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Warning
This assignment is due by Friday October 14 on Blackboard Assignments.
The PDF is useful if you want to print out the problem set and write on it. The R Project is a zipped .zip file which contains a .qmd file to write answers in, and the data, all in a logical working directory. (See this resource for help unzipping files). You can also just write an .R file in the project if you don’t want to use markdown. If you use the cloud project, I have already installed tidyverse and tinytex (to produce pdfs).
# Theory and Concepts
## Question 1
In your own words, describe what exogeneity and endogeneity mean, and how they are related to bias in our regression. What things can we learn about the bias if we know $$X$$ is endogenous?
## Question 2
In your own words, describe what $$R^2$$ means. How do we calculate it, what does it tell us, and how do we interpret it?
## Question 3
In your own words, describe what the standard error of the regression ($$SER$$) means. How do we calculate it, what does it tell us, and how do we interpret it?
## Question 4
In your own words, describe what homoskedasticity and heteroskedasticity mean: both in ordinary English, and in terms of the graph of the OLS regression line.
## Question 5
In your own words, describe what the variation in $$\hat{\beta_1}$$ (either variance or standard error) means, or is measuring. What three things determine the variation, and in what way?
## Question 6
In your own words, describe what a p-value means, and how it is used to establish statistical significance.
## Question 7
A researcher is interested in examining the impact of illegal music downloads on commercial music sales. The author collects data on commercial sales of the top 500 singles from 2017 (Y) and the number of downloads from a web site that allows ‘file sharing’ (X). The author estimates the following model:
$\text{music sales}_i = \beta_0+\beta_1 \text{illegal downloads}_i + u_i$
The author finds a large, positive, and statistically significant estimate of $$\hat{\beta_1}$$. The author concludes these results demonstrate that illegal downloads actually boost music sales. Is this an unbiased estimate of the impact of illegal music on sales? Why or why not? Do you expect the estimate to overstate or understate the true relationship between illegal downloads and sales?
# Theory Problems
For the following questions, please show all work and explain answers as necessary. You may lose points if you only write the correct answer. You may use R to verify your answers, but you are expected to reach the answers in this section “manually.”
## Question 8
A researcher wants to estimate the relationship between average weekly earnings ($$AWE$$, measured in dollars) and $$Age$$ (measured in years) using a simple OLS model. Using a random sample of college-educated full-time workers aged 25-65 yields the following:
$\widehat{AWE} = 696.70+9.60 \, Age$
### Part A
Interpret what $$\hat{\beta_0}$$ means in this context.
### Part B
Interpret what $$\hat{\beta_1}$$ means in this context.
### Part C
The $$R^2=0.023$$ for this regression. What are the units of the $$R^2$$, and what does this mean?
### Part D
The $$SER, \, \hat{\sigma_u}=624.1$$ for this regression. What are the units of the SER in this context, and what does it mean? Is the SER large in the context of this regression?
### Part E
Suppose Maria is 20 years old. What is her predicted $$\widehat{AWE}$$?
### Part F
Suppose the data shows her actual $$AWE$$ is \$430. What is her residual? Is this a relatively good or a bad prediction? Hint: compare your answer here to your answer in Part D.
### Part G
What does the error term, $$u_i$$ represent in this case? What might individuals have different values of $$\hat{u}_i$$?
### Part H
Do you think that $$Age$$ is exogenous? Why or why not? Would we expect $$\hat{\beta_1}$$ to be too large or too small?
## Question 9
Suppose a researcher is interested in estimating a simple linear regression model:
$Y_i=\beta_0+\beta_1X_i+u_i$
In a sample of 48 observations, she generates the following descriptive statistics:
• $$\bar{X}=30$$
• $$\bar{Y}=63$$
• $$\displaystyle\sum^n_{i=1}(X_i-\bar{X})^2= 6900$$
• $$\displaystyle\sum^n_{i=1}(Y_i-\bar{Y})^2= 29000$$
• $$\displaystyle\sum^n_{i=1}(X_i-\bar{X})(Y_i-\bar{Y})=13800$$
• $$\displaystyle\sum^n_{i=1}\hat{u}^2=1656$$
### Part A
What is the OLS estimate of $$\hat{\beta_1}$$?
### Part B
What is the OLS estimate of $$\hat{\beta_0}$$?
### Part C
Suppose the OLS estimate of $$\hat{\beta_1}$$ has a standard error of $$0.072$$. Could we probably reject a null hypothesis of $$H_0: \beta_1=0$$ at the 5% level?
### Part D
Calculate the $$R^2$$ for this model. How much variation in $$Y$$ is explained by our model?
# R Questions
Answer the following questions using R. When necessary, please write answers in the same document (rendered to html or pdf, typed .doc(x), or handwritten) as your answers to the above questions. Be sure to include (email or print an .R file, or show in your rendered quarto document) your code and the outputs of your code with the rest of your answers.
### Question 10
Download the MLBattend dataset. This data contains data on attendance at major league baseball games for all 32 MLB teams from the 1970s-2000. We want to answer the following question:
“How big is home-field advantage in baseball? Does a team with higher attendance at home games over their season have score more runs over their season?”
### Part A
Clean up the data a bit by mutate()-ing a variable to measure home attendance in millions. This will make it easier to interpret your regression later on.
### Part B
Get the correlation between Runs Scored and Home Attendance.
### Part C
Plot a scatterplot of Runs Scored (y) on Home Attendance (x). Add a regression line.
### Part D
We want to estimate a regression of Runs Scored on Home Attendance:
$\text{runs scored}_i = \beta_0 + \beta_1 \, \text{home attendance}_i + u_i$
Run this regression in R.
What are $$\hat{\beta_0}$$ and $$\hat{\beta_1}$$ for this model? Interpret them in the context of our question.
Hint: make sure to save your regression model as an object, and get a summary() of it. This object will be needed later.
### Part E
Write out the estimated regression equation.
### Part F
Make a regression table of the output using modelsummary().
### Part G
Check the goodness of fit statistics. What is the $$R^2$$ and the $$SER$$ of this model? Interpret them both in the context of our question.
### Part H
Now let’s start running some diagnostics of the regression. Make a histogram of the residuals. Do they look roughly normal?
Hint: you will need to use the broom package’s augment() command on your saved regression object to add containing the residuals (.resid), and save this as a new object - to be your data source for the plot in this part and the next part.
### Part I
Make a residual plot.
### Part J
Test the regression for heteroskedasticity. Are the errors homoskedastic or heteroskedastic?
Hint: use the lmtest package’s bptest() command on your saved regression object.
Run another regression using robust standard errors. Hint: use the estimatr package’s lm_robust() command and save the output like the following:
reg_robust <-lm_robust(y ~ x, data = the_data, # change y, x, and data names to yours
se_type = "stata") # we'll use this method to calculate
Now make another regression output table with modelsummary, with one column using regular standard errors (just use your original saved regression object) and another using robust standard errors (use this new saved object)
### Part K
Test the data for outliers. If there are any, identify which team(s) and season(s) are outliers. Hint: use the car package’s outlierTest() command on your saved regression object.
### Part L
Look back at your regression results. What is the marginal effect of home attendance on runs scored? Is this statistically significant? Why or why not?
### Part M
Now we’ll try out the infer package to understand the $$p$$-value for our observed slope in our regression model.
First, save the (value of) our sample $$\hat{\beta}_1$$ from your regression in Part D as an object, I suggest:
our_slope <- 123 # replace "123" with whatever number you found for the slope in part D
Then, using the infer package run the following simulation:
# save our simulations as an object (I called it "sims")
sims <- data %>% # "data" here is whatever you named your dataframe!
specify(y ~ x) %>% # replacing y and x with your variable names
hypothesize(null = "independence") %>% # H_0 is that slope is 0, x and y are independent
generate(reps = 1000,
type = "permute") %>% # make 1000 samples assuming H_0 is true
calculate(stat = "slope") # estimate slope in each sample
# look at it
sims
# calculate p value
sims %>%
get_p_value(obs_stat = our_slope,
direction = "both") # a two-sided H_a: slope =/= 0
Compare to the p-value in your original regression output in previous parts of this question.
### Part N
Make a histogram of the simulated slopes, and plot our sample slope on that histogram, shading the p-value.
You can pipe sims into visualize(obs_stat = our_slope), or use ggplot2 to plot a histogram in the normal way, using sims as the data source and add a geom_vline(xintercept = our_slope) to show our finding on the distribution.
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# How to count (using $\zeta$ functions)
Steffen Højris Pedersen
(Institut for Matematik)
Foredrag for studerende
Fredag, 25 september, 2015, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
There is a delicate relation between counting the number of primes less then a given $x$, and the Riemann $\zeta$ function
$\zeta(s) = \sum_{n=1}^{\infty} \dfrac{1}{n^s}.$
In the talk I will explain how these two things interact with each other. Furthermore I will explain how the framework of $\zeta$ functions can be used in other counting problems of a similar type.
The talk is going to use results from Complex Analysis, but used as black boxes.
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type/with_unit - Maple Programming Help
Home : Support : Online Help : Science and Engineering : Units : Verifications : type/with_unit
type/with_unit
check for a Maple unit
Calling Sequence type(expr, with_unit(typ, coeff, unit))
Parameters
expr - expression typ - (optional) Maple type coeff, unit - (optional) names
Description
• A Maple expression expr is of type with_unit if:
– expr is the function Units[Unit] applied to some arguments;
– expr is a rational power of such a function call;
– expr is a product of such function calls, potentially raised to rational powers, potentially also with another Maple expression;
– a call to the function Units[Unit] is present anywhere in expr (in the sense of the command has) and combine(expr, units) is of one of the three earlier forms described above.
• The function call type(expr, with_unit) returns true if expr is of type with_unit. Otherwise, false is returned.
• In the remainder of this help page, we will refer to the product of the powers of the Units[Unit] function as the unit of the expression, and the product of the remaining factors (if any) as the multiplier. If there are no other factors, the multiplier is $1$. If expr is not of one of the first three forms above and calling combine(expr, units) is necessary to get it in that form, then the unit and multiplier of expr are those of combine(expr, units).
• If an optional type typ is given and expr is of the form described above, it additionally checks that the multiplier is of the type typ. The function returns true if this additional condition is satisfied. Otherwise, false is returned.
• If optional arguments coeff and unit are given and type(expr, with_unit(typ)) returns true, then the unit of expr is assigned to unit and the multiplier is assigned to coeff.
Examples
Notes:
– To enter a unit in 2-D Math input, select the unit from the appropriate Units palette. If the unit you want is not there, select $\mathrm{unit}$ and then enter the unit.
– When you edit a unit, double brackets appear around it.
> $\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right):$
> $3\mathrm{Unit}\left(m\right)$
${3}{}⟦{m}⟧$ (1)
> $\mathrm{type}\left(3\mathrm{Unit}\left(m\right),\mathrm{with_unit}\right)$
${\mathrm{true}}$ (2)
> $\mathrm{type}\left(3\mathrm{Unit}\left(m\right),\mathrm{with_unit}\left(\mathrm{integer}\right)\right)$
${\mathrm{true}}$ (3)
> $\mathrm{type}\left(3{x}^{2}\mathrm{Unit}\left(m\right),\mathrm{with_unit}\left(\mathrm{integer},'a','b'\right)\right)$
${\mathrm{false}}$ (4)
> $a,b$
${a}{,}{b}$ (5)
> $\mathrm{type}\left(3{x}^{2}\mathrm{Unit}\left(m\right),\mathrm{with_unit}\left(\mathrm{anything},'a','b'\right)\right)$
${\mathrm{true}}$ (6)
> $a,b$
${3}{}{{x}}^{{2}}{,}⟦{m}⟧$ (7)
> $\mathrm{type}\left(2\mathrm{Unit}\left(m\right),\mathrm{with_unit}\left(\mathrm{integer},'a','b'\right)\right)$
${\mathrm{true}}$ (8)
> $a,b$
${2}{,}⟦{m}⟧$ (9)
> $\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Natural}\right]\right):$
> $3m$
${3}{}⟦{m}⟧$ (10)
> $\mathrm{type}\left(3m,\mathrm{with_unit}\right)$
${\mathrm{true}}$ (11)
> $\mathrm{type}\left(3m,\mathrm{with_unit}\left(\mathrm{integer}\right)\right)$
${\mathrm{true}}$ (12)
> $\mathrm{type}\left(3{x}^{2}m,\mathrm{with_unit}\left(\mathrm{integer},'a','b'\right)\right)$
${\mathrm{false}}$ (13)
> $a,b$
${2}{,}⟦{m}⟧$ (14)
> $\mathrm{type}\left(3{x}^{2}m,\mathrm{with_unit}\left(\mathrm{anything},'a','b'\right)\right)$
${\mathrm{true}}$ (15)
> $a,b$
${3}{}{{x}}^{{2}}{,}⟦{m}⟧$ (16)
> $\mathrm{type}\left(2m,\mathrm{with_unit}\left(\mathrm{integer},'a','b'\right)\right)$
${\mathrm{true}}$ (17)
> $a,b$
${2}{,}⟦{m}⟧$ (18)
In the following cases, expr is detected as being of type with_unit only after the call combine(expr, units).
> $\mathrm{restart}$
> $\mathrm{expr}≔x\mathrm{Unit}\left(m\right)+y\mathrm{Unit}\left(\mathrm{ft}\right)$
${\mathrm{expr}}{≔}{x}{}⟦{m}⟧{+}{y}{}⟦{\mathrm{ft}}⟧$ (19)
> $\mathrm{type}\left(\mathrm{expr},\mathrm{with_unit}\left(\mathrm{anything},'a','b'\right)\right)$
${\mathrm{true}}$ (20)
> $a,b$
${x}{+}\frac{{381}{}{y}}{{1250}}{,}⟦{m}⟧$ (21)
In this case, the call combine(expr, units) combines the units for expr by converting to the SI system of units. Because this is a necessary step in determining the unit and multiplier for this expression, the unit and multiplier found by with_unit are in the SI system.
> $\mathrm{expr}≔5\mathrm{Unit}\left(\mathrm{ft}\right)+x\mathrm{Unit}\left(\mathrm{ft}\right)$
${\mathrm{expr}}{≔}{5}{}⟦{\mathrm{ft}}⟧{+}{x}{}⟦{\mathrm{ft}}⟧$ (22)
> $\mathrm{combine}\left(\mathrm{expr},\mathrm{units}\right)$
$\left(\frac{{381}}{{250}}{+}\frac{{381}{}{x}}{{1250}}\right){}⟦{m}⟧$ (23)
> $\mathrm{type}\left(\mathrm{expr},\mathrm{with_unit}\left(\mathrm{anything},'a','b'\right)\right)$
${\mathrm{true}}$ (24)
> $a,b$
$\frac{{381}}{{250}}{+}\frac{{381}{}{x}}{{1250}}{,}⟦{m}⟧$ (25)
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# Weinberg angle
(Redirected from Weak mixing angle)
The pattern of weak isospin, T3, and weak hypercharge, YW, of the known elementary particles, showing electric charge, Q, along the Weinberg angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive W and Z bosons.
The Weinberg angle or weak mixing angle is a parameter in the WeinbergSalam theory of the electroweak interaction, and is usually denoted as θW. It is the angle by which spontaneous symmetry breaking rotates the original W0 and B0 vector boson plane, producing as a result the Z0 boson, and the photon.
$\begin{pmatrix} \gamma \\ Z^0 \end{pmatrix} = \begin{pmatrix} \cos \theta_W & \sin \theta_W \\ -\sin \theta_W & \cos \theta_W \end{pmatrix} \begin{pmatrix} B^0 \\ W^0 \end{pmatrix}$
It also gives the relationship between the masses of the W and Z bosons (denoted as mW and mZ):
$m_Z=\frac{m_W}{\cos\theta_W}$
The angle can be expressed in terms of the $SU(2)_L$ and $U(1)_Y$ coupling constants (g and g', respectively):
$\cos\theta_W = \frac{g}{\sqrt{g^2+g'^2}}$ and $\sin\theta_W = \frac{g'}{\sqrt{g^2+g'^2}}$
As the value of the mixing angle is currently determined empirically, it has been mathematically defined as:[1]
$\cos\theta_W=\frac{m_W}{m_Z}$
The value of θW varies as a function of the momentum transfer, Q, at which it is measured. This variation, or 'running', is a key prediction of the electroweak theory. The most precise measurements have been carried out in electron-positron collider experiments at a value of Q = 91.2 GeV/c, corresponding to the mass of the Z boson, mZ.
In practice the quantity sin2θW is more frequently used. The 2004 best estimate of sin2θW, at Q = 91.2 GeV/c, in the MS scheme is 0.23120 ± 0.00015. Atomic parity violation experiments yield values for sin2θW at smaller values of Q, below 0.01 GeV/c, but with much lower precision. In 2005 results were published from a study of parity violation in Møller scattering in which a value of sin2θW = 0.2397 ± 0.0013 was obtained at Q = 0.16 GeV/c, establishing experimentally the 'running' of the weak mixing angle. These values correspond to a Weinberg angle of ~30°.
Note, however, that the specific value of the angle is not a prediction of the standard model: it is an open, unfixed parameter. At this time, there is no generally accepted theory that explains why the measured value is what it is.
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## Saturday, September 8, 2018
### Zurich statement calls for phase-out of non-essential uses of PFAS
Per- and polyfluoroalkyl substances (PFASs) are man-made chemicals that contain at least one perfluoroalkyl moiety, $C sub n F sub 2 n$. To date, over 4,000 unique PFASs have been used in technical applications and consumer products, and some of them have been detected globally in human and wildlife biomonitoring studies. Because of their extraordinary persistence, human and environmental exposure to PFASs will be a long-term source of concern. Some PFASs such as perfluorooctanoic acid (PFOA) and perfluorooctanesulfonic acid (PFOS) have been investigated extensively and thus regulated, but for many other PFASs, knowledge about their current uses and hazards is still very limited or missing entirely. To address this problem and prepare an action plan for the assessment and management of PFASs in the coming years, a group of more than 50 international scientists and regulators held a two-day workshop in November, 2017. The group identified both the respective needs of and common goals shared by the scientific and the policy communities, made recommendations for cooperative actions, and outlined how the science–policy interface regarding PFASs can be strengthened using new approaches for assessing and managing highly persistent chemicals such as PFASs. https://doi.org/10.1289/EHP4158
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Trigonometric Approximation of SO(N) Loops
Open Access Publications from the University of California
Trigonometric Approximation of SO(N) Loops
• Author(s): Shingel, Tatiana
• et al.
Published Web Location
https://doi.org/10.1007/s00365-010-9107-6
Abstract
This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})$ for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.
Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you.
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# Does a quadrupole transition mean emission of one photon with spin 2?
If it's true and spin-2 photons do exist, could you please point to some literature that discusses spin-2 photons?
If not, then how exactly does a selection rule for quadrupole transition make sense in terms of interaction with spin-1 photons?
-
No, photons are always spin-one particles. The state of a photon may be described as a linear superposition of states with well-defined momenta $p$ and polarization vectors $\vec e$ and each of these basis vectors corresponds to a spin-one particle because the polarization is given by a transverse vector.
The selection rules – see this summary table – primarily boil down to the angular momentum conservation law. The conservation law says that the angular momentum of the atom before the transition is equal to the angular momentum of the products after the transition (the atom in the final state plus the photon). So for quadrupole transitions, as the table recalls, we may have $\Delta J = 0,\pm 1,\pm 2$ – depending on the relative angle between the angular momenta.
There isn't any need for the photon itself to have $J=2$ just because the operator responsible for the transition is a spin-two (quadrupole) operator. Such an agreement between the spins of the operator causing the transition on one side and the photon on the other hand would only have to hold if the initial and final states were the same – or at least had the same spin.
But because in the transition, the initial and final state of the atom are different and have different values of the spin in general, the spin-two character of the transition operator (quadrupole) means that the photon plus the difference between the final and initial spin of the atom corresponds to spin-two: this conservation law doesn't constrain the spin of the photon itself.
To summarize, I believe that in your argument which was "nearly" right, you forgot to add the difference between the final and initial angular momentum of the atom which is nonzero.
-
This seems self-contradictory. In the second paragraph you say that momentum of "the atom in the final state plus the photon" conserves. Then you say that "photon plus the difference between the final and initial spin of the atom corresponds to spin-two". These two equations seem to have no commons solutions. Kind of like. 'J1 = J2 + Photon' together 'Photon + (J2 - J1) = 2'. It's equivalent to '0=2'. – Klayman Apr 8 '13 at 14:35
Also, it's confusing that this article in wiki implies that photons may have angular momentum larger than one: "The emitted particle carries away an angular momentum λ, which for the photon must be at least 1" – Klayman Apr 8 '13 at 14:44
No, there's no contradiction. If the atom changes its state from $m_J = 0$ to $m_J = -1$, and the photon has $m_s = +1$, the exchange of angular momentum is still 2$\hbar$, which is all quadrupole transitions need with regards to angular momentum conservation. – Chay Paterson Apr 9 '13 at 1:34
Dear Klayman, the two conditions you mentioned are not only non-contradictory but completely equivalent. $J_{atom,ini}+J_\gamma = J_{atom, final}$ is the same thing as $J_\gamma = J_{atom, final}-J_{atom, initial}$. The spin-two character of the quadrupole operator doesn't really enter to the angular momentum conservation in the general form. Maybe an important thing to realize is that the total angular momentum of a photon in a momentum state isn't really $1$ or $2$ in general? It also has an "orbital part". The spin-two nature of the quadrupole operator imposes a selection on the whole sum. – Luboš Motl Apr 9 '13 at 5:31
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# Disjunction and Implication
## Theorems
### Modus Tollendo Ponens
#### Formulation 1
$p \lor q \dashv \vdash \neg p \implies q$
#### Formulation 2
$\vdash \paren {p \lor q} \iff \paren {\neg p \implies q}$
### Rule of Material Implication
#### Formulation 1
$p \implies q \dashv \vdash \neg p \lor q$
#### Formulation 2
$\vdash \paren {p \implies q} \iff \paren {\neg p \lor q}$
Both of the above come in negative forms:
$\ds \neg \paren {p \implies q}$ $\dashv \vdash$ $\ds \neg \paren {\neg p \lor q}$ $\ds \neg \paren {\neg p \implies q}$ $\dashv \vdash$ $\ds \neg \paren {p \lor q}$
Disjunction is definable through implication:
$p \lor q \dashv \vdash \paren {p \implies q} \implies q$
### Alternative rendition
They can alternatively be rendered as:
$\ds \vdash \ \$ $\ds \paren {\neg \paren {p \implies q} }$ $\iff$ $\ds \paren {\neg \paren {\neg p \lor q} }$ $\ds \vdash \ \$ $\ds \paren {\neg \paren {\neg p \implies q} }$ $\iff$ $\ds \paren {\neg \paren {p \lor q} }$ $\ds \vdash \ \$ $\ds \paren {p \lor q}$ $\iff$ $\ds \paren {\paren {p \implies q} \implies q}$
They can be seen to be logically equivalent to the forms above.
## Proof
By the tableau method of natural deduction:
$\neg \paren {p \implies q} \vdash \neg \paren {\neg p \lor q}$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \paren {p \implies q}$ Premise (None)
2 2 $\neg p \lor q$ Assumption (None) Assume the opposite of what is to be proved ...
3 2 $p \implies q$ Sequent Introduction 2 Rule of Material Implication
4 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 3, 1 ... demonstrating a contradiction
5 1 $\neg \paren {\neg p \lor q}$ Proof by Contradiction: $\neg \II$ 2 – 4 Assumption 2 has been discharged
$\blacksquare$
By the tableau method of natural deduction:
$\neg \paren {\neg p \lor q} \vdash \neg \paren {p \implies q}$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \paren {\neg p \lor q}$ Premise (None)
2 2 $p \implies q$ Assumption (None) Assume the opposite of what is to be proved ...
3 2 $\neg p \lor q$ Sequent Introduction 2 Rule of Material Implication
4 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 3, 1 ... demonstrating a contradiction
5 1 $\neg \paren {p \implies q}$ Proof by Contradiction: $\neg \II$ 2 – 4 Assumption 2 has been discharged
$\blacksquare$
## Law of the Excluded Middle
This theorem depends on the Law of the Excluded Middle, by way of Rule of Material Implication.
This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.
However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.
This in turn invalidates this theorem from an intuitionistic perspective.
By the tableau method of natural deduction:
$\neg \paren {\neg p \implies q} \vdash \neg \paren {p \lor q}$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \paren {\neg p \implies q}$ Premise (None)
2 2 $p \lor q$ Assumption (None) Assume the opposite of what is to be proved ...
3 2 $\neg p \implies q$ Sequent Introduction 2 Modus Tollendo Ponens
4 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 3, 1 ... demonstrating a contradiction
5 1 $\neg \paren {p \lor q}$ Proof by Contradiction: $\neg \II$ 2 – 4 Assumption 2 has been discharged
$\blacksquare$
By the tableau method of natural deduction:
$\neg \paren {p \lor q} \vdash \neg \paren {\neg p \implies q}$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \paren {p \lor q}$ Premise (None)
2 2 $\neg p \implies q$ Assumption (None) Assume the opposite of what is to be proved ...
3 1 $\neg p \land \neg q$ Sequent Introduction 1 De Morgan's Laws: Conjunction of Negations
4 1 $\neg p$ Rule of Simplification: $\land \EE_1$ 3
5 1, 2 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 4 ... from the assumption
6 1 $\neg q$ Rule of Simplification: $\land \EE_2$ 3
7 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 5, 6 ... demonstrating a contradiction
8 1 $\neg \paren {\neg p \implies q}$ Proof by Contradiction: $\neg \II$ 2 – 7 Assumption 2 has been discharged
$\blacksquare$
By the tableau method of natural deduction:
$p \lor q \vdash (p \implies q) \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Premise (None)
2 2 $p \implies q$ Assumption (None)
3 3 $p$ Assumption (None)
4 2, 3 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 2, 3
5 5 $q$ Assumption (None)
6 2 $r$ Proof by Cases: $\text{PBC}$ 1, 3 – 4, 5 – 5 Assumptions 3 and 5 have been discharged
7 1 $\paren {p \implies q} \implies q$ Rule of Implication: $\implies \II$ 2 – 6 Assumption 2 has been discharged
$\blacksquare$
### Comment
Note that this:
• $\neg \paren {\neg p \implies q} \dashv \vdash \neg \paren {p \lor q}$
can be proved in both directions without resorting to the Law of Excluded Middle.
All the others:
• $p \lor q \vdash \neg p \implies q$
• $\neg p \lor q \vdash p \implies q$
• $\neg \paren {p \implies q} \vdash \neg \paren {\neg p \lor q}$
are not reversible in intuitionistic logic.
## Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.
As can be seen by inspection, in all cases the truth values under the main connectives match for all boolean interpretations.
$\begin {array} {|cccc||ccccc|} \hline \neg & (p & \lor & q) & \neg & (\neg & p & \implies & q) \\ \hline \T & \F & \F & \F & \T & \T & \F & \F & \F \\ \F & \F & \T & \T & \F & \T & \F & \T & \T \\ \F & \T & \T & \F & \F & \F & \T & \T & \F \\ \F & \T & \T & \T & \F & \F & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
$\begin {array} {|cccc||ccccc|} \hline \neg & (p & \implies & q) & \neg & (\neg & p & \lor & q) \\ \hline \F & \F & \T & \F & \F & \T & \F & \T & \F \\ \F & \F & \T & \T & \F & \T & \F & \T & \T \\ \T & \T & \F & \F & \T & \F & \T & \F & \F \\ \F & \T & \T & \T & \F & \F & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
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# On average, you eat 8 spiders a year while sleeping
Level 3
Evaluate
where $\tau(N)$ denotes the number of positive integer divisors of N, and, where $\frac { a }{ \frac { b }{ \frac { c }{ \frac { d }{ \frac { e }{ f } } } } } = \left( \begin{array}{c} a \\ \hline \left( \begin{array}{c} b \\ \hline \left( \begin{array}{c} c \\ \hline \left( \begin{array}{c} d \\ \hline \left( \dfrac{e}{f} \right) \end{array} \right) \end{array} \right) \end{array} \right) \end{array} \right)$
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# Centripetal and Tangencial Accelaration
1. Oct 23, 2011
### martcapt
1. The problem statement, all variables and given/known data
English is not my first language from the beginning I apologize for any mistakes in scientific terms, anyhow.
Car with 500kg.
No initial velocity, and from origin or the referential.
The module of the velocity increases in 2m/s
2. Relevant equations
I can't post the damn symbols for some reason but I believe this is a very simple problem.. Sorry
3. The attempt at a solution
Well there are three parts
Indication of the angular position
Indication of the angular velocity after a lap.
Indication of the time in witch the centripetal acceleration is the same as the tangential acceleration
The first two i'm pretty sure of the results it gave me 0.02t(squared) on the first and angular velocity=0.7rad/s on the second, if someone would be as so kind as checking this I would really apreciate it, but the problem in witch i'm having trouble is the third, this is the only thing I have done so far:
at=ac
2=v(squared)/r
v=10
sorry for bothering, but my teacher has the bad habit of giving exercises with no solution and I would really apreciate one for this one :)
Anyways so as to not create another thread if someone happens to know some condensed but thorough material in two dimensional motion, it would be great :)
thanks.
2. Oct 23, 2011
### Andrew Mason
You have to be a little clearer in stating the problem. I gather that the tangential acceleration is 2 m/sec^2. Is the first question: what is the angular position as a function of time? Is the second question: what is the angular velocity after one complete rotation ie through an angle of $2\pi$ radians? and is the third question: at what time will the centripetal acceleration be equal in magnitude to the tangential acceleration?
AM
3. Oct 23, 2011
### martcapt
Correct on all three.
Sorry,I'm not used to writing this stuff in text, and in english
Last edited: Oct 23, 2011
4. Oct 24, 2011
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# Unreachable Real Numbers - Randomness & Computability
I've recently read that there were many real numbers that would never be reachable by humanity. The explanation itself says that we can write as many programs as integers which is infinite, but there are infinite real numbers between two integers so Real numbers infinity is bigger than integer numbers.
Well my question is, couldn't we overpass this limitation by writing a program that prints out a random number in every execution? I know this is not a warranty that every real number would be printed but at least they would all have a chance.
Please point out if my reasoning is wrong and why.
Thanks you all very much!
• for better response plz cite where you read this & the exact stmt. fyi turings early 1936 essay considered the idea. see computable numbers / wikipedia – vzn Mar 25 '15 at 23:00
• @vzn What is the precise idea considered by Turing? Using randomness? No computer program can print an uncomputable number, – babou Mar 26 '15 at 1:37
• @babou basically/ roughly as sketched out in some answers below, a 1-1 correspondence can be made between reals and computable functions to show that not all reals are computable. so Turings result on undecidability can be seen equivalently as a proof of the existence of uncomputable real(s). as to the other angle in the question, random numbers in computers are computed via pseudorandom functions and therefore do not escape this inability to compute uncomputable numbers. – vzn Mar 26 '15 at 15:56
In a nutshell: Printing a random non-computable real is a meaningless task, for precise technical reasons. The meaningful problem is to print non-computable numbers precisely identified by some unique property. But these cannot be printed by any program precisely because they are not computable. Using randomness in the hope of printing by chance the uncomputable number you are interested in is impossible with a program, unless you use a physical random phenomenon as randomness source (depending on physics, with probability zero, and without knowing whether you are succeeding).
I am not sure what you mean by a number that is not "reachable by humanity".
The fact that there is an infinity of real numbers between two integers does not imply that there are more real numbers than integers (though there are more).
For example, there is an infinity of rational number between two integers, but the number (cardinality) of integers and rational numbers is the same. Actually, there is an infinity of rational numbers between any two rational numbers, but it does not imply that there are more rational numbers than rational numbers.
By the way, there is an infinity of rational numbers between any two real numbers, but there are more real numbers than rational numbers.
That should at least tell you that determining which set is bigger is not at all obvious, and is not related to the number of angels who can dance between two numbers.
Then why are there more real numbers than integer? Here is one of the proofs due to Cantor. The reals in the interval $[0.0, 1.0[$ are isomorphic to infinite sequences of the symbols 0 and 1, which is their representation in binary notation (by just adding a decimal/binary point in front). If you consider any enumeration of these infinite strings, it is always possible to define a string that is not in the enumeration, by using for its $n^{th}$ symbol the opposite (1 for 0, and 0 for 1) of the $n^{th}$ symbol of the $n^{th}$ number of the enumeration. This means that if you attempt to define a one-one correspondance between the integers (used to enumerate) and the reals (being enumerated) there are always reals that are not included in the enumeration, which can be interpreted as implying that there are more reals than integers.
This technique is a diagonalization proof, similar to the proof technique used for proving the undecidability of the halting problem.
Now, as remarked in the question, any program texts can be read as an integer, so that one may consider there are no more programs than integers. This is why we may conclude that there are numbers that cannot be computed by a program, hence cannot be printed (else, each real could be mapped to the integer corresponding to the program that prints it.). But this fast reasonning may not be so convincing, so lets get more into details. and also consider an actual example.
Your printing device, assuming it does what you expect, even if it started at the big bang, is only producing digits one by one, and will at any time have produced only a finite number of digits, say $n$. That only defines a rational number, and even if we are actually printing some irrational number. Assuming we are using binary numbers, if we have already printed $n$ digits, they correspond to a rational number $x$, and all we know is that the number we ultimately print will be in the interval $[x, x+2^{-n}[$. But this interval will always contain an infinity of rational numbers, of computable real, and of uncomputable reals. And this will remain so for any future value of $n$. Stating that you want to print an uncomputable number, any uncomputable number, is a meaningless endeavor. It reminds me of people looking for the pot of gold at the foot of the rainbow.
The meaningful problem is not to write any uncomputable number, but to write a specific one, identified by a specific property. Consider for example the binary real $h\in[0.0, 1.0[$ defined as follows: its $n^{th}$ digit is $1$ iff the Turing Machine with Gödel number $n$ halts on empty input, and $0$ otherwise. This number is very precisely defined. But if it were computable, we would have a way to decide the halting problem. Hence, there is no algorithm, no program, that can enumerate the bits of the real number $h$, and thus there is no way to print it. And using random generators will not help.
Of course, it could be that some random generator, controlled by a random physical process such as radioactive decay would produce precisely that number (with a probability closing to zero as time passes). But, at best:
1- you would have at any time only a rational number which is a prefix of the non-computable real you are interested in
2- you would most likely be unable to check wether your prefix is really correct
3- you would have no garantee that the next digit will still be correct.
But, even then, you must assume that the random generator uses some physical "true" randomness source. Even randomly created, or computing with whatever technique known or yet to be invented, no computer program can print an uncomputable real number, such as the number $h$ defined above. That is a result of computability theory that no such program can exist.
Your only hope is that computability theory does not apply to radioactive decay or some other such physical phenomena. But I know of no proof either way. And it would not really help you in any meaningful way, because the probability of success is zero, and because you would have no way of knowing you are succeeding. Note that I cannot say "of knowing you have succeeded", because you never reach the end of that infinite process, so that any proof would have to be an on-going continuous process, which cannot be finitely defined for essentially the same reason a printing program cannot exist for an uncomputable number.
Not all real numbers can be printed.
Consider some system of representing numbers using a finite number of symbols. You can make this system as complex as you want: decimal numbers, fractions, roots, integrals, trig functions, etc.
Because there's a finite number of symbols in any system we have to write or print a number, we can create a bijection from the set of numbers we can write down to the integers.
Cantor proved that there is no 1:1 mapping from the integers to the real numbers. Which means, there are more real numbers than we can print.
Now, there are real numbers which we can't enumerate: that is, there's no algorithm saying "give me the first $n$ symbols of an infinite-length representation of this number.
The thing is, uncomputable numbers do exist. Your random generator could, by chance, generate the first $n$ symbols of an uncomputable number. But there's no way to check if those first $n$ symbols are correct, and there's no way to get the $n+1$th symbol, even if we have the first $n$ symbols.
• Thanks for your answer. But imagine my program has an loops infinitely printing a random sequence of numbers of a random length (and randomly writes or not a point in this sequence). Given any real number possible and enough time my program would print it or maybe not. But theoretically is possible. Isn't this right? – NMO Mar 25 '15 at 16:01
• So, the string is infinite, so at any point it's possible you will have outputted the first $n$ symbols. Given enough time, for any $n$, your program will print the first $n$ symbols of an undecidable number. But 1. you have no way of knowing that you've printed the correct digits, and 2. you never print the "whole" number, since its sequence is infinite. – jmite Mar 25 '15 at 21:37
This is not really an answer to your question, but a direct consequence from the fact that "$\mathcal{R}$ is uncountable", which may interest (or dismay) you.
There are much more unsolvable (in terms of Turing-decidable) problems than solvable ones.
The argument is as follows:
The set of all algorithms (represented as Turing machines) is countably infinite, equinumerous to the set of natural numbers $\mathcal{N}$. We can each Turing machine into a string and count them one by one, e.g., in the order of the increasing string length (note that the alphabet $\Sigma$ is finite).
However, the number of all the problems (represented as languages) is $2^{\mathcal{N}}$ (it is the number of mappings from $\mathcal{N}$ to $\{0,1 \}$), which is uncountably infinite. It is impossible to count/print all the algorithms, just like it is impossible to count all the real numbers.
We conclude that some problems have no corresponding algorithms at all. Even worse, there are much more unsolvable problems than solvable ones.
• This seems pretty close to what the OP is saying, less formally, and with a mathematical error, in his first paragraph. And it also appears in some form in the other answers, including mine. However, I do not feel too confortable with it since I tried to understand the interpretation of Cantor's diagonal argument in Wikipedia. It seems to state that constructivist mathematicians do not infer from that a "difference in size (cardinality)", as classical mathematicians do, but only the non-existence of a bijection. – babou Mar 26 '15 at 11:39
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What does Uncharacterized Protein Uniprot designation mean?
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6 months ago
emalekos ▴ 10
I am blasting some predicted peptides against the Uniprot database, and many of the hits are "Uncharacterized Protein". How is this designation chosen? i.e. what level of evidence is required for a peptide sequence to be added to the database and given this designation rather than be excluded?
I don't see it described on Uniprot website. I tried to read the publication to see where this term is explained, but there are a crazy number of pubs going back to 1997 (and I can't access that one) https://www.uniprot.org/help/publications
Thanks!
uniprot uncharacterized protein • 413 views
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what level of evidence is required for a peptide sequence to be added to the database and given this designation rather than be excluded?
Probably not a lot.
If you look at the history of one such entry https://www.uniprot.org/uniprotkb/Q9H425/history you will see that it was originally added via Trembl. After a number of years it appears to have been seen in a mass spectrometry paper https://rest.uniprot.org/unisave/Q9H425?format=txt&versions=26 Is has stayed in the designation since that time.
You should use the reviewed swiss-prot part of UniProt or better yet use a specific proteome, if possible.
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Generally, in a UniProt entry, it is important to look at the evidence label in order to be able to distinguish
• expert-curated and reviewed annotation
• automatic annotation
• information imported from an external database
There are unfortunately many uncharacterized proteins, in particular in UniProtKB/TrEMBL: https://www.uniprot.org/uniprotkb?query=(protein_name:%22uncharacterized%20protein%22)
Example: https://www.uniprot.org/uniprotkb/H3BNH8/entry : It is imported from Ensembl ENSP00000454861 which says "novel protein".
Regarding expert-curated, reviewed entries (i.e. those in UniProtKB/Swiss-Prot):
Protein naming guidelines are available here: https://ftp.uniprot.org/pub/databases/uniprot/current_release/knowledgebase/complete/docs/International_Protein_Nomenclature_Guidelines.pdf
They do recommend to use "uncharacterized protein" in certain cases as a last resort for novel proteins of unknown function.
See also the news article from UniProt release 2011_04: "The art of defining the unknown", https://www.uniprot.org/help/2011-04-05-release
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# Time of Flight of a Projectile
Written by Jerry Ratzlaff on . Posted in Relativity
A projectile object that is given an initial velocity and is acted on by gravity. The amount of time it spends in the air is called the time of flight. If the ground from which the projectile is launched is level, the time of flight only depends on the initial velocity, the launch angle, and the acceleration due to gravity.
### Time of Flight of a Projectile formula
$$\large{ t = \frac{ 2 \; v_0 \; sin \; \theta }{ g } }$$
Where:
$$\large{ t }$$ = time
$$\large{ g }$$ = gravitational acceleration
$$\large{ v_0 }$$ = launch velocity
$$\large{ \theta }$$ = vertical angle
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# About the operator theory in Hilbert space
Let $H$ be a hilbert space. $L(H)$ be the set of linear operators on $H$.
Suppose that $S,T\in L(H)$ and $S\geq0$, $\|Sx\|=\|Tx\|$ for every $x\in H$.
Can I conclude that $S=\sqrt{T^*T}$?
-
Do I miss something obvious. Why it cannot be that $S=T$? – Fabian Apr 5 '11 at 18:31
I modified it. Thanks. – user8484 Apr 5 '11 at 18:34
Because $\|\sqrt{T^*T}x\|^2=\langle T^*Tx,x\rangle=\|Tx\|^2$ for all $x$, the question amounts to the following: If $S$ and $P$ are positive operators on $H$ such that $\|Sx\|=\|Px\|$ for all $x\in H$, then must $S=P$? The answer is yes.
For all $x$ and $y$ in $H$, comparing the expansions of $\|S(x+y)\|^2$ and $\|P(x+y)\|^2$ in terms of the inner product shows that $\Re\langle S^2x,y\rangle =\Re\langle P^2x,y\rangle$, and comparing the expansions of $\|S(x+iy)\|^2$ and $\|P(x+iy)\|^2$ in terms of the inner product shows that $\Im\langle S^2x,y\rangle=\Im\langle P^2x,y\rangle$. Therefore $\langle S^2x,y\rangle=\langle P^2x,y\rangle$ for all $x$ and $y$, which implies that $S^2=P^2$. By uniqueness of positive square roots, $S=P$.
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Yes. For all $x \in H$, $\langle x, S^2 x \rangle = \|S x\|^2 = \|T x\|^2 = \langle x T^* T x \rangle$. By the polarization identity, $\langle x, S^2 y \rangle = \langle x, T^* T y \rangle$ for all $x$ and $y$, so that $S^2 = T^* T$. By the uniqueness of positive semidefinite square roots, $S = \sqrt{T^* T}$.
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Yes: the condition $\|Sx\| = \|Tx\|$ squares and expands to $(S^*Sx|x) = (T^*Tx|x)$ for all $x$. It follows by Polarisation that $T^*T = S^*S = S^2$. Now positive operators have unique square-roots, so we conclude that $S = \sqrt{T^*T}$ as required.
Polarisation: for an operator $R$ we have that $$4(Rx|y) = \sum_{k=0}^3 i^k (R(x+i^ky)|x+i^ky).$$ So if $(Rx|x)=0$ for all $x$, this shows that also $(Rx|y)=0$, so $R=0$.
I think you're missing a factor of $1/4$ in you polarization identity? – joriki Apr 5 '11 at 19:24
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Lattice-Inspired Broadcast Encryption and Succinct Ciphertext-Policy ABE
Zvika Brakerski and Vinod Vaikuntanathan
Abstract
We propose a candidate ciphertext-policy attribute-based encryption (CP-ABE) scheme for circuits, where the ciphertext size depends only on the depth of the policy circuit (and not its size). This, in particular, gives us a Broadcast Encryption (BE) scheme where the size of the keys and ciphertexts have a poly-logarithmic dependence on the number of users. This goal was previously only known to be achievable assuming ideal multilinear maps (Boneh, Waters and Zhandry, Crypto 2014) or indistinguishability obfuscation (Boneh and Zhandry, Crypto 2014) and in a concurrent work from generic bilinear groups and the learning with errors (LWE) assumption (Agrawal and Yamada, Eurocrypt 2020). Our construction relies on techniques from lattice-based (and in particular LWE-based) cryptography. We analyze some attempts at cryptanalysis, but we are unable to provide a security proof.
Available format(s)
Category
Foundations
Publication info
Preprint. MINOR revision.
Keywords
Contact author(s)
vinodv @ mit edu
History
2021-04-26: revised
See all versions
Short URL
https://ia.cr/2020/191
CC BY
BibTeX
@misc{cryptoeprint:2020/191,
author = {Zvika Brakerski and Vinod Vaikuntanathan},
title = {Lattice-Inspired Broadcast Encryption and Succinct Ciphertext-Policy ABE},
howpublished = {Cryptology ePrint Archive, Paper 2020/191},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/191}},
url = {https://eprint.iacr.org/2020/191}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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TheInfoList
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light.[1] These optically anisotropic materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.
Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it[2] in calcite, a crystal having one of the strongest birefringences. However, it was not until the 19th century that Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector).
## Explanation
Incoming light in the parallel (p) polarization sees a different effective index of refraction than light in the perpendicular (s) polarization, and is thus refracted at a different angle.
Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.
A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.
### Uniaxial materials
The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index no (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one li
Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it[2] in calcite, a crystal having one of the strongest birefringences. However, it was not until the 19th century that Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector).
A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.
### Uniaxial materials
The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index no (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that would be perpendicular to the optic axis, and a ray with that polarization is called an ordinary ray and is governed by the same refractive index value no. However, for a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of the optic axis, and this extraordinary ray will be governed by a different, direction-dependent refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray. The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:[verification needed]
${\displaystyle \Delta n=n_{\mathrm {e} }-n_{\mathrm {o} }\,.}$
The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as if there were no birefringence involved. However, the extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between no and ne). However, its power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.[verification needed]
When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations are perpendicular to the optic axis and see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to ne) but
The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index no (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that would be perpendicular to the optic axis, and a ray with that polarization is called an ordinary ray and is governed by the same refractive index value no. However, for a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of the optic axis, and this extraordinary ray will be governed by a different, direction-dependent refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray. The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:[verification needed]
## Measurement
Birefringence and other polarization-based optical effects (such as optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence. The addition of quarter-wave plates permit examination of circularly polarized light. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids.[11][12]
Birefringence of lipid bilayers can be measured using dual-polarization interferometry. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.
## Applications
optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence. The addition of quarter-wave plates permit examination of circularly polarized light. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids.[11][12]
Birefringence of lipid bilayers can be measured using dual-polarization interferometry. This provides a measure of the degree of order within these fluid layers and how thi
Birefringence of lipid bilayers can be measured using dual-polarization interferometry. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.
Birefringence is used in many optical devices. Liquid-crystal displays, the most common sort of flat-panel display, cause their pixels to become lighter or darker through rotation of the polarization (circular birefringence) of linearly polarized light as viewed through a sheet polarizer at the screen's surface. Similarly, light modulators modulate the intensity of light through electrically induced birefringence of polarized light followed by a polarizer. The Lyot filter is a specialized narrowband spectral filter employing the wavelength dependence of birefringence. Waveplates are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.
Birefringence also plays an important role in second-harmonic generation and other nonlinear optical components, as the crystals used for this purpose are almost always birefringent. By adjusting the angle of incidence, the effective refractive index of the extraordinary ray can be tuned in order to achieve phase matching, which is required for the efficient operation of these devices.
### Medicine
Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed polarizing filters. Light from the source is polarized in the x direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called analyzer) oriented in the y direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. However, areas of the sample possessing birefringence will generally couple some o
Birefringence also plays an important role in second-harmonic generation and other nonlinear optical components, as the crystals used for this purpose are almost always birefringent. By adjusting the angle of incidence, the effective refractive index of the extraordinary ray can be tuned in order to achieve phase matching, which is required for the efficient operation of these devices.
Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed polarizing filters. Light from the source is polarized in the x direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called analyzer) oriented in the y direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. However, areas of the sample possessing birefringence will generally couple some of the x-polarized light into the y polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.
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## Algebraic & Geometric Topology
### The decomposition of the loop space of the mod $2$ Moore space
#### Abstract
In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod $p$ Moore space for primes $p>2$ and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for $p=2$ are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod $2$ Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod $2$ Moore space or to an improvement in the known bounds for the exponent of the $2$–torsion in the homotopy groups of spheres.
#### Article information
Source
Algebr. Geom. Topol., Volume 8, Number 2 (2008), 945-951.
Dates
Revised: 28 March 2008
Accepted: 23 April 2008
First available in Project Euclid: 20 December 2017
https://projecteuclid.org/euclid.agt/1513796851
Digital Object Identifier
doi:10.2140/agt.2008.8.945
Mathematical Reviews number (MathSciNet)
MR2443103
Zentralblatt MATH identifier
1148.55004
Subjects
Primary: 55P35: Loop spaces
Secondary: 16W30
#### Citation
Grbić, Jelena; Selick, Paul; Wu, Jie. The decomposition of the loop space of the mod $2$ Moore space. Algebr. Geom. Topol. 8 (2008), no. 2, 945--951. doi:10.2140/agt.2008.8.945. https://projecteuclid.org/euclid.agt/1513796851
#### References
• J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. $(2)$ 72 (1960) 20–104
• F R Cohen, Applications of loop spaces to some problems in topology, from: “Advances in homotopy theory (Cortona, 1988)”, London Math. Soc. Lecture Note Ser. 139, Cambridge Univ. Press (1989) 11–20
• F R Cohen, J C Moore, J A Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. $(2)$ 110 (1979) 549–565
• F R Cohen, J C Moore, J A Neisendorfer, Torsion in homotopy groups, Ann. of Math. $(2)$ 109 (1979) 121–168
• F R Cohen, J C Moore, J A Neisendorfer, Exponents in homotopy theory, from: “Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983)”, Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 3–34
• I M James, Reduced product spaces, Ann. of Math. $(2)$ 62 (1955) 170–197
• J A Neisendorfer, $3$–primary exponents, Math. Proc. Cambridge Philos. Soc. 90 (1981) 63–83
• P Selick, $2$–primary exponents for the homotopy groups of spheres, Topology 23 (1984) 97–99
• P Selick, S Theriault, J Wu, Functorial decompositions of looped coassociative co–$H$ spaces, Canad. J. Math. 58 (2006) 877–896
• P Selick, J Wu, Some calculations of $\mathrm{Lie}(n)^{\mathrm{max}}$ for low $n$, to appear in Journal of Pure and Applied Algebra
• P Selick, J Wu, On natural coalgebra decompositions of tensor algebras and loop suspensions, Mem. Amer. Math. Soc. 148 (2000) viii+109
• P Selick, J Wu, On functorial decompositions of self-smash products, Manuscripta Math. 111 (2003) 435–457
• P Selick, J Wu, The functor $A^{\mathrm{min}}$ on $p$–local spaces, Math. Z. 253 (2006) 435–451
• S Theriault, Homotopy exponents of mod $2^r$ Moore spaces, to appear in J. Topol.
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# Why do the professional mathematicians believe blindly in so meaningless concepts as Infinity? [closed]
To refute such a concept as Infinity (or many infinities) in mathematics doesn't at all require all that big efforts mainly from its own definition in mathematics.
To explain this very simple fiction in human minds, just consider the natural numbers, where simply they are a continuous chain of "endless" successive integers with no existing largest integer, where this only invalidates strictly the concept of infinity since the later is not any number nor anything else (from its own definition), so how can we truly compare it with numbers? wonder!
So, the so obvious fact of natural numbers being actually "endless" is much stronger term than using any meaningless concept as Infinity, where this concept is just a plain hopeless try to finite the natural numbers fact in order to justify and legalize so many theorems and well- established results in modern mathematics as an agreement and never any real true discovery
Why do I add this question here, because of I found no tolerance at all to all my topics that had been added in many mathematical sections, where simply the trend is always deleting my content without being able to refute me in this very basic issue,
However, many of my proven topics were including true discoveries and so many mathematical challenges that are still standing evidence where simply no one could ever bring a single counterexample (especially in Number theory and Geometry)
And to give a brief idea about those many deleted topics that the reader would simply laugh at when hearing for the first time due to huge incorrect mathematical concepts that had been built and were well-established (based on so naive conclusions or merely were just plain wrong decisions, and not at all any true proved discovery)
However, the science of physics was the main victim of the current and alleged modern mathematical sciences, where also the world economy and intelligent people waste may be regarded as the second victim of so much wrong modern mathematics
In short, I claim (with many public published so rigorous proofs) at sci.math or Quora or at SE-(here-few still undeleted topics), The following famous fallacies:
1) Imaginary numbers were simply and WRONGLY DECIDED and never were any true discovery
2) Infinity concept is a totally fictional concept that doesn't mean anything but was just introduced or fabricated to legalize so many illegal mathematics
3) The fundamental theorem of algebra totally flawed
4) The CONVERGENCE principle is also a flawed concept since the Infinite sum never exists because basically, natural numbers are endless
And if you look more carefully about the divergence or convergence you would certainly find them as the same but with so tinny deference which is that decimal notation denoted by a dot point (.), where it is not any fundamental operation in mathematics or any magical tool that can suddenly and turn the non-numbers to real numbers for sure
And the easiest way for a clever school student or a layperson to understand the deepest theme is to work for a few minutes in fractions and without using that mind blocking notation (decimal point), for sure
https://www.quora.com/When-shall-mathematicians-realize-that-no-existing-theorem-in-mathematics-would-yield-exactly-the-cube-root-of-any-prime-number
5) All real numbers associated with the fictional concept of Infinity such as the non-constructible irrational numbers (real algebraic and trans.) numbers that don't exist on the real number line (but only notations in minds), with so special story of $\pi$
See here in the below link, how is it too elementary to refute the most famous human mind fallacy in mathematics about? $$1 = 0.999...$$ where thence applicable on every alleged real number that is generally assumed with an infinite number of digits after the decimal notation, despite so many alleged proofs for this refuted fallacy https://groups.google.com/forum/#!topic/sci.math/v88rBgVXFrY
6) The impossibility of solving the general polynomials by radicals for degrees higher than fourth was so simple and so naive and beyond one's common beliefs in our modern mathematics
7) A very famous challenging example of the non-existence of many integer degrees angles of the form ($3n +/- 1$), in any existing triangle with exactly known and constructible sides, where this so obvious fact reveals strictly all the legendary real numbers in our current modern mathematics
Of course, one must consider that no Journal or University would accept such closed topics nowadays, therefore It was my duty to make them publically available to keen researchers in future, where absolute facts must be raised above all common fallacies ultimately
For interested philosophers or logicians in those many critical issues in mathematics nowadays, people can simply read many relevant topics in a free spoken site, where simply no professional control on the content of any topic, a reader must distinguish himself the facts from illusions, and not personalizing any self-issue, here: https://groups.google.com/forum/#!forum/sci.math
Note that, if all my deleted questions or answers were recovered at SE, then it would certainly facilitate the so easy task to understand all those many fictions in our modern mathematics, sure
• Comments are not for extended discussion; this conversation has been moved to chat. – user2953 Apr 7 '18 at 11:22
I think you have some fundamental misunderstandings about how the discipline of mathematics operates.
Infinity concept is a totally fictional concept that doesn't mean anything but was just introduced or fabricated to legalize so many illegal mathematics.
Yes, "infinity" is a "totally fictional concept", but so are concepts like "addition", the idea of "less" or "more", negative numbers, fractions and decimals, imaginary numbers, even the concepts of numbers themselves.
Mathematics admits that it simply "makes up stuff" all the time. The questions within mathematics are more likely to take the form:
• Is (new idea) consistent with prior concepts, or does it contradict them in some way?
• Is (new idea) consistent within itself, or does it self-contradict somehow?
• Is it interesting (either to mathematicians, or to people in other disciplines)?
• Is it useful, either internally within mathematics (it makes some calculations easier or something like that), or useful externally (in that it appears in some way to correspond to something in the real world)?
Fractions are useful because it helps us describe how much is left of a partially-eaten pizza. Negative numbers are useful because it helps us understand a "balance" vs a "debt" in business. Calculus is useful because it helps engineers calculate an airliner's or spacecraft's trajectory. Other areas of mathematics may or may not have direct real-world applications, but might still be "interesting" to mathematicians.
Here's (one of several possible) mathematical definitions of an "infinite set":
an infinite set is a set that is not a finite set
Hmm, not so helpful without further context. So what is a finite set?
a set S is called finite if there exists a bijection (a 1-to-1 correspondence) between the set S and the natural numbers {1,...,n}, for some natural number n.
In other words, if you can "count" a set up to a specific number n, then its finite. If you can't, its infinite. (One immediate results is that the set of natural numbers itself is infinite, since there is no largest number n).
You're certainly welcome to think that this definition is strange or confusing or useless or just silly, but it does satisfy all the needs within mathematics: it turns out to be consistent, useful, and interesting (at least to mathematicians).
Note that we can even have areas of mathematics that are not consistent with each other, as long as they are consistent within themselves. Best example I can think of is Euclidean vs non-euclidean geometries. Both are useful, in different ways.
• Most likely, you didn't get my point, then how do you distinguish your finite set from your infinite set? wonder!, since a finite integer can also fill few galaxies with its random sequence digits (if you store say every trillion digits say in only one mm cube) for instance, and of course there is much more to this simple example – Bassam Karzeddin Apr 4 '18 at 16:50
• @bassamkarzeddin Clearly we have notations that allow us to deal with large sets in an abstract way, requiring me to fill galaxies with digits by writing them out entirely is a pretty absurd objection. You seem to think that mathematical things have to be "real" in some way, this is false. "Imaginary numbers" are similarly conceptual; we have a precise definition, and they derive consistent rules, and have useful implications to mathematics and other fields. But as I said above, fractions and decimals and division and geometry aren't any more "real" than imaginary numbers are. – BradC Apr 4 '18 at 17:22
• @bassamkarzeddin Your answer there again contends that mathematical concepts have to be "real" in some way. This is false. They simply have to be consistent and useful. If you believe they aren't useful, you're entitled to your opinion. But if you believe that established ideas like negative numbers or fractions or imaginary numbers or ideal geometric shapes or limits approaching infinity are false because they don't make intuitive sense to you in relation to the real world, then you really don't know what you're talking about. – BradC Apr 4 '18 at 17:57
• @bassamkarzeddin Nobody's claiming you have to "successfully count all the way up to infinity" to imagine an infinite set; that's your misconception. And as I said, I concede that all math is a "human mind-brain fart fabrication", that was the main point of my answer! It is still consistent and interesting and (in many ways) useful. – BradC Apr 5 '18 at 13:42
• @bassamkarzeddin Are you asking whether a "real" infinite set of physical objects exists in our universe? Because that is an empirical question for astrophysics. Yes, evidence leads most astrophysicists to believe that "the set of all particles in our universe" is probably finite, but that has nothing to do with whether mathematics can define and work with abstract infinite sets. True "ideal geometric" objects don't "exist" either. Neither do negative numbers. Or the square root of 2. Or PI. Or the Fibonacci sequence. Or "zero" for that matter. NONE. OF. THAT. MATTERS. – BradC Apr 5 '18 at 16:07
They do not "believe blindly in so meaningless concepts as Infinity." They actually approach the concept of infinity with eyes wide open, aware of all of the difficulties and complications therein.
A mathematician should consider it well within your right to assert that you do not consider these concepts that you describe to be "real." The finitists are a group that do exactly that. However, you should not believe that there is no infinity blindly. You should do your research to understand what sorts of conceptual issues are resolved by the concept of infinity, such as Zeno's paradox.
Mathematicians do not blindly believe in infinity. They came up to the concept from a different angle. They had models of how the world works (natural numbers, real numbers, etc), and they ran into issues where these concepts did not line up with reality. Zeno's paradox, which I mentioned earlier, is one of the easy ones which makes the argument that you can't move anywhere. As mathematicians can obviously see that we appear to move places, they needed to update their models. So they put out concepts which would help make their models function, and tested them thoughtfully. Infinity is one of the concepts which has by and large survived the testing.
Myself, I recommend reading up on what infinity actually means to mathematicians, with an open mind. For example, your wording seems to suggest that you think mathematicians consider "infinity" to be a number. For much of mathematics it is not a number. It is a cardinality. It is a measure of the size of the set of natural numbers. There are areas of mathematics where you do see infinity as a number (transfinite/ordinal numbers) and they take great care to treat it differently from the cardinality concepts. Much of mathematics does not depend on these transfinite concepts. Mathematicians will still generally give credit to them, because the concept is rigorously defined and consistent, but that doesn't imply that they must believe in them as anything more than an intellectual concept. The concept of infinity as a cardinality has much wider acceptance because has been found to lead to useful real world implications (such as calculus, which got us to the moon).
I won't explore your other questions, because stack exchange format prefers to focus on one question at a time, but the argument is the same for all of them: mathematicians did not believe in them blindly. They defined them and carefully analyzed to make sure they seemed reasonable.
If you are interested, I recommend the VSauce video How to Count Past Infinity. It captures the concept well without assuming you have a substantial math background. And he also is willing to dig at the question you are asking without dismissing it outright. (it's roughly 13 minutes in, but I don't recommend skipping. let him make his argument first)
You may also be interested in researching the Axiom of Choice (AoC). It's a funny little axiom which looks benign, but leads to all sorts of funny issues such as the Banach-Tarski paradox. It is an example of a modern day abstract concept which is in the process of being accepted or rejected by the mathematical community, so it's a great way to shed light on how they go about accepting or rejecting these things with eyes wide open.
• – Cort Ammon Apr 5 '18 at 15:05
• @CortAmmon, I truly didn't insult anyone personally nor like to do this at any time but truly emphasizing on pushing the so easy facts that I see with so many pieces of evidence (say at least for me), that might seems for few others as an insult, wonder!, why should I do that if my true intention is to uprise the facts by convincing them about it (instead of insulting), but it seems that people don't feel so comfortable especially if someone seems to abuse other things in mathematics that usually they used to beileave in due to the global education knowledge, so the whole issue seems physioc – Bassam Karzeddin Apr 5 '18 at 17:15
• If your interest is in convincing people of what you believe the facts to be, then I highly recommend studying the art of convincing people. You have a lot of feedback here, and the universal thread between all of them is that nobody considers your argument to be convincing. In fact, many have phrased even stronger opinions. Also, from experience, I know many of these people are fully aware of the arguments you make, and could help you learn to make them more convincingly. You really should reach out and read the arguments in support of finitism... – Cort Ammon Apr 5 '18 at 17:30
• ... and constructivism. You are not the first to make these cases, and it is much easier to reference a body of existing literature than it is to singlehandedly seek to upheave centuries of math with nothing but your own statements that "it's obvious everyone is wrong" to back them up. – Cort Ammon Apr 5 '18 at 19:02
• Also, it may be worth doing some research into philosophy itself. Philosophy has a long history of handling people who argue that it's "obvious everyone is wrong," and there are general patterns you can use to challenge the status quo in a way that shows you are serious and thinking things through. – Cort Ammon Apr 5 '18 at 19:05
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## Key Concepts
• The addition property of equality states, for all real numbers $a, b, \text{ and } c: \text{ if } a=b \text{, then } a+c=b+c$. That is, we may add or subtract the same amount to both entire sides of an equation without changing its value.
• The multiplication property of equality states, for all real numbers $a, b, \text{ and } c: \text{ if } a=b \text{, then } a \cdot c=b \cdot c$. That is, we may multiply or divide the same amount to both entire sides of an equation without changing its value.
• Any point graphed in the coordinate plane is of form $\left(x, y\right)$ where $x$ is called the x-coordinate and $y$ is called the y-coordinate. With these coordinates, any point in the plane may be unambiguously located or identified.
• The coordinates of any ordered pair contained in the graph of an equation satisfies the equation (makes it a true statement when substituted for x and y).
• The slope of a line, of form $m=\dfrac{\text{rise}}{\text{run}}$, is a measure of the steepness of a line.
• Parallel lines have identical slopes; perpendicular lines have opposite, reciprocal slopes.
## Key Expressions, Equations, and Inequalities
• $\displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}$ and $\displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ where $m=\text{slope}$ and $\displaystyle ({{x}_{1}},{{y}_{1}})$ and $\displaystyle ({{x}_{2}},{{y}_{2}})$ are two points on the line.
• For any number x and any integers a and b, $\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}$.
• For any non-zero number x and any integers a and b: $\displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$
• For any positive number x and integers a and b: $\left(x^{a}\right)^{b}=x^{a\cdot{b}}$.
• For any nonzero numbers a and b and any integer x, $\left(ab\right)^{x}=a^{x}\cdot{b^{x}}$.
• For any number a, any non-zero number b, and any integer x, $\displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}$.
• Any number or variable raised to a power of 1 is the number itself. $n^{1}=n$
• Any non-zero number or variable raised to a power of 0 is equal to 1. $n^{0}=1$
• For any nonzero real number $a$ and natural number $n$, the negative rule of exponents states that ${a}^{-n}=\frac{1}{{a}^{n}}$.
• With a, b, m, and n not equal to zero, and and n as integers, the following rules apply: $a^{-m}=\frac{1}{a^{m}}$, $\frac{1}{a^{-m}}=a^{m}$, $\frac{a^{-n}}{b^{-m}}=\frac{b^m}{a^n}$.
## Glossary
absolute value
a number’s distance from zero on the number line, which is always positive
constant
a number whose value always stays the same
coefficient
a number multiplying a variable
equation
two expressions connected by an equal sign
exponent
a number in a superscript position that tells how many times to multiply the base by itself
expression
a number, a variable, or a combination of numbers and variables and operation symbols
reciprocal
two fractions are reciprocals if their product is $1$
term
a single number, variable, or a product or quotient of numbers and/or variables
variable
a symbol that stands for an unknown quantity, often represented with letters, like x, y, or z.
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### How to CTRL+ALT+DEL in RDP
ctrl+alt+end
### Networking training materials
I’ve stumbled upon this great list of materials you can use to get started on IT, and Networking in particular. I have to admit that some are new for me, even after being in this field for over 12 years.
### Udemy offline
Here’s another cool and helpful thing I just discovered that might help you stranger of the internet.
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# Ether and electrons in relativity theory (1900–1911)
Scott A. Walter
To appear in J. Navarro, ed, Ether and Modernity, Oxford: Oxford University Press
###### Abstract
This chapter discusses the roles of ether and electrons in relativity theory. One of the most radical moves made by Albert Einstein was to dismiss the ether from electrodynamics. His fellow physicists felt challenged by Einstein’s view, and they came up with a variety of responses, ranging from enthusiastic approval, to dismissive rejection. Among the naysayers were the electron theorists, who were unanimous in their affirmation of the ether, even if they agreed with other aspects of Einstein’s theory of relativity. The eventual success of the latter theory (circa 1911) owed much to Hermann Minkowski’s idea of four-dimensional spacetime, which was portrayed as a conceptual substitute of sorts for the ether.
## Introduction
The history of twentieth-century physics is scanned by the “classical”, or non-quantum theory of the electron, introduced by H. A. Lorentz, Joseph Larmor and others from the early 1890s. As a research program in physics, electron theory became more promising with the measurement of the mass-to-charge ratio in cathode-ray deflection experiments by J. J. Thomson and Emil Wiechert at the end the decade. It was successful in explaining magneto-optical phenomena, beginning with the Zeeman effect, and soon appeared to point the way to unifying the explanation of all forces – including the force of gravitation – by means of interactions between electrons and the ether. During the first two decades of the twentieth century, the physicists Lorentz, Henri Poincaré, Albert Einstein, Ebenezer Cunningham and Max Laue, all of whom were familiar with the idea of electromagnetic waves and electrons propagating in the ether, were led to engage with new technologies, mathematical formalisms and ideas about relative time and space. This chapter sketches some of the ways in which leading physicists sought to adapt pre-relativistic ether-based electron theories to the physics of relativity, from the emergence of the electromagnetic world-picture in 1900, to the publication of Emil Wiechert’s post-Minkowskian defense of the ether in 1911.
The history of the electron and that of the ether are tightly knit in the early twentieth century, although historians have paid much more attention to the former than the latter, leaving gaps in the literature concerning electron theory.11The emergence of electron theory in the 1890s is described in a monograph by Buchwald (1985), while the development of electron theory has been studied in detail up to the time of Einstein’s and Poincaré’s theories of relativity (1905–1906) by Arthur I. Miller (1981), and Olivier Darrigol (2000). For the period up to 1920, works of reference include R. McCormmach (1970), B. Wheaton (1983), H. Kragh (2001), O. Knudsen (2001), W. Kaiser (2001), and Navarro (2012). Since E. T. Whittaker’s history of theories of “æther” and electricity (1953), historians of physics have considered the emergence of relativity theory in the context of ether and electron theory. A one-time participant in this history, Whittaker created a stir late in his life by attributing the theory of relativity not to Albert Einstein but to Lorentz and Poincaré. Much effort was subsequently expended to discredit Whittaker’s view and reinforce Einstein’s priority of discovery. The historian Stanley Goldberg, for example, argued that the theories of Poincaré and Einstein were distinct, except for the formalism (1970, 91). Many commentators since Goldberg have focused on the contrast between, on the one hand, Einstein’s provocative claim that the ether is a superfluous concept in electrodynamics and, on the other hand, Poincaré’s advocacy of an abstract ether.22For a thoughtful review of the literature, see Darrigol (2004).
Most recently, Peter Galison showed that Poincaré and Einstein shared an interest in the topics of microphysics and clock synchronization (Galison 2003). For Galison, the emergence of relativity theory was the result of a chance confluence of three separate conceptual streams: physics, philosophy and technology. To these three streams I will add a fourth: applied mathematics, which Poincaré and Einstein brought into play in 1905, with profound and far-reaching consequences for conceptions of ether, electrons and relativity.
The first three streams came together in Leiden, in December, 1900. The occasion for this meeting was the jubilee celebration of Lorentz’s defense of his doctoral thesis; Poincaré had been invited to this celebration, along with one of his students and most of Europe’s electrodynamicists. Poincaré declined the invitation, but contributed a paper in which he analyzed the status of Newton’s third law in Lorentz’s electron theory.33On Poincaré’s participation in the Lorentz Festschrift, see his correspondence with Heike Kamerlingh Onnes, published in Walter (2016).
## 1 Lorentz electrons and local time in Leiden
For purely mathematical reasons, Lorentz had introduced a fictive time which he called “Ortszeit” and which Poincaré called “temps local”, or local time, following the translation of his former student, Alfred Liénard (1898b). Poincaré realized that local time could be construed as a measurable quantity: local time is the time measured at relative rest by observers in uniform motion with respect to the ether, provided that they synchronize their watches in a certain way. The watches of the co-moving observers are synchronized, Poincaré explained, via an exchange of light signals, whereby light time-of-flight between observers is taken into account when zeroing the watches, but their common motion with respect to the ether is ignored (Poincaré 1900, 272).
Poincaré knew quite well that he was breaking new ground, as he described his time-synchronization protocol as a definition of local time. In providing this definition, Poincaré turned Lorentz’s electron theory into a regular theory of physics, that is, one having measurable consequences, whereas, before, it had been an abstract model. Lorentz seems not to have appreciated this point, although he thanked Poincaré for his contribution.44See Lorentz to Poincaré, 20 January, 1901, in Walter et al. (2007, § 38.1). According to Lorentz (1895), the local time $t^{\prime}$ was a linear function of the ether time $t$ and the product of the distance $\mathbf{x}$ and the frame velocity $\mathbf{v}$, divided by the square of the velocity of light in the ether, $c$: $t^{\prime}=t-\mathbf{vx}/c^{2}$. Lorentz noted that his formula was valid up to a first-order approximation in $v/c$.
In addition to being a contribution to physics, Poincaré’s paper was a result of philosophical reflection on the nature of distant simultaneity, in relation to technical advances in astronomy and geodesy. As a member of the French Bureau of Longitudes, Poincaré was charged with the scientific oversight of national and colonial mapping projects, and French participation in international collaborations in astronomy and geodesy.
One of the latter projects sought to measure an arc of meridian passing through Peru, Ecuador and Colombia. Longitude measurement at the end of the nineteenth century involved several techniques, including that of fixing the position of celestial objects, and noting the temporal instant of astronomical events such as occultations and zenith passages. To determine the exact time of a given event, geodesists relied on time signals relayed by telegraph, as Poincaré observed in an original essay entitled “The measurement of time” (1898, English translation in Gould 2001, 220ff).
Poincaré’s essay on time measurement and his definition of local time both deeply influenced the later development of the principle of relativity, while Poincaré’s own conceptions evolved in the first years of the twentieth century, partly as a result of new ideas in electron theory, including those of Paul Langevin.
## 2 Langevin’s electron waves in ether
Paul Langevin’s theory of the electron drew on several sources, especially Joseph Larmor’s Æther and Matter (1900), J. J. Thomson’s Notes on Recent Researches (1893) and George Searle’s calculation of the energy of a Heaviside ellipsoid (Searle 1897). He introduced a distinction between the velocity fields and acceleration fields of the electron, and published a graphical depiction of the velocity waves of a spherical electron in motion. This led in turn to a geometrical derivation of the field of a Heaviside ellipsoid, introduced by Heaviside in 1889 and glossed by J. J. Thomson in 1893.
Langevin was concerned, as the title of his paper implies, with the source of electromagnetic radiation and the inertia of the electron. He supposed (following Larmor) that electron radiation was due entirely to acceleration, and he calculated thereby the energy of an electron in uniform motion. The “electromagnetic mass” of such an electron was given to be a function of the “wake” of the electron in motion, represented by Langevin’s velocity waves. The electron’s wake, in turn, depended on the charge distribution of the electron; Langevin considered uniform distributions by surface and volume.
Langevin’s distinction between velocity and acceleration waves recalled the retarded potentials introduced to electrodynamics by Langevin’s former teacher Henri Poincaré in the early 1890s, as well as the formulation of the potentials for a moving point charge due to Alfred Liénard (1898a) and Emil Wiechert (1900). Langevin’s theory, like all contemporary electron theories, assumed that his waves propagated in an ether at absolute rest. According to Langevin’s theory, electrons traveled through the ether at sublight velocity, generating velocity waves and, in the case of non-inertial motion, acceleration waves. These waves propagated in the ether with the speed of light; velocity waves dissipated rapidly, such that only acceleration waves could be detected far from the electron.55For details on Langevin’s paper, see Miller (1973).
## 3 Poincaré on apparent time and the relativistic electron
As a student in the mid-1890s, Langevin had followed Poincaré’s lectures on Sommerfeld’s theory of diffraction, but he did not engage personally with Poincaré until September, 1904, when they were both members of the French delegation to the Congress of Arts and Sciences, held at the World’s Fair in St. Louis. The younger man was flattered by the attention of his former teacher, as he recounted the meeting by letter to his wife back in Paris.66See Langevin’s notebook, box 123, and the letter to his wife of 26 September, 1904, box 3, Fonds Langevin, Library of l’École supérieure de physique et de chimie industrielle, Paris. By that time, the two men had a mutual interest in the theory of electrons, which was the topic of Langevin’s lecture in St. Louis (Langevin, 1906).
We do not know if Langevin discussed with Poincaré his forthcoming paper on the inertia of the electron, but we do know that Poincaré found inspiration from the latter paper for his discovery of the Lorentz group. Under the coordinate transformations of the Lorentz group, Poincaré demonstrated in 1905, the laws of electrodynamics retain their form. What impressed Poincaré most was not Langevin’s constant-volume model of the electron but his explanation of the velocity and acceleration waves produced by an electron: according to Langevin, these waves propagated in free ether at the speed of light. Instead of Langevin’s model, Poincaré preferred the deformable electron model proposed by Lorentz, as this model had the advantage, as Poincaré proved, of preserving the principle of relativity. Poincaré (1906, 149) noticed further that, by applying the Lorentz transformations to Langevin’s acceleration waves, he could recover Hertz’s solution of Maxwell’s equations for an oscillator at rest in the absolute ether.77On Hertz’s solution, see Darrigol (2000, 251).
In June 1905, Poincaré supposed that all laws of physics were, likewise, form-invariant with respect to the transformations of the Lorentz group, including the law of gravitation. In a letter to Lorentz announcing his discovery, Poincaré observed that the requirement of Lorentzian form-invariance spelled the end of what he called the “unity of time” (Poincaré to Lorentz, ca. May, 1905, in Walter 2007, § 2-38-3). Yet, Poincaré was not ready to abandon the traditional definition of time and space in this new theoretical context. He deftly elided the question of time and space deformation in his memoir on the dynamics of the electron (Poincaré 1906) by focusing on active transformations alone (Sternberg 1986).
In his university lectures of 1906–1907, Poincaré explained how, in principle, one could measure Langevin waves and thereby determine the shape of an electromagnetic pulse generated by a source in motion with respect to the ether. According to lecture notes recorded by a student note-taker, Poincaré recalled Langevin’s paper and reproduced the latter’s schematic of an electron in motion (Fig. 1) along with a diagram of his own creation (Fig. 2), which showed how the electromagnetic pulse was related to the Lorentz transformations. The pulse created by the point source had the form of an ellipsoid, which was elongated in the direction of motion of the source and had a focus located at the source. A section through a meridian of the ellipsoid produces the ellipse shown in Fig. 2.
Poincaré’s diagram illustrates the Lorentz contraction, whereby all material objects contract by a Lorentz factor, but only in the direction of their motion with respect to the ether. Electromagnetic waves are immaterial, and are thereby unaffected by this contraction. If we measure the form of an electromagnetic pulse with a material rod at any instant of time, the result will depend on the speed of the rod with respect to the ether. For any nonzero rod velocity, the “true” form of the pulse is then an ellipsoid with an elongation proportional to the velocity of the measuring rod, assumed to be at relative rest with respect to the moving point source.
Poincaré’s ellipse was designed to explain the meaning of the primed variables appearing in the Lorentz transformations, which is to say, in Poincaré’s terminology, the “apparent” time and space coordinates of a reference frame moving uniformly with respect to the ether. While Poincaré affirmed the reality of Lorentz contraction of material bodies, he recoiled from affirming the corresponding reality of temporal deformation. According to student notes, he explained the situation as follows:
So Lorentz assumes that all bodies undergo a contraction in the direction of motion proportional to the square of velocity. Lengths are then altered, and durations are altered by the impossibility of setting watches truly, such that the apparent velocity of light is constant. (Vergne Notebook 2, 52, Viète Center)
Why did Poincaré feel it was impossible to “truly” synchronize mobile timekeepers in a state of relative rest? He did not explain this statement, but his intention is not hard to discern. Poincaré always referred to primed time and space coordinates of a reference frame in motion with respect to the ether as “apparent” coordinates, while the unprimed, “true” coordinates belonged to the frame of the absolute ether. In a manner consistent with this view, Poincaré’s discussion of the electromagnetic ellipse referred only to measurements made with rods and timekeepers at rest with respect to the ether, or to rods in motion, but not to timekeepers in motion. Thus, Poincaré had greater faith in mobile rods than in mobile clocks, even though the rods and clocks in physics laboratories and astronomical observatories around the world were all understood to be in motion with respect to extraterrestrial objects, for example, the center of the Sun. Poincaré was not alone in his distrust of mobile clocks in 1906, although he later had a change of heart, as explained in § 5.
## 4 Einstein exiles the light-ether
Even after relativity theory became a part of mainstream physics, Poincaré remained attached to the notion that the ether was a useful construct for the theorist. When he first affirmed that the laws of physics exhibit Lorentz covariance, electron theorists were agreed that the ether was an essential element of the physical world, with one exception: Albert Einstein. Shortly following Poincaré’s discovery of the Lorentz group, Einstein derived the Lorentz transformations from kinematic arguments, based on two postulates: the relativity of phenomena for inertial frames of motion, and the universal invariance of the speed of light in empty space. One consequence of Einstein’s theory, announced in the second paragraph of his essay “On the electrodynamics of moving bodies”, was that that the ether (or “light-ether”, in Einstein’s terms) was not needed in electrodynamics:
The introduction of a “light-ether” will prove to be superfluous, inasmuch as the view to be developed here will neither introduce an “absolutely resting space” endowed with special properties, nor associate a velocity-vector with a point of empty space at which electromagnetic processes occur.88Einstein (1905b, 892), translation by the author. For line-by-line commentary on Einstein’s paper, see Miller (1981, 392), and Stachel (1989, Doc. 23).
Einstein’s remark was certainly designed to capture physicists’ attention, and so it did.
For a theory of the electrodynamics of moving bodies, refusing to assign a velocity to points of empty space where electromagnetic processes occur would have been an obvious non-starter. However, Einstein had figured out a way around this problem. The physicist needed only to describe electromagnetic processes in a frame with constant rectilinear velocity. Knowledge of the processes in all such frames was then obtained via the Lorentz transformations, which depend on the inter-frame velocity. All measurements are performed in inertial frames, and there is no recourse to an absolute frame of reference, or to a luminiferous ether.
Einstein’s decision to exile the ether from electrodynamics (and from physics in general) came at a high cost to what Poincaré referred to (see § 3) as the unity of time. While physicists at the turn of the twentieth century were familiar with the concept of the luminiferous ether, they were unfamiliar in general with the idea that time and space were relative notions.99Notable exceptions the Cambridge electron theorist Joseph Larmor, who remarked on the “difference of time reckoning” of orbiting electrons (see Larmor 1897, 229), and Poincaré, who gave an operational definition of local time (see Poincaré 1900, 272). Naturally, Einstein sought to render such ideas more plausible. The details of his arguments in favor of the relativity of time and space have been discussed by many authors, and need not be rehearsed here.1010For a clear exposition of Einstein’s kinematics, see Martínez (2009). I will focus instead on how Einstein tried to persuade his readers of the logical compatibility of his two postulates.
The compatibility of Einstein’s postulates of relativity and light-speed invariance followed for Einstein from an argument which may be summarized as follows.1111For a more detailed presentation see Walter (2018a), from which the present account is drawn. Let a spherical light wave be transmitted from the coordinate origin of two inertial frames designated $S$ and $S^{\prime}$ at time $t=\tau=0$. In frame $S$ the light wave spreads with velocity $c$ such that the wavefront is expressed as
$x^{2}+y^{2}+z^{2}=c^{2}t^{2}.$ (1)
To obtain the equation of the wavefront in frame $S^{\prime}$ moving with velocity $v$ with respect to $S$, we apply a certain linear transformation of coordinates from $S$ to $S^{\prime}$, depending on a factor that depends on velocity.1212Let this factor be $\varphi$, and coordinates in the frame $S^{\prime}$ be defined as: $\xi=\varphi(v)\gamma(x-vt)$, $\eta=\varphi(v)y$, $\zeta=\varphi(v)z$, $\tau=\varphi(v)\gamma\left(t-\frac{vx}{c^{2}}\right)$, where $\gamma=(1-v^{2}/c^{2})^{-\frac{1}{2}}$, and the coordinate origins of the two frames are assumed to coincide. If we set $\varphi(v)=1$, these transformations are equivalent to the Lorentz transformations. The result of the transformations, Einstein found, was just
$\xi^{2}+\eta^{2}+\zeta^{2}=c^{2}\tau^{2}.$ (2)
Since (1) goes over to (2) via these transformations, Einstein observed, the light wave that is spherical in $S$ is also spherical in $S^{\prime}$, it propagates with the same velocity $c$, and consequently, “our two basic principles are mutually compatible” (Einstein 1905b, § 3, 901).
Einstein’s compatibility demonstration addressed one of the more immediate objections to be raised against his theory: that the propagation of light implied the existence of a substrate. This substrate, known as the ether, was common to the electron theories of Lorentz, Joseph Larmor, Alfred Bucherer, Paul Langevin and Max Abraham. Some, but not all of these theorists, went on to adopt relativity, but none of them ever saw fit to banish the luminiferous ether from physics.
## 5 Cunningham’s multiple ethers
Among British theorists, relativity theory had few proponents, if any, until Ebenezer Cunningham (1881–1977) took it up. A Cambridge-trained mathematician, senior wrangler in 1902, and lecturer in applied mathematics at University College London in 1907, Cunningham understood Einstein’s theory to be consistent with the existence of multiple ethers, provided that every inertial frame is associated with an ether.1313See Goldberg (1970), and Hunt (1986). Inspired by Larmor’s electron theory, Cunningham’s multiple-ether approach to relativity recalls the mechanics proposed by the Leipzig mathematician Carl Neumann. Newton’s laws of mechanics, Neumann (1870) observed, give one the freedom to consider any inertial frame to be at rest with respect to a fixed set of coordinate axes he called the “Body Alpha”.1414Cunningham (1911) recalled this fact, without mentioning Neumann. Views equivalent to Cunningham’s, but stripped of reference to the ether, were subsequently advanced by Minkowski (1909, 79) and Laue (1911, 33).
Cunningham’s first paper on relativity set out to overturn an objection raised by Max Abraham with respect to Lorentz’s electron theory. Abraham (1905, 205) believed that energy conservation required a fundamental modification of Lorentz’s deformable electron model, in the form of a supplemental internal and non-electromagnetic source of energy. Cunningham challenged Abraham’s (frame-dependent) definition of electromagnetic momentum, and found that, under the same quasi-stationary approximation, and an alternative momentum definition, the problem vanishes. He concluded that no non-electromagnetic energy was required by Lorentz’s electron model, which remained for him a possible foundation for a “purely electromagnetic theory of matter.”1515Cunningham’s conclusion agrees with that reached later by Enrico Fermi; see Rohrlich (2007, 17), and Janssen & Mecklenburg (2006).
Along the way, Cunningham assumed that if Lorentz’s deformable electron is spherical when at rest, when put in motion and measured by comoving observers, it will remain spherical. But, when measured with respect to a frame at rest, the moving electron will have a “spheroidal shape as suggested by Lorentz” (Cunningham 1907, 540). Cunningham took this suggestion a step further, arguing that a light wave would appear spherical to all inertial observers, in agreement with Einstein on this point (and with reference to Einstein’s relativity paper of 1905).
Next, Cunningham took an important step toward the legitimation of the concept of light-sphere covariance, as Goldberg (1970, 114) first noticed. Einstein’s derivation of the Lorentz transformations could be reduced to a handful of steps, Cunningham realized, by requiring the covariance of the light-sphere equation (1) with respect to these transformations. Cunningham’s requirement of covariance of the light-sphere equation entailed the relativity of space and time:
For it is required, among other things, to explain how a light wave traveling outwards in all directions with velocity $C$ relative to an observer $A$, may at the same time be traveling outwards in all directions with the same velocity relative to an observer $B$ moving relative to $A$ with velocity $v$. This can clearly not be done without some transformation of the space and time variables of the two observers. (Cunningham 1907, 544)
Shortly after Cunningham’s derivation of the Lorentz transformations appeared in print, Einstein employed the same method of derivation, making Cunningham the first British contributor to what was later known as Einstein’s theory of relativity. However, Einstein did not acknowledge Cunningham as the source of the derivation, and he may well have come up with it on his own.1616See Einstein (1907, § 3), reed. in Stachel (1989, Doc. 47). Cunningham’s paper appeared in the October 1907 issue of Philosophical Magazine, and Einstein’s review article was submitted for publication in Johannes Stark’s Jahrbuch der Radioaktivität und Elektronik on 4 December, 1907.
