied the value by 1000 to see the numbers.
Amihud (2002) illiquidity ratio, ILLIQ, is one of the most widely used in the industry and is
the daily ratio of absolute stock return to its dollar volume averaged over some period.
0
Notebook previews are currently unavailable.
Than I decided to test ILLIQ as volatility switch on Yulia Malitskaya conventional momentum winners (W_10).
I even left her magic threshold 0.27.
The results are not the best but very similar to backtests with other volatility factors.
0
Notebook previews are currently unavailable.
I see this as a potential test case to see if Q is really interested in funding a lot more itsty-bitsy alphas under the new signal combination paradigm, as they've said they would. It seems that they should publish a list of 50-100 of such little projects, versus trying to find the goose that laid the golden egg. I'd be glad to get in on the action at some low, but consistent monthly payout, versus shooting for the \$50M grand prize.
I used DailyReturns
Thanks for the tip! I think I had the Amihud (2002) illiquidity ratio incorrect.
Now I have:
class ILLIQ(CustomFactor):
inputs = [USEquityPricing.close, USEquityPricing.volume, DailyReturns(window_length=2)]
window_length = 253
window_safe = True
def compute(self, today, assets, out, close, volume, ret):
ilq = np.absolute(ret)/(close*volume)
out[:] = preprocess(np.nansum(ilq,axis=0))
with
from sklearn import preprocessing
def preprocess(a):
a = a.astype(np.float64)
a[np.isinf(a)] = np.nan
not_nan_ind = np.argwhere(~np.isnan(a))
if not_nan_ind.size > 0:
a_win = winsorize(a[not_nan_ind], limits=[WIN_LIMIT,WIN_LIMIT])
a[not_nan_ind] = a_win
else:
a = winsorize(a, limits=[WIN_LIMIT,WIN_LIMIT])
a = np.nan_to_num(a - np.nanmean(a))
return preprocessing.scale(a)
Grant ,
It is not necessary to specify a window length for DailyReturns
DailyReturns() is the same as DailyReturns(window_length=2)
I did not use preprocessing just to see how it looks like originally.
One extension here would be to use the Q minute bar data. I have to think there's more than just thin air in liquidity, if Vanguard (and perhaps others) went to the trouble of launching an ETF. There's probably a bunch of academic and industry research on the topic, or they'd be sticking their necks out pretty far.
Grant -- according to Quantopian's TOS as soon as you've posted an algorithm to the forum they are free to use it license-free -- you basically give up any authorship rights to that code. So I hope for your sake you're keeping the really juicy stuff secret.
@ Viridian Hawk -
It's a hobby so there's really nothing to lose.
Anyway, if you have anything technical to add to this thread, on the topic of liquidity, please contribute.
Glancing at https://investor.vanguard.com/etf/profile/portfolio/vflq suggests that this "factor" ETF may have significant exposure to some "common" factors (e.g. size and value). I'm guessing that even though it is called a "factor" ETF, it is more like a "smart beta" ETF in that it has a constraint to attempt to kinda-sorta track the Russell 3000 benchmark versus actually isolating liquidity while controlling for other common factors.
May be of interest:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2291942
We find that the pricing of the Amihud measure is not attributable to
the construction of the return-to-volume ratio that is intended to
capture price impact, but driven by the trading volume component
@ Antony -
Thanks. I gather that you are an academic finance type. Is there any kind of consensus on liquidity as a factor in academic circles? I imagine that at some point, there was a sort of consensus on the original Fama and French factors being "real" (of course, academics make a living out of arguing over minutiae, so "consensus" is probably too strong of a word); perhaps there is similar acknowledgement that liquidity is not just vaporware.
Not an overly academic thing, @Grant.
If you buy an asset knowing that it's going to be difficult to unwind when you most need to, you'll probably want to pay less for it. And that's where the risk premium is earned.
@ Antony -
O.K., and I guess it works the other way too? Stocks that are easy to trade are a bit over-priced due to the fact that folks are willing to pay extra for the assurance that they can cash out whenever they want.
If you buy an asset knowing that it's going to be difficult to unwind when you most need to, you'll probably want to pay less for it. And that's where the risk premium is earned.
Would it be reasonable then to assume then that heavily shorted stocks would have an offsetting effect, since you'll probably want to short the stock at a higher basis for the same reasons and collect risk premium on the short side?
@ Viridian Hawk -
I think you are suggesting that liquidity may be different from a long versus a short perspective, since there may be a difference in closing out the position. In effect, there could be "long liquidity" and "short liquidity".
For example, the QTradableStocksUS() uses the 200-day median daily dollar volume as a proxy for liquidity (see https://www.quantopian.com/help#quantopian_pipeline_experimental_QTradableStocksUS). However, maybe this tilts the QTU toward long liquidity? But really one needs to be able to go long or short with the same ease, I think, for the kind of fund Q is constructing. It also costs more to short, so there's a barrier to shorting relative to going long.
Does Quantopian have so-called short interest data? Along with relative borrowing rates... Seems like a pretty basic set of data...
I think I'll put this aside for awhile. By the way, I sent an e-mail to Q asking if they'd be interested in a liquidity factor--no response. I guess that means "no" but who knows.
1
Backtest from to with initial capital
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
Returns 1 Month 3 Month 6 Month 12 Month
Alpha 1 Month 3 Month 6 Month 12 Month
Beta 1 Month 3 Month 6 Month 12 Month
Sharpe 1 Month 3 Month 6 Month 12 Month
Sortino 1 Month 3 Month 6 Month 12 Month
Volatility 1 Month 3 Month 6 Month 12 Month
Max Drawdown 1 Month 3 Month 6 Month 12 Month
from quantopian.algorithm import attach_pipeline, pipeline_output
from quantopian.pipeline import Pipeline
from quantopian.pipeline.factors import CustomFactor, DailyReturns
from quantopian.pipeline.data.builtin import USEquityPricing
from quantopian.pipeline.data import Fundamentals, factset
import quantopian.optimize as opt
from scipy.stats.mstats import winsorize
from scipy.stats import rankdata
import numpy as np
import pandas as pd
from quantopian.pipeline.classifiers.morningstar import Sector
WIN_LIMIT = 0.0
def preprocess(a):
a = a.astype(np.float64)
a[np.isinf(a)] = np.nan
not_nan_ind = np.argwhere(~np.isnan(a))
if not_nan_ind.size > 0:
a_win = winsorize(a[not_nan_ind], limits=[WIN_LIMIT,WIN_LIMIT])
a[not_nan_ind] = a_win
else:
a = winsorize(a, limits=[WIN_LIMIT,WIN_LIMIT])
a = np.nan_to_num(a - np.nanmean(a))
a = rankdata(a)
a = a/np.amax(a)
a = a - 0.5
return a/np.sum(np.absolute(a))
def normalize(x):
r = x.rank()
r = r/r.max()
r = r - 0.5
return r/r.abs().sum()
def make_factors():
class ILLIQ(CustomFactor):
inputs = [USEquityPricing.close, USEquityPricing.volume, DailyReturns(), Fundamentals.shares_outstanding]
window_length = 253
def compute(self, today, assets, out, close, volume, ret, n_shares):
ilq = np.absolute(ret)/(close*volume)
a = preprocess(np.nanmean(close,axis=0)*np.nanmean(ilq,axis=0))
st = n_shares/volume
b = preprocess(np.nanmean(st,axis=0))
dt = n_shares/(close*volume)
c = preprocess(np.nanmean(close,axis=0)*np.nanmean(dt,axis=0))
out[:] = preprocess(a+b+c)
factors = [
ILLIQ,
]
return factors
def factor_pipeline():
factors = make_factors()
sectors = [101,102,103,104,205,206,207,308,309,310,311]
pipeline_columns = {}
for k,f in enumerate(factors):
for s in sectors:
pipe = Pipeline(columns = pipeline_columns,
return pipe
def initialize(context):
attach_pipeline(factor_pipeline(), 'factor_pipeline')
# Schedule my rebalance function
schedule_function(func=rebalance,
date_rule=date_rules.every_day(),
time_rule=time_rules.market_close(hours=1),
half_days=True)
# record my portfolio variables at the end of day
schedule_function(func=recording_statements,
date_rule=date_rules.every_day(),
time_rule=time_rules.market_close(),
half_days=True)
def recording_statements(context, data):
record(num_positions=len(context.portfolio.positions))
record(leverage=context.account.leverage)
def rebalance(context, data):
alpha = pipeline_output('factor_pipeline').sum(axis=1)
objective = opt.TargetWeights(normalize(alpha))
order_optimal_portfolio(objective=objective,
constraints=[]
)
There was a runtime error.
|
2019-09-22 16:08:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18453282117843628, "perplexity": 9815.048767293325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575596.77/warc/CC-MAIN-20190922160018-20190922182018-00473.warc.gz"}
|
http://vort.org/tag/life/
|
# Russell's Blog
## Makers do not make weapons
Posted by Russell on December 17, 2012 at 6:58 p.m.
Last Tuesday, I started writing an article about Thing 11770 on Thingiverse, a MakerBot Industries for sharing 3D printable objects. Thing 11770 is a reinforced 3D printable lower receiver for an AR-15 assault rifle. This is the part of the gun that feeds bullets from the magazine into upper receiver, which handles the cycling of the spent round and the insertion of the new round. With the right combination of upper and lower receiver, fresh rounds are cycled into the weapon using a portion of the kinetic energy from firing the previous round. When the trigger is pulled, this process happens continuously, firing one bullet after another. That is what it means to be an "automatic" weapon. Thing 11770 is particularly interesting because, legally speaking, the lower receiver is the gun itself. It is the engine that makes the gun a gun, rather than a movie prop. And you can 3D print it. And it works.
At very the moment I was hemming and hawing over how to articulate my feelings about this development, someone used an AR-15 to murder twenty seven people, including twenty children, ages six and seven at Sandy Hook Elementary School. Now I know exactly how I feel.
I love 3D printing. I love the maker movement. I love the idea of people building home-brew versions of all sorts of devices, and inventing entirely new classes of devices. 3D printing has played, and will continue to play, an important role in that.
When I was fourteen, like many boys at that age, I thought missiles and fighter planes and tanks were pretty awesome. I read a lot of Tom Clancy books, and I indulged in my interest by dragging my family to the Wright Patterson Air Force Base Museum, the Smithsonian Air & Space Museum, the California Science Center’s Air & Space Museum, and the Intrepid Air, Sea & Space Museum. At Wright Patterson, I visited the F-117 Nighthawk as many times as I could. The author of Thing 11770 calls himself "Have Blue," the codename for the Nighthawk demonstrator aircraft.
When I was sixteen, I went to boarding school, where I learned vector calculus and farming. I learned how to grub potatoes out of the freezing ground in the driving rain, how to make maple syrup, how to lay in beets and squash and onions for the winter. I stood on a windy mountain top and learned how to find the orbital ephemera of a comet. I learned how to milk cows, how to care for cows when they are sick, and how to make the most delicious yogurt and mozzarella cheese you could possibly imagine. I learned how to repair a tractor engine with a mallet and a wrench. One freezing night, I found myself covered in blood and shit and urine and fear as I helped bring a new life gasping and staggering into the world.
Farming also means slaughtering and butchering. One morning, I walked into the barn. I was handed a weapon. I was asked to take a life.
I found that I could not.
Not ever.
The instant my shoulders took up the weight of the strange, snub nosed machine, it felt like the weight of the metal hung from my heart, stretching and distorting it. I wanted the weight of it to tear me apart, but I knew it was a weight I could carry, if I wanted to. I quietly handed the gun back to the farm manager, and walked out into the thawing snow, and spent the rest of the black pre-dawn puking into the mud behind the water tower.
Many people have wondered why I do not eat meat. This is why. For the rest of my life, I will feel the weight of that terrible little machine.
There are reasons to make, to have and to use guns. To defend your country, yes. To humanely put down an animal before butchering it, perhaps. For vainglory? For entertainment? No.
Tools are sacred things. We are a tool-using species; our tools are projections of our hopes and aspirations. When we are filled with joy, we pick up our tools and hammer the air into music. We need to understand and to be understood, and so we shape our voices into language. We send our tools delicately probing into the bodies of our loved ones, seeking out cancers and blood clots and infections. We invest huge amounts of effort building and maintaining tools that allow us to speak to one another across great distances. We hurl our tools across the void to other planets to satisfy our craving for knowledge. When we grieve, we take up our tools and carve the names of those we have lost into the living rock of our planet. Our tools are our souls. They are our defining characteristic. Love may be what makes us alive, but our tools are what make us human.
A gun is a tool. It is a simple tool. Any man or woman or child can use one. A gun is not much more complicated than a can opener, and not nearly as sophisticated as cordless screwdriver. Like all tools, a gun reveals something fundamental about its maker, its wielder and its abuser. This is true for all weapons.
As a strong supporter of the maker movement, of free and open source software, of open science, I want people to have as much freedom as possible to make and remake and experiment. I also believe very, very strongly in the responsibly we have to one another. I believe that we each have a responsibility not make things that hurt and kill and destroy.
I am not yet prepared to call for a law to prohibit Have Blue from posting functional 3D printable assault rifle parts on the internet. The law is a blunt instrument, and would cause a great deal of collateral damage. However, I am prepared to say that Have Blue is a fucking asshole. I am prepared to call Justin Halford, who created the original CNC model, a fucking asshole. I am prepared to say that anyone who considers themselves a "gun enthusiast" and is older than about sixteen needs to grow the fuck up. The maker community should not tolerate this behavior. Meditate on the meaning of the word antisocial for a moment. What could be more antisocial than gleefully proliferating machines whose principal function is murder?
The maker community should not tolerate these designs, or the ideas and opinions of their designers until they show evidence of behaving like adults. It's clear that the CNC Gunsmithing community has a lot of talented, clever people. It's clear from reading his blog that Have Blue is neither ignorant nor stupid.
So, I'm calling you folks out. There are twenty children dead in Connecticut. Their bodies were ripped apart by the very machines you are "democratizing." As far as I know, nobody has used your designs to kill anyone. If you continue down this path, some future version of Thing 11770 will be used to murder little children. It's just a matter of time, and probably a lot less time than you think. However, there is still time to take a stand. Do the right thing. Take down the designs. Apologize for what you've done. Find a new project. Use your talents for something good. This will not stop people from murdering children with 3D printed guns, but perhaps you can buy us some time before that day comes. You know that this is true.
If making home-brew assault rifles is really what you want to do, there is perhaps one venue where this might actually make sense. Freight your CNC machine to Istanbul, and smuggle it into Homs or Aleppo. Help the Free Syrian Army get rid of Bashar Assad. Oh wait, what’s that? You don't want to get shot? Fancy that.
It takes courage to admit you are wrong. Show us some courage.
Update : It appears that MakerBot has decided to remove Thing 11770 from Thingiverse. If you follow the link to the item, the files have been removed and a message says, "This Thing is currently under moderation for violating the Thingiverse Terms of Service. Files and images for this Thing are currently unavailable." I'm glad it's no longer up, but I am disappointed in how this was handled. I'm disappointed that MakerBot left it up for so long, but I'm also disappointed that Have Blue didn't just take it down himself.
## A new Prometheus
Posted by Russell on December 08, 2012 at 2:33 a.m.
One year ago, UC Davis law student Megan Glanville was killed a stone's throw from my front door. She was crossing the street for a morning run. It was foggy. The driver didn't see her.
Since then, the intersection where she died has been redesigned. It is now a three-way stop with modern LED lighting. Watching over the scene, there is a new flashing red beacon.
This sort of infrastructure is easy to take for granted. As a Commissioner for the City of Davis, I suppose I pay closer attention to these things that most people do. I've payed particular attention to this little piece of city infrastructure because I pass through it several times a day.
Something has changed there since the red beacon went up. Up and down the boulevard, for almost a mile, there are crossings to access the bicycle path. Drivers now stop and let me cross. They never did that before. I am not exaggerating when I say that wherever the beacon's light falls, the feel of the street has changed. It's no longer the tail end of a lonely country road. It's a neighborhood street, and people act accordingly.
I would like to think that drivers feel the significance of the flashing beacon. I would like to think that they have noticed that the intersection has been redesigned. I would like to think that they know that Megan Glanville died there. In all likelihood, they are oblivious to these things. They stop and smile and waive me through anyway.
Why?
Good design matters. That's why.
The rocky ledge runs far into the sea,
And on its outer point, some miles away,
The Lighthouse lifts its massive masonry,
A pillar of fire by night, of cloud by day.
Even at this distance I can see the tides,
Upheaving, break unheard along its base,
A speechless wrath, that rises and subsides
In the white lip and tremor of the face.
And as the evening darkens, lo! how bright,
Through the deep purple of the twilight air,
Beams forth the sudden radiance of its light
With strange, unearthly splendor in the glare!
Not one alone; from each projecting cape
And perilous reef along the ocean's verge,
Starts into life a dim, gigantic shape,
Holding its lantern o'er the restless surge.
Like the great giant Christopher it stands
Upon the brink of the tempestuous wave,
Wading far out among the rocks and sands,
The night-o'ertaken mariner to save.
-- The Lighthouse, Henry Wadsworth Longfellow
On a superficial level, a flashing red beacon is a utilitarian thing. If you look more closely, you will see that it is also a thing of beauty. It is an avatar of the compulsion we all feel to protect, to warn, to guide. The humble beacon is one of the better angels of our nature, sculpted with massive limbs of galvanized steel and eyes of electrically exuberant gallium phosphide. It sends our message out into the world, again, and again, and again.
be careful
be careful
be careful
be careful
be careful
...
## Moving forward by stopping
Posted by Russell on April 02, 2012 at 4:06 p.m.
Just a three weeks after I was sworn in for my term on the City of Davis Safety and Parking Advisory Commission, UC Davis law student Megan Glanville was killed just a few dozen feet from my doorstep. She was out jogging on a foggy morning, and truck coming into town from the county road ran her down in the crosswalk. I never knew Megan, but her death deeply upsets me.
I've been worrying about pedestrian and bike safety ever since my little sister was nearly killed by a careless driver.
I find it extremely frustrating that most people do not look beyond the (usually imagined) behavior of the people involved in an accident like the one that almost killed my sister, or that did kill Megan Glanville. Either they identify with the frustrating experience of driving, and blame the victim, or they side with the law, and place the responsibility at the feet of the operator of the more dangerous vehicle. I will always side with the person who suffered more, but both views are myopic. When someone has been killed in an accident, the question of who was more "right" in that sliver of time is irrelevant. It is worse than irrelevant; it is an insult to the lives of all the people affected.
There are other, far more urgent questions that need to be raised. If you see a problem, the first question you should always ask is, "In what way am I responsible for this?" We are all bound together by bonds of mutual responsibility, and nothing happens among people, good or bad, for which each of us are not in some sense responsible. That is what words like "society," "community," and "civilization" mean. They describe the fact that the bonds that link us together are fundamentally inescapable. There is such a thing as integrity, but there is no such thing as self-reliance. Interdependence is the very essence of what makes us human. And so, if you see something that upsets you, the first thing you should look at is your own role in causing it. Through our choices, we were all present on morning that George Souza killed Megan Glanville. You. Me. Everyone. We all had a hand in it.
Clearly, we failed. You failed. I failed. Someone is dead as a consequence of that failure.
So, let us set aside the choices of George Souza and Megan Glanville, and look at the choices we made that contributed to this terrible thing. They are easy enough to see :
This is the crosswalk where Megan was killed, which is part of a system of roads that belong to the City of Davis. The arrow on the yellow sign is pointing almost directly at the spot. The laws that govern the design of the road are a kaleidoscopic fugue of local, county, state, federal and international regulations. Within that often contradictory matrix of statutes, the city government has a small keyhole of authority within which it may choose what the road looks like and how it works.
From an engineering point of view, it's pretty clear what the problem is. The road on the left is just a stone's throw from the border of the city. Beyond the border, it is a wide county road that cuts a nearly straight line for miles among orchards and farms. When it crosses into the city, this road suddenly plunges into a dense residential neighborhood with no transition whatsoever. The intersection where Megan was killed is the very first intersection an eastbound driver encounters in the City of Davis. So, drivers come in from the county road going at county road speeds, and roar through this intersection where people are trying to cross to the bike path that parallels the road. Add a little darkness and bit of fog, and the accident was basically inevitable.
Why was this intersection designed this way? I don't know. According to the laws and statutes that regulate its engineering, there is nothing particularly wrong with it. But then again, houses that catch fire and burn people alive inside are often built to code. Compliance with the law is not enough. Only thoughtful design can keep people safe, and the absence of that thoughtfulness killed someone.
So, who is to blame? The legislators who wrote the statues describing how intersections should be designed? The engineers whose designs were constrained by those statutes? The City of Davis Public Works Department that built and maintained it? Surely, some of the responsibility falls to them. But not very much. If you've ever driven, walked or bicycled through the intersection of Lake and Russell, then a great deal of the responsibility falls on you. If you've ever felt uncomfortable or unsafe while passing through it, then you knew someone would get hurt there sooner or later.
The Council Chambers are open to the public. The meetings and agendas are available weeks in advance for all to see, at CityOfDavis.org. You can even submit your concerns in writing if you don't have time to come to the meetings. In other words, you had the reason and the means to get this fixed, or at least play a part in getting it fixed, before Megan Glanville was killed. I share in this responsibility; I serve on the commission charged with advising the City Council on these things, and I did not raise this issue either. And I use this intersection several times a day. And I always feel unsafe. It is my fault too.
So, here is what is going to happen. The City Council was asked, and agreed, to take steps to prevent anyone else from getting killed. The proposed changes will add stop signs on Russell Boulevard in both directions, a blinking red light in case drivers don't see the stop signs in the fog, and four new street lights for better illumination overall. It will cost about \$20,000.
This is a much better design. It's impossible to know if it would have saved Megan's life had it been in place in December, but it seems likely that it would have. I strongly support it.
Roads are not natural phenomena. They are public infrastructure, and they are designed and built and maintained in exactly the way the public asks them to be. Let's try to do a better job of holding up our end of that conversation.
## The Dover Train
Posted by Russell on February 14, 2012 at 4:47 a.m.
Well, it's Valentine's Day again. For all you lucky folks who have someone from whom deserts may be extorted, cheers!
As for me, I'm writing this post for the girl on the Dover train who gave me a funny look. Um... Hi.
Don't know if this will work, but stranger things have happened.
## Happy 60th, Mom and Dad!
Posted by Russell on February 10, 2012 at 2:24 p.m.
As is often the case, odd people had odd children. Readers of this blog have no doubt long noted a certain... peculiarity of the author, and hypothesized about the individuals who raised me. I can proudly confirm that they are every bit as odd as I am. Anyway, they both turned sixty in the last couple of weeks (my mom in December, and my dad a few days ago), and I wanted to post a little celebration of our great family tradition of crossgrainedness.
For my mom's birthday, she and my sister came over and slept on the floor of my apartment in Davis, despite the availability of perfectly comfortable and reasonably priced lodgings downtown. For her birthday, we found a fetid puddle of water near Lake Berryessa with some tadpoles in it. She was delighted.
For my dad's birthday, he's celebrating his election to the National Academy of Engineering (the sister organization to the Institute of Medicine. He also got this birthday card from his longtime college friend Bruce Reznick :
Bruce is a professor of mathematics at the University of Illinois. He studies the identities of high-order polynomials. So, this really is the a birthday card only he would think to send.
## Cat replenishment
Posted by Russell on January 22, 2012 at 3:45 p.m.
I hear the internet is running low on pictures of adorable cats. It's raining outside in Davis today. Buzz seems to be hoping that I'll put down the camera and stop playing with ISO settings, and rub his belly. Either that, or he's mugging for the camera. It's hard to tell.
That is all.
## New Equipment Thursday
Posted by Russell on August 25, 2011 at 7:40 p.m.
My vacuum desiccator arrived today, and so naturally I put it to productive use. You know. For science.
Haw! This thing is cool.
Posted by Russell on August 15, 2010 at 4:59 p.m.
My apologies for getting behind on posting my updates from Uzon. After we returned from Uzon, we rested for a day, and then crammed ourselves and our equipment into a van and went to Peratunka for the Biodiversity, Molecular Biology and Biogeochemistry of Thermophiles international workshop, where I was scheduled to give a 20 minute talk.
The speaking docket got shuffled around a lot, and I ended up having to give my talk much earlier than planned. I suppose this is the inevitable downside of procrastination. While I was scrambling to finish it, I didn't have much time for blog updates!
I survived the talk. There were lots and lots of excellent questions, and I have a lot to think about now. Anyway, back to the updates from Uzon.
## Back from Uzon
Posted by Russell on August 10, 2010 at 8:49 p.m.
Panorama overlooking Orange Fields in Uzon Caldera
We just arrived back in Petropavlovsk after a week in the field. I was very sad to leave Uzon, and it was a privilege and an honor of the highest order to have spent those days there.
The expedition was, I think, a great success. We'll know for sure once we're back at our labs and can use more sophisticated methods to examine our samples. I am very confident, though.
It was a bit touch-and-go right at the end. Our high speed centrifuge crapped out last night, just as Sarah was in the middle of the last big run of DNA extractions. The Russian team brought their own centrifuge, but we couldn't run it on our generator. Much to our relief, Albert was able to magically get the thing working again by holding it at just the right angle. They worked through the night to finish processing the samples; I think Albert must have had his thumb wedged under the centrifuge for the entire run.
I'm sorry I wasn't able to send many Twitter updates toward the end of the expedition. Once I had identified my sampling targets, I suddenly had a lot less free time on my hands (and I didn't have much to begin with). Also, I'm sorry for updating in ALL CAPS. Iridium handsets are essentially 1993 technology. Composing text messages is extremely painful, and the battery only lasts long enough to compose two or three of them. This is a pain when you have to recharge on generator power, and the generator only cranks up for a few hours a night, and even then only to power lab equipment for DNA extractions. Hats off to my dad for relaying the messages!
Right now, I'm sitting in a friendly internet cafe in Petropavlovsk where they've let me use their wireless connection. When we arrived at our crowded little apartment, the hot water was broken, and thus no showers yet. A wide selection of interesting geologic samples are wedged under my fingernails, and I think I have wads of some sort of hardened liquid sulfur caked in my hair. The helicopter arrived ridiculously early, and we just barely get everything aboard. As a result, I'm still wearing my field clothes from yesterday, which are splattered with volcanic mud. I may actually be the worst-smelling person in Petropavlovsk. Perhaps it is fortunate that this internet cafe caters mainly to kids playing StarCraft.
I composed blog entries for each day we were in Uzon, and I'll be posting them as soon as I run them past the rest of the team. I also have almost two thousand photos to sort, tag and upload.
That said, I have a correction for one of my Twitter updates. I wrote :
YERTERDAY ALBERT & TEAM WERE CHASED AWAY FROM A SITE BY A BEAR THAT WAS ACTUALY A BUSH IN THE FOG.
Albert pointed out that they were interrupted for a few minutes, but not actually chased away. He stepped forward and shouted see if he bear (or bears) would go away, with his signal torch uncapped and ready. The bears were revealed to be bushes as the wind shifted and created a channel in the mist. It's funny, but given how foggy it was that day, it wasn't actually that surprising. We were at the same site the next day, and were surprised by an actual bear. It wandered pretty close to us before we could actually see it (the full story will come with the article for that day).
A bear interrupting important EisenLab work at Boiling Spring.
Update : Albert also says that I'm wrong about having to wedge his thumb under the centrifuge the whole time. It started working again after shaking it around in the air a bit, and placing it just so on the table. He only had his thumb wedged underneath it for a minute or two to check to see if it was overheating.
## I'm going to Kamchatka!
Posted by Russell on July 19, 2010 at 5:47 p.m.
I just got the reservations for my flight to Petropavlovsk-Kamchatsky for the International Workshop on Biodiversity, Molecular Biology and Biogeochemistry of Thermophiles, hosted by Moscow State University and Winogradsky Institute of Microbiology.
I've been working on the analysis of environmental samples from two sites at Uzon Caldera (about 10,000 Sanger reads from each sequenced at the JGI), and I'm hoping that I'll be able to reprocess the DNA here at the UC Davis Genome Center using some of our high-throughput machines. Licensing and customs restrictions will probably make it impossible to bring my own samples back, but I may be able to entrust them to a colleague with fancier credentials than my own.
I'll be arriving in Petropavlovsk on the 30th of July, with the help of a generous grant from the Carnegie Institution for Science Deep Carbon Observatory.
## Good luck, whoever you are
Posted by Russell on June 08, 2010 at 3:54 p.m.
Last week, I got an urgent call from the National Marrow Donor Program. Somewhere, there is a girl about to go into chemotherapy, and my tissue type matches hers. They needed to run some more detailed tissue typing, and screen for infections diseases. The NMDP sent a kit to the UC Davis student health center (the new building is right next to my house), and I had my blood drawn this morning.
I hate giving blood. They didn't need very much, but I don't get along very well with steel needles. I count it as a major victory that I didn't barf until I got home.
Now we all wait for the results.
Good luck, whoever you are.
## 20s
Posted by Russell on May 31, 2010 at 9:17 p.m.
To steal the idea from John Scalzi, here is the last photograph of me in my 20s.
My twenties; better than my teens. Some good times, and some pretty awful times, and on average kind of meh. If the trend holds, my thirties should be in the tolerable to nice range. Hopefully the underlying process is geometric, and not linear or logarithmic.
Hence the awkward sort of half-smile.
## Holly Allan-Young
Posted by Russell on May 26, 2010 at 10:57 p.m.
Terry Young: She is gone
Sent: 2:54PM
I will miss her terribly.
Posted by Russell on April 28, 2010 at 11:26 a.m.
After six months of using Google Latitude, I've amassed about 7108 location updates, or about 38 a day. It would probably be a lot more if I hadn't managed on occasion to break the GPS or automatic updating by fiddling with the software.
Not surprisingly, I spent most of my time in California, mostly in Davis and the Bay Area, with a few trips to Los Angeles via I-5, the Coast Starlight, and the San Joaquin (the density of points along those routes is indicative of the data service along the way).
The national map shows my trip to visit my dad's family in New Jersey and Massachusetts, as well as a layover in Denver that I'd completely forgotten about.
I have somewhat mixed feelings about this dataset. On one hand, it's very useful to have, and sharing it with my friends and with Google is very useful. It's also cool to have this sort of quantitative insight into my recent past so easily accessible. On the other hand, I'm not particularly happy with the idea that Google controls this data. I chose the word controls deliberately. I don't mind that they have the data -- after all, I did give it to them. As far as I know, Google has been a good citizen when it comes to keeping personal location data confidential. The Latitude documentation makes their policy pretty clear :
### Privacy
Google Location History is an opt-in feature that you must explicitly enable for the Google Account you use with Google Latitude. Until you opt in to Location History, no Latitude location history beyond your most recently updated location if you aren't hiding is stored for your account. Your location history can only be viewed when you're signed in to your Google Account.
You may delete your location history by individual location, date range, or entire history. Keep in mind that disabling Location History will stop storing your locations from that point forward but will not remove existing history already stored for your Google Account.
...
If I delete my history, does Google keep a copy or can I recover it?
No. When you delete any part of your location history, it is deleted completely and permanently within 24 hours. Neither you nor Google can recover your deleted location history.
So, that's what they'll do with it, and I'm happy with that. What bothers me is this: Who owns this data?
This question leads directly to one of the most scorchingly controversial questions you could ask for, and there are profound legal, social, economic and moral outcomes riding on how we answer it. This isn't just about figuring out what coffee shops I like. If you want to see how high the stakes go, buy one of 23andMe's DNA tests. You're giving them access to perhaps the most personal dataset imaginable. In fairness, 23andMe has a very strong confidentiality policy.
But therein lays the problem -- it's a policy. Ambiguous or fungible confidentiality policies are at the heart of an increasing number of lawsuits and public snarls. For example, there is the case of the blood samples taken from the Havasupai Indians for use in diabetes research that turned up in research on schizophrenia. The tribe felt insulted and misled, and sued Arizona State University (the case was recently settled, the tribe prevailing on practically every item).
You can't mention informed consent and not revisit HeLa, the first immortal human cells known to science. HeLa was cultured from a tissue biopsy from Henrietta Lacks and shared among thousands of researchers -- even sold as a commercial product -- making her and her family one of the most studied humans in medical history. The biopsy, the culturing, the sharing and the research all happened without her knowledge or consent, or the knowledge or consent of her family.
And, of course, there is Facebook -- again. Their new "Instant Personalization" feature amounts to sharing information about personal relationships and cultural tastes with commercial partners on an op-out basis. Unsurprisingly, people are pissed off.
Some types of data are specifically protected by statute. If you hire a lawyer, the data you share with them is protected by attorney-client privilege, and cannot be disclosed even by court order. Conversations with a psychiatrist are legally confidential under all but a handful of specifically described circumstances. Information you disclose to the Census cannot be used for any purpose other than the Census. Nevertheless, there are many types of data that have essentially no statutory confidentiality requirements, and these types of data are becoming more abundant, more detailed, and more valuable.
While I appreciate Google's promises, I'm disturbed that the only thing protecting my data is the goodwill of a company. While a company might be full of a lots of good people, public companies are always punished for altruistic behavior sooner or later. There is always a constituency of assholes among shareholders who believe that the only profitable company is a mean company, an they'll sue to get their way. Managers must be very mindful of this fact as they navigate the ever changing markets, and so altruistic behavior in a public company can never be relied upon.
We cannot rely on thoughtful policies, ethical researchers or altruistic companies to keep our data under our control. The data we generate in the course of our daily lives is too valuable, and the incentives for abuse are overwhelming. I believe we should go back to the original question -- who owns this data? -- and answer it. The only justifiable answer is that the person described by the data owns the data, and may dictate the terms under which the data may be used.
People who want the data -- advertisers, researchers, statisticians, public servants -- fear that relinquishing their claim on this data will mean that they will lose it. I strongly disagree. I believe that people will share more freely if they know they can change their mind, and that the law will back them up.
### Update
The EFF put together a very sad timeline of Facebook's privacy policies as they've evolved from 2005 to now. They conclude, depressingly :
Viewed together, the successive policies tell a clear story. Facebook originally earned its core base of users by offering them simple and powerful controls over their personal information. As Facebook grew larger and became more important, it could have chosen to maintain or improve those controls. Instead, it's slowly but surely helped itself — and its advertising and business partners — to more and more of its users' information, while limiting the users' options to control their own information.
## A desirable extinction
Posted by Russell on March 25, 2010 at 3:19 p.m.
Some weeks ago, Buzz (my cat) escaped out my front door while I was carrying my bicycle into the apartment. For ten or twenty minutes, he romped through the ivy and bushes around my apartment while I followed him around rattling a bag of cat treats. Eventually, he let me pick him up and take him back inside. Naturally, he picked up a few fleas. Naturally, they have multiplied.
Oddly, the fleas don't seem to like Neil very much, nor do they like me. It's just poor Buzz that's beset by the nasty little critters.
### Figure 1: A flea.
As it happens, I've been thinking about endogenous metrics for estimating the sampling quality of an environmental shotgun sequencing dataset, and Buzz's little problem presented an opportunity to play with a simplified problem. So, I have decided to make Buzz, or rather his fleas, into a small experiment in ecology. I am going to try to see if I can drive them into extinction.
Now, this is normally what a pet owner does when they discover their pet has contracted some sort of annoying parasite, but I decided to take a more quantitative approach.
### Figure 2: A cat.
It's simple enough to count fleas on a cat, if the cat is willing to cooperate. Buzz loves the flea comb, and will gleefully hop onto the coffee table and wait to be combed if you show it to him. So, in the interest of science, I convinced my roommate to count the number of passes I made with the flea comb and how many fleas I captured (posterity will remember your efforts, Mehdi). Using his tally, I plotted the cumulative number of passes verses the cumulative number of fleas.
### Figure 3: Fleas captured
As expected, it became somewhat more difficult to capture the next flea as more fleas were captured, suggesting a depletion curve. The value of the asymptote should be the actual number of fleas on Buzz at the time, and reaching that number would imply local extinction for the fleas. Of course, there are probably other fleas lurking about that would recolonize Buzz. In principle, if I were to repeat the exercise frequently enough, Buzz would become a sink for fleas, and their migration to his fur would gradually deplete them from the environment.
There are a couple of different ways to model the impact of the combing on the flea population, with various advantages and disadvantages. All we really want to do here is to estimate the value of the asymptote, and so a simple model is probably sufficient. I showed this data to my fried Sharon Shewmake, an economics graduate student. Sharon, after editorializing on the endeavor ("Ew."), suggested this very simple model.
Assume that Buzz is not going to sit still long enough for the fleas to reproduce, for more fleas to migrate to his fur, and that the fleas already on his fur are going to stay put unless captured. Thus, there is a fixed initial population which only changes as a result of capturing fleas. Next, we assume that any given flea is equally likely to be captured on a single pass of the comb. So, the expectation value for number of fleas captured on a single pass is the product of the current population and the probability of capturing a flea.
$N_{i+1}=N_i-pN_i$
where N is the population of fleas and p is the probability of any particular flea being captured on a single pass. One could tart this up a bit by modeling it as a stochastic process and executing a bunch of Monte Carlo trials until the outcomes converge, but that seems like overkill for a simple single variable problem like this. We will put up with the intellectual inconvenience of capturing fractional fleas.
This is a little easier to see if we let N represent the number of fleas remaining on the cat, rather than the number of fleas captured.
$N_{i+1}=qN_i$
If we stretch our credulity far enough to imagine this as a continuous function, we can express it as a differential equation.
$\frac{d N}{d t}=\alpha N$
Sorry if this bothers you. Not only are we extracting fractional fleas, but we are now modeling the combing process as a sort of flea-killing-combine continuously mowing its way through the fur. This is a model, so you shouldn't be surprised to find massless rope and spherical cows. Anyway, it has a nice easy solution.
$N(t)=e^{\alpha t}$
Well, what the heck. This is a decaying function, so let's pluck a minus sign out of the exponential factor, and maybe tack on a scale factor for the initial population.
$N(t)=N_0 e^{-\alpha t}$
While we're at it, why don't we go back to letting the function stand for the number of fleas captured, rather than the fleas on the cat.
$N(t)=N_0(1-e^{-\alpha t})$
This gives us a nice function to use for a linear regression. A little help from scipy, and we find that the initial population is estimated at 39.7 fleas, and the decay factor is 0.011.
### Figure 4: Flea population
I captured 34 fleas, so that means I missed about five or six. In order to be reasonably confident that I'd captured all 39 fleas, I would have had to continued for about 400 passes with the comb, instead of 173. Buzz is a patient cat, but he started to loose interest around 120 passes, and had to be fetched back onto the coffee table a few time times during the last 50 passes. My guess is that 400 passes would require some kind of sedative. On the other hand, he does seem to like Guinness, so there may be something to that.
Science has been served. I'm going to the pet store to buy some flea collars.
## Espresso
Posted by Russell on March 13, 2010 at 3:56 p.m.
A few weeks ago, my dad sent me this really nice espresso machine to cheer me up. Actually, he sent it to me because it was it was his birthday. He's a pretty awesome dad that way -- I only sent him a book.
I'm still getting the hang of getting a decent pull of espresso out of it. I've found that my burr grinder doesn't quite go fine enough for espresso, so I'm going to have to take it apart and see if I can adjust the grinding wheels so they're closer together. Anyway, here is my latest effort :
## Facepaw
Posted by Russell on February 10, 2010 at 3:42 a.m.
Spent the day writing a piece of code I already wrote six months ago. Not sure how I managed to forget. The new code wasn't very good, so I threw it away. Day down the tube.
Even asleep, Neil seems to understand.
## Loss
Posted by Russell on February 03, 2010 at 11:48 p.m.
I normally don't talk a lot about my personal life on my blog, and except for the occasional announcement, I'd like to keep it that way. People's little triumphs and tragedies are mainly interesting to those directly involved, and are at best kind of boring to everyone else. A lot of my friends and family do read this blog, but by and large most of you are strangers or acquaintances. I try to respect that.
Those of you who are close to me know that I'm going through a sad time in my life right now. Those of you who work or study with me have probably noticed that I've not been my usual cheerful self. In deference to the many people who aren't here to read about that, and the fact that I can barely think about it (never mind write about it), I'm not going to discuss what's happened on my blog.
The one thing that has helped has been hearing about all the cool things that other people are doing. So, even though I'm not exactly Mr. Social right now, please don't take that as a sign that I want to be left alone.
On the contrary. Now would be a great time to tell me about whatever is on your mind, especially if it's cool.
To those of you who've been kind enough to treat me like a normal person over the last two weeks despite my melancholy behavior, I owe you guys. Really.
|
2013-05-21 01:39:31
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 6, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22117012739181519, "perplexity": 1980.5796466597865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368699632815/warc/CC-MAIN-20130516102032-00001-ip-10-60-113-184.ec2.internal.warc.gz"}
|
http://neweuroclub.trustlike.tw1.ru/mrs-dalloway-york-notes/
|
# Mrs Dalloway York Notes
am now 15 years old and a teen age cancer survivor I am a volunteer Despite the tumor shrinking medical miracle that • Iowa • Naidu, M.S.; Kamataru, V. (1982), High Voltage Engineering, Tata McGraw-Hill, ISBN 0-07-451786-4 9781292221328 1292221321 $82,280 Werner Herzog’s documentary “My Best Fiend” ruminates nostalgically on the explosive, complicated relationship between Herzog and Klaus Kinski, an actor with whom he collaborated on five special films. The movie provides footage from the sets and final products of each film, so that Herzog might further consider the significance of Kinski’s career and person. We bear witness to what Herzog has referred to as a “psychic connection” between the two men. Kinski was notoriously difficult to deal with on set, not willing to acquiesce to a director’s instructions, threatening to quit the production or hurt all those who defied him; some of Herzog’s footage of Kinski misbehaving is quite damning for the actor. And yet Herzog was strangely able to tame the beast, at least for long enough to get a tremendous performance out of him. In especially tense moments and ego clashes during the filming of “Fitzcarraldo” or “Aguirre: the Wrath of God,” Herzog and Kinski threatened (quite seriously) to kill one another. Both wanted to be in charge, and neither wanted to surrender their power. Still, they were the most intimate of enemies. Their overpowering bond (and their peculiar obsession with each other) is what made their creative partnership so successful. “My Best Fiend” leaves us unsettled: we are torn between thinking both these men are crazy, and thinking they shared an enviable, uncommon camaraderie. https://nikolaassodachanh1992.files.wordpress.com/2020/06/beginning-powerapps-tim-leung-9106.pdf • December 2009 https://rhinbelgnate1976.files.wordpress.com/2020/05/unleashing-your-dog-2230.pdf https://evelynamcmurray83.files.wordpress.com/2020/06/proust-et-le-roman-5041.pdf https://skyannedster82.files.wordpress.com/2020/05/e-boek-pdf-negqvniscs-6689.pdf To meet these requirements, students should choose from the following IFP pathways: https://genestaugustus.files.wordpress.com/2020/05/e-boek-pdf-padethwbgc-3386.pdf https://www.zigya.com/study/book?class=12&board=hbse&subject=English&book=Flamingo&chapter=The+Rattrap&q_type=&q_topic=The+Rattrap&q_category=&question_id=ENEN12025768 https://search.schlowlibrary.org/Record/398661/TOC The fundamental principles of chemical engineering underlie the operation of processes extending well beyond the boundaries of the chemical industry, and chemical engineers are employed in a range of operations outside traditional areas. Plastics, polymers, and synthetic fibres involve chemical-reaction engineering problems in their manufacture, with fluid flow and heat transfer considerations dominating their fabrication. The dyeing of a fibre is a mass-transfer problem. Pulp and paper manufacture involve considerations of fluid flow and heat transfer. While the scale and materials are different, these again are found in modern continuous production of foodstuffs. The pharmaceuticals industry presents chemical engineering problems, the solutions of which have been essential to the availability of modern drugs. The nuclear industry makes similar demands on the chemical engineer, particularly for fuel manufacture and reprocessing. Chemical engineers are involved in many sectors of the metals processing industry, which extends from steel manufacture to separation of rare metals. https://thistlestrose.files.wordpress.com/2020/06/ebook-zehn-krimis-2000-seiten-4384.pdf https://www.britannica.com/biography/Kate-DiCamillo 25 Jan 2021 – 26 Jan 2021 • Berlin, Germany Camera and Electrical Department Peter Ewens • 22.10: The Other Group 14 Elements: Si, Ge, Sn, and Pb Bookmarks One of the most common terms we discuss in evaluating electricity is voltage. A voltage is the difference in potential between two points in an electric field. Voltage gives us an idea of just how much pushing force an electric field has. • Estados Unidos window.modules[“855”] = [function(require,module,exports){var baseGet=require(801),baseSet=require(856),castPath=require(802);function basePickBy(e,a,t){for(var r=-1,s=a.length,b={};++r n?0:n+r),(a=a>n?n:a)<0&&(a+=n),n=r>a?0:a-r>>>0,r>>>=0;for(var o=Array(n);++l =LARGE_ARRAY_SIZE){var h=r?null:createSet(e);if(h)return setToArray(h);c=!1,t=cacheHas,n=new SetCache}else n=r?[]:i;e:for(;++s r||l&&u&&s&&!m&&!c||n&&u&&s||!e&&s||!o)return 1;if(!n&&!l&&!c&&i 1?r[i-1]:void 0,n=i>2?r[2]:void 0;for(s=e.length>3&&”function”==typeof s?(i–,s):void 0,n&&isIterateeCall(r[0],r[1],n)&&(s=i<3?void 0:s,i=1),t=Object(t);++a -1?a[n?r[s]:s]:void 0}}module.exports=createFind; Consider the general equilibrium reaction: • 25.5: Functional Groups • General Test Prep https://khalibyake.files.wordpress.com/2020/06/ebook-true-lovers-4-mister-bodyguard-7982.pdf Enabled West Virginia 2013 impact factor = 0.8023 2018 impact factor = 0.3357 https://kretzschmarjoshue.files.wordpress.com/2020/05/e-boek-pdf-nlrtzjasxy-2483.pdf https://kalvinwhitelock1982.files.wordpress.com/2020/06/e-boek-pdf-uaufxovore-6601.pdf • Imhotep (3) https://lauriannastowman1984.files.wordpress.com/2020/05/jai-deux-mains-jai-deux-pieds-6713.pdf • save • Fill in the Missing Letter https://talonburdex85.files.wordpress.com/2020/05/the-curse-of-deadmans-forest-1253.pdf • researcher Watch the next lesson: https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/electron-configurations-jay-sal/v/noble-gas-configuration-electronic-structure-of-atoms-chemistry-khan-academy?utm_source=YT&utm_medium=Desc&utm_campaign=chemistry • Property https://sittneranavictoria1999.files.wordpress.com/2020/05/e-boek-pdf-xnxqorwiau-6259.pdf https://kerkelmaddylynn1985.files.wordpress.com/2020/06/revival-the-experiment-of-bolshevism-1930-6638.pdf https://rawlsbenita.files.wordpress.com/2020/06/buch-einfuehrung-in-die-regenerative-energiewirtschaft-2731.pdf • World Poetry Day https://www.jstor.org/stable/1185718 • September 2007 https://lucicrosslyn.files.wordpress.com/2020/06/e-boek-pdf-ysttefntkb-3658.pdf https://shielselrey1994.files.wordpress.com/2020/05/e-boek-pdf-awowzsxoia-4141.pdf https://misumiadamari1993.files.wordpress.com/2020/06/ebook-edition-211-toedliche-jagd-8622.pdf 0.0318 M https://herasmylan.files.wordpress.com/2020/06/buch-batman-graphic-novel-collection-superschwer-1845.pdf https://daylanimoustafa97.files.wordpress.com/2020/06/buch-mogelpackung-work-life-blending-6357.pdf 6. Electronic Structure of Atoms The salt effects [4] ( salting in and salting-out) refers to the fact that the presence of a salt which has no ion in common with the solute, has an effect on the ionic strength of the solution and hence on activity coefficients, so that the equilibrium constant, expressed as a concentration quotient, changes. https://www.sparknotes.com/lit/endersgame/ https://binary-coffee.dev/post/como-aplicar-para-github-education-pack Chemistry majors normally fulfill the mathematics requirement by taking three mathematics courses: Cycling https://mehganmotts.files.wordpress.com/2020/06/le-jeu-du-sicle-1010.pdf • Lamps & Lights https://munizjanson.files.wordpress.com/2020/05/pourquoi-le-christianisme-fait-scandale-7482.pdf https://bofingerdaniya94.files.wordpress.com/2020/05/e-boek-pdf-mcgwmjcbmz-9154.pdf 2018 03 20 Submitted2018 03 26 Under Review2018 05 11 In Revision2018 05 21 Manuscript revised2018 05 25 Manuscript Accepted2018 05 26 Accepted Manuscript OnlineThe questions raised by the reviewers are more in-depth and more professional, thanks to the editors and reviewers!After the article was accepted, the speed of Online exceeded my imagination Show Review in Original Language https://www.bbc.com/culture/article/20151204-the-25-greatest-british-novels https://jayseanagricola.files.wordpress.com/2020/06/e-boek-pdf-xwdhmlerwd-7207.pdf https://vishikanarkevicius1993.files.wordpress.com/2020/05/e-boek-pdf-bozpfsbfvl-1314.pdf https://temilayostableford.files.wordpress.com/2020/06/gerontechnology-7180.pdf • Follow us • cherry$131,806 AUD ($92,406 USD) David Tennant and Michael Sheen were due to put on a play in the West End before the production was put on hold because of the pandemic. The play’s director, Simon Evans, is worried that his big brea… https://ajakclaybaugh90.files.wordpress.com/2020/06/e-boek-pdf-jpczoldrqe-8116.pdf Although the kit is not • SSC CHSL Preparation Plan https://loflinrainier90.files.wordpress.com/2020/06/e-boek-pdf-lxwtgegxwm-4261.pdf Chemical Reactions I Lab https://rominnalesnick96.files.wordpress.com/2020/06/ebook-bubsimaus-nachtwanderung-5677.pdf https://glecklerdaryl.files.wordpress.com/2020/05/lumire-de-lair-7198.pdf कार्बनिक रसायन · प्रकाश रसायन • Social Media • Demonstrations https://okamotomelida1984.files.wordpress.com/2020/06/ebook-kooperatives-lernen-grundschule-evangelische-religion-6027.pdf • Equal Rites An acid dissociation constant is a particular example of an equilibrium constant. The dissociation of a monoprotic acid, HA, in dilute solution can be written as Destination, rates & speeds Frictional force https://www.urdu2eng.com/prose-urdu-meanings.html interesting application of a physics engine, and particles are https://cutreraraylen.files.wordpress.com/2020/06/buch-rother-wanderfuehrer-picos-de-europa-3159.pdf • Lab First Aid Kits https://peplinskitane98.files.wordpress.com/2020/06/ebook-zebra-books-sweet-vengeance-5024.pdf • Dinosaur Exhibit • 5.07 Introduction to Biological Chemistry • English Turkish English – Turkish • The King of Staten Island Returns Pete Davidson to Anonymity The graphical result is similar to the particles animation but now we have shapes moving and rotating. Mendeley Data Repository is free-to-use and open access. It enables you to deposit any research data (including raw and processed data, video, code, software, algorithms, protocols, and methods) associated with your research manuscript. Your datasets will also be searchable on Mendeley Data Search, which includes nearly 11 million indexed datasets. For more information, visit Mendeley Data. https://gailundamieon1991.files.wordpress.com/2020/06/buch-a-mothers-reckoning-6244.pdf • Hotplate & Magnetic Stirrer – ISG https://shaelinfrothingham86.files.wordpress.com/2020/06/buch-top-1000-spanisch-niveau-a1-karteikarten-m-lernbox-3762.pdf • Chillers • Wave Behavior Toolkit https://puciochett.files.wordpress.com/2020/05/franois-dassise-4823.pdf • • All Valley Hotels https://dunlopguiliana84.files.wordpress.com/2020/06/e-boek-pdf-etmurhuwyx-3600.pdf https://margarethlenzini1984.files.wordpress.com/2020/06/ebook-die-bauern-von-tver-1023.pdf https://jaytenlarosa.files.wordpress.com/2020/06/the-prospects-for-increasing-the-reuse-of-digital-training-content-6448.pdf Kids can make fizzy bombs, “rockets”, explosive balloons and more with the Playz Explosive Combustion Lab Kit. The kit includes the ingredients necessary to do 25 fun experiments as well as safety goggles(but don’t worry, these aren’t major explosions). The lab guide is great for helping kids understand the how and why of the explosive reactions. Although they’re reasonably tame, it’s still a good idea to use some of the experiments in this kit outside to avoid a mess! • Martin Balsam https://haspermalaquias1999.files.wordpress.com/2020/05/factor-e-els -van-der-luit-5407.pdf https://www.mse.cornell.edu/mse/research/research-topics • Word of the day –This edition includes completely NEW material on: collision detection, 2D physics, casual game physics for Flash games. Chemistry—A European Journal is a truly international society journal with top quality contributions (2018 ISI Impact Factor: 5.16). It publishes a wide range of outstanding Reviews, Minireviews, Concepts, Full Papers, and Communications from all areas of chemistry and related fields. It is published on behalf of Chemistry Europe, an association of 16 European chemical societies, and supported by the Asian Chemical Editorial Societies. Scientist New Zealand Earthquake https://gustincarasco.files.wordpress.com/2020/06/ebook-gegen-staat-und-kapital-fuer-die-revolution-9591.pdf • NCERT Class 10 Solutions CHEM 391 12th Physics Notes In Hindi Rbse (i) Aspartame: It is the most successful and widely used artificial sweetener. It is roughly 100 times as sweet as cane sugar. It is methyl ester of dipeptide formed from aspartic acid and phenylalanine. • Paleontology & Natural History (10) • Weather • Culture Tubes External links [ edit ] • https://alaznemoten.files.wordpress.com/2020/05/useful-photography-12-9409.pdf https://vaniphorng.files.wordpress.com/2020/06/buch-worauf-du-dich-verlassen-kannst-6521.pdf https://www.bath.ac.uk/research-centres/centre-for-mathematical-biology/ Chemistry and Life Radiation Therapy https://kuryoasis1995.files.wordpress.com/2020/06/ebook-gateway-reverse-universe-9947.pdf • Parker’s body chemistry is drastically altered after the accidental bite. • https://demetrionburgardt.files.wordpress.com/2020/05/e-boek-pdf-xfwnrysrgh-9488.pdf • Custom Reactors • बी॰एन॰एफ॰ अभिज्ञापक वाले विकिपीडिया पृष्ठ • GRE • Lab Bootie & Hair Net Dispensers • USA • Mother Up! (2013–14) https://tinleethibeau.files.wordpress.com/2020/06/buch-hilfe-bei-schilddruesenerkrankungen-6440.pdf • 081 Spanish Parts of a Car and Gas Station https://brielcryderman.files.wordpress.com/2020/06/buch-finn-und-frieda-feiern-den-sommer-4917.pdf I = F Δt = ΔM https://mckenareep.files.wordpress.com/2020/06/ebook-hitlers-ostkrieg-und-die-deutsche-siedlungspolitik-9602.pdf • practical and logical Warning: If you are one of those people who wear a What Would Jesus Do? bracelet and believes that the purple Teletubby really is gay? https://writingexplained.org/grammar-dictionary/round-character https://dillysiaosi1991.files.wordpress.com/2020/06/review-of-the-us-climate-change-science-programs-draft-synthesis-and-assessment-product-24-2562.pdf https://putzermosheh1998.files.wordpress.com/2020/06/spiritualit-du-diaconat-7225.pdf https://rydolphskylin88.files.wordpress.com/2020/05/e-boek-pdf-tgalqppolw-5310.pdf x = xo + vot + 1/2at2 | (Vƒ)2 = (Vo)2 + 2ax (e) 2.86 mol Co(NH 3) 6Cl 3 https://guillamareshard.files.wordpress.com/2020/05/e-boek-pdf-rkgvcjiziy-8936.pdf https://nikolaassodachanh1992.files.wordpress.com/2020/06/le-mystre-de-la-reine-amaya-3014.pdf https://kishawnsturdevant1985.files.wordpress.com/2020/06/tradimenti-harold-pinter.pdf • Company This was the most amazing (but saddest) book ever. I would recommend it for 13+, but I read it when I was 11, and I handled fine. I would not say that most 11 year olds should read it, but I know about the “stuff” that happens so it was okay for me. This was the best book of my life soooo read it • Polymers https://arionnapennigton1995.files.wordpress.com/2020/05/le-persan-par-les-mots-et-les-textes-7043.pdf https://www.investopedia.com/terms/b/blue_ocean.asp Periodic Trends in Electron Affinity • Tales Of Hans Christian Andersen Anti -depressant drugs are needed to counteract the effect of depression. These drugs inhibit enzymes catalysing the degradation of the neurotransmitter, noradrenaline. As a result, the important neurotransmitter is slowly metabolised and then it can activate its receptor for longer periods of time. IT Management View All • Custom Cut-To-Length Tubing https://rahimafivecoat85.files.wordpress.com/2020/06/e-boek-pdf-dwdhghuqlv-7896.pdf • Reviewers https://academicjournals.org/journal/JCEMS https://kaydrakegley92.files.wordpress.com/2020/06/e-boek-pdf-xilzyzacjb-1386.pdf ► Multiply the mass of the object by its specific heat capacity and by the amount of temperature change. This formula is written H = mcΔ T, where ΔT means “change in temperature.” For this example, this would be 500g x 4.19 x 20, or 41,900 joules. • Pharmaceutical https://mccloughankohei87.files.wordpress.com/2020/06/e-boek-pdf-ymrqpagohm-9129.pdf int main (void) https://www.amazon.com/Paulo-Coelho/e/B000AQ3HB8 https://duwhitebelany90.files.wordpress.com/2020/06/e-boek-pdf-mdulztrqmc-9654.pdf Get it here. • • Foundations of Everyday Leadership Lisbon Give some macromolecules which are selected as targets for the drug. “God does not play dice with the universe; He plays an ineffable game of His own devising, which might be compared, from the perspective of any of the other players [i.e. everybody], to being involved in an obscure and complex variant of poker in a pitch-dark room, with blank cards, for infinite stakes, with a Dealer who won’t tell you the rules, and who smiles all the time.” • Blog: Nucleus Occupational employment projections are developed for all states by Labor Market Information (LMI) or individual state Employment Projections offices. All state projections data are available at www.projectionscentral.com . Information on this site allows projected employment growth for an occupation to be compared among states or to be compared within one state. In addition, states may produce projections for areas; there are links to each state’s websites where these data may be retrieved. CareerOneStop Lisa failed chemistry last semester. Premise [ edit ] https://necosgroi.files.wordpress.com/2020/06/ebook-diplom-de-taekwondo-und-karate-im-olympiawettlauf-7121.pdf https://bladesdabria1989.files.wordpress.com/2020/05/e-boek-pdf-bzdohaxxrz-9533.pdf https://jennelleniayla1983.files.wordpress.com/2020/05/e-boek-pdf-ppunhyxpmt-5671.pdf https://samuelkatis.files.wordpress.com/2020/05/e-boek-pdf-unozmuemtw-3524.pdf Once again, another product from Thames & Kosmos that ranks among the best science kits for kids. Unlike its higher-end counterparts, the C1000 has just 125 experiments. However, this is still a very complete set, recommended for ages 10 and up. https://jerikurschner87.files.wordpress.com/2020/05/baudouin-en-filigrane-9037.pdf • Engineering Colleges in India Gutright Instructions https://amyskanes.files.wordpress.com/2020/06/personne-nest-lintrieur-de-rien-9846.pdf https://chattnaythen1990.files.wordpress.com/2020/06/buch-mit-allen-sinnen-9147.pdf Paolo Calabresi __________________________________ https://deyarminshanon1987.files.wordpress.com/2020/05/e-boek-pdf-pvyapytajz-8588.pdf • AI for Everyone https://beaureguardrom1984.files.wordpress.com/2020/06/e-boek-pdf-dxmdokgcyf-5229.pdf https://www.nytimes.com/2018/11/08/magazine/gillian-flynn-women.html https://giegericharlie1999.files.wordpress.com/2020/06/ebook-politische-rhetorik-im-nationalsozialismus-die-aussen-und-6393.pdf https://nikolaassodachanh1992.files.wordpress.com/2020/06/au-pays-des-mille-collines-coll-trimestre–les-maternelles-7001.pdf © Stanford University, Stanford, California 94305. though all are variations on the same theme. In games, you can find https://gouldshady1999.files.wordpress.com/2020/05/the-book-of-honu-1052.pdf 2.5 The Periodic Table https://calicuttjaxom.files.wordpress.com/2020/05/e-boek-pdf-ixwczihdbn-4171.pdf https://alohnamirsch.files.wordpress.com/2020/06/la-voix-de-ma-mre-2023.pdf A Closer Look Charge-Transfer Color “use strict”;var format=require(980);module.exports=pino;var _console=global.console||{},stdSerializers={req:mock,res:mock,err:asErrValue};function pino(e){(e=e||{}).browser=e.browser||{};var r=e.browser.transmit;if(r&&”function”!=typeof r.send)throw Error(“pino: transmit option must have a send function”);var i=e.browser.write||_console;e.browser.write&&(e.browser.asObject=!0);var t=e.serializers||{},s=Array.isArray(e.browser.serialize)?e.browser.serialize.filter(function(e){return”!stdSerializers.err”!==e}):!0===e.browser.serialize&&Object.keys(t),l=e.browser.serialize;Array.isArray(e.browser.serialize)&&e.browser.serialize.indexOf(“!stdSerializers.err”)>-1&&(l=!1);”function”==typeof i&&(i.error=i.fatal=i.warn=i.info=i.debug=i.trace=i),!1===e.enabled&&(e.level=”silent”);var n=e.level||”info”,a=Object.create(i);a.log||(a.log=noop),Object.defineProperty(a,”levelVal”,{get:function(){return”silent”===this.level?1/0:this.levels.values[this.level]}}),Object.defineProperty(a,”level”,{get:function(){return this._level},set:function(e){if(“silent”!==e&&!this.levels.values[e])throw Error(“unknown level “+e);this._level=e,set(o,a,”error”,”log”),set(o,a,”fatal”,”error”),set(o,a,”warn”,”error”),set(o,a,”info”,”log”),set(o,a,”debug”,”log”),set(o,a,”trace”,”log”)}});var o={transmit:r,serialize:s,asObject:e.browser.asObject,levels:[“error”,”fatal”,”warn”,”info”,”debug”,”trace”]};return a.levels=pino.levels,a.level=n,a.setMaxListeners=a.getMaxListeners=a.emit=a.addListener=a.on=a.prependListener=a.once=a.prependOnceListener=a.removeListener=a.removeAllListeners=a.listeners=a.listenerCount=a.eventNames=a.write=a.flush=noop,a.serializers=t,a._serialize=s,a._stdErrSerialize=l,a.child=function(i){if(!i)throw new Error(“missing bindings for child Pino”);var l=i.serializers;if(s&&l){var n=Object.assign({},t,l),a=!0===e.browser.serialize?Object.keys(n):s;delete i.serializers,applySerializers([i],a,n,this._stdErrSerialize)}function o(e){this._childLevel=1+(0|e._childLevel),this.error=bind(e,i,”error”),this.fatal=bind(e,i,”fatal”),this.warn=bind(e,i,”warn”),this.info=bind(e,i,”info”),this.debug=bind(e,i,”debug”),this.trace=bind(e,i,”trace”),n&&(this.serializers=n,this._serialize=a),r&&this._logEvent.bindings.push(i)}return o.prototype=this,new o(this)},r&&(a._logEvent=createLogEventShape()),a}function set(e,r,i,t){var s=Object.getPrototypeOf(r);r[i]=r.levelVal>r.levels.values[i]?noop:s[i]?s[i]:_console[i]||_console[t]||noop,wrap(e,r,i)}function wrap(e,r,i){var t;(e.transmit||r[i]!==noop)&&(r[i]=(t=r[i],function(){for(var s=Date.now(),l=new Array(arguments.length),n=Object.getPrototypeOf&&Object.getPrototypeOf(this)===_console?_console:this,a=0;a -1&&l in i&&(e[s][l]=i[l](e[s][l]))}function bind(e,r,i){return function(){var t=new Array(1+arguments.length);t[0]=r;for(var s=1;s “],[10097,”>”],[65310,”>”],[10100,”{“],[65371,”{“],[10101,”}”],[65373,”}”],[8314,”+”],[8330,”+”],[65291,”+”],[8316,”=”],[8332,”=”],[65309,”=”],[65281,”!”],[8252,”!!”],[8265,”!?”],[65283,”#”],[65284,”$”],[8274,”%”],[65285,”%”],[65286,”&”],[8270,”*”],[65290,”*”],[65292,”,”],[65294,”.”],[8260,”/”],[65295,”/”],[65306,”:”],[8271,”;”],[65307,”;”],[65311,”?”],[8263,”??”],[8264,”?!”],[65312,”@”],[65340,”\\”],[8248,”^”],[65342,”^”],[65343,”_”],[8275,”~”],[65374,”~”]]),module.exports=ASCIIFolder; https://monserratlamana1984.files.wordpress.com/2020/05/cat-215-9918.pdf Although it involves some mature content, it is all rather innocent. What makes it such a phenomenal read is that unlike many books involving cancer, the characters are so alive that the cancer is in some ways irrelevant to the story. They’re so much more than just kids with cancer, because they have so much more depth than that. The journey they go on is one that will leave you laughing through the tears. Contents: • Sam Treiman You may want to contact the webmaster and inform them • Where’s Neil When You Need Him? https://picardkaydyn.files.wordpress.com/2020/06/ebook-struik-lifestyle-cooking-with-the-kosher-butchers-wife-9926.pdf https://emerikwindsheimer1982.files.wordpress.com/2020/06/ebook-interpretation-der-parabel-eine-kaiserliche-botschaft-von-1299.pdf standard heat of formation – calor de formacion estandar • Baby Signing https://jonathanbarshaw86.files.wordpress.com/2020/06/buch-eu-datenschutz-grundverordnung-dsgvo-7620.pdf • Gifts Expand sub-menu information described below to the designated agent listed below. If Varsity Tutors takes action in response to https://grasynkilbert.files.wordpress.com/2020/06/e-boek-pdf-ydqlooowue-2141.pdf Chapter 9 केमिस्ट्री की शब्दोत्पत्ति – भाषा वैज्ञानिक दृष्टि [ संपादित करें ] • Latest News • Weather & Climate https://www.jstor.org/stable/23329416 • IIT JEE https://noaahgrosskopf.files.wordpress.com/2020/05/e-boek-pdf-ewjvwcyggr-1050.pdf • Packaging https://acrabold1992.files.wordpress.com/2020/06/buch-das-matrikelbuch-der-universitaet-ingolstadt-landshut-1434.pdf https://savielposley.files.wordpress.com/2020/05/return-on-learning-4084.pdf • Water decomposition apparatus • Cookie statement https://mehganmotts.files.wordpress.com/2020/06/rebelles-7278.pdf https://mekiyacabon1997.files.wordpress.com/2020/06/ebook-mitternachtsflut-8780.pdf https://daraelfenbein96.files.wordpress.com/2020/06/e-boek-pdf-nujvrgbupz-1235.pdf https://naylaasponaugle1987.files.wordpress.com/2020/05/a-varse-5352.pdf https://dariecarroca98.files.wordpress.com/2020/06/e-boek-pdf-aqzekdrnvf-1277.pdf CareerOneStop includes hundreds of occupational profiles with data available by state and metro area. There are links in the left-hand side menu to compare occupational employment by state and occupational wages by local area or metro area. There is also a salary info tool to search for wages by zip code. • https://idalyzmortensen.files.wordpress.com/2020/06/markt-en-mededingingsrecht-9222.pdf https://gottoilianna.files.wordpress.com/2020/05/kevin-volans-pieces-for-precocious-pianists-piano-3647.pdf https://zeldaschnickel1989.files.wordpress.com/2020/06/e-boek-pdf-kedpsifxni-2806.pdf https://annaleesevelis1991.files.wordpress.com/2020/05/e-boek-pdf-skdjmdcipz-2718.pdf • soil biochemistry 20. Applying Kirchhoff’s Rules: Examples & Problems • research and develop new materials and procedures to help make production processes more efficient • Apps & Games I think this is an honest account of the process of writing Good Omens. It was fairly easy to keep track of because of the way we sent discs to one another, and because I was Keeper of the Official Master Copy I can say that I wrote a bit over two thirds of Good Omens. However, we were on the phone to each other every day, at least once. If you have an idea during a brainstorming session with another guy, whose idea is it? One guy goes and writes 2,000 words after thirty minutes on the phone, what exactly is the process that’s happening? I did most of the physical writing because: • https://trayshawnfellenz.files.wordpress.com/2020/06/la-vieille-guimbarde-3884.pdf • Graphic Novels Read more from the Study Guide This section contains 585 words • A A Milne https://www.amazon.com/Steelheart-Reckoners-Brandon-Sanderson/dp/0385743564 • July 2014, issue 7 CHEM 181L • Molecules https://bajwajosearmando1996.files.wordpress.com/2020/05/deeltafels-1-t-m-10-1455.pdf https://packerfeinen1992.files.wordpress.com/2020/06/9783132410640-palliativmedizin-1000-fragen.pdfhttps://gangeljc.files.wordpress.com/2020/05/journalistieke-verbeelding-2751.pdf https://trojeckiwaldo1999.files.wordpress.com/2020/05/e-boek-pdf-jgcpilmclh-2747.pdf Cites / Doc. (3 years) https://www.routledge.com/The-Routledge-History-of-Literature-in-English-Britain-and-Ireland-3rd/Carter-McRae/p/book/9780415839747 Not exactly! It looks like you forgot to square the strength of the current (4 amperes) in the formula P = I² x R. Multiply the square of 4 amperes by the current’s resistance. Try another answer… https://ihsanschiedler1990.files.wordpress.com/2020/06/ebook-diplom-de-ich-habe-mein-leben-zurueck-2121.pdf Help Center I• ISO 9000 • View All هیچکس نمیخواد تو رو بکشه.؛ • Refraction and Lenses • Magical Mathematics https://dennonylla89.files.wordpress.com/2020/06/e-boek-pdf-qtkwaygici-6989.pdf test tube holder – soporte de tubo de ensayo Warner Bros. Responds To J.K. Rowling’s Shocking Commitment to Being Transphobic https://annarogowski1984.files.wordpress.com/2020/05/the-palgrave-macmillan-dictionary-of-political-thought-6229.pdf The chemicals which are added to food to improve its shelf-life, taste, odour and appearance are called food additives. If you only remember one thing about the mole, let it be this: The mole is a bridge between the life-size stuff we experience every day and the universe of the very small. Step 1: Write out the reaction https://emikobroz.files.wordpress.com/2020/06/la-louve-de-mars-9505.pdf © • आपके लिए रहेंगे खास ये फ्री प्रोजेक्ट मैनेजमेंट कोर्सेज और सर्टिफिकेशन्स10 hrs ago • 156 Spanish Food Preparation Vocabulary • https://mehganmotts.files.wordpress.com/2020/06/objects-and-identity-7826.pdf https://annayahcarmena93.files.wordpress.com/2020/05/e-boek-pdf-oitsyhvrsi-1258.pdf https://zephankanae1992.files.wordpress.com/2020/05/e-boek-pdf-lctuqtivbr-6816.pdf Core modules � Concentration of ions in solution is constant. To read more about the Joule, click here For Further Reading ISO 4 https://buskkacia98.files.wordpress.com/2020/05/lichtjaren-3368.pdf • Political Science https://alesialanzillo1999.files.wordpress.com/2020/06/e-boek-pdf-natzwnypcf-9151.pdf • Kung Fu Panda 3 https://nikolaassodachanh1992.files.wordpress.com/2020/06/e-boek-pdf-wqmleeyyhy-6979.pdf Google Scholar https://palovickcian.files.wordpress.com/2020/06/buch-kommissar-jennerwein-band-3-niedertracht-8637.pdf https://perrinejawaun.files.wordpress.com/2020/06/ebook-vertvolta-press-be-vigilant-but-not-afraid-3668.pdf B = L I = μ 2 r I {\displaystyle B=LI={\frac {\mu }{2r}}I} UK edition https://www.youtube.com/watch?v=cvEJ4eSskKM We assigned X as the solubility of the Ca 2+ which is equal Posters • Green Technology • Blog $pH = pK_w – pOH \nonumber$ • 25. Alcohols, Ethers and Haloalkanes Elementary Quantitative Analysis • Acceptor https://ludlammanveer1983.files.wordpress.com/2020/05/e-boek-pdf-aclhwhcvry-6895.pdf window.modules[“904″] = [function(require,module,exports){var flatten=require(84),overRest=require(866),setToString=require(867);function flatRest(e){return setToString(overRest(e,void 0,flatten),e+””)}module.exports=flatRest; • Food additives such as sweetening agents, preservatives, flavours, edible colours, nutritional supplements and antioxidants are added to the food to make it palatable, attractive and also add nutritive values. https://www.consilium.europa.eu/en/documents-publications/library/library-blog/posts/fire-and-fury-inside-the-trump-white-house-by-michael-wolff/ • ICT ► https://storlievedaant1993.files.wordpress.com/2020/06/le-canard-qui-ne-savait-sur-qui-compter-4492.pdf • Research in Bioanalytical Studies (1 – 12)
|
2020-07-12 02:43:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29786962270736694, "perplexity": 11123.945460611241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657129517.82/warc/CC-MAIN-20200712015556-20200712045556-00428.warc.gz"}
|
https://plainmath.net/29734/consider-binomial-experiment-with-15-trials-and-probability-45-success
|
# Consider a binomial experiment with 15 trials and probability 0.45 of success on
Consider a binomial experiment with 15 trials and probability 0.45 of success on a single trial.