In subsequent years, Cunningham contributed to the generalization of the principle of relativity beyond frames with uniform motion, and published one of the first English-language textbooks on relativity. The latter work introduced Minkowski’s spacetime theory to English readers, and revived Cunningham’s earlier view of multiple mobile ethers, in a chapter entitled “Relativity and an Objective Æther”. The sufficient condition for the objective reality of the mobile ether was announced by Cunningham quite simply to be the conformity of its kinematics to those of the principle of relativity (Cunningham 1914, 193). Cunningham’s view of the mobile ether found employment throughout the 1920s, and well beyond Cambridge, thanks to Sommerfeld’s celebrated textbook Atombau und Spektrallinien (1919, 319) and its translations into French and English.
Contributing to Cunningham’s confidence in the universal validity of the principle of relativity was Minkowski’s spacetime theory, and also Einstein’s notion of light quanta, whereby light energy is not distributed uniformly in space but exists in discrete packets (Einstein 1905a). The theory of radiation appeared thereby to conform to relativity, at least by 1914, thanks in part to Einstein’s light quanta. Even before light quanta came to be accepted, the theory of radiation was central to discussions of relativity and the ether, as shown in the next section.
## 6 Relative time from radio waves
In the fall of 1908, the Göttingen mathematician David Hilbert (1862–1943) extended an invitation to Poincaré on behalf of the Wolfskehl Foundation to deliver a series of lectures. Poincaré accepted the honor, and delivered six lectures in Göttingen in April 1909, before an international audience of mathematicians and physicists.1717For details on the lectures see Gray (2013), Rowe (2017), and Walter (2018b). One of Hilbert’s Göttingen colleagues, and perhaps his closest friend, was Hermann Minkowski (1864–1909), who had recently put forward the theory of spacetime. Minkowski’s spacetime theory borrowed key insights from Poincaré’s study of the Lorentz-covariant law of gravitation, including the idea of a four-dimensional vector space with one imaginary dimension. Minkowski had planned to pursue the theory, but following an attack of appendicitis, he died in January, 1909, just three months before Poincaré’s lecture series. There are indications that Minkowski’s theory inspired the topic of Poincaré’s final Wolfskehl lecture, which was entitled “The New Mechanics”, although the only Göttingen scientist mentioned by name in the published text of the lecture is the electron theorist Max Abraham, whom Poincaré characterized, along with H. A. Lorentz, as one of the “great demolitionists” of Newtonian mechanics (Poincaré 1910, 51).
One of the novelties of Minkowski’s theory was its definition of “proper time” as the parameter of a four-dimensional trajectory in spacetime, which was given to be the time read by an ideal clock describing this same trajectory. According to this theory, there are an infinite number of temporal axes, and as many corresponding three-dimensional spaces, which may be described with reference to a single spacetime extending infinitely in four dimensions. In Poincaré’s original scheme, as noted (§ 3), there was only one three-dimensional space – the ether – and one temporal dimension. Only clocks at rest in the ether were reliable timekeepers. But in Göttingen, Poincaré admitted that ideal clocks in motion with respect to the ether might also tell the right time.
The circumstances of this change are of interest, as they illustrate how technology can shake the foundations of science. Since Heinrich Hertz’s discovery of electromagnetic waves in air in the late 1880s, Poincaré had contributed to the theory of Hertzian waves, and engaged with the emerging technology of wireless telegraphy. When the Eiffel Tower was menaced with destruction, Poincaré backed a plan to employ it as the world’s tallest wireless transmission antenna (Galison 2003, 276), thereby strengthening the tower’s chances for preservation.
Poincaré knew that Hertzian waves propagate in the ether with the speed of light, $c$. In his final Wolfskehl lecture, he imagined an observer $B$ in a vehicle in motion with respect to the ether with speed $2c/3$. Observer $B$ transmits telemetry data to an observer $A$ in a second vehicle moving with equal and opposite speed:
$A$ and $B$ begin by setting their watches, then $B$ sends telegrams to $A$ indicating his successive positions; putting these signals together, $A$ can give an account of $B$’s motion, and trace its curve. Well, the signals propagate at the speed of light; the watches marking apparent time vary at every instant and it will all go down as if $B$’s watch were fast. (Poincaré 1910, 54-55)
As a result of the deformation of time revealed by the telemetry data, Poincaré explained a few months later, observer $B$ would come to believe his vehicle was advancing not at hyperlight velocity but at sublight velocity, so that the principle of relativity would not be violated (Poincaré 1909, 173).
Notice that Poincaré’s watch in motion runs fast (with respect to a watch at rest in the ether), while Einstein’s clocks in motion run slow (with respect to clocks at relative rest). Not only is Einstein’s notion of relativistic time dilation readily admitted in physics, it also underpins the accuracy of global positioning systems.1818Such systems correct for several motional effects, including special-relativistic time dilation, the latter effect being offset by gravitational blueshift; for a clear explanation, see Ashby (2002). What then are we to make of Poincaré’s fast watch? His thought experiment supposes that observer $B$ initially judges his speed relative to $A$ to be $4c/3$, in accordance with Newtonian kinematics. Only after exchanging telemetry data at the speed of light does $B$ revise his initial speed estimate downwards to a sub-light value. If we ignore Lorentz contraction, then time contraction is the only plausible means of obtaining a reduction to sub-light vehicle (and watch) velocity. Simply stated, observer $B$ concludes that his watch runs fast, because his speed is slower than Newtonian kinematics had led him to believe.1919For a more detailed explanation, see Walter (2014).
As for Lorentz, by 1909 he, too, admitted that both rods and clocks in motion with respect to the ether could be used to measure temporal and spatial intervals, respectively. Like Poincaré, he recognized that inertial observers, whether in uniform motion or at rest, would agree on the speed of light, provided that the mobile clocks were optically synchronized in the frame of motion. Lorentz admitted the (Lorentz) contraction of bodies in their direction of motion, and proved that the apparent time of an observer in motion is just that indicated by the Lorentz transformations, such that time is apparently dilated in the observer’s frame of motion and, consequently, that his clocks run slower than identical clocks at rest in the ether. In his authoritative monograph The Theory of Electrons, Lorentz encouraged his reader to keep in mind that,
in doing all that has been said, the observer would remain entirely unconscious of his system moving (with himself) through the ether, and of the errors of his rod and his clocks. (Lorentz 1909, 226)
The “errors” of rod and clocks that Lorentz refers to are understood to be relative to the “true” values indicated by an identical rod and identical clocks at rest in the ether. From the foregoing considerations of Poincaré and Lorentz, we see that their attachment to the ether did not prevent them from admitting that time could be measured in moving frames. It was just not the true time.
## 7 Minkowskian electrons in spacetime
Since 1902, Hermann Minkowski had been a professor of mathematics in Göttingen, where he took up several topics in theoretical physics, including heat radiation and the electrodynamics of moving bodies. Before moving to Göttingen, Minkowski had been on the faculty of Zürich Polytechnic (currently ETH Zürich), where Albert Einstein and Mileva Marić were enrolled in his course in analysis.2020Other students receiving grades from Minkowski included Louis Kollros, Marcel Grossmann, and Jacob Ehrat; undated autograph, Minkowski Papers, Jewish National and University Library. In Göttingen, Minkowski co-led a seminar on electron theory in the summer of 1905, in tandem with David Hilbert. Two years later, in the summer of 1907, Minkowski took notice of Einstein’s and Poincaré’s foundational contributions to relativity theory, and soon thereafter made a fundamental contribution of his own: the theory of spacetime.2121On the electron-theory seminar, see Pyenson (1979); on Minkowski’s career in physics, see Walter (2008).
Minkowski presented his theory of spacetime in a lecture entitled “Space and Time” at the annual meeting of the German Association of Natural Scientists and Physicians in Cologne on 21 September, 1908. Central to his semi-popular lecture was a hand-colored transparency (Fig. 3). The transparency depicts two diagrams: on the left is a model of a two-dimensional Minkowski space and, on the right, a comparison of the trajectories in such a space of two electrons, with one at rest and the other in uniform motion. The transparency features several formulæ which illustrate Minkowski’s demonstration that the contraction of electrons (and Lorentz contraction in general) is a consequence of the fact that space and time are not separate entities, but part of a four-dimensional continuum: spacetime.
The transparency also shows that temporal intervals are dilated for observers in motion, although Minkowski appears not to have explained the details of the interpretation in Cologne. There can be little doubt about Minkowski’s recognition of the reality of time dilation, however, as he credited Einstein with “first clearly recognizing that the time of one electron is just as good as that of the other, which is to say, that $t$ and $t^{\prime}$ are to be treated identically” (Walter 1999). Minkowski thus sided with Einstein on the relative nature of time, rather than with Poincaré.
Minkowski did not take up the question of the form of an electromagnetic pulse for a source in motion, a topic earlier broached by Poincaré (see § 3), but his Cologne lecture carefully illustrated the Liénard-Wiechert potential on a three-dimensional spacetime diagram. In all of these considerations, Minkowski avoided reference to the luminiferous ether; he mentioned it only once in a critical way, saying that the Lorentz contraction should not be thought of “as something like a consequence of ether drag” (Minkowski 1909, 80).
In Minkowski’s view, the ether constituted an obstacle to the general acceptance of his theory of spacetime. He thus concluded his Cologne lecture with an appeal to “those for whom the abolition of familiar views is unappealing or distressful”, suggesting in effect that, although the electromagnetic ether was no longer tenable in modern physics, four-dimensional spacetime was in fact the “true kernel of an electromagnetic world view” (Minkowski, 1909, 88). The idea that the electromagnetic world view could be pursued successfully by means of the spacetime theory was, at best, an instance of wishful thinking on Minkowski’s part but, as a rhetorical ploy, it effectively situated spacetime theory at the forefront of theoretical physics.
## 8 Minkowski spacetime as an ether stand-in
The pursuit of the electromagnetic world view implied a microphysical reduction and, for Minkowski, this reduction was to be carried out in the arena of four-dimensional spacetime. He was unable to advance this project very far himself, as he perished in January 1909, as previously mentioned. Other theorists stepped in to continue Minkowski’s research program, including his assistant in Göttingen, Max Born, and his boyhood friend, Arnold Sommerfeld, the professor of theoretical physics at the Ludwig-Maximilian University of Munich.
Both Born and Sommerfeld were instrumental to the success of spacetime theory among theoretical physicists. Most importantly in this respect, Sommerfeld reformulated Minkowski’s elegant but unfamiliar matrix calculus as a four-dimensional vector analysis, which appealed to physicists familiar with ordinary three-dimensional vector analysis. When presenting the latter formalism to readers of the Annalen der Physik, Sommerfeld formulated an argument that was implicit in Minkowski’s Cologne lecture (see § 7), to the effect that spacetime, or the “absolute world” in the Minkowskian vernacular, was a substitute for the ether:
The absolute world appears in place of the older theory, that is, the connection of space and time via the velocity of light $c$, the immutability of which now constitutes the absolute substrate of electrodynamics. (Sommerfeld 1910, 749)
The “older theory” to which Sommerfeld refers here is that of Maxwell and Hertz, which assumed an ether at rest as the substrate of electromagnetic wave propagation with velocity $c$ in empty space. The immutability of the latter velocity was postulated by Einstein in 1905, as noted (in § 4), and also, in turn, by Minkowski, when he formulated the theory of spacetime.
In the years leading up to Minkowski’s discovery of spacetime, Sommerfeld had established his reputation as a leading electron theorist, and a critic of Einstein’s theory of relativity. The same was true of the Göttingen Privatdozent Max Abraham. A former doctoral student of Max Planck, Abraham was the author of an influential treatise on the electromagnetic theory of radiation. In the 1908 edition, which appeared shortly after Minkowski’s spacetime theory, Abraham adopted an approach midway between Sommerfeld’s enthusiastic embrace of Minkowskian relativity, and outright rejection of the principle of relativity.
For research in the electrodynamics of moving media, Abraham considered that both Lorentz’s theory and that of Minkowski represented attractive alternatives, provided that electrons were considered as point charges, in virtue of the ease of calculation these two theories offered. For electrons in empty space, on the other hand, Abraham felt that Lorentz’s theory was worthless. Not only was this theory inconsistent from a formal standpoint, in that it gave an improper value for electron rest mass (unless, like Poincaré, one admitted a non-electromagnetic binding potential), but it was also disconfirmed by Walter Kaufmann’s cathode-ray deflection experiments.
Earlier we saw how Poincaré had no truck with mobile clocks and preferred to keep time with clocks at rest in the ether (see § 3). Abraham shared this view with Poincaré, and consequently dismissed Einstein’s notion of path-dependent time, although he acknowledged that this was not ruled out by experiment. Time was absolute, in Abraham’s view, such that the difference between the rate of an ideal clock in motion and an equivalent clock at rest in the ether could be detected in principle, if not in practice. Space was also absolute for Abraham and, in light of the null result of the Michelson-Morley experiment (1887), he readily admitted that all material bodies contract in the direction of their motion with respect to the ether (Abraham, 1908, 368–369).
Both Sommerfeld and Abraham began to favor relativity following Minkowski’s contributions, and Alfred Bucherer’s experimental confirmation of the relativistic dynamics of charged particles. Sommerfeld argued that spacetime was a replacement for the ether, while Abraham devised an alternative to Minkowski’s relativistic theory of moving media, giving rise to over a century of debate over which formulation was to be preferred.2222For an overview, see Ramos, Guillermo, and Obukhov (2011). The theory of moving media proposed by Abraham was no less relativistic than Minkowski’s theory. Yet, Abraham held that Einstein’s light postulate was incomprehensible without an ether, arguing that electromagnetic waves and fields could not subsist without a substrate (Abraham 1914).
Max Abraham was not alone in his rejection of Einstein’s relativity, not even in Göttingen. Walter Ritz (1878–1909) had been a former student of Minkowski in Zürich, a former doctoral student of Woldemar Voigt and, from 1909, a Privatdozent in Göttingen. Ritz found fault both with Einstein’s relativity, for its postulation of universal light-speed invariance, and with field theory in general. He also dismissed the ether in favor of an emission theory, which performed well in comparison to Lorentz’s electron theory, although it failed to account for the optics of moving bodies, including Fizeau’s demonstration that light propagation speed in running water changes with flow rate (Fizeau, 1851). Ritz’s theory of electrodynamics was even more radical than Einstein’s in some respects and, while Einstein paid attention to it, most physicists did not. When Ritz succumbed to tuberculosis in 1909, his theory died with him.2323On Ritz’s contributions to electrodynamics see Martínez (2004) and Darrigol (2012).
While Ritz dismissed the luminiferous ether, Emil Wiechert sought to defend it. Director of the Göttingen Institute for Geophysics, Emil Wiechert was, like Ritz, unsatisfied with Einstein’s relativity. At the University of Königsberg in the 1890s, Wiechert developed a theory of charged particles which shared certain features of Lorentz’s theory, including an ether at rest. Unsurprisingly, Wiechert found much to admire in Lorentz’s electron theory and, like Lorentz, he felt that Einstein was wrong to do away with the ether. Along with many of his contemporaries, including Max Abraham (as previously mentioned), Wiechert felt that the electromagnetic field was not self-standing, but required a substrate. In addition, in his paper entitled “The Principle of Relativity and the Ether”, Wiechert deplored Einstein’s neglect of a model of electron shape and charge distribution, which meant for Wiechert that Einstein’s theory was not realistic, by default (Wiechert 1911, 748).
It is hard to imagine that Wiechert’s criticism would have bothered Einstein much. Unlike other electron theorists, Einstein neither needed nor offered a model of electron structure. Nonetheless, following Einstein’s own example, his contemporaries often referred to the “Lorentz-Einstein” theory, in virtue of a common recourse to the principle of relativity, and they considered the theories of Lorentz and Einstein to be empirically equivalent.
This conflation of theories may be explained by recalling a certain conceptual drift of the idea of ether among electrodynamicists, dating from the 1890s. There were those, like Henri Poincaré, who considered the ether to be real but undetectable in principle. Others, like Paul Drude, imagined that the properties of the electromagnetic field were actually properties of space, and not of the ether. And, while Poincaré and Drude both sought a microphysical reduction of electromagnetic phenomena, one theorist, Emil Cohn of the Kaiser-Wilhelm University of Strasbourg, eschewed both electrons and the ether in favor of a phenomenological theory featuring bare field equations, which compared well with that of Lorentz (Darrigol 2000, 366).
Wiechert’s criticism of relativity focused on the ether. The prospect of replacing the luminiferous ether with Minkowski spacetime did not please Wiechert, because spacetime seemed to him to allow bodies to propagate with velocities greater than the speed of light, while no such hyperlight phenomena had been observed. The possibility of hyperlight phenomena destroying the principle of relativity had been examined by Poincaré (1904), who noted that the propagation velocity of gravitational action had been calculated by Laplace to exceed the speed of light a million times over. Once Poincaré had discovered the Lorentz group, however, he showed that, if the speed of gravitation were equal to that of light, this would be no less consistent with astronomical observations than Newtonian gravitation. As one might guess, Wiechert’s defense of the ether did not mention Poincaré’s argument in favor of relativity.
Among those who admired Wiechert’s paper was the geodesist Friedrich Helmert, who cited it approvingly in his letter of nomination of Wiechert as a corresponding member of the Prussian Academy of Sciences; the letter was co-signed by Max Planck and Walther Nernst (Kirsten 1975, 198). One has to wonder how closely these scientists read Wiechert’s paper. In response to Wiechert’s article, Max Laue remarked the logical fallacy involved in deducing the existence of a class of preferred frames of motion (such as that of the luminiferous ether) from the inexistence of a different class of frames (in Wiechert’s case, frames with hyperlight velocity).
Laue went on to suggest that, in the absence of evidence for the existence of hyperlight phenomena, there was “no shame” in banning from physics further discussion of the existence of the ether and absolute time (Laue 1912). Seven years earlier, Einstein had deemed the ether superfluous to physics (as mentioned in § 4), but now Laue was recommending that the topic itself be censored, along with that of absolute time.
Educated in Berlin and Göttingen, Laue was, at the time of his critique of Wiechert’s view, a modest Privatdozent at the Ludwig-Maximilian University of Munich, attached to Sommerfeld’s Institute of Theoretical Physics. He had recently published the first German textbook on the principle of relativity (Laue 1911), and it soon became the standard work of reference in this domain. Six months after Laue pointed out the logical shortcomings of Wiechert’s defense of the ether, Sommerfeld communicated to the Munich Academy a paper written by Walter Friedrich, Paul Knipping and Laue on the interference of X-rays by crystals, a discovery for which Laue was awarded the Nobel Prize in Physics in 1914. That same year, Einstein became a member of the Prussian Academy of Sciences, on the strength of Planck’s letter of nomination, which emphasized Einstein’s revision of the notion of time and how this revision led to Minkowski’s spacetime theory (Kirsten 1975, 201). It is telling that Planck’s letter described Minkowski’s spacetime theory as a consequence of Einstein’s discoveries and that it neglected to mention Einstein’s criticism of the ether. Planck must have known that the members of the Prussian Academy were not ready to renounce the ether. The recognition of the two most vocal critics of the ether and absolute time – Laue and Einstein – by the Royal Swedish Academy of Sciences and the Prussian Academy of Sciences, respectively, sent a strong signal to physicists young and old that they, too, could do without the ether.
Yet, even among physicists who promoted the Einstein-Minkowski theory of relativity, Einstein and Laue’s anti-ether campaign met with resistance. As observed above, Sommerfeld and Max Abraham did not do away with the ether altogether. Likewise, when lecturing on the theories of Einstein, Minkowski, and Sommerfeld at the Collège de France in 1910–1911, Paul Langevin deplored Einstein’s view of the ether. Lecture notes by Léon Brillouin record Langevin’s remark on the subject: “The very notion of the ether loses its meaning, says Einstein – this is an exaggeration”. Langevin went on to point out that, while it is physically impossible to determine velocity with respect to the ether, “we can determine accelerations and rotations.”2424Léon Brillouin, Notebook “Cours de Relativité au Collège de France 1910–1911”, Léon Brillouin Papers, Box 7, folder 8, American Institute of Physics, Niels Bohr Library.
## 9 Conclusion
When Einstein announced that his theory of the electrodynamics of moving bodies had no use for the luminiferous ether and that this ether was consequently “superfluous” to the theoretical domain of electromagnetism, he must have known it would capture the attention of his peers. And so it did, although few physicists were ready to banish talk of the ether from physics.
The electron-theoretical origins of relativity theory guided its development, even as new conceptual tools were brought into play. Poincaré’s discovery of the Lorentz group enabled him to reinterpret Langevin’s conception of velocity waves and acceleration waves propagating in the ether, and to recover Hertz’s demonstration of the production of electromagnetic waves by an oscillator. Such waves, he showed later, could be used to demonstrate in principle the deformation of temporal intervals for observers in frames of reference in motion with respect to the ether. The mechanism for this deformation remained mysterious, although, for Lorentz, at least, it was no more mysterious than the contraction of bodies in their direction of motion with respect to the ether. Both time dilation and length contraction issued from the same velocity-based modification of dynamical laws, in Lorentz’s view.
Directly opposed to the latter “dynamical” approach to relativity, Einstein’s relativistic kinematics had no need of an ether, and did not introduce one. Einstein took care, nonetheless, to provide an argument for the logico-mathematical compatibility of his twin postulates of relativity and universal light-speed invariance. This argument, as we have seen, did not convince electron theorists, including Lorentz, Poincaré, Max Abraham and Wiechert, to forgo the ether.
Soon after the theories of Poincaré and Einstein appeared on the scene, a third approach to relativity proved effective in attracting electron theorists, and many others besides, to the relativistic fold. Minkowski’s spacetime theory featured a powerful blend of ideas drawn from a variety of disparate sources, including Hertz’s theory of the electrodynamics of moving bodies; Lorentz-Poincaré electron theory; Einsteinian kinematics and the theory of continuous transformation groups. Where Einstein offered in exchange for the ether only a pair of postulates, the logical consistency of which was suspect, Minkowski and his followers proposed spacetime as a conceptual substitute.
From the fact that neither Einstein’s theory nor that of Minkowski found immediate success, it may be gathered that, in general, physicists did not find the idea of renouncing the ether to be particularly compelling. Nonetheless, most theoretical physicists adopted spacetime theory, including several theorists encountered in this chapter: Einstein, Cunningham, Abraham, Sommerfeld, Langevin and Lorentz. Many mathematicians, who had no prior attachment to the ether, joined them in this endeavor.2525For quantitative details on the disciplinary reception of relativity theory, see Walter (1999).
By the end of the first decade of the twentieth century, relativity and spacetime theory were poised to dominate theoretical physics. To some extent, the rise of these theories came at the expense of ether theories, which continued nonetheless to appeal to theorists like Wiechert. The Lorentz-Poincaré electron theory survived the onslaught of relativity, via mathematical reformulation and conceptual adaptation to the principle of relativity. In a lecture delivered in Salzburg in September 1909, Einstein remarked that Lorentz’s theory was the only electron theory that was useful and had clear foundations (McCormmach 1970, 79). By the end of 1911, however, electron theory was seen to be incapable of explaining black-body radiation. This incapacity was not due to the theory’s fixed-ether foundation but rather on its assumption that bodies emit and absorb energy continuously (Kuhn 1978, 134).
Acknowledgments: I am grateful to the Dibner Library of the History of Science and Technology for granting me a residence in November, 2016, and to the Niedersächsiche Staats- und Universitätsbibliothek Göttingen for authorizing publication of the diagram in Figure 3.
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Advancements in computational musculoskeletal biomechanics are constrained by a lack of experimental measurement under real-time physiological loading conditions. This paper presents the design, configuration, capabilities, accuracy, and repeatability of The University of Texas at El Paso Joint Load Simulator (UTJLS) by testing four cadaver knee specimens with 47 real-time tests including heel and toe squat maneuvers with and without musculotendon forces. The UTJLS is a musculoskeletal simulator consisting of two robotic manipulators and eight musculotendon actuators. Sensors include eight tension load cells, two force/torque systems, nine absolute encoders, and eight incremental encoders. A custom control system determines command output for position, force, and hybrid control and collects data at 2000 Hz. Controller configuration performed forward-dynamic control for all knee degrees-of-freedom (DOFs) except knee flexion. Actuator placement and specimen potting techniques uniquely replicate muscle paths. Accuracy and repeatability standard deviations across specimen during squat simulations were equal or less than 8 N and 5 N for musculotendon actuators, 30 N and 13 N for ground reaction forces (GRFs), and 4.4 N·m and 1.9 N·m for ground reaction moments. The UTJLS is the first of its design type. Controller flexibility and physical design support axis constraints to match traditional testing rigs, absolute motion, and synchronous real-time simulation of multiplanar kinematics, GRFs, and musculotendon forces. System DOFs, range of motion, and speed support future testing of faster maneuvers, various joints, and kinetic chains of two connected joints.
## Introduction
Joint diseases and disorders, either due to aging or injury, are among the most prevalent, debilitating, and painful medical conditions [16]. Although tremendous advances have been made in musculoskeletal biomechanics, potential breakthroughs are hindered by limitations in acquiring reliable in vivo measurements such as joint contact forces and soft tissue deformations [7,8]. While hip and knee joint contact forces have been measured using instrumented implants [911], these devices are typically implanted in older patients, and the motions collected are limited to relatively slow motions such as gait, stair climbing, and slow jogging. In addition, the behavior may not be representative of what happens in a native joint since geometries and material properties have been altered.
To circumvent these critical barriers related to in vivo studies, investigators have developed computational models and in vitro simulators, the latter to perform measurements on cadaveric specimens. Computational models are very versatile since virtually any type of tissue can be simulated, in parallel to applying a variety of kinematic and kinetic conditions. The main drawback of predictive computational modeling is the need for thorough validation, which in turn relies on experimental measurements. Bates et al. [12] identified 77 anterior cruciate ligament (ACL)-related in vitro studies from 2004 to 2013, which may be categorized as robotic or mechanical impact tests.
Numerous robotic simulators [1321] have been developed to replicate the complexity of in vivo joint biomechanics. Many simulations are static or quasi-static [1315,19,22], but some have applied time-varying kinetics and kinematics to more accurately apply physiological loading conditions [1618,20,21,23,24]. The physiological accuracy of in vitro testing is dependent upon numerous factors. Critical to recreating in vivo conditions is the synchronous application of multiplanar joint kinematics, external forces (e.g., ground reaction forces (GRFs) and body weight), and musculotendon forces. In addition, to obtaining realistic joint loading conditions, muscle moment arms must also be replicated.
In particular, musculotendon force control varies tremendously between existing in vitro simulators: nonexistent musculotendon loading [14,15,17,19,21,22], grouped knee flexor and extensor musculotendon loading [16,20,24], and individual musculotendon force control [13,18,25]. Since each individual musculotendon crossing a specific joint uniquely contributes to its stability and therefore internal loading condition [26], grouping individual muscle forces or failing to represent muscle forces altogether can negatively influence the joint kinematics and introduce measurement errors regarding contact forces, ligament strains, and relative joint movement. Musculotendon force application is also affected by muscle moment arms. Since joint kinematics are governed by optimized muscle forces balanced between agonist and antagonist muscles, differences in muscle moment arms will alter these precisely tuned loading patterns and thus any further measurements performed.
Few simulators attempt to apply multiplanar kinematics and kinetics. Full constraint of a single joint requires control of six degrees-of-freedom (DOFs), where a knee consists of a tibiofemoral and a patellofemoral joint. Many simulators limit control to sagittal DOFs [13,15,16,20] and do not fully constrain tibiofemoral DOFs as demonstrated by other simulators [14,1719,2123].
Current methods of reproducing high strain rate loads lack accurate and repeatable control [12]. The recommended solution to this problem is highly dynamic load introduced through precise robotic manipulation [12]. Cassidy et al. [16] demonstrated high-speed loading but only did so with sagittal DOFs and grouped muscles.
To address the current limitations of existing in vitro simulators with the aim of providing improved physiological experimental data, we sought to develop a versatile in vitro multi-joint simulator capable of replicating highly dynamic activities such as running, jump landing, and cutting. As a critical step toward this goal, this paper introduces The University of Texas at El Paso Joint Load Simulator (UTJLS) and includes an outline of the design, a demonstration of its current configuration, and an assessment of its accuracy and repeatability during real-time squat simulations. From this thorough assessment of simulator performance, opportunities are identified for continued advancements toward simulation of highly dynamic physiological loads, variable joints, and kinetic chain testing.
## Materials and Methods
A custom, multiplane simulator was developed with the range of motion (ROM) and component specifications to support variable joint and kinetic chain testing at high speeds. The associated controller functions on a real-time operating system and is reconfigurable to support a wide ROM and a variety of human joints. The performance evaluation provided in this study determined the standard deviations of controller accuracy across four specimens during four squat loading conditions, trial repeatability across four specimens, and specimen repeatability for repeated loading of the same maneuver on the same specimen.
### Component Design.
The University of Texas at El Paso Joint Load Simulator requirements were determined through analysis of lower extremity kinematics and kinetics during highly dynamic maneuvers including cutting, braking, and drop jumping. These maneuvers are of clinical relevance as they commonly lead to injury and account for some of the highest accelerations and forces occurring in the human body. Subject kinematic and kinetic data for this analysis were collected during preliminary tests using in-house motion capture and verified through literature [2729].
Two 6DOF manipulators were designed into the UTJLS, but the three translations of the lower gantry system were not implemented for this particular study (Fig. 1(a)). For this knee study and for validation with literature, the UTJLS manipulators were configured as surrogate hip and ankle joints, accommodating a full-length lower limb. Components of the simulator include various servomotor types (Table 1), tension load cells (Table 2), multi-axis load cells (Table 3), encoders (Table 4), control computers, data-acquisition boards, supplementary electronics, and a custom specimen-mounting fixture.
#### Linear Actuators.
The current configuration applies relative linear motion through the hip gantry system. The required ROMs were determined through compiling the maximum displacements of a subject's hip during the foot-to-floor contact phases of the previously mentioned maneuvers. Each of the three translation axes utilizes two linear servomotors (RIPPED series, Parker Hannifin Corp., Cleveland, OH) guided by linear bearings (Table 1). These motors have a high power density, an electrical time constant of 3 ms, and no backlash. This selection provides the capacity and loading rate that is necessary to follow real-time position and force profiles.
For the purpose of safety, each axis is equipped with emergency stops (SNALD, Enertrols, Farmington Hills, MI) that can be positioned to limit system ROM and is equipped pneumatic brakes (RB15, Nexen Group, Inc., Vadnais Heights, MN) operated by three-way solenoid valves (S8, Pneumadyne, Inc., Plymouth, MN). Both the brakes and solenoid valves are spring loaded and engage in less than 0.12 s.
#### Rotational Actuators.
Each manipulator contains three rotational axes mounted in series with one another. The upper rotations are mounted on a gantry system, while the lower rotations are mounted on a steel stand centered in the upper manipulators' horizontal ROM (Fig. 1(a)). Each rotational axis is driven by a servomotor (MPP series, Parker Hannifan Corp., Cleveland, OH) driving a gearbox (AB series, Apex Dynamics, Ronkonkoma, NY) (Table 1). On each manipulator, the outermost rotation is oriented along the simulator's X-axis with the innermost rotation aligned with the respective bone's mechanical axis (Fig. 1).
#### Musculotendon Actuators.
Accurate application of joint moments from musculotendon forces requires both accurate force magnitude and moment arms, which are dependent upon multiple factors including insertion sites and muscle path [31]. To retain anatomical insertion sites, musculotendon actuators are connected directly to the cadaver tendon. In the case of a knee joint, while muscle insertions on the femur, tibia, and fibula are specimen-specific, surrogate muscle insertions on the pelvis and calcaneus are simulator-specific, i.e., determined by musculotendon actuator placement and steel cable guides. To represent accurate anatomical muscle insertions, the mounting location of each actuator was derived from previously published three-dimensional models (Fig. 2) [32].
The UTJLS has eight musculotendon actuators configured to replicate the following muscles: vastus medialis (VM), vastus lateralis (VL), rectus femoris (RF), semitendinosus (ST), semimembranosus (SM), biceps femoris (BF), and medial and lateral gastrocnemius (GM and GL). These actuators consist of servomotors (BE series, Parker Hannifan Corp., Cleveland, OH) and gearboxes (AB042, Apex Dynamics, Ronkonkoma, NY) and are connected to pulleys with steel cables providing a linear ROM of 300 mm (Table 1).
#### Sensors.
Various sensors have been implemented in the design of the UTJLS to measure forces and positions (Table 4). Musculotendon forces are measured with tension load cells (XFTC321, Measurement Specialties, Hampton, VA), which are connected at the point of each tendon's attachment (Table 2) (Fig. 2(a)). This sensor placement prevents measurement error due to friction in cable guides, and the 10 mm diameter of the tension load cells allows for the musculotendon line of action to route adjacent to bones without collision.
The rotations of the lower manipulator are mounted on spherical bearings connected to four three-axis load cells (3A120, Interface, Inc., Scottsdale, AZ) (range: 0–1000 N, accuracy: 0.3% full scale, and repeatability: 0.1% rated output) (Table 3) (Fig. 1(a)). Each three-axis load cell is mounted equidistant from the ankle's center of rotation, which provides moment arms used for moment measurement. Through the use of this custom design, the range of vertical force and x-moment measurement is configured for simultaneous, highly dynamic loading.
A six-axis load cell (Omega160, ATI Industrial Automation, Apex, NC) is attached to the hip manipulator to measure hip forces and moments (Table 3). It is located at the center of rotation of the simulated hip with the superior musculotendon actuator system located on the measurement side of the load cell, which prevents musculotendon forces from being included in the measurement (Table 4) (Fig. 1(a)).
All rotations and linear axes are equipped with absolute encoders. Linear axes, Hx and Hz, are equipped with the same model absolute encoders (EMAX 2, ELGO Electronic, Inc., Chicago, IL) (maximum speed: 4 m/s, resolution: 0.01 mm, and accuracy: 0.186 mm), and the Hy linear axis is equipped with another model absolute encoder (LMA10, Renishaw, Inc., Hoffman Estates, IL) (maximum speed: 14 m/s, resolution: 7.8 μm, and accuracy: 0.06 mm). All six MPP servomotors are equipped with an internal absolute encoder (EQI 1331, Heidenhain Corp., Schaumburg, IL).
#### Controller and Data Acquisition.
Two computers, a host computer and a real-time control computer, have been included to support the custom UTJLS software developed in LabVIEWTM version 2014 SP1 (National Instruments, Austin, TX). The host computer provides a user interface, whereas the real-time control computer operates using a LabVIEWTM real-time operating system. Control computation and data collection are performed by the real-time computer with support from a data-acquisition chassis (NI PXI-1036, National Instruments, Austin, TX) including a field-programmable gate array (NI PXI-7811 R, National Instruments, Austin, TX). This configuration provides the computing power to operate the UTJLS in real-time with a controller loop speed of 2000 Hz.
#### Electronic Hardware.
All actuators run off drivers configured for current control. Rotational motors are driven by analog brushless drives providing trapezoidal commutation through Hall state feedback (BE series, Advanced Motion Controls, Camarillo, CA), and linear motors are driven by digital drives providing sinusoidal commutation through combined encoder and Hall state feedback (DP Series, Advanced Motion Controls, Camarillo, CA). All drives receive commands from the UTJLS controller through analog signals.
Custom low voltage hardware has been included on printed circuit boards to assist in data acquisition and interfacing between equipment. Functions served by this equipment include analog signal amplification, low-pass filtering, differential to single-ended signal conversion, and digital signal amplification (Fig. 3).
#### Specimen Mounting.
A custom mounting device was developed to support correct alignment of specimens and to minimize the shape, weight, and inertia of cadaver mounting components. For both the femur and tibia, complex casting molds were developed that accommodate a large range of bone sizes without intersecting muscle paths (Fig. 2(b)). The mounting fixture has four line lasers that project two intersecting planes as guides for bone cutting and casting. During alignment, each bone was adjusted until the line lasers intersected its proximal and distal joint centers of rotation, which ensured that the mechanical axis of each bone was aligned to its respective manipulator. The effective length of each bone was matched to that of the in vivo subject by removing the proximal end of the femur and the distal end of the tibia. While maintaining proper alignment, the bone was then lowered into the mold, and polyurethane (ProtoCast 80 R, Industrial Polymers Corp., Houston, TX) was poured into the mold to secure the bone to the metal base for simulator interfacing.
### Controller Development
#### Controller Architecture and Functions.
During each loop iteration at a rate of 2000 Hz, the controller collects and stores sensor data, loads trajectory data, calculates feedback command outputs, and checks safety limits for all 29 sensors and 17 actuators. In addition, the UTJLS also provides a variety of functions to assist in cadaver testing including data review, manual subsystem control, specimen alignment, trajectory planning, and error reporting.
#### Control Strategy.
To support the accurate application of force and positions, the simulator employs a variety of control strategies that may be applied synchronously. These control strategies include feedforward force, feedforward position, force feedback, position feedback, and compensation for gravity and friction. Feedforward command voltages are determined by multiplying predicted forces by a loading constant and predicted accelerations by a mass constant, where acceleration is calculated from predicted positions using derivative estimation. Gravity compensation is configured for three axes including $θxH$, $θxA$, and Hz. The compensation for Hz is a constant voltage offset, while gravity compensation for $θxH$ and $θxA$ monitors the present rotation and adds a voltage command based on a sinusoidal fit. The friction compensators use position feedback and derivative estimation to determine the current velocity. If velocity magnitude exceeds a specified range, the controller applies a command voltage in the same direction of the measured velocity. Serving as the principal control strategy for the UTJLS, proportional–integral–derivative controllers are employed for force and position feedback, which are operated synchronously to support hybrid control.
#### Simulation Phases.
Each simulation contains multiple phases of control. The seeking phase initiates the procedure by applying tension to all tendons and ramping the specimen to the initial position of the maneuver, where it is locked in place. The simulator applies the following control sequence: ramp force controlled axes to their initial loads, ramp position controlled axes to their initial conditions, perform the maneuver, and undergo a stabilization phase when the maneuver is completed or an error has occurred. The UTJLS controller also supplies an analog signal coinciding with maneuver initiation and completion to synchronize simulator data with cadaver instrumentation.
#### User Interface.
The UTJLS user interface provides numerous features, which are separated into five categories including maneuver selection, parameter configuration, simulation, data analysis, and subcomponent operation. When a maneuver is imported or generated, it is saved on the host computer as an option presented in list form. Selection of one of these maneuvers in the interface will send the associated dataset from the host to the real-time control computer, which then may be reviewed graphically alongside feedforward commands. Parameters may be modified manually or by parameter file upload, and since every simulation outputs a copy of the current parameter file, the UTJLS can always be reverted to a previous configuration to identically recreate a test. If an error occurs during testing, the controller will immediately stabilize the simulator and display a description of the error. When a simulation is completed or an error is encountered, the program then stores the maneuver trajectories, sensor measurements, and parameter configuration to a file on the host computer.
#### Control Scheme.
The UTJLS controller's flexible architecture allows for various control schemes to be implemented. Excluding musculotendon actuators which are only configured for force control, each axis may be selected to operate with position feedback, force feedback, or hybrid feedback. Table 4 identifies the available feedback sources and the control scheme selected for squat testing.
### Squat Simulation
#### Simulator Input Data.
An athletic male (age: 20 yr, height: 1.68 m, and weight: 498 N) participated in a motion capture session where he performed five trials of two different squat techniques. The test was approved by the internal review board with informed consent. The subject was instrumented with a 37-marker set of the lower body (Northern Digital, Waterloo, ON, Canada) and electromyography electrodes on the eight muscles represented on the simulator (Delsys, Boston, MA) with one force platform (Bertec, Columbus, OH) per foot positioned shoulder width apart. The subject was instructed to squat descending for 5 s until the tops of his thighs were parallel to the ground and take 5 s returning to stance. Squats were performed with center of force directly on his heels (heel squat) and on his toes (toe squat) at the apex of the squat. A model was developed in Visaul3D (C-Motion, Inc., Germantown, MD) and allowed for three rotations and three translations at the hip, knee, and ankle joints. Lower extremity rotations, translations, moments, and ground reaction forces (GRFs) were calculated, exported, and used as inputs for the UTJLS in vitro simulations. Individual muscle forces were calculated through the calibrated EMG-informed neuromusculoskeletal modelling Toolbox, a hybrid electromyography-driven model, and input into the simulator [33].
Four lower extremity male specimens (age: 21–55 yr, side: right) were labeled M, K, U, and B. Specimen M, U, and B were partially fixed, whereas specimen K was fully fixed. While skin, adipose tissue, and fascia were removed up to the knee, the joint capsule, tendons, ligaments, and soft tissue inside of the knee were left intact. Tendons were cleaned of muscle tissue and transected 75 mm from their attachment point. Anterior incisions were made along the medial and lateral aspects of the patella to support instrumentation including a differential variable reluctance transducer (LORD Sensing, Williston, VT) and a custom pressure sensor (Teksan, Inc., Boston, MA). Posterior incisions were made superior to the medial and lateral menisci allowing insertions of the custom pressure sensor and placed anteriorly through the joint capsule superior to the meniscus. The sensor was surgically attached both anteriorly and posteriorly. Differential variable reluctance transducer was sutured onto the anteromedial bundle of the ACL. The patella incisions were sutured closed with sensors inside to mimic an intact knee capsule. Specimens were placed in the UTJLS and tendons were connected through freezer clamps (Fig. 2(a)).
Tendons were wrapped in gauze, covered in electrolyte gel, and inserted into extension hulls which were attached to tension load cells and musculotendon actuators. Tendons were frozen in liquid nitrogen to prevent damage and increase tensile strength [34]. Motion capture bone pins (Northern Digital, ON, Canada) were screwed into the femur, patella, and tibia (Fig. 2(a)).
#### Experimental Protocol.
To account for specimen geometry and fixation variability during casting, an alignment protocol was used to manually determine specimen-specific offsets for $θyH$ and $θzA$. During this protocol, the simulator applied a vertical 100 N force at 30 deg and 90 deg flexion with all other GRFs held at zero. Offsets were then iteratively adjusted until the tibial plateau forces were distributed across both the medial and lateral pressure sensor pads to replicate a 1.5:1 force distribution, respectively [35,36].
Prior to calibration testing and squat simulation, an axial preload of 500 N was applied to the specimen for pressure sensor conditioning. Pressure sensor calibration tests applied a vertical 250 N force at 60 deg flexion with all other GRFs held at zero, and the same test was applied at 30 deg flexion to measure offset of internal/external rotation (IE). Each of the four specimens underwent three heel squat tests with musculotendon force, three heel squat tests without musculotendon force, three toe squat tests with musculotendon force, and three toe squat tests without musculotendon force. Only two toe squat tests without musculotendon forces were collected from specimen K due to a labeling error. In total, data from 47 tests were collected to determine the accuracy and repeatability of the UTJLS. The magnitude of the musculotendon force profiles was scaled by 50% of in vivo estimates to prevent tendon failure during testing. Tibiofemoral contact, relative ACL strain, and relative tibiofemoral kinematics were collected for a future study identifying differences between heel and toe squat maneuvers. The VM tendon of specimen M was ruptured during testing and replaced with a threaded insert.
#### Data Conditioning.
For repeatability and accuracy analysis, the signal output from the UTJLS force sensors was low-pass filtered at 58 Hz to eliminate interference from nearby AC power, and all signals including position sensors were resampled and interpolated from 2000 Hz to provide an effective 100 Hz sampling.
#### Calculation of Anatomical Rotation.
The anatomical rotations presented in this study were determined through a vector analysis using the floating axis method described in Ref. [37]; however, the axes were not determined by bony landmarks. Rather, the anatomical axes were identified from simulator sensors assuming that the calibration protocol aligned the FE axis to the simulator's X-axis and achieved zero IE.
#### Statistical Methods.
This study assessed the short-term accuracy and precision errors of the UTJLS during the four squat maneuvers described in the experimental protocol. Standard deviations were determined for the UTJLS position and force sensors identified in Table 4 along with the three linear forces measured by the Omega160 load cell (i.e., $FxH$, $FyH$, and $FzH$). The following equations were used to calculate the standard deviation and upper limit of confidence intervals for accuracy, trial repeatability, and specimen repeatability [38]:
$SD=∑k=1m∑j=1nj∑i=1oi(xijk−y)2df(oi)$
(1)
$df=∑k=1m(nj−1)$
(2)
$σ2
(3)
where SD is standard deviation, xijk is the sensor measurement collected on specimen k during maneuver j for data point i, y is the reference value for the corresponding data point, df is the degrees-of-freedom of the test, oi is the total number of data points collected during trial j, nj is the total number of trials for specimen k, m is the total number of specimen, σ is true error, and χ2 is the chi-square distribution with probability level α/2 with df degrees-of-freedom. The reference value, y, is equal to the target trajectory for control accuracy SD, the mean average of all specimen during the same maneuver type for trial repeatability SD, and the mean average of specimen k during the same maneuver type for specimen repeatability SD. Offsets for $θyH$, $θzA$, and $θzA$ were subtracted from their respective measurements and during calculation of reference trajectories.
## Results
### Position Control Response.
The four rotation motors operating with position control feedback demonstrated excellent accuracy and trial repeatability with SDs less than 0.13 deg and 0.12 deg (Table 5). Alignment offsets for $θyH$ were 3.5 deg, 5 deg, 0.5 deg, and 5 deg, and those for $θzA$ were 2 deg, −2 deg, 2.5 deg, and −12 deg for specimens M, K, U, and B, respectively. The average result for each specimen during heel squat testing including 50% musculotendon forces is shown in Fig. 4(a).
### Force Control Response.
The seven musculotendon actuators utilized force proportional–integral–derivative control without any form of position feedback. Accuracy and trial repeatability for all musculotendon forces were less than 8 N for every axis (Table 5). These results demonstrate an excellent force response as desired for in vitro testing with no need for further force control development. The average musculotendon forces for specimen B during heel squat testing with 50% musculotendon forces are shown in Fig. 5.
### Hybrid Control Response.
Axes utilizing hybrid feedback were tuned to optimize force response. The corresponding GRFs had a maximum accuracy SD of 29 N for forces and 4.4 N·m for moments, and the corresponding SD for specimen repeatability was 12 N and 1.8 N·m (Table 5). The average result for each specimen during heel squat testing including 50% musculotendon forces is shown in Fig. 4(a).
### Dependent Variable Repeatability.
The results for sensors with no targeted output are less predictable and repeatable than those that the UTJLS monitors and controls. The SD for each axis is shown in Table 6, and the average result for each specimen during heel squat testing with 50% musculotendon forces is shown in Fig. 4(b). Alignment offsets for $θzA$ were −0.1 deg, 14.7 deg, 5.6 deg, and 7.7 deg for specimens M, K, U, and B, respectively. Included in this analysis are resultant anatomical rotations, which despite no explicit control maintained FE within an SD of 0.38 deg.
For most sensors, application of musculotendon forces minimally influenced accuracy or repeatability, but for some, the effect was substantial. For example, during heel squat testing, musculotendon forces improved the specimen repeatability RMS of $MxA$ and $FyA$ by 2.6 N·m and 6.7 N. The same change in loading increased IE specimen repeatability RMS by 0.63 deg but reduced RMS accuracy error by 6.2 deg. Anatomical knee rotations of one specimen are shown in Fig. 6.
A similar effect was measured by Tekscan instrumentation, where the introduction of musculotendon forces reduced the range of the COP travel on both sensor pads without increasing specimen repeatability. Specimen U's COP during heel squat testing is shown in Fig. 7.
## Discussion
During our preliminary work, individual actuator tests demonstrated exceptional accuracy and repeatability, but such results are not indicative of performance during physiological loading where interactions among multiple actuators increase the error. An example of such an interaction can be seen in Fig. 5, where sudden changes in flexion caused localized peaks in RMS accuracy error for musculotendon forces. Hence, system performance was assessed during simulations of four physiological maneuvers including synchronous control of up to 16 actuators.
### Control Scheme.
The control scheme of this study effectively performed knee F/E with inverse dynamics while all other knee DOFs operated in forward dynamics. This approach provided accurate and repeatable loads while also accommodating geometric variation among cadaver specimens. The desired loading was achieved in real-time despite changes in maneuver, application of musculotendon forces, and specimen.
The results also present some outcomes that are unique to this method. For example, $θyA$ was driven by force rather than position allowing for specimen-specific paths, and as a result, the moment arms between knee center and GRFs also varied throughout maneuvers and generated specimen-specific varus/valgus (VV) knee moments. Other control schemes in literature do not demonstrate this outcome [14,1719,21]. These simulators maintain constant moment arms between knee center of rotation and GRFs by fixing the tibia to the GRF load cell. The UTJLS controller can be configured to effectively match this control scheme by controlling $MyA$ and $MzA$ using $θyH$ and $θzA$. This control scheme would allow for synchronous control of $θyA$, $θzA$, $MyA$, and $MzA$, and therefore consistent application of knee VV and IE moments by GRFs for all specimen.
### Controller Response.
To date, no study has assessed simulator repeatability and accuracy during a real-time, physiological maneuver that includes musculotendon forces and multiplanar kinematics and kinetics. Time-scaled gait maneuvers of Noble et al. [18] and Maletsky and Hillberry [24], which have similar loading conditions, may serve as a reference for comparison. During simulation of physiological loads, even axes assigned a near zero or constant target experience large disturbances due to direct influence from other axes and inertial effects generated by system accelerations. Despite these disturbances, the position controlled axes and musculotendon actuators performed all tasks with minimal error (Table 5). The controller accuracy SDs of $FxA$, $FyA$, and $FzA$ with a maximum of 29 N are consistent with reported RMS errors in Refs. [18] and [24], but $MyA$ and $MzA$ with a maximum SD of 4.4 N·m were less accurate than RMS errors reported in Ref. [18] (Table 5).
Numerous factors contribute to the accuracy error of $MyA$ and $MzA$ including accuracy of reference trajectories (Fig. 4(b)), high-load capacity (Table 1) operating in a low range of operation (Fig. 4(a)), disturbances developed from other axes, disturbances from friction, limited resolution from feedback sources (Table 3), and backlash in the associated gearboxes (Table 1). These axes demonstrated superior performance during individual actuator testing and with reference trajectories tuned for a specific maneuver. The predicted trajectories for $θyA$ and $θzA$ were referenced by the hybrid controller and feedforward components of the controller, but the references did not reflect the actual paths observed during testing, which were specimen-specific and unpredictable prior to testing. Accordingly, reference trajectories tuned for each specimen and maneuver are expected to improve $MyA$ and $MzA$ performance. In future studies, improved reference trajectories may be obtained through trajectory optimization algorithms as demonstrated elsewhere [14,18,24], which have been able to improve force response by more than 85% for linear GRFs and 50% for GRF moments [14].
The UTJLS demonstrated consistent loading (Table 5), and the specimen response to that loading was repeatable (Table 6). Anatomical rotation repeatability was all less than 1.25 deg, which is reported to be sufficient for investigating passive path kinematics, a simple maneuver when compared to physiological loading [14]. While the current configuration is suitable for knee testing, improvement may be possible through trajectory optimization and addition of encoder feedback to support hybrid control of musculotendon actuators (Table 4).
### Limitations.
If the same parameters and control scheme are used during real-time UTJLS simulation of a faster maneuver, the accuracy and repeatability SDs are expected to increase beyond what was reported here. Additionally, some limitation to these results may exist due to the material properties of the partially and fully fixed specimens used in this study. For maneuvers with similar characteristics and UTJLS settings, the repeatability and accuracy of the UTJLS as presented in Tables 5 and 6 have confidence intervals that were determined with 95% confidence and have an upper limit of +42% for musculotendon actuators and +28% for all other axes and sensors.
### Future Work.
The flexible architecture of the UTJLS and its controller support testing a variety of human joints, highly dynamic maneuvers, and kinetic chains of up to two connected joints (e.g., hip and knee or knee and ankle). The primary systems of the UTJLS provide the DOFs, ROM, speed, precision, and accuracy necessary to accomplish these tasks; however, some tasks will require system adjustment that may include end effector customization, controller refinement, mounting fixture modification, and addition of the lower gantry system shown in Fig. 1(a). Recommended modifications to improve system performance include trajectory optimization and a reconfiguration of the custom GRF load cells for improved $MyA$ and $MzA$ resolution [14,18].
## Conclusion
The UTJLS can simulate physiological loads via in vitro musculoskeletal testing including real-time, synchronous application of musculotendon forces and GRFs during multiplanar kinematics. Being the first design of its kind, our simulator utilizes two separate robotic manipulators that contain a total of eight musculotendon actuators and two six-axis load cells. By adjusting the control scheme configuration, the UTJLS can match the constraints of traditional testing rigs (e.g., Oxford rig or robotic arm), recreate absolute motion to reproduce gravitational and inertial loads, and uniquely investigate joint moment contributions from GRFs as demonstrated in this study. With the necessary DOFs, ROM, and speed, the UTJLS is suitable for future testing of faster maneuvers, a variety of human joints, and a kinetic chain of two connected joints (e.g., hip and knee).
## Acknowledgment
The authors would like to thank Paul Power for his contributions to the design and development of the software and hardware for the user interface and reconfigurable controller of the UTJLS. Numerous students including Herman Cordero, Victor Contreras, Xavier Villarreal, and Javier Ornelas provided invaluable support that made this project possible.
## Funding Data
• National Science Foundation (Grant No. BES 0966398).
• The University of Texas STARS Funding.
• ADVANCED Motion Controls University Outreach Program.
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# Extending absolute values on local fields - what is the 'correct' normalization and the relation to the global theory?
I'm having a tough time figuring out the 'correct' normalization for extending absolute values of local fields. I'm also trying to piece together how this interacts with the global theory, so below is essentially a discussion of my thoughts and a few questions at the end. Am I thinking about this the right way?
If $$K/\mathbb Q$$ is finite then every nonzero ideal $$\mathfrak a\subseteq \mathcal O_K$$ has a unique factorization $$\mathfrak a=\prod_{\mathfrak p: \,\text{prime}}\mathfrak p^{e_\mathfrak p(\mathfrak a)}.$$ The function $$v_\mathfrak p:\mathcal O_K\rightarrow \mathbb Z\cup \infty$$ defined by $$v_\mathfrak p(x)={e_\mathfrak p(x\mathcal O_K)}$$ defines a discrete valuation on $$\mathcal O_K$$. We can extend this to a valuation on $$K$$ by noting that all elements of $$K$$ can be represented by a quotient of algebraic integers. Fixing a prime $$p$$ and a prime ideal $$\mathfrak p\mid p$$, for any $$0 we have an induced nonarchimedean absolute value on $$K$$, given by $$|x|_\mathfrak p=a^{v_\mathfrak p(x)}$$. Completing $$K$$ with respect to this absolute value results in a finite extension $$K_\mathfrak p$$ of $$\mathbb Q_p$$.
If $$e$$ and $$f$$ are the ramification and inertia degrees of $$p$$, by Neukirch Theorem 4.8, I also understand that $$K_\mathfrak p$$ has a unique absolute value extending the $$p$$-adic absolute value $$|\cdot |_p$$ on $$\mathbb Q_p$$ , given by $$|x|_\mathfrak p=|N_{K_\mathfrak p/\mathbb Q_p}(x)|_p^{\frac{1}{ef}}$$ since $$[K_\mathfrak p:\mathbb Q_p]=ef$$. This forces us to fix a value for $$a$$ above: since we want the absolute value on $$K_\mathfrak p$$ to extend $$|\cdot |_p$$, we'd like to have $$|p |_\mathfrak p=p^{-1}$$. Therefore, we need $$a^{v_\mathfrak p(p)}=|N_{K_\mathfrak p/\mathbb Q_p}(p)|^{\frac{1}{ef}},$$ which, since $$N_{K_\mathfrak p/\mathbb Q_p}(p)=p^{ef}$$ (do I need to assume the extension is Galois here?) and $$v_\mathfrak p(p)=e$$, implies that $$a=p^{-1/e}$$. Therefore, we normalize so that $$|x|_\mathfrak p=p^{-v_\mathfrak p(x)/e}$$ is the absolute value extending that of $$\mathbb Q_p$$. All this said, Serre mentions in page 27 of Local Fields, that for a locally compact field (which is the case for $$K_\mathfrak p$$), there is a "canonical way to choose the number $$a$$ [defined the same as I have done above]: one takes $$a=q^{-1}$$, where $$q$$ is the number of elements in the reside field". In our case, I think that $$q=|\mathcal O_{K_\mathfrak p}/\mathfrak p|=p^f$$? If so, then it seems like Serre would have us take $$a=p^{-f}$$... What is the advantage of doing this? Am I making some horrible mistake here? Is there a better way to think about (normalized) valuations on extensions? Is there a way to 'get around' computing the field norm whenever we'd like to take the absolute value of an an element?
I realize I've bunched a few questions together here, and I'd be happy to write them separately if someone with more experience on this site would recommend it. That said, partial answers or even just comments would be welcome.
• I think that the “canonical way” is what makes the Product Formula work for a global field $K$. But it does not extend the canonical $p$-abs. value of $\Bbb Q$. – Lubin Feb 15 at 5:49
• The product formula for number fields says that $\prod_l |\sigma_l(\alpha)|_\infty = |N_{K/\mathbb{Q}}(\alpha)|_\infty = N(\alpha O_K)= \prod_{\mathfrak{p} | (\alpha)} N(\mathfrak{p})^{v_\mathfrak{p}(\alpha)}$ so you'd take $|\alpha|_\mathfrak{p} = N(\mathfrak{p})^{-v_\mathfrak{p}(\alpha)} = p^{-f_\mathfrak{p}v_\mathfrak{p}(\alpha)}$ and make the complex absolute values appear twice to obtain $\prod_v |\alpha|_v = 1$, no ramification index here – reuns Feb 16 at 19:30
## 1 Answer
You take a "local field" as a finite extension of a $$\mathbf Q_p$$, but my discussion below will concern any field which is locally compact w.r.t. a non archimedean absolute value , i.e. which is complete w.r.t. a discrete valuation and has finite residue field.
1) As far as $$K$$ with an absolute value is considered as solely a normed space, only the topology of $$K$$ matters. Recall that two absolute values $${\mid . \mid}_1$$ and $${\mid . \mid}_2$$ are equivalent iff they induce homeomorphic topologies on $$K$$, iff there exists a strictly positive real $$c$$ s.t. $${\mid . \mid}_1={{\mid . \mid}_2}^c$$. An equivalence class of absolute values is called a place. The "normalization" suggested by Serre, as you recalled it, for a field $$K$$ with a discrete valuation $${\mid\mid . \mid\mid}$$ and finite residue field of cardinal $$q$$, consists in putting $${\mid\mid \pi \mid\mid}=q^{-1}$$ for a uniformizer $$\pi$$. It is "canonical" in the following sense. If $$K$$ is locally compact, introduce the Haar measure $$\mu(.)$$ on $$K^+$$ which is uniquely determined by the condition $$\mu(O_K)=1$$. Then $${\mid\mid . \mid\mid}$$ is the unique absolute value which verifies $$\mid\mid y\mid\mid = \mu(x+yO_K)$$ (because of the invariance of $$\mu$$, the RHS is independent of $$x$$).
2) If we consider a finite extension $$L/K$$ of degree $$n$$, a given absolute value $${\mid . \mid}_K$$ admits exactly one extension to an absolute value $${\mid . \mid}_L$$, which is defined by $${\mid x \mid}_L= {{\mid N(x) \mid}_K}^{1/n}$$, where $$N$$ is the norm of $$L/K$$. Here a discrepancy appears with the previous normalized absolute values, in that $${\mid\mid x\mid\mid}_L= {\mid\mid N(x)\mid\mid}_K$$. In some sense, this new relation is more natural w.r.t. to a given $$x$$, because its formulation behaves coherently with a change of extension $$L/K$$.
3) Let us now consider the global-local relations between a global field $$F$$ and its completed local fields $$F_v$$ for all places $$v$$ (including the archimedean ones). For a given $$F$$, a fundamental result is the so called product formula : $$\prod_v {\mid\mid x\mid\mid}_v=1$$ for $$x\in K^*$$, in which the normalized archimedean absolute values are by definition the usual absolute value in $$\mathbf R$$ or the square of the usual modulus in $$\mathbf C$$. Local CFT is expressed in terms of the multiplicative groups of a local field $$K$$ and its extensions. Modern CFT for a global field $$F$$ is expressed in terms of the idèle group $$J_F$$ and the idèle class group $$C_F=J_F/F^*$$ (I don't recall the definitions). The topology on $$J_F$$ is not the product toplogy, but the so called "restricted topological product" of the $${F_v}^*$$ w.r.t. the units $$U_v$$ inside. The subgroup $$F^*$$ is discrete. The (huge) connected component of $$1$$ in $$C_F$$ plays an annoying important role ./.
• So in $\hat{O_K}$ then $\prod_{v \ fin} |x|_v = \mu(a+x\hat{O_K})$ is the Haar measure. But when looking at the whole adeles it is less obvious why we'd want $\prod_v |x|_v = 1,x \in K^*$ and why the complex absolute values appear twice, beside CFT and characters of $\Bbb{A_K}^\times/K^\times$ – reuns Feb 16 at 20:24
• No, it's the product over all places of the global field, and it's $1$. I didn't say that it's equal to the Haar mesure, which is a local thing. What is $\hat O_K$ ? – nguyen quang do Feb 16 at 21:57
• $K$ a number field $\hat{O_K} = \varprojlim O_K/I = \prod_\mathfrak{p} \varprojlim O_K/\mathfrak{p}^n$ the ring of integers of the finite adeles. The complex absolute values appear twice in $\prod_v |x|_v=1$ because they appear twice in the LHS of $\prod_j |\sigma_j(\alpha)| = |N_{K/\mathbb{Q}}(\alpha)| = N(\alpha O_K)= \prod_\mathfrak{p} N(\mathfrak{p})^{v_\mathfrak{p}(\alpha)} = \prod_{v \ fin} |\alpha|_v^{-1}$ – reuns Feb 16 at 22:14
• Ah, OK. I added an explanation about the formula involving the Haar measure. – nguyen quang do Feb 16 at 22:33
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# Data Statistics and Analysis With Java and Python
Java and Python are two of the most popular computer languages in use today. Both are very mature and provide the tools and technology ecosystems to support developing solutions to the challenging problems that arise in the world of data science. Each has its idiosyncrasies. It’s important to understand how they compare tackling different problems, whether they shine or lack the required flexibility to handle the assigned tasks. When one is preferable over the other or when they work in tandem complementing each other.
Python is a dynamically typed language, very straightforward to work with, and is certainly the language of choice to do complex computations if we don’t have to worry about intricate program flows. It provides excellent libraries (Pandas, NumPy, Matplotlib, ScyPy, PyTorch, TensorFlow, etc.) to support logical, mathematical, and scientific operations on data structures or arrays.
Java is a very robust language, strongly typed, and therefore has more stringent syntactic rules that make it less prone to programmatic errors. Like Python provides plenty of libraries to work with data structures, linear algebra, machine learning, and data processing (ND4J, Mahout, Spark, Deeplearning4J, etc.).
In this article, we’re going to focus on a narrow study of how to do simple data analysis of large amounts of tabular data and compute some statistics using Java and Python. We’ll see different techniques on how to do the data analysis on each platform, compare how they scale, and the possibilities to apply parallel computing to improve their performance.
## Problem Layout
We’re going to do a straightforward analysis of a set of prices for a large list of cities in different states. For simplicity, we assume that there is a CSV file that contains this information. We read the file and proceed to filter out some states and group the remaining by city-state pairs to do some basic statistics. We’re interested in finding solutions that can perform efficiently and scale well as the size of the input data grows.
A sample of the data is:
city state basePrice actualPrice La Jose PA 34.17 33.19 Preachers Slough WA 27,46 90.17 Doonan Corners NY 92.0 162.46 Doonan Corners NY 97.45 159.46 Castle Rock WA 162.16 943.21 Marble Rock IA 97.13 391.49 Mineral CA 99.13 289.37 Blountville IN 92.50 557.66 Blountsville IN 122.50 557.66 Coe IN 187.85 943.98 Cecilia KY 92.85 273.61
The purpose is to show how we would approach solving these types of problems using Java and Python. As we can see, the example is very simple and limited in scope, but it will be easy to generalize to more challenging problems.
## Java’s Approach
We start defining a Java record that encapsulates the data entries:
record InputEntry(String city, String state, double basePrice, double actualPrice)
The record is a new kind of type declaration introduced in JDK 14. It’s a concise way to define an immutable class that provides constructors, accessors, equals, and hash implementations.
Next, we read the CVS file and accumulate them in a list:
List<InputEntry> inputEntries = readRecordEntriesFromCSVFile(recordEntries.csv);
To do the grouping of the input entries by city and state we define:
record CityState(String city, String state) ;
We encapsulate the stats for all the entries that belong to a group with the following class:
record StatsAggregation(StatsAccumulator basePrice, StatsAccumulator actualPrice)
StatsAccumulator is part of the Guava library. You can add sets of double values to the class, and it calculates basic statistics like count, mean, variance, or standard deviation. We use the StatsAccumulator to get the statistics for the basePrice and actualPrice of the InputEntry.
Now we have all the ingredients to solve our problem. Java Streams provide a robust framework to implement data manipulation and analysis. Its declarative programming style, support for selection, filtering, grouping, and aggregations, simplify data manipulation and statistical analysis. Its framework also provides a robust implementation that can handle large volumes (even infinite streams) and operates very efficiently through the use of parallelism, laziness, and short-circuit operations. All these features make Java Streams an excellent choice to tackle these types of problems. The implementation is very simple:
Map<CityState, StatsAggregation> stats = inputEntries.stream().
filter(i -> !(i.state().equals("MN") || i.state().equals("CA"))).collect(
groupingBy(entry -> new CityState(entry.city(), entry.state()),
collectingAndThen(Collectors.toList(),
list -> StatsAccumulator sac = new StatsAccumulator();
StatsAccumulator sas = new StatsAccumulator();
return new StatsAggregation(sac, sas);
)));
In line 2 of the code, we use Stream::filter. It’s a Boolean-valued function to test the elements in the list. We implement a lambda expression to remove any entries that contain the states of “MN” or “CA.”
We then proceed to collect the list’s elements and invoke Collectors::groupingBy() (line 3), which takes two parameters:
• A classification function, where we use our CityState record to do the grouping by city and state (line 3).
• A collector for the downstream that contains the items that belong to the same city-state. We use Collectors::collectingAndThen(line 4), which takes two parameters to do the reduction in two steps:
• We use Collectors::toList (line 4), which returns a collector that accumulates all of the elements that belong to the same city-state into a list.
• We apply a finishing transformation to this list. We use a lambda function (lines 5 to 9) to define two StatsAccumulator(s) where we compute the statistics for basePrice and actualPrice entries from the previous list, respectively. Finally, we return a newly created StatsAggregation record that contains these entries.
To summarize, we return a Map<CityState, StatsAggregation> where the keys represent the grouped city-state pairs and their values is a StatsAggregation that contains the statistics for the basePrice and actualPrice for each key.
As we mentioned before, one of the key advantages of using Java Streams is that they provide a simple mechanism to do parallel processing using multithreading. This allows the simultaneous execution of multiple threads utilizing the multicore resources of the CPU. Just adding a “parallel” to the stream as shown:
Map<CityState, StatsAggregation> stats = inputEntries.stream().parallel().
causes the stream framework to subdivide the list of entries into parts and run them in separated threads simultaneously. As all the different threads finish their computation, the framework adds them serially to the resulting Map.
There is an additional optimization using Collectors::groupingByConcurrent in line 4 instead of Collectors:groupingBy. In this case, the framework uses a concurrent map that allows inserting elements from the different threads directly into this Map instead of having to be combined serially.
With these three possibilities, we can check how they perform doing the previous stats calculations (excluding the time to load the data from the CSV file) as the load doubles from five to twenty million entries:
Serial Parallel Parallel & GroupByConcurrent Five Million Entries 3.045 sec 1.941 sec 1.436 sec Ten Million Entries 6.405 sec 2.876 sec 2.785 sec Twenty Million Entries 8.507 sec 4.956 sec 4.537 sec
We see that running in Parallel improves the performance substantially; as the load increases, it almost halves the time. There is also an additional 10% gain using GroupByConcurrent.
Finally, to get the results is trivial; for example, to obtain the stats for Blountsville, IN, we just need to:
StatsAggregation aggreg = stateAggr.get(new CityState("Blountsville ", "IN"));
System.out.println("Blountsville, IN");
System.out.println("basePrice.mean: " + aggreg.basePrice().mean());
System.out.println("basePrice.populationVariance: " + aggreg.basePrice().populationVariance());
System.out.println("basePrice.populationStandardDeviation: " + aggreg.basePrice().populationStandardDeviation());
System.out.println("actualPrice.mean: " + aggreg.basePrice().mean());
System.out.println("actualPrice.populationVariance: " + aggreg.actualPrice().populationVariance());
System.out.println("actualPrice.populationStandardDeviation: " + aggreg.actualPrice().populationStandardDeviation());
The results that we obtain:
Blountsville : IN
basePrice.mean: 50.302588996763795
basePrice.sampleVariance: 830.7527439246837
basePrice.sampleStandardDeviation: 28.822781682632293
basePrice.count: 309
basePrice.min: 0.56
basePrice.max: 99.59
actualPrice.mean: 508.8927831715211
actualPrice.sampleVariance: 78883.35878833274
actualPrice.sampleStandardDeviation: 280.86181440048546
actualPrice.count: 309
actualPrice.min: 0.49
actualPrice.max: 999.33
## Python’s Approach
In Python, we have several libraries that can handle data statistics and analysis. However, we find that the Pandas library is very well suited to processing large amounts of tabular data and provides very efficient filtering, grouping, and statistical analysis methods.
Let’s review how we would analyze the previous data using Python:
import pandas as pd
def group_aggregations(df_group_by):
df_result = df_group_by.agg(
'basePrice': ['count', 'min', 'max', 'mean', 'std', 'var'],
'actualPrice': ['count', 'min', 'max', 'mean', 'std', 'var']
)
return df_result
if __name__ == '__main__':
excluded_states = ['MN', 'CA']
df_st = df.loc[~ df['state'].isin(excluded_states)]
group_by = df_st.groupby(['city', 'state'], sort=False)
aggregated_results = group_aggregations(group_by)
In the main section, we start by invoking pandas.read_csv() (line 11) to load the comma-separated values in the file into a Pandas DataFrame.
In line 13 we use ~df['state'].isin(excluded_states) to get a Pandas Series of Booleans that have False for the excluded states (MN and CA). Finally, we use pandas.loc() on this series to filter them out.
Next, we use DataFrame.groupby() in line 14 to group by city and state. The result is processed by group_aggregations() to get the statistics for each group of the basePrice and actualPrice.
We see that the implementation in Python is very straightforward. To print the results for Blountsville, IN:
print(aggregated_results.loc['Blountsville', 'IN']['basePrice'])
print(aggregated_results.loc['Blountsville', 'IN']['actualPrice'])
This gives us the stats:
base_price:
Name: (Blountsville, IN), dtype: float64
count 309.000000
min 0.560000
max 99.590000
mean 50.302589
std 28.822782
var 830.752744
actual_price:
Name: (Blountsville, IN), dtype: float64
count 309.000000
min 0.490000
max 999.330000
mean 508.892783
std 280.861814
var 78883.358788
To run the previous code in parallel, we have to keep in mind that Python doesn’t support a fine-grained locking mechanism as Java does. We have to contend with the global interpreter lock (GIL) that only allows one thread to execute at a time no matter how many CPU multicores or threads you have. We won’t get into the details.
To support concurrency, we have to consider that we have a CPU-intensive process, therefore, the best approach is to use multiprocessing. In this case, we have to modify our implementation:
from multiprocessing import Pool
import pandas as pd
def aggreg_basePrice(df_group):
ct_st, grp = df_group
return ct_st, grp.basePrice.agg(['count', 'min', 'max', 'mean', 'std', 'var'])
if __name__ == '__main__':
start = time.perf_counter()
excluded_states = ['MN', 'CA']
filtr = ~ df['state'].isin(excluded_states)
df_st = df.loc[filtr]
grouped_by_ct_st = df_st.groupby(['city', 'state'], sort=False)
with Pool() as p:
list_parallel = p.map(aggreg_basePrice, [(ct_st, grouped) for ct_st, grouped in grouped_by_ct_st])
print(f'Time elapsed parallel: round(finish - start, 2) sec')
As we did before, we use Pandas groupby() to get the data grouped by city and state (line 14). In the next line, we use the Pool() provided by the multiprocessing library to map the grouped data using aggreg_basePrice to compute the statistics for each group. The Pool() divides the data and proceeds to compute the stats in several parallel independent processes.
We’ll not review the previous code in detail since, as we’ll show in the table below that, multiprocessing is much slower than running the process serially. Therefore it’s not worth using this approach for these types of problems.
Another possibility to run the code concurrently is to use Modin. Modin provides a seamless way to parallelize your code and is extremely useful when you have to process large amounts of data. Changing the import statement from import pandas as pd to import modin.pandas as pd runs your code in parallel and takes advantage of the cluster of cores that might be available in your environment to speed up the code execution. For more details on how it works, please read the documentation.
As we did with Java, we provide the following table with the runtimes for the different scenarios that we just covered (as before, we exclude the time to read the data from the CSV file):
Serial Multi Process Modin Proc Five Million Entries 1.94 sec 20.25 sec 6.99 sec Ten Million Entries 4.07 sec 25.1 sec 12.88 sec Twenty Million Entries 7.62 sec 36.2 sec 25.94 sec
We see that running the code serially in Python is even slightly faster than in Java. However, using multiprocessing degrades substantially the performance. Using Moding improves the results but still is more advantageous to run the process serially.
It’s worth mentioning that, as we did before, we’re excluding the time to read the data from the CSV file from our time computations.
We see that with CPU-intensive processes in Pandas, there is no advantage in parallelizing the code. In a sense, this is a reflection of how Pandas was originally architected. However, it’s impressive how fast Pandas runs in Serial mode and also scales very well even for large amounts of data.
It’s important to point out that the speed of the calculations for the stats in Python depends on how they are performed. To get fast computations, one needs to be careful in applying the statistical functions that are needed. For example, doing a simple pandas.DataFrame.describe() to get the stats will run very slowly.
We have seen that Java’s Streams or Python’s Pandas are two excellent choices to do analysis and statistics of large amounts of data. Both have very solid frameworks with lots of support to achieve great performance and scaling.
Java provides a very strong infrastructure, ideal to work with complex program flows. It’s very performant, allowing to efficiently run the processes in parallel. This makes it an ideal choice when there is a premium on getting the results quickly.
Python is very well fitted to do math and statistics. It’s very straightforward, reasonably fast, and well-suited to doing complex calculations.
|
{}
|
[rkward] packages/rkwarddev: bookkeeping
m.eik michalke meik.michalke at uni-duesseldorf.de
Mon Nov 30 10:01:45 UTC 2015
Git commit 8664767f384e9994dafa868f9c022b9d51c5d6ff by m.eik michalke.
Committed on 30/11/2015 at 10:01.
Pushed by meikm into branch 'master'.
bookkeeping
- i really have to figure out why/how roxygen2 changes its line-wrap all the time
M +4 -5 packages/rkwarddev/DESCRIPTION
M +1 -1 packages/rkwarddev/R/rkwarddev-package.R
M +4 -2 packages/rkwarddev/man/R.comment.Rd
M +20 -10 packages/rkwarddev/man/XiMpLe-methods.Rd
M +8 -4 packages/rkwarddev/man/echo.Rd
M +8 -4 packages/rkwarddev/man/i18n.Rd
M +14 -7 packages/rkwarddev/man/id.Rd
M +8 -4 packages/rkwarddev/man/idq.Rd
M +4 -2 packages/rkwarddev/man/ite.Rd
M +4 -2 packages/rkwarddev/man/join.Rd
M +16 -8 packages/rkwarddev/man/js.Rd
M +2 -1 packages/rkwarddev/man/qp.Rd
M +2 -1 packages/rkwarddev/man/rk.JS.arr-class.Rd
M +6 -3 packages/rkwarddev/man/rk.JS.array.Rd
M +12 -6 packages/rkwarddev/man/rk.JS.doc.Rd
M +6 -3 packages/rkwarddev/man/rk.JS.opt-class.Rd
M +16 -8 packages/rkwarddev/man/rk.JS.optionset.Rd
M +2 -1 packages/rkwarddev/man/rk.JS.oset-class.Rd
M +10 -5 packages/rkwarddev/man/rk.JS.saveobj.Rd
M +12 -6 packages/rkwarddev/man/rk.JS.scan.Rd
M +8 -4 packages/rkwarddev/man/rk.JS.var-class.Rd
M +22 -11 packages/rkwarddev/man/rk.JS.vars.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.attribute.Rd
M +16 -8 packages/rkwarddev/man/rk.XML.browser.Rd
M +8 -4 packages/rkwarddev/man/rk.XML.cbox.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.component.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.components.Rd
M +12 -6 packages/rkwarddev/man/rk.XML.connect.Rd
M +6 -3 packages/rkwarddev/man/rk.XML.convert.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.copy.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.dependencies.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.dependency_check.Rd
M +10 -5 packages/rkwarddev/man/rk.XML.dialog.Rd
M +12 -6 packages/rkwarddev/man/rk.XML.dropdown.Rd
M +10 -5 packages/rkwarddev/man/rk.XML.embed.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.formula.Rd
M +6 -3 packages/rkwarddev/man/rk.XML.frame.Rd
M +10 -5 packages/rkwarddev/man/rk.XML.input.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.logic.Rd
M +20 -10 packages/rkwarddev/man/rk.XML.matrix.Rd
M +6 -3 packages/rkwarddev/man/rk.XML.option.Rd
M +18 -9 packages/rkwarddev/man/rk.XML.optioncolumn.Rd
M +18 -9 packages/rkwarddev/man/rk.XML.optionset.Rd
M +6 -3 packages/rkwarddev/man/rk.XML.page.Rd
M +16 -8 packages/rkwarddev/man/rk.XML.plugin.Rd
M +28 -14 packages/rkwarddev/man/rk.XML.pluginmap.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.preview.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.require.Rd
M +14 -7 packages/rkwarddev/man/rk.XML.saveobj.Rd
M +12 -6 packages/rkwarddev/man/rk.XML.select.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.set.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.snippets.Rd
M +18 -9 packages/rkwarddev/man/rk.XML.spinbox.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.switch.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.tabbook.Rd
M +18 -9 packages/rkwarddev/man/rk.XML.values.Rd
M +4 -2 packages/rkwarddev/man/rk.XML.valueselector.Rd
M +14 -7 packages/rkwarddev/man/rk.XML.valueslot.Rd
M +38 -19 packages/rkwarddev/man/rk.XML.vars.Rd
M +2 -1 packages/rkwarddev/man/rk.XML.varselector.Rd
M +24 -12 packages/rkwarddev/man/rk.XML.varslot.Rd
M +8 -4 packages/rkwarddev/man/rk.XML.wizard.Rd
M +6 -3 packages/rkwarddev/man/rk.build.plugin.Rd
M +6 -3 packages/rkwarddev/man/rk.get.rkh.prompter.Rd
M +4 -2 packages/rkwarddev/man/rk.local.Rd
M +28 -14 packages/rkwarddev/man/rk.paste.JS.Rd
M +12 -6 packages/rkwarddev/man/rk.paste.JS.graph.Rd
M +10 -5 packages/rkwarddev/man/rk.plotOptions.Rd
M +48 -24 packages/rkwarddev/man/rk.plugin.component.Rd
M +64 -32 packages/rkwarddev/man/rk.plugin.skeleton.Rd
M +2 -1 packages/rkwarddev/man/rk.rkh.doc.Rd
M +8 -4 packages/rkwarddev/man/rk.rkh.scan.Rd
M +2 -1 packages/rkwarddev/man/rk.rkh.section.Rd
M +2 -1 packages/rkwarddev/man/rk.rkh.setting.Rd
M +2 -1 packages/rkwarddev/man/rk.rkh.settings.Rd
M +2 -1 packages/rkwarddev/man/rk.set.comp.Rd
M +2 -1 packages/rkwarddev/man/rk.set.empty.e.Rd
M +2 -1 packages/rkwarddev/man/rk.set.indent.Rd
M +12 -6 packages/rkwarddev/man/rk.set.rkh.prompter.Rd
M +4 -2 packages/rkwarddev/man/rk.updatePluginMessages.Rd
M +1 -1 packages/rkwarddev/man/rkwarddev-package.Rd
M +2 -1 packages/rkwarddev/man/rkwarddev.required.Rd
M +24 -12 packages/rkwarddev/man/tf.Rd
http://commits.kde.org/rkward/8664767f384e9994dafa868f9c022b9d51c5d6ff
diff --git a/packages/rkwarddev/DESCRIPTION b/packages/rkwarddev/DESCRIPTION
index 442d856..dea52e2 100644
--- a/packages/rkwarddev/DESCRIPTION
+++ b/packages/rkwarddev/DESCRIPTION
@@ -8,16 +8,15 @@ Depends:
Suggests:
testthat
Enhances: rkward
-Description: Provides functions to create plugin skeletons and XML
- structures for RKWard.
+Description: Provides functions to create plugin skeletons and XML structures
+ for RKWard.
Encoding: UTF-8
URL: https://rkward.kde.org
-Authors at R: c(person(given="m.eik", family="michalke",
- email="meik.michalke at hhu.de", role=c("aut", "cre", "cph")))
+Authors at R: c(person(given="m.eik", family="michalke", email="meik.michalke at hhu.de", role=c("aut", "cre", "cph")))
Version: 0.08-1
-Date: 2015-11-28
+Date: 2015-11-30
RoxygenNote: 5.0.1
Collate:
'00_class_01_rk.JS.arr.R'
diff --git a/packages/rkwarddev/R/rkwarddev-package.R b/packages/rkwarddev/R/rkwarddev-package.R
index 572eb0b..d980dbc 100644
--- a/packages/rkwarddev/R/rkwarddev-package.R
+++ b/packages/rkwarddev/R/rkwarddev-package.R
@@ -4,7 +4,7 @@
#' Package: \tab rkwarddev\cr
#' Type: \tab Package\cr
#' Version: \tab 0.08-1\cr
-#' Date: \tab 2015-11-28\cr
+#' Date: \tab 2015-11-30\cr
#' Depends: \tab R (>= 2.9.0),methods,XiMpLe (>= 0.03-21),rkward (>= 0.5.7)\cr
#' Enhances: \tab rkward\cr
#' Encoding: \tab UTF-8\cr
diff --git a/packages/rkwarddev/man/R.comment.Rd b/packages/rkwarddev/man/R.comment.Rd
index f41ec41..f211942 100644
--- a/packages/rkwarddev/man/R.comment.Rd
+++ b/packages/rkwarddev/man/R.comment.Rd
@@ -13,9 +13,11 @@ R.comment(..., indent.by = rk.get.indent(escape = TRUE), level = 2,
\item{indent.by}{A character string defining the indentation string to use. Note that
backslashes need to be escaped (e.g. \code{"\\t"} to produce \code{"\t"}).}
-\item{level}{Integer, which indentation level to use in the resulting R code, minimum is 1.}
+\item{level}{Integer, which indentation level to use in the resulting R code,
+ minimum is 1.}
-\item{newline}{Character string, can be set to e.g. \code{"\n"} to force a newline after the call.}
+\item{newline}{Character string,
+ can be set to e.g. \code{"\n"} to force a newline after the call.}
}
\value{
A character string.
diff --git a/packages/rkwarddev/man/XiMpLe-methods.Rd b/packages/rkwarddev/man/XiMpLe-methods.Rd
index 2436807..71f651b 100644
--- a/packages/rkwarddev/man/XiMpLe-methods.Rd
+++ b/packages/rkwarddev/man/XiMpLe-methods.Rd
@@ -25,26 +25,33 @@ plugin2script(obj, prefix = "rkdev", indent = TRUE, level = 1,
level = 1, drop.defaults = TRUE, node.names = TRUE, collapse = ".")
}
\arguments{
-\item{obj}{Either a character vector (path to a plugin XML file to parse), a connection, an already
+\item{obj}{Either a character vector (path to a plugin XML file to parse), a connection,
parsed XML tree (class \code{XiMpLe.doc}) or a single \code{XiMpLe.node} object.}
-\item{prefix}{Character string, used as the prefix for the object names used in the script. Set to \code{""}
+\item{prefix}{Character string,
+ used as the prefix for the object names used in the script. Set to \code{""}
to disable the prefix.}
\item{indent}{Logical, whether the script code should be indented properly.}
\item{level}{Integer number, the starting leven of indentation.}
-\item{drop.defaults}{Logical, whether to check for the default options in function calls. If the
-parsed and translated XML code resulted in default options, they are omitted in the resulting script.}
+\item{drop.defaults}{Logical,
+ whether to check for the default options in function calls. If the
+parsed and translated XML code resulted in default options,
+ they are omitted in the resulting script.}
-\item{node.names}{Logical, whether the node names should become part of the generated R object names.}
+\item{node.names}{Logical,
+ whether the node names should become part of the generated R object names.}
-\item{collapse}{Character string, used to collapse the parts of the generated R object names.}
+\item{collapse}{Character string,
+ used to collapse the parts of the generated R object names.}
}
\value{
Either a character vector (if \code{obj} is a single XML node)
- or a list of character vectors named \code{"logic"}, \code{"dialog"}, and \code{"wizard"},
+ or a list of character vectors named \code{"logic"}, \code{"dialog"},
+ and \code{"wizard"},
(if \code{obj} is a full XML document).