(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)
(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)
You can still ask an expert for help
• Questions are typically answered in as fast as 30 minutes
Solve your problem for the price of one coffee
• Math expert for every subject
• Pay only if we can solve it
coffentw
Step 1
Solution: It is given here that a random variable say x follows the binomial distribution with parameters
The binomial probability function is:
$P\left(X=x\right)=\frac{n!}{\left(n-x\right)!x!}{p}^{x}{\left(1-p\right)}^{n-x};x=0,1,2,..,n$
Step 2
(a) Use the binomial distribution to find the probability of exactly 10 successes.
Answer: It is required to find:
$P\left(x=10\right)$
Using the binomial distribution function:
$P\left(x=10\right)=\frac{15!}{\left(15-10\right)!10!}{0.45}^{10}{\left(1-0.45\right)}^{15-10}$
$=3003×0.000340506×0.050328438$
$=0.051$
Therefore, the probability of exactly 10 successes is 0.051
Step 3
(b) Use the normal distribution to approximate the probability of exactly 10 successes.
The mean and standard deviation of the random variable x is:
$\mu =np=15×0.45=6.75$
$\sigma =\sqrt{np\left(1-p\right)}=\sqrt{15×0.45\left(1-0.45\right)}=1.92678$
It is required to find:
$P\left(x=10\right)$
Using the continuity correction factor, the above probability can be written as:
$P\left(x=10\right)=P\left(10-0.5
$=P\left(9.5
Using the z-score formula:
$P\left(9.5
$=P\left(1.4272
$=P\left(z<1.9462\right)-P\left(z<1.4272\right)$
Now using the excel functions:
$P\left(9.5
The excel functions are:
$=NORMSDIST\left(1.4272\right)=0.9232$
$=NORMSDIST\left(1.9462\right)=0.9742$
Therefore, Using the normal distribution to approximate the probability of exactly 10 successes is 0.051
|
2022-05-22 01:23:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 27, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9295048117637634, "perplexity": 512.8162672201646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543264.49/warc/CC-MAIN-20220522001016-20220522031016-00713.warc.gz"}
|
https://www.hackerearth.com/practice/algorithms/graphs/graph-representation/practice-problems/algorithm/2-way-attack-1/
|
"2" Way Attack
Tag(s):
## Medium
Problem
Editorial
Analytics
The blue army is on the move for the attack on red army.There are infinite number of infinite types of bombs numbered 1,2.... available with blue army to destroy the base camp of the red army.The general of the blue army wants the destruction of the area quick. General ordered his secret services to produce the blue print of the area and necessary details for the attack to take place without any problem.The blue print generated gave the following information: The red army base camp has N tents connected by M gas lines,containing a special gas which prevents explosion to some extent.Each tent has to be targeted with a bomb of some type. General also got a surprising intelligence that the base is designed in such a way that if a gas pipe connects two tents ,then these two tents cannot be bombed with bombs who's type differ by greater than or equal to 2.Also if there is no gas pipe connecting any pair of tents then they have to be bombed with bomb types who's type differ by greater than equal to 2.All this made the task of the general complicated.The general also wants minimum number of his bombs to be used for destruction.Help the general in bombing the blue army in minimum number of types of bombs required. If the base camp is designed in such a way it cannot be bombed print "NO"(without quotes).
Input:
The First line of input consists of T, the number of test cases. The next line has two integers N and M indicating number of tents and gas lines. The next M lines have two integers U and V indicating a gas line between tents U and V.
Output:
For each print "YES"(without quotes) in first line if bombing is possible and then minimum number of bombs required. Else print "NO"(without quotes) if the base is indestructible.
Constraints:
1<=T<=3
1<=N,M,U,V<=100000
SAMPLE INPUT
2
3 2
1 2
2 3
4 4
1 2
2 3
3 4
4 1
SAMPLE OUTPUT
YES
3
NO
Explanation
None
Time Limit: 1.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB
Marking Scheme: Marks are awarded when all the testcases pass.
Allowed Languages: Bash, C, C++, C++14, Clojure, C#, D, Erlang, F#, Go, Groovy, Haskell, Java, Java 8, JavaScript(Rhino), JavaScript(Node.js), TypeScript, Julia, Kotlin, Lisp, Lisp (SBCL), Lua, Objective-C, OCaml, Octave, Pascal, Perl, PHP, Python, Python 3, R(RScript), Racket, Ruby, Rust, Scala, Swift, Swift-4.1, Visual Basic
## CODE EDITOR
Initializing Code Editor...
## This Problem was Asked in
Challenge Name
EPIPHANY 7.1
OTHER PROBLEMS OF THIS CHALLENGE
• Data Structures > Advanced Data Structures
|
2019-01-16 15:53:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2016838937997818, "perplexity": 5207.1069884122035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583657555.87/warc/CC-MAIN-20190116154927-20190116180927-00554.warc.gz"}
|
http://mathhelpforum.com/algebra/155262-scientific-notation-print.html
|
# Scientific notation
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• Sep 5th 2010, 10:07 AM
pychon
Scientific notation
Problem is:
$(5.0028 x 10^-7) + (6.14 x 10^-10)$
If done on the calculator the result is $5.00894 x 10^-7 or 5.0090 x 10^-7$
I must be doing something incorrectly...
-Add $5.0028 + 6.14 = 11.1428$
-Add $10^-7 + 10^-10 = 10^-17$
So I get $1.1143 x 10^-16$
Anyone know what I did incorrectly?
• Sep 5th 2010, 10:24 AM
undefined
Quote:
Originally Posted by pychon
Problem is:
$(5.0028 x 10^-7) + (6.14 x 10^-10)$
If done on the calculator the result is $5.00894 x 10^-7 or 5.0090 x 10^-7$
I must be doing something incorrectly...
-Add $5.0028 + 6.14 = 11.1428$
-Add $10^-7 + 10^-10 = 10^-17$
So I get $1.1143 x 10^-16$
Anyone know what I did incorrectly?
A proper way to do it would be
$(5.0028 \cdot 10^{-7}) + (6.14 \cdot 10^{-10}) = 5002.8 \cdot 10^{-10} + 6.14 \cdot 10^{-10}$
$= 5008.94 \cdot 10^{-10} = 5.00894\cdot10^{-7}$
If working in significance arithmetic this would need to be rounded accordingly.
• Sep 5th 2010, 11:34 AM
pychon
brilliant... :)
another i'm stumbling on...
$\frac {(7.309 x 10^{-1})^2}{5.9843(2.0536 x 10^{-9})}$
doing the math:
$53.421481 x 10^{-2}$
or
$0.53421481$
$5.9843(0.0000000020536) = 1.228935848 x 10^{-8}$
or
$0.00000001228935848$
answer comes out to be $43469706.81$ , sn with my math $4.347 x 10^{7}$
calculator calculates: $4.397 x 10^{-12}$
• Sep 5th 2010, 11:44 AM
undefined
Quote:
Originally Posted by pychon
brilliant... :)
another i'm stumbling on...
$(7.309 x 10^1)^2 / 5.9843(2.0536 x 10^-9)$
doing the math:
$53.421481 x 10^-2$
or
$0.53421481$
$5.9843(0.0000000020536) = 1.228935848 x 10^-8$
or
$0.00000001228935848$
answer comes out to be $43469706.81$ , sn with my math $4.347 x 10^7$
calculator calculates: $4.397 x 10^-12$
Please clarify whether the question is to find
$\displaystyle \frac{(7.309\cdot10^1)^2}{5.9843}\cdot2.0536\cdot1 0^{-9}$
or
$\displaystyle\frac{(7.309\cdot10^1)^2}{5.9843\cdot 2.0536\cdot10^{-9}}$
• Sep 5th 2010, 11:51 AM
pychon
erm, is there a manual for properly coding the math questions? i have no clue... anyway its the second you've listed, but in parentheses
$frac{(7.309 x 10^{1})^{2} / 5.9843(2.0536 x 10^{-9})}$
• Sep 5th 2010, 12:04 PM
undefined
Quote:
Originally Posted by pychon
erm, is there a manual for properly coding the math questions? i have no clue... anyway its the second you've listed, but in parentheses
$frac{(7.309 x 10^{1})^{2} / 5.9843(2.0536 x 10^{-9})}$
What do you mean, "but in parentheses"? Adding parentheses to the expression you indicated has no effect because multiplication is associative.
For help with mathematical typesetting, see LaTeX Help Subforum.
You need to realize that $1.23 \cdot 10^{4}$ means nothing more than 1.23 multiplied by 10^4. Compute as you would any other expression involving multiplication and division. Remember rules for exponents, $x^a \cdot x^b = x^{a+b}$ and $\displaystyle\frac{x^a}{x^b}=x^{a-b}$.
• Sep 5th 2010, 12:24 PM
pychon
Yep, I located LaTex in the faq and finally now in the forum.. mods need to fix the link.
Anyway, the math is being multipled as noted... the work is all there, but suspect your suggesting not to solve the value of the power until the final result.
Also, in parentheses the denomenator is $5.9843(2.0536 x 10^{-9})$, with what you had $5.9843\cdot2.0536\cdot10^{-9}$ wouldn't that be incorrect? If there was a problem of $(10^{-5})^{3}$, wouldn't that be $10^{-15}$ not $10^{-2}$?
$\frac {(7.309 x 10^{-1})^2}{5.9843(2.0536 x 10^{-9})}$
• Sep 5th 2010, 12:43 PM
undefined
Quote:
Originally Posted by pychon
Also, in parentheses the denomenator is $5.9843(2.0536 x 10^{-9})$, with what you had $5.9843\cdot2.0536\cdot10^{-9}$ wouldn't that be incorrect?
$5.9843\cdot2.0536\cdot10^{-9} = 5.9843(2.0536\cdot10^{-9})$
What you wrote here
Quote:
Originally Posted by pychon
If there was a problem of $(10^{-5})^{3}$, wouldn't that be $10^{-15}$ not $10^{-2}$?
does not apply since there is no power raised to another power.
Yes you should set the powers of 10 aside for efficiency purposes.
$\displaystyle\frac{(7.309\cdot10^1)^2}{5.9843\cdot 2.0536\cdot10^{-9}}= \frac{7.309^2}{5.9843\cdot2.0536}\cdot\frac{10^2}{ 10^{-9}}$
Continue from there or say if you didn't see how I did that.
• Sep 5th 2010, 01:13 PM
pychon
I see where you're going, but must be doing something wrong. If the problem is entered entirely into my calculator is comes out to $4.397\cdot10^{-12}$ and I'm not getting anything close to that.
Maybe you could check a previous problem I had, is the answer correct?
$\frac{(5.19\cdot 10^{-6})(8.3\cdot10^{5})}{2.07\cdot10^{4}}
$
$2.1\cdot10^{-4}$
• Sep 5th 2010, 01:16 PM
undefined
Quote:
Originally Posted by pychon
I see where you're going, but must be doing something wrong. If the problem is entered entirely into my calculator is comes out to $4.397\cdot10^{-12}$ and I'm not getting anything close to that.
Maybe you could check a previous problem I had, is the answer correct?
$\frac{(5.19\cdot 10^{-6})(8.3\cdot10^{5})}{2.07\cdot10^{4}}
$
$2.1\cdot10^{-4}$
You are entering it into your calculator wrong. Remember order of operations.
Yes rounded to two significant figures the last calculation is correct.
• Sep 5th 2010, 01:24 PM
pychon
I don't know how I could be entering it into the calculator incorrectly when entering the entire problem into it... as noted before the answer I received was 43469706.81 and doing my math above and following order of operations (para, powers, mult/div)
• Sep 5th 2010, 01:42 PM
undefined
Quote:
Originally Posted by pychon
I don't know how I could be entering it into the calculator incorrectly when entering the entire problem into it... as noted before the answer I received was 43469706.81 and doing my math above and following order of operations (para, powers, mult/div)
The correct answer is indeed 43469706.81 (rounded). I see you had this all along, but using the method outlined above is more efficient (at least for most of these types of problems that you would need to work out on paper). You must be typing something wrong into your calculator because $4.397\cdot10^{-12}$ is not correct.
• Sep 5th 2010, 01:53 PM
pychon
Quote:
Originally Posted by undefined
The correct answer is indeed 43469706.81 (rounded). I see you had this all along, but using the method outlined above is more efficient (at least for most of these types of problems that you would need to work out on paper). You must be typing something wrong into your calculator because $4.397\cdot10^{-12}$ is not correct.
That's what confused me... not sure how $4.397\cdot10^{-12}$ displayed, but I calculated it several times and got that and been trying to figure out how it came out to $10^{-12}$, the math I did came out to $10^{7}$... writing out the long way too... but I like your suggestion... the more efficient the better. Though $4.347\cdot10^{7}$ should be correct?
• Sep 5th 2010, 02:02 PM
undefined
Quote:
Originally Posted by pychon
That's what confused me... not sure how $4.397\cdot10^{-12}$ displayed, but I calculated it several times and got that and been trying to figure out how it came out to $10^{-12}$ over what math I did $10^{7}$... writing out the long way too... but I like your suggestion... the more efficient the better. Though $4.347\cdot10^{7}$ should be correct?
Rounded to four significant figures, yes $4.347\cdot10^{7}$ is correct.
Maybe I'm preaching to the choir but for example if you were presented with this problem
$\displaystyle \frac{6\cdot10^{-60}}{3\cdot10^{-50}}$
you would definitely not want to expand numerator and denominator separately, but rather use the method given above and immediately see that the answer is $2\cdot10^{-10}$.
As to your calculator woes, it's kind of hard to "diagnose over the phone" (meaning it would be much easier if I could see in person), but I suppose if you described what kind of calculator you have and what keystrokes you used we might get to the bottom of it.
• Sep 5th 2010, 02:02 PM
yeKciM
Quote:
Originally Posted by pychon
That's what confused me... not sure how $4.397\cdot10^{-12}$ displayed, but I calculated it several times and got that and been trying to figure out how it came out to $10^{-12}$ over what math I did $10^{7}$... writing out the long way too... but I like your suggestion... the more efficient the better. Though $4.347\cdot10^{7}$ should be correct?
correct answer is $\displaystyle \frac{(5.19\cdot 10^{-6})(8.3\cdot10^{5})}{2.07\cdot10^{4}} = 2.081 \cdot 10^{-4}$
to get it by calculator :
enter 5.16 then pres "exp" than 6 than "-" (+/- probably is written) than multiply "*" then enter 8.3 than "exp" than 5 pres "=" , than divede ( / ) than 2.07 again "exp" and 4.... press "=" again (you can do it without "=" before dividing but if you need to write down on paper .... ) than you will get 0.000208101 so now press "FIX" or something like that (depending on type of calculator , can be F -> E .... or something ) than you will get $2.081 \cdot 10^{-4}$ :D:D:D
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
|
2017-05-24 22:31:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 73, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8457246422767639, "perplexity": 1257.27317219946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607862.71/warc/CC-MAIN-20170524211702-20170524231702-00370.warc.gz"}
|
https://bobthechemist.com/2018/04/throwback-thursday-brockport-chemistry-take-2/
|
# Throwback Thursday – Brockport Chemistry take 2
Last week, I posted an early photo of a Chemistry lab from Brockport. Not to be outdone, my wife Rozenn (historian of the Western Monroe Historical Society at the Morgan Manning House) found this picture in one of her books:The caption for the picture reads:
The [Brockport] Chemistry Laboratory: The 1899 yearbook describes the chemistry laboratory as “one of the best appointed in the state, having ample table room for 50 students at one time … The department has over $2,500 worth of physical apparatus, over 2,500 stereopticon slides and some 3,000 specimens.” That$2,500 in instrumentation would be a bit over 70 thousand in today’s dollars, and I’m happy to say that our department has far more instrumentation than that. The reference to thousands of specimens and stereopticon slides got me thinking about what was taught in Chemistry 118 years ago (hey that’s one year for every element on the periodic table). A quick web search brought me to this article, (which is behind a paywall if you don’t have access to ACS journals) that reviews an historical Chemistry textbook from 1809. It was written by Jane Marcet to “… provide women with a method of educating themselves in chemistry …” and uses a conversational style that is not seen in contemporary instructional materials. This #ThrowbackThursday has me thinking about revisiting some teaching styles (to justify procrastinating on that pile of grading for one more day).
|
2018-06-22 12:54:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24706266820430756, "perplexity": 3193.623057469766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864482.90/warc/CC-MAIN-20180622123642-20180622143642-00078.warc.gz"}
|
https://math.stackexchange.com/questions/2961992/modelling-a-drone-in-3d-finding-the-z-component-of-the-thrust-vector-knowing-pi?noredirect=1
|
# Modelling a drone in 3D: finding the z component of the thrust vector knowing pitch (about x) and roll (about y) angles.
I have been trying to create the model of a drone for fun and when calculating the linear dynamics I need to find out the Z component of the thrust vector.
Basically, in 3D space the thrust vector is always perpendicular to my drone, hence its angle and therefore z component changes with the pitch and the roll of the drone. Pitch is the angle about the x axis and roll is the angle about the y axis with $$0$$ degrees being the vector pointing straight up.
I also know the total magnitude of the thrust vector, basically the total thrust generated by my drone. From there I can easily get the thrust vectors' components in the x and y directions by multiplying the vector's magnitude to, respectively, sin(roll) and sin(pitch).
What I have been doing until now is then using the magnitude to write that: $$M^2=X^2+Y^2+Z^2$$ with M being the magnitude and X, Y Z the components of the thrust vector. From this I rewrote,
$$Z=\sqrt{M^2-X^2-Y^2}$$
Again here: $$X=sin(roll)*M$$ and $$Y=sin(pitch)*M$$ with roll the angle about the y axis and pitch the angle about the X axis.
The problem is that this fails in a lot of cases as $$M^2-X^2-Y^2$$ becomes negative. I am sure I did something basic pretty wrong here but I cannot get my head around it, I would greatly appreciate your help if anyone knows what I am doing wrong.
Thank you!
• Some inspiration for finding the right equations for X and Y can be found here. – Jens Oct 19 '18 at 16:43
A positive pitch creates an angle between Z and the thrust vector (looking in the 2D ZY coordinate system), the z coordinate of this vector then becomes: $$Z=M*cos(Pitch)$$
$$Z=M*cos(Roll)$$
So eventually, putting this together, we get: $$Z=M*cos(Pitch)*cos(Roll)$$
|
2019-09-16 08:16:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6962397694587708, "perplexity": 155.41135325585918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572516.46/warc/CC-MAIN-20190916080044-20190916102044-00056.warc.gz"}
|
https://homework.zookal.com/questions-and-answers/a-194-109-c-charge-has-coordinates-x--0-783345693
|
1. Science
2. Physics
3. a 194 109 c charge has coordinates x 0...
# Question: a 194 109 c charge has coordinates x 0...
###### Question details
A 1.94 10-9 C charge has coordinates x = 0, y = −2.00; a 2.88 10-9 C charge has coordinates x = 3.00, y = 0; and a -5.40 10-9 C charge has coordinates x = 3.00, y = 4.00, where all distances are in cm. Determine magnitude and direction for the electric field at the origin and the instantaneous acceleration of a proton placed at the origin.
(a) Determine the magnitude and direction for the electric field at the origin (measure the angle counterclockwise from the positive x-axis).
(b) Determine the magnitude and direction for the instantaneous acceleration of a proton placed at the origin (measure the angle counterclockwise from the positive x-axis).
|
2021-05-18 05:12:51
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8526421189308167, "perplexity": 643.496966980396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989820.78/warc/CC-MAIN-20210518033148-20210518063148-00235.warc.gz"}
|
https://mathematica.stackexchange.com/questions/73805/goto-and-compile-to-c
|
# Goto and Compile to C
When I use Goto inside compile and use CompilationTarget->"C" I get error message:
fun = Compile[{},
Goto["here"];
Label["here"];
, CompilationTarget -> "C"]
or
fun2 = Compile[{{myt, _Real}, {i, _Integer}},
Module[{t = myt},
Do[If[j >= i, Goto["one"], Goto["two"]];
Label["one"];
t = j;
Break[];
Label["two"];, {j, 1, 10}];
t
], CompilationTarget -> "C"
]
SymbolicCToCCodeString::unk: An unknown element, CCodeGeneratorPrivatebuildLine[10,CCodeGeneratorPrivateobjData\$1791,CompiledFunctionToolsPrivategetInstruction[10,{0,one}]], was found when generating code >>
CCompilerDriverCreateLibrary::cmperr: Compile error: C:\Users\myusername\AppData\Roaming\Mathematica\ApplicationData\CCompilerDriver\BuildFolder\d985623-6852\Working-number-6852-6272-1\compiledFunction0.c(104) : error C2065: 'CUnknownElement' : undeclared identifier >>
Without compiling to C it works well.
• Maybe it doesn't support goto. It should always be possible to rephrase a program without goto though. – Szabolcs Feb 11 '15 at 16:41
• here mathematica.stackexchange.com/questions/1096/… it says it is compilable. How can I for example rephrase fun2 without Goto? – MOON Feb 11 '15 at 16:43
• worth reiterating the issue is specific to the CompilationTarget->"C" option. Works fine for other (or no) target specified. For your example you can just put the code blocks inside the If. – george2079 Feb 11 '15 at 17:08
• @george2079 It works when I put the code block inside If. Thank you. – MOON Feb 11 '15 at 17:22
• @yashar Yes, it was clear from your question that it's compilable to byte code. I meant to say that that doesn't necessarily mean that it's compilable to C. You could ask Wolfram support about it though. – Szabolcs Feb 11 '15 at 18:00
Almost 3 years later.. The problem lies in the Label instruction. What happens when you have code that contains labels and gotos is that the Goto instructions are successfully converted to valid jump op-codes. The labels, however, are just, well, labels. They don't even have a meaning, because jumps in compiled code are converted to jumps to a line number. Therefore, they are turned into no-ops. this can be seen in this example:
fun = Compile[{{a, _Real, 0}},
Module[{x = 1., xp = 0., begin, end},
Label[begin];
If[Abs[xp - x] < 10^-8, Goto[end]];
xp = x;
x = (x + a/x)/2;
Goto[begin];
Label[end];
x]
]
Look at fun // InputForm and you will find {0, begin} which means op-code 0 for "nothing". When a C function is to be constructed from this compiled function, every op-code is converted into its C form. Like this:
<< CompiledFunctionTools
CompiledFunctionToolsPrivategetInstruction[0, {16, 0, 1}]
(* Instruction[Times, Register[Real, 1], {Register[Real, 0]}] *)
Unfortunately, it seems to be an oversight that exactly this is not implemented for the noop code. Since it cannot be turned into a valid instruction, the C-compiler will never get a valid instruction list. Look at the error
Can we fix this? Surely. The question is what is the next best instruction for a noop instruction? Well, jump to the next line sounds reasonable.
To understand why this works you have to know that jumps to labels are turned into jumps to line numbers. So we don't really need the labels at all at this stage anymore.