}
\description{
@@ -52,17 +59,20 @@ These methods try to translate plugin XML objects into \code{rkwarddev} function
}
\details{
They are intended to make it easier to translate previously manually maintained plugin code
-into \code{rkwarddev} scripts. The generated output should not be used as-is, but restructured
+into \code{rkwarddev} scripts. The generated output should not be used as-is,
+ but restructured
properly into a useful script.
-or single (also nested) XiMpLe XML nodes. If you provide a character string, it is assumed to be
+or single (also nested) XiMpLe XML nodes. If you provide a character string,
+ it is assumed to be
the full path to a document to be parsed with \code{parseXMLTree} and then analysed. Connections
are also accepted.
}
\note{
The methods might fail, especially with highly complex plugins. Try to break these
-into sensible chunks and try those speparately. Sometimes, slightly changing the input file
+into sensible chunks and try those speparately. Sometimes,
+ slightly changing the input file
might also do the trick to get some usable results.
}
\examples{
diff --git a/packages/rkwarddev/man/echo.Rd b/packages/rkwarddev/man/echo.Rd
index 8779e89..27b3e00 100644
--- a/packages/rkwarddev/man/echo.Rd
+++ b/packages/rkwarddev/man/echo.Rd
@@ -8,16 +8,20 @@ echo(..., newline = "")
}
\arguments{
\item{...}{One or several character strings and/or \code{XiMpLe.node} objects with plugin nodes,
-and/or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt}, simply separated by comma.}
+and/or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt},
+ simply separated by comma.}
-\item{newline}{Character string, can be set to e.g. \code{"\n"} to force a newline after the call.}
+\item{newline}{Character string,
+ can be set to e.g. \code{"\n"} to force a newline after the call.}
}
\value{
A character string.
}
\description{
-This function will take several elements, either character strings, or objects of class \code{XiMpLe.node}
-which hold an XML node of some plugin GUI definition, or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt}.
+This function will take several elements, either character strings,
+ or objects of class \code{XiMpLe.node}
+which hold an XML node of some plugin GUI definition,
+ or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt}.
From those, it will generate a ready-to-run JavaScript \code{echo();} call from it.
}
\examples{
diff --git a/packages/rkwarddev/man/i18n.Rd b/packages/rkwarddev/man/i18n.Rd
index 41454c7..50c8608 100644
--- a/packages/rkwarddev/man/i18n.Rd
+++ b/packages/rkwarddev/man/i18n.Rd
@@ -7,11 +7,14 @@
i18n(msgid, ..., context = NULL, plural = NULL, newline = "")
}
\arguments{
-\item{msgid}{Either a character string, the message to be translated (if applicable, its singular form),
-or an object of class \code{\link[base:noquote]{noquote}}, which will be pasted as a \code{noquote()} function call.}
+\item{msgid}{Either a character string, the message to be translated (if applicable,
+ its singular form),
+or an object of class \code{\link[base:noquote]{noquote}},
+ which will be pasted as a \code{noquote()} function call.}
\item{...}{Either character string which will be pasted unquoted to be used in conjunctions with
-placeholders in msgid, or XiMpLe.node objects of which the JavaScript variable name will be
+placeholders in msgid,
+ or XiMpLe.node objects of which the JavaScript variable name will be
used.}
\item{context}{Character string, optional context information for this string.}
@@ -19,7 +22,8 @@ used.}
\item{plural}{Character string for plural form of \code{msgid}, must at least include one
placeholder, and the first one has to represent an integer value in the dialog.}
-\item{newline}{Character string, can be set to e.g. \code{"\n"} to force a newline after the call.}
+\item{newline}{Character string,
+ can be set to e.g. \code{"\n"} to force a newline after the call.}
}
\value{
An object of class \code{rk.JS.i18n}.
diff --git a/packages/rkwarddev/man/id.Rd b/packages/rkwarddev/man/id.Rd
index 6e429db..0a0b976 100644
--- a/packages/rkwarddev/man/id.Rd
+++ b/packages/rkwarddev/man/id.Rd
@@ -8,17 +8,22 @@ id(..., quote = FALSE, collapse = "", js = TRUE, .objects = list(...))
}
\arguments{
\item{...}{One or several character strings and/or \code{XiMpLe.node} objects with plugin nodes,
-and/or objects of classes \code{rk.JS.arr}, \code{rk.JS.opt} or \code{rk.JS.var}, simply separated by comma.}
+and/or objects of classes \code{rk.JS.arr}, \code{rk.JS.opt} or \code{rk.JS.var},
+ simply separated by comma.}
-\item{quote}{Logical, if the character strings should be deparsed, so they come out "as-is" when
+\item{quote}{Logical, if the character strings should be deparsed,
+ so they come out "as-is" when
written to files, e.g. by \code{cat}.}
-\item{collapse}{Character string, defining if and how the individual elements should be glued together.}
+\item{collapse}{Character string,
+ defining if and how the individual elements should be glued together.}
-\item{js}{Logical, if \code{TRUE} returns JavaScript varaible names for \code{XiMpLe.node} objects.
+\item{js}{Logical,
+ if \code{TRUE} returns JavaScript varaible names for \code{XiMpLe.node} objects.
Otherwise their actual ID is returned.}
-\item{.objects}{Alternative way of specifying objects, if you already have them as a list.}
+\item{.objects}{Alternative way of specifying objects,
+ if you already have them as a list.}
}
\value{
A character string.
@@ -26,8 +31,10 @@ A character string.
\description{
This function is intended to be used for generating JavaScript code for
RKWard plugins. Its sole purpose is to replace objects of class \code{XiMpLe.node}
-which hold an XML node of some plugin GUI definition, and objects of classes \code{rk.JS.arr},
-\code{rk.JS.opt} or \code{rk.JS.var} with their ID (or JS variable name), and combine these
+which hold an XML node of some plugin GUI definition,
+ and objects of classes \code{rk.JS.arr},
+\code{rk.JS.opt} or \code{rk.JS.var} with their ID (or JS variable name),
+ and combine these
replacements with character strings.
}
\examples{
diff --git a/packages/rkwarddev/man/idq.Rd b/packages/rkwarddev/man/idq.Rd
index e67757e..7805abe 100644
--- a/packages/rkwarddev/man/idq.Rd
+++ b/packages/rkwarddev/man/idq.Rd
@@ -12,10 +12,12 @@ idq(obj, modifiers = NULL, check.modifiers = TRUE, js = TRUE,
\item{modifiers}{A character vector with modifiers you'd like to apply to the XML node property.}
-\item{check.modifiers}{Logical, if \code{TRUE} the given modifiers will be checked for validity. Should only be
+\item{check.modifiers}{Logical,
+ if \code{TRUE} the given modifiers will be checked for validity. Should only be
turned off if you know what you're doing.}
-\item{js}{Logical, if \code{TRUE} returns JavaScript varaible names for \code{XiMpLe.node} objects.
+\item{js}{Logical,
+ if \code{TRUE} returns JavaScript varaible names for \code{XiMpLe.node} objects.
Otherwise their actual ID is returned.}
\item{quote}{Character string to be used for quoting.}
@@ -25,10 +27,12 @@ A character string.
}
\description{
This is actually a convenience wrapper for \code{\link[rkwarddev:id]{id}}
-and returns the XML ID of XiMpLe nodes in quoted format, optionally with an attached modifier.
+and returns the XML ID of XiMpLe nodes in quoted format,
+ optionally with an attached modifier.
}
\details{
-You can use this function to write almost literal JavaScript code, but still be able to extract IDs from
+You can use this function to write almost literal JavaScript code,
+ but still be able to extract IDs from
generated R objects.
}
\note{
diff --git a/packages/rkwarddev/man/ite.Rd b/packages/rkwarddev/man/ite.Rd
index 45bf676..1fd529e 100644
--- a/packages/rkwarddev/man/ite.Rd
+++ b/packages/rkwarddev/man/ite.Rd
@@ -10,12 +10,14 @@ ite(ifjs, thenjs, elsejs = NULL)
\item{ifjs}{Either a character string to be placed in the brackets of an \code{if()} statement,
or an object of class \code{XiMpLe.node}. \code{rk.JS.arr} or \code{rk.JS.opt} (whose identifier will be used).}
-\item{thenjs}{Either a character string, the code to be executed in case the \code{if()} statement is true,
+\item{thenjs}{Either a character string,
+ the code to be executed in case the \code{if()} statement is true,
or an object of class \code{XiMpLe.node}. \code{rk.JS.arr} or \code{rk.JS.opt} (whose identifier will be used).
The latter is especially useful in combination with \code{\link[rkwarddev:rk.JS.options]{rk.JS.options}}.
You can also give another object of class \code{rk.JS.ite}.}
-\item{elsejs}{Like \code{thenjs}, the code to be executed in case the \code{if()} statement is not true.}
+\item{elsejs}{Like \code{thenjs},
+ the code to be executed in case the \code{if()} statement is not true.}
}
\value{
An object of class \code{rk.JS.ite}
diff --git a/packages/rkwarddev/man/join.Rd b/packages/rkwarddev/man/join.Rd
index acdeddc..da47ae8 100644
--- a/packages/rkwarddev/man/join.Rd
+++ b/packages/rkwarddev/man/join.Rd
@@ -18,8 +18,10 @@ An object of class \code{rk.JS.echo}.
}
\description{
This function pastes an object of class \code{rk.JS.arr} similar to \code{\link[rkwarddev:rk.paste.JS]{rk.paste.JS}},
-but was specifically written for elements like \code{<optionset>} or \code{<matrix>}, whose values must be queried
-by \code{getList()} rather than \code{getValue()}. This means, the resulting variable is already an array an merely
+but was specifically written for elements like \code{<optionset>} or \code{<matrix>},
+ whose values must be queried
+by \code{getList()} rather than \code{getValue()}. This means,
+ the resulting variable is already an array an merely
needs to be joined in as R code output (e.g., an \code{<optioncolumn>}).
}
\seealso{
diff --git a/packages/rkwarddev/man/js.Rd b/packages/rkwarddev/man/js.Rd
index c403b2f..89898aa 100644
--- a/packages/rkwarddev/man/js.Rd
+++ b/packages/rkwarddev/man/js.Rd
@@ -18,29 +18,37 @@ JavaScript operators and \code{if} conditions will be kept as-is.}
\item{linebreaks}{Logical, should there be line breaks between the elements in this call?}
-\item{empty.e}{For \code{if} conditions only: Logical, if \code{TRUE} will force to add empty \code{else \{\}} brackets when
-there is no \code{else} statement defined, which is considered to enhance code readability by some.}
+\item{empty.e}{For \code{if} conditions only: Logical,
+ if \code{TRUE} will force to add empty \code{else \{\}} brackets when
+there is no \code{else} statement defined,
+ which is considered to enhance code readability by some.}
-\item{keep.ite}{Logical, if \code{TRUE} returns \code{if/else} conditions in a list of objects of class \code{rk.JS.ite} instead
+\item{keep.ite}{Logical,
+ if \code{TRUE} returns \code{if/else} conditions in a list of objects of class \code{rk.JS.ite} instead
of a pasted character string. Comes in handy if used inside \code{\link[rkwarddev:rk.JS.options]{rk.JS.options}}.}
}
\value{
-A character string (or \code{rk.JS.ite}, if \code{keep.ite=TRUE} and input is an \code{if/else} condition).
+A character string (or \code{rk.JS.ite},
+ if \code{keep.ite=TRUE} and input is an \code{if/else} condition).
}
\description{
that uses \code{eval(substitute(alist(...)))} to preserve the value of \code{...} as-is to be able to
-both keep operators like \code{">="} or \code{"!="} unevaluated in the resulting output, as well as translating
+both keep operators like \code{">="} or \code{"!="} unevaluated in the resulting output,
+ as well as translating
\code{if/else} clauses from R to JavaScript.
}
\details{
-Normally, \code{id} would simply evaluate the condition and then return the result of that evaluation, which
-most of the time is not what you want. With this function, you can test conditions in usual R syntax, yet
+Normally,
+ \code{id} would simply evaluate the condition and then return the result of that evaluation, which
+most of the time is not what you want. With this function,
+ you can test conditions in usual R syntax, yet
the operators and \code{if/else} clauses will end up pasted in the result.
The following operators are supported: +, -, *, /, ==, !=, >, <, >=, <=, || and &&
-These are currently unsupported and still need to be quoted: \%, ++, --, =, +=, -=, *=, /=, \%=, ===, !== and !
+These are currently unsupported and still need to be quoted: \%, ++, --, =, +=, -=, *=,
+ /=, \%=, ===, !== and !
}
\note{
diff --git a/packages/rkwarddev/man/qp.Rd b/packages/rkwarddev/man/qp.Rd
index 4dc0554..9a4bbf9 100644
--- a/packages/rkwarddev/man/qp.Rd
+++ b/packages/rkwarddev/man/qp.Rd
@@ -8,7 +8,8 @@ qp(...)
}
\arguments{
\item{...}{One or several character strings and/or \code{XiMpLe.node} objects with plugin nodes,
-and/or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt}, simply separated by comma.}
+and/or objects of classes \code{rk.JS.arr} or \code{rk.JS.opt},
+ simply separated by comma.}
}
\value{
A character string.
diff --git a/packages/rkwarddev/man/rk.JS.arr-class.Rd b/packages/rkwarddev/man/rk.JS.arr-class.Rd
index 8bf44f3..92df61d 100644
--- a/packages/rkwarddev/man/rk.JS.arr-class.Rd
+++ b/packages/rkwarddev/man/rk.JS.arr-class.Rd
@@ -26,7 +26,8 @@ need to temper with this type of class manually.
\item{\code{option}}{Character string, name of the option to set.}
-\item{\code{opt.sep}}{Character string, separates previous options from the one defined by the array.}
+\item{\code{opt.sep}}{Character string,
+ separates previous options from the one defined by the array.}
}}
\keyword{Classes}
diff --git a/packages/rkwarddev/man/rk.JS.array.Rd b/packages/rkwarddev/man/rk.JS.array.Rd
index b636d00..1b2df14 100644
--- a/packages/rkwarddev/man/rk.JS.array.Rd
+++ b/packages/rkwarddev/man/rk.JS.array.Rd
@@ -14,15 +14,18 @@ constructed from several variables.}
\item{variables}{A list with either character strings (the names of the variables to combine to a vector or list),
or objects of class \code{XiMpLe.node} with plugin XML nodes (whose ID will be extracted and used).}
-\item{funct}{Character string, name of the R function to be called to combine the options, e.g. "list" for \code{list()},
+\item{funct}{Character string, name of the R function to be called to combine the options,
+ e.g. "list" for \code{list()},
or "c" for \code{c()}.}
\item{var.prefix}{A character string. sets a global string to be used as a prefix for the JS variable names.}
-\item{quote}{Logical, if \code{TRUE}, the values will be quoted in the resulting R code (might be neccessary
+\item{quote}{Logical, if \code{TRUE},
+ the values will be quoted in the resulting R code (might be neccessary
for character values).}
-\item{opt.sep}{Character string, will be printed in the resulting R code before the option name.}
+\item{opt.sep}{Character string,
+ will be printed in the resulting R code before the option name.}
}
\value{
An object of class \code{rk.JS.arr}.
diff --git a/packages/rkwarddev/man/rk.JS.doc.Rd b/packages/rkwarddev/man/rk.JS.doc.Rd
index 1ba9d6f..7d6b363 100644
--- a/packages/rkwarddev/man/rk.JS.doc.Rd
+++ b/packages/rkwarddev/man/rk.JS.doc.Rd
@@ -17,7 +17,8 @@ rk.JS.doc(require = c(), variables = NULL, globals = NULL,
or a (list of) objects of class \code{rk.JS.var} which will be coerced into character. These variables will be defined in
the \code{calculate()} and/or \code{doPrintout()} functions.}
-\item{globals}{Like \code{variables}, but these variables will be defined globally. If \code{variables} is set as well,
+\item{globals}{Like \code{variables},
+ but these variables will be defined globally. If \code{variables} is set as well,
the function tries to remove duplicate definitions.}
\item{results.header}{A character string to headline the printed results. Include escapes quotes (\\") if needed.
@@ -33,14 +34,18 @@ pasted as-is, after \code{require} has been evaluated.}
pasted as-is, after \code{variables} has been evaluated.}
\item{printout}{A character string to be included in the \code{printout()} function. This string will be
-pasted as-is, after \code{results.header} has been evaluated. Ignored if \code{doPrintout} is set.}
+pasted as-is,
+ after \code{results.header} has been evaluated. Ignored if \code{doPrintout} is set.}
\item{doPrintout}{A character string to be included in the \code{doPrintout()} function. This string will be
-pasted as-is. You don't need to define a \code{preview()} function, as this will be added automatically.
+pasted as-is. You don't need to define a \code{preview()} function,
+ as this will be added automatically.
Use \code{ite("full", ...)} style JavaScript code to include headers etc.}
-\item{load.silencer}{Either a character string (ID of probably a checkbox), or an object of class \code{XiMpLe.node}.
-This defines a switch you can add to your plugin, to set the \code{require()} call inside \code{suppressMessages()},
+\item{load.silencer}{Either a character string (ID of probably a checkbox),
+ or an object of class \code{XiMpLe.node}.
+ to set the \code{require()} call inside \code{suppressMessages()},
hence suppressing all load messages (except for warnings and errors) of required packages in the output.}
\item{gen.info}{Logical, if \code{TRUE} a comment note will be written into the document,
@@ -50,7 +55,8 @@ which will then also be added to the comment.}
\item{indent.by}{A character string defining how indentation should be done.}
-\item{guess.getter}{Locigal, if \code{TRUE} try to get a good default getter function for JavaScript
+\item{guess.getter}{Locigal,
+ if \code{TRUE} try to get a good default getter function for JavaScript
}
\value{
index b6e1296..720505e 100644
@@ -13,17 +13,20 @@ nested in an i18n() call.}
or \code{addFromUI} -- note that you can use multiple entries with the same name here. Entries named
-\code{add} must be vectors of legth 2, the first being the caption (character), the second its value (either
+\code{add} must be vectors of legth 2, the first being the caption (character),
+ the second its value (either
character or a XiMpLe node from the dialog); if the second value is named \code{noquote} or \code{nq},
the JS output will be nested inside \code{noquote()}. Entries named \code{addFromUI} must have exactly one value
specifying the GUI element to query (either character or a XiMpLe node from the dialog).}
-\item{guess.getter}{Locigal, if \code{TRUE} try to get a good default getter function for JavaScript
+\item{guess.getter}{Locigal,
+ if \code{TRUE} try to get a good default getter function for JavaScript
variable values.}
-\item{.add}{Same as \code{...}, but provided as a single list. If used, values will be appended to \code{...}.}
+\item{.add}{Same as \code{...}, but provided as a single list. If used,
+ values will be appended to \code{...}.}
}
\value{
A character string.
diff --git a/packages/rkwarddev/man/rk.JS.opt-class.Rd b/packages/rkwarddev/man/rk.JS.opt-class.Rd
index 7380771..c5db645 100644
--- a/packages/rkwarddev/man/rk.JS.opt-class.Rd
+++ b/packages/rkwarddev/man/rk.JS.opt-class.Rd
@@ -16,15 +16,18 @@ type of class manually.
\item{\code{opt.name}}{Character string, the name of the option.}
-\item{\code{collapse}}{Character string, used to collapse several options into one string.}
+\item{\code{collapse}}{Character string,
+ used to collapse several options into one string.}
\item{\code{ifs}}{A list with objects of class rk.JS.ite.}
\item{\code{array}}{Logical, whether to use an array for options.}
-\item{\code{funct}}{Character string, name of the R function to be called to combine the options.}
+\item{\code{funct}}{Character string,
+ name of the R function to be called to combine the options.}
-\item{\code{opt.sep}}{Character string, separates previous options from the one defined here.}
+\item{\code{opt.sep}}{Character string,
+ separates previous options from the one defined here.}
}}
\keyword{Classes}
diff --git a/packages/rkwarddev/man/rk.JS.optionset.Rd b/packages/rkwarddev/man/rk.JS.optionset.Rd
index f889eeb..47c6dcb 100644
--- a/packages/rkwarddev/man/rk.JS.optionset.Rd
+++ b/packages/rkwarddev/man/rk.JS.optionset.Rd
@@ -10,29 +10,36 @@ rk.JS.optionset(optionset, ..., loopvar = "i", collapse = ",\\\\n\\\\t",
\arguments{
\item{optionset}{A XiMpLe.node object, the full \code{<optionset>} node.}
-\item{...}{The JavaScript code, optionally including the optioncolumn objects. This will become
+\item{...}{The JavaScript code,
+ optionally including the optioncolumn objects. This will become
the body of the for loop.}
\item{loopvar}{Character, name of the index variable used in the for loop.}
-\item{collapse}{Character string, how all optioncolumns should be concatenated on the R code level
-Hint: To place each one on a new line with tab indentation, set \code{collapse=",\\n\\t"}.}
+\item{collapse}{Character string,
+ how all optioncolumns should be concatenated on the R code level
+Hint: To place each one on a new line with tab indentation, set \code{collapse=",
+ \\n\\t"}.}
-\item{vars}{Logical, if \code{TRUE} all optioncolumn varaibles will be defined first. This is not
+\item{vars}{Logical,
+ if \code{TRUE} all optioncolumn varaibles will be defined first. This is not
-\item{guess.getter}{Logical, if \code{TRUE} try to get a good default getter function for JavaScript
+\item{guess.getter}{Logical,
+ if \code{TRUE} try to get a good default getter function for JavaScript
variable values. Only relevant if \code{vars=TRUE}.}
}
\description{
This function scans an object generated by \code{\link[rkwarddev:rk.XML.optionset]{rk.XML.optionset}},
extract IDs of all optioncolumn objects and nest the JavaScript code you define via \code{...} inside
-a for loop that iterates through all columns. Inside \code{...}, you can use the column objects of
+a for loop that iterates through all columns. Inside \code{...},
+ you can use the column objects of
\code{\link[rkwarddev:rk.XML.optioncolumn]{rk.XML.optioncolumn}} to refer to the respective column,
\code{rk.JS.optionset} will use appropriate variables.
}
\details{
-In case you simply want to define the variables, but not run the loop yet, set \code{vars=TRUE} and
+In case you simply want to define the variables, but not run the loop yet,
+ set \code{vars=TRUE} and
leave \code{...} empty.
}
\examples{
@@ -48,7 +55,8 @@ dep.optionset.packages <- rk.XML.optionset(
)
)), label="Depends on R packages"),
optioncolumn=list(
- dep.optioncol.pckg.name <- rk.XML.optioncolumn(connect=dep.pckg.name, modifier="text"),
+ dep.optioncol.pckg.name <- rk.XML.optioncolumn(connect=dep.pckg.name,
+ modifier="text"),
dep.optioncol.pckg.min <- rk.XML.optioncolumn(connect=dep.pckg.min, modifier="text"),
dep.optioncol.pckg.max <- rk.XML.optioncolumn(connect=dep.pckg.max, modifier="text"),
dep.optioncol.pckg.repo <- rk.XML.optioncolumn(connect=dep.pckg.repo, modifier="text")
diff --git a/packages/rkwarddev/man/rk.JS.oset-class.Rd b/packages/rkwarddev/man/rk.JS.oset-class.Rd
index 6737fcb..d00f045 100644
--- a/packages/rkwarddev/man/rk.JS.oset-class.Rd
+++ b/packages/rkwarddev/man/rk.JS.oset-class.Rd
@@ -20,7 +20,8 @@ type of class manually.
\item{\code{body}}{A list of JavaScript code, the body of the for loop.}
-\item{\code{collapse}}{Character string, how all optioncolumns should be concatenated on the R code level.}
+\item{\code{collapse}}{Character string,
+ how all optioncolumns should be concatenated on the R code level.}
}}
\keyword{Classes}
diff --git a/packages/rkwarddev/man/rk.JS.saveobj.Rd b/packages/rkwarddev/man/rk.JS.saveobj.Rd
--- a/packages/rkwarddev/man/rk.JS.saveobj.Rd
+++ b/packages/rkwarddev/man/rk.JS.saveobj.Rd
@@ -8,15 +8,20 @@ rk.JS.saveobj(pXML, R.objects = "initial", vars = TRUE,
add.abbrev = FALSE, indent.by = rk.get.indent())
}
\arguments{
-\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node}, or path to a plugin XML file.}
+\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node},
+ or path to a plugin XML file.}
-\item{R.objects}{Character vector, the names of the internal R objects to be saved. If not empty must have
-the same length as <saveobject> nodes in the document, or be the keyword "initial", in which case the
+\item{R.objects}{Character vector,
+ the names of the internal R objects to be saved. If not empty must have
+the same length as <saveobject> nodes in the document, or be the keyword "initial",
+ in which case the
\code{intital} attribute values of the nodes are used.}
-\item{vars}{Logocal, whether the variables needed should also be defined in the JavaScript code.}
+\item{vars}{Logocal,
+ whether the variables needed should also be defined in the JavaScript code.}
-\item{add.abbrev}{Logical, if \code{TRUE} the JavaScript variables will all have a prefix with an
+ if \code{TRUE} the JavaScript variables will all have a prefix with an
three letter abbreviation of the XML tag type to improve the readability of the code. But it's
probably better to add this in the XML code in the first place.}
diff --git a/packages/rkwarddev/man/rk.JS.scan.Rd b/packages/rkwarddev/man/rk.JS.scan.Rd
index c81153c..f59a471 100644
--- a/packages/rkwarddev/man/rk.JS.scan.Rd
+++ b/packages/rkwarddev/man/rk.JS.scan.Rd
@@ -8,18 +8,24 @@ rk.JS.scan(pXML, js = TRUE, add.abbrev = FALSE, guess.getter = FALSE,
indent.by = rk.get.indent())
}
\arguments{
-\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node}, or path to a plugin XML file.}
+\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node},
+ or path to a plugin XML file.}
-\item{js}{Logical, if \code{TRUE} usable JavaScript code will be returned, otherwise a character
+\item{js}{Logical, if \code{TRUE} usable JavaScript code will be returned,
+ otherwise a character
vector with only the relevant ID names.}
-\item{add.abbrev}{Logical, if \code{TRUE} the JavaScript variables will all have a prefix with an
+ if \code{TRUE} the JavaScript variables will all have a prefix with an
three letter abbreviation of the XML tag type to improve the readability of the code. But it's
probably better to add this in the XML code in the first place.}
-\item{guess.getter}{Logical, if \code{TRUE} try to get a good default getter function for JavaScript
-variable values. This will use some functions which were added with RKWard 0.6.1, and therefore
-raise the dependencies for your plugin/component accordingly. Nonetheless, it's recommended.}
+\item{guess.getter}{Logical,
+ if \code{TRUE} try to get a good default getter function for JavaScript
+variable values. This will use some functions which were added with RKWard 0.6.1,
+ and therefore
+raise the dependencies for your plugin/component accordingly. Nonetheless,
+ it's recommended.}
\item{indent.by}{Character string used to indent each entry if \code{js=TRUE}.}
}
diff --git a/packages/rkwarddev/man/rk.JS.var-class.Rd b/packages/rkwarddev/man/rk.JS.var-class.Rd
index 9c8b4a8..2d90032 100644
--- a/packages/rkwarddev/man/rk.JS.var-class.Rd
+++ b/packages/rkwarddev/man/rk.JS.var-class.Rd
@@ -20,15 +20,19 @@ type of class manually.
\item{\code{modifiers}}{A list of modifiers to apply to the XML node property.}
-\item{\code{default}}{Logical, whether the default value (no special modifier) of the node should also be defined.}
+\item{\code{default}}{Logical,
+ whether the default value (no special modifier) of the node should also be defined.}
-\item{\code{append.modifier}}{Logical, if a modifier is given, should that become part of the variable name?}
+\item{\code{append.modifier}}{Logical, if a modifier is given,
+ should that become part of the variable name?}
-\item{\code{join}}{Character string, if set is used to collapse multiple values into one string.}
+\item{\code{join}}{Character string,
+ if set is used to collapse multiple values into one string.}
\item{\code{vars}}{A list of objects of class rk.JS.var.}
-\item{\code{getter}}{Character string, the JavaScript function which should be used to fetch the values from the plugin.}
+\item{\code{getter}}{Character string,
+ the JavaScript function which should be used to fetch the values from the plugin.}
}}
\keyword{Classes}
diff --git a/packages/rkwarddev/man/rk.JS.vars.Rd b/packages/rkwarddev/man/rk.JS.vars.Rd
index 93035b6..db7b344 100644
--- a/packages/rkwarddev/man/rk.JS.vars.Rd
+++ b/packages/rkwarddev/man/rk.JS.vars.Rd
@@ -16,26 +16,37 @@ or objects of class \code{XiMpLe.node} with plugin XML nodes (whose ID will be e
\item{modifiers}{A character vector with modifiers you'd like to apply to the XML node property.}
-\item{default}{Logical, if \code{TRUE} the default value (no special modifier) of the node will
+\item{default}{Logical,
+ if \code{TRUE} the default value (no special modifier) of the node will
also be defined. Does nothing if \code{modifiers=NULL}.}
-\item{join}{A character string, useful for GUI elements which accept multiple objects (e.g., multi-varslots).
-If \code{join} is something other than \code{""}, these objects will be collapsed into one string when pasted,
+\item{join}{A character string,
+ useful for GUI elements which accept multiple objects (e.g., multi-varslots).
+If \code{join} is something other than \code{""},
+ these objects will be collapsed into one string when pasted,
joined by this string.}
-\item{check.modifiers}{Logical, if \code{TRUE} the given modifiers will be checked for validity. Should only be
+\item{check.modifiers}{Logical,
+ if \code{TRUE} the given modifiers will be checked for validity. Should only be
turned off if you know what you're doing.}
-\item{getter}{A character string, naming the JavaScript function which should be used to get the values in the
-actual plugin. Depending on the XML element, \code{"getString"}, \code{"getBool"} or \code{"getList"} can be
-useful alternatives. For backwards compatibility, the default is set to \code{"getValue"}.}
+\item{getter}{A character string,
+ naming the JavaScript function which should be used to get the values in the
+actual plugin. Depending on the XML element, \code{"getString"},
+ \code{"getBool"} or \code{"getList"} can be
+useful alternatives. For backwards compatibility,
+ the default is set to \code{"getValue"}.}
-\item{guess.getter}{Logical, if \code{TRUE} try to get a good default getter function for JavaScript
+\item{guess.getter}{Logical,
+ if \code{TRUE} try to get a good default getter function for JavaScript
variable values.}
-\item{object.name}{Logical, if \code{TRUE} the JS variable name will roughly match the R object name. If the
-object name contains dots, they will be removed and the JS name printed in camel code instead. Use this option
-with great caution and do not combine it with \code{\link[rkwarddev:rk.JS.scan]{rk.JS.scan}}, as it will likely result
+\item{object.name}{Logical,
+ if \code{TRUE} the JS variable name will roughly match the R object name. If the
+object name contains dots,
+ they will be removed and the JS name printed in camel code instead. Use this option
+with great caution and do not combine it with \code{\link[rkwarddev:rk.JS.scan]{rk.JS.scan}},
+ as it will likely result
in unusable code. \code{rk.JS.scan} examines XML nodes and therefore does not know any R object names.}
}
\value{
index f2ab7a8..0ffe9ce 100644
@@ -17,9 +17,11 @@ rk.XML.about(name, author, about = list(desc = "SHORT_DESCRIPTION", version =
\item{given}{Author given name}
\item{family}{Author family name}
\item{email}{Author mail address (can be omitted if \code{role} does not include \code{"cre"})}
- \item{role}{This person's specific role, e.g. \code{"aut"} for actual author, \code{"cre"} for maintainer or \code{"ctb"} for contributor.}
+ \item{role}{This person's specific role, e.g. \code{"aut"} for actual author,
+ \code{"cre"} for maintainer or \code{"ctb"} for contributor.}
}
-See \code{\link[utils:person]{person}} for more details on this, especially for valid roles.}
+See \code{\link[utils:person]{person}} for more details on this,
+ especially for valid roles.}
\item{about}{A named list with these elements:
\describe{
@@ -32,11 +34,14 @@ See \code{\link[utils:person]{person}} for more details on this, especially for
\item{long.desc}{A long description (optional, defaults to \code{desc})}
}}
+\item{dependencies}{Deprecated,
+\item{package}{Deprecated,
+\item{pluginmap}{Deprecated,
}
\description{
Create XML node "about" for RKWard pluginmaps
diff --git a/packages/rkwarddev/man/rk.XML.attribute.Rd b/packages/rkwarddev/man/rk.XML.attribute.Rd
index 6cb1f01..1954fb9 100644
--- a/packages/rkwarddev/man/rk.XML.attribute.Rd
+++ b/packages/rkwarddev/man/rk.XML.attribute.Rd
@@ -15,7 +15,8 @@ or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and
\item{label}{Character string, label associated with the attribute.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.browser.Rd b/packages/rkwarddev/man/rk.XML.browser.Rd
index b0bf75b..92f50f5 100644
--- a/packages/rkwarddev/man/rk.XML.browser.Rd
+++ b/packages/rkwarddev/man/rk.XML.browser.Rd
@@ -11,30 +11,38 @@ rk.XML.browser(label, type = "file", initial = NULL, urls = FALSE,
\arguments{
\item{label}{Character string, a text label for this plugin element.}
-\item{type}{Character string, valid values are "dir", "file" and "savefile" (i.e., an non-existing file).}
+\item{type}{Character string, valid values are "dir", "file" and "savefile" (i.e.,
+ an non-existing file).}
-\item{initial}{Character string, if not \code{NULL} will be used as the initial value of the browser.}
+\item{initial}{Character string,
+ if not \code{NULL} will be used as the initial value of the browser.}
\item{urls}{Logical, whether non-local URLs are permitted or not.}
-\item{filter}{Character vector, file type filter, e.g. \code{filter=c("*.txt", "*.csv")} for .txt and .csv files.
+\item{filter}{Character vector, file type filter, e.g. \code{filter=c("*.txt",
+ "*.csv")} for .txt and .csv files.
Try not to induce limits unless absolutely needed, though.}
\item{required}{Logical, whether an entry is mandatory or not.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.cbox.Rd b/packages/rkwarddev/man/rk.XML.cbox.Rd
index 6c3fab1..c963746 100644
--- a/packages/rkwarddev/man/rk.XML.cbox.Rd
+++ b/packages/rkwarddev/man/rk.XML.cbox.Rd
@@ -20,16 +20,20 @@ rk.XML.cbox(label, value = "true", un.value = NULL, chk = FALSE,
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.component.Rd b/packages/rkwarddev/man/rk.XML.component.Rd
index 3b476db..17970eb 100644
--- a/packages/rkwarddev/man/rk.XML.component.Rd
+++ b/packages/rkwarddev/man/rk.XML.component.Rd
@@ -15,14 +15,16 @@ rk.XML.component(label, file, id.name = "auto", type = "standard",
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the label.}
-\item{type}{Character string, type of component. As of now, only "standard" is supported. The option is
+\item{type}{Character string, type of component. As of now,
+ only "standard" is supported. The option is
just implemented for completeness.}
\item{dependencies}{An object of class \code{XiMpLe.node} to define \code{<dependencies>} for this component.
See \code{\link[rkwarddev:rk.XML.dependencies]{rk.XML.dependencies}} for details. Skipped if \code{NULL}.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.components.Rd b/packages/rkwarddev/man/rk.XML.components.Rd
index fa42461..28b6211 100644
--- a/packages/rkwarddev/man/rk.XML.components.Rd
+++ b/packages/rkwarddev/man/rk.XML.components.Rd
@@ -13,7 +13,8 @@ rk.XML.components(...)
A list of objects of class \code{XiMpLe.node}.
}
\description{
-This function will create a components node for a .pluginmap file, with mandatory child nodes "component".
+This function will create a components node for a .pluginmap file,
+ with mandatory child nodes "component".
}
\examples{
test.component <- rk.XML.component("My GUI dialog", "plugins/MyGUIdialog.xml")
diff --git a/packages/rkwarddev/man/rk.XML.connect.Rd b/packages/rkwarddev/man/rk.XML.connect.Rd
index 5f6e591..6a9b031 100644
--- a/packages/rkwarddev/man/rk.XML.connect.Rd
+++ b/packages/rkwarddev/man/rk.XML.connect.Rd
@@ -9,21 +9,27 @@ rk.XML.connect(governor, client, get = "state", set = "enabled",
}
\arguments{
\item{governor}{Either a character string (the \code{id} of the property whose state should control
-the \code{client}), or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted
+the \code{client}),
+ or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted
and used). Usually a \code{<convert>} node defined earlier (see
-\item{client}{Character string, the \code{id} if the element to be controlled by \code{governor}.}
+\item{client}{Character string,
+ the \code{id} if the element to be controlled by \code{governor}.}
-\item{get}{Character string, a valid modifier for the node property of \code{governor}, often
+\item{get}{Character string, a valid modifier for the node property of \code{governor},
+ often
the ".state" value of some apropriate node.}
-\item{set}{Character string, a valid modifier for the node property of \code{client}, usually
+\item{set}{Character string, a valid modifier for the node property of \code{client},
+ usually
one of \code{"enabled"}, \code{"visible"} or \code{"required"}.}
-\item{not}{Logical, if \code{TRUE}, the state of \code{governor} (\code{TRUE/FALSE}) will be inversed.}
+\item{not}{Logical, if \code{TRUE},
+ the state of \code{governor} (\code{TRUE/FALSE}) will be inversed.}
-\item{reconcile}{Logical, forces the \code{governor} to only accept values which are valid for
+\item{reconcile}{Logical,
+ forces the \code{governor} to only accept values which are valid for
the \code{client} as well.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.convert.Rd b/packages/rkwarddev/man/rk.XML.convert.Rd
index b422cda..ac4c322 100644
--- a/packages/rkwarddev/man/rk.XML.convert.Rd
+++ b/packages/rkwarddev/man/rk.XML.convert.Rd
@@ -10,7 +10,8 @@ rk.XML.convert(sources, mode = c(), required = FALSE, id.name = "auto")
\item{sources}{A list with at least one value, either resembling the \code{id} of
an existing element to be queried as a character string, or a previously defined object
of class \code{XiMpLe.node} (whose \code{id} will be extracted and used). If you want
-to examine e.g. the state or string value specificly, just name the value accoringly, e.g.,
+to examine e.g. the state or string value specificly, just name the value accoringly,
+ e.g.,
\code{sources=list("vars0", string="input1", state="chkbx2")}.}
\item{mode}{A named vector with either exactly one of the following elements:
@@ -28,7 +29,8 @@ or at least one of these elemets:
\item{\code{max}}{True if \code{sources} is below this value. They must be numeric.}
}}
-\item{required}{Logical, sets the state of the \code{required_true} attribute. If \code{TRUE},
+\item{required}{Logical,
+ sets the state of the \code{required_true} attribute. If \code{TRUE},
the plugin submit button is only enabled if this property is true.}
\item{id.name}{Character string, a unique ID for this plugin element.
@@ -39,7 +41,8 @@ and \code{mode} value.}
An object of class \code{XiMpLe.node}.
}
\description{
-If \code{sources} holds \code{XiMpLe.node} objects, the validity of modifiers is automatically checked for that tag.
+If \code{sources} holds \code{XiMpLe.node} objects,
+ the validity of modifiers is automatically checked for that tag.
}
\note{
To get a list of the implemented modifiers for \code{sources} in this package see \code{\link[rkwarddev:modifiers]{modifiers}}.
diff --git a/packages/rkwarddev/man/rk.XML.copy.Rd b/packages/rkwarddev/man/rk.XML.copy.Rd
index cd8444d..e85d8df 100644
--- a/packages/rkwarddev/man/rk.XML.copy.Rd
+++ b/packages/rkwarddev/man/rk.XML.copy.Rd
@@ -10,7 +10,8 @@ rk.XML.copy(id, as = NULL)
\item{id}{Either a character string (the \code{id} of the property to be copied),
or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used).}
-\item{as}{A character string resembling the \code{copy_element_tag_name} value. I.e., must be
+\item{as}{A character string resembling the \code{copy_element_tag_name} value. I.e.,
+ must be
a valid tag name. Will cause a change of tag name of the \code{id} (e.g. "tab") to \code{as}
(e.g. "page").}
}
diff --git a/packages/rkwarddev/man/rk.XML.dependencies.Rd b/packages/rkwarddev/man/rk.XML.dependencies.Rd
index a886e35..b98cb04 100644
--- a/packages/rkwarddev/man/rk.XML.dependencies.Rd
+++ b/packages/rkwarddev/man/rk.XML.dependencies.Rd
@@ -32,13 +32,15 @@ rk.XML.dependencies(dependencies = NULL, package = NULL, pluginmap = NULL,
\item{url}{URL to get the pluginmap (required)}
}}
-\item{hints}{Logical, if \code{TRUE}, \code{NULL} values will be replaced with example text.}
+\item{hints}{Logical, if \code{TRUE},
+ \code{NULL} values will be replaced with example text.}
}
\description{
Create XML node "dependencies" for RKWard pluginmaps
}
\note{
-The \code{<dependencies>} node was introduced with RKWard 0.6.1, please set the dependencies
+The \code{<dependencies>} node was introduced with RKWard 0.6.1,
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.dependency_check.Rd b/packages/rkwarddev/man/rk.XML.dependency_check.Rd
index 3edaa8e..6021665 100644
--- a/packages/rkwarddev/man/rk.XML.dependency_check.Rd
+++ b/packages/rkwarddev/man/rk.XML.dependency_check.Rd
@@ -32,13 +32,15 @@ rk.XML.dependency_check(id.name, dependencies = NULL, package = NULL,
\item{url}{URL to get the pluginmap (optional)}
}}
-\item{hints}{Logical, if \code{TRUE}, \code{NULL} values will be replaced with example text.}
+\item{hints}{Logical, if \code{TRUE},
+ \code{NULL} values will be replaced with example text.}
}
\description{
Create XML node "dependency_check" for RKWard pluginmaps
}
\note{
-The \code{<dependency_check>} node was introduced with RKWard 0.6.1, please set the dependencies
+The \code{<dependency_check>} node was introduced with RKWard 0.6.1,
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.dialog.Rd b/packages/rkwarddev/man/rk.XML.dialog.Rd
index 974ba22..e48fba5 100644
--- a/packages/rkwarddev/man/rk.XML.dialog.Rd
+++ b/packages/rkwarddev/man/rk.XML.dialog.Rd
@@ -11,21 +11,26 @@ rk.XML.dialog(..., label = NULL, recommended = FALSE, i18n = NULL)
\item{label}{Character string, a text label for this plugin element.}
-\item{recommended}{Logical, whether the dialog should be the recommended interface (unless the user has configured
-RKWard to default to a specific interface). This attribute currently has no effect, as it is implicitly "true",
+\item{recommended}{Logical,
+ whether the dialog should be the recommended interface (unless the user has configured
+RKWard to default to a specific interface). This attribute currently has no effect,
+ as it is implicitly "true",
unless the wizard is recommended.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a dialog section with optional child nodes "browser", "checkbox",
+This function will create a dialog section with optional child nodes "browser",
+ "checkbox",
"column", "copy", "dropdown", "embed", "formula", "frame", "include", "input", "insert",
-"preview", "radio", "row", "saveobject", "select", "spinbox", "stretch", "tabbook", "text",
+"preview", "radio", "row", "saveobject", "select", "spinbox", "stretch", "tabbook",
+ "text",
"valueselector", "valueslot", "varselector" and "varslot".
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.dropdown.Rd b/packages/rkwarddev/man/rk.XML.dropdown.Rd
index 48907e3..f3b9ce0 100644
--- a/packages/rkwarddev/man/rk.XML.dropdown.Rd
+++ b/packages/rkwarddev/man/rk.XML.dropdown.Rd
@@ -12,24 +12,30 @@ rk.XML.dropdown(label, options = list(label = c(val = "", chk = FALSE, i18n =
\item{label}{Character string, a text label for this plugin element.}
\item{options}{A named list with options to choose from. The names of the list elements will become
-labels of the options, \code{val} defines the value to submit if the option is checked, and
+labels of the options, \code{val} defines the value to submit if the option is checked,
+ and
\code{chk=TRUE} should be set in the one option which is checked by default. You might also provide an \code{i18n}
for this particular option (see \code{i18n}). Objects generated with \code{\link[rkwarddev:rk.XML.option]{rk.XML.option}}
are accepted as well.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.embed.Rd b/packages/rkwarddev/man/rk.XML.embed.Rd
index 77d9f3f..37ff33a 100644
--- a/packages/rkwarddev/man/rk.XML.embed.Rd
+++ b/packages/rkwarddev/man/rk.XML.embed.Rd
@@ -8,17 +8,22 @@ rk.XML.embed(component, button = FALSE, label = "Options",
id.name = "auto", i18n = NULL)
}
\arguments{
-\item{component}{A character string, registered name (\code{id} in pluginmap file) of the component to be embedded.}
+\item{component}{A character string,
+ registered name (\code{id} in pluginmap file) of the component to be embedded.}
-\item{button}{Logical, whether the plugin should be embedded as a button and appear if it's pressed.}
+\item{button}{Logical,
+ whether the plugin should be embedded as a button and appear if it's pressed.}
-\item{label}{A character string, text label for the button (only used if \code{button=TRUE}).}
+\item{label}{A character string,
+ text label for the button (only used if \code{button=TRUE}).}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"}, an ID will be generated automatically from the label and component strings.}
+If \code{"auto"},
+ an ID will be generated automatically from the label and component strings.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.formula.Rd b/packages/rkwarddev/man/rk.XML.formula.Rd
index cd5e024..ae6df7a 100644
--- a/packages/rkwarddev/man/rk.XML.formula.Rd
+++ b/packages/rkwarddev/man/rk.XML.formula.Rd
@@ -12,7 +12,8 @@ rk.XML.formula(fixed, dependent, id.name = "auto")
\item{dependent}{The \code{id} of the varslot holding the selected dependent variable.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"}, an ID will be generated automatically from the \code{fixed} and \code{dependent} value.}
+If \code{"auto"},
+ an ID will be generated automatically from the \code{fixed} and \code{dependent} value.}
}
\value{
An object of class \code{XiMpLe.node}.
diff --git a/packages/rkwarddev/man/rk.XML.frame.Rd b/packages/rkwarddev/man/rk.XML.frame.Rd
index 43d1cc3..4dda697 100644
--- a/packages/rkwarddev/man/rk.XML.frame.Rd
+++ b/packages/rkwarddev/man/rk.XML.frame.Rd
@@ -14,15 +14,18 @@ rk.XML.frame(..., label = NULL, checkable = FALSE, chk = TRUE,
\item{checkable}{Logical, if \code{TRUE} the frame can be switched on and off.}
-\item{chk}{Logical, if \code{TRUE} and \code{checkable=TRUE} the frame is checkable and active by default.}
+\item{chk}{Logical,
+ if \code{TRUE} and \code{checkable=TRUE} the frame is checkable and active by default.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label
if presen, otherwise from the objects in the frame.
If \code{NULL}, no ID will be given.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.input.Rd b/packages/rkwarddev/man/rk.XML.input.Rd
index 85af53a..4b61d1a 100644
--- a/packages/rkwarddev/man/rk.XML.input.Rd
+++ b/packages/rkwarddev/man/rk.XML.input.Rd
@@ -10,7 +10,8 @@ rk.XML.input(label, initial = NULL, size = "medium", required = FALSE,
\arguments{
\item{label}{Character string, a text label for this plugin element.}
-\item{initial}{Character string, if not \code{NULL} will be used as the initial value of the input field.}
+\item{initial}{Character string,
+ if not \code{NULL} will be used as the initial value of the input field.}
\item{size}{One value of either "small", "medium" or "large".}
@@ -19,16 +20,20 @@ rk.XML.input(label, initial = NULL, size = "medium", required = FALSE,
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.logic.Rd b/packages/rkwarddev/man/rk.XML.logic.Rd
index 692a061..667e007 100644
--- a/packages/rkwarddev/man/rk.XML.logic.Rd
+++ b/packages/rkwarddev/man/rk.XML.logic.Rd
@@ -13,8 +13,10 @@ rk.XML.logic(...)
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a logic section with "convert", "connect", "include", "insert", "external" and "set" nodes.
-You can also include JavaScript code to use the locig scripting features of RKWard, if you place it in a comment
+This function will create a logic section with "convert", "connect", "include", "insert",
+ "external" and "set" nodes.
+You can also include JavaScript code to use the locig scripting features of RKWard,
+ if you place it in a comment
with \code{\link[rkwarddev:rk.comment]{rk.comment}}: Its contents will automatically be placed inside a
\code{<script><![CDATA[ ]]></script>} node.
}
diff --git a/packages/rkwarddev/man/rk.XML.matrix.Rd b/packages/rkwarddev/man/rk.XML.matrix.Rd
index 6ee5268..58a8ca4 100644
--- a/packages/rkwarddev/man/rk.XML.matrix.Rd
+++ b/packages/rkwarddev/man/rk.XML.matrix.Rd
@@ -14,7 +14,8 @@ rk.XML.matrix(label, mode = "real", rows = 2, columns = 2, min = NULL,
\arguments{
\item{label}{Character string, a label for the matrix.}
-\item{mode}{Character string, one of "integer", "real" or "string". The type of data that will
+\item{mode}{Character string, one of "integer",
+ "real" or "string". The type of data that will
be accepted in the table (required)}
\item{rows}{Number of rows in the matrix. Has no effect if \code{allow_user_resize_rows=TRUE}.}
@@ -34,17 +35,22 @@ largest representable value.}
\item{allow_missings}{Logical, whether missing (empty) values are allowed in the matrix
(if \code{type} is "string").}
-\item{allow_user_resize_columns}{Logical, if \code{TRUE}, the user can add columns by typing
+\item{allow_user_resize_columns}{Logical, if \code{TRUE},
+ the user can add columns by typing
on the rightmost (inactive) cells.}
-\item{allow_user_resize_rows}{Logical, if \code{TRUE}, the user can add rows by typing on the
+\item{allow_user_resize_rows}{Logical, if \code{TRUE},
+ the user can add rows by typing on the
bottommost (inactive) cells.}
-\item{fixed_width}{Logical, assume the column count will not change. The last (or typically only)
+\item{fixed_width}{Logical,
+ assume the column count will not change. The last (or typically only)
column will be stretched to take up the available width. Do not use in combination with matrices,
-where the number of columns may change in any way. Useful, esp. when creating a vector input element (rows="1").}
+where the number of columns may change in any way. Useful,
+ esp. when creating a vector input element (rows="1").}
-\item{fixed_height}{Logical, force the GUI element to stay at its initial height. Do not use in
+\item{fixed_height}{Logical,
+ force the GUI element to stay at its initial height. Do not use in
combindation with matrices, where the number of rows may change in any way.
Useful, esp. when creating a vector input element (columns="1").}
@@ -55,16 +61,20 @@ Useful, esp. when creating a vector input element (columns="1").}
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
index 6dd7512..a9725ea 100644
@@ -14,16 +14,19 @@ rk.XML.menu(label, ..., index = -1, id.name = "auto", i18n = NULL)
with the last element being the \code{component} value for \code{\link[rkwarddev:rk.XML.entry]{rk.XML.entry}}.}
\item{index}{Integer number to influence the level of menu placement. If \code{...} is a list,
-\code{index} can also be a vector of the same length + 1, so indices will be set in the same order to the
+\code{index} can also be a vector of the same length + 1,
+ so indices will be set in the same order to the
menu levels, the last value is for the entry.}
\item{id.name}{Character, a unique ID for this plugin element.
-If \code{"auto"}, an ID will be generated automatically from the label. Otherwise, if \code{...} is a list,
+If \code{"auto"}, an ID will be generated automatically from the label. Otherwise,
+ if \code{...} is a list,
\code{id.name} must have the same length and will be set in the same order to the menu levels.
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.option.Rd b/packages/rkwarddev/man/rk.XML.option.Rd
index 62c54ce..41f7548 100644
--- a/packages/rkwarddev/man/rk.XML.option.Rd
+++ b/packages/rkwarddev/man/rk.XML.option.Rd
@@ -12,13 +12,16 @@ rk.XML.option(label, val = NULL, chk = FALSE, id.name = NULL,
\item{val}{Character string, defines the value to submit if the option is checked.}
-\item{chk}{Logical, should be set \code{TRUE} in the one option which is checked by default.}
+\item{chk}{Logical,
+ should be set \code{TRUE} in the one option which is checked by default.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.optioncolumn.Rd b/packages/rkwarddev/man/rk.XML.optioncolumn.Rd
index a4756b6..4b4a0fc 100644
--- a/packages/rkwarddev/man/rk.XML.optioncolumn.Rd
+++ b/packages/rkwarddev/man/rk.XML.optioncolumn.Rd
@@ -8,25 +8,34 @@ rk.XML.optioncolumn(connect, modifier = NULL, label = TRUE,
external = FALSE, default = NULL, id.name = "auto", i18n = NULL)
}
\arguments{
-\item{connect}{Either a character string (the \code{id} of the property to connect this optioncolumn to), or an object of
-class XiMpLe.node (whose \code{id} will be extracted and used). For external \code{<optioncolumn>}s, the corresponding value will
-be set to the externally set value. For regular (non-external) \code{<optioncolumn>}s, the corresponding row of the \code{<optioncolumn>} property, will be set
+\item{connect}{Either a character string (the \code{id} of the property to connect this optioncolumn to),
+ or an object of
+class XiMpLe.node (whose \code{id} will be extracted and used). For external \code{<optioncolumn>}s,
+ the corresponding value will
+be set to the externally set value. For regular (non-external) \code{<optioncolumn>}s,
+ the corresponding row of the \code{<optioncolumn>} property, will be set
when the property changes inside the content-area.}
-\item{modifier}{Character string, the modifier of the property to connect to, will be appended to the \code{id} of \code{connect}.}
+\item{modifier}{Character string, the modifier of the property to connect to,
+ will be appended to the \code{id} of \code{connect}.}
-\item{label}{Either logical or a character string. If given, the optioncolumn will be displayed in the \code{<optiondisplay>} in a column by that label.
-If set to \code{TRUE} and you provide a XiMpLe node object to \code{connect}, the label will be extracted from that node.}
+\item{label}{Either logical or a character string. If given,
+ the optioncolumn will be displayed in the \code{<optiondisplay>} in a column by that label.
+If set to \code{TRUE} and you provide a XiMpLe node object to \code{connect},
+ the label will be extracted from that node.}
-\item{external}{Logical, set to \code{TRUE} if the optioncolumn is controlled from outside the optionset.}
+\item{external}{Logical,
+ set to \code{TRUE} if the optioncolumn is controlled from outside the optionset.}
-\item{default}{Character string, only for external columns: The value to assume for this column, if no value is known for an entry. Rarely useful.}
+\item{default}{Character string,
+ only for external columns: The value to assume for this column, if no value is known for an entry. Rarely useful.}
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the \code{connect} object.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.optionset.Rd b/packages/rkwarddev/man/rk.XML.optionset.Rd
index 315d56a..27bcae3 100644
--- a/packages/rkwarddev/man/rk.XML.optionset.Rd
+++ b/packages/rkwarddev/man/rk.XML.optionset.Rd
@@ -13,22 +13,28 @@ rk.XML.optionset(content, optioncolumn, min_rows = 0, min_rows_if_any = 0,
\item{optioncolumn}{A list of \code{<optioncolumn>} XiMpLe.nodes.}
-\item{min_rows}{Numeric (integer), if specified, the set will be marked invalid, unless it has
+\item{min_rows}{Numeric (integer), if specified, the set will be marked invalid,
+ unless it has
at least this number of rows. Ignored if set to 0.}
-\item{min_rows_if_any}{Numeric (integer), like min_rows, but will only be tested, if there is at
+\item{min_rows_if_any}{Numeric (integer), like min_rows, but will only be tested,
+ if there is at
least one row. Ignored if set to 0.}
-\item{max_rows}{Numeric (integer), if specified, the set will be marked invalid, unless it has
+\item{max_rows}{Numeric (integer), if specified, the set will be marked invalid,
+ unless it has
at most this number of rows. Ignored if set to 0.}
\item{keycolumn}{Character}
\item{logic}{A valid \code{<logic>} node.}
-\item{optiondisplay}{Logical value, can be used to automatically add an \code{<optiondisplay>} node on top
-of the \code{<content>} section. Depending on whether it's \code{TRUE} or \code{FALSE}, its \code{index}
-argument will be set to \code{"true"} or \code{"false"}, respectively. Set to \code{NULL} to deactivate.}
+\item{optiondisplay}{Logical value,
+ can be used to automatically add an \code{<optiondisplay>} node on top
+of the \code{<content>} section. Depending on whether it's \code{TRUE} or \code{FALSE},
+ its \code{index}
+argument will be set to \code{"true"} or \code{"false"},
+ respectively. Set to \code{NULL} to deactivate.}
\item{id.name}{Character string, a unique ID for this plugin element.
If \code{"auto"}, an ID will be generated automatically from the <content> nodes.}
@@ -40,13 +46,16 @@ An object of class \code{XiMpLe.node}.
Note that if you want to refer to the optioncolumns in your JavaScript code, the \code{id}
you need is a combination of \code{<optionset id>.<optioncolumn id>.<modifier>}. that is,
you must always prefix it with the sets' \code{id}. For JavaScript code generating with
-\code{rkwarddev}, the easiest way to get to results is to use \code{\link[rkwarddev:rk.JS.optionset]{rk.JS.optionset}}.
+\code{rkwarddev},
+ the easiest way to get to results is to use \code{\link[rkwarddev:rk.JS.optionset]{rk.JS.optionset}}.
It will automatically place your code fragments into a for loop and iterate through all available
rows of the set.
}
\details{
-If this isn't flexible enough for your needs, you can also use the ID that functions like \code{\link[rkwarddev:id]{id}}
-return, because the JavaScript variable name will only contain a constant prefix ("ocol") and the column ID.
+If this isn't flexible enough for your needs,
+ you can also use the ID that functions like \code{\link[rkwarddev:id]{id}}
+return,
+ because the JavaScript variable name will only contain a constant prefix ("ocol") and the column ID.
}
\note{
The \code{<optionset>} node was introduced with RKWard 0.6.1, please set the dependencies
diff --git a/packages/rkwarddev/man/rk.XML.page.Rd b/packages/rkwarddev/man/rk.XML.page.Rd
index 2fc2a26..e7fe824 100644
--- a/packages/rkwarddev/man/rk.XML.page.Rd
+++ b/packages/rkwarddev/man/rk.XML.page.Rd
@@ -17,9 +17,12 @@ If \code{NULL}, no ID will be given.}
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a page node for wizard sections, with optional child nodes "browser", "checkbox",
-"column", "copy", "dropdown", "formula", "frame", "input", "page", "radio", "row", "saveobject",
-"select", "spinbox", "stretch", "tabbook", "text", "valueselector", "valueslot", "varselector" and "varslot".
+This function will create a page node for wizard sections,
+ with optional child nodes "browser", "checkbox",
+"column", "copy", "dropdown", "formula", "frame", "input", "page", "radio", "row",
+ "saveobject",
+"select", "spinbox", "stretch", "tabbook", "text", "valueselector", "valueslot",
+ "varselector" and "varslot".
}
\examples{
# define a checkbox for the actual dialog
diff --git a/packages/rkwarddev/man/rk.XML.plugin.Rd b/packages/rkwarddev/man/rk.XML.plugin.Rd
index d8e59ee..c88b6e8 100644
--- a/packages/rkwarddev/man/rk.XML.plugin.Rd
+++ b/packages/rkwarddev/man/rk.XML.plugin.Rd
@@ -10,7 +10,8 @@ rk.XML.plugin(name, dialog = NULL, wizard = NULL, logic = NULL,
gen.info = TRUE, i18n = NULL)
}
\arguments{
-\item{name}{Character string, the name of the plugin. Will be used for the names of the JavaScript and
+\item{name}{Character string,
+ the name of the plugin. Will be used for the names of the JavaScript and
help files to be included, following the scheme \emph{"<name>.<ext>"}.}
\item{dialog}{An object of class \code{XiMpLe.node} to be pasted as the \code{<dialog>} section. See
@@ -25,18 +26,24 @@ help files to be included, following the scheme \emph{"<name>.<ext>"}.}
\item{snippets}{An object of class \code{XiMpLe.node} to be pasted as the \code{<snippets>} section. See
-\item{provides}{Character vector with possible entries of \code{"logic"}, \code{"dialog"} or \code{"wizard"}, defining what
-sections the document should provide even if \code{dialog}, \code{wizard} and \code{logic} are \code{NULL}.
+\item{provides}{Character vector with possible entries of \code{"logic"},
+ \code{"dialog"} or \code{"wizard"}, defining what
+sections the document should provide even if \code{dialog},
+ \code{wizard} and \code{logic} are \code{NULL}.
These sections must be edited manually and some parts are therefore commented out.}
-\item{help}{Logical, if \code{TRUE} an include tag for a help file named \emph{"<name>.rkh"} will be added to the header.}
+\item{help}{Logical,
+ if \code{TRUE} an include tag for a help file named \emph{"<name>.rkh"} will be added to the header.}
-\item{include}{Character string or vector, relative path(s) to other file(s), which will then be included in the head of the GUI XML document.}
+\item{include}{Character string or vector, relative path(s) to other file(s),
+ which will then be included in the head of the GUI XML document.}
-\item{label}{Character string, a text label for the plugin's top level, i.e. the window title of the dialog.
+\item{label}{Character string, a text label for the plugin's top level,
+ i.e. the window title of the dialog.
Will only be used if \code{dialog} or \code{wizard} are \code{NULL}.}
-\item{clean.name}{Logical, if \code{TRUE}, all non-alphanumeric characters except the underscore (\code{"_"}) will be removed from \code{name}.}
+\item{clean.name}{Logical, if \code{TRUE},
+ all non-alphanumeric characters except the underscore (\code{"_"}) will be removed from \code{name}.}
\item{about}{An object of class \code{XiMpLe.node} with descriptive information on the plugin and its authors,
@@ -52,7 +59,8 @@ You can also provide a character string naming the very rkwarddev script file th
which will then also be added to the comment.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.pluginmap.Rd b/packages/rkwarddev/man/rk.XML.pluginmap.Rd
index e6023f2..6e1c643 100644
--- a/packages/rkwarddev/man/rk.XML.pluginmap.Rd
+++ b/packages/rkwarddev/man/rk.XML.pluginmap.Rd
\item{components}{Either an object of class \code{XiMpLe.node} to be pasted as the \code{<components>} section (see
\code{\link[rkwarddev:rk.XML.components]{rk.XML.components}} for details). Or a character vector with at least
-one plugin component file name, relative path from the pluginmap file and ending with ".xml". Can be set to \code{NULL} if
+one plugin component file name,
+ relative path from the pluginmap file and ending with ".xml". Can be set to \code{NULL} if
\code{require} is used accordingly.}
\item{hierarchy}{Either an object of class \code{XiMpLe.node} to be pasted as the \code{<hierarchy>} section (see
\code{\link[rkwarddev:rk.XML.hierarchy]{rk.XML.hierarchy}} for details). Or a character vector with instructions
-where to place the plugin in the menu hierarchy, one list or string for each included component. Valid single values are
-\code{"file"}, \code{"edit"}, \code{"view"}, \code{"workspace"}, \code{"run"}, \code{"data"},
-\code{"analysis"}, \code{"plots"}, \code{"distributions"}, \code{"windows"}, \code{"settings"} and \code{"help"},
-anything else will place it in a "test" menu. If \code{hierarchy} is a list, each entry represents the label of a menu level.
+where to place the plugin in the menu hierarchy,
+ one list or string for each included component. Valid single values are
+\code{"file"}, \code{"edit"}, \code{"view"}, \code{"workspace"}, \code{"run"},
+ \code{"data"},
+\code{"analysis"}, \code{"plots"}, \code{"distributions"}, \code{"windows"},
+ \code{"settings"} and \code{"help"},
+anything else will place it in a "test" menu. If \code{hierarchy} is a list,
+ each entry represents the label of a menu level.
Can be set to \code{NULL} if \code{require} is used accordingly.}
\item{require}{Either a (list of) objects of class \code{XiMpLe.node} to be pasted as a \code{<require>} section (see
\code{\link[rkwarddev:rk.XML.require]{rk.XML.require}} for details). Or a character vector with at least
one .pluginmap filename to be included in this one.}
-\item{x11.context}{An object of class \code{XiMpLe.node} to be pasted as a \code{<context id="x11">} section, see
+\item{x11.context}{An object of class \code{XiMpLe.node} to be pasted as a \code{<context id="x11">} section,
+ see
-\item{import.context}{An object of class \code{XiMpLe.node} to be pasted as the \code{<context id="import">} section, see
+\item{import.context}{An object of class \code{XiMpLe.node} to be pasted as the \code{<context id="import">} section,
+ see
-\item{clean.name}{Logical, if \code{TRUE}, all non-alphanumeric characters except the underscore (\code{"_"}) will be removed from \code{name}.}
+\item{clean.name}{Logical, if \code{TRUE},
+ all non-alphanumeric characters except the underscore (\code{"_"}) will be removed from \code{name}.}
-\item{hints}{Logical, if \code{TRUE} and you leave out optional entries (like \code{about=NULL}), dummy sections will be added as comments.}
+\item{hints}{Logical,
+ if \code{TRUE} and you leave out optional entries (like \code{about=NULL}), dummy sections will be added as comments.}
\item{gen.info}{Logical, if \code{TRUE} a comment note will be written into the document,
that it was generated by \code{rkwarddev} and changes should be done to the script.
@@ -51,17 +60,22 @@ which will then also be added to the comment.}
\item{dependencies}{An object of class \code{XiMpLe.node} to be pasted as the \code{<dependencies>} section,
See \code{\link[rkwarddev:rk.XML.dependencies]{rk.XML.dependencies}} for details. Skipped if \code{NULL}.}
-\item{namespace}{Character string, the namespace attribute of the \code{<document>} node, defaults to the plugin name (which you probably shouldn't touch...).
+\item{namespace}{Character string, the namespace attribute of the \code{<document>} node,
+ defaults to the plugin name (which you probably shouldn't touch...).
RKWard's internal plugins should use the namespace \code{rkward}. This is taken care of by \code{\link[rkwarddev:rk.plugin.skeleton]{rk.plugin.skeleton}}
if you set \code{internal=TRUE}.}
-\item{priority}{Character string, the priority attribute of the \code{<document>} node. Must be either "hidden", "low", "medium", or "high",
+\item{priority}{Character string,
+ the priority attribute of the \code{<document>} node. Must be either "hidden", "low", "medium", or "high",
defaults to "medium".}
-\item{id.name}{Character string, a unique ID for this plugin element. If \code{"auto"}, an ID will be generated automatically from \code{name}.}
+\item{id.name}{Character string, a unique ID for this plugin element. If \code{"auto"},
+ an ID will be generated automatically from \code{name}.}
-\item{require.defaults}{Logical, if \code{TRUE}, \code{<require map="rkward::menu" />} and \code{<require map="rkward::embedded" />} will be added
-by default, which ensures that the menu structure and embeddable plugins are loaded. It shouldn't hurt to set this.}
+\item{require.defaults}{Logical, if \code{TRUE},
+by default,
+ which ensures that the menu structure and embeddable plugins are loaded. It shouldn't hurt to set this.}
}
\value{
An object of class \code{XiMpLe.node}.
diff --git a/packages/rkwarddev/man/rk.XML.preview.Rd b/packages/rkwarddev/man/rk.XML.preview.Rd
index 8759b9e..8522789 100644
--- a/packages/rkwarddev/man/rk.XML.preview.Rd
+++ b/packages/rkwarddev/man/rk.XML.preview.Rd
@@ -10,7 +10,8 @@ rk.XML.preview(label = "Preview", i18n = NULL)
\item{label}{A character string, text label for the preview checkbox.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
index 5446f7f..b34aa3f 100644
@@ -12,23 +12,28 @@ rk.XML.radio(label, options = list(label = c(val = NULL, chk = FALSE, i18n =
\item{label}{Character string, a text label for this plugin element.}
\item{options}{A named list with options to choose from. The names of the list elements will become
-labels of the options, \code{val} defines the value to submit if the option is checked, and
+labels of the options, \code{val} defines the value to submit if the option is checked,
+ and
\code{chk=TRUE} should be set in the one option which is checked by default. You might also provide an \code{i18n}
for this particular option (see \code{i18n}). Objects generated with \code{\link[rkwarddev:rk.XML.option]{rk.XML.option}}
are accepted as well.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting
node in the .rkh file. Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.require.Rd b/packages/rkwarddev/man/rk.XML.require.Rd
index 1542633..bef8b5c 100644
--- a/packages/rkwarddev/man/rk.XML.require.Rd
+++ b/packages/rkwarddev/man/rk.XML.require.Rd
@@ -7,10 +7,12 @@
rk.XML.require(file = NULL, map = NULL)
}
\arguments{
-\item{file}{Character string, file name of another .pluginmap file to be included. Should be
+\item{file}{Character string,
+ file name of another .pluginmap file to be included. Should be
preferred over \code{map} if that file is in the same package.}
-\item{map}{Character string, should be \code{"namespace::id"} of another .pluginmap to be included.
+\item{map}{Character string,
+ should be \code{"namespace::id"} of another .pluginmap to be included.
Can be used to address plugin maps which are not part of the same plugin package.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.saveobj.Rd b/packages/rkwarddev/man/rk.XML.saveobj.Rd
index 0000f80..2ce01c6 100644
--- a/packages/rkwarddev/man/rk.XML.saveobj.Rd
+++ b/packages/rkwarddev/man/rk.XML.saveobj.Rd
@@ -11,28 +11,35 @@ rk.XML.saveobj(label, chk = FALSE, checkable = TRUE, initial = "auto",
\arguments{
\item{label}{Character string, a text label for this plugin element.}
-\item{chk}{Logical, if \code{TRUE} and \code{checkable=TRUE} the option is checkable and active by default.}
+\item{chk}{Logical,
+ if \code{TRUE} and \code{checkable=TRUE} the option is checkable and active by default.}
\item{checkable}{Logical, if \code{TRUE} the option can be switched on and off.}
\item{initial}{Character string, the default name for the object should be saved to.
-If \code{"auto"} and a label was provided, an name will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an name will be generated automatically from the label.}
\item{required}{Logical, whether an entry is mandatory or not.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.select.Rd b/packages/rkwarddev/man/rk.XML.select.Rd
index baf32a7..427240d 100644
--- a/packages/rkwarddev/man/rk.XML.select.Rd
+++ b/packages/rkwarddev/man/rk.XML.select.Rd
@@ -12,24 +12,30 @@ rk.XML.select(label, options = list(label = c(val = "", chk = FALSE, i18n =
\item{label}{Character string, a text label for this plugin element.}
\item{options}{A named list with options to choose from. The names of the list elements will become
-labels of the options, \code{val} defines the value to submit if the option is selected, and
+labels of the options, \code{val} defines the value to submit if the option is selected,
+ and
\code{chk=TRUE} should be set in the one option which is selected by default. You might also provide an \code{i18n}
for this particular option (see \code{i18n}). Objects generated with \code{\link[rkwarddev:rk.XML.option]{rk.XML.option}}
are accepted as well.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.set.Rd b/packages/rkwarddev/man/rk.XML.set.Rd
index 82f259c..ea190f8 100644
--- a/packages/rkwarddev/man/rk.XML.set.Rd
+++ b/packages/rkwarddev/man/rk.XML.set.Rd
@@ -14,7 +14,8 @@ or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and
\item{to}{Character string or logical, the value the property should be set to.}
-\item{check.modifiers}{Logical, if \code{TRUE} the given modifiers will be checked for validity. Should only be
+\item{check.modifiers}{Logical,
+ if \code{TRUE} the given modifiers will be checked for validity. Should only be
turned off if you know what you're doing.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.snippets.Rd b/packages/rkwarddev/man/rk.XML.snippets.Rd
index 30e3c25..3af5db7 100644
--- a/packages/rkwarddev/man/rk.XML.snippets.Rd
+++ b/packages/rkwarddev/man/rk.XML.snippets.Rd
@@ -13,7 +13,8 @@ rk.XML.snippets(...)
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a snippets node for the document section, with optional child nodes
+This function will create a snippets node for the document section,
+ with optional child nodes
\code{<snippet>} and \code{<include>}.
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.spinbox.Rd b/packages/rkwarddev/man/rk.XML.spinbox.Rd
index da21d03..a9d3d81 100644
--- a/packages/rkwarddev/man/rk.XML.spinbox.Rd
+++ b/packages/rkwarddev/man/rk.XML.spinbox.Rd
@@ -11,31 +11,40 @@ rk.XML.spinbox(label, min = NULL, max = NULL, initial = 0, real = TRUE,
\arguments{
\item{label}{Character string, a text label for this plugin element.}
-\item{min}{Numeric, the lowest value allowed. Defaults to the lowest value technically representable in the spinbox.}
+\item{min}{Numeric,
+ the lowest value allowed. Defaults to the lowest value technically representable in the spinbox.}
-\item{max}{Numeric, the largest value allowed. Defaults to the highest value technically representable in the spinbox.}
+\item{max}{Numeric,
+ the largest value allowed. Defaults to the highest value technically representable in the spinbox.}
\item{initial}{Numeric, will be used as the initial value.}
\item{real}{Logical, whether values should be real or integer numbers.}
-\item{precision}{Numeric, if \code{real=TRUE} defines the default number of decimal places shown in the spinbox.}
+\item{precision}{Numeric,
+ if \code{real=TRUE} defines the default number of decimal places shown in the spinbox.}
-\item{max.precision}{Numeric, maximum number of digits that can be meaningfully represented.}
+\item{max.precision}{Numeric,
+ maximum number of digits that can be meaningfully represented.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"} and a label was provided, an ID will be generated automatically from the label.}
+If \code{"auto"} and a label was provided,
+ an ID will be generated automatically from the label.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.switch.Rd b/packages/rkwarddev/man/rk.XML.switch.Rd
index 0a5b85b..15a611d 100644
--- a/packages/rkwarddev/man/rk.XML.switch.Rd
+++ b/packages/rkwarddev/man/rk.XML.switch.Rd
@@ -18,7 +18,8 @@ as you need, setting a return value for each \code{condition == case} respective
Each list must contain either a \code{fixed_value} or a \code{dynamic_value} element.
In addition, each \code{case} list must also have one \code{standard} element.}
-\item{modifier}{Character string, an optional modifier to be appended to \code{condition}.}
+\item{modifier}{Character string,
+ an optional modifier to be appended to \code{condition}.}
\item{id.name}{Character string, a unique ID for this property.
If \code{"auto"}, IDs will be generated automatically from the condition ID.}
diff --git a/packages/rkwarddev/man/rk.XML.tabbook.Rd b/packages/rkwarddev/man/rk.XML.tabbook.Rd
index ee9a6cd..348a87b 100644
--- a/packages/rkwarddev/man/rk.XML.tabbook.Rd
+++ b/packages/rkwarddev/man/rk.XML.tabbook.Rd
@@ -18,7 +18,8 @@ If \code{"auto"}, IDs will be generated automatically from the labels.
If \code{NULL}, no IDs will be given.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
@@ -28,7 +29,8 @@ An object of class \code{XiMpLe.node}.
Create XML node "tabbook" for RKWard plugins
}
\note{
-If a node in \code{tabs} is \code{<insert>}, it is returned as-is, without being nested in \code{<tab>}.
+If a node in \code{tabs} is \code{<insert>}, it is returned as-is,
+ without being nested in \code{<tab>}.
}
\examples{
test.checkboxes <- rk.XML.row(rk.XML.col(
diff --git a/packages/rkwarddev/man/rk.XML.values.Rd b/packages/rkwarddev/man/rk.XML.values.Rd
index 0aa9840..3ef4a51 100644
--- a/packages/rkwarddev/man/rk.XML.values.Rd
+++ b/packages/rkwarddev/man/rk.XML.values.Rd
@@ -16,7 +16,8 @@ rk.XML.values(label, slot.text, options = list(label = c(val = NULL, chk =
\item{slot.text}{Character string, a text label for the value selection slot.}
\item{options}{A named list with string values to choose from. The names of the list elements will become
-labels of the options, \code{val} defines the value to submit if the value is selected, and
+labels of the options, \code{val} defines the value to submit if the value is selected,
+ and
\code{chk=TRUE} should be set in the one option which is checked by default. You might also provide an \code{i18n}
for this particular option (see \code{i18n}). Objects generated with \code{\link[rkwarddev:rk.XML.option]{rk.XML.option}}
are accepted as well.}
@@ -25,7 +26,8 @@ are accepted as well.}
\item{multi}{Logical, whether the valueslot holds only one or several objects.}
-\item{duplicates}{Logical, if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
+\item{duplicates}{Logical,
+ if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
\item{min}{If \code{multi=TRUE} defines how many objects must be selected.}
@@ -42,16 +44,22 @@ if \code{FALSE} below it.}
\item{frame.label}{Character string, a text label for the whole frame.}
-\item{id.name}{Character vector, unique IDs for the frame (first entry), the valueselector (second entry)
-and valueslot (third entry). If \code{formula.dependent} is not \code{NULL}, a fourth and fifth entry is needed as well,
+\item{id.name}{Character vector, unique IDs for the frame (first entry),
+ the valueselector (second entry)
+and valueslot (third entry). If \code{formula.dependent} is not \code{NULL},
+ a fourth and fifth entry is needed as well,
for the dependent valueslot and the formula node, respectively.
-If \code{"auto"}, IDs will be generated automatically from \code{label} and \code{slot.text}.}
+If \code{"auto"},
+ IDs will be generated automatically from \code{label} and \code{slot.text}.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
}
@@ -61,7 +69,8 @@ An object of class \code{XiMpLe.node}.
\description{
This function will create a <frame> node including a <valueselector> and a <valueslot> node. It is
actually a wrapper for \code{\link[rkwarddev:rk.XML.valueslot]{rk.XML.valueslot}} and
-\code{\link[rkwarddev:rk.XML.valueselector]{rk.XML.valueselector}}, since you usually won't define one
+ since you usually won't define one
without the other.
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.valueselector.Rd b/packages/rkwarddev/man/rk.XML.valueselector.Rd
index 24626a8..f8033e5 100644
--- a/packages/rkwarddev/man/rk.XML.valueselector.Rd
+++ b/packages/rkwarddev/man/rk.XML.valueselector.Rd
@@ -12,7 +12,8 @@ rk.XML.valueselector(label = NULL, options = list(label = c(val = NULL, chk
Must be set if \code{id.name="auto"}.}
\item{options}{A named list with string values to choose from. The names of the list elements will become
-labels of the options, \code{val} defines the value to submit if the value is selected, and
+labels of the options, \code{val} defines the value to submit if the value is selected,
+ and
\code{chk=TRUE} should be set in the one option which is checked by default. You might also provide an \code{i18n}
for this particular option (see \code{i18n}). Objects generated with \code{\link[rkwarddev:rk.XML.option]{rk.XML.option}}
are accepted as well.}
@@ -20,7 +21,8 @@ are accepted as well.}
\item{id.name}{Character vector, unique ID for this element.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.valueslot.Rd b/packages/rkwarddev/man/rk.XML.valueslot.Rd
index 2998572..fd615a4 100644
--- a/packages/rkwarddev/man/rk.XML.valueslot.Rd
+++ b/packages/rkwarddev/man/rk.XML.valueslot.Rd
@@ -12,10 +12,12 @@ rk.XML.valueslot(label, source, property = NULL, required = FALSE,
\item{label}{Character string, a text label for the valueslot.}
\item{source}{Either a character string (the \code{id} name of the \code{valueselector} to select values
-from), or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used). If it is not
+from),
+ or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used). If it is not
a \code{<valueselector>} node, you must also specify a valid property for the node.}
-\item{property}{Character string, valid property for a XiMpLe node defined by \code{source}. In the XML code, it
+\item{property}{Character string,
+ valid property for a XiMpLe node defined by \code{source}. In the XML code, it
will cause the use of \code{source_property} instead of \code{source}. Only used if \code{source} is not a
\code{<valueselector>} node.}
@@ -23,7 +25,8 @@ will cause the use of \code{source_property} instead of \code{source}. Only used
\item{multi}{Logical, whether the valueslot holds only one or several objects.}
-\item{duplicates}{Logical, if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
+\item{duplicates}{Logical,
+ if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
\item{min}{If \code{multi=TRUE} defines how many objects must be selected. Sets \code{multi=TRUE}.}
@@ -36,16 +39,20 @@ are selected at all. Sets \code{multi=TRUE}.}
\item{id.name}{Character vector, unique ID for the valueslot.
If \code{"auto"}, the ID will be generated automatically from \code{label}.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.vars.Rd b/packages/rkwarddev/man/rk.XML.vars.Rd
index 0673642..43182b7 100644
--- a/packages/rkwarddev/man/rk.XML.vars.Rd
+++ b/packages/rkwarddev/man/rk.XML.vars.Rd
@@ -20,7 +20,8 @@ rk.XML.vars(label, slot.text, required = FALSE, multi = FALSE,
\item{multi}{Logical, whether the varslot holds only one or several objects.}
-\item{duplicates}{Logical, if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
+\item{duplicates}{Logical,
+ if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
\item{min}{If \code{multi=TRUE} defines how many objects must be selected.}
@@ -35,14 +36,19 @@ of dimensions is acceptable.}
\item{min.len}{The minimum length, an object needs to have.}
-\item{max.len}{The maximum length, an object needs to have. If \code{NULL}, defaults to the largest
+\item{max.len}{The maximum length, an object needs to have. If \code{NULL},
+ defaults to the largest
integer number representable on the system.}
-\item{classes}{An optional character vector, defining class names to which the selection must be limited.}
+\item{classes}{An optional character vector,
+ defining class names to which the selection must be limited.}
-\item{types}{If you specify one or more variables types here, the varslot will only accept objects of those
-types. Valid types are "unknown", "number", "string", "factor", "invalid". Optional, use with great care,
-the user should not be prevented from making valid choices, and rkward does not always know the type
+\item{types}{If you specify one or more variables types here,
+ the varslot will only accept objects of those
+types. Valid types are "unknown", "number", "string", "factor", "invalid". Optional,
+ use with great care,
+the user should not be prevented from making valid choices,
+ and rkward does not always know the type
of a variable!}
\item{horiz}{Logical. If \code{TRUE}, the varslot will be placed next to the selector,
@@ -52,26 +58,38 @@ if \code{FALSE} below it.}
\item{frame.label}{Character string, a text label for the whole frame.}
-\item{formula.dependent}{Character string, if not \code{NULL} will cause the addition of a second
-varslot for the dependent variable(s), using the text of \code{formula.dependent} as its label. Also
-a \code{<formula>} node will be added, using both varslots for \code{fixed_factors} and \code{dependent}
+\item{formula.dependent}{Character string,
+ if not \code{NULL} will cause the addition of a second
+varslot for the dependent variable(s),
+ using the text of \code{formula.dependent} as its label. Also
+a \code{<formula>} node will be added,
+ using both varslots for \code{fixed_factors} and \code{dependent}
respectively.}
-\item{dep.options}{A named list with optional attributes for the \code{dependent} varslot, if \code{formula.dependent}
-is not \code{NULL}. Valid options are \code{required}, \code{multi}, \code{min}, \code{any}, \code{max},
-\code{dim}, \code{min.len}, \code{max.len}, \code{classes} and \code{types}. If an options is undefined, it defaults
+\item{dep.options}{A named list with optional attributes for the \code{dependent} varslot,
+ if \code{formula.dependent}
+is not \code{NULL}. Valid options are \code{required}, \code{multi}, \code{min},
+ \code{any}, \code{max},
+\code{dim}, \code{min.len}, \code{max.len},
+ \code{classes} and \code{types}. If an options is undefined, it defaults
to the same values like the main options of this function.}
-\item{id.name}{Character vector, unique IDs for the frame (first entry), the varselector (second entry)
-and varslot (third entry). If \code{formula.dependent} is not \code{NULL}, a fourth and fifth entry is needed as well,
+\item{id.name}{Character vector, unique IDs for the frame (first entry),
+ the varselector (second entry)
+and varslot (third entry). If \code{formula.dependent} is not \code{NULL},
+ a fourth and fifth entry is needed as well,
for the dependent varslot and the formula node, respectively.
-If \code{"auto"}, IDs will be generated automatically from \code{label} and \code{slot.text}.}
+If \code{"auto"},
+ IDs will be generated automatically from \code{label} and \code{slot.text}.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
}
@@ -81,7 +99,8 @@ An object of class \code{XiMpLe.node}.
\description{
This function will create a <frame> node including a <varselector> and a <varslot> node. It is
actually a wrapper for \code{\link[rkwarddev:rk.XML.varslot]{rk.XML.varslot}} and
-\code{\link[rkwarddev:rk.XML.varselector]{rk.XML.varselector}}, since you usually won't define one
+ since you usually won't define one
without the other.
}
\examples{
diff --git a/packages/rkwarddev/man/rk.XML.varselector.Rd b/packages/rkwarddev/man/rk.XML.varselector.Rd
index 91ab562..d7453c2 100644
--- a/packages/rkwarddev/man/rk.XML.varselector.Rd
+++ b/packages/rkwarddev/man/rk.XML.varselector.Rd
@@ -13,7 +13,8 @@ Must be set if \code{id.name="auto"}.}
\item{id.name}{Character vector, unique ID for this element.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.varslot.Rd b/packages/rkwarddev/man/rk.XML.varslot.Rd
index 89e5ac1..ff0b26b 100644
--- a/packages/rkwarddev/man/rk.XML.varslot.Rd
+++ b/packages/rkwarddev/man/rk.XML.varslot.Rd
@@ -13,10 +13,12 @@ rk.XML.varslot(label, source, property = NULL, required = FALSE,
\item{label}{Character string, a text label for the varslot.}
\item{source}{Either a character string (the \code{id} name of the \code{varselector} to select variables
-from), or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used). If it is not
+from),
+ or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used). If it is not
a \code{<valueselector>} node, you must also specify a valid property for the node.}
-\item{property}{Character string, valid property for a XiMpLe node defined by \code{source}. In the XML code, it
+\item{property}{Character string,
+ valid property for a XiMpLe node defined by \code{source}. In the XML code, it
will cause the use of \code{source_property} instead of \code{source}. Only used if \code{source} ist not a
\code{<valueselector>} node.}
@@ -24,7 +26,8 @@ will cause the use of \code{source_property} instead of \code{source}. Only used
\item{multi}{Logical, whether the varslot holds only one or several objects.}
-\item{duplicates}{Logical, if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
+\item{duplicates}{Logical,
+ if \code{multi=TRUE} defines whether the same entry may be added multiple times. Sets \code{multi=TRUE}.}
\item{min}{If \code{multi=TRUE} defines how many objects must be selected. Sets \code{multi=TRUE}.}
@@ -39,29 +42,38 @@ of dimensions is acceptable.}
\item{min.len}{The minimum length, an object needs to have.}
-\item{max.len}{The maximum length, an object needs to have. If \code{NULL}, defaults to the largest
+\item{max.len}{The maximum length, an object needs to have. If \code{NULL},
+ defaults to the largest
integer number representable on the system.}
-\item{classes}{An optional character vector, defining class names to which the selection must be limited.}
+\item{classes}{An optional character vector,
+ defining class names to which the selection must be limited.}
-\item{types}{If you specify one or more variables types here, the varslot will only accept objects of those
-types. Valid types are "unknown", "number", "string", "factor", "invalid". Optional, use with great care,
-the user should not be prevented from making valid choices, and rkward does not always know the type
+\item{types}{If you specify one or more variables types here,
+ the varslot will only accept objects of those
+types. Valid types are "unknown", "number", "string", "factor", "invalid". Optional,
+ use with great care,
+the user should not be prevented from making valid choices,
+ and rkward does not always know the type
of a variable!}
\item{id.name}{Character vector, unique ID for the varslot.
If \code{"auto"}, the ID will be generated automatically from \code{label}.}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.
-If set to \code{FALSE}, \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.
+If set to \code{FALSE},
+ \code{\link[rkwarddev:rk.rkh.scan]{rk.rkh.scan}} will ignore this node.
Also needs \code{component} to be set accordingly!}
-\item{component}{Character string, name of the component this node belongs to. Only needed if you
+\item{component}{Character string,
+ name of the component this node belongs to. Only needed if you
want to use the scan features for automatic help file generation; needs \code{help} to be set
accordingly, too!}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.XML.wizard.Rd b/packages/rkwarddev/man/rk.XML.wizard.Rd
--- a/packages/rkwarddev/man/rk.XML.wizard.Rd
+++ b/packages/rkwarddev/man/rk.XML.wizard.Rd
@@ -11,20 +11,24 @@ rk.XML.wizard(..., label = NULL, recommended = FALSE, i18n = NULL)
\item{label}{Character string, a text label for this plugin element.}
-\item{recommended}{Logical, whether the wizard should be the recommended interface (unless the user has configured
+\item{recommended}{Logical,
+ whether the wizard should be the recommended interface (unless the user has configured
RKWard to default to a specific interface).}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
-or \code{comment}, to give some \code{i18n_context} information for this node. If set to \code{FALSE},
+or \code{comment},
+ to give some \code{i18n_context} information for this node. If set to \code{FALSE},
the attribute \code{label} will be renamed into \code{noi18n_label}.}
}
\value{
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a wizard section with optional child nodes "browser", "checkbox",
+This function will create a wizard section with optional child nodes "browser",
+ "checkbox",
"column", "copy", "dropdown", "embed", "formula", "frame", "include", "input", "insert",
-"page", "preview", "radio", "row", "saveobject", "select", "spinbox", "stretch", "tabbook", "text",
+"page", "preview", "radio", "row", "saveobject", "select", "spinbox", "stretch",
+ "tabbook", "text",
"valueselector", "valueslot", "varselector" and "varslot".
}
\examples{
diff --git a/packages/rkwarddev/man/rk.build.plugin.Rd b/packages/rkwarddev/man/rk.build.plugin.Rd
index b3fd4ec..a71f351 100644
--- a/packages/rkwarddev/man/rk.build.plugin.Rd
+++ b/packages/rkwarddev/man/rk.build.plugin.Rd
@@ -7,15 +7,18 @@
rk.build.plugin(plugin, check = FALSE, install = FALSE, R.libs = NULL)
}
\arguments{
-\item{plugin}{A character string, path to the plugin package root directory (hint: it's the directory with
+\item{plugin}{A character string,
+ path to the plugin package root directory (hint: it's the directory with
the DESCRIPTION file in it).}
-\item{check}{Logical, whether the package should be checked for errors. Always do this before you
+\item{check}{Logical,
+ whether the package should be checked for errors. Always do this before you
publish a package!}
\item{install}{Logical, whether the built package should also be installed locally.}
-\item{R.libs}{A character string, path to local R packages, used by \code{install} to figure
+\item{R.libs}{A character string, path to local R packages,
+ used by \code{install} to figure
out where to install to.}
}
\description{
diff --git a/packages/rkwarddev/man/rk.get.rkh.prompter.Rd b/packages/rkwarddev/man/rk.get.rkh.prompter.Rd
--- a/packages/rkwarddev/man/rk.get.rkh.prompter.Rd
+++ b/packages/rkwarddev/man/rk.get.rkh.prompter.Rd
@@ -7,11 +7,14 @@
rk.get.rkh.prompter(component = NULL, id = NULL)
}
\arguments{
-\item{component}{Character string, the name under which you stored information. If \code{NULL},
+\item{component}{Character string,
+ the name under which you stored information. If \code{NULL},
returns all information stored in the internal \code{rkh.prompter} list.}
-\item{id}{Character string, the node ID if a given component to search for. If \code{NULL}, returns
-the full list of the given component, otherwise only the help information for the requested node.}
+\item{id}{Character string,
+ the node ID if a given component to search for. If \code{NULL}, returns
+the full list of the given component,
+ otherwise only the help information for the requested node.}
}
\description{
Get .rkh related information stored internally
diff --git a/packages/rkwarddev/man/rk.local.Rd b/packages/rkwarddev/man/rk.local.Rd
index c51f85f..aed7b10 100644
--- a/packages/rkwarddev/man/rk.local.Rd
+++ b/packages/rkwarddev/man/rk.local.Rd
@@ -13,9 +13,11 @@ rk.local(...)
The result of evaluating the object(s).
}
\description{
-Can be used like \code{\link[base:local]{local}}, but evaluation is being done in a speacial
+ but evaluation is being done in a speacial
local environment of the rkwarddev package. This can be neccessary if you want to call functions
-nested insinde \code{\link[rkwarddev:js]{js}}, because it might not find all objects if they were
+ because it might not find all objects if they were
only defined in a standard local environment.
}
diff --git a/packages/rkwarddev/man/rk.paste.JS.Rd b/packages/rkwarddev/man/rk.paste.JS.Rd
index 5d84737..bd3cf81 100644
--- a/packages/rkwarddev/man/rk.paste.JS.Rd
+++ b/packages/rkwarddev/man/rk.paste.JS.Rd
@@ -10,7 +10,8 @@ rk.paste.JS(..., level = 2, indent.by = rk.get.indent(), funct = NULL,
opt.sep = NULL)
}
\arguments{
-\item{...}{Objects of class \code{rk.JS.ite}, \code{rk.JS.arr}, \code{rk.JS.opt}, \code{rk.JS.oset} or character.
+\item{...}{Objects of class \code{rk.JS.ite}, \code{rk.JS.arr}, \code{rk.JS.opt},
+ \code{rk.JS.oset} or character.
Another special case is XiMpLe nodes created by \code{rk.comment()}, which will be turned
into JavaScript comments (i.e., lines starting with "//").}
@@ -18,10 +19,13 @@ into JavaScript comments (i.e., lines starting with "//").}
\item{indent.by}{A character string defining the indentation string to use.}
-\item{funct}{For \code{rk.JS.arr} and \code{rk.JS.opt} objects only: Character string, name of the R function
-to be called to combine the options, e.g. "list" for \code{list()}, or "c" for \code{c()}.}
+\item{funct}{For \code{rk.JS.arr} and \code{rk.JS.opt} objects only: Character string,
+ name of the R function
+to be called to combine the options, e.g. "list" for \code{list()},
+ or "c" for \code{c()}.}
-\item{array}{For \code{rk.JS.opt} objects only: Logical, whether the options should be collected
+\item{array}{For \code{rk.JS.opt} objects only: Logical,
+ whether the options should be collected
in an array or a concatenated character string.}
\item{var.prefix}{For \code{rk.JS.var} objects only: A character string. will be used as a prefix
@@ -29,24 +33,34 @@ for the JS variable names.}
\item{modifiers}{For \code{rk.JS.var} objects only: A character vector with modifiers you'd like to apply the XML node's property.}
-\item{default}{For \code{rk.JS.var} objects only: Logical, if \code{TRUE} the default value (no special modifier) of the node will
+\item{default}{For \code{rk.JS.var} objects only: Logical,
+ if \code{TRUE} the default value (no special modifier) of the node will
also be defined. Does nothing if \code{modifiers=NULL}.}
-\item{join}{For \code{rk.JS.var} objects only: A character string, useful for GUI elements which accept multiple objects
-(e.g., multi-varslots). If \code{join} is something other than \code{""}, these objects will be collapsed into one string
+\item{join}{For \code{rk.JS.var} objects only: A character string,
+ useful for GUI elements which accept multiple objects
+(e.g., multi-varslots). If \code{join} is something other than \code{""},
+ these objects will be collapsed into one string
when pasted, joined by this string.}
-\item{getter}{For \code{rk.JS.var} objects only: A character string, naming the JavaScript function which should be used to get the
-values in the actual plugin. Depending on the XML element, \code{"getString"}, \code{"getBool"} or \code{"getList"} can be
-useful alternatives. For backwards compatibility, the default is set to \code{"getValue"}.}
+\item{getter}{For \code{rk.JS.var} objects only: A character string,
+ naming the JavaScript function which should be used to get the
+values in the actual plugin. Depending on the XML element, \code{"getString"},
+ \code{"getBool"} or \code{"getList"} can be
+useful alternatives. For backwards compatibility,
+ the default is set to \code{"getValue"}.}
-\item{var}{For \code{rk.JS.var} objects only: Logical, if \code{FALSE} the variable(s) are assumed to already be defined (globally?)
+\item{var}{For \code{rk.JS.var} objects only: Logical,
+ if \code{FALSE} the variable(s) are assumed to already be defined (globally?)
and the JS keyword "var" will be omitted.}
-\item{empty.e}{For \code{rk.JS.ite} objects only: Logical, if \code{TRUE} will force to add empty \code{else \{\}} brackets when
-there is no \code{else} statement defined, which is considered to enhance code readability by some.}
+\item{empty.e}{For \code{rk.JS.ite} objects only: Logical,
+ if \code{TRUE} will force to add empty \code{else \{\}} brackets when
+there is no \code{else} statement defined,
+ which is considered to enhance code readability by some.}
-\item{opt.sep}{For \code{rk.JS.arr} and \code{rk.JS.opt} objects only: Character string, will be printed in the resulting R code
+\item{opt.sep}{For \code{rk.JS.arr} and \code{rk.JS.opt} objects only: Character string,
+ will be printed in the resulting R code
before the option name.}
}
\value{
diff --git a/packages/rkwarddev/man/rk.paste.JS.graph.Rd b/packages/rkwarddev/man/rk.paste.JS.graph.Rd
index bb6c23f..ee3eb20 100644
--- a/packages/rkwarddev/man/rk.paste.JS.graph.Rd
+++ b/packages/rkwarddev/man/rk.paste.JS.graph.Rd
@@ -10,7 +10,8 @@ rk.paste.JS.graph(..., plotOpts = NULL, printoutObj = NULL, level = 2,
\arguments{
\item{...}{The actual plot code, passed through to \code{rk.paste.JS}.}
-\item{plotOpts}{An object generated by \code{rk.XML.embed} or \code{rk.plotOptions}, i.e. embedded plot options.}
+\item{plotOpts}{An object generated by \code{rk.XML.embed} or \code{rk.plotOptions},
+ i.e. embedded plot options.}
\item{printoutObj}{An \code{rk.JS.var} object fetching the \code{"code.printout"} modifier of \code{plotOpts}
(see examples below!). If \code{NULL} and \code{plotOpts} is of class \code{rk.plot.opts} (as returned by \code{rk.plotOptions}),
@@ -20,21 +21,26 @@ will be fetched from \code{plotOpts} automatically.}
\item{indent.by}{A character string defining the indentation string to use.}
-\item{empty.e}{For \code{rk.JS.ite} objects only: Logical, if \code{TRUE} will force to add empty \code{else \{\}} brackets when
-there is no \code{else} statement defined, which is considered to enhance code readability by some.}
+\item{empty.e}{For \code{rk.JS.ite} objects only: Logical,
+ if \code{TRUE} will force to add empty \code{else \{\}} brackets when
+there is no \code{else} statement defined,
+ which is considered to enhance code readability by some.}
}
\value{
A character string.
}
\description{
-This function is similar to \code{rk.paste.JS}, but adds some code parts to its output which
+This function is similar to \code{rk.paste.JS},
+ but adds some code parts to its output which
are commonly used to generate plots with RKWard.
}
\details{
The contents of the \code{...} argument are evaluated by \code{rk.paste.JS} and encapsulated
between \code{if(full)\{rk.graph.on()\} try(\{} and \code{\}) if(full)\{rk.graph.off()\}}. If generic
-plot options are supplied, their \code{"code.preprocess"} and \code{"code.calculate"} modifiers are
-also automatically taken care of, so you only need to include \code{"code.printout"} inside of
+plot options are supplied,
+ their \code{"code.preprocess"} and \code{"code.calculate"} modifiers are
+also automatically taken care of,
+ so you only need to include \code{"code.printout"} inside of
\code{...}.
}
\examples{
diff --git a/packages/rkwarddev/man/rk.plotOptions.Rd b/packages/rkwarddev/man/rk.plotOptions.Rd
index 6a4c6f8..bcc8684 100644
--- a/packages/rkwarddev/man/rk.plotOptions.Rd
+++ b/packages/rkwarddev/man/rk.plotOptions.Rd
@@ -8,21 +8,26 @@ rk.plotOptions(label = "Generic plot options",
embed = "rkward::plot_options", button = TRUE, id.name = "auto")
}
\arguments{
-\item{label}{A character string, text label for the button (only used if \code{button=TRUE}).}
+\item{label}{A character string,
+ text label for the button (only used if \code{button=TRUE}).}
-\item{embed}{A character string, registered name (\code{id} in pluginmap file) of the plot options component to be embedded.}
+\item{embed}{A character string,
+ registered name (\code{id} in pluginmap file) of the plot options component to be embedded.}
-\item{button}{Logical, whether the plot options should be embedded as a button and appear if it's pressed.}
+\item{button}{Logical,
+ whether the plot options should be embedded as a button and appear if it's pressed.}
\item{id.name}{Character string, a unique ID for this plugin element.
-If \code{"auto"}, an ID will be generated automatically from the label and component strings.}
+If \code{"auto"},
+ an ID will be generated automatically from the label and component strings.}
}
\value{
An object of class \code{rk.plot.opts}.
}
\description{
Generates XML and JavaScript code snippets by calling \code{rk.XML.embed} and \code{rk.JS.vars} with useful presets. The
-resulting object can be used inside the dialog XML object (to place the plot options button and disable certain tabs), as
+resulting object can be used inside the dialog XML object (to place the plot options button and disable certain tabs),
+ as
well as in the JS object (to then insert the actual plot options).
}
\examples{
diff --git a/packages/rkwarddev/man/rk.plugin.component.Rd b/packages/rkwarddev/man/rk.plugin.component.Rd
index 1d5668d..ffd16d0 100644
--- a/packages/rkwarddev/man/rk.plugin.component.Rd
+++ b/packages/rkwarddev/man/rk.plugin.component.Rd
@@ -11,51 +11,73 @@ rk.plugin.component(about, xml = list(), js = list(), rkh = list(),
gen.info = TRUE, indent.by = rk.get.indent())
}
\arguments{
-\item{about}{Either a character string with the name of this plugin component, or an object of class \code{XiMpLe.node}
+\item{about}{Either a character string with the name of this plugin component,
+ or an object of class \code{XiMpLe.node}
+with further descriptive information on it,
for details). This is only useful for information that differs from the \code{<about>} section of the \code{.pluginmap} file.}
-\item{xml}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}}, to generate the GUI XML file.
+\item{xml}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}},
+ to generate the GUI XML file.
Not all options are supported because some don't make sense in this context. Valid options are:
\code{"dialog"}, \code{"wizard"}, \code{"logic"} and \code{"snippets"}.
-If not set, their default values are used. See \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}} for details.}
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}} for details.}
-\item{js}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}}, to generate the JavaScript file.
+\item{js}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}},
+ to generate the JavaScript file.
Not all options are supported because some don't make sense in this context. Valid options are:
+ \code{"globals"}, \code{"preprocess"},
-If not set, their default values are used. See \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}} for details.}
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}} for details.}
-\item{rkh}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}}, to generate the help file.
+\item{rkh}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}},
+ to generate the help file.
Not all options are supported because some don't make sense in this context. Valid options are:
-\code{"summary"}, \code{"usage"}, \code{"sections"}, \code{"settings"}, \code{"related"} and \code{"technical"}.
-If not set, their default values are used. See \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}} for details.}
+\code{"summary"}, \code{"usage"}, \code{"sections"}, \code{"settings"},
+ \code{"related"} and \code{"technical"}.
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}} for details.}
-\item{provides}{Character vector with possible entries of \code{"logic"}, \code{"dialog"} or \code{"wizard"}, defining what
-sections the GUI XML file should provide even if \code{dialog}, \code{wizard} and \code{logic} are \code{NULL}.
+\item{provides}{Character vector with possible entries of \code{"logic"},
+ \code{"dialog"} or \code{"wizard"}, defining what
+sections the GUI XML file should provide even if \code{dialog},
+ \code{wizard} and \code{logic} are \code{NULL}.
These sections must be edited manually and some parts are therefore commented out.}
\item{scan}{A character vector to trigger various automatic scans of the generated GUI XML file. Valid enties are:
\describe{
\item{\code{"var"}}{Calls \code{\link{rk.JS.scan}} to define all needed variables in the \code{calculate()} function
- of the JavaScript file. These variables will be added to variables defined by the \code{js} option, if any (see below).}
+ of the JavaScript file. These variables will be added to variables defined by the \code{js} option,
+ if any (see below).}
\item{\code{"saveobj"}}{Calls \code{\link{rk.JS.saveobj}} to generate code to save R results in the \code{printout()}
- function of the JavaScript file. This code will be added to the code defined by the \code{js} option, if any (see below).}
+ function of the JavaScript file. This code will be added to the code defined by the \code{js} option,
+ if any (see below).}
\item{\code{"settings"}}{Calls \code{\link{rk.rkh.scan}} to generate \code{<setting>} sections for each relevant GUI element in
the \code{<settings>} section of the help file. This option will be overruled if you provide that section manually
by the \code{rkh} option (see below).}
}}
-\item{guess.getter}{Logical, if \code{TRUE} try to get a good default getter function for JavaScript
-variable values (if \code{scan} is active). This will use some functions which were added with RKWard 0.6.1, and therefore
-raise the dependencies for your plugin/component accordingly. Nonetheless, it's recommended.}
+\item{guess.getter}{Logical,
+ if \code{TRUE} try to get a good default getter function for JavaScript
+variable values (if \code{scan} is active). This will use some functions which were added with RKWard 0.6.1,
+ and therefore
+raise the dependencies for your plugin/component accordingly. Nonetheless,
+ it's recommended.}
-\item{hierarchy}{A character vector with instructions where to place this component in the menu hierarchy, one list or string.
-Valid single values are \code{"file"}, \code{"edit"}, \code{"view"}, \code{"workspace"}, \code{"run"}, \code{"data"},
-\code{"analysis"}, \code{"plots"}, \code{"distributions"}, \code{"windows"}, \code{"settings"} and \code{"help"},
-anything else will place it in a "test" menu. If \code{hierarchy} is a list, each entry represents the label of a menu level.}
+\item{hierarchy}{A character vector with instructions where to place this component in the menu hierarchy,
+ one list or string.
+Valid single values are \code{"file"}, \code{"edit"}, \code{"view"}, \code{"workspace"},
+ \code{"run"}, \code{"data"},
+\code{"analysis"}, \code{"plots"}, \code{"distributions"}, \code{"windows"},
+ \code{"settings"} and \code{"help"},
+anything else will place it in a "test" menu. If \code{hierarchy} is a list,
+ each entry represents the label of a menu level.}
-\item{include}{Character string or vector, relative path(s) to other file(s), which will then be included in the head of the GUI XML document.}
+\item{include}{Character string or vector, relative path(s) to other file(s),
+ which will then be included in the head of the GUI XML document.}
\item{create}{A character vector with one or more of these possible entries:
\describe{
@@ -68,9 +90,11 @@ anything else will place it in a "test" menu. If \code{hierarchy} is a list, eac
See \code{\link[rkwarddev:rk.XML.dependencies]{rk.XML.dependencies}} for details. Skipped if \code{NULL}.
This is only useful for information that differs from the \code{<dependencies>} section of the \code{.pluginmap} file.}
-\item{hints}{Logical, if \code{TRUE} and you leave optional entries empty (like \code{rkh=list()}), dummy sections will be added.}
+\item{hints}{Logical,
+ if \code{TRUE} and you leave optional entries empty (like \code{rkh=list()}), dummy sections will be added.}
-\item{gen.info}{Logical, if \code{TRUE} comment notes will be written into the genrated documents,
+\item{gen.info}{Logical,
+ if \code{TRUE} comment notes will be written into the genrated documents,
that they were generated by \code{rkwarddev} and changes should be done to the script.
You can also provide a character string naming the very rkwarddev script file that generates this component,
which will then also be added to the comment.}
diff --git a/packages/rkwarddev/man/rk.plugin.skeleton.Rd b/packages/rkwarddev/man/rk.plugin.skeleton.Rd
index e853e4a..a404e14 100644
--- a/packages/rkwarddev/man/rk.plugin.skeleton.Rd
+++ b/packages/rkwarddev/man/rk.plugin.skeleton.Rd
@@ -15,58 +15,80 @@ rk.plugin.skeleton(about, path = tempdir(), provides = c("logic", "dialog"),
}
\arguments{
\item{about}{Either an object of class \code{XiMpLe.node} with descriptive information on the plugin and its authors
+ or a character string with the name of the plugin package.
If the latter, no \code{DESCRIPTION} file will be created.}
-\item{path}{Character sting, path to the main directory where the skeleton should be created.}
+\item{path}{Character sting,
+ path to the main directory where the skeleton should be created.}
-\item{provides}{Character vector with possible entries of \code{"logic"}, \code{"dialog"} or \code{"wizard"}, defining what
-sections the GUI XML file should provide even if \code{dialog}, \code{wizard} and \code{logic} are \code{NULL}.
+\item{provides}{Character vector with possible entries of \code{"logic"},
+ \code{"dialog"} or \code{"wizard"}, defining what
+sections the GUI XML file should provide even if \code{dialog},
+ \code{wizard} and \code{logic} are \code{NULL}.
These sections must be edited manually and some parts are therefore commented out.}
\item{scan}{A character vector to trigger various automatic scans of the generated GUI XML file. Valid enties are:
\describe{
\item{\code{"var"}}{Calls \code{\link{rk.JS.scan}} to define all needed variables in the \code{calculate()} function
- of the JavaScript file. These variables will be added to variables defined by the \code{js} option, if any (see below).}
+ of the JavaScript file. These variables will be added to variables defined by the \code{js} option,
+ if any (see below).}
\item{\code{"saveobj"}}{Calls \code{\link{rk.JS.saveobj}} to generate code to save R results in the \code{printout()}
- function of the JavaScript file. This code will be added to the code defined by the \code{js} option, if any (see below).}
+ function of the JavaScript file. This code will be added to the code defined by the \code{js} option,
+ if any (see below).}
\item{\code{"settings"}}{Calls \code{\link{rk.rkh.scan}} to generate \code{<setting>} sections for each relevant GUI element in
the \code{<settings>} section of the help file. This option will be overruled if you provide that section manually
by the \code{rkh} option (see below).}
}}
-\item{guess.getter}{Logical, if \code{TRUE} try to get a good default getter function for JavaScript
-variable values (if \code{scan} is active). This will use some functions which were added with RKWard 0.6.1, and therefore
-raise the dependencies for your plugin/component accordingly. Nonetheless, it's recommended.}
+\item{guess.getter}{Logical,
+ if \code{TRUE} try to get a good default getter function for JavaScript
+variable values (if \code{scan} is active). This will use some functions which were added with RKWard 0.6.1,
+ and therefore
+raise the dependencies for your plugin/component accordingly. Nonetheless,
+ it's recommended.}
-\item{xml}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}}, to generate the GUI XML file.
+\item{xml}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}},
+ to generate the GUI XML file.
Not all options are supported because some don't make sense in this context. Valid options are:
\code{"dialog"}, \code{"wizard"}, \code{"logic"} and \code{"snippets"}.
-If not set, their default values are used. See \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}} for details.}
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.XML.plugin]{rk.XML.plugin}} for details.}
-\item{js}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}}, to generate the JavaScript file.
+\item{js}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}},
+ to generate the JavaScript file.
Not all options are supported because some don't make sense in this context. Valid options are:
+ \code{"globals"}, \code{"preprocess"},
-If not set, their default values are used. See \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}} for details.}
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.JS.doc]{rk.JS.doc}} for details.}
-\item{pluginmap}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.pluginmap]{rk.XML.pluginmap}}, to generate the pluginmap file.
+\item{pluginmap}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.XML.pluginmap]{rk.XML.pluginmap}},
+ to generate the pluginmap file.
Not all options are supported because some don't make sense in this context. Valid options are:
-If not set, their default values are used. See \code{\link[rkwarddev:rk.XML.pluginmap]{rk.XML.pluginmap}} for details.}
+ \code{"hierarchy"} and \code{"require"}.
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.XML.pluginmap]{rk.XML.pluginmap}} for details.}
-\item{rkh}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}}, to generate the help file.
+\item{rkh}{A named list of options to be forwarded to \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}},
+ to generate the help file.
Not all options are supported because some don't make sense in this context. Valid options are:
-\code{"summary"}, \code{"usage"}, \code{"sections"}, \code{"settings"}, \code{"related"} and \code{"technical"}.
-If not set, their default values are used. See \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}} for details.}
+\code{"summary"}, \code{"usage"}, \code{"sections"}, \code{"settings"},
+ \code{"related"} and \code{"technical"}.
+If not set,
+ their default values are used. See \code{\link[rkwarddev:rk.rkh.doc]{rk.rkh.doc}} for details.}
-\item{overwrite}{Logical, whether existing files should be replaced. Defaults to \code{FALSE}.}
+\item{overwrite}{Logical,
+ whether existing files should be replaced. Defaults to \code{FALSE}.}
\item{tests}{Logical, whether directories and files for plugin tests should be created.
Defaults to \code{TRUE}. A new testsuite file will only be generated if none is present
(\code{overwrite} is ignored).}
+ whether the package should be prepared for lazy loading or not. Should be left \code{TRUE},
unless you have very good reasons not to.}
\item{create}{A character vector with one or more of these possible entries:
@@ -80,7 +102,8 @@ unless you have very good reasons not to.}
}
Default is to create all of these files. Existing files will only be overwritten if \code{overwrite=TRUE}.}
-\item{suggest.required}{Logical, if \code{TRUE} R package dependencies in \code{about} will be added to the \code{Suggests:}
+\item{suggest.required}{Logical,
+ if \code{TRUE} R package dependencies in \code{about} will be added to the \code{Suggests:}
field of the \code{DESCRIPTION} file, otherwise to the \code{Depends:} field.}
\item{components}{A list of plugin components. See \code{\link[rkwarddev:rk.XML.component]{rk.XML.component}} for details.}
@@ -88,25 +111,32 @@ field of the \code{DESCRIPTION} file, otherwise to the \code{Depends:} field.}
\item{dependencies}{An object of class \code{XiMpLe.node} to be pasted as the \code{<dependencies>} section,
See \code{\link[rkwarddev:rk.XML.dependencies]{rk.XML.dependencies}} for details. Skipped if \code{NULL}.}
-\item{edit}{Logical, if \code{TRUE} RKWard will automatically open the created files for editing, by calling \code{rk.edit.files}.
+\item{edit}{Logical,
+ if \code{TRUE} RKWard will automatically open the created files for editing, by calling \code{rk.edit.files}.
This applies to all files defined in \code{create}.}
-\item{load}{Logical, if \code{TRUE} and \code{"pmap"} in \code{create}, RKWard will automatically add the created .pluginmap file
+\item{load}{Logical, if \code{TRUE} and \code{"pmap"} in \code{create},
+ RKWard will automatically add the created .pluginmap file
to its menu structure by calling \code{rk.load.pluginmaps}. You can then try the plugin immediately.}
-\item{show}{Logical, if \code{TRUE} and \code{"pmap"} in \code{create}, RKWard will automatically call the created plugin after
-it was loaded (i.e., this implies and also sets \code{load=TRUE}). This will only work on the main component, though.}
+\item{show}{Logical, if \code{TRUE} and \code{"pmap"} in \code{create},
+ RKWard will automatically call the created plugin after
+ this implies and also sets \code{load=TRUE}). This will only work on the main component, though.}
-\item{gen.info}{Logical, if \code{TRUE} comment notes will be written into the genrated documents,
+\item{gen.info}{Logical,
+ if \code{TRUE} comment notes will be written into the genrated documents,
that they were generated by \code{rkwarddev} and changes should be done to the script.
You can also provide a character string naming the very rkwarddev script file that generates this plugin and its main component,
which will then also be added to the comment.}
-\item{hints}{Logical, if \code{TRUE} and you leave out optional entries (like \code{dependencies=NULL}), dummy sections will be added as comments.}
+\item{hints}{Logical,
+ if \code{TRUE} and you leave out optional entries (like \code{dependencies=NULL}), dummy sections will be added as comments.}
\item{indent.by}{A character string defining the indentation string to use.}
-\item{internal}{Logical, a simple switch to build an internal plugin for official distribution with RKWard. If set to \code{TRUE}:
+\item{internal}{Logical,
+ a simple switch to build an internal plugin for official distribution with RKWard. If set to \code{TRUE}:
\itemize{
\item{The plugin will have its namespace set to \code{"rkward"}.}
\item{The \code{<about>} info will also be available in the main component.}
@@ -118,8 +148,10 @@ which will then also be added to the comment.}
Character string with the path to the plugin root directory.
}
\description{
-With this function you can write everything from a basic skeleton structure to a complete functional plugin, including several
-components/dialogs. You should always define one main component (by \code{xml}, \code{js}, \code{rkh} etc.) before you provide
+With this function you can write everything from a basic skeleton structure to a complete functional plugin,
+ including several
+components/dialogs. You should always define one main component (by \code{xml}, \code{js},
+ \code{rkh} etc.) before you provide
}
\examples{
diff --git a/packages/rkwarddev/man/rk.rkh.doc.Rd b/packages/rkwarddev/man/rk.rkh.doc.Rd
index 9879897..b8eed29 100644
--- a/packages/rkwarddev/man/rk.rkh.doc.Rd
+++ b/packages/rkwarddev/man/rk.rkh.doc.Rd
@@ -31,7 +31,8 @@ Refer to \code{\link{rk.rkh.scan}} for a function to create this from an existin
\item{title}{An object of class \code{XiMpLe.node} to be pasted as the \code{<title>} section. See
-\item{hints}{Logical, if \code{TRUE} and you leave out optional entries (like \code{technical=NULL}), empty dummy sections will be added.}
+\item{hints}{Logical,
+ if \code{TRUE} and you leave out optional entries (like \code{technical=NULL}), empty dummy sections will be added.}
\item{gen.info}{Logical, if \code{TRUE} a comment note will be written into the document,
that it was generated by \code{rkwarddev} and changes should be done to the script.
diff --git a/packages/rkwarddev/man/rk.rkh.scan.Rd b/packages/rkwarddev/man/rk.rkh.scan.Rd
index ae045ca..e654f15 100644
--- a/packages/rkwarddev/man/rk.rkh.scan.Rd
+++ b/packages/rkwarddev/man/rk.rkh.scan.Rd
@@ -7,14 +7,18 @@
rk.rkh.scan(pXML, help = TRUE, captions = TRUE, component = NULL)
}
\arguments{
-\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node}, or path to a plugin XML file.}
+\item{pXML}{Either an object of class \code{XiMpLe.doc} or \code{XiMpLe.node},
+ or path to a plugin XML file.}
-\item{help}{Logical, if \code{TRUE} a list of XiMpLe.node objects will be returned, otherwise a character
+\item{help}{Logical, if \code{TRUE} a list of XiMpLe.node objects will be returned,
+ otherwise a character
vector with only the relevant ID names.}
-\item{captions}{Logical, if \code{TRUE} captions will be generated for all "page", "tab" and "frame" nodes.}
+\item{captions}{Logical, if \code{TRUE} captions will be generated for all "page",
+ "tab" and "frame" nodes.}
-\item{component}{Character string, name of the scanned component. Only needed if you want to search for
+\item{component}{Character string,
+ name of the scanned component. Only needed if you want to search for
}
\value{
diff --git a/packages/rkwarddev/man/rk.rkh.section.Rd b/packages/rkwarddev/man/rk.rkh.section.Rd
index 4456ae6..33301c0 100644
--- a/packages/rkwarddev/man/rk.rkh.section.Rd
+++ b/packages/rkwarddev/man/rk.rkh.section.Rd
@@ -19,7 +19,8 @@ If \code{"auto"}, an ID will be generated automatically from the \code{title} va
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
or \code{comment}, to give some \code{i18n_context} information for this node.
-If set to \code{FALSE}, the attribute \code{title} will be renamed into \code{noi18n_title}.}
+If set to \code{FALSE},
+ the attribute \code{title} will be renamed into \code{noi18n_title}.}
}
\value{
An object of class \code{XiMpLe.node}.
diff --git a/packages/rkwarddev/man/rk.rkh.setting.Rd b/packages/rkwarddev/man/rk.rkh.setting.Rd
index aed8dd7..0d70853 100644
--- a/packages/rkwarddev/man/rk.rkh.setting.Rd
+++ b/packages/rkwarddev/man/rk.rkh.setting.Rd
@@ -17,7 +17,8 @@ of the element will be shown.}
\item{i18n}{Either a character string or a named list with the optional elements \code{context}
or \code{comment}, to give some \code{i18n_context} information for this node.
-If set to \code{FALSE}, the attribute \code{title} will be renamed into \code{noi18n_title}.}
+If set to \code{FALSE},
+ the attribute \code{title} will be renamed into \code{noi18n_title}.}
}
\value{
An object of class \code{XiMpLe.node}.
diff --git a/packages/rkwarddev/man/rk.rkh.settings.Rd b/packages/rkwarddev/man/rk.rkh.settings.Rd
index 3d9ba9c..abcde2c 100644
--- a/packages/rkwarddev/man/rk.rkh.settings.Rd
+++ b/packages/rkwarddev/man/rk.rkh.settings.Rd
@@ -13,7 +13,8 @@ rk.rkh.settings(...)
An object of class \code{XiMpLe.node}.
}
\description{
-This function will create a settings node for the document section, with optional child nodes "setting" and "caption".
+This function will create a settings node for the document section,
+ with optional child nodes "setting" and "caption".
}
\examples{
# define a sample frame
diff --git a/packages/rkwarddev/man/rk.set.comp.Rd b/packages/rkwarddev/man/rk.set.comp.Rd
index a1bdb11..0644663 100644
--- a/packages/rkwarddev/man/rk.set.comp.Rd
+++ b/packages/rkwarddev/man/rk.set.comp.Rd
@@ -14,7 +14,8 @@ subsequently.}
\description{
This small tool let's you set a component name as kind of "active", which simply
means it will be returned by \code{\link[rkwarddev:rk.get.comp]{rk.get.comp}}. This can be
+used by functions like, e.g., \code{\link[rkwarddev:rk.XML.cbox]{rk.XML.cbox}},
for .rkh pages automatically to the current plugin component.
}
diff --git a/packages/rkwarddev/man/rk.set.empty.e.Rd b/packages/rkwarddev/man/rk.set.empty.e.Rd
index 800b5ed..bb61360 100644
--- a/packages/rkwarddev/man/rk.set.empty.e.Rd
+++ b/packages/rkwarddev/man/rk.set.empty.e.Rd
@@ -13,7 +13,8 @@ rk.set.empty.e(empty = FALSE)
\item{empty}{Logical, whether .}
}
\value{
-\code{rk.set.empty.e} returns invisible(NULL), \code{rk.get.empty.e} either \code{TRUE} or \code{FALSE}.
+\code{rk.set.empty.e} returns invisible(NULL),
+ \code{rk.get.empty.e} either \code{TRUE} or \code{FALSE}.
}
\description{
Some JS functions allow to decide whether empty \code{else} statements should be omitted or printed
diff --git a/packages/rkwarddev/man/rk.set.indent.Rd b/packages/rkwarddev/man/rk.set.indent.Rd
index 430a263..eb772dd 100644
--- a/packages/rkwarddev/man/rk.set.indent.Rd
+++ b/packages/rkwarddev/man/rk.set.indent.Rd
@@ -10,7 +10,8 @@ rk.get.indent(escape = FALSE)
rk.set.indent(by = "\\t")
}
\arguments{
-\item{escape}{Logical, if set to \code{TRUE} each occurring "\\t" will be escaped by an additional "\\".}
+\item{escape}{Logical,
+ if set to \code{TRUE} each occurring "\\t" will be escaped by an additional "\\".}
\item{by}{Character string, indentation string to be defined globally.}
}
diff --git a/packages/rkwarddev/man/rk.set.rkh.prompter.Rd b/packages/rkwarddev/man/rk.set.rkh.prompter.Rd
index cd221c0..d4a6546 100644
--- a/packages/rkwarddev/man/rk.set.rkh.prompter.Rd
+++ b/packages/rkwarddev/man/rk.set.rkh.prompter.Rd
@@ -8,27 +8,33 @@ rk.set.rkh.prompter(component = NULL, id = NULL, help = NULL,
rm = FALSE)
}
\arguments{
-\item{component}{Character string, should be a unique name to identify the current plugin/component.
+\item{component}{Character string,
+ should be a unique name to identify the current plugin/component.
If \code{NULL}, this function quits silently without any action.}
\item{id}{Either a character string (the \code{id} of the node to store the help information for),
or an object of class \code{XiMpLe.node} (whose \code{id} will be extracted and used).}
-\item{help}{Character string or list of character values and XiMpLe nodes, will be used as the \code{text} value for a setting node in the .rkh file.}
+\item{help}{Character string or list of character values and XiMpLe nodes,
+ will be used as the \code{text} value for a setting node in the .rkh file.}
-\item{rm}{Logical, If \code{TRUE} will remove all information stored by the name of \code{component} (if
+\item{rm}{Logical,
+ If \code{TRUE} will remove all information stored by the name of \code{component} (if
\code{id=NULL}) or of the given \code{id=NULL}, respectively.}
}
\description{
By using an environment like this, you are able to write information for RKWard help files
-directly into your script code of certrain functions, like for radio buttons or checkboxes.
+directly into your script code of certrain functions,
+ like for radio buttons or checkboxes.
}
\details{
-The information is temporarily stored in an internal environment, using the plugin/component name
+The information is temporarily stored in an internal environment,
+ using the plugin/component name
you specify. Each entry is named after the ID of its respective node. If you later call
\code{\link[rkwarddev:rk.plugin.component]{rk.plugin.component}} (or it is called by other
functions) and you activate the \code{scan} option for rkh files, the scanning process
-will try to find a match for each relevant XML node. That is, the info which is stored in the
+will try to find a match for each relevant XML node. That is,
+ the info which is stored in the
environment will magically be written into the help file.
}
\examples{
diff --git a/packages/rkwarddev/man/rk.updatePluginMessages.Rd b/packages/rkwarddev/man/rk.updatePluginMessages.Rd
index 51221aa..3067a7c 100644
--- a/packages/rkwarddev/man/rk.updatePluginMessages.Rd
+++ b/packages/rkwarddev/man/rk.updatePluginMessages.Rd
@@ -8,9 +8,11 @@ rk.updatePluginMessages(pluginmap, extractOnly = FALSE, default_po = NULL,
outdir = NULL)
}
\arguments{
-\item{pluginmap}{Character string, full path to the main pluginmap file of the plugin to translate.}
+\item{pluginmap}{Character string,
+ full path to the main pluginmap file of the plugin to translate.}
-\item{extractOnly}{Logical, should translatable strings only be extracted? If \code{FALSE}, translatable
+\item{extractOnly}{Logical,
+ should translatable strings only be extracted? If \code{FALSE}, translatable
strings will be updated and installed.}
\item{default_po}{Optional character string, fallback default name for \code{*.pot} file.}
diff --git a/packages/rkwarddev/man/rkwarddev-package.Rd b/packages/rkwarddev/man/rkwarddev-package.Rd
index 389b2fe..7ed2273 100644
--- a/packages/rkwarddev/man/rkwarddev-package.Rd
+++ b/packages/rkwarddev/man/rkwarddev-package.Rd
@@ -12,7 +12,7 @@ A Collection of Tools for RKWard Plugin Development.
Package: \tab rkwarddev\cr
Type: \tab Package\cr
Version: \tab 0.08-1\cr
-Date: \tab 2015-11-28\cr
+Date: \tab 2015-11-30\cr
Depends: \tab R (>= 2.9.0),methods,XiMpLe (>= 0.03-21),rkward (>= 0.5.7)\cr
Enhances: \tab rkward\cr
Encoding: \tab UTF-8\cr
diff --git a/packages/rkwarddev/man/rkwarddev.required.Rd b/packages/rkwarddev/man/rkwarddev.required.Rd
index 1829863..a5b3a51 100644
--- a/packages/rkwarddev/man/rkwarddev.required.Rd
+++ b/packages/rkwarddev/man/rkwarddev.required.Rd
@@ -12,7 +12,8 @@ rkwarddev.required(min = "0.06-5", lib.loc = NULL)
\item{lib.loc}{The \code{lib.loc} argument passed over to \code{\link[utils:packageVersion]{packageVersion}}.}
}
\value{
-The function has no return value, but wil stop with an error if the specified version requirement is not met.
+The function has no return value,
+ but wil stop with an error if the specified version requirement is not met.
}
\description{
Check for rkwarddev package version requirements
diff --git a/packages/rkwarddev/man/tf.Rd b/packages/rkwarddev/man/tf.Rd
index ae3aa2e..29a6b99 100644
--- a/packages/rkwarddev/man/tf.Rd
+++ b/packages/rkwarddev/man/tf.Rd
@@ -8,27 +8,38 @@ tf(cbox, true = TRUE, not = FALSE, ifelse = FALSE, false = FALSE,
opt = NULL, prefix = ",\\n", level = 3, indent.by = rk.get.indent())
}
\arguments{
-\item{cbox}{An object of class \code{XiMpLe.node} containing a \code{<checkbox>} node, as generated
+\item{cbox}{An object of class \code{XiMpLe.node} containing a \code{<checkbox>} node,
+ as generated
-\item{true}{Logical or character, the value the option should get. E.g., if \code{true=TRUE} then the option will be
-set to \code{TRUE} if the box is checked, or in case \code{not=TRUE}, if the box is not checked.}
+\item{true}{Logical or character, the value the option should get. E.g.,
+ if \code{true=TRUE} then the option will be
+set to \code{TRUE} if the box is checked, or in case \code{not=TRUE},
+ if the box is not checked.}
-\item{not}{Logical, inverses the checked status of the checkbox. In other words, set this to \code{TRUE}
+\item{not}{Logical, inverses the checked status of the checkbox. In other words,
+ set this to \code{TRUE}
if you want the option to be set if the box is not checked.}
-\item{ifelse}{Logical, whether the options should be set anyway. By default, the option will only
-be set in one condition. If \code{ifelse=TRUE}, it will get the inverse value in case of the alternative
-condition, e.g. it will be set to either \code{not=TRUE} or \code{not=FALSE} if the box is checked or unchecked.}
+\item{ifelse}{Logical, whether the options should be set anyway. By default,
+ the option will only
+be set in one condition. If \code{ifelse=TRUE},
+ it will get the inverse value in case of the alternative
+condition,
+ e.g. it will be set to either \code{not=TRUE} or \code{not=FALSE} if the box is checked or unchecked.}
-\item{false}{Logical or character, the value the option should, only used get if \code{ifelse=TRUE} as well.
-E.g., if \code{false=FALSE} then the option will be set to \code{FALSE} if the box is not checked,
+\item{false}{Logical or character, the value the option should,
+ only used get if \code{ifelse=TRUE} as well.
+E.g.,
+ if \code{false=FALSE} then the option will be set to \code{FALSE} if the box is not checked,
or in case \code{not=TRUE}, if the box is checked.}
-\item{opt}{A character string, naming the R option to be set. If \code{NULL}, the XML ID of the checkbox node
+\item{opt}{A character string, naming the R option to be set. If \code{NULL},
+ the XML ID of the checkbox node
will be used.}
-\item{prefix}{A character string, what should be pasted before the actual option string. Default is a
+\item{prefix}{A character string,
+ what should be pasted before the actual option string. Default is a
comma and a newline.}
\item{level}{Integer, which indentation level to use, minimum is 1.}
@@ -43,7 +54,8 @@ An object of class \code{rk.JS.ite}.
This function is a basically shortcut for \code{\link[rkwarddev:ite]{ite}} with some assumptions.
It's thought to be used when a checkbox should turn an option of an R function to a specified value,
by default \code{TRUE} or \code{FALSE} (hence the name, abbreviated "true or false").
-The same result can be obtained with \code{ite}, but for most common cases \code{tf} is much quicker.
+The same result can be obtained with \code{ite},
+ but for most common cases \code{tf} is much quicker.
}
\examples{
# an example checkbox XML node
|
{}
|
# Binomial Distribution Formula
The prefix ‘bi’ means two or twice. A binomial distribution is considered as the probability of a trail with only two possible outcomes. It is a type of distribution that has two different outcomes which are ‘success’ and ‘failure’. For example, if we toss the coin then there is an equal chance of outcome it can be heads or tails. Therefore there is a 50% chance of the outcomes. It is applicable to discrete random variables only. The origin of Binomial distribution can be taken as towards Bernoulli’s trial. In this article, we will discuss the Binomial distribution formula with examples. Let us begin learning!
## Binomial Distribution Formula
### What is Binomial Distribution?
The binomial distribution formula helps to check the probability of getting an “x” number of successes in the “n” independent trials of a binomial experiment. As we know that binomial distribution is a type of probability distribution in statistics that has two possible outcomes. In probability theory, the binomial distribution has two parameters n and p.
The probability distribution becomes a binomial probability distribution if it satisfies the following requirements.
1. Each trail can have only two outcomes. These outcomes may be either a success or a failure.
2. The trails should be a fixed number.
3. All the outcomes each trial must be independent of each other.
4. Also, the success of probability must remain the same for each trail.
The binomial distribution summarized the number of trials, survey or experiments conducted. It is very useful when each outcome has an equal chance of attaining a particular value.
### The formula for Binomial Distribution in Probability:
The formula for the binomial probability distribution is given below:
P(x) = [$$\frac{n!}{r!(n-r)!}$$] $$p^r (1-p)^{n-r}$$
Where,
P(x) Binomial probability Distribution X Random variable N Total number of events R The total number of successful events. P Probability of success on a single trial. 1-p Probability of failure on a single trial. $$binomial{n}{r}$$ Combinatorial value
### Mean and Variance of a Binomial Distribution
Calculation of Binomial distribution value sometimes needs mean and variance values. These two terms will give more stability and reliability. Formulas are as given below:
$$\mu = np$$
$$\sigma^2 = npq$$
$$\mu$$ Mean $$\sigma^2$$ Variance p Probability of success q Probability of failure
Also, it should be noted that,
• The variance of a Binomial Variable will always be less than its mean. i.e. npq < np.
• For Maximum Variance, p and q both should be 0.5 and hence
$$\sigma= \frac{1}{4}$$
## Solved Examples
Q.1: Toss a coin for 12 times. Then what will be the probability of getting exactly 7 times head?
Solution: As given,
Number of trails I.e. n = 12
Number of success i.e. r = 7
Also, Probability of single trail i.e. p = $$\frac {1}{2}$$ = 0.5
Formula for Binomial Distribution is,
P(x) = $$binom{n}{r}p^r (1-p)^{n-r}$$
$$binom{n}{r}$$
= $$\frac{n!}{r!(n-r)!}$$
= $$\frac{12!}{7!(5)!}$$
= 792
$$p^r= 0.5 ^ 7$$
= 0.0078125
To Find $$(1−p)^{n-r},$$
calculate (1-p) and (n-r).
1 – p = 1 – 0.5 = 0.5
n – r = 12 – 7 = 5
So, $$(1−p)^{n-r} = 0.5^5 = 0.03125$$
Thus,
P(x) = $$binom{n}{r}p^r (1-p)^{n-r}$$
= 792 x 0.0078125 x 0.03125
= 0.193359375
Therefore, probability of getting exactly 7 heads is 0.19
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KUCKOO B
I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26
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Yashitha
Hi
Same
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virat
yes
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# Multiple Dimensions - Extending Disk Integration
In calculus, disk integration, allows us to calculate the volume of a solid by integrating along an axis. It models the resulting shape as a stack of disks of infinitesimal thickness. Most of the time, we chose an axis of revolution, which results in a circular disk, though this is not necessary. The washer method uses the same idea, and gives us a 'hollow' solid of revolution. We'd push this idea even further.
Let's consider the case of a circle with radius $$r$$. We know that it has an area of $$\pi r^2$$ and a perimeter of $$2 \pi r$$, but how can we show this?
Wiki
One basic approach, is to say that the circle is the set of points such that $$x^2 + y^2 \leq r^2$$, and hence we want to calculate $$\int_R 1, dR$$. Let's use the area element, with a chosen axis of the $$x -$$axis. When $$x = X$$, we see that $$y^2 \leq r^2 - X^2$$ and so $$- \sqrt{ r^2 - X^2} \leq y \leq \sqrt{ r^2 - X^2}$$. Hence, at $$x = X$$, the area contributed is equal to $$\sqrt{ r^2 - X^2} - (-\sqrt{ r^2 - X^2} ) = 2 \sqrt{ r^2 - X^2}$$. As such, this allows us to calculate that:
$\int A(x) dx = \int_{-r}^r 2 \sqrt{ r^2 - x^2} \, dx = \pi r^2$
Wiki
Another way to use the area element, is to think about obtaining the area of the circle, by considering how faraway each of the points are. In this case, our chosen axis would be the radius of the circle. When the radius is $$R$$, the contribution to the area is $$p(R)$$, which denotes the perimeter of the circle. This tells us that
$\int_0^r P(R) \, dR = \pi r^2.$
Applying the fundamental theorem of calculus, we get that $$P(r) = 2 \pi r$$. We see this relationship represented in the previous 2 problems, where we worked on the 2 dimensional circle and the 3 dimensional sphere.
This seems like a torturous way to obtain the area and perimeter of a circle, which are already well known. In the following questions, we will apply this method to find the area and perimeter of a sphere in 4 dimensions!
###### Image credit: Wikipedia Macks, Wikipedia Ksmrq
Note by Calvin Lin
3 years, 4 months ago
Sort by:
Thanks for a nice presentation. What about the hollow cylinder method ? · 3 years, 4 months ago
Yes, there are also extensions to the shell method. I avoided talking about it, because it's not intuitive and most people have some trouble grasping it. It uses the same idea of "area element", which is how I prefer to think about it. Staff · 3 years, 4 months ago
Thank you. In some case the integral is simple by one method or the other. · 3 years, 4 months ago
0/3 · 3 years ago
Nice presentation · 3 years, 3 months ago
i dont know · 3 years, 3 months ago
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Calculating with Rotations edit page
## Rotating Vectors
Let
a certain rotation. Then the rotation of the xvector is computed via
The inverse rotation is computed via the backslash operator
## Concatenating Rotations
Let
be two rotations. Then the rotation defined by applying first rotation one and then rotation two is computed by
## Rotational angle and the rotational axis
Then rotational angle and the axis of rotation can be computed via then commands angle(rot) and axis(rot)
If two rotations are specifies the command angle(rot1,rot2) computes the rotational angle between both rotations
## The inverse Rotation
The inverse rotation you get from the command inv(rot)
## Conversion into Euler Angles and Rodrigues Parametrisation
There are methods to transform rotations in almost any other parameterization of rotations as they are:
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# Double Summation question from Hoffman/Kunze
I've been self-studying Hoffman/Kunze and I'm puzzled by this seemingly easy double summation. I still don't see the jump in logic from the 3rd to the 4th step on pg 118 in Ch4 polynomials.
$[(fg)h]_n=\displaystyle\sum_{i=0}^n(fg)_ih_{n-i}$
$=\displaystyle\sum_{i=0}^n\left(\sum_{j=0}^if_ig_{i-j}\right)h_{n-i}$
$=\displaystyle\sum_{i=0}^n\sum_{j=0}^if_ig_{i-j}h_{n-i}$
$=\displaystyle\sum_{j=0}^nf_j\sum_{i=0}^{n-j}g_ih_{n-i-j}$
I reached the 4th step but got $f_{i+j}$ instead of $f_j$. Have I made a mistake?
P.S : I can't read math format well on chrome browser. Is there an issue?
• In (2) we shift the index $i$ to start from zero.
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The dissolution of ammonium chloride is supplied to cool a container the water inserted in the solution. It"s an endothermic process. What absorbs the heat and what loses it?
When stating this question, one requirements to think around all the processes affiliated (or at least those that account for most of what happens). In general, any procedure will take place spontaneously, if $\Delta G$ is negative. To determine $\Delta G$, equation (1) is used.
You are watching: Why does nh4cl dissolve in water
$$\Delta G = \Delta H - T \Delta S\tag1$$
There is an enthalpic term related to the warmth a reaction produces or requires and also an entropic term the is temperature-dependent. We cannot to speak a lot around the entropic term a priori, yet we can around the enthalpic ax — i m sorry is luckily what the question is about. For that, we an initial need to compose a reaction equation (2):
$$\ceNH4Cl (s) + n H2O > NH4+ (aq) + Cl- (aq) + m H2O\tag2$$
Dissolving of the ammonium chloride can likewise be assumed as two different processes, view equation $(2")$.
$$\ceNH4Cl (s) -> + -> NH4+ (aq) + Cl- (aq)\tag2"$$
Meaning that very first we break up the ammonium chloride crystal framework and 2nd we dissolve the ceiling ions. If we want to create that in single enthalpy terms, we can do the as displayed in equation (3).
$$\Delta H_\mathrmtot = -\Delta H_\mathrmlattice(\ceNH4Cl) + \Delta H_\mathrmsolv(\ceNH4+) + \Delta H_\mathrmsolv(\ceCl-)\tag3$$
All of this values can be looked up. I have given those together calculated by Jenkins and Morris in table 1.<1>
$$\textbfTable 1:\text values of enthalpies supplied in this prize as\\\textquoted indigenous Jenkins and also Morris (reference 1).\\\beginarraycccc\hline\textcompound & \Delta H_\mathrmlattice <\mathrmkJ/mol> & \Delta H_\mathrmsolv <\mathrmkJ/mol> & \Delta H_\mathrmtot <\mathrmkJ/mol> \\\hline\ceNH4Cl & -709.1 & - 694.7 & 14.4 \\ \hline\endarray$$
So us need energy (a lot of it!) to rest up the crystal lattice the $\ceNH4Cl$. We then re-gain energy by creating the hydrated, i.e. Dissolved, ions in solution. If energy is required, it is typically (excluding photochemical reactions — no the case here) heat power which is simply attracted from the surroundings. If energy is released, that is typically (same caveat) exit as heat into the surroundings.
Thus, the answer come your inquiry is:
The warmth is absorbed by the heavy ammonium chloride. The is used to break up the salt crystal according to equation $(2".1)$. Part of it is re-released (unnoticed) by forming solvent–ion hydrogen binding (the solvation device of $\ceNH4Cl$ — equation $(2".2)$).
This heat is drained from the surroundings, which in this case is generally the water in i beg your pardon you want to dissolve the ammonium chloride. (Of course, the equipment will then additional exchange warm with everything is around it, meaning that one of two people the air about a bottle/flask or an outer flask together in your described experimental setup will lose heat at the benefit of the $\ceNH4Cl$-solution.)
Reference:
<1>: H. D. B. Jenkins, D. F. C. Morris, Mol. Phys.
See more: Gas Mileage Of 2004 Pontiac Grand Am Gt Mpg, Gas Mileage Of 2004 Pontiac Grand Am
1976, 32, 231. DOI: 10.1080/00268977600101741.
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# combine two translation/rotation/scale triplets without matrices
Let t1/2, r1/2 and s1/2 be two sets of transformations. Translations are vec3, rotations are quaternions and scales are vec3. Lets assume that all common operations are defined (overloaded operators in c++).
How do I combine (in what order) the two sets of transformations into a third set t3, r3 and s3, such that mat4(t1, r1, s1) * mat4(t2, r2, s2) == mat4(t3, r3, s3), without actually converting to matrices?
Mat4 is composed as M = T * R * S. (OpenGL conventions, if it makes any difference)
Thanks
My original code:
transform transform::operator * (const transform &other) const
{
transform r;
r.orientation = orientation * other.orientation;
r.scale = scale * other.scale;
r.position = position + (other.position * orientation) * scale;
return r;
}
this used to work when scale was just one number (uniform scale). But it does not work with non-uniform scale. I guess that one of the scales have to be rotated, but I am failing to find the correct way.
This is how I test it:
for (uint32 round = 0; round < 10; round++)
{
vec3 pa = randomDirection3() * randomRange(-5, 20);
vec3 pb = randomDirection3() * randomRange(-5, 20);
quat oa = randomDirectionQuat();
quat ob = randomDirectionQuat();
vec3 sa = randomRange3(real(0.1), 10);
vec3 sb = randomRange3(real(0.1), 10);
transform ta(pa, oa, sa), tb(pb, ob, sb);
test(mat4(ta * tb), mat4(ta) * mat4(tb));
}
• This is not possible in the general case when you have both non-90-degree rotations on the "first" transformation and non-uniform scale on the second transformation. This can result in a squash or stretch being applied diagonal to the coordinate axes, which a vec3 of axis-aligned scale values is insufficient to describe. See discussion in this Q&A for more details. Would you be open to adding a second quaternion to your transformation representation to store the basis for scaling? – DMGregory Jan 20 at 14:07
• Oh, I had not thought about that. Thank you very much. I will stay with just uniform scales. – Tomas Jan 20 at 14:37
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×
Get Full Access to Organic Chemistry - 8 Edition - Chapter 10 - Problem 28p
Get Full Access to Organic Chemistry - 8 Edition - Chapter 10 - Problem 28p
×
Give IUPAC names for the following compounds.
ISBN: 9780321768414 33
Solution for problem 28P Chapter 10
Organic Chemistry | 8th Edition
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• 2901 Step-by-step solutions solved by professors and subject experts
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Organic Chemistry | 8th Edition
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Problem 28P
Give IUPAC names for the following compounds.
Step-by-Step Solution:
Step 1 of 3
IUPAC names:-
1. 4-methyl-2-sulfhydrylpentane
2. 2,3-dimethyl-1-sulfhydrylpent-2-ene
3. thiophenol
Step 2 of 3
Step 3 of 3
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Calculus: Early Transcendental Functions : Riemann Sums and Definite Integrals
?In Exercises 3–8, evaluate the definite integral by the limit definition. $$\int_{1}^{2}\left(x^{2}+1\right) d x$$
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?In Problems 1–8, use the Mann–Whitney test to test the given hypotheses at the = 0.05 level of significance. The independent samples were obtained ra
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How to show rownames in densityheatmaps
0
0
Entering edit mode
Do it! ▴ 10
@do-it-23093
Last seen 4 days ago
Germany, München
Hello Dear Biocondactors,
I would like to ask if there is a possibility to put row names in the density heatmap (Complexheatmap package). I want to show which density curve is indicating which gene. Simulated data and the codes are below. Thank you!
set.seed(22)
li.A <- matrix(rnorm(50), nrow = 10)
rownames(li.A) <- LETTERS[1:10]
colnames(li.A) <- paste0("S_", ncol = 1:5)
library(ComplexHeatmap)
densityHeatmap(li.A)
ComplexHeatmap • 125 views
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## coolaidd Group Title You buy a house for $250,000 and it appreciates at a rate of 3% a year. How much will the house be worth after 3 years? Round to the nearest dollar. one year ago one year ago • This Question is Closed 1. tcarroll010 250000(1.03)^3 = 273182 2. coolaidd thanks! I have like 2more questions like this.. you think you can help? 3. tcarroll010 I can tyr! 4. coolaidd 1) Find the amount of time it will take to double an investment of$400 with a rate of 4.5% compounded continuously. Round your answer to the nearest tenth. 2) Find the rate it would take to double an investment of \$600 in 10 years if the rate is compounded continuously. Round your answer to the nearest tenth.
5. coolaidd
@tcarroll010
6. tcarroll010
First, the formula for compounding continuously is: $A = P \times e ^{0.045t}$
7. tcarroll010
$800 = 400 \times e ^{0.045t}$ $2 = e ^{0.045t}$$\ln 2 = \ln e ^{0.045t}$$\frac{ \ln 2 }{ 0.045 } = t$
8. tcarroll010
t = 15.4
9. tcarroll010
For the second one,$1200 = 600 \times e ^{(r)(10)}$ $2 = e ^{10r}$$\ln 2 = \ln e ^{10r}$$\frac{ \ln 2 }{ 10 } = r = 0.0693$
10. coolaidd
Thank you soo soo much!
11. tcarroll010
uw!
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# Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers
The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Kim and Lin. The strength of our approach is its robustness: we enumerate four families F_1 ⊂ F_2 ⊂ F_3 ⊂ F_4 of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family F_i which generalizes the one for the family F_i-1. The second topic of the paper is the enumeration of a fifth family F_5 of pattern-avoiding inversion sequences (containing F_4). This enumeration is also solved via a succession rule, which however does not generalize the one for F_4. The associated enumeration sequence, which we call of powered Catalan numbers, is quite intringuing, and further investigated. We provide two different succession rules for it, denoted Ω_pCat and Ω_steady, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the steady paths, which are naturally associated with Ω_steady. They allow to brigde the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated to Ω_pCat). Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.
## Authors
• 1 publication
• 1 publication
• 1 publication
• 2 publications
• ### On 120-avoiding inversion and ascent sequences
Recently, Yan and the first named author investigated systematically the...
03/26/2020 ∙ by Zhicong Lin, et al. ∙ 0
• ### Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths
In this note, we present constructive bijections from Dyck and Motzkin m...
04/02/2021 ∙ by Jean-Luc Baril, et al. ∙ 0
• ### Descent distribution on Catalan words avoiding a pattern of length at most three
Catalan words are particular growth-restricted words over the set of non...
03/18/2018 ∙ by Jean-Luc Baril, et al. ∙ 0
• ### A Bijection Between Weighted Dyck Paths and 1234-avoiding Up-Down Permutations
Three-dimensional Catalan numbers are a variant of the classical (bidime...
11/30/2020 ∙ by Justine Falque, et al. ∙ 0
• ### On sequences associated to the invariant theory of rank two simple Lie algebras
We study two families of sequences, listed in the On-Line Encyclopedia o...
11/23/2019 ∙ by Alin Bostan, et al. ∙ 0
• ### Catalan and Schröder permutations sortable by two restricted stacks
We investigate the permutation sorting problem with two restricted stack...
04/03/2020 ∙ by J. -L. Baril, et al. ∙ 0
• ### Constrained Type Families
We present an approach to support partiality in type-level computation w...
06/29/2017 ∙ by J. Garrett Morris, et al. ∙ 0
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## 1 Introduction and preliminaries
### 1.1 Context of our work
An inversion sequence of length is any integer sequence satisfying , for all . There is a well-known bijection between the set of all permutations of length (or size) and the set of all inversion sequences of length , which maps a permutation into its left inversion table , where . This bijection is actually at the origin of the name inversion sequences.
The study of pattern-containment or pattern-avoidance in inversion sequences was first introduced in [25], and then further investigated in [15]. Namely, in [25], Mansour and Shattuck studied inversion sequences that avoid permutations of length , while in [15], Corteel et al. proposed the study of inversion sequences avoiding subwords of length . The definition of inversion sequences avoiding words (which may in addition be permutations) is straightforward: for instance, the inversion sequences that avoid the word (resp. the permutation ) are those with no such that (resp. ). Pattern-avoidance on special families of inversion sequences has also been studied in the literature, namely by Duncan and Steingrímsson on ascent sequences – see [17].
The pattern-avoiding inversion sequences of [15] were further generalized in [26], extending the notion of pattern-avoidance to triples of binary relations . More precisely, they denote by the set of all inversion sequences in having no three indices such that , , and , and by . For example, the sets and coincide for every . In [26] all triples of relations in are considered, where “” stands for any possible relation on a set , i.e. for any . Therefore, all the possible triples of relations are examined and the resulting families of pattern-avoiding inversion sequences are subdivided into equivalence classes. Many enumeration results complementing those in [15, 25] have been found in [26]. In addition, several conjectures have been formulated in [26]
. Some (but by far not all!) of them have been proved between the moment a first version of
[26] was posted on the arXiv and its publication, and references to these recent proofs can also be found in the published version of [26].
In this paper we study five families of inversion sequences which form a hierarchy for the inclusion order. The enumeration of these classes – by well-known sequences, such as those of the Catalan, the Baxter, and the newly introduced semi-Baxter numbers [11] – was originally conjectured in the first version of [26]. These conjectures have attracted the attention of a fair number of combinatorialists, resulting in proofs for all of them, independently of our paper. Still, our work reproves these enumeration results. Along the way, we further try to establish bijective correspondences between these families of inversion sequences and other known combinatorial structures. The most remarkable feature of our work is that all the families of inversion sequences are presented and studied in a unified way by means of generating trees. Before proceeding, let us briefly recall some basics about generating trees. Details can be found for instance in [2, 3, 8, 31].
### 1.2 Basics of generating trees
Consider a combinatorial class , that is to say a set of discrete objects equipped with a notion of size such that the number of objects of size is finite, for any . We assume also that contains exactly one object of size . A generating tree for is an infinite rooted tree whose vertices are the objects of each appearing exactly once in the tree, and such that objects of size are at level (with the convention that the root is at level ). The children of some object are obtained by adding an atom (i.e. a piece of the object that makes its size increase by ) to . Since every object appears only once in the generating tree, not all possible additions are acceptable. We enforce the unique appearance property by considering only additions that follow some prescribed rules and call the growth of the process of adding atoms according to these rules.
To illustrate these definitions, we describe the classical growth for the family of Dyck paths, as given by [3]. Recall that a Dyck path of semi-length is a lattice path using up and down unit steps, running from to and remaining weakly above the -axis. The atoms we consider are factors, a.k.a. peaks, which are added to a given Dyck path. To ensure that all Dyck paths appear exactly once in the generating tree, a peak is inserted only in a point of the last descent, defined as the longest suffix containing only letters. More precisely, the children of the Dyck path are , ,…, , .
The first few levels of the generating tree for Dyck paths are shown in Figure 1 (left).
When the growth of is particularly regular, we encapsulate it in a succession rule. This applies more precisely when there exist statistics whose evaluations control the number of objects produced in the generating tree. A succession rule consists of one starting label (axiom) corresponding to the value of the statistics on the root object and of a set of productions encoding the way in which these evaluations spread in the generating tree – see Figure 1(right). The growth of Dyck paths presented earlier is governed by the statistic “length of the last descent”, so that it corresponds to the following succession rule, where each label indicates the number of steps of the last descent in a Dyck path,
ΩCat=⎧⎪⎨⎪⎩(1)(k)⇝(1),(2),…,(k),(k+1).
Obviously, as we discuss in [12], the sequence enumerating the class can be recovered from the succession rule itself, without reference to the specifics of the objects in : indeed, the th term of the sequence is the total number of labels (counted with repetition) that are produced from the root by applications of the set of productions, or equivalently, the number of nodes at level in the generating tree. For instance, the well-know fact that Dyck paths are counted by Catalan numbers (sequence A000108 in [27]) can be recovered by counting nodes at each level in the above generating tree.
### 1.3 Content of the paper
In our study we focus on five different families of pattern-avoiding inversion sequences, which are depicted in Figure 2. As the figure shows, these families are naturally ordered by inclusion, and are enumerated by well-known number sequences.
The objective of our study is twofold. On the one hand we provide (and/or collect) enumerative results about the families of inversion sequences of Figure 2. On the other hand we aim at treating all these families in a unified way. More precisely, in each of the following sections we first provide a simple combinatorial characterization for the corresponding family of inversion sequences, and then we show a recursive growth that yields a succession rule.
The main noticeable property of the succession rules provided in Sections 2, 3, 4, and 5 is that they reveal the hierarchy of Figure 2 at the abstract level of succession rules. Specifically, the recursive construction (or growth) provided for each family is obtained by extending the construction of the immediately smaller family. Moreover, the ways in which these growths are encoded by labels in succession rules are also each a natural extension of the case of the immediately smaller family. Hence, these examples provide another illustration of the idea of generalizing/specializing succession rules that we discussed in details in [6, Section 2.2]. The outcome of the discussion in [6, Section 2.2] is the following proposed definition for generalization/specialization of succession rules. To say that a succession rule specializes (equivalently, that generalizes or extends ), we require
• the existence of a comparison relation “smaller than or equal to” between the labels of and those of , and,
• for any labels of and of with smaller than or equal to , a way of mapping the productions of the label in to a subset of the productions of the label in , such that a label is always mapped to a larger or equal one.
Comparing Propositions 4, 8, 14 and 17, and mapping the labels in the obvious way, it is easy to see that the succession rules in these propositions satisfy this proposed definition (the comparison relation being here just the componentwise natural order on integers).
We conclude our introduction with a few words commenting on the classes of our hierarchy and our results on them.
• We start in Section 2 with , which we call the family of Catalan inversion sequences. We define two recursive growths for this family, one according to (hence proving that is enumerated by the Catalan numbers) and a second one that turns out to be a new succession rule for the Catalan numbers. The fact that this family of inversion sequences is enumerated by the Catalan numbers was conjectured in [26] and it has recently been proved independently of us by Kim and Lin in [23]. Moreover, we are able to relate the family of Catalan inversion sequences to a family of permutations defined by the avoidance of vincular patterns, proving that they are in bijection with a family of pattern-avoiding permutations.
• In Section 3 we consider the family . This class has been considered independently of us by Lin in the article [24], which proves the conjecture (originally formulated in [26]) that these inversion sequences are counted by sequence A108307 on [27] – defining the enumerative sequence of set partitions of that avoid enhanced 3-crossings [10]. We review Lin’s proof, which fits perfectly in the hierarchy of succession rules that we present.
• In Section 4 we study inversion sequences in , which we call Baxter inversion sequences. This family of inversion sequences was originally conjectured in [26] to be counted by Baxter numbers. The proof of this conjecture was provided in [23] by means of a growth for Baxter inversion sequences that neatly generalizes the previous growth for the family .
• In Section 5, we deal with the family , which we call semi-Baxter inversion sequences. Indeed, this family of inversion sequences was originally conjectured in [26] to be counted by the sequence A117106 [27]; these numbers have been thoroughly studied and named semi-Baxter in the article [11], which among other results proves this conjecture of [26].
• Finally, in Section 6 we deal with , which is the rightmost element of the chain of Figure 2. We call the elements of powered Catalan inversion sequences, since the succession rule we provide for them is a “powered version” of the classical Catalan succession rule.
When turning to powered Catalan inversion sequences, the hierarchy of Figure 2 is broken at the level of succession rules. Indeed, although the combinatorial characterization of these objects generalizes naturally that of semi-Baxter inversion sequences, we do not have a growth for powered Catalan inversion sequences that generalizes the one of semi-Baxter inversion sequences. This motivates the second part of the paper, devoted to the study of this “powered Catalan” enumerative sequence from Section 6 on.
The enumeration of powered Catalan inversion sequences (by A113227, [27]) was already solved in [15]. Our first contribution (in Section 6) is to prove that they grow according to the succession rule , which generalizes the classical rule by introducing powers in it. This motivates the name powered Catalan numbers which we have coined for the numbers of sequence A113227.
Many combinatorial families are enumerated by powered Catalan numbers. Some are presented in Section 7. These families somehow fall into two categories. Inside each category, the objects seem to be in rather natural bijective correspondence. However, between the two categories, the bijections are much less clear. Our result of Section 7 is to provide a second succession rule for powered Catalan numbers (more precisely, for permutations avoiding the vincular pattern ), which should govern the growth of objects in one of these two categories, the other category being naturally associated with the rule .
In Section 8, we describe a new occurrence of the powered Catalan numbers in terms of lattice paths. More precisely, we introduce the family of steady paths and prove that they are enumerated by the powered Catalan numbers. This is proved by showing a growth for steady paths that is encoded by (a variant called of) the succession rule for permutations avoiding the pattern . We also provide a simple bijection between steady paths and permutations avoiding the vincular pattern , therefore recovering the enumeration of this family, already known [4] to be enumerated by A113227.
Finally, in Section 9 we bridge the gap between the two types of powered Catalan structures, by showing a bijection between steady paths (representing the succession rule ) and valley-marked Dyck paths (emblematic of the succession rule ).
## 2 Catalan inversion sequences: I(≥,−,≥)
The first family of inversion sequences considered is . It was originally conjectured in [26] to be counted by the sequence of Catalan numbers [27, A000108] (hence the name Catalan inversion sequences) whose first terms we recall:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,…
We note that this conjectured enumeration has recently been proved independently from us by Kim and Lin in [23]. Their proof does not involve generating trees, but displays a nice Catalan recurrence for the filtration of where the additional parameter is the value of the last element of an inversion sequence.
We provide another proof of this conjecture in Proposition 3 by showing that there exists a growth for according to the well-known Catalan succession rule . Moreover, we show a second growth for , thereby providing a new Catalan succession rule, which is appropriate to be generalized in the next sections. In addition, we show a direct bijection between and a family of pattern-avoiding permutations, which thus results to be enumerated by Catalan numbers.
### 2.1 Combinatorial characterization
Let us start by observing that the family of Catalan inversion sequences has a simple characterization in terms of inversion sequences avoiding patterns of length three.
###### Proposition 1.
An inversion sequence is in if and only if it avoids , , , , and .
###### Proof.
The proof is rather straightforward, since containing , , such that , with , is equivalent to containing the listed patterns. ∎
In addition to the above characterization, we introduce the following combinatorial description of Catalan inversion sequences, as it will be useful to define a growth according to the Catalan succession rule .
###### Proposition 2.
Any inversion sequence is a Catalan inversion sequence if and only if for any , with ,
if forms a weak descent, i.e. , then , for all .
###### Proof.
The forward direction is clear. The backwards direction can be proved by contrapositive. More precisely, suppose there are three indices , such that . Then, if , forms a weak descent and the fact that concludes the proof. Otherwise, since , there must be an index , with , such that forms a weak descent and . This concludes the proof as well. ∎
The previous statement means that any of our inversion sequences has a neat decomposition: they are concatenations of shifts of inversion sequences having a single weak descent, at the end. A graphical view of this decomposition is shown in Figure 3.
### 2.2 Enumerative results
###### Proposition 3.
Catalan inversion sequences grow according to the succession rule ,
ΩCat=⎧⎪⎨⎪⎩(1)(k)⇝(1),(2),…,(k),(k+1).
###### Proof.
Given an inversion sequence , we define the inversion sequence as the sequence , where the entry is inserted in position , for some , and the entries are shifted rightwards by one. By definition of inversion sequences, is the largest possible value that the th entry can assume. And moreover, letting , it holds that , for all ; namely the index is the rightmost index such that . For example, if and , then .
Then, we note that given a Catalan inversion sequence of length , by removing from the rightmost entry whose value is equal to its position minus one, we obtain a Catalan inversion sequence of length . Note that for every Catalan inversion sequence, thus such an entry always exists.
Therefore, we can describe a growth for Catalan inversion sequences by inserting an entry in position . By Proposition 2, since the entry forms a weak descent in , the inversion sequence is a Catalan inversion sequence of length if and only if . Then, we call active positions all the indices , with , such that is a Catalan inversion sequence of length . According to this definition, and are always active positions: indeed, both and are Catalan inversion sequences of length .
We label a Catalan inversion sequence of length with , where the number of active positions is . Note that the smallest inversion sequence has label , which is the axiom of rule .
Now, we show that given a Catalan inversion sequence of length with label , the labels of , where ranges over all the active positions, are precisely the label productions of in .
Let be the active positions of from left to right. Note that and . We argue that, for any , the active positions of the inversion sequence are , and . Indeed, on the one hand any position which is non-active in is still non-active in . On the other hand, by Proposition 2, the index becomes non-active in , since by definition. Similarly, any position , with , which is active in becomes non-active in . Thus, the active positions of are , and . Hence, has label , for any .∎
Furthermore, we can provide a new succession rule for generating Catalan inversion sequences: the growth we provide in the following is remarkable as it allows generalizations in the next sections.
###### Proposition 4.
Catalan inversion sequences grow according to the following succession rule
ΩCat2=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(1,1)(h,k)⇝(0,k+1)h,(h+1,k),(h+2,k−1),…,(h+k,1).
###### Proof.
We consider the growth of Catalan inversion sequences that consists of adding a new rightmost entry, and we prove that this growth defines the succession rule . Obviously, this growth is different from the one provided in the proof of Proposition 3.
Let be the maximum value among the entries of . And let be the maximum value of the set of all entries that form a weak descent of ; if has no weak descents, then . By Proposition 1, since avoids , and , the value is or . In particular, if , then .
By Proposition 2, it follows that is a Catalan inversion sequence of length if and only if . Moreover, if , then forms a new weak descent of , and becomes the value ; whereas, if , then since the weak descents of and coincide.
Now, we assign to any Catalan inversion sequence of length the label , where and . In other words, (resp. ) marks the number of possible additions smaller than or equal to (resp. greater than) the maximum entry of .
The sequence has no weak descents, thus it has label , which is the axiom of . Let be a Catalan inversion sequence of length with label . As Figure 4 illustrates, the labels of the inversion sequences of length produced by adding a rightmost entry to are
• , for any ,
• , when ,
which concludes the proof that Catalan inversion sequences grow according to . ∎
It is well worth noticing that although the above succession rule generates the well-known Catalan numbers, we do not have knowledge of this succession rule in the literature.
### 2.3 One-to-one correspondence with AV(1-23,2-14-3)
In this section we show that Catalan inversion sequences are just left inversion tables of permutations avoiding the patterns and , thereby proving that the family of pattern-avoiding permutations forms a new occurrence of the Catalan numbers. We start by recalling some terminology and notation.
A (Babson-Steingrímsson-)pattern of length is any permutation of where two adjacent entries may or may not be separated by a dash – see [1]. Such patterns are also called generalized or vincular. The absence of a dash between two adjacent entries in the pattern indicates that in any pattern-occurrence the two entries are required to be adjacent: a permutation of length contains the vincular pattern , if it contains as pattern, and moreover, there is an occurrence of the pattern where the entries of not separated by a dash are consecutive entries of the permutation ; otherwise, avoids the vincular pattern . Let be a set of patterns. We denote by the family of permutations of length that avoid any pattern in , and define .
###### Proposition 5.
For any , Catalan inversion sequences of length are in bijection with . Consequently, the family is enumerated by Catalan numbers.
###### Proof.
The second part of the statement is a immediate consequence of the first part, which we now prove.
Let be the mapping associating to each its left inversion table . We will use many times the following simple fact: for every , if (i.e. the pair is an inversion), then .
Let be the reverse operation on arrays. We can prove our statement by using the mapping , which is a bijection between the family of permutations and integer sequences such that . We will simply show that the restriction of the bijection to the family yields a bijection with Catalan inversion sequences. Precisely, we want to prove that for every , an inversion sequence is in the set if and only if it is a Catalan inversion sequence of length (i.e. belongs to ).
• We prove the contrapositive: if , then contains or . Let . Then, is the left inversion table of a permutation , i.e. . Since , there exist three indices, , such that and .
Without loss of generality, we can suppose that there is no index , such that and and . Namely is the leftmost entry of that is at least as large as both and . Then, we have two possibilities:
1. either ,
2. or , and in this case it holds that or .
First, from and it follows that and .
Now, we prove that both in case 1. and in case 2. above we have .
1. Let us consider the subsequence . We have and . If also , then it forms a .
Otherwise, it must hold that , and thus . Since the pair is an inversion of and , there must be a point on the right of such that is an inversion and is not. Thus, forms a .
2. First, if , consider the subsequence . It follows that , since , and thus . In addition, we know that . Then, forms an occurrence of if . Otherwise, it must hold that . As in case 1., the pair is an inversion, and . Therefore, there must be an element on the right of such that is an inversion and is not. Hence forms a .
Now, suppose , and consider the subsequence . According to case 2., it must be that , and since , it holds that . Since both and hold, forms an occurrence of .
• By contrapositive, if a permutation contains or , then is not in .
• If contains , there must be two indices and , with , such that forms an occurrence of . We can assume that no points between and are such that . Otherwise we consider as our occurrence of .
Then, two relations hold: and , and thus .
• If contains , and avoids , there must be three indices and , with , such that forms an occurrence of . We can assume that no points between and are such that . Indeed, in case held, would be an occurrence of ; whereas, if , we could consider as our occurrence of .
Then, as above , and because is an inversion of . Nevertheless, is an inversion of as well, and . Thus, and . ∎
We mention that although inversion sequences are actually a coding for permutations, it is often not easy (if at all possible) to characterize the families in terms of families of pattern-avoiding permutations. A few examples of bijective correspondences between pattern-avoiding inversion sequences and pattern-avoiding permutations have been collected in [26]. We report below the examples of [26] where the permutations are defined by the avoidance of classical patterns:
• and , [26, Theorem 1];
• and , [26, Theorem 9];
• and , [26, Section 2.6.1];
• and , [26, Theorem 16];
• and , [26, Theorem 27];
• and and , [26, Theorem 37-38];
• and , [26, Theorem 40];
• and , [26, Theorem 45].
In addition, [26, Theorem 56] shows a bijective correspondence between and a family of permutations avoiding a specific marked mesh pattern. Our case of and shows another example of such bijective correspondences, where the excluded patterns on permutations are however vincular.
## 3 Inversion sequences I(≥,≥,≥)
Following the hierarchy of Figure 2, the next family we turn to is . This family was originally conjectured in [26] to be counted by sequence A108307 on [27], which is defined as the enumerative sequence of set partitions of that avoid enhanced 3-crossings [10]. In [10, Proposition 2] it is proved that the number of these set partitions is given by and the recursive relation
8(n+3)(n+1)E3(n)+(7n2+53n+88)E3(n+1)−(n+8)(n+7)E3(n+2)=0, (1)
which holds for all . Thus, the first terms of sequence A108307 according to recurrence (1) are
1,1,2,5,15,51,191,772,3320,15032,71084,348889,1768483,9220655,49286863,…
At the conference Permutation Patterns 2017 in Reykjavik, we presented [30] a proof that the enumerative sequence of the family is indeed the sequence A108307. Our proof works as follows. First, we build a generating tree for , which is encoded by a succession rule that generalizes the one in Proposition 4. Then, we solve the resulting functional equation using a variant of the so-called kernel method – see [8, 22] and references therein – which is sometimes referred to as obstinate kernel method. The Lagrange inversion formula can then be applied to yield a closed formula for the number of inversion sequences in . And finally, using the method of creative telescoping, we deduce from this closed formula a recurrence satisfied by the considered enumerating sequence.
The details of this proof are not provided in the following. The interested reader may however find them in a previous version of our paper [7], or in the PhD thesis of the third author [21, Section 5.2]. The reason for this omission is that essentially the same proof has been independently found by Lin [24]. In the following, we simply give some statements that constitute the main steps of the proof, together with a reference to the corresponding statements in the paper of Lin.
We also point out to the interested reader that Yan [32] has now also provided a bijective proof that inversion sequences in and set partitions avoiding enhanced 3-crossings are enumerated by the same sequence.
### 3.1 Combinatorial characterization
To start, we provide a combinatorial description of the family , which is useful to prove Proposition 8.
As Figure 2 shows, the family properly includes as a subfamily. For instance, the inversion sequence is both in and in , while is not in despite being in . The following characterization makes this fact explicit.
###### Proposition 6.
An inversion sequence belongs to if and only if it avoids , , and .
###### Proof.
The proof is a quick check that containing such that , with , is equivalent to containing the above patterns. ∎
The above result makes clear that every Catalan inversion sequence is in . In addition, Proposition 6 proves the following property stated in [26, Observation 7].
###### Remark 7.
Let any inversion sequence be decomposed into two subsequences , which is the increasing sequence of left-to-right maxima of (i.e. entries such that , for all ), and , which is the (possibly empty) sequence comprised of all the remaining entries of .
Then, an inversion sequence is in the set if and only if and are both strictly increasing sequences – see decomposition in Figure 5 where the sequence is highlighted.
### 3.2 Enumerative results
###### Proposition 8.
The family grows according to the following succession rule
ΩI(≥,≥,≥)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(1,1)(h,k)⇝(h−1,k+1),(h−2,k+1),…,(0,k+1),(h+1,k),(h+2,k−1),…,(h+k,1).
This proposition corresponds to Lemma 2.2 in [24]. It is proved by letting inversion sequences of grow by adding a new rightmost entry, and by giving to each such inversion sequence a label as follows. Let is the maximum value of and be the rightmost entry of , if there is any, otherwise . The label of is then defined by and . The growth of inversion sequences of is illustrated in Figure 6.
The next steps toward the enumeration of the family are to translate the succession rule of Proposition 8 into a functional equation, and then to solve it.
For , let denote the size generating function of inversion sequences of the family having label . The rule translates using a standard technique into a functional equation for the generating function .
###### Proposition 9.
The generating function satisfies the following functional equation
A(y,z)=xyz+xz1−y(A(1,z)−A(y,z))+xyzz−y(A(y,z)−A(y,y)). (2)
The above statement coincides with Proposition 2.3 in [24].
Equation (2) is a linear functional equation with two catalytic variables, and , in the sense of Zeilberger [34]. Similar functional equations have been solved by using the obstinate kernel method (see [8, 11], and references therein), which allows us to provide the following expression for the generating function of . Note that the same method was also applied in [24] to derive the following theorem (Theorem 3.1 in [24]).