Unprotect[CompiledFunctionToolsPrivategetInstruction];
CompiledFunctionToolsPrivategetInstruction[line_, {0, _}] :=
CompiledFunctionToolsPrivategetInstruction[line, {3, 1}]
Instruction {3,1} is a jump (3) to the line: current line + 1.
fun = Compile[{{a, _Real, 0}},
Module[{x = 1., xp = 0., begin, end},
Label[begin];
If[Abs[xp - x] < 10^-8, Goto[end]];
xp = x;
x = (x + a/x)/2;
Goto[begin];
Label[end];
x], CompilationTarget -> "C"
]
fun[2]
(* 1.41421 *)
`
|
2019-12-14 07:07:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30861344933509827, "perplexity": 4733.561858888198}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540585566.60/warc/CC-MAIN-20191214070158-20191214094158-00096.warc.gz"}
|
https://www.physicsforums.com/threads/formulating-the-kinetic-energy-of-an-object-helical-motion.953647/
|
# Formulating the kinetic energy of an object - helical motion
• I
Hi guys, I need your support to formulate the kinetic energy of an object:
- having mass m [Kg]
- rotating with angular velocity o [rad/sec] referred to an axis t [m] distant (and parallel) to the symmetry axis of the object
- moving along the direction of its symmetry axis with a costant velocity w [m/sec]
We can say that the motion describes an Helix.
Now, is it possible to write the kinetic energy making a sum of the rotating energy plus the translating energy, as Ek = 1/2*I*o^2 + 1/2*m*w^2 ? With I the inertia of the object calculated through the Huygens-Steiner Theorem for a parallel axis.
Please consider any object you like (sphere, cylinder..). It´s clear that the peripheral velocity v=o*t [m/sec] and w [m/sec] are ortogonal to each other, so what is the consequent formulation for the kinetic energy?
Ennio
Last edited by a moderator:
jfizzix
Gold Member
In general, the kinetic energy of any object can always be broken up into two terms. The first is the kinetic energy of the center of mass, and the second is the kinetic energy of the object relative to its center of mass. The reason this is so is because the location of any point in an object can be written as the vector sum of the position of the center of mass of the object, and the position of this point, relative to the center of mass.
In this case, you would first find the velocity of the center of mass $\vec{v}_{CM}$, and use its magnitude square to get the kinetic energy (with the other factors).
$KE_{CM} = \frac{1}{2} m |\vec{v}_{CM}|^{2}$
The kinetic energy of a rigid object relative to its center of mass is described with its angular velocity $\vec{\omega}$ relative to the center of mass, which requires knowing the moment of intertia tensor relative to the center of mass:
$KE_{ relative}=\frac{1}{2} \vec{\omega}\cdot\mathbf{I}_{CM}\cdot\vec{\omega}$
Since the moment of inertia is a tensor (i.e., a matrix), we can take dot products on both the left and right side, where on the left is multiplication by a row vector, and on the right is multiplying by a column vector. Here, $\vec{\omega}$ is a vector pointing along the axis of rotation (in the direction given by the right hand rule), and with magnitude given by angular velocity in radians per second.
Please see my Sketch below. Can we break up the kinetic energy to put it than together? Or it makes no sense.
KE = KEm + KErel
KE = 1/2*m*w^2 + (1/2*m*r^2 + m*r^2) (FYI the rotation axis was not meant the symmetry axis of the object)
Can it represent the total EK, or is it exactly as you have written in your previuous comment?
By the way sorry I do not yet know how to add ormulas
Thanks again
In general, the kinetic energy of any object can always be broken up into two terms. The first is the kinetic energy of the center of mass, and the second is the kinetic energy of the object relative to its center of mass. The reason this is so is because the location of any point in an object can be written as the vector sum of the position of the center of mass of the object, and the position of this point, relative to the center of mass.
In this case, you would first find the velocity of the center of mass $\vec{v}_{CM}$, and use its magnitude square to get the kinetic energy (with the other factors).
$KE_{CM} = \frac{1}{2} m |\vec{v}_{CM}|^{2}$
The kinetic energy of a rigid object relative to its center of mass is described with its angular velocity $\vec{\omega}$ relative to the center of mass, which requires knowing the moment of intertia tensor relative to the center of mass:
$KE_{ relative}=\frac{1}{2} \vec{\omega}\cdot\mathbf{I}_{CM}\cdot\vec{\omega}$
Since the moment of inertia is a tensor (i.e., a matrix), we can take dot products on both the left and right side, where on the left is multiplication by a row vector, and on the right is multiplying by a column vector. Here, $\vec{\omega}$ is a vector pointing along the axis of rotation (in the direction given by the right hand rule), and with magnitude given by angular velocity in radians per second.
#### Attachments
• 15.5 KB Views: 402
A.T.
Ennio said:
KE = 1/2*m*w^2 + (1/2*m*r^2 + m*r^2)
Doesn't look right to me. The parameters o and t don't even show up. The units of the second term are wrong. Is the object a solid cylinder, or just a cylindrical shell?
#### Attachments
• 15.5 KB Views: 358
It´s a solid cylinder and your are right:
KE = 1/2*m*w^2 + 1/2*(1/2*m*r^2 + m*r^2)*o^2
where I = 1/2*m*r^2 + m*r^2 is the inertia
Is the formulation correct? Can we sum up these two kinetic terms?
Doesn't look right to me. The parameters o and t don't even show up. The units of the second term are wrong. Is the object a solid cylinder, or just a cylindrical shell?
A.T.
KE = 1/2*m*w^2 + 1/2*(1/2*m*r^2 + m*r^2)*o^2
where I = 1/2*m*r^2 + m*r^2 is the inertia
I still see no t in there.
I still see no t in there.
typing error --> Now KE = 1/2*m*w^2 + 1/2*(1/2*m*r^2 + m*t^2)*o^2 : )
Please is the formulation/concept correct ?Can we sum up these two kinetic Terms? A.T. thanks in adv.
A.T.
typing error --> Now KE = 1/2*m*w^2 + 1/2*(1/2*m*r^2 + m*t^2)*o^2 : )
Please is the formulation/concept correct ?Can we sum up these two kinetic Terms? A.T. thanks in adv.
Try it out. Dissolve the bracket, and combine the terms related to the two components of the COMs linear velocity. Compare with the formula given by @jfizzix .
Try it out. Dissolve the bracket, and combine the terms related to the two components of the COMs linear velocity. Compare with the formula given by @jfizzix .
Hi @A.T. , acc. to @jfizzix --> Kinetic terms separated: KEm = 1/2*m*(v^2+w^2) = 1/2*m*t^2*o^2 + 1/2*m*w^2 Joule where v=0*t m/sec
and KEr = 1/2 * 1/2*m*r^2 * o^2 = 1/4*m*r^2*o^2 Joule
It´s the same compared to my calculation, except the fact that I sum the terms!
KE = 1/2*m*w^2 + 1/2*(1/2*m*r^2 + m*t^2)*o^2 =
= 1/2*m*w^2 + 1/4*m*r^2*o^2 + 1/2*m*t^2*o^2
My last question is: makes sense to sum the kinetic Terms in order to write KEtot = KEm + KEr ?
Thanks again
A.T.
|
2021-02-26 19:28:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7321287989616394, "perplexity": 492.668947401408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357935.29/warc/CC-MAIN-20210226175238-20210226205238-00358.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-5-section-5-5-dividing-polynomials-exercise-set-page-386/65
|
## Introductory Algebra for College Students (7th Edition)
Divide each term in the polynomial by the monomial. Subtract the powers of the variable to simplify each term. (4x$^2$-6x)$\div$x=$\frac{4x^2}{x}$-$\frac{6x}{x}$=4x$^{(2-1)}$-6(1)=4x-6 Check the answer. x(4x-6)=4x(x)-6(x)=4x$^2$-6x
|
2018-10-16 06:59:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8892616033554077, "perplexity": 2609.7588630432087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510019.12/warc/CC-MAIN-20181016051435-20181016072935-00231.warc.gz"}
|
https://mgi.gov/content/center-theoretical-and-computational-materials-science-ctcms
|
# Center for Theoretical and Computational Materials Science (CTCMS)
### Mission
#### The Center's mission is to support the Material Measurement Laboratory's mission in materials measurement and data delivery by:
• developing, solving, and quantifying materials models using state-of-the-art computational approaches;
• creating opportunities for collaboration where CTCMS can make a positive difference by virtue of its structure, focus, and people;
• developing powerful new tools for materials theory and modeling and accelerating their integration into industrial research.
### Active Working Groups
• Diffusion Working Group on High Throughput Analysis of Multicomponent Multiphase Diffusion Data
• OOF: Object-Oriented Finite Element Analysis of Real Material Microstructures Working Group
• FiPy: A Finite Volume PDE Solver Using Python
• Interatomic Potential Repository Project
• Micromagnetic Modeling ($\mu$MAG) Working Group
• The Materials Digital Library
### Working Group Archives
• Solder Interconnect Geometry and Reactive Wetting code archive
• Phase Field Modeling Tools simulation archive
## Navigate to Other Activities by Strategic Goal
Data and Computational Tools for Advanced Materials Design: Structural Materials Applications - Cobalt Based Superalloys Innovation in High Energy Diffraction Microscopy Adds New Insights to Material Deformation and Failure Rational Design of Advanced Polymeric Capacitor Films Multidisciplinary University Research Initiative (MURI) The Nanoporous Materials Genome Center Center of Excellence on Integrated Materials Modeling (CEIMM) Center for Hierarchical Materials Design (CHiMaD) The Center for Materials in Extreme Dynamic Environments (CMEDE) Center of Materials in Extreme Dynamic Environments (CMEDE) QMCPACK PRedictive Integrated Structural Materials Science (PRISMS) Center DOE EERE Fuel Cell Technologies Office Database Multidisciplinary University Research Initiative: Managing the Mosaic of Microstructure
Innovation in High Energy Diffraction Microscopy Adds New Insights to Material Deformation and Failure Center of Materials in Extreme Dynamic Environments (CMEDE) Joint Center for Artificial Photosynthesis (JCAP) PRedictive Integrated Structural Materials Science (PRISMS) Center Materials Data Curation System Data and Computational Tools for Advanced Materials Design: Structural Materials Applications - Cobalt Based Superalloys DOE EERE Fuel Cell Technologies Office Database AFRL, NIST, and NSF Announce Materials Science and Engineering Data Challenge Awardees Center for Hierarchical Materials Design (CHiMaD) Automatic Flow for Materials Discovery (AFLOW) Development and application of innovative methods for quantification of hexavalent chromium in soils Center for Theoretical and Computational Materials Science (CTCMS) The Materials Project Innovative methods to identify critical and/or strategic elements from unconventional domestic sources
The Materials Project Center of Materials in Extreme Dynamic Environments (CMEDE) Center of Excellence on Integrated Materials Modeling (CEIMM) Multidisciplinary University Research Initiative: Managing the Mosaic of Microstructure Joint Center for Energy Storage Research (JCESR) QMCPACK AFRL, NIST, and NSF Announce Materials Science and Engineering Data Challenge Awardees The Brilliance of Diamonds PRedictive Integrated Structural Materials Science (PRISMS) Center Center for Hierarchical Materials Design (CHiMaD) The Center for Materials in Extreme Dynamic Environments (CMEDE) The Nanoporous Materials Genome Center
Center of Excellence on Integrated Materials Modeling (CEIMM) PRedictive Integrated Structural Materials Science (PRISMS) Center Multidisciplinary University Research Initiative: Managing the Mosaic of Microstructure Automatic Flow for Materials Discovery (AFLOW) Center for Hierarchical Materials Design (CHiMaD) Center of Materials in Extreme Dynamic Environments (CMEDE) Rational Design of Advanced Polymeric Capacitor Films Multidisciplinary University Research Initiative (MURI) Joint Center for Energy Storage Research (JCESR) The Materials Project
|
2019-04-25 16:40:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25125429034233093, "perplexity": 14118.937377669845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578727587.83/warc/CC-MAIN-20190425154024-20190425180024-00198.warc.gz"}
|
http://physics.aps.org/synopsis-for/10.1103/PhysRevB.78.121303
|
# Synopsis: Spin injection with half-metals: A perfect match
The efficient injection of polarized spins from magnetic materials into semiconductors, a prerequisite for spintronics applications, is a formidable challenge. With ferromagnetic ${\text{Co}}_{2}\text{FeSi}$ it is now possible to achieve a spin injection efficiency of close to 50%.
The efficient injection of a spin-polarized current from a magnetic material into a semiconductor is one of the important prerequisites for spintronics. With dilute magnetic semiconductors, or traditional ferromagnets such as Ni, Fe, Co, and their alloys, it is possible to achieve fairly high spin injection efficiencies. However, often the spin polarization is weak or the interfaces are difficult to engineer and control.
In a Rapid Communication appearing in Physical Review B, Manfred Ramsteiner and collaborators from the Paul-Drude-Institut in Berlin discuss an important development in spin injection using another class of materials, called Heusler alloys, some of which have long been predicted to be half-metals [1]. Half-metals, which are extremely rare in nature, have 100% spin polarization at the Fermi level, and therefore only pass a current in which the electrons are polarized either “up” or “down.”
Up to now, attempts to use Heusler alloys as spin injectors have not been promising because the alloy becomes disordered at the interface with the semiconductor and the spin polarization is greatly reduced. By better understanding the alloy-semiconductor interface, Ramsteiner et al. have reached a spin injection efficiency of at least 50% from ${\text{Co}}_{2}\text{FeSi}$ into GaAs. This is the highest efficiency achieved so far with a Heusler alloy.
${\text{Co}}_{2}\text{FeSi}$ is unusual in many respects—there are indications that it indeed might be half-metallic, it remains ferromagnetic up to 1100 K and has the largest magnetic moment among the Heusler compounds. Most importantly, it has a crystal structure that matches perfectly with that of III-V semiconductors, which allows for fabrication of high-quality interfaces. – Ashot Melikyan
[1] de Groot et al., Phys. Rev. Lett. 50, 2024 (1983).
### Announcements
More Announcements »
Spintronics
## Previous Synopsis
Interdisciplinary Physics
## Next Synopsis
Atomic and Molecular Physics
## Related Articles
Magnetism
### Synopsis: Controlling Magnetism by Electricity
Small voltages can control the magnetic properties of thin films at room temperature. Read More »
Semiconductor Physics
### Viewpoint: Spin Transport Goes Ballistic
The injection of spins into a high-mobility two-dimensional electron gas is unexpectedly efficient, suggesting that new theories may be needed to describe spin transport in such systems. Read More »
Spintronics
### Viewpoint: Putting a New Spin on Heat Flow
Researchers demonstrate, for the first time, the spin Peltier effect, in which a spin current in a metal causes heat flow in an adjoining magnet. Read More »
|
2015-05-05 22:23:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.4749344289302826, "perplexity": 2290.453060993171}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430457008123.23/warc/CC-MAIN-20150501051008-00077-ip-10-235-10-82.ec2.internal.warc.gz"}
|
https://en.wikipedia.org/wiki/Hurwitz_problem
|
# Hurwitz problem
In mathematics, the Hurwitz problem, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables.
There are well-known multiplicative relationships between sums of squares in two variables
${\displaystyle (x^{2}+y^{2})(u^{2}+v^{2})=(xu-yv)^{2}+(xv+yu)^{2}\ ,}$
(known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as multiplicativity for the norms on the complex numbers, quaternions and octonions respectively.[1]:1–3[2]
The Hurwitz problem for the field K is to find general relations of the form
${\displaystyle (x_{1}^{2}+\cdots +x_{r}^{2})\cdot (y_{1}^{2}+\cdots +y_{s}^{2})=(z_{1}^{2}+\cdots +z_{n}^{2})\ ,}$
with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xiyj.[3]:127 We call a triple (rsn) admissible for K if such an identity exists.[1]:125 Trivial cases of admissible triples include (rsrs). The problem is uninteresting for K of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition.[1]:137
Hurwitz posed the problem in 1898 in the special case r = s = n and showed that, when coefficients are taken in C, the only admissible values (nnn) were n = 1, 2, 4, 8:[3]:130 his proof extends to any field of characteristic not 2.[1]:3
The "Hurwitz–Radon" problem is that of finding admissible triples of the form (rnn). Obviously (1, nn) is admissible. The Hurwitz–Radon theorem states that (ρ(n), nn) is admissible over any field where ρ(n) is the function defined for n = 2uv, v odd, u = 4a + b, 0 ≤ b ≤ 3, as ρ(n) = 8a + 2b.[1]:137[3]:130
Other admissible triples include (3,5,7)[1]:138 and (10, 10, 16).[1]:137
|
2017-09-22 09:20:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8988635540008545, "perplexity": 515.1533073309942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688926.38/warc/CC-MAIN-20170922074554-20170922094554-00085.warc.gz"}
|
http://math.stackexchange.com/questions/78456/find-inside-a-large-circle-the-maximum-number-of-small-circles-placed-60-degre
|
# Find, inside a large circle, the maximum number of small circles placed 60 degrees to each other and
... starts with a small circle in the center of the large circle.
The above picture shows a program I wrote to actually draw the circles out. But you can see that this method does not yield maximum number of blue circles. There are still spaces around the red circle.
The method i used is to draw blue circle "rings" starting from the center outwards. i.e move out in the blue arrow direction for one circle diameter, then go around in the red arrow direction, then repeat next ring in the blue arrow direction.
Anyone can share a smarter method? Thank you all. I need only to calculate the number, but if there is a systematic way to draw will be better.
-
How about you just fill a grid of circles and compute an algorithm that removes the wrong ones? – Patrick Da Silva Nov 3 '11 at 3:46
Some interesting picures are at www2.stetson.edu/~efriedma/cirincir and at hydra.nat.uni-magdeburg.de/packing/cci Close packing is not always optimal – Ross Millikan Nov 3 '11 at 4:00
@Patrick because when there is large difference in big and small circle diameter, it is not that efficient counting one by one. I was hoping to just have a formula/theory to calculate the max, but I end up drawing (and counting) one by one because it appeared easier to begin solving. – Jake Nov 3 '11 at 4:11
@Ross I saw those links before I posted. The application for the small circles is to house heating elements. So the packing has to be uniform. – Jake Nov 3 '11 at 4:13
@Jake: then I think Carl's solution is a good one. The worry would be that there will be slop between the small circles and the big one that might allow them to shift. – Ross Millikan Nov 3 '11 at 4:40
If your goal is to make a program, then considering you already have that done, it seems like the easiest strategy would just be to: 1. Add on more rings until you know that every circle in a new ring will be outside the main circle. 2. Iterate through the small circles, removing all of the circles who's centers are further than R-r from the main circle, where R and r are the radii of the large and small circles respectively.
That however assumes that you can't fit more circles in by translating your entire set of small circles to the side a bit more. If you want to make sure you have the maximum, you might have to do some more fudging.
-
if the radius between the large and small circle is big, I have to draw a lot more rings since the blue hexagon will have sides further to the red circumference. Using (2), I don't know when to stop drawing rings. A method to calculate the perdicular distance from center to edge of blue hexagon could help, but i don't think it will be conclusive. And imho (1) does not work because rings closer to circumference will not be complete rings, cut at the blue hexagon corners. – Jake Nov 3 '11 at 3:55
in any case, main objective is to calculate the max. The drawing is just to visually verify. – Jake Nov 3 '11 at 3:56
I will try you suggestion later today. Thanks. – Jake Nov 3 '11 at 5:19
I figured how the perpendicular distance from center to the side of hexagon and used your suggestion to cull the blue circles that extends outside of the red circumference. So the drawing works now! Thanks. But I still wonder if there is a direct formula to calcualate this. – Jake Nov 3 '11 at 6:01
|
2014-07-23 16:45:21
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.828592836856842, "perplexity": 709.3871603866932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997880800.37/warc/CC-MAIN-20140722025800-00216-ip-10-33-131-23.ec2.internal.warc.gz"}
|
https://valutabrmt.web.app/40766/16216.html
|
# "Matrix elasticity directs stem cell lineage specification". Cell. 126 (4): 677–89. "Doing good science: Authenticating cell line identity" (PDF). Cell Notes.
Furthermore if you want a concrete example of a matrix whose square is the identity but not itself a simple matrix consider for example this one: $$\begin{bmatrix}\frac{1}{2} & \frac{3}{4} \\ 1 & -\frac{1}{2}\end{bmatrix}$$ These matrices are called involuntory
0.10 matrix inversion lemma (sherman-morrison-woodbury) using the above results for block matrices we can make some substitutions and get the following important results: (A+ XBXT) 1 = A 1 A 1X(B 1 + XTA 1X) 1XTA 1 (10) jA+ XBXTj= jBjjAjjB 1 + XTA 1Xj (11) where A and B are square and invertible matrices but need not be of the Python program to print an identity matrix : In this tutorial, we will learn how to print an identity matrix in python. A matrix is called identity matrix if all of its diagonal elements from the upper left corner to bottom right corner is 1 and all other elements are 0. Identity Matrices. A special kind of matrix tha t has its main diagonal cells filled with ones(1s) and the rest of the cells filled with zeros. Here is what a 3×3 identity matrix looks like: In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.It also serves as the additive identity of the additive group of × matrices, and is denoted by the symbol or —followed by subscripts corresponding to the dimension of the matrix as the context sees fit. If you believe that you have become a victim of identity theft, the Federal Trade Commission (FTC) advises you to take immediate steps to protect yourself from further problems that may arise. These steps include calling the companies where Having a sense of identity is important because it allows people to stand out as individuals, develop a sense of well-being and importance, and fit in with Having a sense of identity is important because it allows people to stand out as ind Identive News: This is the News-site for the company Identive on Markets Insider © 2021 Insider Inc. and finanzen.net GmbH (Imprint).
Hämtad från identity matrix, enhets matris, identitets matris. if and only if, om och endast om. image, bild. inconsistent (system), olösbart (system).
While we say “the identity matrix”, we are often talking about “an” identity matrix. For any whole number $$n$$, there is a corresponding $$n \times n$$ identity matrix.
## numpy.identity¶ numpy.identity (n, dtype=None, *, like=None) [source] ¶ Return the identity array. The identity array is a square array with ones on the main diagonal. Parameters n int. Number of rows (and columns) in n x n output. dtype data-type, optional. Data-type of the output. Defaults to float. like array_like
27 Mar 2019 How many different ways are there to create an identity matrix in R? This was an interesting little challenge set by Guillaume Nicoulaud on Recall that a square matrix with 1's on the diagonal (starting from the top-left down to the bottom-right) and 0's everywhere else is called an identity matrix. The identity matrix is an n x n diagonal matrix with 1's in the diagonal and zeros everywhere else.
### 9 Sep 2014 I would like to assemble an identity matrix to interact with others assembled matrices. Second, How can I create a diagonal matrix that has ones
Such a matrix is of the form given The identity matrix is always a square matrix. While we say “the identity matrix”, we are often talking about “an” identity matrix. For any whole number $$n$$, there is a corresponding $$n \times n$$ identity matrix. These matrices are said to be square since there is always the same number of rows and columns. The identity property of multiplication states that when 1 is multiplied by any real number, the number does not change; that is, any number times 1 is equal to itself. The number "1" is called the multiplicative identity for real numbers.
Grab a 5- to 10-pound medicine ball and kneel on the floor with your knees hip-width apart. Lengthen your spine and pr Helpful tips for discovering your identity. Here are some ways to embark on a journey of self-discovery, from a fellow explorer. Read full profile Whether you’re questioning your identity or just haven’t taken the time to develop your own i There are various causes of identity crisis, including improper upbringing, lack of affirmation and unpleasant past experiences. Other factors that lead to There are various causes of identity crisis, including improper upbringing, lack of The term identity crisis refers to an inability to achieve an identity or to struggle with finding an identity. Many people having an identity crisis feel like they don't know who they are, what they want, or what makes unique. Identity cri Identity politics are broadly defined, but they typically involve an individual who bases his identity on social categories and divisions.
Herbert felix död
matrix-docker-ansible-deploy. Bevaka DOMAIN/fullchain.pem /matrix/ssl/config/live/riot.DOMAIN/privkey.pem proxy /_matrix/identity matrix-msisd:8090 {.
The identity array is a square array with ones on the main diagonal.
Won kursi
filosofie kandidatexamen socialt arbete
holistiskt synsätt 1177
etikett kläder
sykes sveg
### Learn what the Identity and Zero Matrix are by looking at free maths videos and example questions. Study the free resources during your math revision and pass
The identity matrix, denoted , is a matrix with rows and columns. The entries on the Investigation: Multiplying by the identity matrix. Try a few multiplication problems involving the appropriate identity Connections to the real Properties of Identity Matrix 1) It is always a Square Matrix These Matrices are said to be square as it always has the same number of rows and 2) By multiplying any matrix by the unit matrix, gives the matrix itself.
Fruktpasar tyg
kanske så på engelska
### A matrix is said to be an identity matrix if it is a square matrix in which elements of principle diagonal are ones, and the rest of the elements are zeroes. Algorithm.
The identity matrix is a the simplest nontrivial diagonal matrix, defined such that I(X)=X (1) for all vectors X. An identity matrix may be denoted 1, I, E (the latter being an abbreviation for the German term "Einheitsmatrix"; Courant and Hilbert 1989, p.
|
2023-01-30 20:39:59
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6537730097770691, "perplexity": 905.2877228198059}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499829.29/warc/CC-MAIN-20230130201044-20230130231044-00681.warc.gz"}
|
https://cstheory.stackexchange.com/questions/2147/ugc-hardness-of-the-predicate-naex-1-x-ell-for-x-i-in-gfk
|
# UGC hardness of the predicate $NAE(x_1, ..., x_\ell)$ for $x_i \in GF(k)$?
Background:
In Subhash Khot's original UGC paper (PDF), he proves the UG-hardness of deciding whether a given CSP instance with constraints all of the form Not-all-equal(a, b, c) over a ternary alphabet admits an assignment satisfying 1-$\epsilon$ of the constraints or whether there exist no assignments satisying $\frac{8}{9}+\epsilon$ of the constraints, for arbitrarily small $\epsilon > 0$.
I'm curious whether this result has been generalized for any combination of $\ell$-ary constraints for $\ell \ge 3$ and variable domains of size $k \ge 3$ where $\ell \ne k \ne 3$. That is,
Question:
Are there any known hardness of approximation results for the predicate $NAE(x_1, \dots, x_\ell)$ for $x_i \in GF(k)$ for $\ell, k \ge 3$ and $\ell \ne k \ne 3$?
I'm especially interested in the combination of values $\ell = k$; e.g., the predicate Not-all-equal($x_1, \dots, x_k$) for $x_1 \dots, x_k \in GF(k)$.
• Please a reference for case $k=3$? Oct 12, 2010 at 22:13
• @turkistany, after looking at my question further, I decided to remove the sub-question (because I was asking just way too much all at once!). The paper I was originally referring to, though, was this. Oct 12, 2010 at 23:00
• If you do post a question about Bulatov's paper, note that there has been significant simplification of the approach over the last decade. Several of the algorithms have been simplified and merged, see the recent LICS paper by Barto and Kozik for an overview. Oct 13, 2010 at 6:42
• @Andras: I assume you mean this? It's looks interesting; I'll definitely read it, thanks! In any case, I'll probably re-ask the previous sub-question as a new question soon, assuming I don't answer it for myself (plus, I'm short on time to ensure I state it properly at the moment). Oct 13, 2010 at 12:10
• yes, that's the one. The references therein provide a quick tour through the subsequent history. Oct 13, 2010 at 12:43
I realized that what I claimed above is in fact known.
For $\ell = 3$ and arbitrary $k \ge 3$, this is in Khot's FOCS 2002 paper "Hardness of coloring 3-colorable 3-uniform hypergraphs" (the paper actually talks about general $k$, though the title only talks about the 3-colorable case).
For $\ell \ge 4$ and $k \ge 2$, in fact a stronger hardness is known. Even if there is in fact an assignment of just two values to the variables that satisfies all NAE constraints (in other words the $\ell$-uniform hypergraph can be colored using 2 colors without any monochromatic hyperedge), it is still NP-hard to find an assignment from a domain size $k$ which satisfies at least $1-1/k^{\ell-1}+\epsilon$ NAE constraints (for arbitrary constant $\epsilon > 0$). This follows easily from the fact that the known inapproximability result for hypergraph 2-coloring gives a strong density statement in the soundness case. The formal statement appears in my SODA 2011 paper with Ali Sinop "The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number" (Lemma 2.3 in the SODA final version, and Lemma 2.8 in the older version available on ECCC http://eccc.hpi-web.de/report/2010/111/).
• That's quite beautiful. I'll probably end up using this in the very near future. Thank you! Feb 2, 2011 at 22:23
I am pretty sure that for the problem you are asking, it should be NP-hard to tell if the instance is satisfiable, or if at most $1-1/k^{\ell-1}+\epsilon$ fraction of constraints can be satisfied. That is, a tight hardness result (matching what simply picking a random assignment would achieve), for satisfiable instances, and no need for the UGC.
For $k=2$ and $\ell \ge 4$, this follows from Hastad's factor 7/8+epsilon inapproximability result for 4-set-splitting (which can then be reduced to k-set splitting for $k > 4$). If negations are okay, one can also use his tight hardness result for Max ($\ell-1$)-SAT.
For $k=\ell=3$, Khot proved this in a FOCS 2002 paper "Hardness of coloring 3-colorable 3-uniform hypergraphs." (That is, he removed the original UGC assumption.)
For $\ell=3$ and arbitrary $k\ge 3$, Engebretsen and I proved such a result in "Is constraint satisfaction over two variables always easy? Random Struct. Algorithms 25(2): 150-178 (2004)". However, I think our result required "folding" i.e., the constraints will actually be of the form NAE($x_i+a,x_j+b,x_k$) for some constants $a,b$. (This is the analog of allowing negations of Boolean variables.)
For the general case, I don't know if this has been written down anywhere. But if you really need it, I can probably find something or check the claim.
• Thanks for the great answer! I was unaware of the last paper you linked (yours with Engebretsen), and it will definitely help. I am still interested in the general case (and I've encountered a similar situation: it doesn't seem to be written anywhere!). Even something for the $\ell = 4$ and arbitrary $k$ case would very likely provide enough insight. Dec 13, 2010 at 18:47
Prasad Raghavendra in his STOC'08 Best Paper proved, assuming the Unique Games Conjecture, that a simple semidefinite programming algorithm gives the best approximation for any constraint satisfaction problem (including NAE) with constraints on constant number of variables each and with constant alphabet. To actually know what is the hardness factor for NAE, you need to understand how well the simple algorithm does for it, i.e., prove an integrality gap for the program. I don't know whether someone already did that for NAE in its full generality, or not.
• Oh, good! I've spent some reading some other versions of Raghavendra's STOC paper, too. I should've made this connection! I don't know if the NAE values have been computed specifically either, but they'd definitely interest me! Nov 21, 2010 at 23:44
|
2022-08-10 12:01:07
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8288183808326721, "perplexity": 742.6727187177023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571153.86/warc/CC-MAIN-20220810100712-20220810130712-00791.warc.gz"}
|
https://brilliant.org/discussions/thread/how-can-a-post-a-link-to-my-set-of-problems/
|
×
How can a post a link to my set of problems?
Note by Guiseppi Butel
3 years, 8 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
Sort by:
Hi Guiseppi, when you open your set there should be a link on the right hand side of your set called "Direct Link".
Staff - 3 years, 8 months ago
I found that and tried to post it but it wasn't a direct hot link.
- 3 years, 8 months ago
To get a hyper link to display, you should type it as
[reference name](url link)
For example,
[JB's posted problems](https://brilliant.org/profile/guiseppi-fwubxj/sets/jbs-posted-problems/)
is displayed as
Staff - 3 years, 6 months ago
You can click the "Share" or "Reshare" button to share the set to the feed.
You can access your set directly by going to https://brilliant.org/profile/guiseppi-fwubxj/sets/jbs-posted-problems/.
Staff - 3 years, 8 months ago
|
2018-03-22 06:24:24
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9979085922241211, "perplexity": 13450.6369582835}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647777.59/warc/CC-MAIN-20180322053608-20180322073608-00238.warc.gz"}
|
https://electronics.stackexchange.com/questions/416989/time-it-takes-for-b-to-go-to-zero-with-r-and-l-given
|
# Time it takes for B to go to zero with R and L given
Given that your solenoid has inductance L and resistance R with initial current I such that there is uniform B inside the coil, I would like to figure out how much time will pass before B goes to zero.
Using the equations shown here, I think I have some idea of how to solve for $$\\Delta t\$$.
First, we know that the energy stored in the inductor is $$\ E = \frac{1}{2}LI^2\$$. Then, power, the rate of energy change, is $$\ P = L I \dot{I} \$$. Also, we know that $$\ P = I^2 R \$$.
From the link above, given the energy density inside the coil spring (solenoid), the energy stored can also be written as $$\ E =\frac{B^2 V}{\mu} \$$, where $$\V \$$ is the volume.
Take its time derivative, $$\P = \frac{BV}{\mu} \frac{dB}{dt} = I^2 R \$$.
Since $$\ I = \frac{Bl}{\mu N} \$$, $$\P = \frac{BV}{\mu} \frac{dB}{dt} = (\frac{Bl}{\mu N})^2 R \$$.
Can I now just solve for $$\ dt \$$ to figure out the time lapse for the magnetic field to go to zero?
• Er... Coil spring?. What do springs have to do with this? – Edgar Brown Jan 14 at 23:32
• @EdgarBrown: I suspect he means a typical 1-layer solenoid coil. OP: please edit your question. – TimWescott Jan 14 at 23:33
• "with current I" If the current is present, the magnetic field will never go to zero. Again: edit your question. – Oldfart Jan 14 at 23:38
• Vdrop across R rises with I, and V drop on L controls LdI/dt – Sunnyskyguy EE75 Jan 15 at 1:52
If you are talking about a coil with internal resistance that is shorted, or if you are talking about an explicit LR circuit, then it probably matches the schematic below. The equation that controls the circuit is way simpler than you make out: $$R i_L = -L\frac{di_L}{dt}$$
This quickly simplifies to $$\\frac{d i_L}{dt} = -\frac{R}{L} i_L\$$
This is a simple 1st-order linear ordinary differential equation with solution $$i_L(t) = A e^{-t/\tau}$$ where $$\\tau = \frac{L}{R}\$$ and $$\A\$$ is found from some boundary condition on the differential equation. For simplicity's sake, just say that we don't know what all else happened before $$\t = 0\$$, but at $$\ t = 0\$$ $$\ i_L(0) = i_0\$$. Then $$i_L(t) = i_0 e^{-t/\tau}$$
Now note that $$\ e^x \$$ is strictly positive for any finite value of $$\x\$$, and the answer to your question is -- never. That's it!
|
2019-08-23 07:22:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6984776854515076, "perplexity": 347.5274110472949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318011.89/warc/CC-MAIN-20190823062005-20190823084005-00272.warc.gz"}
|
http://mathhelpforum.com/pre-calculus/62262-equation-circle-help-print.html
|
# Equation of a Circle. HELP!
• Nov 29th 2008, 07:58 PM
meiyukichan
Equation of a Circle. HELP!
How do you write the equation of the circle in standard form? and what is the center and radius of it?
$4x^2 + 4y^2 + 24x + 16y + 43 = 0$
A step-by-step view would really help to understand for future problems.
Thanks!
• Nov 29th 2008, 10:40 PM
mr fantastic
Quote:
Originally Posted by meiyukichan
How do you write the equation of the circle in standard form? and what is the center and radius of it?
$4x^2 + 4y^2 + 24x + 16y + 43 = 0$
A step-by-step view would really help to understand for future problems.
Thanks!
Re-arange:
$4x^2 + 24x + 4y^2 + 16y + 43 = 0$
$\Rightarrow 4(x^2 + 6x) + 4(y^2 + 4y) + 43 = 0$.
Now complete the square in each bracket. You should see where to go from there.
• Nov 29th 2008, 11:04 PM
meiyukichan
Quote:
Originally Posted by mr fantastic
Re-arange:
$4x^2 + 24x + 4y^2 + 16y + 43 = 0$
$\Rightarrow 4(x^2 + 6x) + 4(y^2 + 4y) + 43 = 0$.
Now complete the square in each bracket. You should see where to go from there.
What do you mean by complete the square in each bracket? I don't understand...
• Nov 29th 2008, 11:35 PM
andreas
Quote:
Originally Posted by meiyukichan
What do you mean by complete the square in each bracket? I don't understand...
I will continue from here $4(x^2 + 6x) + 4(y^2 + 4y) + 43 = 0$ (*)
$x^2+6x=x^2+6x+9-9=(x+3)^2 -9$ (1)
$y^2+4y=y^2+4y+4-4=(y+2)^2-4$ (2)
Substitute (1) and (2) into (*):
you get $4*(x+3)^2-4*9+4*(y+2)^2-4*4+43=4(x+3)^2+4(y+2)^2=9$ Center is at $x=-3$ and $y=-2$ ,Radius is 3
• Nov 29th 2008, 11:36 PM
mr fantastic
Quote:
Originally Posted by mr fantastic
Re-arange:
$4x^2 + 24x + 4y^2 + 16y + 43 = 0$
$\Rightarrow 4(x^2 + 6x) + 4(y^2 + 4y) + 43 = 0$.
Now complete the square in each bracket. You should see where to go from there.
Quote:
Originally Posted by meiyukichan
What do you mean by complete the square in each bracket? I don't understand...
The first bracket is $x^2 + 6x = (x + 3)^2 - 9$. Get it?
• Nov 30th 2008, 12:48 AM
meiyukichan
Quote:
Originally Posted by mr fantastic
The first bracket is $x^2 + 6x = (x + 3)^2 - 9$. Get it?
Then is this correct?
$4 (x - 3)^2 - 9 + 4 (y + 2)^2 - 4 + 43 = 0$
Rearrange: $4 (x - 3)^2 + 4 (y + 2)^2 - 4 + 43 -9 = 0$
$4 (x - 3)^2 + 4 (y + 2)^2 = -29$
Does that mean that the center would be (3,2)?
• Nov 30th 2008, 01:58 AM
acevipa
$
4x^2 + 4y^2 + 24x + 16y + 43 = 0
$
Rearrange the equation:
$4x^2 + 24x + 4y^2 + 16y = -43$
Try and make two perfect squares on the left side of the eqn.
$4x^2 + 24x + 36 + 4y^2 + 16y + 16 = -43 + 36 + 16$
$(2x+6)^2 + (2y+4)^2 = 9$
Then find the centre and radius of the circle
• Nov 30th 2008, 02:36 AM
mr fantastic
Quote:
Originally Posted by meiyukichan
Then is this correct?
$4 {\color{red}[} (x {\color{red}+} 3)^2 - 9{\color{red}]} + 4 {\color{red}[} (y + 2)^2 - 4 {\color{red}]}+ 43 = 0$
Rearrange: $4 (x {\color{red}+} 3)^2 + 4 (y + 2)^2 - {\color{red}16} + 43 - {\color{red}36} = 0$
$4 (x {\color{red}+} 3)^2 + 4 (y + 2)^2 = -29$ Mr F says: This should have told you that there were mistakes ..... How can the square of the radius be less than zero?
Does that mean that the center would be (3,2)?
Note the corrections I have made in red. The correction of the other mistake in the last line is left for you.
• Nov 30th 2008, 09:43 AM
meiyukichan
Quote:
Originally Posted by mr fantastic
Note the corrections I have made in red. The correction of the other mistake in the last line is left for you.
Then I should get:
$4 (x + 3^2) + 4(y + 2)^2 = 3^2$
...?
With the center being (3,2) and the radius 3...?
• Nov 30th 2008, 11:06 AM
mr fantastic
Quote:
Originally Posted by meiyukichan
Then I should get:
$4 (x + 3)^{\color{red}2} + 4(y + 2)^2 = 3^2$
...?
With the center being (3,2) and the radius 3...?
Apart from the typo, you're almost correct. The centre is at (-3, -2).
By the way, there was an insulting post which I have deleted. My apologies on behalf of MHF if you had already read it.
• Nov 30th 2008, 12:12 PM
meiyukichan
Quote:
Originally Posted by mr fantastic
Apart from the typo, you're almost correct. The centre is at (-3, -2).
By the way, there was an insulting post which I have deleted. My apologies on behalf of MHF if you had already read it.
Oh! Thank you. I forgot that the equation was like this $(x - h)^2 + (y - k)^2 = r^2$
meaning that since it was a positive, it is a negative then...
I didn't read the insulting post, but if it was something like how can I make such a simply mistake and saying how stupid I am, then I don't care because I'm too young to understand this stuff fully anyways...
I'm only in the 8th grade...
|
2017-07-20 20:49:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 29, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8965958952903748, "perplexity": 1610.4181068742525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423486.26/warc/CC-MAIN-20170720201925-20170720221925-00171.warc.gz"}
|
http://xxicla.dm.uba.ar/sessionAbs.php?session=99ecb7cc71&order=1
|
# Session S05 - Rings and Algebras
Chair: Iryna Kashuba (USP, Brasil)
Session website
View session abstracts PDF
## Talks
July 28, 15:00 ~ 15:45
## Octonions in low characteristics
### Universidad de Zaragoza, Spain - elduque@unizar.es
Some special features of Cayley algebras, and their Lie algebras of derivations, over fields of low characteristics will be presented. As an example, over fields of characteristic two, the isomorphism class of the Lie algebra of derivations of a Cayley algebra does not depend on the Cayley algebra itself.
View abstract PDF
July 28, 15:45 ~ 16:10
## Polynomial identities, codimensions and a conjecture of Regev
### Università di Palermo, Italy - antonio.giambruno@unipa.it
Let $A$ be an algebra over a field $F$ of characteristic zero and $Id(A)$ its T-ideal of identities. The space of multilinear polynomials in $n$ fixed variables modulo $Id(A)$ is a representation of the symmetric group $S_n$ and its degree is called the $n$th codimension of $A$. As soon as $A$ is associative and satisfies a non-trivial identity, its sequence of codimensions is exponentially bounded and, following a conjecture of Amitsur regarding its exponential growth, Regev made a conjecture about the precise asymptotics of such sequence. I will talk about the results around this conjecture also in the case of non associative algebras.
View abstract PDF
July 28, 16:10 ~ 16:35
## Essencial idempotents in group algebras and coding theory
### Universidade de São Paulo, Brazil - polcinomilies@gmail.com
We introduce the concept of essencial idempotents in group algebras, a notion inspired in coding theory. We shall give some criteria to identify which primitive idempotents are essential, and discuss some applications. Among these, we show that every minimal non-cyclic abelian code is a repetition code, and that every minimal abelian code is combinatorially equivalent to a cyclic code of the same lenth. Also, we shall give an example showing that a non minimal abelian code of lenth $p^2$ with $p$ a prime integer, can be more convenient than any cyclic code of that length.
View abstract PDF
July 28, 16:35 ~ 17:00
## Free groups in a normal subgroup of the field of fractions of a skew polynomial ring
### University of Sao Paulo, Brazil - jz.goncalves@usp.br
Let $k(t)$ be the field of rational functions over the field $k$, let $\sigma$ be a $k$-automorphism of $K=k(t)$, let $D=K(X; \sigma)$ be the ring of fractions of the skew polynomial ring $K[X; \sigma]$, and let $D^{\bullet}$ be the multiplicative group of $D$. We show that if $N$ is a non central normal subgroup of $D^{\bullet}$, then $N$ contains a free subgroup.
View abstract PDF
July 28, 17:30 ~ 17:55
## Non-commutative algebraic geometry of semi-graded rings
### Universidad Nacional de Colombia, Bogotá, Colombia, Colombia - jolezamas@unal.edu.co
In this short talk we introduce the semi-graded rings, which generalize graded rings and skew PBW extensions. For this new type of non-commutative rings we will study some basic problems of non-commutative algebraic geometry. In particular, we will discuss the Serre-Artin-Zhang-Verevkin theorem about non-commutative schemes.
Joint work with Edward Latorre (Universidad Nacional de Colombia, Bogotá, Colombia).
View abstract PDF
July 28, 18:00 ~ 18:25
## On finite generation and presentation of algebras
### King Abdulaziz University, Saudi Arabia - adelnife2@yahoo.com, analahmadi@kau.edu.sa
The talk will focus on finite generation and presentation of associative and Lie algebras with idempotent conditions.
Joint work with Hamed Alsulami.
View abstract PDF
July 28, 18:30 ~ 18:55
## Automorphisms of ideals of polynomial rings
### Universidade Federal de São Paulo, Brasil - castilho.thiago@gmail.com
Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. Such ideal may be considered as an $R$-algebra. In this talk we present recent results obtained with T. Macedo [arXiv:1604.08531], concerning the group $Aut(I_f)$, of $R$-algebra automorphisms of $I_f$. We will show that $Aut(I_f)$ can be obtained by analyzing some symmetries of the roots of $f$ in the algebraic closure of the quotient field of $R$ (counted with multiplicities). In particular, we show that, under certain mild hypothesis, if $f$ has at least two different roots in the algebraic closure of the quotient field of $R$, then $Aut(I_f)$ is a cyclic group and its order can be completely determined by analyzing the roots of $f$.
Supported by Fapesp and CNPq
Joint work with Tiago Macedo (Universidade Federal de São Paulo).
View abstract PDF
July 29, 15:00 ~ 15:45
## Finitely presented Lie and Jordan algebras
### University of California, USA - efim.zelmanov@gmail.com
We will consider important examples of Lie and Jordan algebras and address the question when they can be presented by finitely many defining relations.
View abstract PDF
July 29, 15:45 ~ 16:10
## Graded algebras and polynomial identities
### Technion, Haifa, Israel - elialjadeff@gmail.com
Connections (or bridges'') between PI theory (polynomial identities) and group gradings on associative algebras are quite well known for more than 30 years. For instance, Kemer applied the theory of super algebras'' in order to solve the famous Specht problem for nonaffine PI algebras. Our interest is in the opposite direction. We apply PI theory in order to solve a conjecture of Bahturin and Regev on regular G-gradings'' on associative algebras where G is a finite abelian group. Moreover, we show how to extend it to nonabelian groups. As a second application, we present a Jordan's like theorem on G-gradings on associative algebras.
Joint work with Ofir David (Technion, Israel).
View abstract PDF
July 29, 16:10 ~ 16:35
## Color involutions of primitive graded rings.
### University of Brasilia, Brazil - I.Sviridova@mat.unb.br
Kaplansky's Theorem [2] characterizes involutions of primitive rings with a nonzero socle in terms of hermitian and alternate forms. In 1997 M.L.Racine [3] constructed similar structure theory for primitive associative superalgebras. And Yu.A. Bakhturin, M. Bresar, M. Kochetov [1] obtained similar results for graded rings with graded involutions.
We present analogous characterizations of primitive graded rings in terms of twisted pairing. This implies the extension of Kaplansky's Theorem for primitive graded rings with a color involution in case of a grading by a cyclic group of a prime order. We also obtain some corollaries on color involutions of finite dimensional simple graded algebras. In particular, these results generalise the corresponding theorems of [2].
The work is partially supported by CNPq, CAPES.
[1] Yu.A. Bakhturin, M. Bresar, M. Kochetov, Group gradings on finitary simple Lie algebras, Int. J. Algebra Comp., 22(2012), 125-146.
[2] N. Jacobson, Structure of Rings, AMS Colloquium Publication 37, AMS, Providence, R.I., 1964.
[2] M.L. Racine, Primitive Superalgebras with Superinvolution, J. Algebra 206(2)(1998), 588-614.
Joint work with Keidna Cristiane Oliveira Souza (University of Brasilia, Brazil).
View abstract PDF
July 29, 16:35 ~ 17:00
## Lie algebras of slow growth
### University of Brasilia, Brazil - petrogradsky@rambler.ru
We discuss rather old and recent constructions of Lie algebras and superalgebras of slow growth. In particular, we obtain examples of finitely generated (self-similar) (restricted) Lie (super)algebras of slow polynomial growth with a nil p-mapping.
By their properties, these restricted Lie (super)algebras resemble Grigorchuk and Gupta-Sidki groups. We discuss different properties of these algebras and their associative hulls.
View abstract PDF
July 29, 17:30 ~ 17:55
## On $D$ algebras.
### Wayne State University, USA - lml@wayne.edu
Consider an algebraic function $z$ of $n$ variables $x_1, x_2, \dots, x_n$. Denote by $D(z)$ a subalgebra of the field ${\Bbb C}(x_1, x_2, \dots, x_n)[z]$ which is generated by $x_1, x_2, \dots, x_n; \ z$ and all partial derivatives of $z$. I am interested in properties of algebras $D(z)$.
In my talk I will discuss the following conjectural dichotomy:
If $z \in {\Bbb C}[x_1, \dots, x_n]$ then (obviously) $D(z) = {\Bbb C}[x_1, \dots, x_n]$, but if $z \not\in {\Bbb C}[x_1, \dots, x_n]$ then $D(z)$ cannot be embedded into a polynomial ring.
View abstract PDF
July 29, 18:00 ~ 18:25
## Partial actions and subshifts
### Universidade de São Paulo, Brazil - dokucha@gmail.com
An arbitrary (one-sided) subshift $X$ over a finite alphabet $\Lambda$ with $n$ letters can be naturally endowed with a partial action $\theta$ of the free group ${\mathbb F}_n$ with $n$ free generators $g_{\lambda}, (\lambda \in \Lambda ),$ such that $g_{\lambda }$ maps $x$ to $\lambda x,$ where $x$ is an element in $X$ for which $\lambda x \in X.$ Naturally $g^{-1}_{\lambda }$ removes $\lambda$ from $\lambda x.$ We call $\theta$ the standard partial action, and it is a starting point to constract a $C^*$-algebra ${\mathcal O}^*_X$ associated with $X$, as well as an abstract algbera ${\mathcal O}^K_X$ over an arbitrary field $K$ of characteristic $0.$ Both ${\mathcal O}^K_X$ and ${\mathcal O}^*_X$ are defined in a fairly similar way: using the standard partial action we construct a partial representation $u$ of ${\mathbb F}_n$ into an appropriate algebra (which depends on whether the case is abstract or $C^*$) and then define ${\mathcal O}^K_X$ (or ${\mathcal O}^*_X$) as the subalgebra (respectively, a $C^*$-subalgebra) generated by $u({\mathbb F}_n ).$ Then using a general procedure (see [4,Proposition 10.1] we obtain a partial action $\tau$ of ${\mathbb F}_n$ on a commutative subalgebra ${\mathcal A}$ and prove that ${\mathcal O}^K_X$ is isomorphic to the crossed product ${\mathcal A}$ $\rtimes _{\tau} {\mathbb F}_n.$ In the $C^*$ case (see [3,Theorem 9.5]), due to an amenability property, ${\mathcal O}^*_X$ is isomorphic to both the full and the reduced crossed product: ${\mathcal O}^*_X \cong {\mathcal D} \rtimes _{\tau} {\mathbb F}_n \cong {\mathcal D} \rtimes^{\rm red} _{\tau} \, {\mathbb F}_n,$ where $\mathcal D$ is a commutative $C^*$-algebra defined in a similar way as $\mathcal A$. This gives a possibility to study algebras related to subshifts using crossed products by partial actions. It turns out that ${\mathcal O}^*_X$ is isomorphic to the $C^*$-algebra defined by T. M. Carlsen in [1] in a somewhat different way (see [3,Theorem 10.2]). In particular, if $X$ is a Markov subshift, then ${\mathcal O}^*_X$ is isomorphic to the Cuntz-Krieger algebra defined in [2]. The $C^*$ version is elaborated in the preprint [3], in which, amongst several related results, a criterion is given for simplicity of ${\mathcal O}^*_X$ (see [3,Theorem 14.5]).
[1] T. M. Carlsen, Cuntz-Pimsner, $C^*$-algebras associated with subshifts, Internat. J. Math., 19 (2008), 47–70.
[2] J. Cuntz, W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 63 (1981), 25–40.
[3] M. Dokuchaev, R. Exel, Partial actions and subshifts, Preprint, arXiv:1511.00939v1 (2015).
[4] R. Exel, Partial Dynamical Systems, Fell Bundles and Application, to be published in a forthcoming NYJM book series. Available from http://mtm.ufsc.br/?exel/papers/pdynsysfellbun.pdf.
Joint work with Ruy Exel (Universidade Federal de Santa Catarina, Brazil).
View abstract PDF
July 29, 18:30 ~ 18:55
## Identities of finitely generated alternative and Malcev algebras
### University of São Paulo , Brazil - shestak@ime.usp.br
We prove that for every natural number $n$ there exists a natural number $f(n)$ such that every multilinear skew-symmetric polynomial on $f(n)$ variables which vanishes in the free associative algebra vanishes as well in any $n$-generated alternative algebra over a field of characteristic $0$. Similarly, for any $n$ there exists $g(n)$ such that every multilinear skew-symmetric polynomial on $g(n)$ variables vanishes in any $n$-generated Malcev algebra over a field of characteristic $0$. Before a similar result was known only for a series of skew-symmetric polynomials of special type on $2m+1$ variables constructed by the author, where $m>\tfrac{C^1_n+C^2_n+C^3_n}{2}$.
View abstract PDF
## Abelian Group Codes
### Universidade Federal da Bahia , Salvador - argentina.ale@gmail.com
Let $F$ be a finite field and $n$, a non negative integer. A linear code $C$ of length $n$ is a subspace of $F^n$. A (left) group code of length $n$ is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism $FG \rightarrow F^n$ for any $G$, a finite group with $\vert G \vert=n$. In this case $C$, is a (left) $G$-code. In [1], Bernal, del R\'io and Sim\'on obtain a criterion to decide when a linear code is a group code in terms of the group of permutation automorphisms of $C$, $PAut(C)$. Sabin and Lomonaco, in [4], have proved that if $C$ a $G$-code with $G$ a semidirect product of cyclic groups, then $C$ is an abelian group code. As an application of criterion and extending the result of Sabin and Lomonaco, in [1], they provide a family of groups for which every two-sided group code is an abelian group code. Pillado, González, Martínez, Markov e Nechaev describe some classes of groups and fields for which all group codes are abelian in [2]. Motivated by [3], they have shown that there exist a non-Abelian $G$-code over $F$. In order to extend the result on groups with abelian decompotition, we explore some conditions to determine a group $G$ which can written as a product of abelian subgroups, such that the $G$-codes with $G \in \mathcal{G}$ will be abelian group code.
\\
[1] \textsc{J. J. Bernal, Á. del Río and J.J. Simón, An intrinsical description of group of codes, \textit{Des. Codes Cryptogr. } \textbf{51}(3) 289-300 (2009).}\\
[2] \textsc{ C. García Pillado, S. González, C. Martínez, V. Markov and A. Nechaev, Group codes over non-abelian groups, \textit{ J. Algebra Appl.} \textbf{ 12 }(7) (2013).} \\
[3] \textsc{C. García Pillado, S. González, C. Martínez, V. Markov and A. Nechaev, When all group codes of a noncommutative group are groups abelian (a computational approach)?, \textit{ J. Math. Sci.} \textbf{186}(5) 578-585 (2012).} \\
[4] \textsc{ R.E. Sabin and S.J. Lomonaco}, \textit{Metacyclic Error-Correcting Codes}, AAECC, 6, 191-210 (1995).
Joint work with Thierry Petit Lob\~ao (Universidade Federal da Bahia, Brasil).
View abstract PDF
## Semiclean Rings
### Universidade Federal da Bahia, Brasil - assiselen@yahoo.com.br
A ring $R$ with unity is said to be clean if every element in the ring can be written as the sum of a unit and an idempotent of the ring. These rings were introduced by Nicholson, \cite{1}, in his study of lifting idempotents and exchange rings. The division rings, boolean rings and local rings are examples of clean rings.
In the article \cite{6}, a new class of rings is defined; semiclean rings . A ring $R$ with unity is called semiclean if, every $x \in R$, $x= u + a$ with $u \in \mathcal{U}(R)$ where $a$ is periodic element, i.e., $a^k=a^l$ with $k,z \in \mathbb{Z}$ and $k\neq z$. Therefore, every semiclean ring is a clean ring, because the idempotents elements of ring are periodics. Nicholson e Han, \cite{5}, demonstrated that group ring $Z_{(7)}C_3$ is not a clean ring. Yuanqing Ye showed, in the article \cite{6}, that the group ring $Z_{(p)}C_3$ is an semiclean ring. This result assures that the two classes, clean and semiclean, are different.
Motivated by the article \cite{6}, we intend to investigate if the Yuanging Ye's demonstration can be generalized, as in the cases $Z_{(11)}C_5$ and $Z_{(p)}C_5$, in search of a possible answer about the ring $Z_{(p)}C_q$ with $p$ and $q$ relatively primers.
Joint work with Elen Deise Assis Barbosa(Universidade Federal da Bahia, Brasil).
View abstract PDF
## The Normalizer Property and its Relation with Extensions of Groups
### UFBA - Universidade Federal da Bahia, Brazil - jcintra@ufba.br
The determination of the normalizer of the basis group in the group of units of the associated group ring is a question that naturally imposes by itself. In integral group rings, in particular, it has been observed that, for important classes of finite groups, this normalizer is minimal, in other words, $\mathcal N_\mathcal U(G)=G\cdot Z$. When this occurs, we say that the group in question and its integral group ring satisfy the normalizer property. This property, also known as (Nor), has recently gained great importance when Mazur, in [Ma95], noticed an interesting relation with the famous problem of isomorphism in integral group rings also known as (Iso). Exploring this connection, Hertweck in [He01] found an example of a finite group that does not satisfy (Nor), and indirectly, by the relation mentioned above, obtained a counterexample to (Iso). Given that the counter example of Hertweck to (Nor) consists of an extension given by a semidirect product, but [LPS99] proves that extensions given by direct products are solutions (Nor), it is important to investigate which other other extensions of finite groups answer the property. Recently, Petit Lobão e Sehgal in [PeS03] demonstrated the validity of (Nor) for the class of complete monomial groups; in other words, a wreath extension of a finite nilpotent group with the symmetric group on m letters. Zhengxing Li e Jinke Hai in a series of articles, among which we have [HL12], [HL12b], HL11], also obtained interesting solutions of this property. The purpose of this work is to verify the relation between (Nor) and extensions of groups, where such component groups are solutions (Nor), in order to obtain necessary and sufficient conditions to find positive solutions to the property in question.
\begin{thebibliography}{C}
\bibitem[HL12]{jz}{HAI, J.; Li, Z. {The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups}, {\it Communications in Algebra}, v. 40, n. 7, p. 2613-2627, 2012.}
\bibitem[HL12b]{jz1}{HAI, J.; LI, Z.{The Normalizer Property for Integral Group Rings of Finite Solvable T-Groups}, {\it Journal of Group Theory}, v. 15, n. 2, p. 237-243, 2012.}
\bibitem[HL11]{zj1}{HAI, J.; LI, Z. {The normalizer property for integral group rings of wreath products of finite nilpotent groups by some 2-groups}, {\it Journal of Group Theory}, v. 14, n. 2, p. 299-306, 2011.}
\bibitem[He01]{hertweck}{HERTWECK, M. {A counterexample to the isomorphism problem for integral group rings}, {\it Annals of Mathematics}, v. 154, n. 1, p. 115-138, 2001.}
\bibitem[LPS99]{lps}{LI, Y.; PARMENTER, M. M.; SEHGAL, S. K. On the normalizer property for integral group rings. {\it Communications in Algebra}, v. 27, n. 9, p. 4217-4223, 1999.}
\bibitem[Ma95]{mazur}{MAZUR, M. {On the isomorphism problem for integral group rings of infinite groups}, {\it Expositiones Mathematicae}, v. 13, n. 5, p. 433-445, 1995.}
\bibitem[PeS03]{thierry}{PETIT LOBÃO, T.; SEHGAL, S. K. The Normalizer Property for Integral Group Rings of Complete Monomial Groups, {\it Communications in Algebra}, v. 31, n. 6, p. 2971-2983, 2003.}
\end{thebibliography}
Joint work with Thierry Petit Lobão (Universidade Federal da Bahia).
View abstract PDF
## Involution inverting gradings on matrix algebras
### Universidade Federal de São Paulo, Brasil - castilho.thiago@gmail.com
Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $M_n(F)$ is an algebra with involution $*$, we describe $G$-gradings $M_n(F)=A=\oplus_{g\in G} A_g$ on $A$, satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$.
Joint work with Luís Felipe Gonçalves Fonseca (Universidade Federal de Viçosa).
View abstract PDF
## GENERATION OF SULLIVAN DECOMPOSABLE ALGEBRAS VIA CERTAIN PDEs
### Cauca University, Colombia - sicbravo@gmail.com
In this work we investigate properties of certain commutative differential graded algebras naturally associated to submanifolds of a infinite Jet manifold determined by finite systems of finite-order PDEs, particularly those inspired by the study of linear gauge complexes and by one-forms associated to equations of pseudo-spherical type. More explicitly, we identify linear gauge complexes as a particular type of certain twisted complexes, and we generate Sullivan decomposable algebras using hierarchies of equations of pseudo-spherical type.
Joint work with Enrique Reyes García (Universidad de Santiago de Chile, Chile).
View abstract PDF
## Anticommutativity of Symmetric and Skew-symmetric Elements under Generalized Oriented Involutions
### UFBA - Universidade Federal da Bahia, Brazil - edward.landi@ufba.br
Given an involution $*$ in a group ring $RG$, we can define the sets $(RG)_*=\left\{\alpha\in RG:\alpha^*=\alpha\right\}$ and $(RG)_*^-=\left\{\alpha\in RG:\alpha^*=-\alpha\right\}$, called the set of symmetric and skew-symmetric elements, respectively. Under certain conditions in $R$, $G$, or the involution in $RG$, many authors proved that some identities satisfied in these sets could be lifted to the entire group ring, and, in some cases, given the impossibility of such lifting, they describe the basic structures of the group ring $RG$.
Generalizing the results found in [GP13a, GP13b, GP14], using a group homomorphism $\sigma:G\rightarrow \mathcal{U}(R)$, we will define and explore the involution $\sigma*:RG\rightarrow RG$, called generalized oriented involution, exposing the group structures, as well as the ring conditions, such that $(RG)_{\sigma*}$ or $(RG)_{\sigma*}^-$ be anticommutative.
\begin{thebibliography}{5} \bibitem[BP06]{BP06}{BROCHE CRISTO, O.; POLCINO MILIES, C. Symmetric elements under orientated involutions in group rings, \begin{itshape}Communications in Algebra\end{itshape}, v. 34, n. 9, p. 3347-3356, 2006.}
\bibitem[GP13a]{GP13a}{GOODAIRE, E. G.; POLCINO MILIES, C. Involutions and Anticommutativity in Group Rings. Em: {\it Canadian Mathematical Bulletin}, v. 52, n. 2, p. 344-353, 2013.}
\bibitem[GP13b]{GP13b}{GOODAIRE, E. G.; POLCINO MILIES, C. Oriented Involutions and Skew-symmetric Elements in Group Rings. Em: {\it Journal of Algebra and Its Applications}, v. 12, n. 1, 2013.}
\bibitem[GP14]{GP14}{GOODAIRE, E. G.; POLCINO MILIES, C. Oriented Group Involutions and Anticommutativity in Group Rings. Em: {\it Communications in Algebra}, v. 42, n. 4, p. 1657-1667, 2014.}
\bibitem[JM06]{JM06}{JESPERS, E.; RUIZ MARÍN, M. On symmetric elements and symmetric units in group rings, {\it Communications in Algebra}, v. 34, n. 2, p. 727-736, 2006.}
\bibitem[PT15a]{PT15a}{PETIT LOBAO, T. C.; TONUCCI, E. L. Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions, preprint (2015) arXiv:1511.06907v1 [math.RA].}
\bibitem[PT15b]{PT15b}{PETIT LOBAO, T. C.; TONUCCI, E. L. Anticommutativity of Symmetric Elements under Generalized Oriented Involutions, preprint (2015) arXiv:1510.06004v1 [math.RA].}
\bibitem[V13]{V13}{VILLA, A. H. {\it Involuções de grupo orientadas em álgebras de grupo}. 2013. 76f. Ph.D thesis - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2013.} \end{thebibliography}
Joint work with Thierry Petit Lobao (Universidade Federal da Bahia, Brazil).
View abstract PDF
## $*$ - Clean Group Algebras
### Universidade Federal Fluminense, Brasil - gian1427@gmail.com
An element of a(n associative) ring (with $1$) is clean if it is the sum of a unit and an idempotent. A ring is clean if every element in it is clean. The property of cleanness was formulated by Nicholson [4] in the course of his study of exchange rings. From then on, several related concepts were proposed: uniquely clean rings, strongly clean rings, weakly clean rings, $*$ - clean rings, r - clean rings, nil - clean rings, to cite a few. In the realm of group rings, these properties have been studied from 2001 [2] on with the aim of characterizing the rings $R$ and groups $G$ such that the group ring $RG$ is clean.
In 2010 Vas proposed the definition of a $\ast$ - clean ring (star''- clean) [5]: a $\ast$ - ring (i.e., rings with an involution) in which every element may be written as a sum of a unit and a projection. Clearly, every $\ast$-clean ring is a $star$ - ring and is a clean ring. In [5], Vas asked: when is a $\ast$ - ring clean, but not $\ast$-clean?