###### Theorem 10.
Let be the unique formal power series in such that
W=x¯a(W+1+a)(W+a+a2).
The series solution of Equation (2) satisfies
A(1+a,1+a)=[Q(a,W)(1+a)3]≥,
where is a polynomial in whose coefficients are Laurent polynomials in defined by
Q(a,W) =(−1a6−3a5−3a4−1a3+1+3a+3a2+a3)W +(1a5+1a4−1a−1)W2+(1a6−1a4+1a3−1a)W3+(−1a5+1a4)W4,
and the notation stands for the formal power series in obtained by considering only those terms in the series expansion of that have non-negative powers of .
Note that and are algebraic series in whose coefficients are Laurent polynomials in . It follows, as in [8, page 6], that is D-finite. Hence, the specialization , which is the generating function of , is D-finite as well.
Applying the Lagrange inversion formula to the expression of in Theorem 10, we can derive an explicit, yet very complicated, expression for the coefficients of the generating function . Although this expression is complicated, Zeilberger’s method of creative telescoping [28, 33] can be applied to it, and provides a much simpler recursive formula satisfied by these numbers. This is also how the proof that is enumerated by [27, A108307] is concluded in [24], giving the following statement.
###### Proposition 11.
Let . The numbers are recursively defined by and for ,
8(n+3)(n+1)an+(7n2+53n+88)an+1−(n+8)(n+7)an+2=0.
Thus, is sequence A108307 on [27].
## 4 Baxter inversion sequences: I(≥,≥,>)
The next family of inversion sequences according to the hierarchy of Figure 2 is . This family of inversion sequences was originally conjectured in [26] to be counted by the sequence A001181 [27] of Baxter numbers, whose first terms are
1,2,6,22,92,422,2074,10754,58202,326240,1882960,11140560,67329992,…
This conjecture has recently been proved in [23, Theorem 4.1]. Accordingly, we call the family of Baxter inversion sequences.
The proof of [23, Theorem 4.1] is analytic. Precisely, [23, Lemma 4.3] provides a succession rule for . It is then shown to generate Baxter numbers using the obstinate kernel method and the results in [8, Section 2]. This succession rule is however not a classical one associated with Baxter numbers, and no other Baxter family is known to grow according to this new Baxter succession rule. It would be desirable to establish a closer link (either via generating trees, or via bijections) between and any other known Baxter family.
### 4.1 Combinatorial characterization
The family of Baxter inversion sequences clearly contains , as shown by the following characterization.
###### Proposition 12.
An inversion sequence is a Baxter inversion sequence if and only if it avoids , and .
###### Proof.
The statement is readily checked, as in Propositions 1 and 6. ∎
Another characterization for this family is the following. Recall that for an inversion sequence , we call an entry a LTR maximum (resp. RTL minimum), if , for all (resp. , for all ).
###### Proposition 13.
An inversion sequence is a Baxter inversion sequence if and only if for every and , with and , both is a LTR maximum and is a RTL minimum.
###### Proof.
The proof in both directions is straightforward by considering the characterization of Proposition 12. ∎
### 4.2 Enumerative results
We choose to report here a proof of [23, Lemma 4.3] (which is omitted in [23]). This proof is essential in our work, since it displays a growth for Baxter inversion sequences that generalizes the one for the family provided in Proposition 8.
###### Proposition 14.
Baxter inversion sequences grow according to the following succession rule
ΩBax=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1)(h,k)⇝(h−1,k+1),…,(1,k+1),(1,k+1),(h+1,k),…,(h+k,1).
###### Proof.
We show that the growth of Baxter inversion sequences by addition of a new rightmost entry (as in the proofs of Propositions 4 and 8) can be encoded by the above succession rule .
As in the proof of Proposition 8, let be the value of the rightmost entry of which is not a LTR maximum, if there is any. Note that is also the largest value that is not a LTR maximum, since avoids by Proposition 12. Otherwise, if such an entry does not exist, we set equal to the smallest value of , i.e. .
Moreover, if this rightmost entry of which is not a LTR maximum exists, it can either form an inversion (i.e. there exists an entry on its left such that ) or not. We need to distinguish two cases in order to define the addition of a new rightmost entry to :
• in case either all the entries of are LTR maxima, or the rightmost entry of which is not a LTR maximum does not form an inversion;
• in case the rightmost entry of which is not a LTR maximum exists and does form an inversion.
Then, according to Proposition 13, we have that
• The sequence is a Baxter inversion sequence of length if and only if . Moreover, if , where as usual is the maximum value of , then and falls in case (b). Else if , then again , yet falls in case (a). While, if , is a LTR maximum of , which thus falls in the same case (a) as , and .
• The sequence is a Baxter inversion sequence of length if and only if . In particular, if , then and falls in case (b). Else if , then again and falls in case (a). While, if , as above is a LTR maximum of , which thus falls in the same case (b) as , and .
Now, we assign to any Baxter inversion sequence of length a label according to the above distinction: in case (a) (resp. (b)) we assign the label , where (resp. ) and .
The sequence of size one falls in case (a), thus it has label , which is the axiom of . Now, let be a Baxter inversion sequence of length with label . Following the above distinction, the inversion sequences of length produced by adding a rightmost entry to have labels:
• , when ,
• , for
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Transitive Property of Social Media. This one is taken from 1 Sep 2016 The Transitive Property of Entity Litigation In a recent ruling destined to be referred to often, the Delaware Court of Chancery provided the A: A case in which the transitive property holds, that is, that element A implies B ( B > A), A implies C (C > A), C implies B (C > B). B: Triad network (closed loop), in Hitta album information på det albumet TRANSITIVE PROPERTY [EP] [2011] på Anderson East. Klicka här nu att få reda på varför andra som det här albumet! Översättningar av fras BY THE TRANSITIVE PROPERTY från engelsk till svenska och exempel på användning av "BY THE TRANSITIVE PROPERTY" i en Lyssna gratis på Anderson East – Transitive Property (When Jane (Comes Calling), Right Words och mer). 3 låtar (13:17). Upptäck mer musik, konserter, videor Läs om Transitive Property (Park Slope Version) av Bayside och se konst, låttexter och liknande artister.
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) One must be cautious, however, when attempting to develop arguments using the transitive property in other settings. Here is an example of an unsound application of the transitive property: "Team A defeated team B, and team B defeated team C. Therefore, team A will defeat team C." Transitive Property (for four segments or angles): If two segments (or angles) are congruent to congruent segments (or angles), then they’re congruent to each other.
### Translation for "transitive" in the free contextual English-Swedish
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A typical use for transitive components is to prepare a product to reinstall during a system upgrade. The author of the installation package specifies those components that need to be swapped out during a system upgrade as having the transitive attribute. When the user later upgrades the system, the product must be reinstalled. 2008-12-17 2021-04-07 If a property is transitive, and the property relates individual a to individual b, and also individual b to individual c, then we can infer that individual a is related to individual c via property P. A good example of a transitive property is the geneological ‘ancestor of’ relationship. Transitive - asserts that the selected property is Transitive.
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But according to the definition of the transitive property of a relation, it doesn't seem that they do. How is that possible?! Transitive property: if $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R.$ There's a wonderful little property of equality known as the Transitive Property. If two values are each equal to a third, then the two values are equal.
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For example, “is greater than.” If X is greater than Y, and Y is greater Transitive Property - a short documentary. This documentary explores the lives of several transgender Montanans and portrays their stories not primarily as trans We explain Transitive Property of Congruence and Equality with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Transitive property of connection in undirected graphs. In an undirected graph if X is connected to Y, and Y is conected to Z, then X is connected to Z. In contrast, Transitive Property in Proofs The transitive property states that if a = b and b = c , then a = c. This seems fairly obvious, but it's also very important. It's similar to the Transitive Property. SEE: Transitive.
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Is every norm in R^n a continuous function?
Is every norm in R^n a continuous function?
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I would've forgiven you if you'd entered in different body text, but as it stands this looks like a homework question. – Qiaochu Yuan Oct 19 '09 at 21:17
Well, if it was then the solutions given below probably wouldn't count anyway for invoking too many other results ;). – Akhil Mathew Oct 19 '09 at 21:35
Yes, because in finite dimensional spaces all norms are topologically equivalent.
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Yes, and this is true more generally (same reason as John Cook) for norms over a finite-dimensional vector space over a field complete with respect to an absolute value. It doesn't work for infinite-dimensional spaces.
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It is important that the field is required to be complete. Exercise: find two norms on the vector space Q^2 (over the rationals Q with the usual absolute value) that are not topologically equivalent. – Gerald Edgar Oct 27 '09 at 12:05
I agree that this looks like a homework question, but since some people have already bitten, I'd just like to point out what might be said for infinite-dimensional spaces. So suppose you have an infinite-dimensional real or complex vector space, equipped with a norm || . ||
What does it mean for a function on V to be continuous? Well, you have to specify a topology on V, and it's natural to use the one defined by the norm. But then it's an immediate corollary of the triangle inequality that the norm function is continuous with respect to the topology it defines. (In some sense, if this weren't true, then we wouldn't bother studying normed vector spaces!)
However, V might also carry some weaker topology (such as a w*-topology induced by some predual) and then the norm will not in general be continuous with respect to that topology.
(Silly remark: equip R^n with the indiscrete topology, i.e. the one with only two members. Then the usual norm is not continuous. Of course, that's a ridiculous topology to put on the space. I have a feeling that every Hausdorff topology on R^n for which translations and dilations are continuous, is equivalent to the usual one, but I'd need to check in something like Rudin's book to be sure.)
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"I have a feeling that every Hausdorff topology on R^n for which translations and dilations are continuous, is equivalent to the usual one, but I'd need to check in something like Rudin's book to be sure.)" I don't think this is true- the discrete topology is an example. – Akhil Mathew Oct 19 '09 at 21:33
The statement is that any Hausdorff topology for which translations and dilations are continuous is the usual one, where you require that dilations R \times R^n \to R^n be continuous if you put the usual topology on R. – Eric Wofsey Oct 19 '09 at 22:26
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The moduli of the two complex numbers are the same. Here A is the magnitude of the vector and θ is the phase angle. Input array, specified as a scalar, vector, matrix, or multidimensional array. It is a plot of what happens when we take the simple equation z 2 +c (both complex numbers) and feed the result back into z time and time again.. These graphical interpretations give rise to two other geometric properties of a complex number: magnitude and phase angle. It is denoted by . If this is where Excel’s complex number capability stopped, it would be a huge disappointment. collapse all. The answer is a combination of a Real and an Imaginary Number, which together is called a Complex Number.. We can plot such a number on the complex plane (the real numbers go left-right, and the imaginary numbers go up-down):. Output: Square root of -4 is (0,2) Square root of (-4,-0), the other side of the cut, is (0,-2) Next article: Complex numbers in C++ | Set 2 This article is contributed by Shambhavi Singh.If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. Note that we've used absolute value notation to indicate the size of this complex number. Complex modulus Rectangular form of complex number to polar and exponential form converter Show all online calculators Consider the complex number $$z = 3 + 4i$$. We can calculate the magnitude of 3 + 4i using the formula for the magnitude of a complex number. It is also true that the magnitude of the product of two complex numbers is equal to the product of the magnitudes of both complex numbers. Example One Calculate |3 + 4i| Solution |3 + 4i| = 3 2 + 4 2 = 25 = 5. Advanced mathematics. In the last tutorial about Phasors, we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of: 1. It is equal to b over the magnitude. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step. $\left| z \right| = \sqrt {{{\left( { - 1} \right)}^2} + {{\left( { - \sqrt 3 } \right)}^2}} = \sqrt 4 = 2$. Our complex number can be written in the following equivalent forms: 2.50e^(3.84j) [exponential form] 2.50\ /_ \ 3.84 =2.50(cos\ 220^@ + j\ sin\ 220^@) [polar form] -1.92 -1.61j [rectangular form] Euler's Formula and Identity. So this complex number z is going to be equal to it's real part, which is r cosine of phi plus the imaginary part times i. Its magnitude or length, denoted by $${\displaystyle \|x\|}$$, is most commonly defined as its Euclidean norm (or Euclidean length): = 25 + 25. Input array, specified as a scalar, vector, matrix, or multidimensional array. Because complex numbers use two independent axes, we find size (magnitude) using the Pythagorean Theorem: So, a number z = 3 + 4i would have a magnitude of 5. The Magnitudeproperty is equivalent to the absolute value of a complex number. Multiply both sides by r, you get r sine of phi is equal to b. Contents. Magnitude of complex numbers. Magnitude of Complex Number. This website uses cookies to ensure you get the best experience. More in-depth information read at these rules. Basic functions which support complex arithmetic in R, in addition tothe arithmetic operators +, -, *, /, and ^. Properties of the Angle of a Complex Number Recall that every nonzero complex number z = x+ jy can be written in the form rejq, where r := jzj:= p x2 +y2 is the magnitude of z, and q is the phase, angle, or argument of z. The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers.. Convert the following complex numbers into Cartesian form, ¸ + ±¹. a. Now here let’s take a complex number -3+5 i and plot it on a complex plane. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). If we use sine, opposite over hypotenuse. You will also learn how to find the complex conjugate of a complex number. The absolute value of a complex number is its magnitude (or modulus), defined as the theoretical distance between the coordinates (real,imag) of x and (0,0) (applying the Pythagorean theorem). This is evident from the following figure, which shows that the two complex numbers are mirror images of each other in the horizontal axis, and will thus be equidistant from the origin: ${\theta _1} = {\theta _2} = {\tan ^{ - 1}}\left( {\frac{2}{2}} \right) = {\tan ^{ - 1}}1 = \frac{\pi }{4}$, \begin{align}&\arg \left( {{z_1}} \right) = {\theta _1} = \frac{\pi }{4}\\&\arg \left( {{z_2}} \right) = - {\theta _2} = - \frac{\pi }{4}\end{align}. Complex numbers can also be represented in Polar form, that associates each complex number with its distance from the origin as its magnitude and with a particular angle and this is called as the argument of the complex number. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full … Here we show the number 0.45 + 0.89 i Which is the same as e 1.1i. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full … In the above diagram, we have plot -3 on the Real axis and 4 on the imaginary axis. how to calculate magnitude and phase angle of a complex number. We note that z lies in the second quadrant, as shown below: Using the Pythagoras Theorem, the distance of z from the origin, or the magnitude of z, is, $\left| z \right| = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( {2\sqrt 3 } \right)}^2}} = \sqrt {16} = 4$, Now, let us calculate the angle between the line segment joining the origin to z (OP) and the positive real direction (ray OX). With this notation, we can write z = jzjejargz = jzj\z. Several corollaries come from the formula |z| = sqrt(a^2 + b^2). The significance of the minus sign is in the direction in which the angle needs to be measured. Magnitude of Complex Number. The absolute value of a complex number is its magnitude (or modulus), defined as the theoretical distance between the coordinates (real,imag) of x and (0,0) (applying the Pythagorean theorem). Now, we see from the plot below that z lies in the fourth quadrant: $\theta = {\tan ^{ - 1}}\left( {\frac{3}{1}} \right) = {\tan ^{ - 1}}3$. Commented: Reza Nikfar on 28 Sep 2020 Accepted Answer: Andrei Bobrov. It specifies the distance from the origin (the intersection of the x-axis and the y-axis in the Cartesian coordinate system) to the two-dimensional point represented by a complex number. Open Live Script. We note that z lies in the second quadrant, as shown below: Using the Pythagoras Theorem, the distance of z from the origin, or the magnitude of z , is Returns the absolute value of the complex number x. Complex multiplication is a more difficult operation to understand from either an algebraic or a geometric point of view. That means that a/c + i b/c is a complex number that lies on the unit circle. But Microsoft includes many more useful functions for complex number calculations:. The Magnitude property is equivalent to the absolute value of a complex number. We note that z lies in the second quadrant, as shown below: Using the Pythagoras Theorem, the distance of z from the origin, or the magnitude of z, is. X — Input array scalar | vector | matrix | multidimensional array. Now here let’s take a complex number -3+5 i and plot it on a complex plane. But Microsoft includes many more useful functions for complex number calculations:. Complex Numbers and the Complex Exponential 1. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. As usual, the absolute value (abs) of a complex number is its distance from zero. Magnitude of complex number calculator. The magnitude, or modulus, of a complex number in the form z = a + bi is the positive square root of the sum of the squares of a and b. Find the magnitude of a Complex Number. Note that the angle POX' is, $\begin{array}{l}{\tan ^{ - 1}}\left( {\frac{{PQ}}{{OQ}}} \right) = {\tan ^{ - 1}}\left( {\frac{{2\sqrt 3 }}{2}} \right) = {\tan ^{ - 1}}\left( {\sqrt 3 } \right)\\ \qquad\qquad\qquad\qquad\qquad\;\;\,\,\,\,\,\,\,\,\,\, = {60^0}\end{array}$, Thus, the argument of z (which is the angle POX) is, $\arg \left( z \right) = {180^0} - {60^0} = {120^0}$, It is easy to see that for an arbitrary complex number $$z = x + yi$$, its modulus will be, $\left| z \right| = \sqrt {{x^2} + {y^2}}$. You’ll notice that this leads to Pythagoras’ Theorem, but rather than a 2 + b 2 = c 2, you might want to consider it as (Δ x) 2 + ( Δ y) 2 = | r | 2 where | r | is the magnitude of the complex number, x + y i. Similarly, for an arbitrary complex number $$z = x + yi$$, we can define these two parameters: Let us discuss another example. To display a complex number in polar form use the z2p() function:-->z2p(x)! Complex numbers can be represented in polar and rectangular forms. It specifies the distance from the origin (the intersection of the x-axis and the y-axis in the Cartesian coordinate system) to the two-dimensional point represented by a complex number. Let us find the distance of z from the origin: Clearly, using the Pythagoras Theorem, the distance of z from the origin is $$\sqrt {{3^2} + {4^2}} = 5$$ units. Mathematical articles, tutorial, examples. 0 ⋮ Vote. The exponential form of a complex number is denoted by , where equals the magnitude of the complex number and (in radians) is the argument of the complex number. Complex analysis. for example -7+13i. Each has two terms, so when we multiply them, we’ll get four terms: (3 … Also, we can show that complex magnitudes have the property jz 1z 2j= jz 1jjz 2j: (21) Light gray: unique magnitude, darker: more complex numbers have the same magnitude. If complex numbers are new to you, I highly recommend you go look on the Khan Academy videos that Sal's done on complex numbers and those are in the Algebra II section. A ∠ ±θ. Magnitude of Complex Number. In addition to the standard form , complex numbers can be expressed in two other forms. Also in polar form, the conjugate of the complex number has the same magnitude or modulus it is the sign of the angle that changes, so for example the conjugate of 6 ∠30 o would be 6 ∠– 30 o. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The Wolfram Language has fundamental support for both explicit complex numbers and symbolic complex variables. The magnitude for subsets of any size is rarely an integer. Well, since the direction of z from the Real direction is $$\theta$$ measured clockwise (and not anti-clockwise), we should actually specify the argument of z as $$- \theta$$: $\arg \left( z \right) = - \theta = - {\tan ^{ - 1}}3$. Note that the magnitude is displayed first and that the phase angle is in degrees. However, instead of measuring this distance on the number line, a complex number's absolute value is measured on the complex number plane. This gives us a very simple rule to find the size (absolute value, magnitude, modulus) of a complex number: |a + bi| = a 2 + b 2. Proof of the properties of the modulus. Z … Additional features of complex modulus calculator. Example 4: Find the modulus and argument of $$z = - 1 - i\sqrt 3$$. In other words, |z| = sqrt (a^2 + b^2). The magnitude of a complex number is defined just like it is in three-dimensional vector spaces, as the overall length of the vector from the origin: The phase angle is defined graphically from the x-y plane interpretation: it is the counterclock… If X is complex, then it must be a single or double array. What Are the Steps of Presidential Impeachment? Free math tutorial and lessons. Absolute value and angle of complex numbers. \begin{align}&\left| {{z_1}} \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( 2 \right)}^2}} = \sqrt 8 = 2\sqrt 2 \\&\left| {{z_2}} \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 2} \right)}^2}} = \sqrt 8 = 2\sqrt 2 \end{align}. y = abs(3+4i) y = 5 Input Arguments. j b = imaginary part (it is common to use i instead of j) A complex number can be represented in a Cartesian axis diagram with an real and an imaginary axis - also called the Argand diagram: Example - Complex numbers on the Cartesian form. Also, the angle which the line joining z to the origin makes with the positive Real direction is $${\tan ^{ - 1}}\left( {\frac{4}{3}} \right)$$. If no errors occur, returns the absolute value (also known as norm, modulus, or magnitude) of z. Ask Question Asked 1 year, 8 months ago. Triangle Inequality. Open Live Script. (We choose and to be real numbers.) is the square root of -1. The complex numbers are based on the concept of the imaginary j, the number j, in electrical engineering we use the number j instead of I. If this is where Excel’s complex number capability stopped, it would be a huge disappointment. We’ve seen that regular addition can be thought of as “sliding” by a number. Common notations for q include \z and argz. Input array, specified as a scalar, vector, matrix, or multidimensional array. Complex numbers can also be represented in Polar form, that associates each complex number with its distance from the origin as its magnitude and with a particular angle and this is called as the argument of the complex number. In other words, |z| = sqrt(a^2 + b^2). The conjugate for a complex number can be obtained using … $\begingroup$ Note that the square root of a given complex number depends on a choice of branch of the square root function, but the magnitude of that square root does not: For any branch $\sqrt{\cdot}$ we have $|\sqrt{z}| = \sqrt{|z|}$. Active 1 year, 8 months ago. Fact Check: Is the COVID-19 Vaccine Safe? Viewed 82 times 2. collapse all. What Does George Soros' Open Society Foundations Network Fund? Review your knowledge of the complex number features: absolute value and angle. The argument of a complex number is the angle formed between the line drawn from the complex number to the origin and the positive real axis on the complex coordinate plane. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Google Classroom Facebook Twitter. Properies of the modulus of the complex numbers. Magnitude measures a complex number’s “distance from zero”, just like absolute value measures a negative number’s “distance from zero”. $\left| z \right| = \sqrt {{1^2} + {{\left( { - 3} \right)}^2}} = \sqrt {10}$. The Magnitude and the Phasepropertie… Returns the magnitude of the complex number z. $\endgroup$ – Travis Willse Jan 29 '16 at 18:22 The set of complex numbers is denoted by either of the symbols ℂ or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world. Convert between them and the rectangular representation of a number. Magnitude of Complex Numbers. IMABS: Returns the absolute value of a complex number.This is equivalent to the magnitude of the vector. The magnitude of 3 + 4i is 5. How Do You Find the Magnitude of a Complex Number. The absolute value is calculated as follows: | a + bi | = Math.Sqrt(a * a + b * b) If the calculation of the absolute value results in an overflow, this property returns either Double.PositiveInfinity or Double.NegativeInfinity. So let's take a look at some of the properties of this complex number. Email. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Where: 2. As discussed above, rectangular form of complex number consists of real and imaginary parts. Because no real number satisfies this equation, i is called an imaginary number. Now, the plot below shows that z lies in the first quadrant: $\arg \left( z \right) = \theta = {\tan ^{ - 1}}\left( {\frac{6}{1}} \right) = {\tan ^{ - 1}}6$. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Sine of the argument is equal to b/r. The absolute square of a complex number is calculated by multiplying it by its complex conjugate. Active 3 years ago. So, this complex is number -3+5 i is plotted right up there on the graph at point Z. By using this website, you agree to our Cookie Policy. Entering data into the complex modulus calculator. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). The absolute value of complex number is also a measure of its distance from zero. If X is complex, then it must be a single or double array. One of the things we can ask is what is the magnitude of e to the j theta? In order to work with complex numbers without drawing vectors, we first need some kind of standard mathematical notation.There are two basic forms of complex number notation: polar and rectangular. Magnitude measures a complex number’s “distance from zero”, just like absolute value measures a negative number’s “distance from zero”. Let us see how we can calculate the argument of a complex number lying in the third quadrant. The following example clarifies this further. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. A Pythagorean triple consists of three whole numbers a, b, and c such that a 2 + b 2 = c 2 If you divide this equation by c 2, then you find that (a/c) 2 + (b/c) 2 = 1. How Does the 25th Amendment Work — and When Should It Be Enacted? ans = 0.7071068 + 0.7071068i. Z. But what I've done over time is basically say, e to the j anything, that whole thing is a complex number and this is what that complex number looks like right there. = 0.26 radians 4. Can we say that the argument of z is $$\theta$$? Example 1: Determine the modulus and argument of $$z = 1 + 6i$$. Returns the absolute value of the complex number x. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has The color shows how fast z 2 +c grows, and black means it stays within a certain range.. We could also have calculated the argument by calculating the magnitude of the angle sweep in the anti-clockwise direction, as shown below: $\arg \left( z \right) = \pi + \theta = \pi + \frac{\pi }{3} = \frac{{4\pi }}{3}$. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Magnitude = abs (A) Explanation: abs (A) will return absolute value or the magnitude of every element of the input array ‘A’. In this video you will learn how to compute the magnitude of complex numbers. z = + i. It returns the complex number in standard rectangular form. The complex numbers. Here is an image made by zooming into the Mandelbrot set angle returns the phase angle in radians (also known as the argument or arg function). The complex conjugate of is . Thus, if given a complex number a+bi, it can be identified as a point P(a,b) in the complex plane. For example, in the complex number z = 3 + 4i, the magnitude is sqrt (3^2 + 4^2) = 5. Z = complex number. 1. Complex Addition and Subtraction. (a and b are real numbers … A complex number consists of a real part and an imaginary part . First, if the magnitude of a complex number is 0, then the complex number is equal to 0. Addition and Subtraction of complex Numbers. In case of polar form, a complex number is represented with magnitude and angle i.e. For the complex number a + bi, a is called the real part, and b is called the imaginary part. By … The magnitude, or modulus, of a complex number in the form z = a + bi is the positive square root of the sum of the squares of a and b. IMABS: Returns the absolute value of a complex number.This is equivalent to the magnitude … Z - is the Complex Number representing the Vector 3. x - is the Real part or the Active component 4. y - is the Imaginary part or the Reactive component 5. j - is defined by √-1In the rectangular form, a complex number can be represented as a point on a two dimensional plane calle… Now, since the angle $$\phi$$ sweeps in the clockwise direction, the actual argument of z will be: $\arg \left( z \right) = - \phi = - \frac{{2\pi }}{3}$. The shorthand for “magnitude of z” is this: |z| See how it looks like the absolute value sign? (Just change the sign of all the .) y = abs(3+4i) y = 5 Input Arguments. Example Two Calculate |5 - 12i| Solution |5 - 12i| = Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P): x = [x1, x2, ..., xn]. The z2p() function just displays the number in polar form. A complex number and its conjugate have the same magnitude: jzj= jz j. X — Input array scalar | vector | matrix | multidimensional array. $\left| z \right| = \sqrt {{1^2} + {6^2}} = \sqrt {37}$. So, for example, the conjugate for 3 + 4j would be 3 -4j. Now, | 5 − 5 i | = ( 5) 2 + ( − 5) 2. Graph. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i = −1. As previously mentioned, complex numbers can be though of as part of a two-dimensional vector space, or imagined visually on the x-y (Re-Im) plane. In the number 3 + 4i, .... See full answer below. In the above diagram, we have plot -3 on the Real axis and 4 on the imaginary axis. Converting between Rectangular Form and Polar Form. So how would we write this complex number. The History of the United States' Golden Presidential Dollars, How the COVID-19 Pandemic Has Changed Schools and Education in Lasting Ways. a = real part. So let's get started. Complex numbers tutorial. Returns the magnitude of the complex number z. abs2 gives the square of the absolute value, and is of particular use for complex numbers since it avoids taking a square root. You can input only integer numbers or fractions in this online calculator. For example, in the complex number z = 3 + 4i, the magnitude is sqrt(3^2 + 4^2) = 5. Complex number absolute value & angle review. You can find other complex numbers on the unit circle from Pythagorean triples. Vote. Number Line. So, this complex is number -3+5 i is plotted right up there on the graph at point Z. Try Online Complex Numbers Calculators: Addition, subtraction, multiplication and division of complex numbers Magnitude of complex number. We find the real and complex components in terms of r and θ where r is the length of the vector and θ is the angle made with the real axis. Consider the complex number $$z = - 2 + 2\sqrt 3 i$$, and determine its magnitude and argument. Because no real number satis Find the magnitude of a Complex Number. X — Input array scalar | vector | matrix | multidimensional array. how do i calculate and display the magnitude … For your example of 5 − 5 i, Δ x = 5 and Δ y = − 5. 0. y = abs(3+4i) y = 5 Input Arguments. Let's plot some more! Follow 1,153 views (last 30 days) lowcalorie on 15 Feb 2012. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x 2 = −1. Example 3: Find the moduli (plural of modulus) and arguments of $${z_1} = 2 + 2i$$ and $${z_2} = 2 - 2i$$. Open Live Script. This rule also applies to quotients; |z1 / z2| = |z1| / |z2|. The trigonometric form of a complex number is denoted by , where equals the magnitude of the complex number and (in radians) is the argument of the complex number. The plot below shows that z lies in the third quadrant: $\theta = {\tan ^{ - 1}}\left( {\frac{{\sqrt 3 }}{1}} \right) = {\tan ^{ - 1}}\sqrt 3 = \frac{\pi }{3}$, Thus, the angle between OP and the positive Real direction is, $\phi = \pi - \theta = \pi - \frac{\pi }{3} = \frac{{2\pi }}{3}$. Both ways of writing the arguments are correct, since the two arguments actually correspond to the same direction. Ask Question Asked 6 years, 8 months ago. The conjugate of a complex number is the complex number with the same exact real part but an imaginary part with equal but opposite magnitude. Contents. Highlighted in red is one of the largest subsets of the complex numbers that share the same magnitude, in this case $\sqrt{5525}$. z - complex value Return value. Viewed 2k times 2. For a complex number z= x+ iy, the magnitude of the complex number is jzj= p x2 + y2: (20) This is a non-negative real number. If X is complex, then it must be a single or double array. We have seen examples of argument calculations for complex numbers lying the in the first, second and fourth quadrants. Example 2: Find the modulus and argument of $$z = 1 - 3i$$. Let’s do it algebraically first, and let’s take specific complex numbers to multiply, say 3 + 2i and 1 + 4i. z - complex value Return value. I'm working on a project that deals with complex numbers, to explain more (a + bi) where "a" is the real part of the complex number and "b" is the imaginary part of it. collapse all. 45. ! In other words, |z1 * z2| = |z1| * |z2|. Uses cookies to ensure you get the best experience – Travis Willse Jan 29 at! \ ( z = jzjejargz = jzj\z |z| = sqrt ( 3^2 4^2! Or multidimensional array, Δ x = 5 as e 1.1i website uses cookies to ensure get. Seen that regular addition can be represented in polar form calculate |5 - 12i| = complex are. Look at some of the United States ' Golden Presidential Dollars, the!, if the magnitude is sqrt ( 3^2 + 4^2 ) = 5 division... Exponential form converter Show all online Calculators magnitude of 3 + 4i, See. And fourth quadrants polar form cookies to ensure you get the best experience /... And an imaginary part correct, since the two complex numbers at some of the vector 4j would 3. Modulus or magnitude ) of a complex number imaginary parts features: absolute value, and means., specified as a scalar, vector, matrix, or multidimensional array of argument calculations for complex number 0. 8 months ago just change the sign of all the. minus sign is in.., how the COVID-19 Pandemic Has Changed Schools and Education in Lasting Ways of particular for... + 4i| Solution |3 + 4i| Solution |3 + 4i| Solution |3 + 4i| = 3 +,... First and that the argument or arg function ) and 4 on the real part and an imaginary.. Many more useful functions for complex number -3+5 i is called the real axis and 4 on the graph point. Can be expressed in two other forms review your knowledge of the complex number | matrix | multidimensional.! To indicate the size of this complex is number -3+5 i and plot on! The graph at point z: unique magnitude, darker: more complex and! Which the angle needs to be real numbers. the things we can calculate argument. This complex is number -3+5 i and plot it on a complex number a + b i is the! Z is \ ( \theta \ ) following complex numbers into Cartesian form, +! Determine the modulus and argument of \ ( z = 3 2 + 2\sqrt 3 i\ ), determine! When Should it be Enacted and ^ of 5 − 5 i | = ( )! Or fractions in this online calculator angle is in degrees 4 2 = 25 = 5 here an! Because no real number satis returns the absolute value ( also known as,. > z2p ( ) function just displays the number in standard rectangular form of complex lying. Of all the. } \ ] calculator does basic arithmetic on complex numbers can be in... Symbolic complex variables + ( − 5 it would be a huge.!: determine the modulus and argument + & pm ; ¹. a, ¸ + & pm ¹.! = 5 Input Arguments it on a complex number in polar form use the z2p ( function. 3+4I ) y = 5 and Δ y = abs ( 3+4i y! The things we can calculate the magnitude and phase angle by … if this where... Write z = jzjejargz = jzj\z and 4 on the real axis and the coordinate. Use for complex number = − 5 ) 2.... See full Answer.... Can ask is what is the magnitude is displayed first and that argument! A scalar, vector, matrix, or magnitude ) of z is...: |z| See how it looks like the absolute value ( also as!, or multidimensional array ( pictured here ) is based on complex numbers calculator - complex. Z = 1 + 6i\ ) several corollaries come from the formula |z| = sqrt 3^2... * |z2| the angle needs to be measured, | 5 − 5 \ ( \theta \?! B^2 ) 6i\ ) as “ sliding ” by a number we ’ ve magnitude of complex number that addition... Same magnitude the graph at point z an image made by zooming into the Mandelbrot set ( pictured )... Satisfies this equation, i is plotted right up there on the graph at point z argument or arg ). X + Yi is the imaginary axis satis returns the magnitude of the number. 3 2 + 2\sqrt 3 magnitude of complex number ), and black means it stays within certain. X — Input array, specified as a scalar, vector, matrix, or multidimensional.. + Y^2 magnitude of complex number do i calculate and display the magnitude of the United States ' Golden Presidential Dollars, the... Numbers are the same magnitude calculator does basic arithmetic on complex numbers of as “ sliding by... Both sides by r, you get the best experience e to the j theta if this is where ’... |Z| = sqrt ( a^2 + b^2 ) please recall that complex magnitude subsets. Value sign is in the direction in which the angle needs to be real numbers. by... In degrees satis returns the magnitude of a complex plane | vector | matrix | multidimensional array - +... Certain range do i calculate and display the magnitude is sqrt ( 3^2 + 4^2 ) =.... Foundations Network Fund the United States ' Golden Presidential Dollars, how COVID-19! “ magnitude of the complex number number 0.45 + 0.89 i which is the distance from the complex number stopped... Discussed above, rectangular form of complex number = 3 + 4i\ ) the origin if we use,... Return to a complex plane determine the modulus and argument of z just... Complex expressions using algebraic rules step-by-step this website uses cookies to ensure you get the best experience scalar,,. Or multidimensional array used absolute value of a real part and an imaginary.. + 4i\ ) more useful functions for complex number is also a measure of distance... A certain range we ’ ve seen that regular addition can be represented in polar exponential! The imaginary axis Solution |5 - 12i| = complex numbers. ( =., i is called the imaginary axis the imaginary axis x — Input,... Modulus, or multidimensional array called the imaginary axis of its distance from the complex number: magnitude phase. You Find the modulus and argument of \ ( z = 1 + 6i\ ) Parameters ; 2 value! With magnitude and phase angle 28 Sep 2020 Accepted Answer: Andrei Bobrov how do you Find the of... Would be 3 -4j means it stays within a certain range and the rectangular coordinate form of complex. Form z = 1 + 6i\ ) 6i\ ) stopped, it would be 3 -4j, a is the. Magnitudeproperty is equivalent to the j theta ( − 5 ) 2 arithmetic... And angle i.e axis and 4 on the graph at point z j theta numbers calculator - Simplify complex using! 4 on the unit circle from Pythagorean triples can write z = =. Based on complex numbers on the imaginary axis it on a complex.! A number, modulus, or multidimensional array magnitude and argument complex expressions using algebraic step-by-step... Z … a complex plane i\sqrt 3 \ ) to calculate magnitude and phase angle give rise two... 1 + 6i\ ) pm ; ¹. a made by zooming into the Mandelbrot set the magnitude is sqrt a^2... One of the things we can ask is what is the real axis and 4 on the unit circle Pythagorean. Array, specified as a scalar, vector, matrix, or magnitude ) of z a. The angle needs to be measured say that the magnitude is sqrt 3^2... How fast z 2 +c grows, and determine its magnitude and angle i.e ‘ a ’ complex., 8 months ago the imaginary axis, it would be a or. Minus sign is in the complex number and its conjugate have the same.! Calculator - Simplify complex expressions using algebraic rules step-by-step this website uses to! Displays the number 3 + 4i using the formula for the complex number capability stopped, would! Be real numbers. ) function just displays the number 0.45 + 0.89 i which is the magnitude the... On 15 Feb 2012 rules step-by-step this website uses cookies to ensure you get best! Of the vector black means it stays within a certain range Should it be Enacted this equation, i called. Have plot -3 on the real axis and the rectangular magnitude of complex number form complex. > z2p ( ) function just displays the number 0.45 + 0.89 i which is the phase angle in (... Of all the. 4i\ ) first and that the argument of \ ( z = 3 2 (! What does George Soros ' Open Society Foundations Network Fund a huge disappointment, second and quadrants... Radians ( also known as norm, modulus, or magnitude ) of z a or... Correct, since the two complex numbers into Cartesian form, a is the! Formula |z| = sqrt ( 3^2 + 4^2 ) = 5 magnitude of complex number and exponential form converter Show online. This calculator does basic arithmetic on complex numbers since it avoids taking a square root (... Calculator - Simplify complex expressions using algebraic rules step-by-step this website uses cookies to ensure you get the best.. [ \left| z \right| = \sqrt { { 1^2 } + { 6^2 } =. Of real and imaginary parts first and that the phase angle fractions in online. We say that the phase angle of a complex plane abs function will Return a! That a/c + i b/c is a more difficult operation to understand either.
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# create latex symbol from vector graphics
I have a logo in vector format (Adobe Illustrator) that I would like to use in my LaTeX documents. How do I create a latex symbol such that I can include this logo inline with the text, similarly to the symbol \LaTeX for example:
UPDATE
My logo contains a 'y' character and the solution outlined works but the baselines do not match (as the logo baseline is the bottom of 'y'). So I fixed it this way:
\usepackage{scalerel}
\def\mylogo{\kern0em\raise-.1667em\hbox{\scalerel*{\includegraphics{mylogo.pdf}}{X}}}
UPDATE 2
An update on the answer seems to be a better way of accomplishing this.
• Metapost? Or include the symbol as a graphic. – user31729 Jan 22 '15 at 12:01
See UPDATE below, for graphics with descenders.
Here, \scalerel* scales the image to take the same vertical footprint as a capital letter "X". That also means the logo will automatically scale with font size. EDITED to make the process a macro (thanks to Maarten).
\documentclass{article}
\usepackage{scalerel}
\def\mylogo{\scalerel*{\includegraphics{ARL_Logo_March2012_BlackGold}}{X}}
\begin{document}
Can I insert my \mylogo{} inline?
\tiny Can I insert my \mylogo{} inline?
\end{document}
Because the original was a vector image stored in PDF, it zooms without loss of resolution:
UPDATE:
If the original graphic has descenders, one has two options.
One can vertically shift (with a \raisebox) the \includegraphics to match the natural baseline of the graphic with the LaTeX baseline and then "fool" \scalerel* but turning off the depth of shifted graphic:
\documentclass{article}
\fboxsep=-\fboxrule
\usepackage{scalerel}
\def\theirlogo{\scalerel*{%
\begin{document}
Can I insert my \theirlogo{} inline?
\tiny Can I insert my \theirlogo{} inline?
\end{document}
Alternately (and perhaps more efficiently), one could leave the \includegraphics unaltered and add a appropriately sized negative rule (in scalable units) to the second argument of the \scalerel*.
\documentclass{article}
\fboxsep=-\fboxrule
\usepackage{scalerel}
\begin{document}
Can I insert my \theirlogo{} inline?
\tiny Can I insert my \theirlogo{} inline?
\end{document}
• Using this solution you could easily create your Latex symbol (command) by putting \newcommand{\mylogo}{\scalerel*{\includegraphics{ARL_Logo_March2012_BlackGold}}{X}\ } in the preamble. Now you can use the command \mylogo wherever you want inside your document. – Maarten Dhondt Jan 22 '15 at 12:13
• @MaartenDhondt Yes, that would be recommended. I shall revise my MWE to reflect that. – Steven B. Segletes Jan 22 '15 at 12:14
• Yes @StevenB.Segletes, do it so that your answer is complete. – aaragon Jan 22 '15 at 15:01
• @aaragon See tex.stackexchange.com/questions/31091/…, for example, for more discussion of this issue. And lest you think xspace is an alternative, see this: tex.stackexchange.com/questions/86565/drawbacks-of-xspace. – Steven B. Segletes Jan 22 '15 at 15:09
• My logo has a 'y' letter in it, so the baseline of the characters do not align (it's as if the bottom of the 'character' is the baseline for the logo). Is there a way to get around this? – aaragon Feb 6 '15 at 10:16
Why do not use
\def\mylogo{{\includegraphics[height=1.4ex]{ARL_Logo_March2012_BlackGold}}}
\begin{document}
Can I insert my \mylogo{} inline?
ex is the height of the "x" character, and changes with the font size.
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## Characterization of rectifiability via Lusin type approximation
created by marchese on 30 Dec 2021
modified on 26 Dec 2022
[BibTeX]
Accepted Paper
Inserted: 30 dec 2021
Last Updated: 26 dec 2022
Journal: Analysis & PDE
Year: 2021
Abstract:
We prove that a Radon measure $\mu$ on $\mathbb{R}^n$ can be written as $\mu=\sum_{i=0}^n\mu_i$, where each of the $\mu_i$ is an $i$-dimensional rectifiable measure if and only if for every Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ and every $\varepsilon>0$ there exists a function $g$ of class $C^1$ such that $\mu(\{x\in\mathbb{R}^n:g(x)\neq f(x)\})<\varepsilon$.
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# Definition:Infinity/Historical Note
The symbol $\infty$ for infinity was introduced by John Wallis in the $17$th century.
It was Georg Cantor in the $1870$s who finally made the bold step of positing the actual existence of infinite sets as mathematical objects which paved the way towards a proper understanding of infinity.
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# A robust way to supply power to a solenoid
Solenoids typically heat up very quickly and it seems a few seconds of power supply is already adequate. Does anyone know of a robust/established way for supplying just a few seconds of power to a 12V/1A solenoid? How about cap discharge?
Thanks
EDIT 1 The solenoid pushes and pulls a lock for my rabbit house, so continuous operation isn't needed. In fact, my solenoid needs a fair bit of cooling time.
EDIT 2 My current plan is to drive it with 8051 and a transistor, but I'm open to alternatives. I could get the 8051 to give the pulses, but I fear that my code is buggy and frying my rabbits as a result. So I was hoping to come up with a relatively fail-safe method.
The solenoid is spring loaded, so that the pin retracts when the solenoid is engaged (spring is compressed) and releases when it's not (spring is released). With this motion, I'm planning to get it to unblock and block the gate (thus, a lock).
The rabbits will be locked up at night.
Yes. This is easily achieved with a switch and a battery. No need for a capacitor.
simulate this circuit – Schematic created using CircuitLab
Operation
• Press the button.
• Hold for required number of seconds.
• Release the button.
As you suspected, a capacitor could be used to provide a pulse to operate the solenoid.
simulate this circuit
Here we'll charge up the capacitor through a resistor. If the switches are left on the resistor will limit the current to a safe value that won't overheat the solenoid. Let's figure out some values. First the solenoid.
$$R = \frac{V}{I} = \frac{12}{1} = 12 Ω$$
The power (heat) in the solenoid is given by
$$P = V·I = 12·1 = 12 W$$
Let's limit the current to 0.25 A. This will require a total resistance of 48 Ω (12 V / 0.25 A) so we'll need a 36 Ω resistor. We'll use a 39 Ω as this is a standard value.
Let's say we need a 0.5 s pulse to move the solenoid. We don't have a figure for the inductance of the solenoid so we'll just treat it as a resistor for now.
The time constant for an R-C circuit is simply R·C. Therefor
$$C = \frac{t}{R} = \frac{0.5}{12} = 0.041 F = 41,000 uF$$
This is a lot but not impossible. 4,700 uF capacitors are readily available. Assuming your rabbit is on a tight budget you might find capacitors in old power-supplies, etc. You need to make sure that they are rated for 12V or higher. Simply parallel all the capacitors making sure to keep the polarity correct.
I'm suggesting a 3-way, centre-off switch. This way the solenoid can be reversed for open and closed.
If finding the capacitors proves to be a problem we'll have to add some electronics to do the timing.
Edit after Update 2.
The requirements are simpler now because the solenoid is known to be spring return.
simulate this circuit
I tried to come up with an alternative but it's hard to beat the old 555 timer. This configuration is a monostable. When Q1 turns on it triggers the 555 which turns on Q2 and the relay. Time delay is set by $R4 \cdot C2$.
• I'm not sure if it'd be wise to trust the user. A solenoid can heat up rapidly, so surely something more robust than a switch may be desirable. Perhaps a cap, at least? – WKleinberg Jan 3 '16 at 14:27
• Many solenoids are designed for continuous operation. What sort of solenoid are you talking about? What is the application? Please post the additional details in your question and not here in the comments. – Transistor Jan 3 '16 at 14:35
• That's a three carrot answer if ever I saw one. – Brian Drummond Jan 3 '16 at 15:52
• @BrianDrummond: Ho, ho. Very bunny. – Transistor Jan 3 '16 at 15:56
• Thanks for the great answer. 41,000uF, that's very high indeed. I don't think I can get my hands on something that size. Perhaps, cap discharge won't fit my case so well... Voted up for the great write-up though! – WKleinberg Jan 3 '16 at 20:00
If you want something really simple you could consider a PTC .This is a resistor that increases its resistance greatly when it heats up.Place the PTC in series with your solenoid .Select a PTC rated at 12V or better that has a cold resistance of less than say 5% of the DCR of your solenoid.This means that the solenoid will pull in normally.You can tape the PTC around the solenoid winding to get good thermal contact .The PTCs that are designed for loudspeaker protection would be a good starting point to make your selection .PTCs are sometimes marketed under different names like "PolySwitch"You could place a 680 ohm resistor and a red led across the PTC to indicate that the system is in a high resistance state due to getting hot .
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# Definition of “solution” [closed]
In my textbook “solutions” are defined as follows:
Homogenous mixtures of two or more substances are known as solutions.
Should the two substances always be non-reacting?
The definition has no indication of this, buts it’s just something that came to my mind. Won’t the solution still be homogeneous so shouldn’t we still call it a solution?
• Your textbook definition is good enough to answer your question. If the substance reacts with the solvent yielding soluble product, then you got a solution. Of course, there is no definite answer as to wheter the product(s) will always be soluble, you need to look at the concrete example. Jun 6 '20 at 18:46
• Please never use MathJax's math mode for emphasis. Jun 6 '20 at 18:47
• @andselisk Thanks! I completely neglected the possibility of precipitate formation after the reaction! So yes...it should depend on the reaction taking place. Jun 6 '20 at 18:50
• The reasons against using MathJax's math mode for emphasis: 1. It's just wrong; 2. It's semantically incorrect to use math mode for highlighting the text; 3. Even if you were to use MathJax for emphasis you should've used $\textit{…}$ instead; 4. Every time MathJax is introduced, the web page loads corresponding JS scripts which are in general resource-heavy; 5. Did I mention it's wrong? Also, see Is it OK to abuse Mathjax for emphasis? Jun 6 '20 at 18:58
• Jun 7 '20 at 3:11
This is what Encyclopaedia Britannica definition of a solution in chemistry:
Solution, in chemistry, a homogeneous mixture of two or more substances in relative amounts that can be varied continuously up to what is called the limit of solubility. The term solution is commonly applied to the liquid state of matter, but solutions of gases and solids are possible. Air, for example, is a solution consisting chiefly of oxygen and nitrogen with trace amounts of several other gases, and brass is a solution composed of copper and zinc.
A solution consists of solutes (at least one) and a solvent. The solute is define as the substance that is dissolved in the solvent. In other words, for solutions with components in the same phase, the substance(s) present in lower concentration are solutes while the major substance present in highest abundance is the solvent. For example, 190-proof alcohol is a liquid-liquid solution, which is the mixture of 95% ethanol and 5% water by volume. The major substance in this mixture is ethanol. Therefore, it is the solvent. The 5% water is the solute. Brass is a solid-solid solution of copper and zinc metals. General composition of brass is 65% copper and 35% zinc by weight. Hence, in brass, copper is the solvent and zinc is the solute. Air is a solution of multi-gases. In air, 72% nitrogen and 20% oxygen, and rest is other gases including $$\ce{CO2}$$ and $$\ce{H2}$$. Thus, the solvent of air is $$\ce{N2}$$ while $$\ce{O2}$$ is one of its solutes. Saline solution, on the other hand, is a solid-liquid solution. In a saline solution, solid salt ($$\ce{NaCl}$$) is the solute dissolved in water, which is the solvent (liquid).
• A few days ago a senior member was arguing with me here that air is not a solution. I disagreed. Now Encyclopedia Britannica says the same that air $is$ a solution. Thanks for sharing. Jun 7 '20 at 1:36
• @M.Farooq Na, sorry. A solution must be a condensed phase. Britannica is inconsequent in its definition, because it is per se impossible that two gases have any kind of solubility limit. You change temperature or pressure, the componentes in a gas mixture condense totally independent of each other. No interaction -> not a solution.
– Karl
Jun 7 '20 at 7:52
• @Karl, It is ultimately semantics. There is no restriction on miscibility. ACN and water are miscible in all proportions. If we freeze a solution of water and NaCl, water freezes separately to some extent. Sorry, I don't buy this argument. Can you provide a solid reference which says that a solution must be a condensed phase? In microwave spectroscopy, there is a technique called chiral tagging where a enantiomer is exposed to a chiral alcohol in the gas phase and neon is used a "reaction medium". They form weak complexes in gas phase. Who says there can be no interaction in gases? Jun 7 '20 at 13:19
• @Karl, just a clarification, chiral tagging in microwave spectroscopy is done at millitorr pressures in a chamber. It is neither supercritical state nor a high pressure experiment. Do solid solutions have colligative properties? I am saying that this is all semantics. You must read the poem "Five Blind Men of Indostan" en.wikipedia.org/wiki/Blind_men_and_an_elephant Jun 7 '20 at 17:13
• That's a valid point. Perhaps worth bringing up in meta if no one has already. Jun 7 '20 at 18:37
Should the two substances always be non-reacting?
Short answer yes. If they two substances react, you cannot recover the original components which defies the definition of a mixture.
However, if you think deeply, what is the meaning of non-reacting? Sometimes words are not enough they can only be an approximation of a reality.
Let us take the example of a $$\ce{I2}$$ (iodine solid) and dissolve it in hexane or carbon tetrachloride, we get a purple solution. If I dissolve it in acetone, we get a brown solution. Chemists are okay to call it a solution. Thinking deeper, the purple color or the brown color indicates that the "species" of iodine is not the same into solvents. It means somehow the solvent is indeed interacting with iodine molecules. That interaction must be very weak, because the moment we evaporate the solution, we will get the exact amount of iodine and the solvent back!
So you can see there is no such situation where there is no interaction. Even there is some interaction between two strangers by gravitational forces, but it is extremely extremely weak.
• You don't need to get into subtleties. Like every reaction you run in lab is in solution. Does it stop being solution when reaction starts? No, it doesn't. Jun 6 '20 at 21:08
• From a physical chemist's perspective, those reaction mixture will no longer be a binary component solution, it will have multiple solutes (=reactants and products). Jun 6 '20 at 21:15
• Chemists tend to loosely use the word solution e.g., dissolving HCl gas in water yields HCl solution. Is it a mixture of HCl gas and water- Nope! Jun 6 '20 at 21:16
• Come on, no chemist says "HCl solution"! It´s hydrochloric acid, acide chlorhydrique, Salzsäure! :-)
– Karl
Jun 6 '20 at 23:10
• @Karl, Yes, but the usage HCl solution for hydrochloric acid solutions is not uncommon. I just did quick Google Scholar search scholar.google.com/… Jun 6 '20 at 23:40
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### 2. Ground Distance to Grid
#### a. Ground to Ellipsoid
Going from ground to the ellipsoid is independent of the grid system.
Figure K-2 Ground to Geodetic
Use Equation H-2 to reduce ground to the ellipsoid
Equation H-2
Te equation uses a line's average elevation and the average geoid height to determine the geodetic length on the ellipsoid.
RE is 20.906 x 106 ft.
Jerry's geoid height is -33.902 meters computed using the GEOID18 model. It is negative because the geoid is below the ellipsoid in Wisconsin.
Since we're working in feet, the geoid height must be converted from meters:
Geoid height generally does not vary significantly over an area of this size. Since Jerry is centrally located, -111.2 ft can be used as a project average.
#### b. Ellipsoid to Grid
Figure K-3 Geodetic to Grid
To go from ellipsoid to grid, the geodetic distance multiplied by the grid scale factor, k, Equation H-3.
Equation H-3
The grid scale in Equation H-3 can be:
Jerry's scale used for the entire project
Average of the endpoint scales for each line
A weighted average scale, Equation H-4, for each line based on scales at the mid- and endpoints
Equation H-4
We'll compute the grid distance each way and compare their results.
##### (1) Using Jerry's scale: k=0.99996 957.
Line Grid Distance Jerry-A1: Jerry-B7:
##### (2) Average scale for each line
Using software, scale at points A1 and B7 can be determined using their approximate coordinates. NGS's NCAT can be used for this as well as the NAD 83 Coordinate Conversion workbook.
Point Scale A1 0.99996 8421 B7 0.99996 8894
Multiplying each line's geodetic distance by its average scale:
Line Average k Grid Dist Jerry-A1 0.99996 8996 4281.768 Jerry-B7 0.99996 9232 5145.760
##### (3) Weighted average scale for each line
Use the same software to determine the scale at each line's midpoint; midpoint coordinates are the average of the endpoint coordinates.
Line Mid-point k Jerry-A1 0.99996 8991 Jerry-B7 0.99996 9230
Multiplying each line's geodetic length by its weighted average scale:
Line Weighted k Grid Dist Jerry-A1 0.99996 8992 4281.768 Jerry-B7 0.99996 9231 5145.760
##### (4) Combined Factor
Grid distance can also determined from multiplying ground distance by Jerry's Combined Factor: 0.99991 863, Equation H-6.
Equation H-6
This simplified method does not require computing geodetic distances - the entire project is scaled by a single CF.
Line Grid Dist Jerry-A1 Jerry-B7
##### (5) Comparing results
Theoretically, the weighted average scale gives the best grid distance. In Table K-2 it is used as the base to which the others are compared.
Table K-2 Grid Reduction Comparisons Jerry-A1 Jerry-B7 DE x k Grid Dist, ft Diff, ft Grid Grid, ft Diff, ft Wtd Ave k 4281.768 - - 5145.760 - - Average k 4281.768 0.000 5145.760 0.000 Jerry's k 4281.771 +0.003 5145.761 +0.001 Combined Factor DHxCF 4281.762 -0.006 5145.781 +0.021
The largest differences are for the CF grid distances. Considering there is a 255 foot elevation variation across the project, that's not surprising. The other differences are 0.003 ft or less. An acceptable level of accuracy might be achieved with Jerry's scale factor for all lines - it simplifies computations somewhat although using line averages increases the accuracy without too much more effort.
Depending on accuracy requirement, a worse-case scenario should be examined. Consider a line furthest from the control in direction of scale variation and/or at largest elevation difference. Any method acceptable for that line will be acceptable for the entire project.
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# Probability of having at least 'k' marbles specific to each of 'm' bags filled by sampling with replacement
I'm going to rewrite my original question to make it a bit clearer:
Assume I have some set $P$ with $||P|| = N$ unique elements. I also have $S$ multisets, $(m_1, ..., m_S)$, of cardinality $L$, consisting of elements in $P$ chosen with uniform probability. We call a multiset, $m_i$, 'distinguishable' if it contains at least $k$ elements, though not necessarily distinct elements, that exist in no other multiset.
What is the probability of all $M$ multisets being 'distinguishable' according to this definition?
While sampling $S$ times with replacement from the set $P$, we can state the probability of never choosing the same element twice as:
Prob( $S$ unique selections from $P$ ) = $\prod \frac{(N - i)}{N}$ for $i = 0$ to $(S - 1)$
Or equivalently, we can calculate the probability that the multiset of $S$ sampled elements contains all unique elements as:
Prob( $S$ unique selections from $P$ ) = $\prod ((1-(\frac{1}{N - i}))^{(S - 1 - i)})$ for $i = 0$ to $(S - 1)$
-
There was a question yesterday that was formulated differently, about "pruning" of multisets, but AFAIR was effectively the same question. I can't find it anymore; it may have been deleted. Was that question by you, too? – joriki Apr 8 '11 at 5:28
@joriki: didn't you remember the number of the user? ;-) – Fabian Apr 8 '11 at 5:59
@joriki, yes, alas, that was me... number 8861. I wanted to spend more time thinking about the question before posting it here, so I put up the 'pruning' example, then decided to take it down after about 20 minutes. Sorry if that was a faux pas. – user8861 Apr 8 '11 at 6:10
@user8861: I wouldn't go so far as to call it a faux pas, but you could have mentioned it briefly in the question so that people who'd seen the other question wouldn't go looking for it to mark this as a duplicate. – joriki Apr 8 '11 at 6:40
@joriki, fair enough, and thanks for looking at the original version! The reason I didn't mention it was mostly because I didn't want to confuse things, and the site told me that ~5 people looked at the original. I'm actually a bit surprised anyone noticed this as a rephrasing. – user8861 Apr 8 '11 at 6:55
This seems like an interesting problem even for $k=1$, which I would solve before attacking the more general version.
Your first calculation for the probability that a multiset has distinct elements is correct, although you are using $P$ rather than $N$ to mean $\#P$.
Your second calculation is unjustified and incorrect. You can compare it with the correct one. Suppose $N=10$ and $\#S=4$. The probability that $4$ draws are distinct is $\frac{10}{10} \times \frac{9}{10} \times \frac{8}{10} \times \frac {7}{10} = \frac{504}{1000} = 0.504$. Your second expression says $(1-\frac{1}{10}^3) \times (1-\frac{1}{9}^2) \times (1-\frac{1}{8}^1) \times (1-\frac{1}{7}^0)$. That last term is $0$, which makes the whole product $0$. If we leave it out, we get $\frac{259}{300} = 0.86333...$ which is quite different from the correct value.
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# 1.7 Inverse functions (Page 2/10)
Page 2 / 10
$\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left(y\right)=x$
This holds for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.
Given a function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ we can verify whether some other function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is the inverse of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ by checking whether either $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=x\text{\hspace{0.17em}}$ is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example, $\text{\hspace{0.17em}}y=4x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\frac{1}{4}x\text{\hspace{0.17em}}$ are inverse functions.
$\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(4x\right)=\frac{1}{4}\left(4x\right)=x$
and
$\left({f}^{}\circ {f}^{-1}\right)\left(x\right)=f\left(\frac{1}{4}x\right)=4\left(\frac{1}{4}x\right)=x$
A few coordinate pairs from the graph of the function $\text{\hspace{0.17em}}y=4x\text{\hspace{0.17em}}$ are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function $\text{\hspace{0.17em}}y=\frac{1}{4}x\text{\hspace{0.17em}}$ are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.
## Inverse function
For any one-to-one function $\text{\hspace{0.17em}}f\left(x\right)=y,\text{\hspace{0.17em}}$ a function $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)\text{\hspace{0.17em}}$ is an inverse function of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}{f}^{-1}\left(y\right)=x.\text{\hspace{0.17em}}$ This can also be written as $\text{\hspace{0.17em}}{f}^{-1}\left(f\left(x\right)\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ It also follows that $\text{\hspace{0.17em}}f\left({f}^{-1}\left(x\right)\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ is the inverse of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$
The notation ${f}^{-1}$ is read $\text{“}f$ inverse.” Like any other function, we can use any variable name as the input for ${f}^{-1},$ so we will often write $\text{\hspace{0.17em}}{f}^{-1}\left(x\right),$ which we read as $“f$ inverse of $x.”$ Keep in mind that
${f}^{-1}\left(x\right)\ne \frac{1}{f\left(x\right)}$
and not all functions have inverses.
## Identifying an inverse function for a given input-output pair
If for a particular one-to-one function $\text{\hspace{0.17em}}f\left(2\right)=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=12,\text{\hspace{0.17em}}$ what are the corresponding input and output values for the inverse function?
The inverse function reverses the input and output quantities, so if
Alternatively, if we want to name the inverse function $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}g\left(4\right)=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(12\right)=5.$
Given that $\text{\hspace{0.17em}}{h}^{-1}\left(6\right)=2,\text{\hspace{0.17em}}$ what are the corresponding input and output values of the original function $\text{\hspace{0.17em}}h?\text{\hspace{0.17em}}$
$h\left(2\right)=6$
Given two functions $\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right),\text{\hspace{0.17em}}$ test whether the functions are inverses of each other.
1. Determine whether $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x.$
2. If both statements are true, then $g={f}^{-1}$ and $f={g}^{-1}.\text{\hspace{0.17em}}$ If either statement is false, then both are false, and $\text{\hspace{0.17em}}g\ne {f}^{-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\ne {g}^{-1}.$
## Testing inverse relationships algebraically
If $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x+2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{x}-2,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$
$\begin{array}{l}\begin{array}{l}\hfill \\ g\left(f\left(x\right)\right)=\frac{1}{\left(\frac{1}{x+2}\right)}-2\hfill \end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x+2-2\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x\hfill \end{array}$
so
This is enough to answer yes to the question, but we can also verify the other formula.
$\begin{array}{l}\begin{array}{l}\\ f\left(g\left(x\right)\right)=\frac{1}{\frac{1}{x}-2+2}\end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{\frac{1}{x}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x\hfill \end{array}$
If $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt[\text{\hspace{0.17em}}3]{x+4},\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$
Yes
## Determining inverse relationships for power functions
If $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ (the cube function) and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{3}x,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$
$f\left(g\left(x\right)\right)=\frac{{x}^{3}}{27}\ne x$
No, the functions are not inverses.
If $\text{\hspace{0.17em}}f\left(x\right)={\left(x-1\right)}^{3}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}g\left(x\right)=\sqrt[3]{x}+1,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$
Yes
## Finding domain and range of inverse functions
The outputs of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ are the inputs to $\text{\hspace{0.17em}}{f}^{-1},\text{\hspace{0.17em}}$ so the range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is also the domain of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ Likewise, because the inputs to $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ are the outputs of $\text{\hspace{0.17em}}{f}^{-1},\text{\hspace{0.17em}}$ the domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the range of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ We can visualize the situation as in [link] .
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
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# 18.9.2 Algorithms (Matrix Smoothing)
Matrix smoothing is performed by shrinking and then expanding the matrix.
If the number of columns or rows is less than 32, the matrix is first expanded so that the row number and the column number are both twice that of the original. Then the expanded matrix is shrunk to the original size. The matrix expansion is implemented by the mexpand X-Function. Thus, Biquadratic interpolation is used. The matrix shrinking is done via the Mshrink X-Function. The average value of 4 adjacent cells is calculated to obtain each cell value of the output matrix. Through this process of expanding and shrinking, the size of the output matrix will be exactly the same as the original matrix. However, the data will be much smoother.
If both the number of columns and the number of rows in the original matrix are greater than 31, the matrix is first shrunk and then expanded to obtain the smoothed matrix. Shrinking and expanding are done in the same way as described above.
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# cohen
/Tag: cohen
## Sotah – Suspected Adulteress as a Schrödinger Cat
In quantum mechanics, the state of a physical system is described by the so-called wave function (or the "wavefunction"). What is the wavefunction? All attempts by Schrödinger, who first introduced the wavefunction, and others to interpret it as a scalar potential of some physical field, or as the de Broglie wave (as in particle-wave dualism) were not successful. In 1926, Max Born noticed that the squared amplitude of the wavefunction of a particle in a given region gives the probability of finding the particle in this region. He suggested that the wavefunction represented not a physical reality but rather our knowledge of the quantum state of an object. The wavefunction represents our knowledge of all possible quantum-mechanical states of an object. In other words, the quantum-mechanical state of a physical system is a linear [...]
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Warning
This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.
# square_clustering¶
square_clustering(G, nodes=None)[source]
Compute the squares clustering coefficient for nodes.
For each node return the fraction of possible squares that exist at the node [1]
$C_4(v) = \frac{ \sum_{u=1}^{k_v} \sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v} \sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]},$
where $$q_v(u,w)$$ are the number of common neighbors of $$u$$ and $$w$$ other than $$v$$ (ie squares), and $$a_v(u,w) = (k_u - (1+q_v(u,w)+\theta_{uv}))(k_w - (1+q_v(u,w)+\theta_{uw}))$$, where $$\theta_{uw} = 1$$ if $$u$$ and $$w$$ are connected and 0 otherwise.
Parameters: G (graph) – nodes (container of nodes, optional (default=all nodes in G)) – Compute clustering for nodes in this container. c4 – A dictionary keyed by node with the square clustering coefficient value. dictionary
Examples
>>> G=nx.complete_graph(5)
>>> print(nx.square_clustering(G,0))
1.0
>>> print(nx.square_clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
Notes
While $$C_3(v)$$ (triangle clustering) gives the probability that two neighbors of node v are connected with each other, $$C_4(v)$$ is the probability that two neighbors of node v share a common neighbor different from v. This algorithm can be applied to both bipartite and unipartite networks.
References
[1] Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005 Cycles and clustering in bipartite networks. Physical Review E (72) 056127.
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# Do Curtmola et al.'s IND-CKA1/2 security definitions protect against search pattern leakage?
In the article Searchable Symmetric Encryption: Improved Definitions and Efficient Constructions, Curtmola et al. propose adaptive and non-adaptive (indistinguishability and simulator-based) security definitions for searchable encryption schemes, conventionally called IND-CKA1 and IND-CKA2.
My question is: Do the IND-CKA1/2 security definitions guarantee that the search pattern is not leaked (i.e., that an attacker can not distinguish whether two issued trapdoors are generated with the same keywords)?
In their article, they mention that
[...] the security notion achieved for SSE is that nothing is leaked beyond the access pattern and the search pattern [...]
Further, in their article, Bösch et al. state that
Curtmola et al. review existing security definitions for searchable encryption and propose new indistinguishability and simulation-based definitions that address the shortcomings of the existing definitions. At the same time they loosen the character of SSE by allowing the leakage of a user's search pattern.
So it seems pretty obvious that the definition should not guarantee search pattern hiding. Nevertheless, taking a look at the IND-CKA2 definition:
$\mathbf{\mathrm{Ind}}^{*}_{\mathrm{SSE},\mathcal{A}}(k) \\ \;K\leftarrow \mathrm{Gen}(1^k)\\ \;b\overset{\$}{\leftarrow}\{0,1\}\\ \;(\mathrm{st}_{\mathcal{A}},\mathbf{D}_{0},\mathbf{D}_{1})\leftarrow\mathcal{A}_{0}(1^k)\\ \;(I_{b},\mathbf{c}_{b})\leftarrow\mathrm{Enc}_{K}(\mathbf{D}_{b})\\ \;(\mathrm{st}_{\mathcal{A}},w_{0,1},w_{1,1})\leftarrow\mathcal{A}_{1}(\mathrm{st}_\mathcal{A},I_{b})\\ \;t_{b,1}\leftarrow\mathrm{Trpdr}_{K}(w_{b,1})\\ \;\mbox{for }2\le i\le q,\\ \quad(\mathrm{st}_{\mathcal{A}},w_{0,i},w_{1,i})\leftarrow\mathcal{A}_{i}(\mathrm{st}_{\mathcal{A}},I_{b},c_{b},t_{b,1},\ldots,t_{b,i-1})\\ \quad t_{b,i}\leftarrow\mathrm{Trpdr}_{K}(w_{b,i})\\ \;\mbox{let }\mathbf{t}_{b}=(t_{b,1},\ldots,t_{b,q})\\ \;b'\leftarrow \mathcal{A}_{q+1}(\mathrm{st}_{\mathcal{A}},I_{b},\mathbf{c}_{b},\mathbf{t}_{b})\\ \;\mbox{if }b'=b\mbox{, output }1\\ \;\mbox{otherwise output }0$one sees that any algorithm able to distinguish whether two trapdoors encode the same words or not directly breaks IND-CKA2, which would mean that IND-CKA2 protects against search pattern leakage. For instance, and very informally, in the first trapdoor query$\mathcal{A}$can set$w_{0,1}=w_{1,1}=w_{1}$for some fixed$w_{1}$and receive$t_{b,1}=\mathrm{Trpdr}_{K}(w_{1})$. In the second query, it can issue$w_{0,2}=w_{1}$and$w_{1,2}\neq w_{1}$. Then$t_{b,2}$either encodes$w_{1}$or some other word, and$\mathcal{A}$can guess$b$by guessing whether$t_{b,1}$and$t_{b,2}$encode the same word ($b=0$) or not ($b=1$). Continuing in this fashion, since$\mathcal{A}$breaks the "search pattern challenge",$\mathcal{A}$has a non-negligible advantage in distinguishing$b$. ## 1 Answer The condition$\tau(\mathbf{D}_{0}, w_{0,1},\ldots,w_{0,q})=\tau(\mathbf{D}_{1}, w_{1,1},\ldots,w_{1,q})$rules out the possibility that$\mathcal{A}$issues the keywords as I pointed in the question, since the output of$\tau\$ includes the search pattern matrix defined in page 9. This has the effect of weakening the definition to allow search pattern leakage.
As a note, the stronger definition in Shen et al.'s 1 does not include a restriction of this type, and so it captures the search pattern protection property.
1 Shen, Emily, Elaine Shi, and Brent Waters. “Predicate Privacy in Encryption Systems.” In Theory of Cryptography, edited by Omer Reingold, 5444:457–73. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2009.
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Article Text
Time is money: quantifying savings in outpatient appendectomy
1. Elise Taylor Bernard,
2. Daniel L Davenport,
3. Courtney M Collins,
4. Bethany A Benton,
5. Andrew C Bernard
1. Department of Surgery, Acute Care Surgery, College of Medicine, University of Kentucky, Lexington, Kentucky, USA
1. Correspondence to Elise Taylor Bernard, Department of Surgery, Acute Care Surgery, College of Medicine, University of Kentucky, Lexington, KY 40506, USA; 16bernarde{at}gmail.com
## Abstract
Background Laparoscopic appendectomy can be performed on a fast-track, short-stay, or outpatient basis with high success rates, low morbidity, low readmission rates, and shorter length of hospital stay. Cost savings from outpatient appendectomy have not been well described. We hypothesize that outpatient laparoscopic appendectomy is associated with cost savings.
Methods We performed an original retrospective cohort analysis of patients undergoing laparoscopic appendectomy between June 2013 and April 2017 at our academic medical center before and after implementation of an outpatient protocol which began on January 1, 2016. We assessed appendicitis grade, length of stay (LOS), cost, net revenue, and profit margin.
Results After protocol implementation, the percentage of patients discharged from the the postanesthesia care unit (PACU) increased from 3.7% to 29.7% (χ2 p<0.001). The proportion of inpatient admissions and admissions to observation decreased by 5.7% and 20.3%, respectively. On average, PACU-to-home patients had a total hospital cost of $4734 compared with$5781 in patients admitted to observation, for an estimated savings of $1047 per patient (p<0.001). Comparing the time periods, the mean LOS decreased for all groups (p<0.001). Appendicitis grade was higher in those who required inpatient admission, but could not distinguish which patients required an observation bed. Discussion Outpatient appendectomy saves approximately$1000 per patient. Adoption of an outpatient appendectomy pathway is likely to be gradual, but will result in incremental improvement in resource utilization immediately. Grade does not predict which patients should be observed. Considering established safety, our data support widespread implementation of this protocol.
Level of evidence III.
• cost
• observation
• value
• appendicitis
This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0/.
## Introduction
Laparoscopic appendectomy is the most common treatment of appendicitis. Recent evidence suggests laparoscopic appendectomies can be performed on a fast-track, short-stay, or even outpatient basis.1 This approach was described a decade ago and has been used in adults and children.2 This outpatient appendectomy protocol provides high success rates, low morbidity, low readmission rates, and shorter length of hospital stay.3
Financial benefits of outpatient protocol have not been established as definitively as clinical outcomes.4 In one study, a fast-track protocol for laparoscopic appendectomy reduced hospital costs; however, the average length of stay for fast-track patients was still longer than 1 day, so this study did not fully represent the true effect of outpatient appendectomy.4 In a similar study in children with appendicitis treated on a fast-track protocol, the average savings was approximately $350 compared with children who stayed overnight.5 In another study, total hospital charges were reduced by around$900 in patients successfully managed on a fast-track protocol designed for appendicitis, but costs were not reported. In a study from Canada showing true outpatient appendectomy where patients were discharged from the postanesthesia care unit (PACU), cost savings were $323.46 per patient.6 Cost savings in adult patients undergoing outpatient appendectomy have not been well reported in the USA. Length of stay is an important determinant of hospital costs, and therefore an important potential financial opportunity.7 For example, in patients with acute cholecystitis, costs increase incrementally for each additional hospital day.8 We hypothesize that an outpatient appendectomy protocol where patients go home from the PACU will result in cost savings. ## Patients and methods After institutional review board approval, we performed a retrospective cohort analysis (level III evidence) of data from the University of Kentucky healthcare discharge database, Hospital Finance, and medical records of patients admitted for the time period of June 1, 2013 to April 30, 2017. We included subjects 15 years or older with the admission diagnosis of “appendicitis.” Elective appendectomies, open cases, non-resective operations, and those involving intestinal resection more than the appendix were excluded. Individual medical records were analyzed to determine whether patients went home from the PACU or were admitted to an observation or inpatient bed. For patients not discharged directly from the PACU, the indication for observation or inpatient admission was determined. Patients were categorized as inpatient admission, observation admission, or PACU-to-home. Patients were divided into two time periods, before and after the outpatient appendectomy protocol was implemented on January 1, 2016. Outpatient protocol was not restricted by age. Patients were admitted to the emergency department (ED) observation area and taken to the operating room on an emergency basis. The criteria for identifying patients eligible for outpatient appendectomy include laparoscopic procedure, could tolerate liquids in PACU without vomiting, did not require prolonged intravenous antibiotics, and had adequate pain control. The process for the outpatient appendectomy protocol was defined by a discharge order for the patient to leave directly from the PACU without any diet order, no planned observation bed, and no plan for interval examination by the team. PACU discharge was based on standard PACU criteria for other outpatient general surgery (eg, cholecystectomy, inguinal hernia repair). Faculty and residents were encouraged to use this protocol based on their clinical judgment. One concern about the use of outpatient appendectomy is short-term readmission rates. Short-term readmission was defined as within 3 days of discharge, which was based on the mean length of stay for inpatients, found to be slightly above 3 days. Outpatient, observation bed, and inpatient admission decision were analyzed by time of case completion to determine the effect of night-time on discharge decision. We also analyzed post-PACU disposition by appendicitis grade using the American Association for the Surgery of Trauma (AAST) grading scale, American Society of Anesthesiologists (ASA) class, and age. Net revenue, direct and indirect costs, profit margin (revenue − total costs), and specific resource costs (operating room, ED, diagnostics, and pharmacy) were obtained from the hospital cost accounting system. Revenue was analyzed according to payer grouping. Each cost center was examined separately. Changes in proportions between periods were calculated using χ2 tests. Differences in financial parameters were calculated using t-tests or analyses of variance. Significance was set at p<0.05 for all comparisons. Statistical tests were performed using SPSS V.24. ## Results There were 453 patients who underwent emergency laparoscopic appendectomy on the acute care surgery service at our academic medical center during the study period, 295 before protocol implementation and 158 after. The median patient age was 31 years (IQR 21–45 years) and 44.6% were female. One hundred and eighty-six (41.1%) were admitted to an inpatient bed, 209 (46.1%) to an observation bed, and 58 (12.8%) were discharged home from the PACU. The percentage of PACU-to-home discharges increased from 3.7% during the pre-implementation period to 29.7% in the postimplementation period, whereas inpatient admissions decreased from 48.1% to 27.8% (table 1, χ2 p<0.001). The proportion of patients assigned to an observation bed also decreased with an absolute reduction of 5.7% and a relative reduction of 11.8%. Table 1 Number and percent of patients by PACU discharge destination preimplementation and postimplementation of protocol Of the 58 patients discharged from the PACU during the study period, three patients returned to the ED, two of whom fell within the 3-day threshold to be classified as a readmission. One patient presented to the ED the day after discharge with complaints of difficulty voiding and blurry vision. This patient was discharged from the ED with no abnormality found. A second patient, a 60-year-old woman, presented 2 days after discharge with a chronic obstructive pulmonary disease exacerbation and a type II non-ST-elevation myocardial infarction. She was discharged 2 days later. One patient returned to the ED on day 10 with constipation, and again on day 29 concerned with wound healing. At the clinic follow-up appointment this patient was well. This patient was not classified as a readmission, as they simply chose the ED for follow-up care. Two of these three patients who returned to the ED after discharge were not English-speaking. PACU-to-home patients had a mean total hospital cost of$4734 compared with $5781 for those placed into an observation bed, for an estimated savings of$1047 (95% CI $718 to$1377) per patient discharged from the PACU under the protocol (p<0.05, table 2). The total cost for inpatients was dramatically higher than patients discharged from the PACU and those admitted for observation (p<0.05). When analyzed by cost center, operating room services and supplies were $241 higher in patients admitted to observation and$654 higher in those admitted to an inpatient bed (p<0.05). Inpatients had significantly higher pharmacy, imaging and lab, and ED service and supply costs than patients discharged from the PACU or admitted for observation (p<0.05).
Table 2
Financial performance by PACU discharge destination
Revenue was not significantly different for patients discharged home compared with patients admitted for observation; however, as a result of decreased costs, patients discharged home had a higher profit margin than patients admitted for observation (p<0.05, table 2). Revenue was higher for inpatients than either patients discharged home or admitted for observation (p<0.05, table 2). Significantly increased costs for inpatients paralleled increased net revenue resulting in a profit margin comparable with patients discharged home and significantly greater than patients admitted for observation. There were less managed care and more Medicare in the inpatient group compared with the other two groups (p<0.001, table 3).
Table 3
PACU discharge destination by payer group
After implementation of the protocol, the total average time from operation completion to discharge home decreased by 7.5 hours for those discharged from the PACU and those placed in an observation bed (table 4, p<0.001). In addition, the average time from operation to discharge decreased for each group individually, with a 2.5-hour decrease for patients discharged from the PACU and a 4.2-hour decrease for patients admitted to observation.
Table 4
Hours to discharge decreased in both patients discharged to home and admitted to observation after implementation of the protocol
Patients whose appendectomy procedure ended between 21:00 and 05:00 were more likely to go to an observation bed (p=0.036) or an inpatient bed (p=0.014) than to be sent home from the PACU (table 5). AAST operative grade was predictive of who would require an inpatient bed (p<0.001), but could not distinguish patients who would require an observation bed from those discharged from the PACU (p=0.171). Similarly, AAST clinical grade predicted who would be admitted to an inpatient bed (p<0.001), but did not identify those who would require an observation bed rather than being discharged home from the PACU (p=0.345). Age and ASA classification were both higher in inpatients, but did not differ between observation and outpatient cases.
Table 5
Predictive factors for post-PACU disposition
## Discussion
Outpatient appendectomy has now been clearly established as safe and effective, with complication rates, readmission rates, and satisfaction equivocal to those patients kept in observation.9 We were successful in implementing an outpatient appendectomy protocol that increased the percentage of patients discharged from the PACU. Implementation of an outpatient appendectomy protocol was not successful in facilitating discharge home of all eligible patients. By comparing the preimplementation and postimplementation periods, we are able to show gradual adoption of this protocol. The most important difference between the two time periods is a significant increase in patients who were sent home from the PACU. One reason that this protocol is expected to yield a slow adoption rate is that individual surgeons varied in the rate at which they embraced outpatient appendectomy. During data analysis, we found numerous patients admitted to observation in the postimplementation period who seemed to be appropriate to be discharged home, but who were treated by surgeons who had not yet chosen to adopt to the protocol. Conversely, there were also surgeons who were discharging patients home from the PACU even before formal implementation of the protocol. Evidence suggests that more than half of new perioperative protocols meet barriers to initiation, including logistical issues, time constraints, and opposition from colleagues.10 A greater comfort level with outpatient appendectomy developed among our providers over time. Implementing outpatient appendectomy protocol requires a culture change and effects are likely to appear gradually.
Appendicitis severity can be graded using numerous methods.11 We sought to determine whether appendicitis grading would determine which patients could be discharged home from the PACU. The AAST grading system has been the most extensively studied, and we have shown AAST grade correlates with cost and operative duration.12 Data shown here indicate that patients with high-grade appendicitis are more likely to require an inpatient bed. High-grade appendicitis is characterized clinically by abdominal tenderness, mass, and peritonitis, and operatively by perforation, phlegmon, or generalized peritonitis. It is therefore not surprising that patients with high-grade appendicitis more often require inpatient admission. However, 80% of patients in our study had clinical or operative grade I or II appendicitis. Within these low grades, we could not discriminate who required an observation bed based on grading. This lack of discrimination leads us to conclude that in the majority of patients, the decision to discharge home from the PACU is more dependent on the surgeon than the patient.
Culture change in the postoperative management of appendicitis occurred at our academic medical center, evidenced by fewer patients admitted to observation and inpatient beds. Despite gradual and incomplete adoption of the protocol, we still report an eightfold increase in patients discharged from the PACU and this required no bed at all. Even those patients admitted to observation stayed fewer hours. Gurien et al 3 reported increased hospital charges in patients not discharged directly from the PACU, and these charges appear primarily related to longer hours of hospital stay.3
Cost savings data reported here are on a per-case basis. For each patient discharged from the PACU, there will be incremental cost savings of over $1000 compared with patients admitted to observation. Considering data shown in table 2, the greatest cost savings appears to be related to bed cost, since costs related to ED, pharmaceuticals, and imaging were the same, and operating room services and supply costs were only slightly greater in patients admitted to observation. To estimate total opportunity cost savings at our institution, we applied the percentage increase in patients discharged home after the protocol was implemented, or 26.0%, to the number of patients treated before implementation, 295. Based on this estimate, 77 patients would have avoided being admitted to observation, resulting in a total cost savings of$80 619 during the 30-month period. Other authors have estimated that nationwide implementation of an outpatient appendectomy protocol could save \$921 500 000 in annual direct healthcare costs; therefore, our work, combined with that of others, suggests a potential significant cost savings for the healthcare community.13 Total cost savings may be underestimated by only considering objective cost data. For example, additional cost savings could be realized by avoiding bed turnover and cleaning, and the downstream effects of bed occupancy, such as ED boarding and lost transfers.
Revenue was highest in the inpatient group. However, indirect and direct costs for inpatients were also higher, resulting in a margin slightly less than for outpatient appendectomy group. Increased direct cost was multifactorial. Imaging costs were greater in the PACU-to-home and admitted to observation groups, probably because more complex cases of appendicitis required more complex imaging to treat complications. Pharmacy costs were higher because of prolonged antimicrobial therapy in complicated cases. As with the other two groups, the majority of direct costs were related to bed cost, determined directly by length of stay. Increased revenue was likely due to higher illness severity and additional or alternative diagnosis-related groups (DRGs) in the inpatient group. It is unlikely that significant opportunity cost savings lies in this higher acuity group; however, based on our data, we think strongly that there is a definite opportunity cost savings that lies in the low acuity appendicitis population by avoiding admitting patients to observation beds. Outpatient appendectomy cases outperformed both cases placed in an observation bed and inpatients.
Cost is a relevant and objective financial target, and cost reduction is a viable financial strategy. Performance of the acute care surgery service includes financials and can influence organization resource allocation. Reimbursement is primarily based on DRG and payer type, both of which are difficult to control. The simple outpatient appendectomy protocol described here results in savings greater than those reported in a same-day discharge protocol in children and an outpatient protocol performed in Canada.3 5 Our data corroborate previous reports showing that shorter length of stay, even by a few hours, results in cost savings. These results should be easily reproducible.
A larger population of Medicare payer type in the inpatient group was likely due to a greater majority of elderly patients in this population, with greater comorbidity that were more likely to require an inpatient stay. In the postimplementation period, the decrease in self-pay/charity payer type was likely due to Kentucky being an early adopter of the Affordable Care Act, resulting in more patients with Medicaid. Payer status will definitely affect profit margin. Our focus in this study is cost, which we think to be the most relevant financial target to try to influence because it is most directly modifiable.
Readmission rates are a concern in outpatient appendectomy; however, readmission rates are low using this protocol, reported at 3% in a multicenter trial.9 In our study, only two patients returned within 3 days of discharge. No patient managed with the outpatient protocol returned to the ED within 3 days as a result of being treated on an outpatient basis. One patient who returned to the ED with constipation related to his appendectomy was not English-speaking, a factor known to be associated with increased use of the ED for access to care.14 The 3-day threshold determined for the readmission period in this study is arbitrary. The median length of stay for observation and inpatient groups were both 2 days, which are both less than the 3-day threshold. From this, we think that that 3-day threshold is a valid defining point for readmission rates in this study.
This was a retrospective study in which the decision to discharge the patient home was left to the discretion of the operating surgeon. There is inevitable selection bias in the outpatient appendectomy group; however, the dramatic increase in patients discharged from the PACU after protocol implementation affirms a significant opportunity to avoid admission to an observation bed. A recent multicenter trial sponsored by the Southwestern Surgical Congress affirmed that a laparoscopic appendectomy protocol can be successfully generalized to multiple institutions, with low rates of readmission and complication rates.9 We were not able to measure PACU flow in these patients and whether the protocol negatively effected throughput; however, no PACU flow issues were reported anecdotally during either period.
## Conclusion
Development of an outpatient laparoscopic appendectomy protocol will be associated with significant cost savings in eligible patients. The exact amount of total cost savings from implementing this protocol will depend on the percentage of patients who are clinically appropriate to be discharged home, surgeon comfort, organizational efficiency, and appendicitis grade. Profit margin will ultimately depend on total cost savings and payer mix.
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View Abstract
## Footnotes
• Presented at Presented at the 2018 Academic Surgical Congress, Jacksonville, Florida, January 30, 2018. Formatted for Trauma Surgery & Acute Care Open .
• Contributors ETB designed the study, collected the data, analyzed the data, and wrote the article. DLD designed the study, analyzed the data, and edited the article. CMC collected and analyzed the AAST grading data. BAB designed the study, collected the data, and edited the article. ACB designed the study, analyzed the data, edited the article, and supervised the project.
• Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.
• Competing interests None declared.
• Patient consent for publication Not required.
• Provenance and peer review Not commissioned; externally peer reviewed.
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## Subsection5.1.3What you will learn
This week is all about solving nonsingular linear systems via LU (with or without pivoting) and Cholesky factorization. In practice, solving $A x = b$ is not accomplished by forming the inverse explicitly and then computing $x = A^{-1} b \text{.}$ Instead, the matrix $A$ is factored into the product of triangular matrices and it is these triangular matrices that are employed to solve the system. This requires fewer computations.
Upon completion of this week, you should be able to
• Link Gaussian elimination to LU factorization.
• View LU factorization in different ways: as Gaussian elimination, as the application of a sequence of Gauss transforms, and the operation that computes $L$ and $U$ such that $A = L U \text{.}$
• State and prove necessary conditions for the existence of the LU factorization.
• Extend the ideas behind Gaussian elimination and LU factorization to include pivoting.
• Derive different algorithms for LU factorization and for solving the resulting triangular systems.
• Employ the LU factorization, with or without pivoting, to solve $A x= b \text{.}$
• Identify, prove, and apply properties of Hermitian Positive Definite matrices.
• State and prove conditions related to the existence of the Cholesky factorization.
• Derive Cholesky factorization algorithms.
• Analyze the cost of the different factorization algorithms and related algorithms for solving triangular systems.
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A little boy wrote the numbers $1,2,3,...,2011$ on a blackboard. He picks any two numbers $x,y$ , erases them with a sponge and writes the number $|x-y |$. This process continues until only one number is left. Prove that the number left is even
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# Discrete Math concepts E-book Paperwork – DM Address Notes Pdf
Combinatorics, solid induction,bird golf hole rule, permutation and blend, repeat relationships, straight line non homogeneous recurrence relationship having regular, the principle of addition and exclusion. Logic along with resistant, propositions about statement, connectives, simple connectives, simple fact family table regarding essential connectives,Plus,Disjunction,conditional point out,bisexual depending declare,tautology,contradiction,misconception,contigency,plausible equialances,idempotent rules,associtative laws,commutative legislation,demorgans legislation,distributive regulation,matches regulation,importance legislations,personality regulation.Your praposition involving with assertion is often a declarative heading that often genuine (or even) bogus definitely not both,conective can be an functioning utilized to plug 2 (or perhaps) over 2 statements.easy is known as sentencal connective. F’ (by, ymca, z .) Equates to x’ b unces + x’ y’ z + x’ p oker z’ + times y’ z’ This can be a function of diploma A couple of from your group of ordered pairs associated with Boolean specifics to your established F ree p(Zero,1)=1,P oker(0,1)=0,F ree p(A single,1)=0 and P oker(One,1)=0
#### Discrete Mathematics Information pdf – DM paperwork pdf
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Let F(times, ful, unces) Equates to (back button + p oker + z .) . Your extensive concept is the one about a certain form of geometry, analogous on the statistical prospect of a group. Your buy essay paper two-element BA displays your lead connection with simple judgement. The actual $$B$$-valued arena is the correct type $$/(M)$$ the actual un famous these $$V$$ohydrates. Your demanding idea is that of a definite form of algebra, analogous towards mathematical notion of a bunch.
### Очередь просмотра
F’ (listing of factors) = ? (list of 0-minterm indices) Make it possible for $$A$$ function as the collection of all equivalence instruction underneath this particular equivalence connection. Boolean always the same (True or maybe False), Boolean specifics and also realistic connectives incorporate alongside one another to create a Boolean expression. This specific confirms the fundamental Rock rendering theorem, and also makes clear the fundamental cause with BAs while cement algebras regarding pieces.
The particular specifics which often can have got 2 under the radar prices Zero (Phony) as well as Just one (True) as well as procedures associated with sensible value are taken care of Boolean algebra. Then $$A$$ in addition to $$\oplus$$ and $$\cdot$$, in addition to 3 as well as One particular, varieties an engagement ring by using However a particular event, Boolean-valued products with regard to arranged basic principle, is extremely at the forefront of present-day research in fixed idea. M) Equals Your + B G) Is equal to (A new . Example: Consider the Boolean geometry D70 whose Hasse plan is revealed around fig: It is Sikorski’s expansion theorem.
### Distributive Law
Theory Isomorphic to be able to span geometry on (A person) essentially undecidable theory $$\mathbb$$, your rationals (3) BAs $$\mathbb \times \mathbb$$, rectangular on the positive integers, ordered lexicographically (Three or more) linear orders $$\mathbf \times \mathbb$$ directed antilexicographically, the place $$\mathbf$$ is $$\mathbb^\mathbb$$ in its common order (4) abelian groups $$(\mathbb + \mathbf) \times \mathbb$$
The purpose coming from A”to Any is named a Boolean Performance in case your Boolean Phrase regarding n parameters can easily identify the item. That projects that will help students grasp the critical aspects regarding let loose numbers. A+(Some sort of.W)=A Then (The,2 ., +,Wi, 2,One particular) is termed a sub-algebra or even Sub-Boolean Algebra of W in case a itself is the Boolean Algebra i actually.age., Your provides the aspects 1 and 1 and is finished underneath the businesses 3 ., + as well as A. This kind of refers inside our most important case in point so that you can $$\subseteq$$. A + A person Is equal to 1 (Or maybe Sort)
The BA with merely your id automorphism is named inflexible. The following video clip tackles the essential notion of Boolean geometry for just a lattice.
Notes:
Boolean Algebra: A lattice ‘L’ is considered to get Boolean algebra should it be both associated plus distributive.
? For the lattice to become either distributive in addition to complemented it is vital that each and every take into account the lattice ought to have only one complement.
Some practice complications:
Determine in case the supplied lattice is actually Boolean geometry or you cannot?
1. Of special importance is the two-element BA, established by taking a set $$X$$ to own an individual ingredient. \sim (Some sort of . A vital ( is required to exhibit a Or even functioning.
#### Distributive Law
Although not immediately clear, this can be the same as the ring-theoretic notion. Or, F(x, ful, z .) Equates to M_3 . M_5 . Another regular algebraic constructions are more strange in order to BAs. This could be concretely created for BAs. Following, most of us specify $$times \oplus ymca Is equal to (a \cdot -y) + (y \cdot -x)$$. F(directory of specifics) Equates to \pi (set of 0-maxterm indices).
• A part $$X$$ involving $$A$$ is definitely irredundant in the event absolutely no aspect of $$X$$ consistantly improves subalgebra earned with the people.
• A subset $$X$$ of an BA $$A$$ will be separate in the event that $$X$$ is really a range of cost-free machines with the subalgebra that it builds. A freedom of $$A$$ would be the supremum regarding cardinalities connected with independent subsets with $$A$$.
• Every changing name is usually a Boolean phrase.
• Hyperlinks for that training books that any of us advocate with regard to Discrete Maths tend to be:
1. A bestessay4u non-empty established B along with a couple binary experditions ? along with ?, any unary operation ?, and 2 particular aspects 3 and also I is named a Boolean Algebra when the adhering to axioms keeps for almost any elements a, b, c ? B: Example: Here are not one but two different Boolean algebras with a pair of factors which are isomorphic. Associated with particular relevance is definitely the two-element BA, created by taking a established $$X$$ to acquire just one part. An Or maybe checkpoint is usually a judgement gates which provides large end result if a minumum of one on the information is definitely high.
August 20, 2019
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A group of $15$ routers is interconnected in a centralized complete binary tree with a router at each tree node. Router $i$ communicates with router $j$ by sending a message to the root of the tree. The root then sends the message back down to router $j$. The mean number of hops per message, assuming all possible router pairs are equally likely is
1. $3$
2. $4.26$
3. $4.53$
4. $5.26$
edited | 4.4k views
0
Just find the expectation for each level of tree
OPTION is C.
Here, we have to count average hops per message.
Steps:
1) Message goes up from sender to root
2) Message comes down from root to destination
1) Average hops message goes to root - $\dfrac{(3\times 8)+(2\times 4)+(1\times 2)+(0\times 1) }{15}=2.267$
Here $3\times 8$ represents $3$ hops & $8$ routers for Bottommost level & So on..
2) Similarly average hops when message comes down - $\dfrac{(3\times 8)+(2\times 4)+(1\times 2)+(0\times 1)}{15}$ {Same as above}
So, Total Hops $= 2\times 2.267 =4.53$ (Answer)
by Boss (15.6k points)
edited
+2
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Ur welcome.
+4
It seems that Source and Destination can be same, by your answer. Isn't it? That results in a very small offset in your answer, from Arjun's. :p
0
@arjun sir @ manoj,@shresta,@kapil why it cant be like this:
1 leaf node can send message to other 7 leafnode and another leaf node can send to other 7 and so on
so it will be 8*8*3(considering leaf to root)
0
yes that is
you r counting how a leaf can send a message.rt?
And he is calculating how many ways a message can be send.
See the last solution , it is what u r telling rt?
0
Excellent!!
0
beautifully explained ...
0
Are source and destination nodes too calculated as hops? For a leaf node, source to root will have 3 hops and root to the destination will have 2 hops. But you are calculating root to the destination as 3 hops. Why are you counting destination as hop in the second step viz., root node to destination calculation?
Explanation: Consider Complete tree in Figure
If H wants to communicate with router at level 3 then it first sends packet to node A, then A forward packet to the router at Level 3; total 6 hops are required if A wants to communicate with any level 3 node.
Similarly, 5 hops are required if H wants to communicate with any level 2 node , 4 hops are required if H wants to communicate with any level 1 node and 3 hops are required if H wants to communicate with any level 0 node . Hops required if H wants to communicates with all other nodes = (8-1)*6 + 4*5+2*4 +1*3 = 73 If all 8 level 3 nodes communicates with all other nodes then hops required=73*8=584
Similarly, Hops required if D wants to communicates with all other nodes = 8*5 + (4-1)*4+2*3 +1*2 = 60 If all 4 level 2 nodes communicates with all other nodes then hops required=60*4=240 Hops required
if B wants to communicates with all other nodes = 8*4 + 4*3+(2-1)*2 +1*1 = 47 If all 4 level 2 nodes communicates with all other nodes then hops required=47*2=94 Hops required
if A wants to communicates with all other nodes = 8*3 + 4*2+2*1 = 34 Total hops required
when all nodes communicate with all other nodes=584+240+94+34= 952
Total number of message is 2 * 15C2 =2 * (15*14/2)=2*105=210 Here 2 is multiplied with 15C2 because in communication between A and B, A sends message to B and B sends message to A. The mean number of hops per message= 952/210= 4.53
by (305 points)
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in your solution assumption is that a router can't send a message to itself. anyway, it's a self understood thing that i is not equal to j.
great explanation!
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Nice Explanation !!
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Great explanation ..thanks.
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@Iceberg
Nice explanation !!
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i also initially thought the same way but did some calculation mistake. very nice answer. thanks
0
Great Explanation!!
The path length differs for nodes from each level. For a node in level $4,$
we have maximum no. of hops as follows,
Level Max. no. of hops
1 3 (3-2-1)
2 3+1 = 4 (3-2-1-2)
3 3 + 2 = 5 (3-2-1-2-3)
4 3 + 3 = 6 (3-2-1-2-3-4)
So, mean no. of hops for a node in level $4$
$= \dfrac{1.3 + 2.4 + 4.5 + 7.6}{14} =\dfrac{73}{14}$, as we have $1, 2, 4$ and $8$ nodes
respectively in levels $1, 2, 3$ and $4$ and we discard the source one in level $4.$
Similarly, from a level $3$ node we get mean no. of hops,
$= \dfrac{1.2 + 2.3 + 3.4+ 8.5}{14} = \dfrac{60}{14}$
From level $2,$ we get mean no. of hops
$= \dfrac{1.1 +1.2 + 4.3 + 8.4}{14} = \dfrac{47}{14}$
And from level $1,$ we get, mean no. of hops
$= \dfrac {0 + 2.1 + 4.2 + 8.3}{14} = \dfrac{34}{14}$.
So, now we need to find the overall mean no. of hops which will be
$= \dfrac{\text{Sum of mean no. of hops for each node}}{\text{No. of nodes}}$
$= \dfrac{ \dfrac{73}{14} \times 8 + \dfrac{60}{14} \times 4 + \dfrac{47}{14} \times 2 + \dfrac{34}{14} \times 1}{15}$
$= \dfrac{68}{15}$
$= 4.53$
by Veteran (425k points)
edited
0
@Arjun , Can you give me reference to more problems of this kind ?
This is very lengthy problem of averaging !
0
@Akash , check my answer..
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how to count the number of hops ?.
0
See now..
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Thanks Arjun Sir....
0
not able to understand. plss sm 1 help me with representation of tree, how everything is being calculated?
+1
@Arjun Why did you taken destination in hop count?
https://en.wikipedia.org/wiki/Hop_(networking) diagram here says you shouldn't
Put n = 4 in the formula given below:
by Boss (33.8k points)
0
How you decide the n=4 ?
I am not getting plz tell ..
+2
15 nodes are there in a complete binary tree. So, $n=4$ as $2^n-1$ is the no. of nodes.
0
what book or e-book is this ?
+3
+1 vote
by Loyal (7.8k points)
+1 vote
And Now comes my answer , hope atleast someone will find it helpful
I am still so not know why to include destination in hop count. as wiki says we should not
https://en.wikipedia.org/wiki/Hop_(networking) but solving this question by including destination
Level No of node 1 1 2 2 3 4 4 8
path from level path to level no of such path length of path calculation
4 4
8C2 = 28
(path from any of level 4 node to level 4 node)
6
4 3 8*4 5
4 2 8*2 4
4 1 8*1 3
3 3 4C2=6 4
3 2 4*2=8 3
3 1 4 2
2 2 1 2
2 1 2 1
1 1 1 0
Now multiply 'no of such path' with respective 'length of path' and divide by total of 'length of path' ANS is C
by Active (4.9k points)
explanation by Amitabh Tiwari:-
You have 8 leaves.
If a leaf wants to communicate to other 7 leaves ...each such communication would need 7 hops. So 7*7 hops for leaf to leaf communication.
Now each of these leaves can communicate with the 4 nodes in the level above them in 5 hops. So 4*5.
Each of these leaves can communicate with 2 nodes who are the children of root in 4 hops. So 2*4.
Each of these leaves could also communicate with root in 3 hops.
So on total for a single leaf average number of hops to communicate with all other nodes in tree is : (7*7 + 4*5 +2*4 + 3)/14
There are 8 such leaves:
So multiply the above expression by 8 to get average number of hops for communication of leaves with all other nodes in tree.
Now the way we did it for leaves.
Repeat the same procedure for nodes at level 2,level 1 and root.
by Active (4.8k points)
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# Spectral density estimation
(Redirected from Periodogram)
For the statistical concept, see probability density estimation.
For a broader coverage related to this topic, see Spectral density.
In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.
SDE should be distinguished from the field of frequency estimation, which assumes that a signal is composed of a limited (usually small) number of generating frequencies plus noise and seeks to find the location and intensity of the generated frequencies. SDE makes no assumption on the number of components and seeks to estimate the whole generating spectrum.
## Overview
Example of voice waveform and its frequency spectrum
A triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental frequency component is at 220 Hz (A2).
Spectrum analysis, also referred to as frequency domain analysis or spectral density estimation, is the technical process of decomposing a complex signal into simpler parts. As described above, many physical processes are best described as a sum of many individual frequency components. Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities, or phases), versus frequency can be called spectrum analysis.
Spectrum analysis can be performed on the entire signal. Alternatively, a signal can be broken into short segments (sometimes called frames), and spectrum analysis may be applied to these individual segments. Periodic functions (such as $sin (t)$) are particularly well-suited for this sub-division. General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis.
The Fourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form. This means that the original function can be completely reconstructed (synthesized) by an inverse Fourier transform. For perfect reconstruction, the spectrum analyzer must preserve both the amplitude and phase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude (amplitude) and phase in polar coordinates (i.e., as a phasor). A common technique in signal processing is to consider the squared amplitude, or power; in this case the resulting plot is referred to as a power spectrum.
In practice, nearly all software and electronic devices that generate frequency spectra apply a fast Fourier transform (FFT), which is a specific mathematical approximation to the full integral solution. Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal.
Because of reversibility, the Fourier transform is called a representation of the function, in terms of frequency instead of time; thus, it is a frequency domain representation. Linear operations that could be performed in the time domain have counterparts that can often be performed more easily in the frequency domain. Frequency analysis also simplifies the understanding and interpretation of the effects of various time-domain operations, both linear and non-linear. For instance, only non-linear or time-variant operations can create new frequencies in the frequency spectrum.
The Fourier transform of a stochastic (random) waveform (noise) is also random. Some kind of averaging is required in order to create a clear picture of the underlying frequency content (frequency distribution). Typically, the data is divided into time-segments of a chosen duration, and transforms are performed on each one. Then the magnitude or (usually) squared-magnitude components of the transforms are summed into an average transform. This is a very common operation performed on digitally sampled time-domain data, using the discrete Fourier transform. This type of processing is called Welch's method. When the result is flat, it is commonly referred to as white noise. However, such processing techniques often reveal spectral content even among data which appears noisy in the time domain.
## Periodogram
Suppose that a signal is sampled at $N$ different times, with the samples uniformly spaced by $\Delta t$, giving values $x_n$. Since the power spectral density of a continuous function defined on the entire real line is the modulus squared of its Fourier transform, the simplest technique to estimate the spectrum is the periodogram, given by the modulus squared of the discrete Fourier transform,
$S(f)=\frac{\Delta t}{N} \left| \sum_{n=0}^{N-1} x_n e^{-i2\pi n f} \right|^2, \qquad -\frac{1}{2\Delta t} < f \le \frac{1}{2\Delta t}$
where $1/(2\Delta t)$ is the Nyquist frequency. The name "periodogram" was coined by Arthur Schuster in 1898.[1]
Despite the simplicity of the periodogram, the method suffers from severe deficiencies. It is an inconsistent estimator because it does not converge to the true spectral density as $N\rightarrow\infty$. It exhibits very high spectral leakage although this can be reduced by multiplying $x_n$ by a window function. In the presence of additive noise, the estimate has a positive bias.
## Techniques
Many different techniques for spectral estimation have been developed to overcome the problems of the naive periodogram. These techniques can generally be divided into non-parametric and parametric methods. The non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure. The periodogram itself is a non-parametric approach, and is essentially equivalent to the Fourier transform of the biased autocovariance convolved with a Fejér kernel. Some of the most common estimators in use for basic applications (e.g. Welch's method) are non-parametric estimators closely related to the periodogram. By contrast, the parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an auto-regressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process.
Following is a partial list of non-parametric spectral density estimation techniques:
Below is a partial list of parametric techniques:
## Parametric estimation
In parametric spectral estimation, one assumes that the signal is modeled by a stationary process which has a spectral density function (SDF) $S(f;a_1,\ldots,a_p)$ that is a function of the frequency $f$ and $p$ parameters $a_1,\ldots,a_p$.[2] The estimation problem then becomes one of estimating these parameters.
The most common form of parametric SDF estimate uses as a model an autoregressive model $AR(p)$ of order $p$.[2]:392 A signal sequence $\{Y_t\}$ obeying a zero mean $AR(p)$ process satisfies the equation
$Y_t = \phi_1Y_{t-1} + \phi_2Y_{t-2} + \cdots + \phi_pY_{t-p} + \epsilon_t,$
where the $\phi_1,\ldots,\phi_p$ are fixed coefficients and $\epsilon_t$ is a white noise process with zero mean and innovation variance $\sigma^2_p$. The SDF for this process is
$S(f;\phi_1,\ldots,\phi_p,\sigma^2_p) = \frac{\sigma^2_p\Delta t}{\left| 1 - \sum_{k=1}^p \phi_k e^{-2i\pi f k \Delta t}\right|^2} \qquad |f| < f_N,$
with $\Delta t$ the sampling time interval and $f_N$ the Nyquist frequency.
There are a number of approaches to estimating the parameters $\phi_1,\ldots,\phi_p,\sigma^2_p$ of the $AR(p)$ process and thus the spectral density:[2]:452-453
• The Yule-Walker estimators are found by recursively solving the Yule-Walker equations for an $AR(p)$ process
• The Burg estimators are found by treating the Yule-Walker equations as a form of ordinary least squares problem. The Burg estimators are generally considered superior to the Yule-Walker estimators.[2]:452 Burg associated these with maximum entropy spectral estimation.[3]
• The forward-backward least-squares estimators treat the $AR(p)$ process as a regression problem and solves that problem using forward-backward method. They are competitive with the Burg estimators.
• The maximum likelihood estimators assume the white noise is a Gaussian process and estimates the parameters using a maximum likelihood approach. This involves a nonlinear optimization and is more complex than the first three.
Alternative parametric methods include fitting to a moving average model (MA) and to a full autoregressive moving average model (ARMA).
## Frequency estimation
Frequency estimation is the process of estimating the complex frequency components of a signal in the presence of noise given assumptions about the number of the components.[4] This contrasts with the general methods above, which do not make prior assumptions about the components.
### Finite number of tones
A typical model for a signal $x(n)$ consists of a sum of $p$ complex exponentials in the presence of white noise, $w(n)$
$x(n) = \sum_{i=1}^p A_i e^{j n \omega_i} + w(n)$.
The power spectral density of $x(n)$ is composed of $p$ impulse functions in addition to the spectral density function due to noise.
The most common methods for frequency estimation involve identifying the noise subspace to extract these components. These methods are based on eigen decomposition of the autocorrelation matrix into a signal subspace and a noise subspace. After these subspaces are identified, a frequency estimation function is used to find the component frequencies from the noise subspace. The most popular methods of noise subspace based frequency estimation are Pisarenko's method, the multiple signal classification (MUSIC) method, the eigenvector method, and the minimum norm method.
Pisarenko's method
$\hat P_{PHD}(e^{j \omega}) = \frac{1}{|\mathbf{e}^{H}\mathbf{v}_{min}|^2}$
MUSIC
$\hat P_{MU}(e^{j \omega}) = \frac{1}{\sum_{i=p+1}^{M} |\mathbf{e}^{H} \mathbf{v}_i|^2}$,
Eigenvector method
$\hat P_{EV}(e^{j \omega}) = \frac{1}{\sum_{i=p+1}^{M}\frac{1}{\lambda_i} |\mathbf{e}^H \mathbf{v}_i|^2}$
Minimum norm method
$\hat P_{MN}(e^{j \omega}) = \frac{1}{|\mathbf{e}^H \mathbf{a}|^2} ; \mathbf{a} = \lambda \mathbf{P}_n \mathbf{u}_1$
### Single tone
If one only wants to estimate the single loudest frequency, one can use a pitch detection algorithm. If the dominant frequency changes over time, then the problem becomes the estimation of the instantaneous frequency as defined in the time–frequency representation. Methods for instantaneous frequency estimation include those based on the Wigner-Ville distribution and higher order ambiguity functions.[5]
If one wants to know all the (possibly complex) frequency components of a received signal (including transmitted signal and noise), one uses a discrete Fourier transform or some other Fourier-related transform.
## References
1. ^ Schuster, A., "On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena," Terrestrial Magnetism, 3, 13-41, 1898.
2. ^ a b c d Donald B. Percival and Andrew T. Walden (1992). Spectral Analysis for Physical Applications. Cambridge University Press. ISBN 9780521435413.
3. ^ Burg, J.P. (1967) "Maximum Entropy Spectral Analysis", Proceedings of the 37th Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma.
4. ^ Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.
5. ^ Lerga, Jonatan. "Overview of Signal Instantaneous Frequency Estimation Methods" (PDF). University of Rijeka. Retrieved 22 March 2014.
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How to make a platformer character go DOWN a 315 degree slope?
Alright, so I got this code I'm trying to write, but the player won't go down the slope, it goes down and up perfectly on the 45 degree slope, but the 315 degree one is a mess. it goes "up" the downward slope, get's stuck and some other stuff. Anyway here's the code:
//you're in a double for loop going through every tile in the map, the int's being used are x, and y
//check if the tile is a slope
if (lv.type[lv.tile[x, y]] == Tiles.SLOPE)
{
//create a rectangle collision box
Rectangle tileCol = new Rectangle(x * lv.tileSize, (y * lv.tileSize), lv.tileSize, lv.tileSize + 1);
//if player collision "col" collides with "tileCol" and you haven't done this before this itteration (only happens once per full double loop)
if (col.Intersects(tileCol) && !onSlope)
{
//get the angle of the tile
float angle = lv.angle[lv.tile[x, y]];
//get the x distance of how far away the player's right is inside the tile
float dist = (col.X + col.Width) - tileCol.X;
//constructs the opposite of a right triangle
float opposite = (float)(Math.Tan(MathHelper.ToRadians(angle)) * (dist));
if (angle < 90)
{
//if player's right is less then or equal to tile's right
if (col.X + col.Width <= tileCol.X + tileCol.Width)
{
//place player on slope. this works properly
pos.Y = tileCol.Y - opposite;
//tell the program we don't wanna go through this again until the next full loop starts.
onSlope = true;
}
}
else if (angle > 90){
if ((col.X + col.Width) >= tileCol.X)
{
//this is where the error is. the player goes "up" a slope that's 315 degrees, instead of down it.
//how do I make the player go down the slope that's 315 degrees!?
pos.Y = tileCol.Y + opposite;
onSlope = true;
}
}
}
}
currently using this code, it makes the player move in a 45 degree angle on the 315 degree blocks.
else if (angle > 90)
{
if (col.X >= tileCol.X)
{
pos.Y = tileCol.Y + lv.tileSize + (dist * -1);
onSlope = true;
}
}
• do you have a picture or movie to show what is happening? Sep 30, 2011 at 5:46
• youtube.com/watch?v=ASYbyA2G2Ug Sep 30, 2011 at 6:01
Handle all slope tiles as a rise/run ratio. Each slope tile will have a ratio, and assuming 45 degrees means going up as you go right, 315 degrees will have a ratio of -1. Think back to basic coordinate algebra. Using the player's X position local to the tile, which is the run, solve for pos.Y by multiplying the X position by -1.
However since y = 0 is likely at the bottom of the tiles, you'll have to offset the height by adding the height of the tile.
So the player is colliding with a downwards sloping tile, the slope ratio is negative and the formula should be
pos.Y = lv.tileSize + (pos.X * slopeRatio)
Otherwise
pos.Y = pos.X * slopeRatio
if the slope is going up. Determine if the slope is upwards or downards by the sign of the slope ratio.
• Thank you, but how do I convert the angle to a rise/run ratio? Sep 30, 2011 at 6:21
• @CyanPrime ratio = tan(angle). Sep 30, 2011 at 6:25
• Sorry, but this doesn't seem to be working. Can you think of any other ways to maybe solve this? Sep 30, 2011 at 6:41
• wait, it almost works. it's going to 45 degree angle on the 315 tiles. I'll post what I got so far. Sep 30, 2011 at 6:47
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# 8.2. Amortized Analysis¶
Our analysis of the efficiency of hash table operations concluded that find runs in expected constant time, where the modifier “expected” is needed to express the fact the performance is on average and depends on the hash function satisfying certain properties.
We also concluded that insert would usually run in expected constant time, but that in the worst case it would require linear time because of needing to rehash the entire table. That kind of defeats the goal of a hash table, which is to offer constant-time performance, or at least as close to it as we can get.
It turns out there is another way of looking at this analysis that allows us to conclude that insert does have “amortized” expected constant time performance—that is, for excusing the occasional worst-case linear performance. Right away, we have to acknowledge this technique is just a change in perspective. We’re not going to change the underlying algorithms. The insert algorithm will still have worst-case linear performance. That’s a fact.
But the change in perspective we now undertake is to recognize that if it’s very rare for insert to require linear time, then maybe we can “spread out” that cost over all the other calls to insert. It’s a creative accounting trick!
Sushi vs. Ramen. Let’s amuse ourselves with a real-world example for a moment. Suppose that you have $20 to spend on lunches for the week. You like to eat sushi, but you can’t afford to have sushi every day. So instead you eat as follows: • Monday:$1 ramen
• Tuesday: $1 ramen • Wednesday:$1 ramen
• Thursday: $1 ramen • Friday:$16 sushi
Most of the time, your lunch was cheap. On a rare occasion, it was expensive. So you could look at it in one of two ways:
• My worst-case lunch cost was $16. • My average lunch cost was$4.
Both are true statements, but maybe the latter is more helpful in understanding your spending habits.
Back to Hash Tables. It’s the same with hash tables. Even though insert is occasionally expensive, it’s so rarely expensive that the average cost of an operation is actually constant time! But, we need to do more complicated math (or more complicated than our lunch budgeting anyway) to actually demonstrate that’s true.
## 8.2.1. Amortized Analysis of Hash Tables¶
“Amortization” is a financial term. One of its meanings is to pay off a debt over time. In algorithmic analysis, we use it to refer to paying off the cost of an expensive operation by inflating the cost of inexpensive operations. In effect, we pre-pay the cost of a later expensive operation by adding some additional cost to earlier cheap operations.
The amortized complexity or amortized running time of a sequence of operations that each have cost $$T_1, T_2, \ldots, T_n$$, is just the average cost of each operation:
$\frac{T_1 + T_2 + ... + T_n}{n}.$
Thus, even if one operation is especially expensive, we could average that out over a bunch of inexpensive operations.
Applying that idea to a hash table, suppose the table has 8 bindings and 8 buckets. Then 8 more inserts are made. The first 7 are (expected) constant-time, but the 8th insert is linear time: it increases the load factor to 2, causing a resize, thus causing rehashing of all 16 bindings into a new table. The total cost over that series of operations is therefore the cost of 8+16 inserts. For simplicity of calculation, we could grossly round that up to 16+16 = 32 inserts. So the average cost of each operation in the sequence is 32/8 = 4 inserts.
In other words, if we just pretended each insert cost four times its normal price, the final operation in the sequence would have been “pre-paid” by the extra price we paid for earlier inserts. And all of them would be constant-time, since four times a constant is still a constant.
Generalizing from the example above, let’s suppose that the the number of buckets currently in a hash table is $$2^n$$, and that the load factor is currently 1. Therefore, there are currently $$2^n$$ bindings in the table. Next:
• A series of $$2^n - 1$$ inserts occurs. There are now $$2^n + 2^n - 1$$ bindings in the table.
• One more insert occurs. That brings the number of bindings up to $$2^n + 2^n$$, which is $$2^{n+1}$$. But the number of buckets is $$2^n$$, so the the load factor just reached 2. A resize is necessary.
• The resize occurs. That doubles the number of buckets. All $$2^{n+1}$$ bindings have to be reinserted into the new table, which is of size $$2^{n+1}$$. The load factor is back down to 1.
So in total we did $$2^n + 2^{n+1}$$ inserts, which included $$2^n$$ inserts of bindings and $$2^{n+1}$$ re-insertions after the resize. We could grossly round that quantity up to $$2^{n+2}$$. Over a series of $$2^n$$ insert operations, that’s an average cost of $$\frac{2^{n+2}}{2^n}$$, which equals 4. So if we just pretend each insert costs four times its normal price, every operation in the sequence is amortized (and expected) constant time.
Doubling vs. Constant-size Increasing. Notice that it is crucial that the array size grows by doubling (or at least geometrically). A bad mistake would be to instead grow the array by a fixed increment—for example, 100 buckets at time. Then we’d be in real trouble as the number of bindings continued to grow:
• Round 1. Insert 100 bindings. There are now 200 bindings and 100 buckets. The load factor is 2.
• Increase the number of buckets by 100 and rehash. That’s 200 more insertions. The load factor is back down to 1.
• The average cost of each insert is so far just 3x the cost of an actual insert (100+200 insertions / 100 bindings inserted). So far so good.
• Round 2. Insert 200 more bindings. There are now 400 bindings and 200 buckets. The load factor is 2.
• Increase the number of buckets by 100 and rehash. That’s 400 more insertions. There are now 400 bindings and 300 buckets. The load factor is 400/300 = 4/3, not 1.
• The average cost of each insert is now 100+200+200+400 / 300 = 3. That’s still okay.
• Round 3. Insert 200 more bindings. There are now 600 bindings and 300 buckets. The load factor is 2.
• Increase the number of buckets by 100 and rehash. That’s 600 more insertions. There are now 600 bindings and 400 buckets. The load factor is 3/2, not 1.
• The average cost of each insert is now 100+200+200+400+200+600 / 500 = 3.2. It’s going up.
• Round 4. Insert 200 more bindings. There are now 800 bindings and 400 buckets. The load factor is 2.
• Increase the number of buckets by 100 and rehash. That’s 800 more insertions. There are now 800 bindings and 500 buckets. The load factor is 8/5, not 1.
• The average cost of each insert is now 100+200+200+400+200+600+200+800 / 700 = 3.7. It’s continuing to go up, not staying constant.
After $$k$$ rounds we have $$200k$$ bindings and $$100k$$ buckets. We have called insert to insert $$100+200k$$ bindings, but all the rehashing has caused us to do $$100+200(k-1)+\sum_{i=1}^{k} 200i$$ actual insertions. That last term is the real problem. It’s quadratic:
$\sum_{i=1}^{k} 200i \quad = \quad \frac{200k (200 (k+1))}{2} \quad = \quad 20,000 (k^2 + k)$
So over a series of $$n$$ calls to insert, we do $$O(n^2)$$ actual inserts. That makes the amortized cost of insert be $$O(n)$$, which is linear! Not constant.
That’s why it’s so important to double the size of the array at each rehash. It’s what gives us the amortized constant-time performance.
## 8.2.2. Amortized Analysis of Batched Queues¶
The implementation of batched queues with two lists was in a way more efficient than the implementation with just one list, because it managed to achieve a constant time enqueue operation. But, that came at the tradeoff of making the dequeue operation sometimes take more than constant time: whenever the outbox became empty, the inbox had to be reversed, which required an additional linear-time operation.
As we observed then, the reversal is relatively rare. It happens only when the outbox gets exhausted. Amortized analysis gives us a way to account for that. We can actually show that the dequeue operation is amortized constant time.
To keep the analysis simple at first, let’s assume the queue starts off with exactly one element 1 already enqueued, and that we do three enqueue operations of 2, 3, then 4, followed by a single dequeue. The single initial element would end up in the outbox. All three enqueue operations would cons an element onto the inbox. So just before the dequeue, the queue looks like:
{o = [1]; i = [4; 3; 2]}
and after the dequeue:
{o = [2; 3; 4]; i = []}
It required
• 3 cons operations to do the 3 enqueues, and
• another 3 cons operations to finish the dequeue by reversing the list.
That’s a total of 6 cons operations to do the 4 enqueue and dequeue operations. The average cost is therefore 1.5 cons operations per queue operation. There were other pattern matching operations and record constructions, but those all took only constant time, so we’ll ignore them.
What about a more complicated situation, where there are enqueues and dequeues interspersed with one another? Trying to take averages over the series is going to be tricky to analyze. But, inspired by our analysis of hash tables, suppose we pretend that the cost of each enqueue is twice its actual cost, as measured in cons operations? Then at the time an element is enqueued, we could “prepay” the later cost that will be incurred when that element is cons’d onto the reversed list.
The enqueue operation is still constant time, because even though we’re now pretending its cost is 2 instead of 1, it’s still the case that 2 is a constant. And the dequeue operation is amortized constant time:
• If dequeue doesn’t need to reverse the inbox, it really does just constant work, and
• If dequeue does need to reverse an inbox with $$n$$ elements, it already has $$n$$ units of work “saved up” from each of the enqueues of those $$n$$ elements.
So if we just pretend each enqueue costs twice its normal price, every operation in a sequence is amortized constant time. Is this just a bookkeeping trick? Absolutely. But it also reveals the deeper truth that on average we get constant-time performance, even though some operations might rarely have worst-case linear-time performance.
## 8.2.3. Bankers and Physicists¶
Conceptually, amortized analysis can be understood in three ways:
1. Taking the average cost over a series of operations. This is what we’ve done so far.
2. Keeping a “bank account” at each individual element of a data structure. Some operations deposit credits, and others withdraw them. The goal is for account totals to never be negative. The amortized cost of any operation is the actual cost, plus any credits deposited, minus any credits spent. So if an operation actually costs $$n$$ but spends $$n-1$$ credits, then its amortized cost is just $$1$$. This is called the banker’s method of amortized analysis.
3. Regarding the entire data structure as having an amount of “potential energy” stored up. Some operations increase the energy, some decrease it. The energy should never be negative. The amortized cost of any operation is its actual cost, plus the change in potential energy. So if an operation actually costs $$n$$, and before the operation the potential energy is $$n$$, and after the operation the potential energy is $$0$$, then the amortized cost is $$n + (0 - n)$$, which is just $$0$$. This is called the physicist’s method of amortized analysis.
The banker’s and physicist’s methods can be easier to use in many situations than a complicated analysis of a series of operations. Let’s revisit our examples so far to illustrate their use:
• Banker’s method, hash tables: The table starts off empty. When a binding is added to the table, save up 1 credit in its account. When a rehash becomes necessary, every binding is guaranteed to have 1 credit. Use that credit to pay for the rehash. Now all bindings have 0 credits. From now on, when a binding is added to the table, save up 1 credit in its account and 1 credit in the account of any one of the bindings that has 0 credits. At the time the next rehash becomes necessary, the number of bindings has doubled. But since we’ve saved 2 credits at each insertion, every binding now has 1 credit in its account again. So we can pay for the rehash. The accounts never go negative, because they always have either 0 or 1 credit.
• Banker’s method, batched queues: When an element is added to the queue, save up 1 credit in its account. When the inbox must be reversed, use the credit in each element to pay for the cons onto the outbox. Since elements enter at the inbox and transition at most once to the outbox, every element will have 0 or 1 credits. So the accounts never go negative.
• Physicist’s method, hash tables: At first, define the potential energy of the table to be the number of bindings inserted. That energy will therefore never be negative. Each insertion increases the energy by 1 unit. When the first rehash is needed after inserting $$n$$ bindings, the potential energy is $$n$$. The potential goes back down to $$0$$ at the rehash. So the actual cost is $$n$$, but the change in potential is $$n$$, which makes the amortized cost $$0$$, or constant. From now on, define the potential energy to be twice the number of bindings inserted since the last rehash. Again, the energy will never be negative. Each insertion increases the energy by 2 units. When the next rehash is needed after inserting $$n$$ bindings, there will be $$2n$$ bindings that need to be rehashed. Again, the amortized cost will be constant, because the actual cost of $$2n$$ re-insertions is offset by the $$2n$$ change in potential.
• Physicist’s method, batched queues: Define the potential energy of the queue to be the length of the inbox. It therefore will never be negative. When a dequeue has to reverse an inbox of length $$n$$, there is an actual cost of $$n$$ but a change in potential of $$n$$ too, which offsets the cost and makes it constant.
The two methods are equivalent in their analytical power:
• To convert a banker’s analysis into a physicist’s, just make the potential be the sum of all the credits in the individual accounts.
• To convert a physicist’s analysis into a banker’s, just designate one distinguished element of the data structure to be the only one that will ever hold any credits, and have each operation deposit or withdraw the change in potential into that element’s account.
So, the choice of which to use really just depends on which is easier for the data structure being analyzed, or which is easier for you to wrap your head around. You might find one or the other of the methods easier to understand for the data structures above, and your friend might have a different opinion.
## 8.2.4. Amortized Analysis and Persistence¶
Amortized analysis breaks down as a technique when data structures are used persistently. For example, suppose we have a batched queue q into which we’ve inserted $$n+1$$ elements. One element will be in the outbox, and the other $$n$$ will be in the inbox. Now we do the following:
# let q1 = dequeue q
# let q2 = dequeue q
...
# let qn = dequeue q
Each one of those $$n$$ dequeue operations requires an actual cost of $$O(n)$$ to reverse the inbox. So the entire series has an actual cost of $$O(n^2)$$. But the amortized analysis techniques only apply to the first dequeue. After that, all the the accounts are empty (banker’s method), or the potential is zero (physicist’s), which means the remaining operations can’t use them to pay for the expensive list reversal. The total cost of the series is therefore $$O(n^2 - n)$$, which is $$O(n^2)$$.
The problem with persistence is that it violates the assumption built-in to amortized analysis that credits (or energy units) are spent only once. Every persistent copy of the data structure instead tries to spend them itself, not being aware of all the other copies.
There are more advanced techniques for amortized analysis that can account for persistence. Those techniques are based on the idea of accumulating debt that is later paid off, rather than accumulating savings that are later spent. The reason that debt ends up working as an analysis technique can be summed up as: although our banks would never (financially speaking) allow us to spend money twice, they would be fine with us paying off our debt multiple times. Consult Okasaki’s Purely Functional Data Structures to learn more.
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## Four-Fermion Interaction Model on $\mathcal{M}^{D-1} \otimes S^1$
10 Apr 2019 · Inagaki Tomohiro, Matsuo Yamato, Shimoji Hiromu ·
Four-fermion interaction models are often used as simplified models of interacting fermion fields with the chiral symmetry. The chiral symmetry is dynamically broken for a larger four-fermion coupling... It is expected that the broken symmetry is restored under extreme conditions. In this paper, the finite size effect on the chiral symmetry breaking is investigated in the four-fermion interaction model. We consider the model on a flat spacetime with a compactified spatial coordinate, $\mathcal{M}^{D-1} \otimes S^1$ and obtain explicit expressions of the effective potential for arbitrary spacetime dimensions in the leading order of the $1/N$ expansion. Evaluating the effective potential, we show the critical lines which divide the symmetric and the broken phase and the sign-flip condition for the Casimir force. read more
PDF Abstract
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# Categories
High Energy Physics - Theory
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POSTER SESSION "MOLECULAR MOTORS AND COLLECTIVE MOTIONS IN BIOLOGY" JIANG Rui School of Engineering Science, University of Science and technology of China, Hefei , China Numerical investigations on coupling of asymmetric exclusion process with zero range process Rui Jiang, Bin Jia, Mao-Bin Hu, Ruili Wang, and Qing-Song Wu In recent years, the driven diffusive system has attracted the interests of physicists because it shows a variety of nonequilibrium effects. Some prominent examples are asymmetric simple exclusion processes (ASEPs), and zero-range process (ZRP). The coupling of ASEPs with other processes has led to many unusual and unexpected phenomena. See, e.g., the investigations on the interplay of ASEP with the creation and annihilation of particles, and the coupling of ASEPs with symmetric diffusive process. Inspired by the previous works, we study the coupling of ASEPs with ZRP in this paper. Our model is defined in a closed ring system consisting of two equally sized compartments. For the lower compartment, the dynamics is governed by TASEP; for the upper compartment, the dynamics is governed by ZRP. Moreover, particle exchange happens between the two compartments in both ends and in the bulk. We employ Monte Carlo (MC) simulations to characterize the emerging nonequilibrium steady states, and various interesting nonlinear effects are revealed. In our simulations, the density profiles in both compartments are investigated. Four thresholds for $\omega_{out}$ are identified. In the upper compartment, the density in the left bulk is high and the density in the right bulk is very small when $\omega_{out}=0$. The different densities are separated by a domain wall. With the increase of $\omega_{out}$, the density increases in the right bulk and the domain wall moves left. At the first threshold $\omega_{out}=\omega_{out,1}$, the domain wall reaches the left boundary. For $\omega_{out} > \omega_{out,1}$, a very small density appears in the left bulk. With the increase of $\omega_{out}$, the domain wall moves right and the density in the right bulk continues to increase. When $\omega_{out}$ is larger than a second threshold $\omega_{out,2}$, a zero density will appear in the left bulk and the density in the right bulk reaches one. The state remains if $\omega_{out}$ is smaller than a third threshold $\omega_{out,3}$. When $\omega_{out} > \omega_{out,3}$, the density begins to increase with $\omega_{out}$ in the left bulk and the domain wall moves right. The density in the right bulk is still one. At the fourth threshold $\omega_{out,4}$, the domain wall reaches the right boundary. When $\omega_{out} > \omega_{out,4}$, a zero density appears in the right bulk. The domain wall moves left and the density in the left bulk still increases with the increase of $\omega_{out}$ until $\omega_{out}=1$. In the lower compartment, the density is very high when $\omega_{out} < \omega_{out,1}$, despite a small density jump exists in the bulk. When $\omega_{out,1} < \omega_{out} < \omega_{out,2}$, the density is monotonically increasing with $x$ in the left bulk and it is high in the right bulk. When $\omega_{out,2} < \omega_{out} < \omega_{out,3}$, a zero density is reached in the left bulk and density one is reached in the right bulk as in the upper compartment. Then similar results as in upper compartment is observed with the increase of $\omega_{out}$. MULLER Melanie Max Planck Institute of Colloids and Interfaces, Potsdam, Germany Bidirectional cargo transport by two species of molecular motors Melanie J.I. Muller, Stefan Klumpp and Reinhard Lipowsky Long-range intracellular transport is based on molecular motors that pull cargos along cytoskeletal filaments. One type of motor always moves in one direction, e.~g. conventional kinesin moves to the microtubule plus end, while cytoplasmic dynein moves to the microtubule minus end. However, many cellular cargos are observed to move bidirectionally, involving both plus-end and minus-end directed motors. We present a stochastic 'tug-of-war' model for this scenario, in which motors work independently and are coupled only via the mechanical interaction with their common cargo. Depending on the motor parameters (such as microtubule affinity or stall force), the cargo displays stochastic switching between fast plus end motion,fast minus end motion and / or no significant motion. In the parameter range which leads to switches between fast plus and minus end motion, the motors appear to act in a cooperative way despite the underlying tug-of-war. SCHADSCHNEIDER Andreas University of Cologne, Germany Traffic flow on ant trails: Empirical results A. John, A. Schadschneider, D. Chowdhury, K. Nishinari We report results of an empirical study of traffic flow on ant trails. In analogy to vehicular traffic "single-vehicle data" like individual velocities, time-headways etc. have been measured for uni-directional and bi-directional trails. Apart from velocity and headway distributions also fundamental diagrams have been calculated. In contrast to highway traffic no jammed regime is observed and the average velocity is constant over a large density regime. We discuss our findings in terms of the spatio-temporal organisation of the ants on their trails and compare with theoretical predictions of a simple cellular automaton approach. YANG Xian-qing College of Science, China University of Mining and Technology, Xuzhou, China Effects of Detachment and size of particles in Totally Asymmetric Simple Exclusion Processes Xian-qing Yang, Kang Qiu,Lin Ren, Wen-tao Xu In this article, the effects of irreversible detachments of particles in totally asymmetric simple exclusion processes (TASEPs) with extended particles which occupy more than one lattice site, are investigated. First, an approximate mean-field theory is used to calculate phase diagrams and density profiles. The results show that the detachment and the size of the particles have distinct effects on the stationary phases in the two sublattices divided by the detachment, especially in the mc/hd and the hd/hd phase. Here, symbols ''mc'', ''hd'', and ''ld'' are initials of maximal current, high density, and low density, respectively, and ''mc/hd'' represents the stationary state that the left sublattice is in the maximum current (mc) phase while the right sublattice is in the high density (hd) phase. When the detachment rate is very large, there are four stationary phases, including ld/ld, ld/hd, mc/ld, and mc/hd phases. When the value of the detachment rate is in the middle range, the hd/hd phase occurs, and hence there exist five stationary phases. When the rate is very small, the mc/hd phase disappears, and there are only four phases again. These theoretical results qualitatively and quantitatively agree with computer Monte Carlo simulations especially in the case of large value of the detachment rate.
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Public Group
# why does this work
This topic is 4426 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
And now for something completely different: This works while I think it should crash: why? OK, i have an array containing items of the class test. The array is called "proef" and has 40 elements. test contains a bool that stores if a test was started. "isgestart" returns this bool. Now I want to start the first test that was not started, so I write: void starttest(){ int i=0; system("cls"); while (proef.isgestart()){i++;} cout << "Proefpersoon" << i+1 << endl <<endl; proef.start(); } works fine, but when I start more than 40 tests, it still seems to work... PS I know system("cls") is a dirty way of cleaning the screen, but i need this program by wednesday.
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It's because C++ doesn't perform bounds checking. It'll quite happily write to any index you tell it to. If the number is quite large, that'll probably cause your program to crash, as it attempts to write to memory it doesn't have permission to write to. But for 41, 42, 43, etc. it may well "work". That doesn't mean it's a good idea, though: it might work today and not work tomorrow.
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Short answer: it doesn't really. This invokes one instance of "undefined behaviour", and anything the computer can do is technically allowed by the standard to happen - including seeming to work, crashing the computer, or emailing the contents of your pr0n folder to the government. (Admittedly, that last one is rather less likely.)
Longer answer: like Sneftel said, C++ doesn't provide (generate in the compiled code) bounds checking. So when 'i' reaches 40, the program is happy to consider a chunk of memory just past the end of your array, the same size as an element, as if the data there represented a valid element. And then perform the 'isgestart()' code, using a pointer to that memory as the this-pointer.
At that point, what happens depends on what values happened to be stored in the memory there, and whether or not the system will allow you to access that memory . Modern OSes provide memory protection so that if you try to use memory that doesn't "belong" to your program, it will just bail out right away. However, memory close to your array is likely to "belong" anyway - often it is holding some other variable in your program, or something that was pushed on the stack, etc. Sometimes it becomes possible to change other variables that way, and cause a mysterious crash sometime later - very difficult to debug.
This is one of the many reasons to avoid arrays - and also null-terminated strings - as much as possible, preferring the standard library containers, or when appropriate other wrappers like Boost::Array; and to make sure to iterate in "safe" ways. In the sample code, a for-loop is more appropriate, since you have an easy way to specify where the loop bounds are. You can also make use of a standard library 'algorithm':
#include <algorithm>// in addition to your other includes.// Here I am using 'persoon' as the type of the proef[] elementsvoid starttest(){ system("cls"); persoon* first_not_started = find_if(proef, proef + 40, mem_fun_ref(&persoon::isgestart)); // find_if is in namespace std of course // The '40' here is simply the number of elements in your array. // What you gain by doing it this way is not having to worry about a counter // variable like 'i'; when you are nesting loops that can prevent errors like // for (int i = 0; i < 10; ++i) { // for (int j = 0; j < 10; ++i) { <-- oops! // and it also means you don't have to think so much about the logic of the // iteration. // Also, if you later change 'proef' to be a std::vector for example, then // all you need to do is change the bounds to (proef.begin(), proef.end() // and it will work again - plus, then you are not limited in the number of // elements :) if (first_not_started == proef + 40) { // There was nothing found :( cout << "I don't really know Dutch" << endl; // ;) // With the std::vector, you could handle this by push_back()ing another // persoon and starting it. But then, you probably wouldn't need all this // isgestart/start logic then anyway; just hold as many things as you need // to start, and not keep any unstarted instances. } else { cout << "Proefpersoon" << first_not_started - proef + 1 << endl <<endl; first_not_started->start(); }}
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thanks a lot guys!
I just thought there was bouds checking in C++...
I'll look into the algorithm library. Sorry about the dutch in the code by the way, but I'm working together on this thing with someone who knows little about programming and less about Englisch (yes, they exist), so it's easier to explain with dutch variables...
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# Square Root of 851929
The square root of 851929 is the number, which multiplied by itself, is 851929. In other words, the square of this number equals eight hundred and fifty-one thousand, nine hundred and twenty-nine. If you have been looking for square root of eight hundred and fifty-one thousand, nine hundred and twenty-nine, then you are right here, too. On this page you can also find what the parts of √851929 are called, and in addition to the terminology of √851929, we also have a calculator you don’t want to miss. Read on to learn everything about the sqrt 851929.$\sqrt{851929}= \pm 923$
Extracting the root is the inverse operation of ^2:$\sqrt[2]{851929}\times \sqrt[2]{851929}= \sqrt[2]{851929}^{2}= 851929$
The term can be written as $\sqrt{851929} \hspace{3 mm}or\hspace{3 mm} \sqrt[2]{851929}$
Like any positive number, the number 851929 has two square roots: $\sqrt[2]{851929}$, which is positive and called principal square root of 851929, and −$\sqrt[2]{851929}$, which is negative.
Together, they are denominated as ± $\sqrt[2]{851929}$.
Although the principal square root of eight hundred and fifty-one thousand, nine hundred and twenty-nine is only one of the two square roots, the term “square root of 851929” usually refers to the positive number.
If you want to know how to find the square root of 851929, then read our article square root which you can find in the header menu.
Here’s the cube root of 851929.
## What is the Square Root of 851929
You already have the answer to what is the square root of 851929, and you also know about the inverse operation of 851929 square root. Keep reading to learn what the parts of √851929 are called.
$\sqrt[n]{a}= b$
n = index, 2 is the index.
b = root = ±923
$\sqrt[2]{851929}= \pm 923$
Now you really know all about √851929, including its values, parts and the inverse. If you like to learn the square root of any other number use our calculator below. Simply insert the number of which you like to find the square root (e.g. 851929); the calculation is done automatically.
### Calculate Square Root
Number:
Value:
Bookmark this calculator now!
Besides √851929, other square roots on this site include, for example:
## Square Root of Eight Hundred And Fifty-One Thousand, Nine Hundred And Twenty-Nine
If you have been searching for whats the square root of 851929 or square root 851929, then you have visited the right post as well.
The same is true if you typed sq root of 851929 or 851929 root in the search engine of your preference, just to name a few similar terms.
To sum up, $\sqrt{851929}= \pm 923$
The negative square root of 851929 is -923, and the positive sqrt 851929 is 923. Make sure to understand that √851929 and 851929 squared, 851929 x 851929 = 725783021041, are not the same.
Finding the square root of the number 851929 is the inverse operation of squaring the √851929. In other words (±923)2 = 851929.
Further information related to √851929 can be found on our page square root. Note that you can also obtain roots like √851929 by means of the search form in the menu and sidebar of this site.
If our article about sqrt 851929 has been useful to you, then press some of the share buttons. If you have any questions about the square roots of 851929, fill in the comment form below.
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# Stellar Dirac constant
The stellar Dirac constant, denoted as ħs, is a physical constant, a natural unit of angular momentum and action for the objects of the stellar level of matter.
## Origin
The introduction of the stellar Dirac constant was one of the consequences of the development of the theory of Infinite Hierarchical Nesting of Matter. In 1999, while Sergey Fedosin studied similarity of matter levels and SPФ symmetry, he determined the values of the stellar Planck constant hs, which was related to the stellar Dirac constant by a factor 2π : hs = 2π ħs.
At each level of matter we can distinguish objects that have similar mass, but have different sizes and matter densities. This is possible if the stability of matter is maintained by various mechanisms. Thus, in the main sequence stars the stability is maintained by the pressure of non-relativistic plasma, in white dwarfs – by the pressure of electrons, and in neutron stars – by the pressure of degenerate nucleon gas. Hence it follows that in order to establish similarity between the stars and elementary particles, depending on the types of stars, different sets of similarity coefficients can be used. In addition, the stars of different types must have noncoincident values of characteristic angular momentum.
For ordinary stars and for planets, revolving around them, it is assumed that ħs = 2.8∙1041 J∙s. For degenerate objects, such as neutron stars, the stellar Dirac constant is greater in magnitude: ħ’s = 5.5∙1041 J∙s. [1] [2]
The values of the corresponding stellar Dirac constant can be obtained with the help of the known coefficients of similarity between the levels of stars and elementary particles. At the level of elementary particles the standard unit of angular momentum is the Dirac constant ħ. Taking into account the dimensional analysis, in order to determine ħs and ħ’s we must multiply ħ by the corresponding coefficients of similarity in mass, size and speeds (more detailed information about it can be found in the articles similarity of matter levels, discreteness of stellar parameters, stellar constants, hydrogen system).
In the hydrogen atom the orbital angular momentum of the electron is quantized and is proportional to ħ, and the nuclear spin is assumed to be equal to the value ħ/2. Similarly, the value ħs for planetary systems specifies the characteristic orbital angular momentum of a typical planet, [3] and the value ħs/2 is close to the limiting angular momentum of the low-mass main-sequence stars. [1] At the same time, the value ħ’s/2 describes the angular momentum of rapidly rotating neutron stars, such as PSR 1937+214, for which the angular momentum, as the product of their inertia moment by the angular speed of rotation, can reach L = 4∙1041 J∙s. [4] In white dwarfs the proper angular momenta also do not exceed the value ħ’s/2.
The analysis of the orbital rotation of moons near planets shows that their angular momentum was determined by the angular momentum of protoplanets during the formation of the Solar system. The same applies to the orbital angular momentum of planets, which have obtained their angular momentum from the rapidly rotating shell of the Sun at the stage of compression of a gas-dust cloud into a star. The discovered quantization of the specific orbital and spin angular momenta of planets and planets’ moons supports the fact that quantization in atomic and stellar systems has the same mechanism that is associated with equilibrium of energy fluxes in the matter of electrons and protoplanetary clouds, respectively, at certain distances from the central objects. [2]
## The stellar Dirac constant in various relations
1) For elementary particles, the Chew-Frautschi plots are known, [5] which correspond to Regge trajectories in quantum mechanics and relate the spin of particles in units of Dirac constant and the squared mass-energy of these particles. Passing from the nucleon Chew-Frautschi trajectory to the corresponding trajectory for neutron stars, taking into account the data on the limiting rotation of neutron stars, [6] we obtain the following estimate: ħ’s < 1.2∙1042 Дж∙с. [1]
2) The coefficient of similarity in size Р’ can be found as the ratio of the neutron star radius to the proton radius. If we now multiply the Bohr radius (this is the most probable location of the electron in the hydrogen atom) by Р’, we will obtain the value of the order of 109 m. The Roche limit (the distance, within which any planet near a neutron star must disintegrate due to the gravitational force gradient) has the same value. Observations show that at the given radius disks of scattered matter are found near a number of neutron stars. [7] In the theory of Infinite Hierarchical Nesting of Matter, such disks are assumed to be the analogues of electrons in atoms. If we calculate the angular momentum of these disks, it appears to be close in value to the stellar Dirac constant ħ’s, similarly to the angular momentum of the electron in the hydrogen atom, which is equal to ħ.
3) The stellar Planck constant and the stellar Dirac constant are related by a numerical factor, therefore, in order to estimate the latter constant the methods can be used, which are described in the article stellar Planck constant. They include:
a) The ratio based on the de Broglie waves:
${\displaystyle ~h_{s}^{'}=2M_{s}R_{s}C_{s}=4.4\cdot 10^{42}}$ J∙s,
where Ms and Rs are the mass and radius of the neutron star, Cs is the characteristic speed of the particles in the neutron star.
b) The statistical angular momentum for a black hole as a measure of ħ’s/2.
c) Calculation of ħ’s as a coefficient of proportionality between the natural oscillation frequency and the excitation energy in the black hole.
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# Predicate logic deciding whether atomic formulae hold in interpretations
Consider the formula $$\varphi$$ of First-order logic defined as
$$\forall x\forall y((B(x,y) \land B(y,x)) \rightarrow (A(x)\land C(y)))$$
State whether it holds in the following interpretations:
1. The domain of discourse is the set of natural numbers $$\mathbb{N}$$ with $$B=\{(a,b) \in \mathbb{N} \times \mathbb{N}: a \le b\}$$ and $$A=C=\mathbb{N}$$
2. The domain of discourse is a class of 30 schoolchildren, 2 of whom are twins, with $$B$$ being the set of pairs of distinct schoolchildren of the same age (in whole months), $$A$$ being the complete set of 30 schoolchildren, and $$C$$ being a subset of 29 schoolchildren with one of the twins removed.
3. The domain of discourse is the set of rational numbers $$\mathbb{Q}$$ with $$B = \{(a,b) \in\mathbb{Q} \times \mathbb{Q}: a < b \}, \ A = \emptyset,$$ and $$C = \{0\}$$
My main confusion comes from understanding how the sub-formulae work and how should they be interpreted in the situation. For example, what exactly does $$A(x)$$ mean in $$(ii)$$, likewise with $$C(y)$$. What is the best way to understand and approach questions like these?
• It appears some deficient writer is using A(x) to mean x in A. – William Elliot May 13 at 21:38
• @WilliamElliot Huh? That's not "deficient" at all - we identify $n$-ary predicates with subsets of the $n$th Cartesian power as standard. – Noah Schweber May 13 at 21:50
• Collectively standard deficiency. – William Elliot May 14 at 3:31
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# Automobile Power Generator
1. Mar 19, 2009
### ruko
Most are familiar with dynamometers that measure torque and horsepower output of the automobile drive train. The auto's drive wheels are driven onto a set of rollers and the output of the system is measured at various speeds. I am wondering if an electric power generator could be driven in this manner? The auto cruise control perhaps could be used to maintain a constant voltage and frequency at various load levels.
2. Mar 19, 2009
### mgb_phys
It could but it would be a lot of effort for no obvious reason!
Tractors and some 4x4 vehicles have power-take-off shafts ( a drive shaft coming out of the front) that let you connect equipement. such as a generator powered by the engine.
3. Mar 19, 2009
### ruko
A lot of effort? Driving your car onto a set of rollers is a lot of effort? For no obvious reason? The very plain obvious reason for this is to generate power for the home. Not everybody has a tractor with a power take off but most everybody has a car. Besides, before I would consider buying a tractor or a 4x4 to power my home I would buy a generator from Home Depot. Much simpler and cheaper. My question was one of engineering curiosity not necessarily something I am advocating to do. By the way, power takeoffs are never on the front of tractors.
4. Mar 19, 2009
### Staff: Mentor
He's not talking about the effort of using it he's talking about the effort of building it. The main benefit would be as backup power in the event of an emergency, but the effort (and money) to build it is almost certainly more than it would cost to just buy a stand-alone generator.
In addition, a stand-alone generator would be vastly more efficient.
5. Mar 19, 2009
### Danger
I'm not sure about tractors (although I know that they can come off of the side and I think that I recall a front one on a Cockshutt from my youth). Mgb mentioned 4x4's as well. Many of them have PTO winches that can be on the front or the back. Front is favoured.
6. Mar 19, 2009
### xxChrisxx
^^This^^
Why build something that has inherent losses through the transmission and contact to a rolling road, making it less efficient and more expensive than a diesel generator?
Yes it can be done, but it shouldnt.
7. Mar 19, 2009
### Averagesupernova
I hate to sound like Yoda, but, wrong you are.
8. Mar 19, 2009
### Equate
Last edited by a moderator: May 4, 2017
9. Mar 20, 2009
### ruko
OK, I'm wrong. Sorry.
Last edited by a moderator: May 4, 2017
10. Mar 20, 2009
### Ranger Mike
no you are not wrong...don't ever give up an idea becuase of what is said
give it up once you do a thorugh analysis and the facts determine the cousre of action
your idea of a back up plan for power generation has merit...roller aparatus can be fabricated relatively cheap..will have effciency less than 100% due to HP loss thru drive train and roller contact..but..beats buying a $4000 Diesel generator used once a year but will take up a lot of floor space..require major reqire to close off power supplied by electric company..the line workes really get miffed whe nthey tap into a power line that is supposed to be dead..ZZZZZZAAAAPPPPPPPP ouch!! hey.. go for it... 11. Mar 20, 2009 ### xxChrisxx Thinking about it, its acutally probably not as outlandish as it first sounds. With a diesel engine car (front wheel drive only) and a flange attached straight to the wheel hub (assuming no PTO) there would be only fairly small losses from the transmission. All you'd need to go would be to jack the front of the car up. So it wouldnt be majorly expensive either. I'll never be as good (or as cheap probably) as a generator designed to do this but in a pinch it'd work just fine. It couldnt feasibly be done with a petrol though. EDIT: damn it mike!!! you beat me to it. was looking at diesel generators as you were replying :P 12. Mar 20, 2009 ### Ranger Mike two kinds of people . Amigo..the quick and the dead.. old racing habits are hard to break...too many fast pit stops 13. Mar 20, 2009 ### mgb_phys For regions that have frequent power loss then generators are already pretty common - more common than cars. In most developed countries power losses are very rare, enough that having a flashlight is probably the most preparation people need. But when (if) electric cars become popular there is an idea that they will be used to store grid electricity, not so much for power cuts, but also to level out peaks of supply and demand. 14. Mar 20, 2009 ### russ_watters ### Staff: Mentor Here's a 3200W gas generator for$500: http://www.electricgeneratorsdirect.com/Generac-5724/p2681.html
That's plenty to power everything in a home minus air conditioning and electric heating applications (heat, stove, water heater, etc). That's pretty friggin cheap for that much capacity.
Now if you wanted to integrate it (or the car-based one) with your home electrical system, you'd need an umbilical and auto or manual transfer switch, plus automatic load shedding or the discipline/knowledge to manually load shed when you turn it on (flipping circuit breakers for large loads). You could probably build/wire something like that yourself for \$1000 or so.
Or you could do no installation at all and just run a bunch of large extension cords to where you need the power.
If I lived in a natural disaster prone area, I'd do the manual approach. If I robbed a bank and retired to a non-extradition country with a beach, I'd do the automatic one. Since I live in a thunderstorm prone area with virtually no natural disaster history, long power outages are exceedingly rare. I've had one power outage that wasn't momentary in the 3 years I've lived in my house. It lasted about 4 hours and during that time, I watched a dvd on my laptop while listening to the news on my radio and using my 17 a-h power tank to keep the laptop charged and run a couple of compact fluorescent lamps. I was putting on my shoes to go find the nearest open Wawa with an ice dispenser or dry ice when the power came back on.
Last edited: Mar 20, 2009
15. Mar 22, 2009
### Bob S
Conventional dynamometers have two rollers about a foot apart for each wheel, so the car sits easily between them. Be sure to tie the car frame to a tree. The car's engine can easily develop 50 HP at 1500 RPM, so this is probably the best speed to run it. Higher RPMs just wastes gasoline. If possible get a car with a manual transmission- they are more efficient.
16. Mar 22, 2009
### xxChrisxx
Bob the engine is operating at its least efficient at no load and closed throttle. The increase in sfc makes getting a manual a bit of a non issue. For this to be feasable it'd have to be a diesel or nothing.
I was going to have a WTF moment at 50hp at 1500rpm, but I keep having to remind myself that you all have collossal V8's over there.
17. Mar 23, 2009
### Staff: Mentor
50 HP is 37 kW. The OP didn't say what we're powering here, but you could power an all-electric heat/appliance house on Thanksgiving with 20! Under normal circumstances a house with gas appliances could be powered with 3 if you aren't using the air conditioning.
18. Mar 23, 2009
### Averagesupernova
Been there, done that. A 5Kw ran the oven that baked the turkey and something else stuffed in there, can't remember what it was though. 3Kw lightly loaded ran a crock pot or 2. Coleman camp stove for mashed potatoes. As soon as the oven went off, fridge and freezer got plugged back in and the furnace turned back up and lights were used more liberally. I'm not sure some of the relatives really believed the comercial power was off. Getting by without commercial power isn't that difficult at all for less than a couple of days. Makes you appreciate what you pay for commercial power when you have to generate your own power.
19. Mar 23, 2009
### Ranger Mike
We took a 5 k generator to the race track once
Barely ran the small air compressor
And battery charger
Forget it
I think min of 10 k
For house
20. Mar 23, 2009
### Averagesupernova
Starting an air compressor and running electric heating such as an oven and crock pots or electric lights are 2 different things. I've used the same 5KW generator to start a small compressor too and it would barely do it. It just depends on what you want to do in your house. Not too likely there will be any hard starting loads other than an automatic washing machine.
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# inductive proof of Fermat’s little theorem proof
We will show
$a^{p}\equiv a\pmod{p}$
with $p$ prime. The equivalent statement
$a^{p-1}\equiv 1\pmod{p}$
when $p$ does not divide $a$ follows by cancelling $a$ both sides (which can be done since then $a,p$ are coprime).
When $a=1$, we have
$1^{p}\equiv 1\pmod{p}$
Now assume the theorem holds for some positive $a$ and we want to prove the statement for $a+1$. We will have as a direct consequence that
$a^{p}\equiv a\pmod{p}$
Let’s examine $a+1$. By the binomial theorem, we have
$\displaystyle(a+1)^{p}$ $\displaystyle\equiv$ $\displaystyle{{p}\choose{0}}a^{p}+{{p}\choose{1}}a^{p-1}+\cdots+{{p}\choose{p-% 1}}a+1$ $\displaystyle\equiv$ $\displaystyle a+pa^{p-1}+p\frac{(p-1)}{2}a^{p-2}+\cdots+pa+1$ $\displaystyle\equiv$ $\displaystyle(a+1)+[pa^{p-1}+p\frac{(p-1)}{2}a^{p-2}+\cdots+pa]$
However, note that the entire bracketed term is divisible by $p$, since each element of it is divsible by $p$. Hence
$(a+1)^{p}\equiv(a+1)\pmod{p}$
Therefore by induction it follows that
$a^{p}\equiv a\pmod{p}$
for all positive integers $a$.
It is easy to show that it also holds for $-a$ whenever it holds for $a$, so the statement works for all integers $a$.
Title inductive proof of Fermat’s little theorem proof InductiveProofOfFermatsLittleTheoremProof 2013-03-22 11:47:46 2013-03-22 11:47:46 mathcam (2727) mathcam (2727) 17 mathcam (2727) Proof msc 11A07 FermatsTheoremProof
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# Math Help - fifth grade problem of the week
1. ## fifth grade problem of the week
I am a fifth grader and am having trouble solving the problem below. Any help would be appreciated. I have worked on it and come up with an answer of 33, but my teacher says the answer is a number greater than 50 so I am stuck. Here it is:
In how many ways can seven basketball players of different heights line up in a single row so that no player is standing between two people taller than she is?
2. Hello, Niko!
Are you sure it's ethical for us to be helping you?
I can get you started . . .
In how many ways can seven basketball players of different heights line up in a row
so that no player is standing between two people taller than she is?
Suppose the players are: $A,B,C,D,E,F,G$ from shortest to tallest.
We have 7 spaces to fill: . _ _ _ _ _ _ _
We note that $A$ (the shortest) cannot be in the interior of the row.
. . She will be between two taller players.
Hence, she must be on the end . . . 2 choices.
We have the other 6 players $(B,C,D,E,F,G)$ to place in 6 positions:. _ _ _ _ _ _
We note that $B$ (the shortest of this group) cannot be in the interior of this row.
. . She will be between two taller players.
Hence, she must be on the end . . . 2 choices.
And so on . . . get the idea?
[I get an answer of 64.]
3. Thank you for your help, I wonder if it's ethical for my teacher to assign me problems that are way beyond my math level, but that you can't help me with. The advice you gave me was the way I had started the problem before e-mailing. I used the numbers 1-7 instead of letters. The shortest, #1 could be on the ends, so in two positions. The next tallest #2 could be in two positions, and then I had the following:
Player positions
3 4
4 5
5 6
6 7
7 7
So, I'm still only getting 33 ways. This is where I'm confused since you, like my teacher, say there are more.
4. Hello, Niko!
You and I agree on the first two players.
But I don't understand your reasoning for the others.
Call the players: . $1,2,3,4,5,6,7$ . . . from shortest to tallest.
There are seven places to fill: . _ _ _ _ _ _ _
"1" cannot be in an interior position . . . She must be on one end.
. . She has 2 choices for her end.
No matter which end she chooses, the other 6 positions are like this: ._ _ _ _ _ _
Then "2" cannot be in an interior position . . . She must be on one end.
. . She has 2 choices for her end.
No matter which end she chooses, the other 5 positions are like this:. _ _ _ _ _
Then "3" cannot be in an interior positon . . . She must be on one end.
. . She has 2 choices for her end.
No matter which end she chooses, the other 4 positions are like this: ._ _ _ _
Then "4" cannot be in an interior positon . . . She must be on one end.
. . She has 2 choices for her end.
No matter which end she chooses, the other 3 positions are like this: . _ _ _
Then "5" cannot be in the middle . . . She must be on one end.
. . She has 2 choices for her end.
No matter which end she chooses, the other 2 positions are like this: ._ _
Then "6" can choose either position: .2 choices.
Then "7" takes the remaining position: .1 choice.
Therefore, the number of choices is: . ${\color{red}2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot1} \;=\;{\color{blue}64}$
5. Hi Soroban- thanks for the additional help. I now understand the problem better. I had to turn in the assignment this morning and I didn't do it correctly, but that's okay. I really wanted to know how to solve it so if I have to do something like this in the future I have a clearer idea of the steps to take.
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# Linear Algebra Matrix Limits/Stochastic Process
by lutheranian
Tags: linear algebra, matrix limits, probability
P: 1 1. The problem statement, all variables and given/known data A diaper liner is placed in each diaper worn by a baby. If, after a diaper change, the liner is soiled, then it is discarded and replaced by a new liner. Otherwise, the liner is washed with the diapers and reused, except that each liner is discarded and replaced after its third use (even if it has never been soiled). The probability that the baby will soil any diaper liner is one third. If there are only new diaper liners at first, eventually what proportions of the diaper liners being used will be new, once used, and twice used? Hint: Assume that a diaper liner ready for use is in one of the three states: new, once used, or twice used. After its use, it then transforms into one of the three states described 2. Relevant equations If A is a transition matrix and v is the initial state vector and lim Am = L as m -->$\infty$ then eventual state is Lv 3. The attempt at a solution I set up the transition matrix with the first column/row corresponding to new liners, the second to once-used, and the third to twice-used, resulting in the following: A= (1/3, 1/3, 1| 2/3, 0, 0 | 0, 2/3, 0) The initial vector is v= (1, 0, 0) I tried finding the limit of Am as m --> $\infty$ using wolframalpha (which is allowed because the homework problems have messy numbers) but the computation times out every time.
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# OpenMath Content Dictionary: plangeo1
Canonical URL:
http://www.win.tue.nl/~amc/oz/om/cds/plangeo1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
plangeo1.ocd
CD as XML Encoded OpenMath:
plangeo1.omcd
Defines:
point, are_on_line, assertion, configuration, incident, line, type
Date:
2004-06-01
Version:
0 (Revision 5)
Review Date:
2006-06-01
Status:
experimental
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work. Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society. In particular, if there is a Content
Dictionary Group whose name is, for example, math' containing
Content Dictionaries named math1', math2' etc., then you should
not name a derived Content Dictionary mathN' where N is an integer.
However you are free to name it private_mathN' or some such. This
is because the names mathN' may be used by the OpenMath Society
for future extensions.
compilation of derived works, but keep paragraphs a) and b)
intact. The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
society at http://www.openmath.org.
Author: Arjeh Cohen
This CD defines symbols for planar Euclidean geometry.
## point
Description:
The symbol is used to indicate a point of planar Euclidean geometry by a variable. The point may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
Example:
Given two lines l and m, a point A on l and m is defined by:
$\mathrm{point}\left(A,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(A,m\right)\right)$
Signatures:
sts
[Next: line] [Last: are_on_line] [Top]
## line
Description:
The symbol is used to indicate a line of planar Euclidean geometry by a variable. The line may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
Example:
Given points A and B, a line l through A and B is defined by:
$\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(B,l\right)\right)$
Signatures:
sts
[Next: incident] [Previous: point] [Top]
## incident
Description:
The symbol represents the logical incidence function which is a binary function taking arguments representing geometric objects like points and lines and returning a boolean value. It is true if and only if the first argument is incident to the second.
Example:
That a point A is incident to a line l is given by:
$\mathrm{incident}\left(A,l\right)$
Signatures:
sts
[Next: configuration] [Previous: line] [Top]
## configuration
Description:
The symbol represents a configuration in Euclidean planar geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations.
Example:
The configuration of a point A and a line l incident to A is defined by:
$\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right)\right)\right)$
Example:
The prevous configuration of a point A and a line l incident with A can be extended by adding a second point B incident with l:
$\mathrm{configuration}\left(\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right)\right)\right),\mathrm{point}\left(B,\mathrm{incident}\left(B,l\right)\right)\right)$
Example:
We describe a triangle on the distinct points A, B, C and lines a, b, c:
$\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{point}\left(B,¬\left(A=B\right)\right),\mathrm{line}\left(c,\mathrm{incident}\left(c,A\right),\mathrm{incident}\left(c,B\right)\right),\mathrm{point}\left(C,¬\mathrm{incident}\left(C,c\right)\right),\mathrm{line}\left(a,\mathrm{incident}\left(a,B\right),\mathrm{incident}\left(a,C\right)\right),\mathrm{line}\left(b,\mathrm{incident}\left(b,A\right),\mathrm{incident}\left(b,C\right)\right)\right)$
Signatures:
sts
[Next: type] [Previous: incident] [Top]
## type
Description:
The symbol represents the type of the basic geometric objects: points, lines, configuration.
Commented Mathematical property (CMP):
If A and B are objects of the same type, then they are not incident.
Formal Mathematical property (FMP):
$\mathrm{type}\left(A\right)=\mathrm{type}\left(B\right)⇒¬\mathrm{incident}\left(A,B\right)$
Signatures:
sts
[Next: assertion] [Previous: configuration] [Top]
## assertion
Description:
The symbol is a constructor with two arguments. Its first argument should be a configuration, its second argument a statement about the configuration, called thesis. When applied to a configuration C and a thesis T, the OpenMath object assertion(C,T) expresses the assertion that T holds in C.
Example:
The assertion that two distinct lines meet in only one point can be expressed as follows using the assertion symbol.
$\mathrm{assertion}\left(\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{point}\left(B\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(B,l\right)\right),\mathrm{line}\left(m,\mathrm{incident}\left(A,m\right),\mathrm{incident}\left(B,m\right),¬\left(l=m\right)\right),A=B\right)\right)$
Signatures:
sts
[Next: are_on_line] [Previous: type] [Top]
## are_on_line
Description:
The statement that a set of points is collinear.
Example:
This example states that A, B, C, and D are collinear.
$\mathrm{are_on_line}\left(A,B,C,D\right)$
Signatures:
sts
[First: point] [Previous: assertion] [Top]
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Article | Open | Published:
# Dynamic changes in binding interaction networks of sex steroids establish their non-classical effects
## Abstract
Non-classical signaling in the intracellular second messenger system plays a pivotal role in the cytoprotective effect of estradiol. Estrogen receptor is a common target of sex steroids and important in mediating estradiol-induced neuroprotection. Whereas the mechanism of genomic effects of sex steroids is fairly understood, their non-classical effects have not been elucidated completely. We use real time molecular dynamics calculations to uncover the interaction network of estradiol and activator estren. Besides steroid interactions, we also investigate the co-activation of the receptor. We show how steroid binding to the alternative binding site of the non-classical action is facilitated by the presence of a steroid in the classical binding site and the absence of the co-activator peptide. Uncovering such dynamic mechanisms behind steroid action will help the structure-based design of new drugs with non-classical responses and cytoprotective potential.
## Introduction
Estrogens are responsible for a wide range of biological actions from the regulation of fertility to cytoprotection1,2,3. Gonadal 17β-estradiol (E2) has a remarkable neuroprotective potential4. Besides slow, classical, genomic effects5,6 (Fig. 1) E2 also exerts rapid, non-classical effects on intracellular second messenger molecules7,8,9,10, via estrogen receptors (ERs, Fig. 1).
Importantly, neuroprotection of E2 is attributed to such rapid actions11,12,13,14 and its binding to estrogen receptor alpha (ERα)15. Previously we have shown that a single dose of E2 as well as Activators of Non-Classical Estrogen-Like Signaling (ANCELS) such as estren-3α,17β-diol (EN)16 induce ERα-dependent neuroprotection via intracellular signaling pathways in neurodegenerative animal model17,18. The protective effect was also observed after traumatic brain injuries4 in rodents. Clinical studies showed that hormone replacement therapy with estrogen and progestin1 decreases the incidence of neurodegenerative diseases such as Alzheimer’s disease, but it also increases risks of stroke and breast cancer. However, structural dynamics of biding events establishing non-classical E2 action on ERs has not been fully elucidated. The lack of such details of molecular mechanisms of neuroprotective actions of estrogens hinders the exploitation of their therapeutic potential.
Estrogen binding to the classical binding site (CBS) of human estrogen receptor alpha (hERα) is well-explained by atomic resolution structures of the Protein Databank (PDB)5,19. The CBS is located between helices H3, H4, H6, H8 and H1120 (Supplementary Video S1) of the ligand-binding domain (LBD) of hERα, and it is known to mediate the slow, genomic actions of ligands, such as the native agonist E2 and antagonist 4-OH-tamoxifen selectively modulate gene expression21.
Besides slow, genomic actions (Fig. 1 top) ANCELS such as EN22, substance A and substance B23 exhibit weak transcriptional activity, selectively activating the non-classical E2 signaling as validated by functional assays22,23. Such non-classical actions of E2 on the signaling system have been known for more than forty years24. However, the underlying mechanism has not been understood due to the lack of atomic resolution structures of the complexes of effector ligands and ERs. An interesting study25 proposed an alternative binding site (ABS) of E2 and EN on hERα, further discussed by Norman and co-workers26, conveying the non-classical actions, analogously to vitamin D receptor25. The proposed ABS is located at the C terminus of H1 and N terminus of H3 helices, with a conserved R residue (R274 in vitamin D receptor and R394 in hERα) in the site. E2 binding to ABS26 does not directly alter gene expression, but rapidly activates the mitogen-activated protein kinase/extracellular-signal regulated kinase (MAPK/ERK) signaling pathway instead (Fig. 1, top)8,9.
Previous studies25,27 identified R394 and E353 as key E2-binding residues of ABS, located at the proximity of 3-hydroxyl group of E2, while the other, 17-hydroxyl group is oriented to R33525. From these results, a conformational ensemble model was constructed26 to explain the different behaviour of the nuclear and membrane associated forms of hERα. In this model, a “concurrent occupancy” was also proposed, when both ABS and CBS sites are simultaneously occupied by two copies of E2. However, the dynamics of simultaneous occupancy has not been investigated yet.
Besides ABS and CBS, there is a binding site for different transcriptional co-activator proteins. A conserved, LXXLL binding motif can be found in the amino acid sequences of these proteins28. Receptors are often co-crystallized with a peptide fragment of the co-activator (CA) protein containing the above conserved sequence bound to the activation function site 2 (AF2 site, Fig. 1 top part)5,29,30. In these structures, CA bridges between helices H3 and H1220,31 via hydrogen bonding at residues K362 on the H3 side and E542 on the H12 side. Furthermore, if E2 binds, and hER is activated (Fig. 1 top), the CA bridge fixes H12 in a position covering the E2-bound CBS20,26 and shielding it from the bulk solvent. Y537 plays an important role in the activation, and it was demonstrated that it is very prone to mutations (Y537S) which make the receptor resistant to estrogen antagonist drugs30. H3 residues E353, H356, M357 and W360 are proposed to form the ABS, and therefore, any perturbation of the conformation of H3 at these residues by CA can influence the binding of ligands to ABS, as well. Despite the importance of the above effects of the CA-bridge on E2 binding, the dynamics of the underlying mechanism, and the route of structural communication between the proposed ABS25,26 and CA has not been elucidated at atomic level.
Although the current cutting edge super-resolution imaging techniques such as single molecule fluorescence resonance energy transfer or stimulated emission depletion microscopy are capable to produce sequence of images in given time frame they have limited temporal (5 μs) and spatial (1 nm)32,33 resolution. Due to the limitations of current structure determination techniques34,35 investigation of the above questions is fairly challenging and “new techniques may be required to study the formation of such transient, though potentially biologically meaningful complexes” of sex steroids with hERα16. At present, molecular dynamics (MD) calculation is the only approach available for investigation of such real time binding events in a receptor-ligand system at atomic resolution. Consequently, several research groups apply MD calculations and present their results on conformational changes of various proteins36,37,38 and binding events of ligands20,39,40,41,42,43 at atomic resolution.
Accordingly, the present study also applies up-to-date, extensive MD calculations to investigate the real time changes of interaction networks of hERα and its ligands at atomic level. The structural dynamics of steroid binding was investigated at both ABS and CBS, taking into account the role of the CA, as well. For this, blind docking of E2 and EN to hERα was performed for an unbiased mapping of available sites. Subsequent MD of the docked complexes surrounded by several thousand of explicit water molecules was applied mimicking the natural dissociation route of the sexual steroids from hERα. The present study also aims at an MD-based elucidation of atomic resolution history of structural changes of ER accompanying non-classical steroid actions.
## Results and Discussion
### Interaction networks in the steroid-free receptor
To study the effect of CA binding on structural dynamics of hERα (Fig. 2a,b and Supplementary Video S1), both the CA bound (CA+) and free (CA−) structures of the steroid-free LBD were investigated (Fig. 2c) for comparison. The p160-type CA44,45 with crucial role in gene transcription was invloved in the present study. The C-terminus of the LBD was completed with a region called F domain extending the crystallographic structure using a modeling procedure described in Methods. In both cases, 1μs-long molecular dynamics (MD) calculations were performed to study the structural evolution of the LBD. Evaluations of the resulted trajectories showed (Fig. 2c) high root mean squared fluctuations (RMSF) of amino acid heavy atoms over the entire 1-μs domain at loops L1, L2, and in the F domain. Since loops are naturally flexible regions, and the F domain is a disordered region such fluctuations were expected. The flexibility of L1 can be explained mostly by its high exposition to the bulk. This loop is of high structural importance, as it has an indirect contact with the CBS through S329, and is also closely connected to helix H3, which is covering both the ABS and CBS (Supplementary Video S1).