Every group $G$ having an element g $\neq$ 1 , with $| \langle g \rangle | \neq 2$, is endowed with the classical involution $g \mapsto g^{-1}$. Because of that, group rings $RG$ are almost always $\ast$ - rings: if $R$ is a commutative rings, for instance, an involution in $RG$ is obtained from the $R$ - linear extension of the classical involution in $G$ (and is also called the classical involution in $RG$ ). The $\ast$-cleanness of group rings was first approached in 2011 [3]. Even though some instances of group rings are answers to Vas's question [1], very little is still known about conditions under which a group ring with the classical involution is $\ast$-clean (not even the case of the group ring $RG$, where $R$ is a commutative ring and $G$ is a cyclic group, is fully stablished!).
In this talk, I present some recent results [1]. Let $R$ be a commutative local ring. I will present $R S_3$ as an answer to Vas 's question, and I will provide necessary and sufficient conditions for the group ring $R Q_8$ to ber $\ast$ - clean, where $Q_8$ is the quaternion group of $8$ elements.
REFERENCES :
[1] Y. Gao, J. Chen, Y. Li, em Some $\ast$-clean Group Rings, Algebra Colloquium 22 (2015) 169--180.
[2] J. Han, W. K. Nicholson, em Extensions of clean rings, Communications in Algebra 29 (2001), 2589--2595.
[3] C. Li, Y. Zhou, em On strongly $\ast$-clean rings, Journal of Algebra and Its Applications 6 (2011), 1363--1370.
[4] W. K. Nicholson, em Lifting idempotents and exchange rings, Transactions of the AMS 229 (1977), 269 -- 278.
[5] L. Vas, em $\ast$-Clean rings; some clean and almost clean Baer $\ast$-rings and von Neumann algebras, J. Algebra 324 (2010), 3388--3400.
View abstract PDF
## Irreducible components of varieties of Jordan algebras
### Universidade de São Paulo, Brazil - eugeniamartin@gmail.com
In 1968, F. Flanigan proved that every irreducible component of a variety of structure constants must carry an open subset of nonsingular points which is either the orbit of a single rigid algebra or an infinite union of orbits of algebras which differ only in their radicals.
In the context of the variety $Jor_n$ of Jordan algebras, it is known that, up to dimension four, every component is dominated by a rigid algebra. In this work, we show that the second alternative of Flanigan's theorem does in fact occur by exhibiting a component of $JorN_5$ which consists of the Zariski closure of an infinite union of orbits of five-dimensional nilpotent Jordan algebras, none of them being rigid.
Joint work with Iryna Kashuba (Universidade de São Paulo, Brazil).
View abstract PDF
## Construction of $\text{Rota}^{m}\text{-Algebras}$ and $\text{Ballot}^{m}\text{-Algebras}$ from Associative Algebras with a Rota-Baxter morphism and a Rota-Baxter Operator of Weights Three and Two
### Universidad del Cauca, Colombia - wamartinez@unicauca.edu.co
We give a generalization of Rota-Baxter Operators and introduce the notion of a Ballot$^{m}$-algebra. Free Rota-Baxter algebras on a set can be constructed from a subset of planar rooted forests with decorations on the angles. We give similar constructions for obtaining an associative algebra in terms of planar binary trees with a modified Rota-Baxter Operator, and so we construct a Ballot$^{m}$-algebra.
We introduce the concepts of a Rota-Baxter Morphism, Dyck$^{m}$-algebra and Rota$^{m}$-algebra. An element $u$ is said to be idempotent with respect to product $\cdot$ in the algebra if: $u \,\cdot\, u = u,$ and it is a left identity if $x \,\cdot\, u = x$ for all element $x$ in the algebra. Associative algebras with a left identity that simultaneously is a element idempotent, permit us to present examples of a Rota-Baxter Morphism and so we can construct a Rota$^{m}$-algebra.
We stress that the construction of Ballot$^{m}$-algebras and Rota$^{m}$-algebras from associative algebras with a generalitation of Rota-Baxter Operators are some of the main results of this work.
View abstract PDF
## Clean rings and clean group rings
### Universidade Federal do Ceará, Brazil - danielsonfilho@hotmail.com
A ring is said to be clean if each element in the ring can be written as the sum of a unit and an idempotent of the ring. The notion of a clean ring was introduced in 1977 by Nicholson in his study of lifting idempotents and exchange rings, and these rings have since been studied by many different authors.
In this poster, we present some properties and examples of clean rings, and then we classify the rings that consist entirely of units, idempotents, and quasiregular elements and we also consider the problems of classifying the groups $G$ whose group rings $RG$ are clean for any clean ring $R$.
Joint work with Rodrigo Lucas Rodrigues (Universidade Federal do Ceará).
View abstract PDF
## On the max-plus algebra of non-negative exponent matrices
### postdoctoral researcher (University of S\~ao Paulo), Brasil - makar.plakhotnyk@gmail.com
An integer $n\times n$-matrix $A=(\alpha_{pq})$ is called exponent if all its diagonal entries are equal to zero and for all possible $i,\, j$ and $k$ the inequality $\alpha_{ij} +\alpha_{jk}\geqslant \alpha_{ik}$ holds. The study of exponent matrices is important because of their crucial role in the theory of tiled orders.
We show that the set $\mathcal{T}$ of minimal non-negative exponent $n\times n$-matrices can be described as follows. The matrix $T=(t_{ij})\in \mathcal{E}_n$ belongs to $\mathcal{T}$ if and only if $t_{ij}\in \{0,\, 1\}$ for all $i,\, j$ and there exists a proper subset $\mathcal{I}$ of $\{1,\ldots,\, n\}$ such that $t_{ij}=1$ is equivalent to $i\in \mathcal{I}$ and $j\not\in \mathcal{I}$.
Let $\oplus$ be the element-wise maximum of matrices and let $\otimes$ be a sum of matrices. Clearly, $A\otimes (B\oplus C) = (A\otimes B) \oplus (A\otimes C)$ for all $A,\, B,\, C\in \mathcal{E}_n$, whence $\mathcal{E}_n$ can be considered as an algebra $(\mathcal{E}_n,\, \oplus, \otimes)$, with respect to operations $\oplus$ and $\otimes$.
We prove the following result.
\textbf{Theorem}. \emph{For any non-zero $A\in \mathcal{E}_n$ there exist a decomposition $$A = B_1\otimes \ldots \otimes B_l\oplus \ldots \oplus C_1\otimes \ldots\otimes C_m,$$ where all matrices $B_1,\ldots,\, C_m$ belong to $\mathcal{T}$ and as usual $\otimes$ performed prior to $\oplus$}.
Thus, $\mathcal{T}$ can be considered as a basis of $(\mathcal{E}_n,\, \oplus, \otimes)$. This basis is unique. Nevertheless, there is no uniqueness of the decomposition of $A\in (\mathcal{E}_n,\, \oplus, \otimes)$ into the max-plus expression of matrices from $\mathcal{T}$.
The work is supported by FAPESP.
Joint work with Mikhailo Dokuchaev (University of S\~ao Paulo, Brasil), Volodymyr Kirichenko (Taras Shevchenko National University of Kyiv, Ukraine) and Ganna Kudryavtseva (University of Ljubljana, Slovenia).
View abstract PDF
## Commutative power-associative nilalgebras and Albert's problem
### IME - USP, Brasil - eoquinro@ime.usp.br
Albert's problem ask if every commutative power-associative nilalgebra is solvable. We proof that commutative power-associative nilalgebras of dimension $n$ and nilindex $n-3$ over a field algebraically closed of characteristic zero are solvable. Finally, we study commutative power-associative nilalgebras of dimension 9 and we proof that they are solvable too.
Joint work with Juan Carlos Gutierrez Fernandez (IME - USP).
View abstract PDF
## A study on clean rings
### Universidade Federal Fluminense, Brasil - laizvalim@gmail.com
A ring is said to be clean if every element can be written as sum of a unit and an idempotent. These rings were defined by Nicholson [5], while studying exchange rings. The class of clean rings is located among other well known classes of rings [3]. In the realm of group rings, these properties have been studied from 2001 [2] on with the aim of characterizing the rings $R$ and groups $G$ such that the group ring $RG$ is clean.
The study of $\ast$-clean rings was motivated by a question made by T. Y. Lam at the Conference on Algebra and Its Applications, in March 2005, at the Ohio University: which von Neumann algebras are clean as rings? Since von Neumann algebras are $\ast$-rings (i.e., rings with an involution), it is more natural to work with projections (idempotents that are symmetric under the ring involution) than with idempotents.
So, in 2010 Vaš defined $\ast$-clean rings [6]: a $\ast$-ring in which every element may be written as a sum of a unit and a projection. Clearly, every $\ast$-clean ring is a $\ast$-ring and is a clean ring.
Every group $G$ is endowed with the classical involution $g \mapsto g^{-1}$. If $R$ is a commutative ring, for instance, the $R$-linear extension of the classical involution in $G$ is the classical involution in $RG$. $\ast$-clean group rings were first studied in 2011 [4]. However very little is still known about when a group ring is $\ast$-clean (not even the case of the group ring $RG$, where $R$ is a commutative ring and $G$ is a cyclic group, is fully stablished!).
In this talk, we present clean rings, their relationship with other types of rings [3] and some recent results [1]. Let $R$ be a commutative local ring. I will provide necessary and sufficient conditions for the group rings $R C_3$ and $R C_4$ to be $\ast$-clean, where $C_n$ denote the cyclic group with $n$ elements.
REFERENCES
[1] Y. Gao, J. Chen, Y. Li. Some $\ast$-clean Group Rings, Algebra Colloquium 22 (2015) 169--180.
[2] J. Han, W. K. Nicholson. Extensions of clean rings, Communications in Algebra 29 (2001), 2589--2595.
[3] N. A. Immormino. Some Notes On Clean Rings, Bowling Green State University, 2012.
[4] C. Li, Y. Zhou. On strongly $\ast$-clean rings, Journal of Algebra and Its Applications 6 (2011), 1363--1370.
[5] W. K. Nicholson. Lifting idempotents and exchange rings}, Transactions of the AMS 229 (1977), 269 -- 278.
[6] L. Vaš. $\ast$-Clean rings; some clean and almost clean Baer $\ast$-rings and von Neumann algebras, J. Algebra 324 (2010), 3388--3400.
View abstract PDF
|
2017-06-23 10:26:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8145628571510315, "perplexity": 967.1801665620216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320049.84/warc/CC-MAIN-20170623100455-20170623120455-00492.warc.gz"}
|
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=410&option_lang=eng
|
RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS
Trudy MIAN: Year: Volume: Issue: Page: Find
Personal entry: Login: Password: Save password Enter Forgotten password? Register
Tr. Mat. Inst. Steklova, 2003, Volume 242, Pages 123–135 (Mi tm410)
This article is cited in 1 scientific paper (total in 1 paper)
On Prenex Fragment of Provability Logic with Quantifiers on Proofs
R. È. Yavorskii
Steklov Mathematical Institute, Russian Academy of Sciences
Abstract: We consider a fragment of provability logic with quantifiers on proofs that consists of formulas with no occurrences of quantifiers in the scope of the proof predicate. By definition, a logic ql is the set of formulas that are true in the standard model of arithmetic under every interpretation based on the standard Gödel proof predicate. We describe Kripke-style semantics for the logic ql and prove the corresponding completeness theorem. For the case of injective arithmetical interpretations, the decidability is proved.
Full text: PDF file (227 kB)
References: PDF file HTML file
English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 242, 112–124
Bibliographic databases:
UDC: 510.6
Received in October 2002
Citation: R. È. Yavorskii, “On Prenex Fragment of Provability Logic with Quantifiers on Proofs”, Mathematical logic and algebra, Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov, Tr. Mat. Inst. Steklova, 242, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 123–135; Proc. Steklov Inst. Math., 242 (2003), 112–124
Citation in format AMSBIB
\Bibitem{Yav03} \by R.~\E.~Yavorskii \paper On Prenex Fragment of Provability Logic with Quantifiers on Proofs \inbook Mathematical logic and algebra \bookinfo Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 242 \pages 123--135 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm410} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054490} \zmath{https://zbmath.org/?q=an:1079.03054} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 242 \pages 112--124 `
Linking options:
• http://mi.mathnet.ru/eng/tm410
• http://mi.mathnet.ru/eng/tm/v242/p123
SHARE:
Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
This publication is cited in the following articles:
1. Yavorskiy R., “On Kripke-style semantics for the provability logic of Gödel's proof predicate with quantifiers on proofs”, J. Logic Comput., 15:4 (2005), 539–549
• Number of views: This page: 302 Full text: 103 References: 42
Contact us: math-net2020_01 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020
|
2020-01-22 17:52:30
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27709993720054626, "perplexity": 13124.011222732886}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607314.32/warc/CC-MAIN-20200122161553-20200122190553-00308.warc.gz"}
|
http://www.zentralblatt-math.org/ioport/en/?q=an%3A01574796
|
History
Please fill in your query. A complete syntax description you will find on the General Help page.
Fuzzy set-theoretic operators and quantifiers. (English)
Dubois, Didier (ed.) et al., Fundamentals of fuzzy sets. Foreword by Lotfi A. Zadeh. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 7, 125-193 (2000).
This chapter in Kluwer’s handbook series devoted to fuzzy sets brings an overview of fuzzy set-theoretic operators and quantifiers. First of all, basic operations for fuzzy sets as well as connectives for fuzzy logics are discussed, such as negation (complementation), conjunction (intersection), disjunction (union), implication, coimplication, equivalence, etc. The authors present here basic characterization results for several unary and binary operations, including most important examples. Note that t-norms (models for fuzzy conjunction) and related operations are, in an exhaustive form, discussed also in a recent monograph of {\it E. P. Klement}, {\it R. Mesiar} and {\it E. Pap} [Triangular norms, Kluwer Academic Publishers, Dordrecht (2000; Zbl 0972.03002)]. Several other operators frequently applied in applications of fuzzy set theory (e.g., in fuzzy control) are also recalled, such as uninorms, OWA operators, root-power mean operators, symmetric sums, etc. Also, several types of quantifiers, especially linguistic quantifiers (“most”, “few”, “almost all”, etc.) are included. Finally, prioritized fuzzy operations are discussed. This contribution is a good state-of-the-art overview with more than 120 relevant references and it can be recommended to any user or person interested in fuzzy sets and their applications.
Reviewer: R.Mesiar (Bratislava)
|
2013-06-19 09:37:49
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.910007119178772, "perplexity": 4697.284122243241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368708664942/warc/CC-MAIN-20130516125104-00095-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://physics.stackexchange.com/questions/98306/why-dont-x-rays-gamma-rays-ionize-all-the-atoms-at-the-surface-of-a-material
|
# Why don't X-rays/Gamma rays ionize all the atoms at the surface of a material?
Recently I've been wondering why certain materials are transparent or opaque to different wavelengths of light. The most common explanation for why a material, like glass, is transparent (in the visible spectrum) is because the photons of those wavelengths don't have enough energy to excite electrons to a higher energy state, so the photons move through the material unaffected. Once you reach a critical frequency, however, the photons will begin to be absorbed because they have enough energy to excite the electrons (which is why glass is opaque in ultra-violet). By that logic, one would expect that ALL frequencies higher than the critical point would be opaque. That is not what happens, however, as x-rays/gamma-rays penetrate deeply into certain surfaces (like your skin).
Furthermore, x-rays/gamma-rays can ionize atoms by providing an electron with enough energy to escape the atom (which I believe is the cause of the photoelectric effect). Finally, an escaped electron can release some of its remaining energy through Compton Scattering, producing a lower frequency wave than that of the incident one.
Compton Scattering seems to offer a candidate solution; i.e. that it's the scattered, lower frequency waves that make it through. However, according to this article, and more specifically this graphic (shown below), Compton Scattering starts occurring at a certain energy threshold, so it doesn't explain the relatively high penetration depth of lower frequencies. Furthermore, the graph also demonstrates that the penetration depth is proportional to the photon's energy, which again contradicts the trend discussed in the first paragraph, where one would expect the penetration depth to decrease as the photon's energy increases.
I have looked for an answer on several sites, but none have provided me with a fully satisfying answer. These sites tend to either simplify the material, or simply gloss over this particular aspect of the more general question: "Why are materials transparent?". So, to summarize, my question is:
Why does the penetration depth of high frequency waves increase for higher energies, and why is this not (apparently) applicable to lower frequencies? Given a high enough frequency, but still lower than the Compton Scattering frequency, why aren't all surface atoms ionized, effectively making the material opaque? (i.e. photoelectric absorption)
You are thinking in terms of atoms and molecules and you are mainly talking of solid state matter .
Solid state is another quantum mechanical phase, it has lattice structure with much smaller energies than atomic and molecular transition structures. Lattices have vibrational levels which are mainly responsible for the black body radiation solids emit, infrared is also photons.
A rule of thumb with radiation impinging on solids is that if the wavelength is smaller than the lattice dimensions the photons can penetrate easily the lattice, interacting only with direct scatters hence the higher penetration of X rays and gamma rays. Here is an article that discusses the penetration of radiation, X rays and higher.
For glass and optical frequencies there is a good answer here in this site., essentially the structure of the transparent materials is such that the photons pass through without loosing energy in the visible.
For infrared where the wavelengths are large in comparison with lattices or distances between molecules in liquids, the photon can give up its energy in collective excitations at the surface gradually heating up the material.
For ultraviolet, glass, depending on the type, has some absorptive bands, the photon energy transferred at the surface to collective modes or breaking molecular bonds and transformed to heat ( infrared) further in.
So your
Once you reach a critical frequency, however, the photons will begin to be absorbed because they have enough energy to excite the electrons (which is why glass is opaque in ultra-violet).
has small probability to happen until x-ray energies are reached which are the energies of bound electrons, and the link above gives the dependence in a simplified manner.
I don't think there's any simple answer to your question that encompasses all materials. Scattering and absorption in matter has many contributions and isn't just Compton scattering. To begin with, particles that penetrate matter will generically scatter multiple times before being stopped and the cross-section will change when they get slower. The absorbtion of low frequencies depends on the shell structure of the atoms, on the geometry of the molecules and the symmetries of the lattice if you have a crystal. Not to mention that it depends on the type of particle that scatters. For the general question of penetration depth of light in matter, you might want to read up on the Beer–Lambert law.
All photon absorptions happen "on the surface" of a material. However the depth of "the surface" depends on the cross sections for the photon absorption processes involved. For visible and UV photons these cross sections depend on the atomic structures and molecular structures and crystal structures in quite complicated ways. For low-energy x-rays, you may have photon wavelengths comparable to the angstrom-scale spacing between atoms, which permits coherent scattering and diffraction. Once the wavelength of the light becomes smaller that the spacing between atoms, the photon essentially interacts with each atom separately, and the probability that a photon travels a distance $\ell$ without scattering is given by $\exp(-n\ell\sigma)$, where $n$ is the number density of scatterers and $\sigma$ is the cross section.
For very high-energy photons, the mean free path between collisions in matter does in fact level off. You just needed a bigger figure:
(This image is from the Particle Data Group.)
• For sub-MeV photons the dominant contribution to photon scattering is electronic transitions in the atoms. As the photons become energetic enough to knock out electrons from the next shell down, the mean free path suddenly decreases, which gives the curves their "spiky" shape (also present in your example).
• For MeV-scale photons the dominant effect is Compton scattering, and so the most important parameter in the mean free path is the number of electrons that the photon passes. This is why all the curves except H come together around 2 MeV: most materials have roughly one electron for every two nucleons, except hydrogen which has roughly one electron for every nucleon.
• For higher-energy photons the dominant scattering process is electron-positron pair creation, which occurs when the high-energy photon scatters off of "virtual photons" in the electric field between the atomic electrons and their nucleus. The internal electric field is stronger for heavier nuclei, so a kilogram of heavy nuclei will stop more high-energy photons than a kilogram of lighter nuclei.
Note that while the mean free path between photon-material interactions levels off for photons somewhere above 100 MeV, the amount of material that's required to "stop" the photon continues to grow, more or less without any upper bound. This is because the photon doesn't stop at its first interaction, but divides its momentum among the electron-positron pairs, scattered electrons, and lower-energy photons that it creates. This "shower" of electromagnetic leftovers propagates forward in a cone, and can penetrate quite a bit of material.
|
2019-08-19 02:14:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7257364988327026, "perplexity": 375.5633705667821}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314638.49/warc/CC-MAIN-20190819011034-20190819033034-00288.warc.gz"}
|
https://math.stackexchange.com/questions/391144/generalized-eigenvector-for-3x3-matrix-with-1-eigenvalue-2-eigenvectors
|
# generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors
I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.)
I have found eigenvectors $\vec {u_1}$ and $\vec {u_2}.$
When I try $u_1$ and $u_2$ as $u_3$ into this equation: $$(A - I)u_4 = u_3$$ I get systems which are inconsistent.
How can I find the $u_3$? I've been told it has something to do with $(A - I)^3 = 0$, but that's about it.
• What is the prime doing in the "equation" $\bf x'=Ax$? What is $t$? There is context missing here, impossible to make much sense out of this. – Marc van Leeuwen May 14 '13 at 8:29
We are given the matrix:
$$\begin{bmatrix}2 & 1 & 1\\1 & 2 & 1\\-2 & -2 & -1\\\end{bmatrix}$$
We want to find the characteristic polynomial and eigenvalues by solving
$$|A -\lambda I| = 0 \rightarrow -\lambda^3+3 \lambda^2-3 \lambda+1 = -(\lambda-1)^3 = 0$$
This yields a single eigenvalue, $\lambda = 1$, with an algebraic multiplicity of $3$.
If we try and find eigenvectors, we setup and solve:
$$[A - \lambda I]v_i = 0$$
In this case, after row-reduced-echelon-form, we have:
$$\begin{bmatrix}1 & 1 & 1\\0 & 0 & 0\\0 & 0 & 0\\\end{bmatrix}v_i = 0$$
This leads to the two eigenvectors as he shows, but the problem is that we cannot use that to find the third as we get degenerate results, like you showed.
Instead, let's use the top-down chaining method to find three linearly independent generalized eigenvectors.
Since the RREF of
$$[A - 1 I] = \begin{bmatrix}1 & 1 & 1\\0 & 0 & 0\\0 & 0 & 0\\\end{bmatrix}$$
We have $E_3 = kernel(A - 1I)$ with dimension $= 2$, so there will be two chains.
Next, since
$$[A - 1 I]^2 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\\\end{bmatrix}$$
the space Kernel $(A-1I)^2$ has dimension $=3$, which matches the algebraic multiplicity of $\lambda=1$.
Thus, one of the chains will have length $2$, so the other must have length $1$.
We now form a chain of $2$ generalized eigenvectors by choosing $v_2$ in kernel $(A-1I)^2$ such that $v_2$ is not in the kernel $(A-1I)$.
Since every vector is in kernel $(A-1I)^2$, and the third column of $(A-1I)$ is non-zero, we may choose:
$$v_2 = (1, 0, 0) \implies v_1 = (A-1I)v_2 = (1,1,-2)$$
To form a basis for $\mathbb R^3$, we need one additional chain of one generalized eigenvector. This vector must be an eigenvector that is independent from $v_1$. Since
$$E_3 = ~\text{span}~ \left(\begin{bmatrix}0\\1\\-1\\\end{bmatrix}, \begin{bmatrix}-1\\0\\1\\\end{bmatrix}\right).$$
and neither of these spanning vectors is itself a scalar multiple of $v1$, we may choose either one of them. So let
$$w_1 = (0, 1, -1).$$
Now we have two chains:
$$v_2 \rightarrow v_1 \rightarrow 0$$
$$w_1 \rightarrow 0$$
So, to write the solution, we have:
$\displaystyle 1^{st}$ Chain
$$x_1(t) = e^t \begin{bmatrix}1\\1\\-2\\\end{bmatrix}$$
$$x_2(t) = e^t\left(t \begin{bmatrix}1\\1\\-2\\\end{bmatrix} + \begin{bmatrix}1\\0\\0\\\end{bmatrix}\right)$$
$\displaystyle 2^{nd}$ Chain
$$x_3(t) = e^t \begin{bmatrix}0\\1\\-1\\\end{bmatrix}$$
Thus:
$$x(t) = x_1(t) + x_2(t) + x_3(t)$$
Note, you can use this linear combination of $x(t)$ and verify that indeed it is a solution to $x' = Ax$.
• Nice work!! (as usual!), and accepted, so I'm sure it "took". congrats on a job well done! – Namaste May 15 '13 at 2:50
• You too! Enjoy work (presentation) and relaxation! ;-) – Namaste May 15 '13 at 3:17
• @Amzoti If $w_1$ is to be an eigenvector, shouldn't it be $\text{col} (-1, 1, 0)$ instead of $\text{col} (0, 1, -1)$ – jaynp May 15 '13 at 3:35
• @user1850672: We could have chosen either ot the two in the span. I chose the first one. There may be other choices too. regards – Amzoti May 15 '13 at 3:37
• @Azmoti No no, I see that you could choose either in $E_3$'s span. Look at the eigenvectors given in the problem. The first is different from your first. Is this correct? Forgive me if I'm misunderstanding something. – jaynp May 15 '13 at 4:33
|
2019-07-22 13:45:09
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.938550591468811, "perplexity": 361.98421030420207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528037.92/warc/CC-MAIN-20190722133851-20190722155851-00362.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-10-review-exercises-page-1125/46
|
## Precalculus (6th Edition) Blitzer
$\frac{27}{2}=13.5$
The sum of a infinite geometric sequence is given as: $S=\dfrac{a_1}{1-r}$ Here, $a_1=9$ and Common Ratio: $r=\dfrac{1}{3}$ $S=\dfrac{a_1}{1-r}$ Now, $S=\dfrac{9}{1-\dfrac{1}{3}}=\dfrac{27}{3-1}=\frac{27}{2}=13.5$
|
2021-04-23 05:24:57
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9859155416488647, "perplexity": 951.0209939477375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00440.warc.gz"}
|
https://www.nature.com/articles/s41598-022-15291-7?error=cookies_not_supported&code=fe95d4a3-8b57-44dc-b054-b30ac7e8f15d
|
Introduction
Food insecurity has been defined by the Food and Agriculture Organization (FAO) as the lack of a sustainable amount of nutritious and safe food to support development, growth, and lead a healthy and active lifestyle1. Globally, nearly one in ten people were exposed to severe levels of food insecurity in 2019. The latest estimates indicate that over 8.8% of the world’s population suffers from hunger. This implies that the number of food-insecure individuals increased by more than 10 million people over the past year and by almost 60 million in the past five years2. Rapid industrialization and urbanization have set a substantial drawback on conventional farming by increasing greenhouse gas emissions, negatively affecting agricultural production and shrinking the available cultivable land. To ensure a secure food supply to the world’s constantly growing population, there is a dire need to develop other sustainable food growing techniques that are more productive, efficient, and environmental friendly.
Urban farming techniques such as greenhouse hydroponics production, aquaponics, sandponics, and many more are among the most efficient solutions. They feature soilless cultivation of crops, less freshwater usage with an increased crop yield per area, and most importantly, they produce healthy and chemical-free nutrient-diverse foods3. Aquaponics (AP), the cultivation of plants and fish together in a recirculating, manufactured ecosystem that uses natural bacterial cycles to utilize fish waste as nutrition for plants4. Aquaponics can be a potential solution for huge global challenges, including water pollution, water scarcity, long food transportation mileages and high energy use, and most importantly, food security and malnutrition. As cited in Li et al.5, AP is especially promising for urban areas to meet the additional food demands because of higher crop yields per area6.
On the other hand, sandponics (SP), which is also referred to as an Integrated Aqua-Vegeculture system (IAVS) is an aquaponic-related growing technique for cultivating plants that utilize sand as a primary medium for mechanical filtration, biofilter, and as well growing media for crops. It is a promising sustainable production option for several crops, including vegetables, vines, and fruits. The system can easily be applied since sand is readily available in most areas, can be easily sterilized, versatile, easily recycled, and cheaper than soil. This makes SP a more efficient, affordable, and low-risk technology. However, some limitations to the SP system need to be addressed. They include operators requiring specialized training, crop nutritional deficiencies due to insufficient fertilizers, finding suitable sand for crops that require cooler climates, and expensive heated systems. Most importantly, very few works in the literature report the system's functionality since it’s not yet a commonly practiced technique7. The system is tailored to enhance productivity by enabling year-round crop organic production within a controlled environment8. Relying solely on such farming systems to solve the food security issue may not be entirely sufficient. However, SP can produce healthy local crops to support a healthy urban/peri-urban lifestyle and eventually build a giant leap towards more nutritious and more food secure communities9.
The nutrient composition of water effluent in SP plays a major role in the system's overall performance. The fish wastes contain nitrogen which is presented mainly as total ammonia. Total ammonia consists of both ammonium (NH4+) and ammonia (NH3); both undergo the nitrification process to oxidize to nitrate (NO3) Nitrogen10. The amount of ammonia produced in the system depends on several factors such as fish biomass, fish size, and the amount/nature of food fed to the fish. Environmental changes like water temperature, salinity, and oxygen levels can also affect all system components' activities and growth, including microbes, fish, and plants11.
Swiss chard Bright lights (Beta vulgaris subsp. cicla) is a multi-colored leafy plant first recorded in the Canary Islands and could be historically traced back to 350 BC12. The plant is also known for a variety of other names, where a popular name in Australia is “Silver Beet”; a location where it is more preferred than regular spinach. The plant is prized for its high yield and high nutritional value that it was the subject of a 13-week study that is geared towards selecting candidate cultivars for space applications as a salad crop for manned missions; the selection was not only based on high yield but also the sensory properties and growth conditions and suitability for vertical farming13. Swiss chard (Beta vulgaris) requires well-aerated crumbly soils with an ample supply of organic matter to retain adequate moisture levels. It would be sensitive to damping-off effects if the soil is not well-drained or properly ventilated14.
According to the US Department of Agriculture nutrient analysis, a cup containing a 36-g amount of cooked Swiss Chard has 18 mg of Calcium (Ca), 0.65 mg of Iron (Fe), 136 mg of Potassium (K), 10.8 mg of Vitamin C, 5 mg of folate, 298 mg of Vitamin K, 29 mg of Magnesium (Mg), 17 mg of Phosphorus (P), 110 mg of Vitamin A and 0.68 mg of Vitamin E. Due to the high folate content within Swiss chard, research has been conducted, making it a filler ingredient in bread making to increase its nutritional makeup. They fortified 20 g of Swiss chard per 100 g of bread increasing its folate content from 19.9 to 57.9 mg in white bread and from 37.4 to 75.5 mg in whole grain bread15.
There is very little or no scientific literature about growing crops in sandponics systems hence, creating so many questions related to the operation, functionality, optimization, sand suitability, and system productivity. The current study's objective was to investigate the productivity of different IAVS (Integrated Aqua-Vegeculture system) sand types and quality from various locations in Egypt on the performance, growth, and yield of Swiss chard crop compared to the deep-water culture aquaponics system.
Materials and methods
Study site
The study was conducted at the Center for Applied Research on the Environment and Sustainability (CARES) at The American University in Cairo, New Cairo, Egypt (30°01′11.7″N 31°29′59.8″E) from 12/Nov/2019 until 31st/March/2020. The experiment was carried out in a greenhouse-controlled environment with temperatures ranging from 18 to 23 °C and relative humidity between 60 and 70% during the growing period.
Experimental design
The proposed design starts by treating brackish water using RO membrane separation technology, powered by an on-grid 10 kW photovoltaic solar panel as shown in Fig. 1. The permeate (freshwater) from the RO facility is directed to the aquaculture units of capacity of 1 m3, where the fish effluents are used as irrigation water and as the sole source of fertilizers for the crops.
The study followed a completely randomized design with four variants, i.e., an aquaponic deep-water culture system (T1) and three sandponics systems (T2–T4). The three sandponics systems were established with different sand collected from different sand locations in Egypt during the period between September and October 2019.
Initially, an exploratory field trip was set to six different locations in Egypt to collect sand samples for lab analysis aimed at sourcing the most suitable sand for the system under study with regards to both the physical and chemical parameters. These areas include Ismailia Governorate; 30°34′55.2″N 31°50′08.1″E, 6th October governorate; 29°54′49.8″N 31°05′51.5″E, Benu Suef governorate; 28°53′18.4″N 30°45′12.9″E, Al-Minya governorate; 28.725799, 30.630305, and two sites from Fayoum governorate; 29°05′07.4″N 30°49′39.9″E.