Having MD results on both CA+ and CA− LBD structures, the influence of CA binding on the LBD was structurally analyzed paying special attention to the CA-connected helices H12, H3 and regions around the binding sites. Both termini of CA are connected to the LBD by salt bridges to E542 of H12, and by an H-bond to K362 of H3. In addition, CA forms hydrophobic contacts with I358 and M357 of H3 and L539 of H12 (Fig. 3a). The hydrophobic contacts with H3 are of particular interest, as I358 is in the vicinity of M357, which is part of the ABS. Therefore, comparison of their movement in CA+/CA− simulations may help to elucidate the mechanism of influence of CA on the process of ligand binding or dissociation to or from ABS. Accordingly, the movements of amino acids H356, M357, and I358 were quantified by calculating the distances between actual and initial positions of their side-chains (Fig. 3 bottom parts) along the MD simulations.
In order to maintain the hydrophobic interactions between the hERα, and CA (Fig. 3a top), I358 situated on H3 fluctuates between a distance of 2–3 Å measured from its initial position as a reference point (Fig. 3 bottom). The fluctuation (Fig. 3a bottom) is higher in the first part (0–400 ns) than in the second part of the simulation (400–1000 ns). The resulted 3 Å shift of I358 from its initial position causes the flipping of M357 into the ABS after 400 ns to further initiate the shift of H356 (Fig. 3a, bottom). In the CA− scenario (Fig. 3b top and bottom), it can be observed, that I358 highly fluctuates during the entire simulation. However, the above-mentioned shift of M357 into the ABS was not observed in the CA− scenario. Thus, the presence of CA can be perceived as a restricting factor, especially on M357. In contrast to M357, the orientation of H356 was not dependent on the presence or absence of CA at the end of the simulations. This can be explained by the contact between H356 (H3) and L327 (L1) through which L327 (L1) transfers its high mobility (Fig. 2c and Supplementary Video S1), to H356 (H3), then M357(H3). It was also found that in both CA+ and CA− simulations, H356 was oriented inside the ABS binding site by the end of 1µs simulation, but this switch occurs faster in the CA+ simulations (400 ns), than in CA− simulations (800 ns, Fig. 3 bottom parts) due to the movement of M357.
In the nucleus, sex steroids45 bind to the CBS activating hERα which results in the occupancy of AF2 binding site46 by CA (Fig. 1 top part). Such activation does not occur if hERα resides in the membrane and the AF2 site is left unoccupied. The membrane bound form of the estrogen receptor is involved47,48 in non-classical effects such as antiapoptotis16, cytoprotection, and neuroprotection11 Kousteni and colleagues have also reported rapid, non-classical effect of E216, which require the extra-nuclear localization of the hERalpha, confirmed by confocal laser scanning microscopy studies49. The above-mentioned antiapoptotis is resulted by targeting50 an ABS outside the CBS of hERα. Fluorescence experimental studies51 also indicated the presence of ABS. Thus, ABS is linked to non-classical effects attributed to the membrane-bound form of hERα.
We found that ABS is available for ligand binding if AF2 is not occupied, otherwise it is dynamically blocked by both M357 and H356 side-chains. Thus, receptor dynamics at these two amino acids is responsible for the availability of ABS in membrane surrounding for certain ligands. In agreement with the herein presented results, experimental studies showed16 that E2, EN and other sex steroids are capable to produce non-classical effects22, occupying the ABS27. For this, sex steroids require extranuclear, membrane-bound localization of the estrogen receptor22, where the AF2 binding site is not occupied by CA.
### Binding sites of sexual steroids
Following structural dynamics investigations on the steroid-free receptor, a complete exploration of binding sites of sex steroids was performed on the entire surface of the apo LBD. Blind docking52,53,54 was used for the search as this method does not require previous knowledge of the location of the binding sites. A representative structure of LBD was produced by MD simulation with subsequent clustering (Methods) and used as a target in the blind docking calculations (Fig. 4). The target structure was validated by blind docking of E2 (Fig. 4, magenta). The docking result was compared with the crystallographic ligand conformation in the CBS (Supplementary Fig. S1). One-hundred blind docking trials were performed with random initial positions of E2 around the target. The results were evaluated as described in previous works52,53. Briefly, the docked steroid copies were clustered and ranked by energy, resulting in a list of explored binding sites and ligand poses with the strongest steroid-site interaction in the first rank. Besides E2, blind docking of EN (Fig. 4, teal) was also performed on the LBD. From the blind docking calculations 11 ranks were identified for E2 and 6 for EN (Fig. 4, Supplementary Table S1).
The CBS was found in the first rank of blind docking by both steroids. Reproduction of the binding mode of E2 in the CBS was successful as a root mean squared deviation (RMSD) of 2.1 Å (Supplementary Fig. S1) was measured between the heavy atoms of the blind docked and crystallographic (reference) steroid conformations. Such a good fit of the docked E2 to the experimental conformation shows that the target LBD structure is valid and blind docking predictions provide accurate results at atomic resolution. Analysis of docked molecules in CBS revealed that binding modes of E2 and EN are very similar to each-other (Supplementary Fig. S2). Both steroids occupy the same orientation with H-bonds formed between 3-hydroxyl of E2 and EN and hERα residues (F404, E353, and R394). Topologically, CBS is separated from the bulk by loop L1, ß-turn T1, H3 and H12. At the same time, structural differences between the steroids influence their hydrophobic interactions with the amino acids in the surroundings (Supplementary Fig. S2). For example, aromatic ring A of E2 (Fig. 2d) forms a perpendicular π-stacking with F404 situated on T1. The lack of aromatic ring in EN results its increased flexibility and weak hydrophobic interactions with F404, if compared to the π-stacking, observed at E2. This could also be part of the reason, why E2 is considered primarily as a CBS-binding ligand26,44 and is selective for the classical pathway16.
The ABS was found in the second rank during blind docking of both E2 and EN in a region proposed by previous studies26,27. ABS is located between H8 and H3 in the vicinity of the CBS (Supplementary Video S1). Exposition of ABS towards the bulk is higher than that of CBS as it is covered only by the highly flexible L1 (Fig. 2a). Similarly to the “ensemble model” of previous studies26,27, the BD calculations showed that R394 and E353 separate the two sites (Supplementary Fig. S3). Furthermore, EN is bound to the ABS, with its 3-hydroxyl group oriented towards R394, which also agrees with previous studies25,26. Lipophilic residues (P324, L327, M357, W360, I386, P406) dominate this site, K449 is the only amino acid with polar side chain. Comparing the binding modes of the two analyzed steroids (E2 and EN) to the ABS, a head-tail swap can be observed between them (Fig. 4, bottom). Accordingly, a hydrogen bond is formed with the backbone amide of L327 with different groups of the steroids (17-hydroxyl of E2, and 3-hydroxyl of EN). In addition, 17-hydroxyl of EN forms another hydrogen bond with K449. This bond was not observed in the complex with E2. The H-bond with the backbone amide of L327 is common for the two ligands. As L327 is on loop L1 it is exposed to the bulk, mobile and susceptible to the thermal motion of the surrounding water molecules (see also Section Interaction networks in the steroid-bound receptor and results on simulations with different velocity distributions). At the same time, the second H-bond specific for EN is formed with H8, buried in the pocket, inaccessible from the bulk stabilizing the interaction of EN with the LBD at the ABS. Concerning the location of ABS and CBS the results are in good agreement with previous studies25,26,27. A previous comparison of the binding interaction energies of E2 and EN produced by manual docking25, showed that binding of EN is stronger to the ABS than that of E2 (Table 1). For the CBS, an opposite trend was observed (Table 1). Other docking25,27 studies also confirmed E2 selectivity towards CBS. Experimental binding studies demonstrated22,55 that E2 has a higher affinity towards ERα than EN. In vitro experiments22,55 showed that E2 plays a role in classical effects associated with its CBS50,51 binding. At the same time, despite the moderate binding affinity of EN23 in vivo studies22 also confirmed that it has a selectivity towards the non-classical pathway, lacking an effect on the reproductive organs which was confirmed by histological analysis of the uterus, and did not stimulate transcription of the C3 gene in the uterus22. In the present study, interaction energies were calculated using the docked and energy-minimized ligand structures. The differences in the energy values show good agreement with those obtained in previous docking (Table 1) and the affinity/selectivity preferences demonstrated by the above-mentioned in vitro and in vivo experimental studies.
Table 1 shows that EN binds 4 kcal/mol stronger to ABS than to CBS. At the same time, the binding of EN to ABS is 5 kcal/mol stronger than that of E2. This is in agreement with the above structural findings, and also with previous results25, showing that EN has a larger affinity to ABS than CBS. These results suggest different binding modes at ABS and CBS which is consistent with the structural observations described above (Fig. 4).
All-in-all, for the top two ranks blind docking gave consensus results identifying the binding sites of both steroids as the CBS and the ABS, respectively. Both steroids bind to both sites with significant interaction energies, with E2 a classical effector on CBS, and EN preferring ABS as a non-classical effector26. In Rank 3 and beyond, steroids found different sites without a consensus result. Notably, binding of E2 to CBS had been precisely described5,20 and the position of ABS was proposed in previous studies25,27. However, steroid binding to ABS has not been fully characterized. Here, atomic resolution structures of the complexed sites with both investigated ligands bound to ABS were provided (Fig. 4), highlighting crucial amino acids, for non-classical activity, and the binding difference between them. Moreover, binding mode of EN to CBS was also provided (Supplementary Fig. S2) and analyzed. Atomic resolution complex structures from the above blind docking calculations were piped in the investigations of the next Section dealing with the molecular dynamics of interaction networks of steroid binding.
## Interaction networks in the steroid-bound receptor
### Interaction dynamics
To effect the transcriptional activity in the classical, genomic pathway, a “long-lived”16 steroid-CBS contact is needed in order to produce the specific conformational changes of hERα. At the same time, steroid ligands form “transient complexes” with the ABS, via a brief association to hERα in the non-classical pathway. However, investigation of such rapid effects of the non-classical pathway requires new approaches and techniques16.
In the present study, we apply molecular dynamics calculations of the steroid-bound hERα surrounded by several thousand (explicit) water molecules. To investigate the interaction dynamics, docked steroid-bound receptor structures were adopted from Section Binding sites of sexual steroids as starting points. Besides singly occupied binding sites, additional complex structures were constructed (Methods) with both ABS and CBS simultaneously occupied for both EN and E2. All versions were produced both in the presence and absence of CA which yielded altogether twelve different complexes for the two sexual steroids (Fig. 1, bottom). For all complex structures, five parallel 100-ns-long MD calculations were performed to follow their trajectories. Thermal dissociation of the steroid ligands was expected by acquiring kinetic energy from its water and protein surrounding. The calculations were repeated five times using different initial velocity distributions resulting in a total of 6 µs MD calculation. Applying more than one starting initial velocity distribution for a starting structure is important to obtain statistically relevant, unbiased conclusions. In other words, five, independent dissociation trials were performed resulting in five, independent dissociation trajectories of the steroids in all twelve complexes.
From the dissociation trajectories (Fig. 5a, Supplementary Video S2), residence frequency (RF) values were calculated to quantify kinetic stability of the complexes in each trial of Fig. 1 (bottom). In drug discovery, assessment of kinetic stability described by the residence of a ligand in the binding site is crucial factor similarly to thermodynamic stability56,57. To calculate RF, the movement of the ligand was described by the distance between the centre of mass (dCOM) of its actual and starting positions at each time frame during the simulation time resulting in a COM-plot. The RF value of a binding site was directly obtained (Equation 3, Methods) from the COM-plots (Fig. 5bc) using a dLIM = 5 Å for dissociation limit.
Results of the merged trajectories of a total of 500 ns simulation time per trial are listed in Tables 2 and 3. Per-trial and RMSD-based evaluations are presented in Supplementary Tables S2S5. In the present study, the theoretical upper limit of RF was 10.0 ns−1, which corresponds to the highest kinetic stability. The mean CBS RF values of E2 and EN (last column in Table 2, average of first four columns), are 10.0 ns−1 and 9.0 ns−1, respectively. Experimental results are in agreement with our calculations (Table 2) affirming that E2 has a stronger affinity to CBS than EN22,25. Results in Table 1 are also in line with a key review by Norman and co-workers26 presuming that steroids such as EN and E2 could have different “fractional occupancies” in the ABS and CBS pockets. Whereas both ligands show good binding stability at the CBS, a drop in RF values can be observed at ABS (Table 3) if compared with those at CBS (Table 2). In the case of ABS, the mean RF of EN is markedly higher than that of E2.
For structural interpretation of the results in Tables 2 and 3, representative individual trajectories were selected with RFs closest to that of the merged trajectory (bold in Supplementary Tables S25). As it was described in Section Binding sites of sexual steroids, EN has H-bonds with both hydroxyl groups, and is stabilized in the ABS at its both ends (Supplementary Video S2 and Fig. 5a). The two H-bonds are formed at the entrance with L327 of loop L1, and K449 of H8 helix, at the bottom of the pocket. Loop L1 is highly exposed to the bulk having a susceptibility to the thermal motion of the hydrating water molecules and it tends to pull out EN from the binding site. At the same time, forming an H-bond with K449, H8 acts as a counter balance and keeps EN in ABS. If the H-bond with K449 is broken, EN will be easily pulled out towards the bulk by the loop. After the breakage of the H-bond between EN and K449, a series of conformational changes are initiated by L327. Firstly, as L327 interacts with the side chain of H356 through hydrophobic interactions and H356 starts to move towards the ABS binding site, as fluctuation of L1 intensifies. Secondly, a conformational change is induced on M357 by H356. Here, the side chain of M357 flips into the ABS binding site, similarly to the apo simulations (see Secion 1). As M357 flips inside de binding site, sterically perturbs EN leading to its expulsion from the site. The above conformational changes were not observed in case of E2, and therefore, no role can be attributed to M357 in its dissociation. As the H-bond with K449 is missing in case of E2, the above described counter balancing effect does not take place. Hence, E2 is pulled out more easily than EN from ABS by the thermal motion of the loop.
The above analyses of the simulation trajectories highlighted that the conformational changes of hERα (Supplementary Video S2 and S3) have crucial role in the dissociation process of EN. In order to quantify the relationship between conformational changes of the receptor and dissociation of EN, dCOM was correlated with the movement of three residues (L327, H356, and M357) in the dCOM < 5 dLIM interval. Correlation results are shown for the CA+/CBS+/ABS+ (Fig. 6) case with representative residue movements. Notably, similar correlations were observed for the CA−/CBS+/ABS+ case (Supplementary Fig. S4), as well. The obtained correlations show that all three proposed residues are important in inducing EN dissociation. Due to their characteristic interaction networks there is a considerable difference in the dynamics of the three side-chains (Fig. 6). While M357 enters the ABS, which results in pushing EN out of its binding pose, L327 exerts a pulling effect on EN from the other side. H356 continuously fluctuates rotating inside the ABS.
In this Section, dissociation mechanisms of sexual steroids from ABS and CBS were uncovered by extensive molecular dynamics calculations. Differences in binding affinities16,25 (Table 1) and kinetic stability (Tables 2 and 3) of steroid-hERα complexes was correlated with the differences in the dynamics of the corresponding interaction networks.
### Effects of co-binders
The steroid-free MD simulations uncovered an interference between ABS and AF2. It was found that the ABS is available for ligand binding only if AF2 is not occupied by CA (Section Interaction networks in the steroid-free receptor). The results of Table 3 also show that occupancy of ABS is influenced by the presence of other ligands co-bound to hERα. Binding of CA to site AF2 or an additional steroid molecule to CBS has considerable effect on steroid binding to ABS (Table 3). In order to investigate the structural background of these effects, we examine how CA affects the binding dynamics of E2 and EN to hERα.
In the CA− scenario, remarkably high stability of E2 and EN binding to ABS was found especially if an additional E2 or EN copy was present in the CBS (CBS+, third column in Table 2). This situation is of particular importance as non-classical effects happen in the absence of CA (Section Interaction networks in the steroid-free receptor). Various experimental studies have suggested that fast, non-classical activity of streoids is exerted by their binding to the ABS16,50. It is also known that binding to ABS is not probable in the presence of CA30,46,58. Consequently, the presence of CA (CA+) would hinder steroid binding to ABS and facilitate dissociation. Our MD approach allowed the investigation of such a non-natural CA+ situation and the analysis of the reasons of the hindering effect of CA binding to AF2, as well. This finding is consistent with the effect of CA over the ABS binding site (Section Interaction networks in the steroid-free receptor) where the effect of CA bridge connecting helices H3 and H12 was demonstrated. As both helices are very close to the binding sites, they interfere with the CA bridge and ligand binding. CA binds to the LBD via ionic and hydrophobic interactions with H3 at M357 and I359, which are part of the ABS. It was also demonstrated (Section Interaction networks in the steroid-free receptor) that M357 tends to occupy ABS in presence of CA. The same mechanism was observed also in the ABS+ simulations, but only in case of EN, which indicates a dependency on the ligand type (see also Section Interaction dynamics, Fig. 5). If CA is present, M357 tends to move towards the center of the ABS, and I359 assists this process providing a steric restraint and keeping a hydrophobic contact with L693 of CA. On the other hand, if CA is missing from the above interaction networks, its influence on I359 and M357 is not there. Thus, I359 can move freely, and therefore, M357 can maintain its orientation towards the bulk, and it does not influence the stability of EN binding. All-in-all, CA changes the dynamic interaction network of the ABS leading to kinetic stability differences presented as RFs in Table 3. Although H356 has no direct contact with CA it also plays an important role in the dissociation mechanism as it is in the vicinity of M357, and also occupies the ABS promoting the dissociation of EN (Fig. 5a and Supplementary Video S2).
The above structural effects are also reflected by the velocity of EN during the dissociation process from ABS as calculated from the COM-plot (Fig. 5bc). The overall dissociation velocity of EN increased from 0.11 to 0.25 Å/ns (Supplementary Tables S6 and S7) in the CA+ case, due to the described destabilizing effect of CA binding to hERα. The dissociation process of EN can be divided into an initial (dCOM ≤ dLIM) and a terminal (dLIM < dCOM ≤ 10 Å) phase. Velocity of EN in the terminal phase is larger than it was in the initial phase, which is specific to EN. EN has higher v2 values than E2 suggesting that final dissociation of E2 from ABS occurs slower than in case of EN. The characteristic, abrupt movement of EN in the terminal phase can be explained by the sudden of breakage of the second stabilizing H-bond, the one with K449.
The effect of the presence of an additional steroid molecule in the CBS (CBS+) is coupled to that of the absence of CA and this CA−/CBS+ case shows the highest stability of the ABS-bound ligands (Table 3). To understand the effect of occupancy of the CBS simulations on EN were analysed for both CBS+ and CBS− cases (Supplementary Table S3, seed 1). Three structual elements T1, L1 and H3 were of particular interest, in analyzing the stabilizing effect of CBS over the ABS. These elements can be considered as parts of “flickering gate” (Fig. 7a and Supplementary Video S3) as proposed by a previous study25. T1 plays the role of the flickering wing, whereas L1 and H3 constitute the stable frame of the gate. Our calculations show that the gate is closed when CBS is occupied (Supplementary Video S3, blue), and opened when CBS is unoccupied (Supplementary Video S3, red). When EN binds to the CBS it is able to keep the “flickering gate” in a closed state, as it interacts with T1 (flickering wing) via a hydrophobic interaction with F404 (Fig. 7b). Therefore, in the closed state, stabilization of T1, by EN in the CBS, will further maintain an H-bond between T1 (N407) and L1 (S329). Stabilized by this H-bonding between T1 and L1 (Fig. 7b), L1 becomes less flexible, and its rigidity will further increase RF of EN in the ABS (Fig. 7c). This happens as L1 binds to EN in ABS via L327 (Fig. 7). See also Section Interaction dynamics showing that the movement of L327 correlates with ligand dissociation (Fig. 6). We found that the flickering gate adopts an opened state if CBS is not occupied (Fig. 7b). This is a consequence of the lack of hydrophobic interaction between T1 (F404) and the CBS-bound EN. The H-bonding between L1 (S329) and T1 (N407) becomes disrupted, and therefore, flexibility of L1 increases. As discussed above, a flexible L1 promotes dissociation of EN from ABS and lowers the corresponding RF (Table 3).
The above dynamic interaction network, especially between CBS, T1, L1, and ABS describe the working mechanism of the “flickering gate”25. Beside providing a detailed description of the opened and closed state of the gate, we were also able to detect two dissociation pathways of EN during exiting ABS. The first dissociation pathway towards F404 and P406 of T1 is shown in Supplementary Video S3, and the second pathway towards P323 of L1 can be followed in Supplementary Video S2. Both dissociation pathays require the opened state of the “flickering gate”25. In addition to the kinetic stability data of Tables 2 and 3, MD allowed the above in-depth analyses of changes of interaction networks at the ABS. The present approach provides a structural background of stability differences pointing to key residues of hERα affecting non-classical steroid action.
## Conclusions
In the present study, elucidation of structural dynamics of non-classical effects of sex steroids was presented. Both classical and alternative binding modes were exhaustively mapped on the ligand-binding domain of human estrogen receptor alpha. Kinetic stability of the steroid –receptor complexes was investigated by molecular dynamics calculations. Real-time investigations of the complete interaction network at atomic resolution pointed to key residues of steroid binding mechanism. We showed how steroid binding to the alternative binding site of non-classical action is facilitated by the presence of a ligand in the classical binding site and the absence of the co-activator peptide. Uncovering such dynamic mechanisms behind steroid action will help the structure-based design of new drugs with rapid, non-classical responses.
## Methods
### Steroid-free systems
#### Selection of target structure
There are 137 hERα LBD entries available in the Protein Databank (PDB, Supplementary Table S8) and among them structure 3q95 has the most amino acids solved with a good resolution of 2.05 Å. The 3q95 structure is co-crystallized with the native ligand (estriol) and CA. As 3q95 is the most complete structure, it was chosen to represent hERα LBD. The ligand-free hERα (2b23) is also available (Supplementary Table S8) and superimposing 2b23 and 3q95 on their backbone atoms with PyMol59 has an excellent overall structural fit quantified by a root mean squared deviation (RMSD) of 0.5 Å. The RMSD was calculated between the two conformations according to Equation (1).
$${\rm{RMSD}}=\sqrt{\frac{1}{{\rm{NH}}}\sum _{{\rm{i}}=1}^{{\rm{NH}}}{| \vec{{\rm{C}}{1}_{{\rm{i}}}}-\vec{{\rm{C}}{2}_{{\rm{i}}}}| }^{2}}$$
(1)
where NH is the number of heavy atoms, C1 and C2 are space vectors of the ith heavy atom of conformations 1 (C1) and 2 (C2), respectively.
Secondary structure prediction was performed on the amino acid sequence of the missing F-domain, the sequence was accessed from UniProt with accession ID of P03372, multiple sequence alignment was performed with Clustal Omega60. Prediction was performed on the PsiPred server61, with the last two amino acids from X-ray structure added, to facilitate the fitting onto the protein after MD. Based on this prediction, the tertiary structure of the polypeptide chain was modelled with Tinker and equilibrated by a 10-ns-long molecular dynamics simulation. After equilibration, further 100 ns, unrestrained MD trajectory was generated for production (see next Section for details). After clustering, the representative structure of the C-terminal region, was merged with both X-ray structures of HERα (3q95 and 2b23) and these extended proteins were used throughout this study.
Both the ligand free and ligand bound PDB entries are appropriate representations of the LBD structure as E2 and estriol do not induce significant changes in the protein structure. In Section Interaction networks in the steroid-free receptor, the extended 2b23 was used, the holo simulations of Section Interaction networks in the steroid-bound receptor were performed with 3q95. The RMSF plot of 3q95 (1 µ simulation, without ligand, Supplementary Fig. S5) shows that overall dynamics of this protein structure is similar to that of 2b23.
#### Preparation of systems for energy minimization
Structures were solvated with the gmx solvate module of GROMACS 5.0.262 in a dodecahedral box with box edges 1 nm from the solute. Missing residues of 2b23 (except the C-terminal region) were not modelled. The box was filled with explicit TIP3P waters63. Parameters from the Amber99SB-ILDN64 force field were used. Sodium or chloride counter ions were added to neutralize the system. The N-terminal region of the receptor proteins was capped; the co-activator peptide was modelled with charged termini.
#### Energy minimization
The optimization of the simulation boxes prior MD and docking calculations were done in two steps. This procedure was applied for all cases. In the first step a steepest descent minimization was performed on the solvated box, with convergence threshold set to 103 kJmol−1nm−1. It was followed by a conjugate gradient minimization, in this step, the convergence was set to 10 kJmol−1nm−1. Position restraints were applied on solute heavy atoms at a force constant of 103 kJmol−1nm−2 in both steps.
#### Molecular dynamics (MD)
After minimization, prior to the productive GROMACS MD calculations, a uniform equilibration procedure was performed. The optimized structure was equilibrated under NPT conditions for 10 ns (with 2 fs time step). The solvent and the solute was coupled separately to 300 K with the velocity-rescaling algorithm65, with time constant of 0.1 ps. Pressure was kept at 1 bar with the Berendsen barostat66 with time constant of 0.5 ps, and compressibility of 4.5 × 10–5 bar−1. Long range interactions were cut off at 1.1 nm. Position restraints of 1000 kJmol−1nm−1 were applied on all protein heavy atoms. After equilibration, productive NPT MD calculations were started using GROMACS, with position restraints removed. Pressure was coupled with the Parrinello-Rahman barostat67 with time constant of 0.5 ps, and compressibility of 4.5 × 10−5 bar−1. The temperature was coupled to 300 K with the velocity-rescaling algorithm65, with time constant of 0.1 ps, with solvent and solute coupled separately. Coordinates were saved at regular time-intervals, at every 10 ps. Simulation on the ligand free structures were 1 μs-long, the terminal loop was simulated for 100 ns. Periodic boundary conditions were treated after the finish of the calculations.
#### Evaluation of MD results
A ligand free simulation of 1µs length contains 105 frames. RMSF calculation was performed with GROMACS gmx rmsf program. RMSF values of 462–471 and 297–300 residues in Fig. 2c were obtained from simulations with 3q95 (Supplementary Fig. S5). Distance calculations from the initial position of M357 (SD), H356 (CE1) and L327 (CG) sidechain atoms was followed throughout the 1µs steroid-free simulation. The distance was calculated using GROMACS rms program, having an alignment of heavy atoms on the initial structure, over H3 (341–361) and H12 (539–545) residues. For efficient presentation in Fig. 3 (bottom part) and Supplementary Fig. S6 average distances were plotted for every 200 frames.
### Binding site search with blind docking
#### Preparation of the target
The most populous cluster from the 3q95 simulation after 100 ns with the modelled C-terminal was used as the target structure. Clustering was performed with Gromacs program cluster using the gromos method, and a 2 Å cut-off RMSD was set between clusters. Only polar hydrogens were treated explicitly, non-polar hydrogens were merged. Gasteiger-Marsili charges68 were added to the protein.
#### Preparation of the ligand
The first step was a steepest descent optimization, with 104 steps. The next step was a conjugate gradient minimization, with a maximum of 104 steps, the with convergence threshold set to 10−7 kcalmol−1Å−1. MMFF94 force field69 was used in both steps. The third and last step was performed on semi-empirical quantum mechanical level with MOPAC201270 with PM7 parametrization71. Gradient norm was set to 0.01 kcalmol−1Å−1. After optimization, force calculations were carried out, ensuring that in all cases, the force constant matrices were positive definite. This optimized structure was used in the dockings with Gasteiger-Marsili charges added.
#### Calculation of grid maps
The grid box around the protein was generated with AutoGrid 4.272. The box was centred to cover the whole protein with 200 grid points along all axes, with a spacing of 0.375 Å.
#### Blind docking
Blind docking calculations52,53,54 of the two steroids (E2 and EN) were performed. Docking calculations were performed with AutoDock 4.2.372, Lamarckian genetic algorithm with Solis-Wets local search was used in geometrical search. Dockings started with a population size of 250, the number evaluations were 107, and the number of generations was set to 107. 100 runs were performed in one docking. For RMSD calculation, between the crystallized and the docked estradiol, 1gwr hHERα was used, where estradiol is the co-crystallized ligand (Supplementary Fig. S1). The estradiol structure from 1gwr was taken after superimposing the Cα atoms of 1gwr on to the Cα atoms of hERα structure used for docking.
#### Calculation of interaction energy (Einter)
Calculation Einter between docked steroids and hERα (Section Binding sites of sexual steroids) was performed after energy minimization with Gromacs (see previous Section, Energy minimization) of the docked complexes. A Lennard-Jones potential (Equation 2) was used with Amber parameters73.
$${{\rm{E}}}_{{\rm{inter}}}=\sum _{{\rm{i}},{\rm{j}}}^{{{\rm{N}}}_{{\rm{T}}}{{\rm{N}}}_{{\rm{L}}}}(\frac{{{\rm{A}}}_{{\rm{ij}}}}{{{\rm{r}}}_{{\rm{ij}}}^{12}}\,-\,\frac{{{\rm{B}}}_{{\rm{ij}}}}{{{\rm{r}}}_{{\rm{ij}}}^{6}})$$
(2)
$${{\rm{A}}}_{{\rm{ij}}}={{\rm{\varepsilon }}}_{{\rm{ij}}}{{\rm{R}}}_{{\rm{ij}}}^{12}$$ $${{\rm{B}}}_{{\rm{ij}}}=2{{\rm{\varepsilon }}}_{{\rm{ij}}}{{\rm{R}}}_{{\rm{ij}}}^{6}$$ $${{\rm{R}}}_{{\rm{ij}}}={{\rm{R}}}_{{\rm{i}}}+{{\rm{R}}}_{{\rm{j}}}$$ $${{\rm{\varepsilon }}}_{{\rm{ij}}}=\sqrt{{{\rm{\varepsilon }}}_{{\rm{i}}}{{\rm{\varepsilon }}}_{{\rm{j}}}}$$
where NT: number of target atoms, NL: number of ligand atoms, rij: actual inter-nuclear distance, εij = potential well depth at equilibrium between i and j atoms types combined from individual well depths, Rij = inter-nuclear distance at equilibrium between i and j atom types combined from individual radii.
### Interaction dynamics of steroid-bound systems
#### Molecular dynamics
The conditions of MD simulations were the same as described at the steroid-free calculations, except that the present steroid-bound trajectories were 100-ns-long each, and 1001 frames were sampled per trajectory. After each trajectory the periodic boundary effects were handled, the system was centred in the box and target molecules in subsequent frames were fit on the top of the first frame. In order to compare the “Open” and the “Closed” state between each other (Fig. 7a), after handing the periodic boundary effects, the first frame of “Open” state was superposed on the “Closed” state by their Cα atoms.
#### Kinetic stability
Residence frequency (RF, Equation 3) was calculated as a measure of kinetic stability. The movement of the ligand was described by the distance between the centre of mass (dCOM) of its actual and starting positions at each time frame during the total simulation time.
$${\rm{RF}}=\frac{{\rm{Count}}\,{\rm{of}}\,{\rm{time}}\,{\rm{frames}}\,{\rm{with}}\,{{\rm{d}}}_{{\rm{COM}}}\le \,{{\rm{d}}}_{{\rm{LIM}}}}{{\rm{Simulation}}\,{\rm{time}}\,({\rm{ns}})}$$
(3)
The value of dissociation limit dLIM was set to 5 Å. The RF values were calculated for the five individual trajectories and also for a merged trajectory of 500 ns including all five trajectories. The theoretical upper limit of RF was 10.0 ns−1 (=1001/100 ns) in the present study which corresponds to the highest kinetic stability.
#### Correlation of movements of M357, H356, L327 residues, with dissociation of EN
The distance of the side chain atoms from the initial positions were calculated using Gromacs rms program, having an alignment of heavy atoms, on the initial structure, over H3 (341–361) and H12 (539–545) residues. Using the same technique as in the steroid-free evaluations (Methods), for efficient presentation, average distance values were calculated for every 10 frames, resulting in 100 distances for 100 ns of simulation. Correlation of movements of M357, H356, L327 residues, with dissociation of EN was followed when dCOM ≤ dLIM. The dLIM corresponded to 27.2 ns, and correlation points (Fig. 6), were taken from 0 to 28 ns. Average distances of the investigated time interval (0–28 ns) resulted in 1.2 Å initial, and 4.9 Å final dCOM values, shown as abscissa in Fig. 6. Up until this frame, the ligand dissociation could be correlated with all three residues, and this is also the point when EN starts to abruptly dissociate from ABS (Fig. 5). The structural representations in Fig. 6, were taken from 18 ns. This was the frame when the movement of all three residues was the most representative.
#### Velocity calculations
In order to characterize the dissociation patterns of both E2 and EN, three types of velocities were calculated, and presented in Supplementary Table S67. The v1 measures the ligand velocity in the initial dissociation phase, until dLIM is reached. The second type of velocity (v2) takes into account the necessary time for the ligand to reach total dissociation after reaching the dLIM. The limit for final dissociation was set to 10 Å, and the time when this limit is reached, was collected in Supplementary Table S6. The v2 characterizes the best, the differences between EN and E2 dissociation mode. In CA− simulations, dLIM was not was not reached for E2, and therefore, v2 was not calculated. The third type of velocity (v3) describes the ligand velocity for the total dissociation, from the start of the simulation.
### Data availability statement
The datasets generated during and/or analysed during the current study are included in this published article (and its Supplementary Information files) or available from the corresponding author on reasonable request.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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## Acknowledgements
We acknowledge a grant of computer time from CSCS Swiss National Supercomputing Centre, and NIIF Hungarian National Information Infrastructure Development Institute. We acknowledge that the results of this research have been achieved using the DECI resource Archer based in the UK at the National Supercomputing Service with support from the PRACE aisbl. The work was supported by the K123836, K112807, K120391 grants from the National Research, Development, and Innovation Office, Hungary and KTIA_NAP_13-2014-0001. The University of Pécs is acknowledged for a grant PTE ÁOK_KA/2017 and also support in the frame of “Viral Pathogenesis” Talent Centre program. We are thankful to the Gedeon Richter Pharmaceutical Plc. for a pre-doctoral scholarship to N.J.
## Author information
### Author notes
1. Mónika Bálint and Norbert Jeszenői contributed equally to this work.
### Affiliations
1. #### Department of Pharmacology and Pharmacotherapy, University of Pécs, Szigeti út 12, 7624, Pécs, Hungary
• Mónika Bálint
• & Csaba Hetényi
2. #### Department of Biochemistry, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary
• Mónika Bálint
3. #### MTA NAP-B Molecular Neuroendocrinology Group, Institute of Physiology, Szentágothai Research Center, Center for Neuroscience, University of Pécs, Szigeti út 12, 7624, Pécs, Hungary
• Norbert Jeszenői
• & István M. Ábrahám
4. #### Chemistry Doctoral School, University of Szeged, Dugonics tér 13, 6720, Szeged, Hungary
• István Horváth
### Contributions
M.B., N.J., and I.H. performed research. M.B., C.H., N.J., and I.M.A. designed research. C.H. and I.M.A. organized research. M.B., C.H., I.M.A., and N.J. wrote the paper.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding authors
Correspondence to István M. Ábrahám or Csaba Hetényi.
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# Go Math Grade 5 Chapter 6 Answer Key Pdf Add and Subtract Fractions with Unlike Denominators
## Add and Subtract Fractions with Unlike Denominators Go Math Grade 5 Chapter 6 Answer Key Pdf
The Go Math Grade 5 Answer Key Add and Subtract Fractions with Unlike Denominators pdf covers the material presented in Chapter 6 Review/Test answers. You can expect the answers for homework and exercise problems in our Go Math Answer Key for Grade 5 Chapter 6 Add and Subtract Fractions with Unlike Denominators. Check out the topics covered in this chapter from the below section.
Lesson 1: Investigate • Addition with Unlike Denominators
Lesson 2: Investigate • Subtraction with Unlike Denominators
Lesson 3: Estimate Fraction Sums and Differences
Lesson 4: Common Denominators and Equivalent Fractions
Lesson 5: Add and Subtract Fractions
Mid-Chapter Checkpoint
Lesson 6: Add and Subtract Mixed Numbers
Lesson 7: Subtraction with Renaming
Lesson 8: Algebra • Patterns with Fractions
Lesson 9: Problem Solving • Practice Addition and Subtraction
Lesson 10: Algebra • Use Properties of Addition
Chapter 6 Review/Test
### Share and Show – Page No. 244
Use fraction strips to find the sum. Write your answer in simplest form.
Question 1.
$$\frac{1}{2}+\frac{3}{8}=$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{8}$$
Explanation:
Step 1:
Place three $$\frac{1}{8}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{2}$$ fraction strip beside the three $$\frac{1}{8}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{2}$$ and $$\frac{3}{8}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{2}$$ = $$\frac{1}{2}$$ × $$\frac{4}{4}$$ = $$\frac{4}{8}$$
$$\frac{3}{8}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{4}{8}$$ + $$\frac{3}{8}$$ = $$\frac{7}{8}$$
Question 2.
$$\frac{1}{2}+\frac{2}{5}=$$
$$\frac{□}{□}$$
Answer: $$\frac{9}{10}$$
Explanation:
Step 1:
Place two $$\frac{1}{5}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{2}$$ fraction strip beside the two $$\frac{1}{5}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{2}$$ and $$\frac{2}{5}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{2}$$ = $$\frac{1}{2}$$ × $$\frac{5}{5}$$ = $$\frac{5}{10}$$
$$\frac{2}{5}$$ = $$\frac{2}{5}$$ × $$\frac{2}{2}$$ = $$\frac{4}{10}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{5}{10}$$ + $$\frac{4}{10}$$ = $$\frac{9}{10}$$
Thus, $$\frac{1}{2}$$ + $$\frac{2}{5}$$ = $$\frac{9}{10}$$
### Page No. 245
Use fraction strips to find the sum. Write your answer in simplest form.
Question 3.
$$\frac{3}{8}+\frac{1}{4}=$$
$$\frac{□}{□}$$
Answer: $$\frac{5}{8}$$
Explanation:
Step 1:
Place three $$\frac{1}{8}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{4}$$ fraction strip beside the three $$\frac{1}{8}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{4}$$ and $$\frac{3}{8}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{4}$$ × $$\frac{2}{2}$$ = $$\frac{2}{8}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{2}{8}$$ + $$\frac{3}{8}$$ = $$\frac{5}{8}$$
Question 4.
$$\frac{3}{4}+\frac{1}{3}=$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{12}$$
Explanation:
Step 1:
Place three $$\frac{3}{4}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{3}$$ fraction strip beside the three $$\frac{1}{4}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{3}$$ and $$\frac{3}{4}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{3}$$ × $$\frac{4}{4}$$ = $$\frac{4}{12}$$
$$\frac{3}{4}$$ × $$\frac{3}{3}$$ = $$\frac{9}{12}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{4}{12}$$ + $$\frac{9}{12}$$ = $$\frac{13}{12}$$ = 1 $$\frac{1}{12}$$
Use fraction strips to find the sum. Write your answer in simplest form.
Question 5.
$$\frac{2}{5}+\frac{3}{10}=$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{10}$$
Explanation:
Step 1:
Place three $$\frac{1}{10}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a two $$\frac{2}{5}$$ fraction strip beside the three $$\frac{1}{10}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{2}{5}$$ and $$\frac{3}{10}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{2}{5}$$ • $$\frac{2}{2}$$ = $$\frac{4}{10}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{4}{10}$$ + $$\frac{3}{10}$$ = $$\frac{7}{10}$$
Question 6.
$$\frac{1}{4}+\frac{1}{12}=$$
$$\frac{□}{□}$$
Answer: $$\frac{4}{12}$$
Explanation:
Step 1:
Place $$\frac{1}{12}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{4}$$ fraction strip beside the $$\frac{1}{12}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{12}$$ and $$\frac{1}{4}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{4}$$ • $$\frac{3}{3}$$ = $$\frac{3}{12}$$
$$\frac{1}{12}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{3}{12}$$ + $$\frac{1}{12}$$ = $$\frac{4}{12}$$
Question 7.
$$\frac{1}{2}+\frac{3}{10}=$$
$$\frac{□}{□}$$
Answer: $$\frac{8}{10}$$
Explanation:
Step 1:
Place three $$\frac{1}{10}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{2}$$ fraction strip beside the three $$\frac{1}{10}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{2}$$ and $$\frac{3}{10}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{2}$$ • $$\frac{5}{5}$$ = $$\frac{5}{10}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{5}{10}$$ + $$\frac{3}{10}$$ = $$\frac{8}{10}$$
Question 8.
$$\frac{2}{3}+\frac{1}{6}=$$
$$\frac{□}{□}$$
Answer: $$\frac{5}{6}$$
Explanation:
Step 1:
Place two $$\frac{1}{3}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{6}$$ fraction strip beside the two $$\frac{1}{3}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{6}$$ and $$\frac{2}{3}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{2}{3}$$ = $$\frac{2}{3}$$ • $$\frac{2}{2}$$ = $$\frac{4}{6}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{4}{6}$$ + $$\frac{1}{6}$$ = $$\frac{5}{6}$$
Question 9.
$$\frac{5}{8}+\frac{1}{4}=$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{8}$$
Explanation:
Step 1:
Place five $$\frac{1}{8}$$ fractions strips under the 1 whole strip on your Mathboard. Then place a $$\frac{1}{4}$$ fraction strip beside the five $$\frac{1}{8}$$ strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{4}$$ and $$\frac{5}{8}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{4}$$ • $$\frac{2}{2}$$ = $$\frac{2}{8}$$
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{2}{8}$$ + $$\frac{5}{8}$$ = $$\frac{7}{8}$$
Question 10.
$$\frac{1}{2}+\frac{1}{5}=$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{10}$$
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{1}{5}$$ and $$\frac{1}{2}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{5}$$ • $$\frac{2}{2}$$ = $$\frac{2}{10}$$
$$\frac{1}{2}$$ • $$\frac{5}{5}$$ = $$\frac{5}{10}$$
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{2}{10}$$ + $$\frac{5}{10}$$ = $$\frac{7}{10}$$
Question 11.
$$\frac{3}{4}+\frac{1}{6}=$$
$$\frac{□}{□}$$
Answer: $$\frac{11}{12}$$
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{3}{4}$$ and $$\frac{1}{6}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{3}{4}$$ • $$\frac{3}{3}$$ = $$\frac{9}{12}$$
$$\frac{1}{6}$$ • $$\frac{2}{2}$$ = $$\frac{2}{12}$$
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{9}{12}$$ + $$\frac{2}{12}$$ = $$\frac{11}{12}$$
Question 12.
$$\frac{1}{2}+\frac{2}{3}=$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{6}$$
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{2}{3}$$ and $$\frac{1}{2}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{2}{3}$$ • $$\frac{2}{2}$$ = $$\frac{4}{6}$$
$$\frac{1}{2}$$ • $$\frac{3}{3}$$ = $$\frac{3}{6}$$
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{4}{6}$$ + $$\frac{3}{6}$$ = $$\frac{7}{6}$$
$$\frac{7}{6}$$ is greater than 1.
Convert the fraction into the mixed fraction
$$\frac{7}{6}$$ = 1 $$\frac{1}{6}$$
Question 13.
$$\frac{7}{8}+\frac{1}{4}=$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{8}$$
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to $$\frac{7}{8}$$ and $$\frac{1}{4}$$. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
$$\frac{1}{4}$$ • $$\frac{2}{2}$$ = $$\frac{2}{8}$$
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
$$\frac{7}{8}$$ + $$\frac{2}{8}$$ = $$\frac{9}{8}$$
Convert $$\frac{9}{8}$$ into the mixed fraction.
$$\frac{9}{8}$$ = 1 $$\frac{1}{8}$$
Question 14.
Explain how using fraction strips with like denominators makes it possible to add fractions with unlike denominators.
Type below:
_________
Answer: The strips for both fractions need to be the same size. Finding like denominators is done by trying smaller strips so they can all be the same size.
### Problem Solving – Page No. 246
Question 15.
Maya makes trail mix by combining $$\frac{1}{3}$$ cup of mixed nuts and $$\frac{1}{4}$$ cup of dried fruit. What is the total amount of ingredients in her trail mix?
$$\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$$
Maya uses $$\frac{1}{12}$$ cup of ingredients.
Write a new problem using different amounts for each ingredient. Each amount should be a fraction with a denominator of 2, 3, or 4. Then use fraction strips to solve your problem.
Pose a problem Solve your problem. Draw a picture of the
fraction strips you use to solve the problem.
Explain why you chose the amounts you did for your problem.
Type below:
_________
$$\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$$
Maya uses $$\frac{1}{12}$$ cup of ingredients.
Maya makes trail mix by combining $$\frac{1}{2}$$ cup of mixed nuts and $$\frac{1}{3}$$ cup of dried fruit and $$\frac{1}{4}$$ cup of chocolate morsels. What is the total amount of ingredients in her trail mix?
$$\frac{1}{2}$$ + $$\frac{1}{3}$$ + $$\frac{1}{4}$$ = x
2 • $$\frac{1}{2}$$ + 2 • $$\frac{1}{3}$$ + 2 • $$\frac{1}{4}$$ = 2 • x
1 + $$\frac{2}{3}$$ + $$\frac{1}{2}$$ = 2x
Now multiply with 3 on both sides
3 • 1 + 3 • $$\frac{2}{3}$$ + 3 • $$\frac{1}{2}$$ = 3 • 2x
3 + 2 + $$\frac{3}{2}$$ = 6x
6 + 4 + 1 = 12 x
11 = 12x
x = $$\frac{11}{12}$$
$$\frac{1}{2}$$ + $$\frac{1}{3}$$ + $$\frac{1}{4}$$ = $$\frac{11}{12}$$
### Share and Show – Page No. 248
Use fraction strips to find the difference. Write your answer in simplest form.
Question 1.
$$\frac{7}{10}-\frac{2}{5}=$$
$$\frac{□}{□}$$
$$\frac{7}{10}$$ – $$\frac{2}{5}$$
$$\frac{7}{10}$$ – $$\frac{2}{5}$$ • $$\frac{2}{2}$$
$$\frac{7}{10}$$ – $$\frac{4}{10}$$ = $$\frac{3}{10}$$
Question 2.
$$\frac{2}{3}-\frac{1}{4}=$$
$$\frac{□}{□}$$
$$\frac{2}{3}$$ – $$\frac{1}{4}$$
Now we have to make the fractions like denominators
$$\frac{2}{3}$$ • $$\frac{4}{4}$$ – $$\frac{1}{4}$$ • $$\frac{3}{3}$$
$$\frac{8}{12}$$ – $$\frac{3}{12}$$ = $$\frac{5}{12}$$
### Page No. 249
Use fraction strips to find the difference. Write your answer in simplest form.
Question 3.
$$\frac{5}{6}-\frac{1}{4}=$$
$$\frac{□}{□}$$
Step 1:
Find fraction strips, all with the same denominator, that fit exactly under the difference $$\frac{5}{6}-\frac{1}{4}$$
Step 2:
Find another set of fraction strips, all with the same the denominator, that fit exactly under the difference $$\frac{5}{6}-\frac{1}{4}$$
Step 3:
Find other fraction strips, all with the same denominator, that fit exactly under the difference $$\frac{5}{6}-\frac{1}{4}$$
$$\frac{5}{6}$$ • $$\frac{4}{4}$$ – $$\frac{1}{4}$$ • $$\frac{6}{6}$$
$$\frac{20}{24}$$ – $$\frac{6}{24}$$ = $$\frac{14}{24}$$ = $$\frac{7}{12}$$
Thus, $$\frac{5}{6}-\frac{1}{4}$$ = $$\frac{7}{12}$$
Question 4.
$$\frac{1}{2}-\frac{3}{10}=$$
$$\frac{□}{□}$$
$$\frac{1}{2}-\frac{3}{10}$$
$$\frac{1}{2}$$ • $$\frac{5}{5}$$ – $$\frac{3}{10}$$
$$\frac{5}{10}$$ – $$\frac{3}{10}$$ = $$\frac{2}{10}$$
Question 5.
$$\frac{3}{8}-\frac{1}{4}=$$
$$\frac{□}{□}$$
$$\frac{3}{8}-\frac{1}{4}$$
$$\frac{3}{8}$$ – $$\frac{1}{4}$$ • $$\frac{2}{2}$$
= $$\frac{3}{8}$$ – $$\frac{2}{8}$$ = $$\frac{1}{8}$$
Question 6.
$$\frac{2}{3}-\frac{1}{2}=$$
$$\frac{□}{□}$$
$$\frac{2}{3}-\frac{1}{2}$$
$$\frac{2}{3}$$ • $$\frac{2}{2}$$ – $$\frac{1}{2}$$ • $$\frac{3}{3}$$
$$\frac{4}{6}-\frac{3}{6}$$ = $$\frac{1}{6}$$
Use fraction strips to find the difference. Write your answer in simplest form.
Question 7.
$$\frac{3}{5}-\frac{3}{10}=$$ $$\frac{□}{□}$$
$$\frac{3}{5}-\frac{3}{10}$$
$$\frac{3}{5}$$ • $$\frac{2}{2}$$ – $$\frac{3}{10}$$
= $$\frac{6}{10}$$ – $$\frac{3}{10}$$ = $$\frac{3}{10}$$
Question 8.
$$\frac{5}{12}-\frac{1}{3}=$$ $$\frac{□}{□}$$
$$\frac{5}{12}-\frac{1}{3}$$
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
$$\frac{5}{12}$$ – $$\frac{1}{3}$$ • $$\frac{4}{4}$$
$$\frac{5}{12}$$ – $$\frac{4}{12}$$ = $$\frac{1}{12}$$
Question 9.
$$\frac{1}{2}-\frac{1}{10}=$$ $$\frac{□}{□}$$
$$\frac{1}{2}-\frac{1}{10}$$
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
$$\frac{1}{2}$$ • $$\frac{5}{5}$$ – $$\frac{1}{10}$$
$$\frac{5}{10}$$ – $$\frac{1}{10}$$ = $$\frac{4}{10}$$
Question 10.
$$\frac{3}{5}-\frac{1}{2}=$$ $$\frac{□}{□}$$
$$\frac{3}{5}-\frac{1}{2}$$
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
$$\frac{3}{5}$$ • $$\frac{2}{2}$$ – $$\frac{1}{2}$$ • $$\frac{5}{5}$$
$$\frac{6}{10}-\frac{5}{10}$$ = $$\frac{1}{10}$$
Question 11.
$$\frac{7}{8}-\frac{1}{4}=$$ $$\frac{□}{□}$$
$$\frac{7}{8}-\frac{1}{4}$$
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
$$\frac{7}{8}$$ – $$\frac{1}{4}$$ • $$\frac{2}{2}$$
$$\frac{7}{8}$$ – $$\frac{2}{8}$$ = $$\frac{5}{8}$$
Question 12.
$$\frac{5}{6}-\frac{2}{3}=$$ $$\frac{□}{□}$$
$$\frac{5}{6}-\frac{2}{3}$$
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
$$\frac{5}{6}$$ – $$\frac{2}{3}$$ • $$\frac{2}{2}$$
$$\frac{5}{6}$$ – $$\frac{4}{6}$$
$$\frac{1}{6}$$
Question 13.
$$\frac{3}{4}-\frac{1}{3}=$$ $$\frac{□}{□}$$
$$\frac{3}{4}-\frac{1}{3}$$
$$\frac{3}{4}$$ • $$\frac{3}{3}$$ – $$\frac{1}{3}$$ • $$\frac{4}{4}$$
$$\frac{9}{12}$$ – $$\frac{4}{12}$$ = $$\frac{5}{12}$$
Question 14.
$$\frac{5}{6}-\frac{1}{2}=$$ $$\frac{□}{□}$$
$$\frac{5}{6}-\frac{1}{2}$$
$$\frac{5}{6}$$ – $$\frac{1}{2}$$ • $$\frac{3}{3}$$
$$\frac{5}{6}$$ – $$\frac{3}{6}$$ = $$\frac{2}{6}$$
$$\frac{5}{6}-\frac{1}{2}=$$ $$\frac{2}{6}$$
Question 15.
$$\frac{3}{4}-\frac{7}{12}=$$ $$\frac{□}{□}$$
$$\frac{3}{4}-\frac{7}{12}$$
$$\frac{3}{4}$$ • $$\frac{3}{3}$$ – $$\frac{7}{12}$$
$$\frac{9}{12}$$ – $$\frac{7}{12}$$ = $$\frac{2}{12}$$
$$\frac{3}{4}-\frac{7}{12}=$$ $$\frac{2}{12}$$
Question 16.
Explain how your model for $$\frac{3}{5}-\frac{1}{2}$$ is different from your model for $$\frac{3}{5}-\frac{3}{10}$$.
Type below:
_________
$$\frac{3}{5}-\frac{3}{10}$$
$$\frac{3}{5}$$ • $$\frac{2}{2}$$ – $$\frac{3}{10}$$
$$\frac{6}{10}$$ – $$\frac{3}{10}$$ = $$\frac{3}{10}$$
### UNLOCK the Problem – Page No. 250
Question 17.
The picture at the right shows how much pizza was left over from lunch. Jason eats $$\frac{1}{4}$$ of the whole pizza for dinner. Which subtraction sentence represents the amount of pizza that is remaining after dinner?
a. What problem are you being asked to solve?
Type below:
_________
Answer: I am asked to solve which subtraction sentence represents the amount of pizza that is remaining after dinner.
Question 17.
b. How will you use the diagram to solve the problem?
Type below:
_________
Answer: I will use number of slices left in the pizza to solve the problem.
Question 17.
c. Jason eats $$\frac{1}{4}$$ of the whole pizza. How many slices does he eat?
______ slices
Explanation:
Given that, Jason eats $$\frac{1}{4}$$ of the whole pizza.
The pizza is cut into 8 slices.
So, 8 × $$\frac{1}{4}$$ = 2 slices.
Thus Jason ate 2 slices.
Question 17.
d. Redraw the diagram of the pizza. Shade the sections of pizza that are remaining after Jason eats his dinner.
Type below:
_________
Question 17.
e. Write a fraction to represent the amount of pizza that is remaining.
$$\frac{□}{□}$$ of a pizza
Answer: $$\frac{3}{8}$$ of a pizza
Explanation:
The fraction of pizzz Jason ate = $$\frac{1}{4}$$
Number of slices left = $$\frac{5}{8}$$
Now subtract $$\frac{5}{8}$$ – $$\frac{1}{4}$$
= $$\frac{3}{8}$$
Thus the fraction to represent the amount of pizza that is remaining is $$\frac{3}{8}$$
Question 17.
f. Fill in the bubble for the correct answer choice above.
Options:
a. 1 – $$\frac{1}{4}$$ = $$\frac{3}{4}$$
b. $$\frac{5}{8}$$ – $$\frac{1}{4}$$ = $$\frac{3}{8}$$
c. $$\frac{3}{8}$$ – $$\frac{1}{4}$$ = $$\frac{2}{8}$$
d. 1 – $$\frac{3}{8}$$ = $$\frac{5}{8}$$
The fraction of pizzz Jason ate = $$\frac{1}{4}$$
Number of slices left = $$\frac{5}{8}$$
Now subtract $$\frac{5}{8}$$ – $$\frac{1}{4}$$ = $$\frac{3}{8}$$
Thus the correct answer is option B.
Question 18.
The diagram shows what Tina had left from a yard of fabric. She now uses $$\frac{2}{3}$$ yard of fabric for a project. How much of the original yard of fabric does Tina have left after the project?
Options:
a. $$\frac{2}{3}$$ yard
b. $$\frac{1}{2}$$ yard
c. $$\frac{1}{3}$$ yard
d. $$\frac{1}{6}$$ yard
Answer: $$\frac{1}{3}$$ yard
Explanation:
The original yard of fabric is 6
Tina uses $$\frac{2}{3}$$ yard of fabric for a project.
$$\frac{1}{1}$$ – $$\frac{2}{3}$$
$$\frac{3}{3}$$ – $$\frac{2}{3}$$ = $$\frac{1}{3}$$ yard
### Share and Show – Page No. 253
Estimate the sum or difference.
Question 1.
$$\frac{5}{6}+\frac{3}{8}$$
a. Round $$\frac{5}{6}$$ to its closest benchmark. ____
b. Round $$\frac{3}{8}$$ to its closest benchmark. ____
c. Add to find the estimate. ____ + ____ = ____
_____ $$\frac{□}{□}$$
a. Round $$\frac{5}{6}$$ to its closest benchmark. $$\frac{6}{6}$$ or 1.
b. Round $$\frac{3}{8}$$ to its closest benchmark. $$\frac{4}{8}$$ or $$\frac{1}{2}$$
c. Add to find the estimate. ____ + ____ = ____
1 + $$\frac{1}{2}$$ = $$\frac{3}{2}$$ = 1 $$\frac{1}{2}$$
Question 2.
$$\frac{5}{9}-\frac{3}{8}$$
_____
Explanation:
Step 1: Place a point at $$\frac{5}{9}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
The fraction rounded to $$\frac{5}{9}$$ is $$\frac{1}{2}$$
Step 2: Place a point at $$\frac{3}{8}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
The fraction rounded to $$\frac{3}{8}$$ is $$\frac{1}{2}$$.
$$\frac{1}{2}$$ – $$\frac{1}{2}$$ = 0
Question 3.
$$\frac{6}{7}+2 \frac{4}{5}$$
_____
Explanation:
Step 1: Place a point at $$\frac{6}{7}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{4}{5}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
1 + 3 = 4
Question 4.
$$\frac{5}{6}+\frac{2}{5}$$
_____ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{5}{6}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{2}{5}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
1 + $$\frac{1}{2}$$ = $$\frac{3}{2}$$ = 1 $$\frac{1}{2}$$
Question 5.
$$3 \frac{9}{10}-1 \frac{2}{9}$$
_____
Explanation:
Step 1: Place a point at $$\frac{9}{10}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{2}{9}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
3 × 1 – 1 × 0 = 3 – 0 = 3
$$3 \frac{9}{10}-1 \frac{2}{9}$$ = 3
Question 6.
$$\frac{4}{6}+\frac{1}{9}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{4}{6}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{1}{9}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
So, $$\frac{1}{2}$$ + 0 = $$\frac{1}{2}$$
$$\frac{4}{6}+\frac{1}{9}$$ = $$\frac{1}{2}$$
Question 7.
$$\frac{9}{10}-\frac{1}{9}$$
_____
Explanation:
Step 1: Place a point at $$\frac{9}{10}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{1}{9}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
1 – 0 = 1
$$\frac{9}{10}-\frac{1}{9}$$ = 1
Estimate the sum or difference.
Question 8.
$$\frac{5}{8}-\frac{1}{5}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{5}{8}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{1}{5}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
1 – $$\frac{1}{2}$$ = $$\frac{1}{2}$$
Question 9.
$$\frac{1}{6}+\frac{3}{8}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{1}{6}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
Step 2: Place a point at $$\frac{3}{8}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
0 + $$\frac{1}{2}$$ = $$\frac{1}{2}$$
Question 10.
$$\frac{6}{7}-\frac{1}{5}$$
_____
Explanation:
Step 1: Place a point at $$\frac{6}{7}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{1}{5}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
1 – 0 = 1
$$\frac{6}{7}-\frac{1}{5}$$ = 1
Question 11.
$$\frac{11}{12}+\frac{6}{10}$$
_____ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{11}{12}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1
Step 2: Place a point at $$\frac{6}{10}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1
1 + $$\frac{1}{2}$$ = $$\frac{3}{2}$$ = 1 $$\frac{1}{2}$$
$$\frac{11}{12}+\frac{6}{10}$$ = 1 $$\frac{1}{2}$$
Question 12.
$$\frac{9}{10}-\frac{1}{2}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{9}{10}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{1}{2}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
1 – $$\frac{1}{2}$$ = $$\frac{1}{2}$$
$$\frac{9}{10}-\frac{1}{2}$$ = $$\frac{1}{2}$$
Question 13.
$$\frac{3}{6}+\frac{4}{5}$$
_____ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{3}{6}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
Step 2: Place a point at $$\frac{4}{5}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1
$$\frac{1}{2}$$ + 1 = $$\frac{3}{2}$$ = 1 $$\frac{1}{2}$$
$$\frac{3}{6}+\frac{4}{5}$$ = 1 $$\frac{1}{2}$$
Question 14.
$$\frac{5}{6}-\frac{3}{8}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{5}{6}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place a point at $$\frac{3}{8}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
1 – $$\frac{1}{2}$$ = $$\frac{1}{2}$$
$$\frac{5}{6}-\frac{3}{8}$$ = $$\frac{1}{2}$$
Question 15.
$$\frac{1}{7}+\frac{8}{9}$$
_____
Explanation:
Step 1: Place a point at $$\frac{1}{7}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
Step 2: Place a point at $$\frac{8}{9}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
0 + 1 = 1
$$\frac{1}{7}+\frac{8}{9}$$ = 1
Question 16.
$$3 \frac{5}{12}-3 \frac{1}{10}$$
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
Step 1: Place a point at $$\frac{5}{12}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
Step 2: Place a point at $$\frac{1}{10}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$
$$\frac{1}{2}$$ – 0 = $$\frac{1}{2}$$
$$3 \frac{5}{12}-3 \frac{1}{10}$$ = $$\frac{1}{2}$$
### Problem Solving – Page No. 254
Question 17.
Lisa and Valerie are picnicking in Trough Creek State Park in Pennsylvania. Lisa has brought a salad that she made with $$\frac{3}{4}$$ cup of strawberries, $$\frac{7}{8}$$ cup of peaches, and $$\frac{1}{6}$$ cup of blueberries. About how many total cups of fruit are in the salad?
_____ cups
Explanation:
Lisa and Valerie are picnicking in Trough Creek State Park in Pennsylvania.
Lisa has brought a salad that she made with $$\frac{3}{4}$$ cup of strawberries, $$\frac{7}{8}$$ cup of peaches, and $$\frac{1}{6}$$ cup of blueberries.
Step 1: Place $$\frac{3}{4}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 2: Place $$\frac{7}{8}$$ on the number line.
The fraction is between $$\frac{1}{2}$$ and 1.
Step 3: Place $$\frac{1}{6}$$ on the number line.
The fraction is between 0 and $$\frac{1}{2}$$.
1 + 1 + 0 = 2
Thus 2 cups of fruit are in the salad.
Question 18.
At Trace State Park in Mississippi, there is a 25-mile mountain bike trail. If Tommy rode $$\frac{1}{2}$$ of the trail on Saturday and $$\frac{1}{5}$$ of the trail on Sunday, about what fraction of the trail did he ride?
$$\frac{□}{□}$$
Answer: $$\frac{1}{2}$$
Explanation:
At Trace State Park in Mississippi, there is a 25-mile mountain bike trail.
If Tommy rode $$\frac{1}{2}$$ of the trail on Saturday and $$\frac{1}{5}$$ of the trail on Sunday
Step 1: Place $$\frac{1}{2}$$ on the number line.
$$\frac{1}{2}$$ lies between 0 and $$\frac{1}{2}$$
Step 2: Place $$\frac{1}{5}$$ on the number line.
$$\frac{1}{5}$$ 0 and $$\frac{1}{2}$$
The number closer to $$\frac{1}{5}$$ is 0
$$\frac{1}{2}$$ – 0 = $$\frac{1}{2}$$
The estimated fraction of the trail he ride is $$\frac{1}{2}$$
Question 19.
Explain how you know that $$\frac{5}{8}+\frac{6}{10}$$ is greater than 1.
Type below:
__________
Step 1: Place $$\frac{5}{8}$$ on the number line.
$$\frac{5}{8}$$ is closer to $$\frac{1}{2}$$
Step 2: Place $$\frac{6}{10}$$ on the number line.
$$\frac{6}{10}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{6}{10}$$ is closer to $$\frac{1}{2}$$
$$\frac{1}{2}$$ + $$\frac{1}{2}$$ = 1
Question 20.
Nick estimated that $$\frac{5}{8}+\frac{4}{7}$$ is about 2.
Explain how you know his estimate is not reasonable.
Type below:
__________
Step 1: Place $$\frac{5}{8}$$ on the number line.
$$\frac{5}{8}$$ is closer to $$\frac{1}{2}$$
Step 2: Place $$\frac{4}{7}$$ on the number line.
$$\frac{4}{7}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{1}{2}$$ + $$\frac{1}{2}$$ = 1
By this, we can say that Nick’s estimation was wrong.
Question 21.
Test Prep Jake added $$\frac{1}{8}$$ cup of sunflower seeds and $$\frac{4}{5}$$ cup of banana chips to his sundae. Which is the best estimate of the total amount of toppings Jake added to his sundae?
Options:
a. about $$\frac{1}{2}$$ cup
c. about 1 $$\frac{1}{2}$$ cups
Explanation:
Given, Test Prep Jake added $$\frac{1}{8}$$ cup of sunflower seeds and $$\frac{4}{5}$$ cup of banana chips to his sundae
Step 1: Place $$\frac{1}{8}$$ on the number line.
$$\frac{1}{8}$$ lies between 0 and $$\frac{1}{2}$$
Step 2: Place $$\frac{4}{5}$$ on the number line.
$$\frac{4}{5}$$ lies between $$\frac{1}{2}$$ and 1.
0 + 1 = 1
The best estimate of the total amount of toppings Jake added to his sundae is about 1 cup.
### Share and Show – Page No. 256
Question 1.
Find a common denominator of $$\frac{1}{6}$$ and $$\frac{1}{9}$$ . Rewrite the pair of fractions using the common denominator.
• Multiply the denominators.
A common denominator of $$\frac{1}{6}$$ and $$\frac{1}{9}$$ is ____.
• Rewrite the pair of fractions using the common denominator.
Type below:
_________
Common denominator is 18.
$$\frac{1}{6}$$ × $$\frac{3}{3}$$ = $$\frac{3}{18}$$
$$\frac{1}{9}$$ × $$\frac{2}{2}$$ = $$\frac{2}{18}$$
The pair of fractions using the common denominator is $$\frac{3}{18}$$, $$\frac{2}{18}$$
Use a common denominator to write an equivalent fraction for each fraction.
Question 2.
$$\frac{1}{3}, \frac{1}{5}$$
common denominator: _________
Type below:
_________
Explanation:
Multiply the denominators of the fraction.
$$\frac{1}{3}$$ × $$\frac{1}{5}$$ = $$\frac{1}{15}$$
Thus the common denominator is 15.
Question 3.
$$\frac{2}{3}, \frac{5}{9}$$
common denominator: _________
Type below:
_________
Explanation:
Multiply the denominators
$$\frac{2}{3}$$ × $$\frac{5}{9}$$
= 3 × 9 = 27
Thus the common denominator of $$\frac{2}{3}, \frac{5}{9}$$ is 27.
Question 4.
$$\frac{2}{9}, \frac{1}{15}$$
common denominator: _________
Type below:
_________
Explanation:
Multiply the denominators
$$\frac{2}{9}$$ × $$\frac{1}{15}$$
The least common denominator of 15 and 9 is 45.
So, the common denominator of $$\frac{2}{9}, \frac{1}{15}$$ is 45.
### Page No. 257
Use the least common denominator to write an equivalent fraction for each fraction.
Question 5.
$$\frac{1}{4}, \frac{3}{8}$$
least common denominator: ______
Type below:
_________
Explanation:
First multiply the denominators of the fractions $$\frac{1}{4}, \frac{3}{8}$$
4 × 8 = 32
The least common denominator is 8
The equivalent fractions with LCD
$$\frac{1}{4}$$ = $$\frac{2}{8}$$
$$\frac{3}{8}$$ = $$\frac{3}{8}$$
Question 6.
$$\frac{11}{12}, \frac{5}{8}$$
least common denominator: ______
Type below:
_________
Explanation:
First, multiply the denominators of the fractions.
12 × 8 = 96
The least common denominator of 12 and 8 is 24.
The equivalent fractions with LCD
$$\frac{11}{12}$$ × $$\frac{2}{2}$$= $$\frac{22}{24}$$
$$\frac{5}{8}$$ × $$\frac{3}{3}$$ = $$\frac{15}{24}$$
Question 7.
$$\frac{4}{5}, \frac{1}{6}$$
least common denominator: ______
Type below:
_________
Explanation:
First, multiply the denominators of the fractions.
5 × 6 = 30
The least common denominator (LCD) = 30
$$\frac{4}{5}$$ × $$\frac{6}{6}$$= $$\frac{24}{30}$$
$$\frac{1}{6}$$ × $$\frac{5}{5}$$ = $$\frac{5}{30}$$
Use a common denominator to write an equivalent fraction for each fraction.
Question 8.
$$\frac{3}{5}, \frac{1}{4}$$
common denominator: ______
Type below:
_________
Explanation:
Multiply the denominators of the fractions to find the common denominator.
5 × 4 = 20
So, the common denominator of $$\frac{3}{5}, \frac{1}{4}$$ is 20.
Question 9.
$$\frac{5}{8}, \frac{1}{5}$$
common denominator: ______
Type below:
_________
Explanation:
Multiply the denominators of the fractions to find the common denominator.
8 × 5 = 40
So, the common denominator of $$\frac{5}{8}, \frac{1}{5}$$ is 40.
Question 10.
$$\frac{1}{12}, \frac{1}{2}$$
common denominator: ______
Type below:
_________
Explanation:
Multiply the denominators of the fractions to find the common denominator.
12 × 2 = 24
The common denominator of $$\frac{1}{12}, \frac{1}{2}$$ is 24.
Practice: Copy and Solve Use the least common denominator to write an equivalent fraction for each fraction.
Question 11.
$$\frac{1}{6}, \frac{4}{9}$$
Type below:
_________
Answer: $$\frac{3}{18}, \frac{8}{18}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 18
Now rewrite the fractions
$$\frac{1}{6}$$ × $$\frac{3}{3}$$ = $$\frac{3}{18}$$
$$\frac{4}{9}$$ × $$\frac{2}{2}$$ = $$\frac{8}{18}$$
Question 12.
$$\frac{7}{9}, \frac{8}{27}$$
Type below:
_________
Answer: $$\frac{21}{27}, \frac{8}{27}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 27
Now rewrite the fractions
$$\frac{7}{9}$$ × $$\frac{3}{3}$$ = $$\frac{21}{27}$$
$$\frac{8}{27}$$ × $$\frac{1}{1}$$ = $$\frac{8}{27}$$
Question 13.
$$\frac{7}{10}, \frac{3}{8}$$
Type below:
_________
Answer: $$\frac{28}{40}, \frac{15}{40}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 40
Now rewrite the fractions
$$\frac{7}{10}$$ × $$\frac{4}{4}$$ = $$\frac{28}{40}$$
$$\frac{3}{8}$$ × $$\frac{5}{5}$$ = $$\frac{15}{40}$$
Question 14.
$$\frac{1}{3}, \frac{5}{11}$$
Type below:
_________
Answer: $$\frac{11}{33}, \frac{15}{33}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 33
Now rewrite the fractions
$$\frac{1}{3}$$ × $$\frac{11}{11}$$ = $$\frac{11}{33}$$
$$\frac{5}{11}$$ × $$\frac{3}{3}$$ = $$\frac{15}{33}$$
Question 15.
$$\frac{5}{9}, \frac{4}{15}$$
Type below:
_________
Answer: $$\frac{25}{45}, \frac{12}{45}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator of $$\frac{5}{9}, \frac{4}{15}$$= 45
Now rewrite the input fractions
$$\frac{5}{9}$$ × $$\frac{5}{5}$$ = $$\frac{25}{45}$$
$$\frac{4}{15}$$ × $$\frac{3}{3}$$ = $$\frac{12}{45}$$
Question 16.
$$\frac{1}{6}, \frac{4}{21}$$
Type below:
_________
Answer: $$\frac{7}{42}, \frac{8}{42}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 42
Now rewrite the fractions
$$\frac{1}{6}$$ × $$\frac{7}{7}$$ = $$\frac{7}{42}$$
$$\frac{4}{21}$$ × $$\frac{2}{2}$$ = $$\frac{8}{42}$$
Question 17.
$$\frac{5}{14}, \frac{8}{42}$$
Type below:
_________
Answer: $$\frac{15}{42}, \frac{8}{42}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 42
Now rewrite the fractions
$$\frac{5}{14}$$ × $$\frac{3}{3}$$ = $$\frac{15}{42}$$
$$\frac{8}{42}$$ × $$\frac{1}{1}$$ = $$\frac{8}{42}$$
Question 18.
$$\frac{7}{12}, \frac{5}{18}$$
Type below:
_________
Answer: $$\frac{21}{36}, \frac{10}{36}$$
Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 36
Now rewrite the fractions
$$\frac{7}{12}$$ × $$\frac{3}{3}$$ = $$\frac{21}{36}$$
$$\frac{5}{18}$$ × $$\frac{2}{2}$$ = $$\frac{10}{36}$$
Algebra Write the unknown number for each ■.
Question 19.
$$\frac{1}{5}, \frac{1}{8}$$
least common denominator: ■
■ = ______
Explanation:
Multiply the denominators of the fractions.
5 × 8 = 40
Therefore, ■ = 40
Question 20.
$$\frac{2}{5}, \frac{1}{■}$$
least common denominator: 15
■ = ______
Explanation:
Multiply the denominators of the fractions.
5 × ■ = 15
■ = 15/5 = 3
Thus ■ = 3
Question 21.
$$\frac{3}{■}, \frac{5}{6}$$
least common denominator: 42
■ = ______
Explanation:
$$\frac{3}{■}, \frac{5}{6}$$
■ × 6 = 42
■ = 42/6
■ = 7
### UNLOCK the Problem – Page No. 258
Question 22.
Katie made two pies for the bake sale. One was cut into three equal slices and the other into 5 equal slices. She will continue to cut the pies so each one has the same number of equal-sized slices. What is the least number of equal-sized slices each pie could have?
a. What information are you given?
Type below:
_________
Answer: I have the information about the two pies for the bake sale. One was cut into three equal slices and the other into 5 equal slices. She will continue to cut the pies so each one has the same number of equal-sized slices.
Question 22.
b. What problem are you being asked to solve?
Type below:
_________
Answer: I am asked to solve the least number of equal-sized slices each pie could have.
Question 22.
c. When Katie cuts the pies more, can she cut each pie the same number of times and have all the slices the same size? Explain.
Type below:
_________
Answer: Yes she can cut into more equal pieces. Katie can cut the pie into 6 equal pieces and 10 equal pieces. But the least number of equal-sized slices each pie could have is 3 and 5.
Question 22.
d. Use the diagram to show the steps you use to solve the problem.
Type below:
_________
There are 2 pies. One pie is cut into 3 equal pieces and the second pie is cut into 5 equal pieces.
So, there are 15 pieces of pies.
Question 22.
e. Complete the sentences.
The least common denominator of $$\frac{1}{3}$$ and $$\frac{1}{5}$$ is ____.
Katie can cut each piece of the first pie into ____ and each piece of the second pie into ____ .
That means that Katie can cut each pie into pieces that are ____ of the whole pie.
Type below:
_________
The least common denominator of $$\frac{1}{3}$$ and $$\frac{1}{5}$$ is 15
5 × 3 = 15
Katie can cut each piece of the first pie into three and each piece of the second pie into five.
That means that Katie can cut each pie into pieces that are 15 of the whole pie.
Question 23.
A cookie recipe calls for $$\frac{1}{3}$$ cup of brown sugar and $$\frac{1}{8}$$ cup of walnuts. Find the least common denominator of the fractions used in the recipe.
____
Explanation:
A cookie recipe calls for $$\frac{1}{3}$$ cup of brown sugar and $$\frac{1}{8}$$ cup of walnuts.
We can calculate the LCD by multiplying the denominators of the fraction.
3 × 8 = 24.
Question 24.
Test Prep Which fractions use the least common denominator and are equivalent to $$\frac{5}{8}$$ and $$\frac{7}{10}$$ ?
Options:
a. $$\frac{10}{40} \text { and } \frac{14}{40}$$
b. $$\frac{25}{40} \text { and } \frac{28}{40}$$
c. $$\frac{25}{80} \text { and } \frac{21}{80}$$
d. $$\frac{50}{80} \text { and } \frac{56}{80}$$
Answer: $$\frac{50}{80} \text { and } \frac{56}{80}$$
Explanation:
The least common denominator of $$\frac{5}{8}$$ and $$\frac{7}{10}$$ is 80.
$$\frac{5}{8}$$ × $$\frac{10}{10}$$ and $$\frac{7}{10}$$ × $$\frac{8}{8}$$
= $$\frac{50}{80} \text { and } \frac{56}{80}$$
Thus the correct answer is option D.
### Share and Show – Page No. 260
Question 1.
$$\frac{5}{12}+\frac{1}{3}$$
$$\frac{□}{□}$$
Find a common denominator by multiplying the denominators.
$$\frac{5}{12}+\frac{1}{3}$$
$$\frac{5}{12}$$ + $$\frac{1}{3}$$ × $$\frac{4}{4}$$
$$\frac{5}{12}$$ + $$\frac{4}{12}$$
$$\frac{9}{12}$$
Question 2.
$$\frac{2}{5}+\frac{3}{7}$$
$$\frac{□}{□}$$
Find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
$$\frac{2}{5}+\frac{3}{7}$$
$$\frac{2}{5}$$ × $$\frac{7}{7}$$ + $$\frac{3}{7}$$ × $$\frac{5}{5}$$
$$\frac{14}{35}+\frac{15}{35}$$
= $$\frac{29}{35}$$
$$\frac{2}{5}+\frac{3}{7}$$ = $$\frac{29}{35}$$
Question 3.
$$\frac{1}{6}+\frac{3}{4}$$
$$\frac{□}{□}$$
Find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
$$\frac{1}{6}$$ × $$\frac{2}{2}$$ + $$\frac{3}{4}$$ × $$\frac{3}{3}$$
$$\frac{2}{12}+\frac{9}{12}$$ = $$\frac{11}{12}$$
So, $$\frac{1}{6}+\frac{3}{4}$$ = $$\frac{11}{12}$$
Question 4.
$$\frac{3}{4}-\frac{1}{8}$$
$$\frac{□}{□}$$
First, find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
$$\frac{3}{4}-\frac{1}{8}$$
$$\frac{3}{4}$$ × $$\frac{2}{2}$$ – $$\frac{1}{8}$$
$$\frac{6}{8}$$ – $$\frac{1}{8}$$ = $$\frac{5}{8}$$
Thus $$\frac{3}{4}-\frac{1}{8}$$ = $$\frac{5}{8}$$
Question 5.
$$\frac{1}{4}-\frac{1}{7}$$
$$\frac{□}{□}$$
First, find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
$$\frac{1}{4}-\frac{1}{7}$$
$$\frac{1}{4}$$ × $$\frac{7}{7}$$ – $$\frac{1}{7}$$ × $$\frac{4}{4}$$
$$\frac{7}{28}$$ – $$\frac{4}{28}$$ = $$\frac{3}{28}$$
$$\frac{1}{4}-\frac{1}{7}$$ = $$\frac{3}{28}$$
Question 6.
$$\frac{9}{10}-\frac{1}{4}$$
$$\frac{□}{□}$$
First, find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
$$\frac{9}{10}-\frac{1}{4}$$
$$\frac{9}{10}$$ × $$\frac{4}{4}$$ – $$\frac{1}{4}$$ × $$\frac{10}{10}$$
$$\frac{36}{40}$$ – $$\frac{10}{40}$$ = $$\frac{26}{40}$$
$$\frac{9}{10}-\frac{1}{4}$$ = $$\frac{26}{40}$$
### On Your Own – Page No. 261
Question 7.
$$\frac{3}{8}+\frac{1}{4}$$
$$\frac{□}{□}$$
Answer: $$\frac{5}{8}$$
Explanation:
$$\frac{3}{8}+\frac{1}{4}$$ = $$\frac{3}{8}$$ + $$\frac{1}{4}$$
LCD = 8
$$\frac{3}{8}$$ + $$\frac{1}{4}$$ × $$\frac{2}{2}$$
$$\frac{3}{8}$$ + $$\frac{2}{8}$$ = $$\frac{5}{8}$$
Thus $$\frac{3}{8}+\frac{1}{4}$$ = $$\frac{5}{8}$$
Question 8.
$$\frac{7}{8}+\frac{1}{10}$$
$$\frac{□}{□}$$
$$\frac{7}{8}+\frac{1}{10}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 40
$$\frac{7}{8}$$ × $$\frac{5}{5}$$ + $$\frac{1}{10}$$ × $$\frac{4}{4}$$
$$\frac{35}{40}$$ + $$\frac{4}{40}$$ = $$\frac{39}{40}$$
$$\frac{7}{8}+\frac{1}{10}$$ = $$\frac{39}{40}$$
Question 9.
$$\frac{2}{7}+\frac{3}{10}$$
$$\frac{□}{□}$$
$$\frac{2}{7}+\frac{3}{10}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 70
$$\frac{2}{7}$$ × $$\frac{10}{10}$$ + $$\frac{3}{10}$$ × $$\frac{7}{7}$$
$$\frac{20}{70}$$ + $$\frac{21}{70}$$ = $$\frac{41}{70}$$
$$\frac{2}{7}+\frac{3}{10}$$ = $$\frac{41}{70}$$
Question 10.
$$\frac{5}{6}+\frac{1}{8}$$
$$\frac{□}{□}$$
$$\frac{5}{6}+\frac{1}{8}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
$$\frac{5}{6}$$ + $$\frac{1}{8}$$
LCD = 24
$$\frac{5}{6}$$ × $$\frac{4}{4}$$ + $$\frac{1}{8}$$ × $$\frac{3}{3}$$
$$\frac{20}{24}$$ + $$\frac{3}{24}$$ = $$\frac{23}{24}$$
$$\frac{5}{6}+\frac{1}{8}$$ = $$\frac{23}{24}$$
Question 11.
$$\frac{5}{12}+\frac{5}{18}$$
$$\frac{□}{□}$$
$$\frac{5}{12}+\frac{5}{18}$$ = $$\frac{5}{12}$$ + $$\frac{5}{18}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 36
$$\frac{5}{12}$$ × $$\frac{3}{3}$$ + $$\frac{5}{18}$$ × $$\frac{2}{2}$$
$$\frac{15}{36}$$ + $$\frac{10}{36}$$ = $$\frac{25}{36}$$
$$\frac{5}{12}+\frac{5}{18}$$ = $$\frac{25}{36}$$
Question 12.
$$\frac{7}{16}+\frac{1}{4}$$
$$\frac{□}{□}$$
$$\frac{7}{16}+\frac{1}{4}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 16
$$\frac{7}{16}$$ + $$\frac{1}{4}$$ = $$\frac{7}{16}$$ + $$\frac{1}{4}$$ × $$\frac{4}{4}$$
$$\frac{7}{16}$$ + $$\frac{4}{16}$$ = $$\frac{11}{16}$$
Question 13.
$$\frac{5}{6}+\frac{3}{8}$$
$$\frac{□}{□}$$
$$\frac{5}{6}+\frac{3}{8}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
$$\frac{5}{6}$$ + $$\frac{3}{8}$$
LCD = 24
$$\frac{5}{6}$$ × $$\frac{4}{4}$$ + $$\frac{3}{8}$$ × $$\frac{3}{3}$$
= $$\frac{20}{24}$$ + $$\frac{9}{24}$$ = $$\frac{29}{24}$$
$$\frac{5}{6}+\frac{3}{8}$$ = $$\frac{29}{24}$$
Question 14.
$$\frac{3}{4}+\frac{1}{2}$$
$$\frac{□}{□}$$
$$\frac{3}{4}+\frac{1}{2}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
$$\frac{3}{4}$$ + $$\frac{1}{2}$$
LCD = 4
$$\frac{3}{4}$$ + $$\frac{1}{2}$$ × $$\frac{2}{2}$$
= $$\frac{3}{4}$$ + $$\frac{2}{4}$$ = $$\frac{5}{4}$$
The miced fractiion of $$\frac{5}{4}$$ is 1 $$\frac{1}{4}$$
Question 15.
$$\frac{5}{12}+\frac{1}{4}$$
$$\frac{□}{□}$$
$$\frac{5}{12}+\frac{1}{4}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
$$\frac{5}{12}$$ + $$\frac{1}{4}$$
LCD = 12
$$\frac{5}{12}$$ + $$\frac{1}{4}$$ × $$\frac{3}{3}$$
$$\frac{5}{12}$$ + $$\frac{3}{12}$$ = $$\frac{8}{12}$$ = $$\frac{2}{3}$$
Practice: Copy and Solve Find the sum or difference. Write your answer in simplest form.