From the six locations in Egypt, preliminary sand analysis was carried out, and sand samples were also collected for both physical and chemical lab analysis at the Soil and Water Lab at the Agricultural Research Center in Dokki, Egypt. Following a thorough technical, field, mechanical, and lab chemical evaluation of the six sand samples from six locations, three sand locations/types were selected for experimentation that seemed fit and suitable for the current study. The criteria parameters for the shortlisting of sand included water retention potential of the sand by the percolation process, testing the carbonates level in the soil, the turbidity of the sand, porosity percentage and drainage potential of the sand. The three locations included 6th October (T2), Benu Suef (T3), and Fayoum site 2 (T4). In the second week of November 2019, ten cubic meter tracks of sand from the three above locations were set to collect sand from these areas to the research facility at CARES where the experiment was carried out.
The study was carried out with two systems/setups, i.e., an aquaponic Deep Water Culture (DWC) and SP systems. The DWC model comprises a 1 m3 fish tank, a settlement tank, a mechanical filter, a biological filter, three grow beds, and a drainage tank. This system being the most practiced aquaponics technique was considered as the control. Fish effluent water flowed from the fish tank to the settlement tank to filter big solid wastes through the mechanical filter to remove the smaller solid wastes and the biological filter for the nitrification process. Then filtered water continues to the grow beds, where overflow drains into the drainage tank and back to the fish tank in a closed system.
On the other hand, the variable in the three IAVS systems is the sand source. This system comprises three independent set-ups: a 1 m3 fish tank, three grow beds, and a drainage tank. Fish effluents flowed from the fish tank directly to the sand grow beds where water was supplied through irrigation drip lines using diaghram emitters connected with valves to ensure uniformity of water application to each grow bed.
All the fish tanks were installed with the same fish stock size of 30 Nile tilapia (Oreochromis niloticus) from an existing fish stock at the research center with an average initial weight of 244 g and the same amount of water, initially 850L per tank. The fish was sourced from an already existing aquaponics system at the research center to avoid any transportation stress effects and related shocks on the small fish, leading to a lot of mortality cases. The fish were fed 3–4 times daily with commercial pellets containing 30% proteins, 5% crude lipid, 6% crude fiber, 13% Ash, and 9% moisture content supplied by Skretting Egypt. The feeding pattern and frequency were according to the fish body biomass percentage of 2–3% depending on the growth stage and upon reaching satiation.
Desalination
The experiment was entirely run with desalinated water produced from a desalination facility at the center. The desalination technology used was Reverse Osmosis (RO); in batch mode; using a Sea Water Pump with Energy Recovery Unit (model Danfoss-APP1.0/APM1.2). The RO membrane used is Hydraunatic SWC5-4040, from Lenntech company with an average salt rejection of 99.7%. Three modules were connected in a series arrangement (3 Pressure Vessels each equipped with a single module). Synthesized brackish water was prepared by dissolving industrial grade sodium chloride (sea salt) from El-Arish Governorate, Egypt. The salt chemical properties are presented in Table 1. Feedwater salinity was 10 mg/L, with an equivalent osmotic pressure equal to 8.61 bars. The osmotic pressure was calculated using Van’t Hoff relation. Permeate Total Dissolved Solids (TDS) was 192 mg/L, and brine TDS was 13.1 g/L as shown in Table 2.
The average pure water flux is 9.5 LMH and was calculated by dividing the permeate volume by the product of membrane surface area and time. Each batch run produced around 4 m3 of permeate, which was enough to irrigate the designated plant beds. The estimated average permeate recovery for the RO process is 22% and salt rejection exceeded 98.7%. The differential pressure between membrane inlet and outlet was equal to 1 bar, where membrane inlet pressure was 16 bars, and the outlet was 15 bars. The RO process operated at an average transmembrane pressure equal to 16 bars and an average permeate and brine flow rates equivalent to 3.49 and 12.41 Lpm, respectively. All experiment runs were performed at 25 °C.
Plant materials and cultivation practice
Swiss chard bright lights (Beta vulgaris subsp. cicia) seeds were imported from Seed kingdom seed company in the USA. Seeds were sown in ¼ inch holes in a seed starting mix containing perlite and vermiculite and irrigated with a hand mist sprayer daily to keep the growing media always moist. Sowing was done on the 12th of November 2019, and seedlings were transplanted when they were 40 days old. Seedlings were transplanted into raised grow beds made of fiberglass material measuring 1.8 × 1.2 × 0.6 m for each of the four systems. The beds were raised off the ground by 0.5 m to allow drainage water from the bed to be collected and circulated back to the fish tank. Each bed was constructed with a drainage pipe at the bottom covered with a mesh net to prevent water blockage by the sand. Also, a 5 cm layer of small gravel was uniformly laid at the bottom of the beds to facilitate drainage, followed by sand with a height of 50 cm.
In the IAVS systems, plants were irrigated using manually punched diaphragm emitters, and the irrigation flow rate was controlled using small plastic valves at the start of every irrigation tube. Emitters were installed in drip tubing at a 30 cm distance as well the tubing lines were also placed 30 cm between each other. Seedlings were transplanted 5 cm away from the emitters at 30 cm between rows and 30 cm within the row. Since the water was pumped with submersible pumps to the grow beds, regulatory pressure valves were installed in between the pump and the main irrigation line, and then water flows through the emitters into the row furrows. Water would then saturate in the sand and eventually drain at the bottom into drainage tanks and pumped back to the fish tanks.
To maintain the water quality, two full cycles of water recirculation were run every day. Every irrigation cycle recirculated 25% of the fish tank, and complete drainage was allowed for a maximum of two hours. Plants were harvested upon reaching maturity for three cuts, except with the T1, which could not grow back after the second cut. Plants took 52 days from transplanting to reach the first cut, 20 days from cut 1 to cut 2, and as well 23 days from cut 2 to reach cut 3. Measurable crop parameters included plant height at harvesting/cutting, leaf area, number of leaves per plant, chlorophyll content, fresh weight per plant, and nutrient composition. Since the focus of SP is on the crops, fish were only measured to monitor their relative growth in terms of weight gained at harvesting/cutting time.
Measurement of crop parameters
Plants were cut 5 cm above the soil surface, and agronomical trait measurements from a representative sample of 12 plants per replicate were taken as follows.
Plant heights were taken using a foot ruler and averages determined. Leaf number was obtained as the number of leaves counted per plant and averages determined. Leaf area was calculated according to the equation reported by Yeshitila and Taye16.
$${\text{Leaf}} \, {\text{ Area }}\left( {{\text{cm}}^{{2}} } \right) = \, - {422}.{973} + { 22}.{752}0{\text{L }}\left( {{\text{cm}}} \right) \, + { 8}.{\text{31W }}\left( {{\text{cm}}} \right)$$
where L and W represent the leaf length and Leaf width respectively, − 422.973 is a constant relating to the shape of the leaf of Swiss chard developed by the author under citation.
Chlorophyll content was measured using MC-100 chlorophyll meter from Apogee Instruments, Inc, and data was expressed as SPAD averages. Fresh weight was measured using a digital weighing balance and data expressed as g/plant.
Sand test
Sand samples were obtained and sent for analysis at the Soils, Water and Environment Research Institute, Agricultural Research Center, Giza, Egypt. The Electrical conductivity (EC) values were measured from the sand paste extract; pH values were taken from sand suspensions at ratio of 1:2.5 as described by Estefan17. The available nitrogen in the sand sample was extracted using potassium chloride (KCl) as an extractable solution with the ratio of (5gm sand to 50 ml KCl) and determined using the micro- kjeldahl method. Available potassium was determined using a flame photometer, and the other elements in the sand sample were determined by using inductively coupled plasma (ICP) Spectrometry (model Ultima 2 JY Plasma)18,19. The physical and chemical characteristics of the used sand are presented in Table 3.
Water analysis
Every 15 days, a measured amount of desalinated water was added to a standard mark of 850L in the fish tanks to compensate for the consumed amount of water in the system. Fish water quality parameters such as water temperature, pH, and dissolved oxygen (DO) was closely monitored using automated digital Nilebot technologies by Conative labs to fit the ideal required levels as reported by Somerville et al.20. In contrast, ammonia, nitrite, and nitrate were adjusted using an API test kit every week. These parameters' recorded values were as follows: water temperature ranged between 25 and 28 °C, DO range between 6–7 mg/L, and pH between 6.5 and 7.0. Ammonia levels were kept below 1 mg/L. Elements in water samples were determined according to EPA methods18 using inductively coupled plasma (ICP) Spectrometry (model Ultima 2 JY Plasma) as presented in Table 4.
Nutritive composition analysis
According to Official methods of analysis from the association of official analytical chemists (A.O.A.C) (1990), moisture content and Vitamin C were determined. Vitamin A was determined according to the procedures described by Aremu and Nweze21. Briefly, 100 g of the sample were homogenized, from which 1 g was obtained and soaked in 5 mL methanol for two hours at room temperature in the dark for complete extraction of a pro-vitamin A carotenoid, β-carotene. Separation of the β-carotene layer was achieved through the addition of hexane to the sample, and moisture was removed using sodium sulphonate. The absorbance of the layer was measured at 436 nm using hexane as a blank. β-carotene was calculated using the formula:
$$\beta {\text{-carotene }}\left( {{\mu g}/{1}00{\text{ g}}} \right) \, = {\text{ Absorbance }}\left( {\text{436 nm}} \right) \, \times {\text{ V }} \times {\text{ D }} \times { 1}00 \, \times { 1}00/{\text{W }} \times {\text{ Y}}$$
where: V = total volume of the extract; D = Dilution factor; W = Sample weight; Y = Percentage dry matter content of the sample.
Vitamin A was then determined according to the concept of Retinol Equivalent (RE) of the β-carotene content of the vegetables using the standard conversion formula. Total hydrolyzable carbohydrates were determined as glucose using phenol–sulfuric acid reagent as described by Michel22.
Vitamin C content was determined using dichlorophenol indophenol reagent. As such, 10 g of fresh leaf tissues, were crushed using a motor and pestle in the presence of 10 ml metaphosphoric acid 6% (Merck). This was followed by centrifugation at 4000×g for 5 min at 4 °C. Five mL of the supernatant were transferred into an Erlenmeyer flask, and 20 mL of 3% metaphosphoric acid were added. The extract was titrated by dichlorophenol indophenol (Sigma-Aldrich) until a rose color was observed. Vitamin C (mg/100 g FW) was then calculated and based on the standard curve of l-Ascorbic acid (Merck) concentrations.
For the determination of protein and mineral content, 0.5 g of dried samples were digested using sulfuric acid (H2SO4) and hydrogen peroxide (H2O2) as described by Cottenie23. From the extracted sample, the following minerals were determined:
Nitrogen was determined according to the procedures described by Plummer24. Briefly, 5 mL of the digestive solution was distilled with 10 mL of sodium hydroxide (NaOH) for 10 min to obtain ammonia. Back titration was then used to determine the amount of nitrogen present in ammonia. Protein content was calculated by multiplying total nitrogen by 6.25 according to methods of AOAC25.
Phosphorus content was determined calorimetrically (660 nm) according to the procedures described by Jackson26. Potassium, Calcium, and Sodium were determined against a standard using a flame-photometer (JEN way flame photometer) as described by Piper27. Magnesium (Mg), Copper (Cu), Manganese (Mn), Zinc (Zn), and Iron (Fe) content were determined using Atomic Absorption Spectrophotometer, Pyeunican SP1900, according to methods described by Liu28.
The moisture percentage of leaf samples was determined by weighing the fresh weight for each sample (Fw), then dried for 72 h at 80 °C. The dry matter weight was record as Dw. The leaf water content was then calculated as the following:
$${\text{Moisture}}\;{\text{ content }}\left( \% \right) \, = \, \left( {{\text{Fw}} - {\text{Dw}}} \right) \, /{\text{ Fw}} * {1}00$$
Statistical analysis
Statistical comparisons among means of more than two groups were performed with analysis of variance (ANOVA) using SPSS V22, and the difference in means was analyzed by Tukey’s test at α = 0.05. Statistical differences were considered significant at P ≤ 0.05 in triplicates and data expressed as mean ± S.D.
Plant material
All plant materials and related procedures in this study were done in accordance with the guidelines of the Institutional Review Board of the American University in Cairo and the Ministry of Agriculture and Land Reclamation in Egypt.
Ethics approval
This study followed the guidelines and approval of Committee of Animal Welfare and Research Ethics, Faculty of Agriculture, Kafrelsheikh University, Egypt.
Results
Effect of different treatments on vegetative growth parameters of Swiss chard
The effect of different treatments on plant height, leaf number, leaf area, and chlorophyll content at different cut numbers is shown in Fig. 2. Results from cut 1 show that plants in the T1 significantly recorded the lowest plant heights (27.83 cm) compared to other treatments (P < 0.0001). Likewise, T2 significantly had a higher plant height (41.05 cm) compared to T3 (34.93 cm) (P < 0.0001). No significant difference in plant height was noted between T2 and T4 during cut 1. In cut 2, still, T1 significantly recorded the lowest plant height (19.18 cm) compared to T2, T3, and T4 (51.2 cm, 49.9 cm, and 53.5 cm, respectively) (P < 0.0001). However, the results of the third cut showed no significant difference in plant height among the three treatments T2, T3, and T4 (Fig. 2a). A significant interaction between treatments and the cut number was observed with respect to plant heights (P < 0.0001). Generally, plant heights increased with the increase in the number of cuts across treatments except for T1.
Data on the effect of different treatments on leaf numbers are shown in Fig. 2b. The first and third cut results indicate no significant difference in leaf number among all treatments even though T2 recorded the highest with 10.8 leaves/plant. For the second cut, however, plants in the T1 significantly had the lowest leaf number (3 leaves/plant) compared to other treatments (P < 0.0001). Furthermore, there was a statistically significant interaction between treatments and cut number, P < 0.0001. Overall, leaf number decreased across all treatments between cut one and two but was maintained at almost eight leaves per plant in both the second and third cut.
Results on the effect of different treatments on leaf area are presented in Fig. 2c. For cut one, leaf area varied across different treatments, and plants in the T1 significantly had a smaller leaf area (92.92 cm2) compared to T2 and T4 (195.30 cm2 and 196.95 cm2, respectively) (P < 0.0001). In the second cut, T2, T3, and T4 treatments showed no significant difference in their leaf area per plant. However, data obtained in the third cut indicated that plants in the T2 treatment significantly had a smaller leaf area (417.63 cm2) compared to T4 (530.85 cm2) except for T3 (P < 0.05). No significant interaction was noted between treatments and cut number regarding leaf area. Overall, it was noted that leaf area significantly increased with the increase in the number of cuts across all treatments.
Figure 2d shows the effect of different treatments on the chlorophyll content of plant leaves. For cut one, plants in the T1 significantly recorded the lowest chlorophyll content (176.33 SPAD) compared to T2, T3, and T4 (399.70, 375.95, and 380.81 SPAD, respectively) (P < 0.0001). However, no significant difference in chlorophyll content of plant leaves was noted across treatments for both cut 2 and cut 3. A statistically significant interaction between treatments and the cut number was noted (P < 0.0001). Generally, the chlorophyll content decreased across all treatments in the first and second cut but slightly increased during the third cut.
Effect of different treatments on biomass yield of Swiss chard
The effect of different treatments on the fresh weight of Swiss chard at different cut numbers is presented in Fig. 3. There was a variation in fresh weights across all treatments in the first cut. The data shows that T1 significantly had the lowest fresh weight (88.29 g/plant) compared to T2 and T4 (169.75 and 166.38 g/plant respectively) (P < 0.0001). However, no significant difference in the average fresh weight was noted among T2, T3, and T4, respectively. Likewise, results of the second cut show that T1 significantly had the lowest fresh weight (156.13 g/plant) compared to other treatments (P < 0.0001). No significant difference in the average fresh weight was noted among T2, T3, and T4 in the third cut. There was a significant interaction between treatments and the cut number, P < 0.05. Overall, the average fresh weight increased with the increasing number of cuts across all treatments.
Results of the average yield for cut 1 indicate that T1 had the lowest yield (14.13 ton/ha) compared to T2 (27.16 ton/ha), T3 (18.56 ton/ha), and T4 (26.62 ton/ha). For cut 2, T1 had the lowest yield (24.96 ton/ha) compared to T2 (36.81 ton/ha), T3 (40.49 ton/ha), and T4 (41.63 ton/ha). Results for cut 3 shows that T2 had a slightly lower yield (41.99 ton/ha) compared to T3 (45.77 ton/ha) and T4 (46.56 ton/ha).
Water consumption of Swiss chard grown under different sand media based IVAS system
The water consumption of Swiss chard grown under different sand media based IVAS systems was calculated. Figure 4 shows the cumulative water consumption in all systems during all the three cuts. Since the fish water quantity was fixed at 850 L, during the first cut, T1 used the least amount of water (372 L) followed by T3 (529 L), T2 (582 L), and T4 (587 L), respectively. In the second cut, T1 still used the least amount of water (113 L) followed by T3 (116 L), T2 (98 L), and T4 (120 L), respectively. In the third cut, T2 and T4 used a lower amount of water (87 L and 99 L, respectively) compared to T3 (120 L). Considering crop water consumption during the three cuts or period of 95 days of experimentation, except for T1, which was only harvested for two cuts; T1 consumed 1.56 L/m2/day, T2 and T3 with 1.87 L/m2/day, and T4 with 1.96 L/m2/day.
Effect of different treatments on the nutritive composition of Swiss chard
In all cut numbers of all treatment groups, the moisture content (%) ranged between 92 to 93% thus indicating no significant difference among all treatments. However, data shows that cut three significantly had a lower moisture (%) compared to cut one and two (P < 0.0001). A highly significant interaction between treatments and the cut number was noted (P < 0.0001).
The protein content of plants sampled from all the treatment groups ranged from 11.84 to 16.79 mg/100 g DW in cut one. No significant difference was noted among all treatment groups. For cut two, plant samples obtained from the T4 treatment group significantly had the highest protein content (18.72 mg/100 g DW) compared to T3 (13.67 mg/100 g DW) and T1 (13.35 mg/100 g DW) (P < 0.05). No significant difference was noted in protein percentage from all plant samples obtained from all treatment groups (T2, T3, and T4) in the third cut. However, a highly significant interaction between treatments and the cut number was noted (P < 0.0001). Overall, cut three had the least protein content than cut one and two, which was significant at P < 0.05.
Data obtained for total carbohydrate content indicates no significant difference among all treatments at different cut numbers. However, treatments combined, the third cut significantly had the lowest total carbohydrate content compared to other cuts (P < 0.05). Also, there was a significant interaction between treatments and the cut number (P < 0.0001).
The vitamin content of plant samples was assessed at different cuts too. Results of cut one indicate that plant samples obtained from the T2 treatment group significantly contained higher vitamin A (2.36 mg/100 g FW) and vitamin C (19.55 mg/100 g FW) compared to those obtained from T4 and T1, and T4 treatments groups respectively (P < 0.05). However, no significant difference in vitamin A and C was noted for cut two and cut three among all treatment groups. Overall, treatments combined, no significant difference in vitamin A content were noted among all cut numbers. Cut one significantly had a lower vitamin C content for vitamin C content than cut two (P < 0.05).
The mineral composition of plant samples (Table 5) shows no significant difference in Fe, Mn, and Cu contents among all treatment groups in all cut numbers. However, Mg content of plant samples obtained from the T1 treatment group in cut one was significantly higher (1841.36 mg/100 g DW) compared to T4, T3, and T2 (1151.28, 959.07, and 863.66 mg/100 g DW respectively) (P < 0.05). No significant difference was noted among all treatment groups in cut two and cut three. The Zn content of plant samples obtained from T3 in cut one was significantly lower (99.08 mg/100 g DW) than T1, T2, and T4 (109.95, 103.78, and 102.27 mg/100 g DW respectively). No significant difference was noted among all treatment groups in cut two and cut three.
Results for Ca content show that plant samples obtained from the T2 treatment group in cut one significantly had a higher Ca content (1134.75 mg/100 g DW) compared to T4 and T3 (747.18 and 695.72 mg/100 g DW, respectively) (P < 0.05). However, data obtained in cut three indicates that plant samples obtained from the T4 treatment group significantly contained higher Ca content (353.08 mg/100 g DW) followed by T3 (344.52 mg/100 g DW) and T2 (263.56 mg/100 g DW) respectively (P < 0.05). There was a significant interaction between treatments and the cut number (P < 0.05) except for Fe. Overall, treatments combined, the mineral composition (Fe, Mg, Ca, and Zn) in cut one was significantly higher compared to cut two and cut three (P < 0.05). However, no significant difference was noted in Mn and Cu contents in all cut numbers.
Discussion
According to some authors29,30,31, the primary sources of nutrients in aquaponics or sandponics are the water source added (containing Mg, Ca, S), uneaten fish feed, and fish waste. The fish effluents contain a significant level of nitrogen and phosphorous, although with fewer quantities of microelements. Both macro and microelements are vital for plant’s growth in varying amounts. The main essential elements include N, K, Ca, Mg, P, S, and the microelements Fe, Mn, B, Zn, Cu, and Mo20,32. The primary required nutrient for the system is nitrates. The exact concentrations of these nutrients in our effluent water are presented in Table 4. Although all of these nutrients exist in solid fish waste, some nutrients, especially Ca, K, and Fe may be limited in aquaponics and may result in plant nutritional deficiencies32,33. Our results for nitrates concentration conform to a range of 5–150 mg/L recommended by FAO20.
Yang and Kim34 have recently experimented on the yield and nutrient management in aquaponics using leafy vegetable crops, including Swiss chard. They reported a maximum plant height of 27 cm. This study results showed that edible Swiss chard could reach more than 53 cm height in the second cut, almost twice the height recorded by the authors above. In this current study, Yang and Kim also reported approximately the same number of leaves obtained per plant. The discrepancy in results could be attributed to the differences in growth media used in both experiments. In our study, the different sand-based IVAS systems supplemented plants with Ca, Mg, Na, and S nutrients which enhanced their growth in contrast to the T1 system solely used in the study conducted by Yang and Kim34. However, our study's plants grown in the DWC system wilted during the third cut, probably due to very low EC values and lack of enough nutrients. Previous studies have shown that very low EC limits plant growth due to nutrient deficiency35,36.
Crop water consumption directly affects the vegetative performance and growth of the plants. Less vegetative structure can also mean less water usage. Since plants in the T1 are partially submerged in the effluent water at all times, not like in sandponic systems, they consumed less water of 1.56 L/m2/day than T2 and T3 with 1.87 L/m2/day, and T4 with 1.96 L/m2/day. Water consumption was significantly different between DWC and SP systems.
Regarding biomass production in an aquaponics experiment with Swiss chard, Kaburagi et al.37 investigated growing Swiss chard (Beta vulgaris L. spp. Cicla) using saline fish wastewater with micronutrient supplementation. Their results reported approximately 68, 42, and 18 g/plant leaves fresh weight during the first, second, and third cuts, respectively, using fish wastewater and micronutrient supplements. Irrigating plants with fish wastewater alone reported 65, 25, and 10 g/plant of fresh leaf weight during the first, second, and third cut. Their total yields amounted to 138 g/plant and 100 g/plant for fish wastewater plus micronutrients and fish wastewater alone. In our study, however, we recorded higher yields from the range of 88–170 g/plant for the first cut, then 156–255 g/plant and 260–280 g/plant in the third cut without supplementary application of micronutrients. The difference in results could be attributed to the growth media used, type of fish, and other parameters such as feeding rate for fish, etc. It’s hypothesized that the sand-based IVAS systems used in our study provided micronutrients and a desirable EC that enhanced nutrient uptake in plants, improving their growth.
Mineral elements are significantly essential for any plant’s growth. Calcium is the plant nutrient most often associated with tissue firmness. This is due to its ability to form cross-linkages with pectins by the ionic association between C’6 carboxyl groups of intra and inter galacturonosyl residues38. In principle, high calcium levels maintain tissue integrity and increase tissue elasticity rather than tissue rigidity. The increasing leaf area and total fresh weights recorded in T2, T3, and T4 SP treatments could be attributed to high calcium contents in the irrigation water compared to those grown in T1. Likewise, previous studies have shown that a high magnesium supply and EC increase tissue firmness39,40. This study shows that the magnesium supply and EC of irrigation water in SP treatments were higher than the DWC. This could also explain larger leaf areas and higher fresh weights in plants grown in SP treatments than in DWC.
On the other hand, magnesium is an important nutrient required for plant growth. It is not only the central core of the chlorophyll molecule in plant tissues but also aids in the activation of specific enzymes. Magnesium deficiency in plants results in stunted growth41. In this study, we recorded higher quantities of magnesium (575–184.36 mg/100 g DW) which was higher than that reported in previous studies42,43.
Microelements such as Fe, Cu, Mn, and Zn play a vital role in redox processes, and cofactors activate approximately 35 different enzymes44. Concentrations of Fe and Zn were within those of Swiss chard reported by Bozokalfa et al.45 except for Cu. Likewise, this study's Cu and Mn contents were lower than those reported in previous studies43,46. This could be attributed to the difference in cultivation practice and variety of the plant.
Vitamin C (ascorbic acid and dehydroascorbic acid) is one of the most important quality parameters of fruits and vegetable crops. It is an essential vitamin for humans, and 90% of it is obtained from fruits and vegetables. However, its content in plant tissues depends on the EC of irrigation water and levels of antioxidant enzyme activity. Ascorbic acid is one of the major antioxidants which could be stimulated under abiotic stress. Likewise, antioxidant enzyme activity is important for determining whether the plant is suffering from biotic or abiotic stress35. This study observed an increase in vitamin C content with an increasing cut number. Therefore, we anticipate that a combination of high EC (2.6–2.8 ds/m) and plant cuts increased plant tissue stress levels, resulting in high antioxidant enzyme activity and increasing vitamin C content. Similar results have been reported on tomatoes38 and Brassica campestris L. ssp. Chinensis35. Moreover, the vitamin C concentrations in this study ranged from 11.22 to 28.88 mg/100 g fresh weight, which is lower to those reported by Ivanović et al.46 and higher than that reported by Daiss et al.47 and Agüero et al.48. We suggest that the difference in results could be attributed to differences in cultivation practices and plant species.
Beta carotene is a precursor for vitamin A and one of the most important vitamins for human health. Vegetables from the Brassicaceae family contain the highest beta carotene amounts, ranging from 0.50 to 19.60 mg/100 g FW49. In our study, vitamin A ranged from 1.30 to 2.36 mg/100 FW, higher than that reported by Mzoughi et al.44. However, our results are not in agreement with those reported by Ivanović et al.46. They reported higher amounts of beta carotene (4.41 to 13.40 mg/100 g fresh weight). Likewise, Reif et al.49 reported higher amounts of beta carotene (6.82 mg/100 g FW) in cv. Carrot bolero of the Daucus carota L. Swiss chard species. The difference could be attributed to plant species or cultivars used, cultivation practices, and chemicals used in the extraction protocol. We compared the values for the Swiss chard variety used in this study with those referenced by the USDA data (3.64 mg/100 g FW) and found a minimal difference hence making this variety a good source of vitamin A.
High levels of soluble sugars are a desirable parameter in terms of food quality. Soluble sugar contents are influenced by several factors such as salinity, EC of irrigation water, and cutting time point. In this study, high EC values (2.6–2.8 dm/s) of irrigation water were noted in the studied SP treatments, which could cause declining contents of total carbohydrates. Similar results have been previously reported by Ding et al.35. High EC increases the respiration rate of tissues which might reduce sugar content50. The quantified total proteins in this study were higher than those earlier reported by Daiss et al.47 and lower than those reported by Ivanovic et al.46. This could be attributed to differences in variety, cultivation practices, and extraction methods. In this study, we reported similar crop water contents ranging from 92.14 to 93.54%, which was in the same range as that reported by Moreira et al.51 and Daiss et al.47.
Conclusion
This study found that sandponics can be an alternative way to produce an increased crop yield in poor sand soils if the soil chemical composition does no harm to the fish. Sandponics uses synergies since the fish wastewater acts as a nutrient source to the crops, and sand acts as a bio-filter to clean the fish water. Sandponics is a promising sustainable crop production technique, efficient and low-risk application. This experiment found that sourcing good quality sand with low levels of carbonates, EC, Na, K, heavy metals and low levels of salts is essential for an efficient functionality of the sandponics system. Additionally, a neutral pH, sand grain size and good drainage are other key factors for an efficient system. Sand quality composition like that from T4 and T3 would be a great sand component/unit of the sandponics system. It drains efficiently, low levels of heavy metals plus carbonates. The study recommends that T4 and T3 sand types would be able to produce sufficient crop yields in terms of biomass and nutritive composition. As compared to DWC, sandponics also minimize water usage to as low as 1.96 L/m2/day. Further research is needed to investigate similar sand qualities with different crops and different fish types (Supplementary Tables 1 and 2).
|
2023-03-22 08:46:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37829330563545227, "perplexity": 4335.34451229981}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00444.warc.gz"}
|
https://gitee.com/openeuler/blog/blob/master/design/content_posts.md
|
## openEuler / blogMIT .gitee-modal { width: 500px !important; }
Explore and code with more than 5 million developers，Free private repositories ！：）
Blog project is used to house the blogs that related openEuler community. spread retract
• HTML 84.5%
• CSS 15.5%
content_posts.md 2.80 KB
Fred_Li authored 2020-03-03 16:06 . typo fixed, s/summuary/summary
# Posts
This file is to explain in which way the content of the blogs are stored and read by the blog system.
## What is supported in the blog
A blog can include many formats of information, like text, pictures, videos, animations or others.
openEuler Blog is designed to support the following formats:
1. text
2. static picture
4. animation
## Folder design
The content of blogs are under ./content
|__ _example --list the some blog examples
|__ guidance --house the guidance to post and maintain the blogs
|__ post --house all the final posts
|__ author_1 --house the blogs by authors' gitee ID, and each author need create your own foler by your id.
|__ author_2 --house the blogs by authors' gitee ID
## Post content design
### File name
To create a post, add a file to your post/author_1/ directory with the following format:
YEAR-MONTH-DAY-title.MARKUP
Where YEAR is a four-digit number, MONTH and DAY are both two-digit numbers, and MARKUP is the file extension representing the format used in the file. For example, the following are examples of valid post filenames:
2020-01-01-new-years-is-coming.md
2020-02-15-how-to-write-a-blog.md
Functionally, the post should support categories, archives, title, date, brief description, thus the file headers should be as below.
+++
title = ""
date = "yyyy-mm-dd"
tags = ["aaaa", "bbbb", "cccc"]
archives = "yyyy-mm" //by months
author = "name of author"
summary = ""
+++
### Including resources
At some point, you’ll want to include images, downloads, or other digital assets along with your text content.
You can put the resources in the same folder as your text file's, and name the resources as
YEAR-MONTH-DAY-title-NN.MARKUP
Where the YEAR, MONTH, DAY, and title are the same as your blog file, and NN is the serial number of the pictures, like 01, 02 and so on. The MARKUP is the file extension, and for pictures it is recommended to use png. The following are one example.
2020-01-01-new-years-is-coming.md
2020-01-01-new-years-is-coming-01.png
2020-01-01-new-years-is-coming-02.gif
2020-01-01-new-years-is-coming-03.pdf
Then, from within any post, they can be linked to using the site’s root as the path for the asset to include. Here are some simple examples in Markdown:
Including an image asset in a post:
... which is shown in the screenshot below:
... you can [get the PDF](/content/post/yyyymm/2020-01-01-new-years-is-coming-03.pdf) directly.
... you can [read more](<https://gitee.com/openeuler/>).
## Thanks
The content above refered to https://jekyllrb.com/docs/posts/#the-posts-folder.
1
https://gitee.com/openeuler/blog.git
git@gitee.com:openeuler/blog.git
openeuler
blog
blog
master
|
2020-07-10 20:20:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25922057032585144, "perplexity": 9411.765474889055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655911896.73/warc/CC-MAIN-20200710175432-20200710205432-00123.warc.gz"}
|
https://www.bloombergprep.com/gmat/practice-question/1/1740/quantitative-section-algebra-inequalities-overview/
|
We cover every section of the GMAT with in-depth lessons, 5000+ practice questions and realistic practice tests.
## Up to 90+ points GMAT score improvement guarantee
### The best guarantee you’ll find
Our Premium and Ultimate plans guarantee up to 90+ points score increase or your money back.
## Master each section of the test
### Comprehensive GMAT prep
We cover every section of the GMAT with in-depth lessons, 5000+ practice questions and realistic practice tests.
## Schedule-free studying
### Learn on the go
Study whenever and wherever you want with our iOS and Android mobile apps.
# Inequalities: Overview
If $$4 < 7 - 2x < 12$$, how many integer solutions of $$x$$ are there?
Incorrect.
[[Snippet]]
Correct. [[Snippet]] Subtract $$7$$ from the inequality to isolate $$-2x$$. >$$4 - 7 < 7 - 2x - 7 < 12 - 7$$ >$$-3 < -2x < 5$$ Divide the inequality by $$2$$ to isolate $$-x$$. >$$\frac{-3}{2} < \frac{-2x}{2} < \frac{5}{2}$$ Multiply the inequality by $$-1$$ to isolate $$x$$. Do not forget to flip signs due to multiplication by $$-1$$. >$$-\left(\frac{-3}{2}\right) > -1(-x) > -\frac{5}{2}$$ >$$\frac{3}{2} > x > -\frac{5}{2}$$ Based on this, $$x$$ is more than $$-\frac{5}{2}$$ and less than $$\frac{3}{2}$$ (i.e., the integer values of $$x$$ are $$-2$$, $$-1$$, $$0$$, and $$1$$). Hence, there are four possible integer values for $$x$$.
Incorrect.
[[Snippet]]
Incorrect.
[[Snippet]]
Incorrect.
[[Snippet]]
6
4
2
1
0
|
2020-10-21 18:12:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8564894795417786, "perplexity": 1867.0013173689458}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107877420.17/warc/CC-MAIN-20201021180646-20201021210646-00444.warc.gz"}
|
https://hsm.stackexchange.com/questions/8167/how-did-newton-and-leibniz-interpret-the-integral/8168
|
How did Newton and Leibniz interpret the integral?
How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?
• They thought of it both ways. In fact this is their main discovery: that anti-derivative is related to the area. – Alexandre Eremenko Jan 15 '19 at 19:25
|
2020-08-12 13:20:28
|
{"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": 0.9209463000297546, "perplexity": 353.57678770362656}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738892.21/warc/CC-MAIN-20200812112531-20200812142531-00555.warc.gz"}
|
http://openstudy.com/updates/4d5de0f73147b7641e19fc6f
|
## anonymous 5 years ago How to solve a fraction when solving literal equations
1. anonymous
The problem is A = (B+K) over G, and it is a literal equation
2. anonymous
solving for K
So you have: $A = \frac{B + K}{G}$ Let's say the $$G$$ and the $$B$$ were numbers: $A = \frac{2 + K}{3}$ How would you solve for K then?
|
2016-09-30 05:00:41
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8431938290596008, "perplexity": 1482.4532771216873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662022.71/warc/CC-MAIN-20160924173742-00215-ip-10-143-35-109.ec2.internal.warc.gz"}
|
https://www.physicsforums.com/threads/converting-triple-integral-coordinates.363036/
|
Homework Help: Converting triple integral coordinates
1. Dec 13, 2009
619313
1. consider the triple integral (x^2 +Y^2) dV where it is bounded by a solid sphere of radius R. Set up the integral using rectangular coordinates
I tried setting this up with the bounds [ -sqrt(R^2-x^2-Y^2) <= Z <= sqrt(R^2-x^2-Y^2) ,
-R <= X <= R , -sqrt(R^2-x^2) <= Y <= sqrt(R^2-x^2) ] am I on the right path???
Last edited: Dec 13, 2009
2. Dec 13, 2009
LCKurtz
Yes, but make sure you have them in the correct order when you write down the integrals.
3. Dec 13, 2009
619313
Fantastic! Yea I know that it goes dz dy dx
now How about if I rewrite to cylindrical coordinates?
0<= theta <= 2pi , 0<=r<=R , -sqrt(R^2-r^2)<=z<= sqrt(R^2-r^2) with the integrand being r^3 dz dr dtheta
4. Dec 13, 2009
LCKurtz
Yes, you've got it.
5. Dec 13, 2009
619313
first, thanks so much for the help
just to make sure I fully understand all of these triple integrals, to set it up in spherical I should get
0<= theta<= 2pi, 0<= phi <= pi, 0<= p <= R,
where the integrand is (p^4) (sin(FI))^3 dp dphI dtheta
6. Dec 13, 2009
LCKurtz
Yes. Very nice. And welcome to the Physics forum.
|
2018-12-17 14:24:34
|
{"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": 0.8580162525177002, "perplexity": 3247.5033272165356}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376828507.84/warc/CC-MAIN-20181217135323-20181217161323-00249.warc.gz"}
|
https://nerdylab.me/how-to-square-a-fraction-exponent/
|
# How To Square A Fraction Exponent Ideas
How To Square A Fraction Exponent. $\sqrt[n] x = x ^ {\frac 1 n}$ $$a^\frac{b}{c} = \sqrt[c]{a^b}$$.
52 = 5 × 5 = 25step 2, realize that squaring fractions works the same way. = polynomial fraction can be simplified with the polynomial present in the numerator or denominator by facotrising and reducing them to the lowest terms.
### 10 Free Low Prep Math Activities Just Print And Go
A radical is simply a fractional exponent: Adding, subtracting, dividing, multiplying exponents.
### How To Square A Fraction Exponent
Change the expression with the fractional exponent back to radical form.Combine the b factors by adding the exponents.Enter
1 at the top of your fraction.Exponent multiplication, solve quadratic equations in matlab, free printout for basic maths, mcdougal littell algebra 2 answers, simplifying square roots calculator, fraction as a decimal worksheet, intermediate algebra calculator.
For cube roots, raise to the 1/3 power.Fractional exponent (shown as b) the denominator of the fractional exponent is the root :From fractions import fraction var1 = input (‘please enter number:’) var1 = fraction (var1) expo = fraction (‘1/2’) //put your fractional exponent here var2 = var1**expo print var2.Glencoe mathematics, algebra 2, tests california edition tests, how to find the square root of a number easily.
Hard integral fraction on exponent and fraction multiplying.Here the square root of 5 is irrational and can be left as the square root of 5, however, the square root of 9 = 3.Hot network questions what is the point of c3 in the.How fractional exponents are related to roots.
How to solve a system of equations with 3 variables in calculator.How to work out this easy fraction?If you get something like (5/9)^ (1/2), take the square root of the numerator and denominator separately.In this lesson we’ll work with both positive and negative fractional exponents.
It shows how many times the base (shown as z) will be multiplied by itself.Let p(x) and q(x), where q(x) cannot be zero.Now you will square the fraction 3 4.Once you have the power in the denominator, you have a positive exponent in the form of a fraction, which you can transform into a root, but keep in mind that it remains in the denominator:
Polynomial fraction is an expression of a polynomial divided by another polynomial.Product and quotient of roots.Raising to a fraction takes the root.Remember that when a a a is a positive real number, both of these equations are true:
Rewrite the fraction as a series of factors in order to cancel factors (see next step).Rewrite the radical using a fractional exponent.Right from convert square root of x to exponent form to multiplying and dividing fractions, we have got all the pieces covered.Simplify the constant and c factors.
Simplifying square root with fraction.Since (1/2) is a rational number, you could alternatively use =d2^0.5.So your answer would be (the square root of 5)/3.Square root of a fraction exponent.
Square the fraction as you would normally by multiplying the numerator by itself and then multiplying the denominator by itself.Step 1, understand how to square whole numbers.The first thing you have to do is pass the exponent from negative to positive and this is done by passing the power to the denominator.The following property will probably be the one we will use the most:
The product of two roots with the same degree is the root (of same degree) of the product of the radicands, this is, $$\sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{a\cdot b}$$.The square (2nd) root of x is just x 1/2, the cube (3rd) root is just x 1/3, and so on.The steps of simplifying a fraction?The symbol is also called the radical sign.
This feature was designed to encourage users to write their own functions.This is a re upload.To calculate exponents such as 2 raised to the power of 2 you would enter 2 raised to the fraction power of (2/1) or 2 2 1.To calculate radicals such as the square root.
To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical.To square a fraction, you multiply the fraction.To square a whole number, you multiply it by itself.[2] x research source for example:Under equation tools on the design tab in the structures group click the radical button.
When changing from radical form to fractional exponents, remember these basic forms:When you find square roots, the symbol for that operation is a radical, which looks like this:When you see an exponent of two, you know that you need to square the number.With this fact at your disposal, you’re in good shape.
You know that the square root of x is x 1/2 and the cube root of that is (x 1/2) 1/3.
|
2021-10-27 12:10:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.8695263266563416, "perplexity": 895.248052654273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588153.7/warc/CC-MAIN-20211027115745-20211027145745-00338.warc.gz"}
|
https://arcsecond.wordpress.com/2011/03/25/visualizing-elementary-calculus-introduction/
|
## Visualizing Elementary Calculus: Introduction
Recently I’ve been trying to be more geometrical when discussing elementary calculus with high school students. I don’t want to write an entire introduction to calculus, but the next few posts will outline some ways I think the geometric view can be helpful.
This series
I – Introduction
II – Trigonometry
You know about $\Delta$, which means “the change in”. For example, if $w$ represents my weight, then $-\Delta w$ represents the weight of the poop I just took.
Let’s say $h$ is your height above sea level. $\Delta h$ is the change in that height, but what change? The change when you climb the stairs? When you jump out of a plane? When you step on a banana peel?
When we think about change, we usually think about two things changing together. You get higher when you climb another stair on the staircase. $h$ is changing, and so is $s$, the number of stairs climbed.
These two changes are related to each other. Say the stairs are 10 cm high. Then you gain 10 cm of height for each stair. We can write that as $\Delta h = 10 {\rm cm} \hspace{.5em} \Delta s$. We can also write it $\Delta h / \Delta s = 10 \hspace{.5em}{\rm cm}$. This says, “the height per stair is ten centimeters.”
This is the goal of calculus – to study the relationships between changing quantities. Let’s do a real example.
### The Area of a Square
Let’s say we have a square whose sides lengths are $x$. Its area is $x^2$. What is the relationship between changes in its area and changes in the length of a side? Draw the square, then expand the sides some. The amount the sides have expanded is $\Delta x$. The new area that’s been added is $\Delta (x^2)$.
We begin with the red square on the left, whose area is x^2. We add an extra amount Delta(x) to the sides, creating all the new green area.
From the picture we see
$\Delta(x^2) = 2x\Delta x + (\Delta x)^2$
This formula relates $\Delta (x^2)$, the change in the area, to $\Delta x$, the change in the length of a side.
### The Derivative of $x^2$
In the picture of the square, there is a little piece in the upper-right corner whose area is $(\Delta x)^2$. It is the smallest bit of area in the whole picture.
Look what happens when we make $\Delta x$ even smaller.
We shrink Delta(x) and observe what happens to the different areas being added on.
In the first picture, $\Delta x$ (no longer marked) is a quarter of $x$. $(\Delta x)^2$ is the dark green area, and it is one quarter as large as $x \Delta x$, the light green area. We see this because the dark patch fits inside the light one four times.
In the second picture, we shrink $\Delta x$ to one eighth of $x$. All the green areas shrink, but the dark patch shrinks on two sides while the light patches shrink on only one. As a result, the dark $(\Delta x)^2$ is now only one eighth the size of the light $x \Delta x$.
If we continued to shrink $\Delta x$, this ratio would continue to decrease. Eventually we could tile the dark patch a million times into the light one. So, as long as $\Delta x$ is very small, we can get a good estimate of the entire green area by ignoring the dark part $(\Delta x)^2$. Thus
$\Delta(x^2) \approx 2x\Delta x$
This approximation becomes better and better as $\Delta x$ shrinks, becoming perfect as $\Delta x$ becomes infinitesimally small.
When we want to indicate these infinitely small changes, we trade in the $\Delta$ for a ${\rm d}$ and write
$\textrm{d}(x^2) = 2x \textrm{d}x$
The terms $\textrm{d}(x^2)$ and $\textrm{d}x$ are called “differentials”. The equation expresses the relationship between two infinitely-small changes, one in $x$ and one in $x^2$.
Frequently, we divide by $\textrm{d}x$ on both sides to get
$\frac{\textrm{d}(x^2)}{\textrm{d}x} = 2x$
This is called “the derivative of $x^2$ with respect to $x$“.
#### Example 1: Estimating Squares
$20^2 = 400$. What is $21^2$?
Here $x$ = 20, and we’re looking at $x^2$. When $x$ goes from 20 to 21, it changes by 1, so $\textrm{d}x = 1$. Our formula tells us
$\textrm{d}(x^2) = 2x \textrm{d}x = 2*20*(1) = 40$
Hence, $x^2$ increases by about 40, from 400 to 440.
The real value is 441. We got the change in $x^2$ wrong by about 2%. That’s because $\textrm{d}x$ wasn’t infinitely small.
Let’s try again, this time estimating the square of 20.00458. Now $\textrm{d}x$ = .00458, so
$\textrm{d}(x^2) = 2 x \textrm{d}x = 2*20*.00458 = .1832$
The estimate is 400.1832. The real value is 400.183221. We did much better, under-estimating the change by only 0.01% this time. Also, it was not much harder to do this problem than the last, but squaring out 20.00458 by hand would be a pain. We saved some work.
#### Example 2: How Far Is the Horizon?
The beach is a good place to think about calculus. If you look out at the ocean, the horizon appears perfectly flat. Nonetheless, we know the Earth is really curved. In fact, we can deduce the curvature of the Earth by standing on the beach and enlisting the help of a friend in a boat.
It works like this: You stand on the beach with your head two meters above the water. Your friend sails away until the boat begins to disappear from sight. The reason the bottom of the boat is disappearing is that it is hidden behind the curvature of Earth.
When the bottom of the boat disappears, measure the distance to some part of the boat you can still see. What’s the relationship between your height, the distance to the boat, and the radius of Earth?
A picture will help. We’ll call your height $h$ and the distance to the horizon $z$.
You are the vertical stick on top, height h. The boat is the brown circle. It's at the horizon, a distance z away. The dotted line shows your line of sight. When the bottom of the boat begins disappearing, a right triangle forms.
Your height, the radius of Earth, and the distance to the horizon are related by the Pythagorean theorem to give
$R^2 + z^2 = (R+h)^2$
this is equivalent to
$z^2 = 2Rh + h^2$
As we have seen, if your height $h$ is small compared to the size of the Earth (and it is), the term $h^2$ drops away and the distance to the horizon is
$z = \sqrt{2Rh}$
You can see about $5 {\rm km}$ at the beach, making the radius of Earth about $6,000 {\rm km}$. (It’s actually $6378.1 {\rm km}$).
Next we want to know how much further you can see if you stand on your tiptoes. That would be a small change $\textrm{d}h$ to your height. It would let you see a small amount $\textrm{d}z$ further. How is $\textrm{d}h$ related to $\textrm{d}z$?
$\textrm{d}(x^2) = 2x\textrm{d}x$
So let $x^2 = h$, or $x = \sqrt{h}$, and we have
$\textrm{d}h = 2\sqrt{h}\hspace{.3em}\textrm{d}(\sqrt{h})$
But we also know
$\sqrt{h} = \frac{z}{\sqrt{2R}}$
so we can substitute that in to $\textrm{d}(\sqrt{h})$ and get
$\textrm{d}h = 2\sqrt{h}\hspace{.3em}\textrm{d}\left(\frac{z}{\sqrt{2R}}\right)$
or
$\frac{\textrm{d}z}{\textrm{d}h} = \sqrt{\frac{R}{2h}}$
This tells us how much further you can see if you get a little higher up. The interesting thing is it depends on $h$. The higher you go, the smaller $\textrm{d}z$. When you’re only two meters up, you get to see almost ten meters further out for every centimeter higher you go. However, if you’re 100m up on top a carousel, you get only 1 meter for each centimeter you rise.
It makes sense that the extra distance you see gets smaller and smaller the higher you go, and eventually shrinks down to zero. No matter how high you go, you can never see more than a quarter way around the globe.
(In reality, light bends due to refraction in the atmosphere, so you can sometimes see a bit further.)
### Circles
Suppose we have a circle with radius $r$. It has a certain area (you undoubtedly know the formula already, but play along). Suppose we increase $r$ by a small amount $\textrm{d}r$. What is the change $\textrm{d}A$ in the area?
The original circle is dark blue with area A and radius R. The radius increases an amount dR, increasing the area by the light blue ring with area dA.
$\textrm{d}A$ is the thin, light-blue ring. Imagine taking that ring and peeling it off the edge of the circle and laying it flat. We’d have a rectangle with width $\textrm{d}R$. Its length comes from the outside edge of the entire circle – the circumference. The circumference is $2 \pi R$, so
$\textrm{d}A = 2\pi R \textrm{d}R$
We saw earlier that $\textrm{d}(x^2) = 2x\textrm{d}x$, so let $x = R$ and we have
$\textrm{d}A = \pi \textrm{d}(R^2)$
Thus the quantities $A$ and $\pi R^2$ change in exactly the same way. Since they also start out the same (both zero when R is zero), we have
$A = \pi R^2$
### Next Post
We’ll look at trigonometry. Geometric arguments about the derivatives of trig functions are very simple ways of visualizing what’s going one, and are usually not introduced in a basic calculus course.
### Exercises
• Draw a cube with sides $x$ and show that $\textrm{d}(x^3) = 3x^2\textrm{d}x$. Thus the derivative of $x^3$ with respect to $x$ is $3x^2$.
• Draw a line with length $x$ and show that $\textrm{d}(x) = \textrm{d}x$, which is of course algebraically obvious. Thus the derivative of $x$ with respect to itself is 1.
• Draw a rectangle with width $w$ and length $c*w$ and show that $\textrm{d}(c*w^2) = 2cw\textrm{d}w = c\textrm{d}(w^2)$. Thus, whenever you have the differential of a variable multiplied by a constant, the constant can pop outside. Where was this property used implicitly in this post?
• Now that you know $\textrm{d}(x^3) = 3x^2\textrm{d}x$, let $x^3 = u$ and find the derivative of $u^{1/3}$ with respect to $u$. (Answer: $\frac{1}{3} u^{-2/3}$)
• What is $\textrm{d}(x^3)/\textrm{d}(x^2)$? Let $u = x^2$ and find the derivative of $u^{3/2}$ with respect to $u$. (Answer: $\frac{3}{2}u^{1/2}$).
• Examine $\textrm{d}(x^4)$ by letting $u = x^2$, so we’re looking at $\textrm{d}(u^2)$. Find the derivative of $x^4$ with respect to $x$. (Answer: $4x^3$)
• Draw an equilateral triangle with sides of length $s$. Increase the sides a small amount $\textrm{d}s$ and relate this to the change in area $\textrm{d}A$. Does this agree with our previous findings?
• Draw an ellipse with a fixed with semi-major axis $a$ and semi-minor axis $b$. Starting with a unit circle, argue by thinking about stretching that the area of the ellipse is $\pi ab$. Increase $a$ by a small amount $\textrm{d}a$ and increase $b$ proportionately. This adds a small area $\textrm{d}A$ to the ellipse. Show that this area is $\pi(a^2+b^2)/b\hspace{.3em}\textrm{d}a$. Does this let us find the circumference of the ellipse by the same thought process as we used for the circle? (Answer: no). Why not?
• Draw a sphere with radius $R$. Use the relationship between $\textrm{d}R$ and $\textrm{d}A$ to find the volume of a sphere, given its surface area is $4\pi R^2$. Check your answer against this post.
|
2019-09-23 01:08:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 136, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8158850073814392, "perplexity": 312.9372574580595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575844.94/warc/CC-MAIN-20190923002147-20190923024147-00433.warc.gz"}
|
http://www.physicsforums.com/showthread.php?p=3756810
|
# Acousto optical modulators
by Niles
Tags: acousto, modulators, optical
P: 1,818 Hi Say I am looking at an AOM working in the Bragg regime (i.e., only a single diffracted beam). It is easy to show using Bragg's law that the frequency-shift Ω of the diffracted wave is given by $$\Omega = 2n\omega \frac{v}{c}\sin(\theta)$$ Here Ω is *also* the frequency the AOM is driven with, in other words the LHS is constant in the sense that in does not depend on the incoming light (so the frequency-shift imparted on the wave is constant). However, the RHS does depend on the incoming light, since the angle θ of the diffracted beam is equal to the angle of incidence of the incoming beam, so I can change it easily by e.g. turning the AOM. In my book it says that the shift Ω is zero for forward scattering and maximum for backscattering. This is what I don't understand: The shift Ω is the same as the frequency of the phonons in the material, which is *constant*. So how can I change the frequency shift of the diffracted wave by changing the angle on incidence? Best, Niles.
P: 1,818 OK, I understand my error now.
Sci Advisor P: 5,401 What book are you using? I have one by Korpel, and it's not working for me...
P: 1,818
## Acousto optical modulators
I am using Boyd's Nonlinear Optics, it has a nice chapter on spontaneous light-scattering including Acoustooptics. I hope it works out.
Best,
Niles.
Related Discussions Biology 5 Classical Physics 2 Classical Physics 2 Electrical Engineering 0
|
2013-12-08 09:44:35
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7330103516578674, "perplexity": 924.9710313709252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163059081/warc/CC-MAIN-20131204131739-00038-ip-10-33-133-15.ec2.internal.warc.gz"}
|
https://math.stackexchange.com/questions/3092495/orientation-of-a-manifold-with-trivial-tangential-bundle
|
# Orientation of a Manifold with Trivial Tangential Bundle
Let $$M$$ be a smooth (eg $$C^{\infty}$$) manifold. Let assume that $$M$$ has trivial, oriented tangent bundle $$TM$$, so $$TM \cong M \times \mathbb{R}^n$$ for appropriate $$n$$ and orientable.
How to conclude that in this case $$M$$ is also orientable?
My considerations:
I use the orientability criterion with charts: eg a manifold $$N$$ is orientable iff there exists an atlas $$(\phi_i:U_i \to V_i)_{i \in I}$$ with $$U_i \subset N, V_i \subset \mathbb{R}^n$$ with following property:
for all $$i,j \in I$$ with $$U_i \cap U_j \neq \varnothing$$ the differential $$D_p(\phi_j \circ \phi_j^{-1}) =T_p(\phi_j \circ \phi_j^{-1})$$ of restricted map $$\phi_j \circ \phi_j^{-1} \vert _{U_i \cap U_j}$$ at every $$p \in U_i \cap U_j$$ has positive determinant.
An attempt is to start with product charts for the oriented (by assumption) tangent bundle of the shape $$\phi_i \times id_{\mathbb{R}^n}$$ which form an oriented atlas for $$TM \cong M \times \mathbb{R}^n$$ and trying to restrict them to charts $$\phi_i$$ is a sophisticated way (modyfying them by multiplying if neccessary with $$\pm 1$$ in appropriate cases ) to get an induces oriented atlas on $$M$$. But I'm not sure how and if that could work. Does anybody have a better idea?
Remark: Since every tangent bundle of a manifold is oriented I think that the triviality of $$TM$$ is the main ingredient here.
• Use the criterion in terms of a volume form. – Moishe Kohan Jan 30 at 3:44
• @MoisheCohen: Hi, do you mean the criterion that if $\Lambda^n TM$ (= volume form) is trivial then $M$ orientable? Or do you mean the other one? By the way: o you maybe see a more "elementary" argument (just using the positive determant criterion for transition maps) to verify that $M$ is orientable if $TM$ is trivial. Maybe considering the local structure of charts of $TM$? – KarlPeter Jan 30 at 18:01
• Yes, this is what I mean. The point is that if $E\to B$ is a trivial vector bundle, so are its exterior powers (and other tensor powers of course), hence, there is really nothing to be proven. – Moishe Kohan Jan 30 at 18:09
• @MoisheCohen: hmmm I'm just a bit curious if there is a more "elementary" argument using only the characterisation for orientability with transition charts as above. Maybe in some kind by analysing the local stucture of transition maps between charts and their differentials (triangle structure) for $TM$ using the triviality assumption. Do you know if it could work? – KarlPeter Jan 30 at 18:17
• @MoisheCohen: This argument with vanishing volume form is essentially the same as the overkill with Stiefel Whitney classes (if we accept that De Rham and simplicial cohomology provide the same theory then the argument essentially the same and taking into account the concrete construction for De Rham cohomology). Do you see some not too "high tech" argument? – KarlPeter Jan 30 at 18:22
Hint: $$(dx_1 \wedge ... \wedge dx_n)(v_1,...,v_n)= det(V),$$ where $$V$$ is the matrix whose column vectors are $$v_1,...,v_n$$. This simply comes from the definition of the wedge product $$u_1\wedge ... \wedge u_n$$ as the alternation of the tensor product $$u_1\otimes ... \otimes u_n$$: $$u_1\wedge ... \wedge u_n= Alt(u_1\otimes ... \otimes u_n)= \sum_{\sigma} sign(\sigma) u_{\sigma(1)} \otimes ... \otimes u_{\sigma(n)}$$ where the sum is taken over all permutations $$\sigma$$ of $$\{1,...,n\}$$. You also use the definition of pairing of covariant and contravariant tensors: $$dx_{\sigma(1)}\otimes ... \otimes dx_{\sigma(n)}(v_1,...,v_n)= (dx_{\sigma(1)}(v_1))...(dx_{\sigma(n)}(v_n))= v_{{\sigma(1)}1}\cdots v_{{\sigma(n)}n}.$$
If $$A$$ is a (linear) endomorphism of $$R^n$$, then $$(A^*(dx_1 \wedge ... \wedge dx_n))(v_1,...,v_n)= dx_1 \wedge ... \wedge dx_n(w_1,...,w_n),$$ where $$w_i=A v_i, i=1,...,n$$. Hence, if $$W$$ is the matrix with column-vectors $$w_1,...,w_n$$ then $$det(W)= det(A) det(V)$$.
Edit. For those who did not read the exchange in the comments section, here are relevant details. Since $$TM$$ is a trivial bundle, so is its $$n$$-th exterior power, $$\wedge^n TM$$. In particular, $$\wedge^n TM$$ admits a nonvanishing section, a volume form $$\eta$$. This is all what is needed to construct an orientation-preserving atlas on $$M$$.
As a general remark, if $$E\to B$$ is a rank $$n$$ vector bundle then $$\wedge^n E\to B$$ is naturally isomorphic to the determinant bundle $$det(E)$$, i.e. the rank one bundle over $$B$$ whose transition maps (between the fibers) are the determinants of the transition maps of the original bundle. This is what the above hint is about. Applying this to the bundle $$E=TM$$, we obtain that if $$\eta$$ is a degree $$n$$ form on $$M$$ and $$\eta_i=\nu_i(x)dx_1\wedge ... \wedge dx_n$$ are the expressions of $$\eta$$ in local coordinate charts $$\phi_i: U_i\to M$$, then for any two charts $$\phi_i, \phi_j$$ the local expressions are related by the formula $$\eta_i= f_{ij}^* \eta_j, f_{ij}= \phi_j^{-1}\circ \phi_i,$$ or $$\nu_i= det(D f_{ij}) \nu_j.$$ In particular, if $$\eta$$ and the charts are chosen so that $$\nu_i(x)>0$$ for all $$x$$ and all $$i$$ (i.e. the forms $$\eta_i$$ are "positive") then the transition maps $$f_{ij}$$ are orientation-preserving, i.e. $$M$$ is oriented. Assuming that charts are chosen so that $$U_i\cap f_{ij}^{-1}(U_j)$$ is connected for each $$i, j$$ and the form $$\eta$$ is a volume form (i.e. is nowhere zero), we obtain that the charts $$\phi_i$$ can be "corrected" (by composing them with reflections in $$R^n$$ if necessary) so that each $$\eta_i$$ is positive. Thus, if $$M$$ admits a volume form then it is orientable in the sense that it admits an orientation atlas.
Your assumption of a trivialization $$f: M\times \Bbb R^n \to TM$$ yields, for each $$p\in M$$, a linear isomorphism $$f_p:\Bbb R^n \to T_pM$$ given by the composition of $$f|_{\{p\}\times \Bbb R^n}$$ with the identification $$\{p\}\times \Bbb R^n \approx \Bbb R^n$$.
Now let $$\cal A$$ be the maximal atlas for $$M$$. For any chart $$(\phi_j,U_j) \in \cal A$$, say that $$\phi_j$$ is "good" if for all $$p\in U_j$$, $$\det\left[(D_p \phi_j) \circ f_p\right] > 0,$$ and that $$\phi_j$$ is "bad" otherwise. Now remove all of the "bad" charts from $$\cal A$$ to get a collection $$\cal{B}$$. Since any good chart can be obtained from a bad chart (and vice versa) by multiplying one of the coordinate functions by $$-1$$, $$\cal{B}$$ is an (nonmaximal) atlas.
You can check that the atlas $$\cal{B}$$ satisfies your specified orientation-preserving criterion (use the chain rule).
Edit (more details): for any pair of "good" charts, by the chain rule it follows that $$D_{\phi_j(p)}(\phi_i\circ \phi_j)^{-1} = \left[(D_{p}\phi_i)\circ f_{p}\right] \circ \left[(D_{p}\phi_j)\circ f_{p}\right]^{-1},$$ which has positive determinant as desired.
• Ah ok so the point is that for every chart $\phi_j: U_j \to V_j$ on the whole $U_j$ we have only positive or negative $\det\left[(D_p \phi_j) \circ f_p\right]$ and modyfying all $\phi_j \circ f$ with $\det\left[(D_p \phi_j) \circ f_p\right] <0$ by reflection map provides the desired result? – KarlPeter Feb 2 at 15:01
• @KarlPeter: Yes, exactly. And by the chain rule, $D_{\phi_j(p)}(\phi_i\circ \phi_j)^{-1} = \left[(D_{p}\phi_i)\circ f_{p}\right] \circ \left[(D_{p}\phi_j)\circ f_{p}\right]^{-1}$, which has positive determinant. This directly shows that the resulting atlas satisfies your specifically requested criterion. – Matthew Kvalheim Feb 2 at 22:33
|
2019-07-23 07:07:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 90, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.956376314163208, "perplexity": 209.68308908841442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529007.88/warc/CC-MAIN-20190723064353-20190723090353-00171.warc.gz"}
|
https://mathoverflow.net/questions/422209/difference-between-stabilizer-and-automorphism-group-of-subvariety-of-an-abelian
|
# Difference between stabilizer and automorphism group of subvariety of an abelian variety
Let $$X$$ be a smooth closed subvariety of a complex abelian variety $$A$$. Assume $$X$$ is of general type and of codimension one with $$\omega_X$$ ample.
Often, people speak about the stabilizer $$\mathrm{Stab}_A(X)$$ of $$X$$ in $$A$$. This is the group of $$a$$ in $$A$$ such that $$X+a = X$$.
What is the relation of $$\mathrm{Stab}_A(X)$$ to $$\mathrm{Aut}(X)$$?
They are both finite. Are they equal? If the stabilizer is trivial, does that imply $$\mathrm{Aut}(X)$$ is trivial? What about vice versa? Does $$\mathrm{Stab}_a(X)$$ inject into $$\mathrm{Aut}(X)$$?
Crossposted from stackexchange, because I didn't get any replies there unfortunately: https://math.stackexchange.com/questions/4446811/difference-between-stabilizer-and-automorphism-group
• It might be more interesting to ask whether $Aut(X)$ is the subgroup of $Aut(A)$ (automorphisms as a variety) which preserves $X$.
– naf
May 11, 2022 at 5:27
• @naf Yes, that's interesting. I've posted it here now mathoverflow.net/questions/422387/… May 12, 2022 at 15:28
They have absolutely no reason to be equal. Consider the case where $$A$$ is the Jacobian of a genus 2 curve $$C$$, and $$X=C$$ embedded in $$A$$ by $$x\mapsto [x]-[p]$$ for some fixed point $$p\in C$$. Then $$X$$ is a Theta divisor, so $$\operatorname{Stab}_A(X)$$ is trivial. But $$X$$ has always a nontrivial automorphism, the hyperelliptic involution, and it may have more in some cases.
Of course $$\operatorname{Stab}_A(X)$$ injects into $$\operatorname{Aut}(X)$$, since a nontrivial translation does not fix any point of $$A$$. But that is all you can say.
|
2023-01-30 13:56:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 27, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.83827805519104, "perplexity": 154.41891332673944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00095.warc.gz"}
|