Question 16.
$$\frac{1}{3}+\frac{4}{18}$$
$$\frac{□}{□}$$
$$\frac{1}{3}+\frac{4}{18}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 18
$$\frac{1}{3}$$ + $$\frac{4}{18}$$
$$\frac{1}{3}$$ × $$\frac{6}{6}$$ + $$\frac{4}{18}$$
$$\frac{6}{18}$$ + $$\frac{4}{18}$$ = $$\frac{10}{18}$$ = $$\frac{5}{9}$$
$$\frac{1}{3}+\frac{4}{18}$$ = $$\frac{5}{9}$$
Question 17.
$$\frac{3}{5}+\frac{1}{3}$$
$$\frac{□}{□}$$
$$\frac{3}{5}+\frac{1}{3}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 15
$$\frac{3}{5}$$ + $$\frac{1}{3}$$
$$\frac{3}{5}$$ × $$\frac{3}{3}$$ + $$\frac{1}{3}$$ × $$\frac{5}{5}$$
$$\frac{9}{15}$$ + $$\frac{5}{15}$$ = $$\frac{14}{15}$$
$$\frac{3}{5}+\frac{1}{3}$$ = $$\frac{14}{15}$$
Question 18.
$$\frac{3}{10}+\frac{1}{6}$$
$$\frac{□}{□}$$
$$\frac{3}{10}+\frac{1}{6}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 30
$$\frac{3}{10}$$ + $$\frac{1}{6}$$
$$\frac{3}{10}$$ × $$\frac{3}{3}$$ + $$\frac{1}{6}$$ × $$\frac{5}{5}$$
$$\frac{9}{30}$$ + $$\frac{5}{30}$$ = $$\frac{14}{30}$$
$$\frac{3}{10}+\frac{1}{6}$$ = $$\frac{14}{30}$$
Question 19.
$$\frac{1}{2}+\frac{4}{9}$$
$$\frac{□}{□}$$
$$\frac{1}{2}+\frac{4}{9}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 18
$$\frac{1}{2}$$ + $$\frac{4}{9}$$
$$\frac{1}{2}$$ × $$\frac{9}{9}$$ + $$\frac{4}{9}$$ × $$\frac{2}{2}$$
= $$\frac{9}{18}$$ + $$\frac{8}{18}$$ = $$\frac{17}{18}$$
$$\frac{1}{2}+\frac{4}{9}$$ = $$\frac{17}{18}$$
Question 20.
$$\frac{1}{2}-\frac{3}{8}$$
$$\frac{□}{□}$$
$$\frac{1}{2}-\frac{3}{8}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 8
$$\frac{1}{2}$$ – $$\frac{3}{8}$$
$$\frac{1}{2}$$ × $$\frac{4}{4}$$ – $$\frac{3}{8}$$
$$\frac{4}{8}$$ – $$\frac{3}{8}$$ = $$\frac{1}{8}$$
$$\frac{1}{2}-\frac{3}{8}$$ = $$\frac{1}{8}$$
Question 21.
$$\frac{5}{7}-\frac{2}{3}$$
$$\frac{□}{□}$$
$$\frac{5}{7}-\frac{2}{3}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 21
$$\frac{5}{7}$$ – $$\frac{2}{3}$$
$$\frac{5}{7}$$ × $$\frac{3}{3}$$ – $$\frac{2}{3}$$ × $$\frac{7}{7}$$
$$\frac{15}{21}$$ – $$\frac{14}{21}$$ = $$\frac{1}{21}$$
$$\frac{5}{7}-\frac{2}{3}$$ = $$\frac{1}{21}$$
Question 22.
$$\frac{4}{9}-\frac{1}{6}$$
$$\frac{□}{□}$$
$$\frac{4}{9}-\frac{1}{6}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 18
$$\frac{4}{9}$$ – $$\frac{1}{6}$$
$$\frac{4}{9}$$ × $$\frac{2}{2}$$ – $$\frac{1}{6}$$ × $$\frac{3}{3}$$
$$\frac{8}{18}$$ – $$\frac{3}{18}$$ = $$\frac{5}{18}$$
$$\frac{4}{9}-\frac{1}{6}$$ = $$\frac{5}{18}$$
Question 23.
$$\frac{11}{12}-\frac{7}{15}$$
$$\frac{□}{□}$$
$$\frac{11}{12}-\frac{7}{15}$$
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 60
$$\frac{11}{12}$$ – $$\frac{7}{15}$$
$$\frac{11}{12}$$ × $$\frac{5}{5}$$ – $$\frac{7}{15}$$ × $$\frac{4}{4}$$
$$\frac{55}{60}$$ – $$\frac{28}{60}$$ = $$\frac{27}{60}$$
$$\frac{11}{12}-\frac{7}{15}$$ = $$\frac{27}{60}$$ = $$\frac{9}{20}$$
Algebra Find the unknown number.
Question 24.
$$\frac{9}{10}$$ − ■ = $$\frac{1}{5}$$
■ = $$\frac{□}{□}$$
$$\frac{9}{10}$$ – $$\frac{1}{5}$$ = ■
■ = $$\frac{9}{10}$$ – $$\frac{1}{5}$$
■ = $$\frac{9}{10}$$ – $$\frac{2}{10}$$ = $$\frac{7}{10}$$
■ = $$\frac{7}{10}$$
Question 25.
$$\frac{5}{12}$$ + ■ = $$\frac{1}{2}$$
■ = $$\frac{□}{□}$$
$$\frac{5}{12}$$ + ■ = $$\frac{1}{2}$$
$$\frac{5}{12}$$ − $$\frac{1}{2}$$ = – ■
– ■ = $$\frac{5}{12}$$ − $$\frac{1}{2}$$
– ■ = $$\frac{5}{12}$$ − $$\frac{1}{2}$$ × $$\frac{6}{6}$$
– ■ = $$\frac{5}{12}$$ − $$\frac{6}{12}$$ = – $$\frac{1}{12}$$
■ = $$\frac{1}{12}$$
### Problem Solving – Page No. 262
Use the picture for 26–27.
Question 26.
Sara is making a key chain using the bead design shown. What fraction of the beads in her design are either blue or red?
$$\frac{□}{□}$$
Answer: $$\frac{11}{15}$$
Explanation:
Total number of red beads = 6
Total number of blue beads = 5
Total number of beads = 6 + 5 = 11
The fraction of beads = $$\frac{11}{15}$$
Question 27.
In making the key chain, Sara uses the pattern of beads 3 times. After the key chain is complete, what fraction of the beads in the key chain are either white or blue?
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{4}{5}$$
Explanation:
In making the key chain, Sara uses the pattern of beads 3 times.
Given that Sara uses the pattern of beads 3 times.
Total number of blue beads = 5
5 × 3 = 15
Number of white beads = 4
4 × 3 = 12
15 + 12 = 27
Actual number of beads = 15
So, the fraction is $$\frac{27}{15}$$ = $$\frac{9}{5}$$
The mixed fraction of $$\frac{9}{5}$$ is 1 $$\frac{4}{5}$$
Question 28.
Jamie had $$\frac{4}{5}$$ of a spool of twine. He then used $$\frac{1}{2}$$ of a spool of twine to make friendship knots. He claims to have $$\frac{3}{10}$$ of the original spool of twine left over. Explain how you know whether Jamie’s claim is reasonable.
Type below:
_________
Explanation:
Jamie had $$\frac{4}{5}$$ of a spool of twine. He then used $$\frac{1}{2}$$ of a spool of twine to make friendship knots. He claims to have $$\frac{3}{10}$$ of the original spool of twine left over.
To know whether his estimation is reasonable or not we have to subtract the total spool of twine from used spool of twine.
$$\frac{4}{5}$$ – $$\frac{1}{2}$$
LCD = 10
$$\frac{4}{5}$$ × $$\frac{2}{2}$$ – $$\frac{1}{2}$$ × $$\frac{5}{5}$$
$$\frac{8}{10}$$ – $$\frac{5}{10}$$ = $$\frac{3}{10}$$
By this is can that Jamie’s claim is reasonable.
Question 29.
Test Prep Which equation represents the fraction of beads that are green or yellow?
Options:
a. $$\frac{1}{4}+\frac{1}{8}=\frac{3}{8}$$
b. [atex]\frac{1}{2}+\frac{1}{4}=\frac{3}{4}[/latex]
c. $$\frac{1}{2}+\frac{1}{8}=\frac{5}{8}$$
d. $$\frac{3}{4}+\frac{2}{8}=1$$
Explanation:
Number of green beads = 4 = [atex]\frac{1}{2}[/latex]
Number of blue beads = 3 = [atex]\frac{3}{4}[/latex]
Number of yellow beads = 1 [atex]\frac{1}{4}[/latex]
The fraction of beads that are green or yellow is [atex]\frac{1}{2}+\frac{1}{4}=\frac{3}{4}[/latex]
The correct answer is option B.
### Mid-Chapter Checkpoint – Vocabulary – Page No. 263
Choose the best term from the box.
Question 1.
A ________ is a number that is a multiple of two or more numbers.
________
A Common Multiple is a number that is a multiple of two or more numbers.
Question 2.
A ________ is a common multiple of two or more denominators.
________
A Common denominator is a common multiple of two or more denominators.
Concepts and Skills
Estimate the sum or difference.
Question 3.
$$\frac{8}{9}+\frac{4}{7}$$
about ______ $$\frac{□}{□}$$
Answer: 1 $$\frac{1}{2}$$
Place $$\frac{8}{9}$$ on the number line.
$$\frac{8}{9}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{8}{9}$$ is closer to 1.
Place $$\frac{4}{7}$$ on the number line.
$$\frac{4}{7}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{4}{7}$$ is closer to $$\frac{1}{2}$$.
1 + $$\frac{1}{2}$$ = 1 $$\frac{1}{2}$$
Question 4.
$$3 \frac{2}{5}-\frac{5}{8}$$
Explanation:
Place $$\frac{2}{5}$$ on the number line.
$$\frac{2}{5}$$ lies between 0 and $$\frac{1}{2}$$
$$\frac{2}{5}$$ is closer to $$\frac{1}{2}$$
Place $$\frac{5}{8}$$ on the number line.
$$\frac{5}{8}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{5}{8}$$ is closer to $$\frac{1}{2}$$
3 + $$\frac{1}{2}$$ – $$\frac{1}{2}$$ = 3
$$3 \frac{2}{5}-\frac{5}{8}$$ = 3
Question 5.
$$1 \frac{5}{6}+2 \frac{2}{11}$$
Explanation:
Place $$\frac{5}{6}$$ on the number line.
$$\frac{5}{6}$$ lies between $$\frac{1}{2}$$ and 1.
$$\frac{5}{6}$$ is closer to 1.
Place $$\frac{2}{11}$$ on the number line.
$$\frac{2}{11}$$ lies between $$\frac{1}{2}$$ and 0.
$$\frac{2}{11}$$ is closer to 0
1 + 1 + 2 + 0 = 4
$$1 \frac{5}{6}+2 \frac{2}{11}$$ = 4
Use a common denominator to write an equivalent fraction for each fraction.
Question 6.
$$\frac{1}{6}, \frac{1}{9}$$
common denominator:
Type below:
__________
Multiply the denominators
6 × 9 = 54
Thus the common denominator of $$\frac{1}{6}, \frac{1}{9}$$ is 54
Question 7.
$$\frac{3}{8}, \frac{3}{10}$$
common denominator:
Type below:
__________
Multiply the denominators
8 × 10 = 80
The common denominator of $$\frac{3}{8}, \frac{3}{10}$$ is 80
Question 8.
$$\frac{1}{9}, \frac{5}{12}$$
common denominator:
Type below:
__________
Multiply the denominators
9 × 12 = 108
The common denominator of $$\frac{1}{9}, \frac{5}{12}$$ is 108
Use the least common denominator to write an equivalent fraction for each fraction.
Question 9.
$$\frac{2}{5}, \frac{1}{10}$$
least common denominator: ______
Explain:
__________
Explanation:
Multiply the denominators
5 × 10 = 50
The least common denominators of $$\frac{2}{5}, \frac{1}{10}$$ is 10.
Question 10.
$$\frac{5}{6}, \frac{3}{8}$$
least common denominator: ______
Explain:
__________
Explanation:
Multiply the denominators
The least common denominator of 6 and 8 is 24
Thus the LCD of $$\frac{5}{6}, \frac{3}{8}$$ is 24
Question 11.
$$\frac{1}{3}, \frac{2}{7}$$
least common denominator: ______
Explain:
__________
Explanation:
Multiply the denominators
The least common denominator of 3 and 7 is 21.
Thus the LCD of $$\frac{1}{3}, \frac{2}{7}$$ is 21.
Question 12.
$$\frac{11}{18}-\frac{1}{6}$$
$$\frac{□}{□}$$
Answer: $$\frac{8}{18}$$
Explanation:
Make the fractions like denominators.
$$\frac{11}{18}$$ – $$\frac{1}{6}$$
$$\frac{1}{6}$$ × $$\frac{3}{3}$$ = $$\frac{3}{18}$$
$$\frac{11}{18}$$ – $$\frac{3}{18}$$ = $$\frac{8}{18}$$
Question 13.
$$\frac{2}{7}+\frac{2}{5}$$
$$\frac{□}{□}$$
Answer: $$\frac{24}{35}$$
Explanation:
Make the fractions like denominators.
$$\frac{2}{7}$$ × $$\frac{5}{5}$$ = $$\frac{10}{35}$$
$$\frac{2}{5}$$ × $$\frac{7}{7}$$ = $$\frac{14}{35}$$
$$\frac{10}{35}$$ + $$\frac{14}{35}$$ = $$\frac{24}{35}$$
Thus $$\frac{2}{7}+\frac{2}{5}$$ = $$\frac{24}{35}$$
Question 14.
$$\frac{3}{4}-\frac{3}{10}$$
$$\frac{□}{□}$$
Answer: $$\frac{18}{40}$$
Explanation:
Make the fractions like denominators.
$$\frac{3}{4}$$ × $$\frac{10}{10}$$ = $$\frac{30}{40}$$
$$\frac{3}{10}$$ × $$\frac{4}{4}$$ = $$\frac{12}{40}$$
$$\frac{30}{40}$$ – $$\frac{12}{40}$$ = $$\frac{18}{40}$$
### Mid-Chapter Checkpoint – Page No. 264
Question 15.
Mrs. Vargas bakes a pie for her book club meeting. The shaded part of the diagram below shows the amount of pie left after the meeting. That evening, Mr. Vargas eats $$\frac{1}{4}$$ of the whole pie. What fraction represents the amount of pie remaining?
$$\frac{□}{□}$$
Answer: $$\frac{1}{4}$$
Explanation:
Mrs. Vargas bakes a pie for her book club meeting. The shaded part of the diagram below shows the amount of pie left after the meeting.
So, the fraction of the pie is $$\frac{1}{2}$$
That evening, Mr. Vargas eats $$\frac{1}{4}$$ of the whole pie.
$$\frac{1}{2}$$ – $$\frac{1}{4}$$ = $$\frac{1}{4}$$
Thus the fraction represents the amount of pie remaining is $$\frac{1}{4}$$
Question 16.
Keisha makes a large sandwich for a family picnic. She takes $$\frac{1}{2}$$ of the sandwich to the picnic. At the picnic, her family eats $$\frac{3}{8}$$ of the whole sandwich. What fraction of the whole sandwich does Keisha bring back from the picnic?
$$\frac{□}{□}$$
Answer: $$\frac{1}{8}$$
Explanation:
Keisha makes a large sandwich for a family picnic. She takes $$\frac{1}{2}$$ of the sandwich to the picnic.
At the picnic, her family eats $$\frac{3}{8}$$ of the whole sandwich.
$$\frac{1}{2}$$ – $$\frac{3}{8}$$
$$\frac{1}{2}$$ × $$\frac{4}{4}$$ – $$\frac{3}{8}$$
$$\frac{4}{8}$$ – $$\frac{3}{8}$$ = $$\frac{1}{8}$$
Thus Keisha brought $$\frac{1}{8}$$ of the sandwich from the picnic.
Question 17.
Mike is mixing paint for his walls. He mixes $$\frac{1}{6}$$ gallon blue paint and $$\frac{5}{8}$$ gallon green paint in a large container. What fraction represents the total amount of paint Mike mixes?
$$\frac{□}{□}$$
Answer: $$\frac{19}{24}$$
Explanation:
Mike is mixing paint for his walls. He mixes $$\frac{1}{6}$$ gallon blue paint and $$\frac{5}{8}$$ gallon green paint in a large container.
$$\frac{1}{6}$$ + $$\frac{5}{8}$$
$$\frac{1}{6}$$ × $$\frac{8}{8}$$ + $$\frac{5}{8}$$ × $$\frac{6}{6}$$
$$\frac{8}{48}$$ + $$\frac{30}{48}$$
$$\frac{38}{48}$$ = $$\frac{19}{24}$$
Therefore the total amount of paint Mike mixes is $$\frac{19}{24}$$
### Share and Show – Page No. 266
Question 1.
Use a common denominator to write equivalent fractions with like denominators and then find the sum. Write your answer in simplest form.
7 $$\frac{2}{5}$$ = ■
+ 4 $$\frac{3}{4}$$ = + ■
—————————
_____ $$\frac{□}{□}$$
Answer: 12 $$\frac{3}{20}$$
Explanation:
First convert the mixed fraction to proper fraction.
7 $$\frac{2}{5}$$ = $$\frac{37}{5}$$
4 $$\frac{3}{4}$$ = $$\frac{19}{4}$$
$$\frac{37}{5}$$ + $$\frac{19}{4}$$
= $$\frac{37}{5}$$ × $$\frac{4}{4}$$ = $$\frac{148}{20}$$
$$\frac{19}{4}$$ × $$\frac{5}{5}$$ = $$\frac{95}{20}$$
$$\frac{148}{20}$$ + $$\frac{95}{20}$$ = $$\frac{243}{20}$$
Now convert it into mixed fraction = 12 $$\frac{3}{20}$$
Question 2.
$$2 \frac{3}{4}+3 \frac{3}{10}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{1}{20}$$
Explanation:
First convert the mixed fraction to proper fraction.
$$2 \frac{3}{4}$$ = $$\frac{11}{4}$$
3 $$\frac{3}{10}$$ = $$\frac{33}{10}$$
Now make the common denominators of the above fractions.
$$\frac{11}{4}$$ × $$\frac{10}{10}$$ = $$\frac{110}{40}$$
$$\frac{33}{10}$$ × $$\frac{4}{4}$$ = $$\frac{132}{40}$$ = $$\frac{121}{20}$$
Now convert the fraction into mixed fraction.
$$\frac{121}{20}$$ = 6 $$\frac{1}{20}$$
Question 3.
$$5 \frac{3}{4}+1 \frac{1}{3}$$
_____ $$\frac{□}{□}$$
Answer: 7 $$\frac{1}{12}$$
Explanation:
First convert the mixed fraction to proper fraction.
5 $$\frac{3}{4}$$ = $$\frac{23}{4}$$
1 $$\frac{1}{3}$$ = $$\frac{4}{3}$$
$$\frac{23}{4}$$ + $$\frac{4}{3}$$
$$\frac{23}{4}$$ × $$\frac{3}{3}$$ = $$\frac{69}{12}$$
$$\frac{4}{3}$$ × $$\frac{4}{4}$$ = $$\frac{16}{12}$$
$$\frac{69}{12}$$ + $$\frac{16}{12}$$ = $$\frac{85}{12}$$
The mixed fraction of $$\frac{85}{12}$$ = 7 $$\frac{1}{12}$$
Question 4.
$$3 \frac{4}{5}+2 \frac{3}{10}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{1}{10}$$
Explanation:
First convert the mixed fraction to proper fraction.
3 $$\frac{4}{5}$$ = $$\frac{19}{5}$$
2 $$\frac{3}{10}$$ = $$\frac{23}{10}$$
$$\frac{19}{5}$$ + $$\frac{23}{10}$$
Now make the common denominators of the above fractions.
$$\frac{19}{5}$$ × $$\frac{2}{2}$$ = $$\frac{38}{10}$$
$$\frac{38}{10}$$ + $$\frac{23}{10}$$ = $$\frac{61}{10}$$
The mixed fraction of $$\frac{61}{10}$$ = 6 $$\frac{1}{10}$$
### Page No. 267
Question 5.
$$9 \frac{5}{6}-2 \frac{1}{3}$$
_____ $$\frac{□}{□}$$
Answer: 7 $$\frac{1}{2}$$
Explanation:
$$9 \frac{5}{6}-2 \frac{1}{3}$$ = $$\frac{59}{6}$$ – $$\frac{14}{6}$$
= $$\frac{45}{6}$$ = $$\frac{15}{2}$$ = 7 $$\frac{1}{2}$$
Question 6.
$$10 \frac{5}{9}-9 \frac{1}{6}$$
_____ $$\frac{□}{□}$$
Answer: 1 $$\frac{7}{18}$$
Explanation:
$$10 \frac{5}{9}-9 \frac{1}{6}$$ = $$\frac{95}{9}$$ – $$\frac{55}{6}$$
= $$\frac{190}{18}$$ – $$\frac{165}{18}$$ = $$\frac{25}{18}$$
= 1 $$\frac{7}{18}$$
$$10 \frac{5}{9}-9 \frac{1}{6}$$ = 1 $$\frac{7}{18}$$
Question 7.
$$7 \frac{2}{3}-3 \frac{1}{6}$$
_____ $$\frac{□}{□}$$
Answer: 4 $$\frac{1}{2}$$
Explanation:
$$7 \frac{2}{3}-3 \frac{1}{6}$$
$$\frac{23}{3}$$ – $$\frac{19}{6}$$ = $$\frac{46}{6}$$ – $$\frac{19}{6}$$
= $$\frac{27}{6}$$ = 4 $$\frac{1}{2}$$
$$7 \frac{2}{3}-3 \frac{1}{6}$$ = 4 $$\frac{1}{2}$$
Question 8.
$$1 \frac{3}{10}+2 \frac{2}{5}$$
_____ $$\frac{□}{□}$$
Answer: 3 $$\frac{7}{10}$$
Explanation:
$$1 \frac{3}{10}+2 \frac{2}{5}$$
$$\frac{13}{10}$$ + $$\frac{12}{5}$$ = $$\frac{13}{10}$$ + $$\frac{24}{10}$$
= $$\frac{37}{10}$$ = 3 $$\frac{7}{10}$$
Thus $$1 \frac{3}{10}+2 \frac{2}{5}$$ = 3 $$\frac{7}{10}$$
Question 9.
$$3 \frac{4}{9}+3 \frac{1}{2}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{17}{18}$$
Explanation:
$$3 \frac{4}{9}+3 \frac{1}{2}$$
$$\frac{31}{9}$$ + $$\frac{7}{2}$$ = $$\frac{62}{18}$$ + $$\frac{63}{18}$$
$$\frac{125}{18}$$ = 6 $$\frac{17}{18}$$
$$3 \frac{4}{9}+3 \frac{1}{2}$$ = 6 $$\frac{17}{18}$$
Question 10.
$$2 \frac{1}{2}+2 \frac{1}{3}$$
_____ $$\frac{□}{□}$$
Answer: 4 $$\frac{5}{6}$$
Explanation:
$$2 \frac{1}{2}+2 \frac{1}{3}$$ = $$\frac{5}{2}$$ + $$\frac{7}{3}$$
$$\frac{15}{6}$$ + $$\frac{14}{6}$$= $$\frac{29}{6}$$
The mixed fraction of $$\frac{29}{6}$$ is 4 $$\frac{5}{6}$$
Question 11.
$$5 \frac{1}{4}+9 \frac{1}{3}$$
_____ $$\frac{□}{□}$$
Answer: 14 $$\frac{7}{12}$$
Explanation:
$$5 \frac{1}{4}+9 \frac{1}{3}$$ = $$\frac{21}{4}$$ + $$\frac{28}{3}$$
$$\frac{63}{12}$$ + $$\frac{112}{12}$$ = $$\frac{175}{12}$$
The mixed fraction of $$\frac{175}{12}$$ is 14 $$\frac{7}{12}$$
Question 12.
$$8 \frac{1}{6}+7 \frac{3}{8}$$
_____ $$\frac{□}{□}$$
Answer: 15 $$\frac{13}{24}$$
Explanation:
$$8 \frac{1}{6}+7 \frac{3}{8}$$ = $$\frac{49}{6}$$ + $$\frac{59}{8}$$
$$\frac{196}{24}$$ + $$\frac{177}{24}$$ = $$\frac{373}{24}$$
The mixed fraction of $$\frac{373}{24}$$ is 15 $$\frac{13}{24}$$
Question 13.
$$14 \frac{7}{12}-5 \frac{1}{4}$$
_____ $$\frac{□}{□}$$
Answer: 9 $$\frac{1}{3}$$
Explanation:
$$14 \frac{7}{12}-5 \frac{1}{4}$$ = $$\frac{175}{12}$$ – $$\frac{21}{4}$$
$$\frac{175}{12}$$ – $$\frac{63}{12}$$ = $$\frac{112}{12}$$
The mixed fraction of $$\frac{112}{12}$$ is 9 $$\frac{1}{3}$$
Question 14.
$$12 \frac{3}{4}-6 \frac{1}{6}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{7}{12}$$
Explanation:
$$12 \frac{3}{4}-6 \frac{1}{6}$$ = $$\frac{51}{4}$$ – $$\frac{37}{6}$$
$$\frac{153}{12}$$ – $$\frac{74}{12}$$ = $$\frac{79}{12}$$
The mixed fraction of $$\frac{79}{12}$$ is 6 $$\frac{7}{12}$$
Question 15.
$$2 \frac{5}{8}-1 \frac{1}{4}$$
_____ $$\frac{□}{□}$$
Answer: 1 $$\frac{3}{8}$$
Explanation:
$$2 \frac{5}{8}-1 \frac{1}{4}$$
$$\frac{21}{8}$$ – $$\frac{5}{4}$$ = $$\frac{21}{8}$$ – $$\frac{10}{8}$$
= $$\frac{11}{8}$$
The mixed fraction of $$\frac{11}{8}$$ is 1 $$\frac{3}{8}$$
Question 16.
$$10 \frac{1}{2}-2 \frac{1}{5}$$
_____ $$\frac{□}{□}$$
Answer: 8 $$\frac{3}{10}$$
Explanation:
$$10 \frac{1}{2}-2 \frac{1}{5}$$ = $$\frac{21}{2}$$ – $$\frac{11}{5}$$
$$\frac{105}{10}$$ – $$\frac{22}{10}$$ = $$\frac{83}{10}$$
The mixed fraction of $$\frac{83}{10}$$ is 8 $$\frac{3}{10}$$
Practice: Copy and Solve Find the sum or difference. Write your answer in simplest form.
Question 17.
$$1 \frac{5}{12}+4 \frac{1}{6}$$
_____ $$\frac{□}{□}$$
Answer: 5 $$\frac{7}{12}$$
Explanation:
$$1 \frac{5}{12}+4 \frac{1}{6}$$ = $$\frac{17}{12}$$ + $$\frac{25}{6}$$
$$\frac{17}{12}$$ + $$\frac{50}{12}$$ = $$\frac{67}{12}$$
The mixed fraction of $$\frac{67}{12}$$ is 5 $$\frac{7}{12}$$
Question 18.
$$8 \frac{1}{2}+6 \frac{3}{5}$$
_____ $$\frac{□}{□}$$
Answer: 15 $$\frac{1}{10}$$
Explanation:
$$8 \frac{1}{2}+6 \frac{3}{5}$$ = $$\frac{17}{2}$$ + $$\frac{33}{5}$$
$$\frac{85}{10}$$ + $$\frac{66}{10}$$ = $$\frac{151}{10}$$
The mixed fraction of $$\frac{151}{10}$$ is 15 $$\frac{1}{10}$$
$$8 \frac{1}{2}+6 \frac{3}{5}$$ = 15 $$\frac{1}{10}$$
Question 19.
$$2 \frac{1}{6}+4 \frac{5}{9}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{13}{18}$$
Explanation:
$$2 \frac{1}{6}+4 \frac{5}{9}$$ = $$\frac{13}{6}$$ + $$\frac{41}{9}$$
$$\frac{39}{18}$$ + $$\frac{82}{18}$$ = $$\frac{121}{18}$$
The mixed fraction of $$\frac{121}{18}$$ is 6 $$\frac{13}{18}$$
$$2 \frac{1}{6}+4 \frac{5}{9}$$ = 6 $$\frac{13}{18}$$
Question 20.
$$20 \frac{5}{8}+\frac{5}{12}$$
_____ $$\frac{□}{□}$$
Answer: 21 $$\frac{1}{24}$$
Explanation:
$$20 \frac{5}{8}+\frac{5}{12}$$ = $$\frac{165}{8}$$ + $$\frac{5}{12}$$
$$\frac{495}{24}$$ + $$\frac{10}{24}$$ = $$\frac{505}{24}$$
The mixed fraction of $$\frac{505}{24}$$ is 21 $$\frac{1}{24}$$
$$20 \frac{5}{8}+\frac{5}{12}$$ = 21 $$\frac{1}{24}$$
Question 21.
$$3 \frac{2}{3}-1 \frac{1}{6}$$
_____ $$\frac{□}{□}$$
Answer: 2 $$\frac{1}{2}$$
Explanation:
$$3 \frac{2}{3}-1 \frac{1}{6}$$ = $$\frac{11}{3}$$ – $$\frac{7}{6}$$
$$\frac{22}{6}$$ – $$\frac{7}{6}$$ = $$\frac{15}{6}$$ = $$\frac{5}{2}$$
The mixed fraction of $$\frac{5}{2}$$ is 2 $$\frac{1}{2}$$
$$3 \frac{2}{3}-1 \frac{1}{6}$$ = 2 $$\frac{1}{2}$$
Question 22.
$$5 \frac{6}{7}-1 \frac{2}{3}$$
_____ $$\frac{□}{□}$$
Answer: 4 $$\frac{4}{21}$$
Explanation:
$$5 \frac{6}{7}-1 \frac{2}{3}$$ = $$\frac{41}{7}$$ – $$\frac{5}{3}$$
$$\frac{123}{21}$$ – $$\frac{35}{21}$$ = $$\frac{88}{21}$$
The mixed fraction of $$\frac{88}{21}$$ is 4 $$\frac{4}{21}$$
Question 23.
$$2 \frac{7}{8}-\frac{1}{2}$$
_____ $$\frac{□}{□}$$
Answer: 2 $$\frac{3}{8}$$
Explanation:
$$2 \frac{7}{8}-\frac{1}{2}$$ = $$\frac{23}{8}$$ – $$\frac{1}{2}$$
= $$\frac{23}{8}$$ – $$\frac{4}{8}$$ = $$\frac{19}{8}$$
The mixed fraction of $$\frac{19}{8}$$ is 2 $$\frac{3}{8}$$
So, $$2 \frac{7}{8}-\frac{1}{2}$$ = 2 $$\frac{3}{8}$$
Question 24.
$$4 \frac{7}{12}-1 \frac{2}{9}$$
_____ $$\frac{□}{□}$$
Answer: 3 $$\frac{13}{36}$$
Explanation:
$$4 \frac{7}{12}-1 \frac{2}{9}$$ = $$\frac{55}{12}$$ – $$\frac{11}{9}$$
$$\frac{165}{36}$$ – $$\frac{44}{36}$$ = $$\frac{121}{36}$$
The mixed fraction of $$\frac{121}{36}$$ is 3 $$\frac{13}{36}$$
### Problem Solving – Page No. 268
Use the table to solve 25–28.
Question 25.
Gavin is mixing a batch of Sunrise Orange paint for an art project. How much paint does Gavin mix?
_____ $$\frac{□}{□}$$ ounces
Answer: 5 $$\frac{7}{8}$$ ounces
Explanation:
Gavin is mixing a batch of Sunrise Orange paint for an art project.
2 $$\frac{5}{8}$$ + 3 $$\frac{1}{4}$$
Solving the whole numbers
2 + 3 = 5
$$\frac{5}{8}$$ + $$\frac{1}{4}$$
LCD = 8
$$\frac{5}{8}$$ + $$\frac{2}{8}$$ = $$\frac{7}{8}$$
5 + $$\frac{7}{8}$$ = 5 $$\frac{7}{8}$$ ounces
Question 26.
Gavin plans to mix a batch of Tangerine paint. He expects to have a total of 5 $$\frac{3}{10}$$ ounces of paint after he mixes the amounts of red and yellow. Explain how you can tell if Gavin’s expectation is reasonable.
Type below:
_________
Gavin plans to mix a batch of Tangerine paint. He expects to have a total of 5 $$\frac{3}{10}$$ ounces of paint after he mixes the amounts of red and yellow.
To mix a batch of Tangerine paint he need 3 $$\frac{9}{10}$$ red and 2 $$\frac{3}{8}$$ yellow paint.
3 + $$\frac{9}{10}$$ + 2 + $$\frac{3}{8}$$
Solving the whole numbers
3 + 2 = 5
$$\frac{9}{10}$$ + $$\frac{3}{8}$$
LCD = 40
$$\frac{9}{10}$$ + $$\frac{3}{8}$$ = $$\frac{36}{40}$$ + $$\frac{15}{40}$$ = $$\frac{51}{40}$$ = 1 $$\frac{11}{40}$$
5 + 1 $$\frac{11}{40}$$ = 6 $$\frac{11}{40}$$
Question 27.
For a special project, Gavin mixes the amount of red from one shade of paint with the amount of yellow from a different shade. He mixes the batch so he will have the greatest possible amount of paint. What amounts of red and yellow from which shades are used in the mixture for the special project? Explain your answer.
Type below:
_________
Gavin used red paint from mango and yellow paint from Sunrise Orange.
5 $$\frac{5}{6}$$ + 3 $$\frac{1}{4}$$
Solving the whole numbers parts
5 + 3 = 8
Solving the fraction part
$$\frac{5}{6}$$ + $$\frac{1}{4}$$
LCD = 12
$$\frac{10}{12}$$ + $$\frac{3}{12}$$ = $$\frac{13}{12}$$
$$\frac{13}{12}$$ = 1 $$\frac{1}{12}$$
Question 28.
Gavin needs to make 2 batches of Mango paint. Explain how you could find the total amount of paint Gavin mixed.
Type below:
_________
Gavin used Red paint and Yellow Paint to make Mango shade.
For one batch he need to add 5 $$\frac{5}{6}$$ + 5 $$\frac{5}{6}$$
Foe 2 batches
5 $$\frac{5}{6}$$+ 5 $$\frac{5}{6}$$ + 5 $$\frac{5}{6}$$ + 5 $$\frac{5}{6}$$
Solving the whole numbers
5 + 5 + 5 + 5 = 20
Solving the fractions part
$$\frac{5}{6}$$ + $$\frac{5}{6}$$ + $$\frac{5}{6}$$ + $$\frac{5}{6}$$ = $$\frac{20}{6}$$
= $$\frac{10}{3}$$
Gavin mixed $$\frac{10}{3}$$ of paint to make 2 batches of Mango Paint.
Question 29.
Test Prep Yolanda walked 3 $$\frac{6}{10}$$ miles. Then she walked 4 $$\frac{1}{2}$$ more miles. How many miles did Yolanda walk?
Options:
a. 7 $$\frac{1}{10}$$ miles
b. 7 $$\frac{7}{10}$$ miles
c. 8 $$\frac{1}{10}$$ miles
d. 8 $$\frac{7}{10}$$ miles
Answer: 8 $$\frac{1}{10}$$ miles
Explanation:
Test Prep Yolanda walked 3 $$\frac{6}{10}$$ miles.
Then she walked 4 $$\frac{1}{2}$$ more miles.
3 $$\frac{6}{10}$$ + 4 $$\frac{1}{2}$$ = 3 + $$\frac{6}{10}$$ + 4 + $$\frac{1}{2}$$
3 + 4 = 7
$$\frac{6}{10}$$ + $$\frac{1}{2}$$
LCD = 10
$$\frac{6}{10}$$ + $$\frac{5}{10}$$ = $$\frac{11}{10}$$
$$\frac{11}{10}$$ = 8 $$\frac{1}{10}$$ miles
Thus the correct answer is option C.
### Share and Show – Page No. 270
Estimate. Then find the difference and write it in simplest form.
Question 1.
Estimate: ______
1 $$\frac{3}{4}-\frac{7}{8}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 1
Difference: $$\frac{7}{8}$$
Explanation:
Estimation: 1 + $$\frac{3}{4}$$ – $$\frac{7}{8}$$
$$\frac{7}{8}$$ is close to 1.
$$\frac{3}{4}$$ is close to 1.
1 + 1 – 1 = 1
Difference: 1 $$\frac{3}{4}-\frac{7}{8}$$
1 + $$\frac{3}{4}$$ – $$\frac{7}{8}$$
$$\frac{3}{4}$$ – $$\frac{7}{8}$$
$$\frac{3}{4}$$ × $$\frac{8}{8}$$ – $$\frac{7}{8}$$ × $$\frac{4}{4}$$
$$\frac{24}{32}$$ – $$\frac{28}{32}$$ = – $$\frac{1}{8}$$
1 – $$\frac{1}{8}$$ = $$\frac{7}{8}$$
Question 2.
Estimate: ______
$$12 \frac{1}{9}-7 \frac{1}{3}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 5
Difference: 4 $$\frac{7}{9}$$
Explanation:
Estimate: 12 + 0 – 7 – 0 = 5
Difference:
12 + $$\frac{1}{9}$$ – 7 – $$\frac{1}{3}$$
12 – 7 = 5
$$\frac{1}{9}$$ – $$\frac{1}{3}$$ = $$\frac{1}{9}$$ – $$\frac{3}{9}$$ = – $$\frac{2}{9}$$
5 – $$\frac{2}{9}$$ = 4 $$\frac{7}{9}$$
### Page No. 271
Estimate. Then find the difference and write it in simplest form.
Question 3.
Estimate: ________
$$4 \frac{1}{2}-3 \frac{4}{5}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: $$\frac{1}{2}$$
Difference: $$\frac{7}{10}$$
Explanation:
$$4 \frac{1}{2}-3 \frac{4}{5}$$
4 – $$\frac{1}{2}$$ – 3 – 1
= $$\frac{1}{2}$$
Difference:
$$4 \frac{1}{2}-3 \frac{4}{5}$$
4 $$\frac{1}{2}$$ – 3 $$\frac{4}{5}$$
Solving the whole number parts
4 – 3 = 1
Solving the fraction parts
$$\frac{1}{2}$$ – $$\frac{4}{5}$$
LCD = 10
$$\frac{5}{10}$$ – $$\frac{8}{10}$$ = – $$\frac{3}{10}$$
1 – $$\frac{3}{10}$$ = $$\frac{7}{10}$$
Question 4.
Estimate: ________
$$9 \frac{1}{6}-2 \frac{3}{4}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 6
Difference: 6 $$\frac{5}{12}$$
Explanation:
$$9 \frac{1}{6}-2 \frac{3}{4}$$
9 + 0 – 2 – 1 = 6
Difference:
$$9 \frac{1}{6}-2 \frac{3}{4}$$
9 + $$\frac{1}{6}$$ – 2 – $$\frac{3}{4}$$
9 – 2 = 7
$$\frac{1}{6}$$ – $$\frac{3}{4}$$
LCD = 12
$$\frac{2}{12}$$ – $$\frac{9}{12}$$ = – $$\frac{7}{12}$$
7 – $$\frac{7}{12}$$ = 6 $$\frac{5}{12}$$
$$9 \frac{1}{6}-2 \frac{3}{4}$$ = 6 $$\frac{5}{12}$$
Estimate. Then find the difference and write it in simplest form.
Question 5.
Estimate: ________
$$3 \frac{2}{3}-1 \frac{11}{12}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 2
Difference: 1 $$\frac{3}{4}$$
Explanation:
Estimate:
$$3 \frac{2}{3}-1 \frac{11}{12}$$
$$\frac{2}{3}$$ is close to 1.
$$\frac{11}{12}$$ is close to 1.
3 + 1 – 1 – 1 = 2
Difference:
$$3 \frac{2}{3}-1 \frac{11}{12}$$
3 + $$\frac{2}{3}$$ – 1 – $$\frac{11}{12}$$
3 – 1 = 2
Solving the fractions part
$$\frac{2}{3}$$ – $$\frac{11}{12}$$
LCD = 12
$$\frac{8}{12}$$ – $$\frac{11}{12}$$ = – $$\frac{3}{12}$$ = – $$\frac{1}{4}$$
3 – $$\frac{1}{4}$$ = 1 $$\frac{3}{4}$$
$$3 \frac{2}{3}-1 \frac{11}{12}$$ = 1 $$\frac{3}{4}$$
Question 6.
Estimate: ________
$$4 \frac{1}{4}-2 \frac{1}{3}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 2
Difference: 1 $$\frac{11}{12}$$
Explanation:
$$4 \frac{1}{4}-2 \frac{1}{3}$$
$$\frac{1}{4}$$ is close to 0.
$$\frac{1}{3}$$ is close to 0.
4 – 2 = 2
Solving the fractions part
$$\frac{1}{4}$$ – $$\frac{1}{3}$$
LCD = 12
$$\frac{1}{4}$$ × $$\frac{3}{3}$$ – $$\frac{1}{3}$$ × $$\frac{4}{4}$$
$$\frac{3}{12}$$ – $$\frac{4}{12}$$ = – $$\frac{1}{12}$$
2 – $$\frac{1}{12}$$ = 1 $$\frac{11}{12}$$
Question 7.
Estimate: ________
$$5 \frac{2}{5}-1 \frac{1}{2}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 4
Difference: 3 $$\frac{9}{10}$$
Explanation:
Estimate:
$$5 \frac{2}{5}-1 \frac{1}{2}$$
5 + $$\frac{1}{2}$$ – 1 – $$\frac{1}{2}$$
5 – 1 = 4
Solving the fractions part
$$5 \frac{2}{5}-1 \frac{1}{2}$$
LCD = 10
$$\frac{4}{10}$$ – $$\frac{5}{10}$$ = – $$\frac{1}{10}$$
4 – $$\frac{1}{10}$$ = 3 $$\frac{9}{10}$$
Question 8.
$$7 \frac{5}{9}-2 \frac{5}{6}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 4 $$\frac{1}{2}$$
Difference: 4 $$\frac{13}{18}$$
Explanation:
Estimate:
$$7 \frac{5}{9}-2 \frac{5}{6}$$
$$\frac{5}{9}$$ is close to $$\frac{1}{2}$$
$$\frac{5}{6}$$ is close to 1.
7 + $$\frac{1}{2}$$ – 2 – 1
4 $$\frac{1}{2}$$
Difference:
$$7 \frac{5}{9}-2 \frac{5}{6}$$
7 + $$\frac{5}{9}$$ – 2 – $$\frac{5}{6}$$
Solving the whole numbers
7 – 2 = 5
Solving the fraction part
$$\frac{5}{9}$$ – $$\frac{5}{6}$$
LCD = 18
$$\frac{10}{18}$$ – $$\frac{15}{18}$$ = – $$\frac{5}{18}$$
5 – $$\frac{5}{18}$$ = 4 $$\frac{13}{18}$$
Question 9.
Estimate: ________
$$7-5 \frac{2}{3}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 1
Difference: 1 $$\frac{1}{3}$$
Explanation:
Estimate:
$$7-5 \frac{2}{3}$$
7 – 5 – $$\frac{2}{3}$$
7 – 5 – 1 = 1
Difference:
$$7-5 \frac{2}{3}$$
7 – 5 = 2
2 – $$\frac{2}{3}$$ = 1 $$\frac{1}{3}$$
Thus $$7-5 \frac{2}{3}$$ = 1 $$\frac{1}{3}$$
Question 10.
Estimate: ________
$$2 \frac{1}{5}-1 \frac{9}{10}$$
Estimate: _____ $$\frac{□}{□}$$
Difference: _____ $$\frac{□}{□}$$
Estimate: 0
Difference: $$\frac{3}{10}$$
Explanation:
Estimate:
$$2 \frac{1}{5}-1 \frac{9}{10}$$
2 + 0 – 1 – 1 = 0
Difference:
$$2 \frac{1}{5}-1 \frac{9}{10}$$
2 $$\frac{1}{5}$$ – 1 $$\frac{9}{10}$$
2 + $$\frac{1}{5}$$ – 1 – $$\frac{9}{10}$$
Solving the whole number parts
2 – 1 = 1
$$\frac{1}{5}$$ – $$\frac{9}{10}$$
LCD = 10
$$\frac{2}{10}$$ – $$\frac{9}{10}$$ = – $$\frac{7}{10}$$
1 – $$\frac{7}{10}$$ = $$\frac{3}{10}$$
Practice: Copy and Solve Find the difference and write it in simplest form.
Question 11.
$$11 \frac{1}{9}-3 \frac{2}{3}$$
_____ $$\frac{□}{□}$$
Answer: 7 $$\frac{4}{9}$$
Explanation:
Rewriting our equation with parts separated
11 + $$\frac{1}{9}$$ – 3 – $$\frac{2}{3}$$
Solving the whole number parts
11 – 3 = 8
Solving the fraction parts
LCD = 9
$$\frac{1}{9}$$ – $$\frac{2}{3}$$
$$\frac{1}{9}$$ – $$\frac{6}{9}$$ = – $$\frac{5}{9}$$
8 – $$\frac{5}{9}$$ = 7 $$\frac{4}{9}$$
Question 12.
$$6-3 \frac{1}{2}$$
_____ $$\frac{□}{□}$$
Answer: 2 $$\frac{1}{2}$$
Explanation:
Rewriting our equation with parts separated
6 – 3 – $$\frac{1}{2}$$
3 – $$\frac{1}{2}$$ = 2 $$\frac{1}{2}$$
Question 13.
$$4 \frac{3}{8}-3 \frac{1}{2}$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{8}$$
Explanation:
Rewriting our equation with parts separated
4 + $$\frac{3}{8}$$ – 3 – $$\frac{1}{2}$$
Solving the whole number parts
4 – 3 = 1
Solving the fraction parts
$$\frac{3}{8}$$ – $$\frac{1}{2}$$ = $$\frac{3}{8}$$ – $$\frac{4}{8}$$
= – $$\frac{1}{8}$$
1 – $$\frac{1}{8}$$ = $$\frac{7}{8}$$
Question 14.
$$9 \frac{1}{6}-3 \frac{5}{8}$$
_____ $$\frac{□}{□}$$
Answer: 5 $$\frac{13}{24}$$
Explanation:
Rewriting our equation with parts separated
9 + $$\frac{1}{6}$$ – 3 – $$\frac{5}{8}$$
Solving the whole number parts
9 – 3 = 6
Solving the fraction parts
$$\frac{1}{6}$$ – $$\frac{5}{8}$$
$$\frac{4}{24}$$ – $$\frac{15}{24}$$ = – $$\frac{11}{24}$$
6 – $$\frac{11}{24}$$ = 5 $$\frac{13}{24}$$
Question 15.
$$1 \frac{1}{5}-\frac{1}{2}$$
$$\frac{□}{□}$$
Answer: $$\frac{7}{10}$$
Explanation:
Rewriting our equation with parts separated
1 + $$\frac{1}{5}$$ – $$\frac{1}{2}$$
Solving the whole number parts
1 + 0 = 1
Solving the fraction parts
$$\frac{1}{5}$$ – $$\frac{1}{2}$$
LCD = 10
$$\frac{2}{10}$$ – $$\frac{5}{10}$$ = – $$\frac{3}{10}$$
1 – $$\frac{3}{10}$$ = $$\frac{7}{10}$$
Question 16.
$$13 \frac{1}{6}-3 \frac{4}{5}$$
_____ $$\frac{□}{□}$$
Answer: 9 $$\frac{11}{30}$$
Explanation:
Rewriting our equation with parts separated
13 + $$\frac{1}{6}$$ – 3 – $$\frac{4}{5}$$
Solving the whole number parts
13 – 3 = 10
Solving the fraction parts
$$\frac{1}{6}$$ – $$\frac{4}{5}$$
LCD = 30
$$\frac{5}{30}$$ – $$\frac{24}{30}$$ = – $$\frac{19}{30}$$
10 – $$\frac{19}{30}$$ = 9 $$\frac{11}{30}$$
Question 17.
$$12 \frac{2}{5}-5 \frac{3}{4}$$
_____ $$\frac{□}{□}$$
Answer: 6 $$\frac{13}{20}$$
Explanation:
Rewriting our equation with parts separated
12 + $$\frac{2}{5}$$ – 5 – $$\frac{3}{4}$$
Solving the whole number parts
12 – 5 = 7
Solving the fraction parts
$$\frac{2}{5}$$ – $$\frac{3}{4}$$
LCD = 20
$$\frac{8}{20}$$ – $$\frac{15}{20}$$ = – $$\frac{7}{20}$$
7 – $$\frac{7}{20}$$ = 6 $$\frac{13}{20}$$
Question 18.
$$7 \frac{3}{8}-2 \frac{7}{9}$$
_____ $$\frac{□}{□}$$
Answer: 4 $$\frac{43}{72}$$
Explanation:
7 + $$\frac{3}{8}$$ – 2 – $$\frac{7}{9}$$
7 – 2 = 5
$$\frac{3}{8}$$ – $$\frac{7}{9}$$ = $$\frac{27}{72}$$ – $$\frac{56}{72}$$
– $$\frac{29}{72}$$
5 – $$\frac{29}{72}$$ = 4 $$\frac{43}{72}$$
### Page No. 272
Summarize
An amusement park in Sandusky, Ohio, offers 17 amazing roller coasters for visitors to ride. One of the roller coasters runs at 60 miles per hour and has 3,900 feet of twisting track. This coaster also has 3 trains with 8 rows per train. Riders stand in rows of 4, for a total of 32 riders per train.
The operators of the coaster recorded the number of riders on each train during a run. On the first train, the operators reported that 7 $$\frac{1}{4}$$ rows were filled. On the second train, all 8 rows were filled, and on the third train, 5 $$\frac{1}{2}$$ rows were filled. How many more rows were filled on the first train than on the third train?
When you summarize, you restate the most important information in a shortened form to more easily understand what you have read.
Summarize the information given.
______________________
Use the summary to solve.
Question 19.
Solve the problem above.
Type below:
_________
On the first train, the operators reported that 7 $$\frac{1}{4}$$ rows were filled.
On the third train, 5 $$\frac{1}{2}$$ rows were filled.
7 $$\frac{1}{4}$$ – 5 $$\frac{1}{2}$$
Solving the whole numbers
7 – 5 = 2
Solving the fractions
$$\frac{1}{4}$$ – $$\frac{1}{2}$$ = – $$\frac{1}{4}$$
2 – $$\frac{1}{4}$$ = 1 $$\frac{3}{4}$$
1 $$\frac{3}{4}$$ more rows were filled on the first train than on the third train.
Question 20.
How many rows were empty on the third train? How many additional riders would it take to fill the empty rows? Explain your answer.
Type below:
_________
The coaster also has 3 trains with 8 rows per train.
The third train has 8 rows.
On the third train, 5 $$\frac{1}{2}$$ rows were filled.
8 – 5 $$\frac{1}{2}$$
8 – 5 – $$\frac{1}{2}$$ = 2 $$\frac{1}{2}$$
2 $$\frac{1}{2}$$ rows are empty.
So, it takes 10 additional riders to fill the empty rows on the third train.
### Share and Show – Page No. 275
Write a rule for the sequence.
Question 1.
$$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \cdots$$
Think: Is the sequence increasing or decreasing?
Rule: _________
Type below:
_________
Answer: The sequence is increasing order with difference $$\frac{1}{4}$$
Question 2.
$$\frac{1}{9}, \frac{1}{3}, \frac{5}{9}, \ldots$$
Type below:
_________
Answer: The sequence is increasing order with difference 2 in numerataor.
Write a rule for the sequence. Then, find the unknown term.
Question 3.
$$\frac{3}{10}, \frac{2}{5}$$, $$\frac{□}{□}$$ , $$\frac{3}{5}, \frac{7}{10}$$
Answer: The sequence is increasing order with difference $$\frac{1}{2}$$
LCD = 10
Add $$\frac{1}{2}$$ to each term
Let the unknown fraction be x
$$\frac{3}{10}$$, $$\frac{4}{10}$$, x, $$\frac{6}{10}$$, $$\frac{7}{10}$$
x = $$\frac{5}{10}$$ = $$\frac{1}{2}$$
Question 4.
$$10 \frac{2}{3}, 9 \frac{11}{18}, 8 \frac{5}{9}$$, ______ $$\frac{□}{□}$$ , $$6 \frac{4}{9}$$
Answer: 7 $$\frac{1}{2}$$
Explanation:
$$\frac{32}{3}$$, $$\frac{173}{18}$$, $$\frac{77}{9}$$, x, $$\frac{58}{9}$$
LCD = 54
$$\frac{576}{54}$$, $$\frac{519}{54}$$, $$\frac{462}{54}$$, x, $$\frac{348}{54}$$
According to the series x = $$\frac{405}{54}$$ = $$\frac{15}{2}$$
The mixed fraction of $$\frac{15}{2}$$ is 7 $$\frac{1}{2}$$
Question 5.
$$1 \frac{1}{6}$$, ______ $$\frac{□}{□}$$ , $$1, \frac{11}{12}, \frac{5}{6}$$
Answer: 1 $$\frac{1}{12}$$
Explanation:
$$1 \frac{1}{6}$$, ______ $$\frac{□}{□}$$ , $$1, \frac{11}{12}, \frac{5}{6}$$
The LCD of the above fractons is 12
Convert them into improper fractions
$$\frac{14}{12}$$, x, $$\frac{12}{12}$$, $$\frac{11}{12}$$, $$\frac{10}{12}$$
According to the series x = $$\frac{13}{12}$$
The mixed fraction of $$\frac{13}{12}$$ is 1 $$\frac{1}{12}$$
Question 6.
$$2 \frac{3}{4}, 4,5 \frac{1}{4}, 6 \frac{1}{2}$$, ______ $$\frac{□}{□}$$
Answer: 7 $$\frac{3}{4}$$
Explanation:
$$2 \frac{3}{4}, 4,5 \frac{1}{4}, 6 \frac{1}{2}$$, ______ $$\frac{□}{□}$$
Convert the mixed fractions into improper fractions
$$\frac{11}{4}$$, $$\frac{4}{1}$$, $$\frac{21}{4}$$, $$\frac{13}{2}$$, x
$$\frac{11}{4}$$, $$\frac{16}{4}$$, $$\frac{21}{4}$$, $$\frac{26}{4}$$, x
According to the series x = $$\frac{31}{4}$$
The mixed fraction of $$\frac{31}{4}$$ is 7 $$\frac{3}{4}$$
Write a rule for the sequence. Then, find the unknown term.
Question 7.
$$\frac{1}{8}, \frac{1}{2}$$, $$\frac{□}{□}$$ , $$1 \frac{1}{4}, 1 \frac{5}{8}$$
Answer: $$\frac{7}{8}$$
Explanation:
$$\frac{1}{8}, \frac{1}{2}$$, $$1 \frac{1}{4}, 1 \frac{5}{8}$$, x
LCD = 8
$$\frac{1}{8}, \frac{4}{8}$$, $$\frac{10}{8}, \frac{26}{8}$$, x
$$\frac{1}{8}$$, $$\frac{4}{8}$$, x, $$\frac{10}{8}$$, $$\frac{26}{8}$$
The difference between the series is 3 in numerator.
x = $$\frac{7}{8}$$
Question 8.
$$1 \frac{2}{3}, 1 \frac{3}{4}, 1 \frac{5}{6}, 1 \frac{11}{12}$$, ______
Explanation:
1 $$\frac{2}{3}$$, 1 $$\frac{3}{4}$$, 1 $$\frac{5}{6}$$, 1 $$\frac{11}{12}$$
Convert the mixed fractions into improper fractions
$$\frac{5}{3}$$, $$\frac{7}{4}$$, $$\frac{11}{6}$$, $$\frac{23}{12}$$, x
The LCD is 12
$$\frac{20}{12}$$, $$\frac{21}{12}$$, $$\frac{22}{12}$$, $$\frac{23}{12}$$, x
x = $$\frac{24}{12}$$ = 2
Question 9.
$$12 \frac{7}{8}, 10 \frac{3}{4}$$, ______ $$\frac{□}{□}$$ , $$6 \frac{1}{2}, 4 \frac{3}{8}$$
Answer: 8 $$\frac{5}{8}$$
Explanation:
$$12 \frac{7}{8}, 10 \frac{3}{4}$$, x , $$6 \frac{1}{2}, 4 \frac{3}{8}$$
Convert the mixed fractions into improper fractions
$$\frac{103}{8}$$, $$\frac{43}{4}$$, x, $$\frac{13}{2}$$, $$\frac{35}{8}$$
The LCD is 8
$$\frac{103}{8}$$, $$\frac{86}{8}$$, x, $$\frac{52}{8}$$, $$\frac{35}{8}$$
x = $$\frac{69}{8}$$
The mixed fraction of $$\frac{69}{8}$$ is 8 $$\frac{5}{8}$$
Question 10.
$$9 \frac{1}{3}$$, ______ $$\frac{□}{□}$$ , $$6 \frac{8}{9}, 5 \frac{2}{3}, 4 \frac{4}{9}$$
Answer: 8 $$\frac{1}{9}$$
Explanation:
$$9 \frac{1}{3}$$, x , $$6 \frac{8}{9}, 5 \frac{2}{3}, 4 \frac{4}{9}$$
Convert the mixed fractions into improper fractions
$$\frac{28}{3}$$, x, $$\frac{62}{9}$$, $$\frac{17}{3}$$, $$\frac{40}{9}$$
LCD = 9
$$\frac{84}{9}$$, x, $$\frac{62}{9}$$, $$\frac{51}{9}$$, $$\frac{40}{9}$$
According to the series x = $$\frac{73}{9}$$ = 8 $$\frac{1}{9}$$
Write the first four terms of the sequence.
Question 11.
Rule: start at 5 $$\frac{3}{4}$$, subtract $$\frac{5}{8}$$
First term: ______ $$\frac{□}{□}$$
Second term: ______ $$\frac{□}{□}$$
Third term: ______ $$\frac{□}{□}$$
Fourth term: ______ $$\frac{□}{□}$$
Let the first term be 5 $$\frac{3}{4}$$
Second term = 5 $$\frac{3}{4}$$ – $$\frac{5}{8}$$ = $$\frac{41}{8}$$ = 5 $$\frac{1}{8}$$
Third term = 5 $$\frac{1}{8}$$ – $$\frac{5}{8}$$ = $$\frac{36}{8}$$ = 4 $$\frac{1}{2}$$
Fourth term = $$\frac{36}{8}$$ – $$\frac{5}{8}$$ = $$\frac{31}{8}$$ = 3 $$\frac{7}{8}$$
Question 12.
Rule: start at $$\frac{3}{8}$$, add $$\frac{3}{16}$$
Type below:
_________
Let the first term be $$\frac{3}{8}$$
Second term = $$\frac{3}{8}$$ + $$\frac{3}{16}$$ = $$\frac{9}{16}$$
Third term = $$\frac{9}{16}$$ + $$\frac{3}{16}$$ = $$\frac{12}{16}$$
Fourth term = $$\frac{12}{16}$$ + $$\frac{3}{16}$$ = $$\frac{15}{16}$$
Question 13.
Rule: start at 2 $$\frac{1}{3}$$, add 2 $$\frac{1}{4}$$
First term: ______ $$\frac{□}{□}$$
Second term: ______ $$\frac{□}{□}$$
Third term: ______ $$\frac{□}{□}$$
Fourth term: ______ $$\frac{□}{□}$$
Let the first term be 2 $$\frac{1}{3}$$
Second term = 2 $$\frac{1}{3}$$ + 2 $$\frac{1}{4}$$ = $$\frac{7}{3}$$ + $$\frac{9}{4}$$
= $$\frac{55}{12}$$ = 4 $$\frac{7}{12}$$
Third term = 4 $$\frac{7}{12}$$ + 2 $$\frac{1}{4}$$ = 6 $$\frac{5}{6}$$
Fourth term = 6 $$\frac{5}{6}$$ + 2 $$\frac{1}{4}$$ = 9 $$\frac{1}{12}$$
Question 14.
Rule: start at $$\frac{8}{9}$$, subtract $$\frac{1}{18}$$
Type below:
_________
Let the first term be $$\frac{8}{9}$$
Second term = $$\frac{8}{9}$$ – $$\frac{1}{18}$$ = $$\frac{15}{18}$$ = $$\frac{5}{6}$$
Third term = $$\frac{15}{18}$$ – $$\frac{1}{18}$$ = $$\frac{14}{18}$$ = $$\frac{7}{9}$$
Fourth term = $$\frac{14}{18}$$ – $$\frac{1}{18}$$ = $$\frac{13}{18}$$
### Problem Solving – Page No. 276
Question 15.
When Bill bought a marigold plant, it was $$\frac{1}{4}$$ inch tall. After the first week, it measured 1 $$\frac{1}{12}$$ inches tall. After the second week, it was 1 $$\frac{11}{12}$$ inches. After week 3, it was 2 $$\frac{3}{4}$$ inches tall. Assuming the growth of the plant was constant, what was the height of the plant at the end of week 4?
______ $$\frac{□}{□}$$ inches
Answer: 3 $$\frac{7}{12}$$ inches
The sequence is the increasing where the first term is $$\frac{1}{4}$$
LCD = 12
First week is $$\frac{3}{12}$$
Second week = $$\frac{13}{12}$$ = 1 $$\frac{1}{12}$$
Third week = 1 $$\frac{11}{12}$$ = $$\frac{23}{12}$$
Fourth week = $$\frac{33}{12}$$ = 2 $$\frac{3}{4}$$
At the end of fourth week = $$\frac{43}{12}$$ = 3 $$\frac{7}{12}$$ inches
The height of the plant at the end of the week is 3 $$\frac{7}{12}$$ inches.
Question 16.
What if Bill’s plant grew at the same rate but was 1 $$\frac{1}{2}$$ inches when he bought it? How tall would the plant be after 3 weeks?
______ inches
Explanation:
The sequence is increasing.
First week 1 $$\frac{1}{2}$$
Let the first term is $$\frac{6}{12}$$
Second term is 1 $$\frac{16}{12}$$
Third term is 1 $$\frac{26}{12}$$
Fourth week is 1 $$\frac{36}{12}$$
1 $$\frac{36}{12}$$ = 1 $$\frac{3}{1}$$ = 1 + 3 = 4
After 4 weeks the plant grew 4 inches.
Question 17.
Vicki wanted to start jogging. The first time she ran, she ran $$\frac{3}{16}$$ mile. The second time, she ran $$\frac{3}{8}$$ mile, and the third time, she ran $$\frac{9}{16}$$ mile. If she continued this pattern, when was the first time she ran more than 1 mile? Explain.
Type below:
_________
Explanation:
Vicki wanted to start jogging. The first time she ran, she ran $$\frac{3}{16}$$ mile. The second time, she ran $$\frac{3}{8}$$ mile, and the third time, she ran $$\frac{9}{16}$$ mile.
The difference is $$\frac{3}{16}$$
First time = $$\frac{3}{16}$$ mile
Second time = $$\frac{3}{16}$$ + $$\frac{3}{16}$$ = $$\frac{3}{8}$$ mile
Third time = $$\frac{3}{8}$$ + $$\frac{3}{16}$$ = $$\frac{9}{16}$$ mile
Fourth time = $$\frac{9}{16}$$ + $$\frac{3}{16}$$ = $$\frac{12}{16}$$ mile
Fifth time = $$\frac{12}{16}$$ + $$\frac{3}{16}$$ = $$\frac{15}{16}$$ mile
Sixth time = $$\frac{15}{16}$$ + $$\frac{3}{16}$$ = $$\frac{18}{16}$$ mile
$$\frac{18}{16}$$ = 1 $$\frac{2}{16}$$ = 1 $$\frac{1}{8}$$
Question 18.
Mr. Conners drove 78 $$\frac{1}{3}$$ miles on Monday, 77 $$\frac{1}{12}$$ miles on Tuesday, and 75 $$\frac{5}{6}$$ miles on Wednesday. If he continues this pattern on Thursday and Friday, how many miles will he drive on Friday?
______ $$\frac{□}{□}$$ miles
Given that,
Mr. Conners drove 78 $$\frac{1}{3}$$ miles on Monday, 77 $$\frac{1}{12}$$ miles on Tuesday, and 75 $$\frac{5}{6}$$ miles on Wednesday.
The sequence is the decreasing where the first term is 78 $$\frac{4}{12}$$
78 $$\frac{4}{12}$$ – 77 $$\frac{1}{12}$$ = 1 $$\frac{3}{12}$$
The difference between the term is 1 $$\frac{3}{12}$$
On thursday, 75 $$\frac{5}{6}$$ – 1 $$\frac{3}{12}$$ = 74 $$\frac{7}{12}$$
On friday, 74 $$\frac{7}{12}$$ – 1 $$\frac{3}{12}$$ = 73 $$\frac{4}{12}$$ = 73 $$\frac{1}{3}$$
Question 19.
Test Prep Zack watered his garden with 1 $$\frac{3}{8}$$ gallons of water the first week he planted it. He watered it with 1 $$\frac{3}{4}$$ gallons the second week, and 2 $$\frac{1}{8}$$ gallons the third week. If he continued watering in this pattern, how much water did he use on the fifth week?
Options:
a. 2 $$\frac{1}{2}$$ gallons
b. 2 $$\frac{7}{8}$$ gallons
c. 3 $$\frac{1}{4}$$ gallons
d. 6 $$\frac{7}{8}$$ gallons
Answer: 2 $$\frac{7}{8}$$ gallons
Explanation:
First term = 1 $$\frac{3}{8}$$
The difference is $$\frac{3}{4}$$ – $$\frac{3}{8}$$ = $$\frac{3}{8}$$
Second term is 1 $$\frac{3}{8}$$ + $$\frac{3}{8}$$ = 1 $$\frac{3}{4}$$
Third term = 1 $$\frac{3}{4}$$ + $$\frac{3}{8}$$ = 1 + 1 $$\frac{1}{8}$$ = 2 $$\frac{1}{8}$$
Fourth term = 2 $$\frac{1}{8}$$ + $$\frac{3}{8}$$ = 2 $$\frac{1}{2}$$
Fifth term = 2 $$\frac{1}{2}$$ + $$\frac{3}{8}$$ = 2 $$\frac{7}{8}$$ gallons
Thus the correct answer is option B.
### Share and Show – Page No. 279
Question 1.
Caitlin has 4 $$\frac{3}{4}$$ pounds of clay. She uses 1 $$\frac{1}{10}$$ pounds to make a cup, and another 2 pounds to make a jar. How many pounds are left?
First, write an equation to model the problem.
Type below:
_________
Answer: 4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2
Explanation:
Subtract the total pound of clay from used clay.
So, the equation of the clay leftover is 4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2
Question 1.
Next, work backwards and rewrite the equation to find x.
Type below:
_________
Answer: 4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2 = x
Explanation:
Let the leftover clay be x
4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2 = x
x = 4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2
Question 1.
Solve.
_____________________
So, ________ pounds of clay remain.
Type below:
_________
Answer: 1 $$\frac{13}{20}$$ pounds
Explanation:
4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2
4 + $$\frac{3}{4}$$ – 1 – $$\frac{1}{10}$$ – 2
4 – 3 = 1
$$\frac{3}{4}$$ – $$\frac{1}{10}$$ = $$\frac{13}{20}$$
1 + $$\frac{13}{20}$$ = 1 $$\frac{13}{20}$$ pounds
Question 2.
What if Caitlin had used more than 2 pounds of clay to make a jar? Would the amount remaining have been more or less than your answer to Exercise 1?
Type below:
_________
Let us assume that Catlin used 2 $$\frac{1}{4}$$ pounds of clay to make a jar and 1 $$\frac{1}{10}$$ pounds to make a cup.
4 $$\frac{3}{4}$$ – 1 $$\frac{1}{10}$$ – 2 $$\frac{1}{4}$$ = 2 $$\frac{1}{20}$$
Question 3.
A pet store donated 50 pounds of food for adult dogs, puppies, and cats to an animal shelter. 19 $$\frac{3}{4}$$ pounds was adult dog food and 18 $$\frac{7}{8}$$ pounds was puppy food. How many pounds of cat food did the pet store donate?
______ $$\frac{□}{□}$$ pounds of cat food
Answer: 11 $$\frac{3}{8}$$ pounds of cat food
Explanation:
A pet store donated 50 pounds of food for adult dogs, puppies, and cats to an animal shelter.
19 $$\frac{3}{4}$$ pounds was adult dog food and 18 $$\frac{7}{8}$$ pounds was puppy food.
19 $$\frac{3}{4}$$ + 18 $$\frac{7}{8}$$ = 38 $$\frac{5}{8}$$
50 – 38 $$\frac{5}{8}$$ = 11 $$\frac{3}{8}$$ pounds of cat food
Thus the pet store donate 11 $$\frac{3}{8}$$ pounds of cat food
Question 4.
Thelma spent $$\frac{1}{6}$$ of her weekly allowance on dog toys, $$\frac{1}{4}$$ on a dog collar, and $$\frac{1}{3}$$ on dog food. What fraction of her weekly allowance is left?
$$\frac{□}{□}$$ of her weekly allowance
Answer: $$\frac{1}{4}$$
Explanation:
Given that, Thelma spent $$\frac{1}{6}$$ of her weekly allowance on dog toys, $$\frac{1}{4}$$ on a dog collar, and $$\frac{1}{3}$$ on dog food.
$$\frac{1}{6}$$ + $$\frac{1}{4}$$ + $$\frac{1}{3}$$ = $$\frac{3}{4}$$
1 – $$\frac{3}{4}$$ = $$\frac{1}{4}$$
$$\frac{1}{4}$$ of her weekly allowance.
### On Your Own – Page No. 280
Question 5.
Martin is making a model of a Native American canoe. He has 5 $$\frac{1}{2}$$ feet of wood. He uses 2 $$\frac{3}{4}$$ feet for the hull and 1 $$\frac{1}{4}$$ feet for the paddles and struts. How much wood does he have left?
______ $$\frac{□}{□}$$ feet
Answer: 1 $$\frac{1}{2}$$ feet
Explanation:
Martin is making a model of a Native American canoe.
He has 5 $$\frac{1}{2}$$ feet of wood.
He uses 2 $$\frac{3}{4}$$ feet for the hull and 1 $$\frac{1}{4}$$ feet for the paddles and struts.
2 $$\frac{3}{4}$$ + 1 $$\frac{1}{4}$$
2 + $$\frac{3}{4}$$ + 1 + $$\frac{1}{4}$$
2 + 1 = 3
$$\frac{3}{4}$$ + $$\frac{1}{4}$$ = 1
3 + 1 = 4
5 $$\frac{1}{2}$$ – 4 = 1 $$\frac{1}{2}$$
Question 6.
What if Martin makes a hull and two sets of paddles and struts? How much wood does he have left?
Answer: 1 $$\frac{1}{4}$$
Explanation:
He has 5 $$\frac{1}{2}$$ feet of wood.
If Martin makes a hull and two sets of paddles and struts
1 $$\frac{1}{4}$$ + 1 $$\frac{1}{4}$$ = 2 $$\frac{1}{2}$$
2 $$\frac{1}{2}$$ + 2 $$\frac{3}{4}$$ = 4 $$\frac{1}{4}$$
5 $$\frac{1}{2}$$ – 4 $$\frac{1}{4}$$
5 + $$\frac{1}{2}$$ – 4 – $$\frac{1}{4}$$
1 + $$\frac{1}{4}$$ = 1 $$\frac{1}{4}$$
Question 7.
Beth’s summer vacation lasted 87 days. At the beginning of her vacation, she spent 3 weeks at soccer camp, 5 days at her grandmother’s house, and 13 days visiting Glacier National Park with her parents. How many vacation days remained?
______ days
Explanation:
Given,
Beth’s summer vacation lasted 87 days.
At the beginning of her vacation, she spent 3 weeks at soccer camp, 5 days at her grandmother’s house, and 13 days visiting Glacier National Park with her parents.
87 – 21 – 5 – 13 = 48 days
The remaining vacation days are 48.
Question 8.
You can buy 2 DVDs for the same price you would pay for 3 CDs selling for $13.20 apiece. Explain how you could find the price of 1 DVD.$ ______
Answer: $19.8 Explanation: To find what is the price of 1 DVD we will find what is the price of 3 DVDs and then because 2 DVDs price is the same than 3 CDs we can easily find the price of 1 DVD.$13.20 × 3 = $39.6 We will divide$39.6 by 2.
$39.6 ÷ 2 =$19.8
The price of 1 DVD is \$19.8
Question 9.
Test Prep During the 9 hours between 8 A.M. and 5 P.M., Bret spent 5 $$\frac{3}{4}$$ hours in class and 1 $$\frac{1}{2}$$ hours at band practice. How much time did he spend on other activities?
Options:
a. $$\frac{3}{4}$$ hour
b. 1 $$\frac{1}{4}$$ hour
c. 1 $$\frac{1}{2}$$ hour
d. 1 $$\frac{3}{4}$$ hour
Answer: 1 $$\frac{3}{4}$$ hour
Explanation:
Test Prep During the 9 hours between 8 A.M. and 5 P.M., Bret spent 5 $$\frac{3}{4}$$ hours in class and 1 $$\frac{1}{2}$$ hours at band practice.
5 $$\frac{3}{4}$$ + 1 $$\frac{1}{2}$$ = 7 $$\frac{1}{4}$$ hour
9 – 7 $$\frac{1}{4}$$ hour
8 + 1 – 7 – $$\frac{1}{4}$$
1 $$\frac{3}{4}$$ hour
The correct answer is option D.
### Share and Show – Page No. 283
Use the properties and mental math to solve. Write your answer in simplest form.
Question 1.
$$\left(2 \frac{5}{8}+\frac{5}{6}\right)+1 \frac{1}{8}$$
______ $$\frac{□}{□}$$
$$\left(2 \frac{5}{8}+\frac{5}{6}\right)+1 \frac{1}{8}$$
2 $$\frac{5}{8}$$ + $$\frac{5}{6}$$
2 + $$\frac{5}{8}$$ + $$\frac{5}{6}$$
LCD = 24
$$\frac{15}{24}$$ + $$\frac{20}{24}$$ = $$\frac{35}{24}$$
$$\frac{35}{24}$$ = 1 $$\frac{11}{24}$$
2 + 1 $$\frac{11}{24}$$ = 3 $$\frac{11}{24}$$
3 $$\frac{11}{24}$$ + 1 $$\frac{1}{8}$$ = 4 $$\frac{7}{12}$$
Question 2.
$$\frac{5}{12}+\left(\frac{5}{12}+\frac{3}{4}\right)$$
______ $$\frac{□}{□}$$
$$\frac{5}{12}+\left(\frac{5}{12}+\frac{3}{4}\right)$$
$$\frac{5}{12}$$ + $$\frac{3}{4}$$
LCD = 12
$$\frac{5}{12}$$ + $$\frac{3}{4}$$ × $$\frac{3}{3}$$
$$\frac{5}{12}$$ + $$\frac{9}{12}$$ = $$\frac{14}{12}$$
$$\frac{5}{12}$$ + $$\frac{14}{12}$$ = $$\frac{19}{12}$$
$$\frac{19}{12}$$ = 1 $$\frac{7}{12}$$
Question 3.
$$\left(3 \frac{1}{4}+2 \frac{5}{6}\right)+1 \frac{3}{4}$$
______ $$\frac{□}{□}$$
$$\left(3 \frac{1}{4}+2 \frac{5}{6}\right)$$
2 + $$\frac{5}{6}$$ + 3 + $$\frac{1}{4}$$
2 + 3 = 5
$$\frac{5}{6}$$ + $$\frac{1}{4}$$
LCD = 12
$$\frac{5}{6}$$ × $$\frac{2}{2}$$ + $$\frac{1}{4}$$ × $$\frac{3}{3}$$
$$\frac{10}{12}$$ + $$\frac{3}{12}$$ = $$\frac{13}{12}$$ = 1 $$\frac{1}{12}$$
5 + 1 $$\frac{1}{12}$$ = 6 $$\frac{1}{12}$$
6 $$\frac{1}{12}$$ + 1 $$\frac{3}{4}$$
6 + $$\frac{1}{12}$$ + 1 + $$\frac{3}{4}$$
6 + 1 = 7
$$\frac{1}{12}$$ + $$\frac{3}{4}$$
$$\frac{1}{12}$$ + $$\frac{9}{12}$$ = $$\frac{10}{12}$$ = $$\frac{5}{6}$$
7 + $$\frac{5}{6}$$ = 7 $$\frac{5}{6}$$
Use the properties and mental math to solve. Write your answer in simplest form.
Question 4.
$$\left(\frac{2}{7}+\frac{1}{3}\right)+\frac{2}{3}$$
______ $$\frac{□}{□}$$
$$\left(\frac{2}{7}+\frac{1}{3}\right)+\frac{2}{3}$$
$$\left(\frac{2}{7}+\frac{1}{3}\right)$$
LCD = 21
$$\left(\frac{6}{21}+\frac{7}{21}\right)$$ = $$\frac{13}{21}$$
$$\frac{13}{21}$$ + $$\frac{2}{3}$$
LCD = 21
$$\frac{13}{21}$$ + $$\frac{14}{21}$$
$$\frac{27}{21}$$ = $$\frac{9}{7}$$
= 1 $$\frac{2}{7}$$
Question 5.
$$\left(\frac{1}{5}+\frac{1}{2}\right)+\frac{2}{5}$$
______ $$\frac{□}{□}$$
$$\left(\frac{1}{5}+\frac{1}{2}\right)$$
$$\frac{1}{5}$$ + $$\frac{1}{2}$$
LCD = 10
$$\frac{2}{10}$$ + $$\frac{5}{10}$$ = $$\frac{7}{10}$$
$$\frac{7}{10}$$ + $$\frac{2}{5}$$
$$\frac{7}{10}$$ + $$\frac{4}{10}$$ = $$\frac{11}{10}$$
$$\frac{11}{10}$$ = 1 $$\frac{1}{10}$$
Question 6.
$$\left(\frac{1}{6}+\frac{3}{7}\right)+\frac{2}{7}$$
$$\frac{□}{□}$$
$$\left(\frac{1}{6}+\frac{3}{7}\right)$$
LCD = 42
$$\left(\frac{7}{42}+\frac{18}{42}\right)$$ = $$\frac{25}{42}$$
$$\frac{25}{42}$$ + $$\frac{2}{7}$$
LCD = 42
$$\frac{25}{42}$$ + $$\frac{12}{42}$$ = $$\frac{37}{42}$$
$$\left(\frac{1}{6}+\frac{3}{7}\right)+\frac{2}{7}$$ = $$\frac{37}{42}$$
Question 7.
$$\left(2 \frac{5}{12}+4 \frac{1}{4}\right)+\frac{1}{4}$$
______ $$\frac{□}{□}$$
$$\left(2 \frac{5}{12}+4 \frac{1}{4}\right)$$
2 $$\frac{5}{12}$$ + 4 $$\frac{1}{4}$$
2 + $$\frac{5}{12}$$ + 4 + $$\frac{1}{4}$$
2 + 4 = 6
$$\frac{5}{12}$$ + $$\frac{1}{4}$$ = $$\frac{8}{12}$$
6 $$\frac{8}{12}$$ = 6 $$\frac{2}{3}$$
6 $$\frac{2}{3}$$ + $$\frac{1}{4}$$ = 6 $$\frac{11}{12}$$
Question 8.
$$1 \frac{1}{8}+\left(5 \frac{1}{2}+2 \frac{3}{8}\right)$$
______
5 $$\frac{1}{2}$$ + 2 $$\frac{3}{8}$$
5 + 2 = 7
$$\frac{1}{2}$$ + $$\frac{3}{8}$$
LCD = 8
$$\frac{4}{8}$$ + $$\frac{3}{8}$$ = $$\frac{7}{8}$$
= 7 $$\frac{7}{8}$$
1 $$\frac{1}{8}$$ + 7 $$\frac{7}{8}$$ = 9
Question 9.
$$\frac{5}{9}+\left(\frac{1}{9}+\frac{4}{5}\right)$$
______ $$\frac{□}{□}$$
$$\frac{1}{9}$$ + $$\frac{4}{5}$$
LCD = 45
$$\frac{5}{45}$$ + $$\frac{36}{45}$$ = $$\frac{41}{45}$$
$$\frac{41}{45}$$ + $$\frac{5}{9}$$
LCD = 45
$$\frac{41}{45}$$ + $$\frac{25}{45}$$ = $$\frac{66}{45}$$
$$\frac{66}{45}$$ = 1 $$\frac{7}{15}$$
### Problem Solving – Page No. 284
Use the map to solve 10–12.
Question 10.
In the morning, Julie rides her bike from the sports complex to the school. In the afternoon, she rides from the school to the mall, and then to Kyle’s house. How far does Julie ride her bike?
______ $$\frac{□}{□}$$ miles
Answer: 1 $$\frac{13}{15}$$ miles
Explanation:
Julie rides her bike from the sports complex to the school = $$\frac{2}{3}$$ mile
In the afternoon, she rides from the school to the mall, and then to Kyle’s house. = $$\frac{2}{5}$$ + $$\frac{4}{5}$$ = $$\frac{6}{5}$$ = 1 $$\frac{1}{5}$$
1 $$\frac{1}{5}$$ + $$\frac{2}{3}$$ mile = 1 $$\frac{13}{15}$$ miles
Question 11.
On one afternoon, Mario walks from his house to the library. That evening, Mario walks from the library to the mall, and then to Kyle’s house. Describe how you can use the properties to find how far Mario walks.
______ $$\frac{□}{□}$$ miles
Mario walks from his house to the library = 1 $$\frac{3}{5}$$ miles
Mario walks from the library to the mall, and then to Kyle’s house = 1 $$\frac{1}{3}$$ and $$\frac{4}{5}$$
1 $$\frac{3}{5}$$ + (1 $$\frac{1}{3}$$ + $$\frac{4}{5}$$)
1 $$\frac{3}{5}$$ + 2 $$\frac{2}{15}$$ = 3 $$\frac{11}{15}$$ miles
Question 12.
Pose a Problem Write and solve a new problem that uses the distances between four locations.
Type below:
_________
In the evening Kyle rides his bike from the sports complex to school. Then he rides from School to the mall and then to his house. How far does Kyle ride his bike?
The distance from Sports complex to School is $$\frac{2}{3}$$ mile
The distance from School to the mall is $$\frac{2}{5}$$
The distance from the mall to Kyle house is $$\frac{4}{5}$$
$$\frac{2}{3}$$ + ($$\frac{2}{5}$$ + $$\frac{4}{5}$$)
$$\frac{2}{3}$$ + $$\frac{6}{5}$$ = 1 $$\frac{13}{15}$$ miles
Question 13.
Test Prep Which property or properties does the problem below use?
$$\frac{1}{9}+\left(\frac{4}{9}+\frac{1}{6}\right)=\left(\frac{1}{9}+\frac{4}{9}\right)+\frac{1}{6}$$
Options:
a. Commutative Property
b. Associative Property
c. Commutative Property and Associative Property
d. Distributive Property
The associative property states that you can add or multiply regardless of how the numbers are grouped. By ‘grouped’ we mean ‘how you use parenthesis’. In other words, if you are adding or multiplying it does not matter where you put the parenthesis.
### Chapter Review/Test – Vocabulary – Page No. 285
Choose the best term from the box.
Question 1.
A _________ is a number that is a common multiple of two or more denominators.
_________
Concepts and Skills
Use a common denominator to write an equivalent fraction for each fraction.
Question 2.
$$\frac{2}{5}, \frac{1}{8}$$
common denominator: ______
Explain:
_________
Multiply the denominators of the fractions
5 × 8 = 40
Question 3.
$$\frac{3}{4}, \frac{1}{2}$$
common denominator: ______
Explain:
_________
Multiply the denominators of the fractions
4 × 2 = 8
Question 4.
$$\frac{2}{3}, \frac{1}{6}$$
common denominator: ______
Explain:
_________
Multiply the denominators of the fractions
3 × 6 = 18
Question 5.
$$\frac{5}{6}+\frac{7}{8}$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{17}{24}$$
Explanation:
$$\frac{5}{6}+\frac{7}{8}$$ = $$\frac{20}{24}$$ + $$\frac{21}{24}$$
= $$\frac{41}{24}$$ = 1 $$\frac{17}{24}$$
Question 6.
$$2 \frac{2}{3}-1 \frac{2}{5}$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{4}{15}$$
Question 7.
$$7 \frac{3}{4}+3 \frac{7}{20}$$
______ $$\frac{□}{□}$$
Answer: 11 $$\frac{1}{10 }$$
Estimate. Then find the difference and write it in simplest form.
Question 8.
$$1 \frac{2}{5}-\frac{2}{3}$$
Type below:
________
Estimate: $$\frac{1}{2}$$
Difference:
Rewriting our equation with parts separated
1 + $$\frac{2}{5}$$ – $$\frac{2}{3}$$
$$\frac{7}{5}$$ – $$\frac{2}{3}$$
$$\frac{7}{5}$$ × $$\frac{3}{3}$$ – $$\frac{2}{3}$$ × $$\frac{5}{5}$$
= $$\frac{21}{15}$$ – $$\frac{10}{15}$$
= $$\frac{11}{15}$$
Question 9.
$$7-\frac{3}{7}$$
Type below:
________
Answer: 6 $$\frac{4}{7}$$
Explanation:
$$7-\frac{3}{7}$$ = $$\frac{49}{7}$$ – $$\frac{3}{7}$$
$$\frac{46}{7}$$ = 6 $$\frac{4}{7}$$
$$7-\frac{3}{7}$$ = 6 $$\frac{4}{7}$$
Question 10.
$$5 \frac{1}{9}-3 \frac{5}{6}$$
Type below:
________
Answer: 1 $$\frac{5}{18}$$
Explanation:
$$5 \frac{1}{9}-3 \frac{5}{6}$$ = 5 + $$\frac{1}{9}$$ – 3 – $$\frac{5}{6}$$
5 – 3 = 2
$$\frac{1}{9}$$ – $$\frac{5}{6}$$ = $$\frac{2}{18}$$ – $$\frac{15}{18}$$ = – $$\frac{13}{18}$$
2 – $$\frac{13}{18}$$ = 1 $$\frac{5}{18}$$
Use the properties and mental math to solve. Write your answer in simplest form.
Question 11.
$$\left(\frac{3}{8}+\frac{2}{3}\right)+\frac{1}{3}$$
______ $$\frac{□}{□}$$
Answer: 1 $$\frac{3}{8}$$
Explanation:
$$\frac{3}{8}$$ + $$\frac{2}{3}$$ = $$\frac{9}{24}$$ + $$\frac{16}{24}$$ = $$\frac{25}{24}$$
$$\frac{25}{24}$$ + $$\frac{1}{3}$$
= $$\frac{25}{24}$$ + $$\frac{8}{24}$$ = $$\frac{33}{24}$$ = $$\frac{11}{8}$$
The mixed fraction of $$\frac{11}{8}$$ is 1 $$\frac{3}{8}$$.
Question 12.
$$1 \frac{4}{5}+\left(2 \frac{3}{20}+\frac{3}{5}\right)$$
______ $$\frac{□}{□}$$
Answer: 4 $$\frac{11}{20}$$
Explanation:
Rewriting our equation with parts separated
2 $$\frac{3}{20}$$ + $$\frac{3}{5}$$ = $$\frac{43}{20}$$ + $$\frac{3}{5}$$
$$\frac{43}{20}$$ + $$\frac{3}{5}$$ = $$\frac{215}{100}$$ + $$\frac{60}{100}$$
= $$\frac{275}{100}$$ = 2 $$\frac{3}{4}$$
2 $$\frac{3}{4}$$ + 1 $$\frac{4}{5}$$ = 2 + $$\frac{3}{4}$$ + 1 + $$\frac{4}{5}$$
2 + 1 = 3
$$\frac{3}{4}$$ + $$\frac{4}{5}$$ = $$\frac{15}{20}$$ + $$\frac{16}{20}$$ = $$\frac{31}{20}$$
$$\frac{31}{20}$$ = 4 $$\frac{11}{20}$$
Question 13.
$$3 \frac{5}{9}+\left(1 \frac{7}{9}+2 \frac{5}{12}\right)$$
______ $$\frac{□}{□}$$
Answer: 7 $$\frac{3}{4}$$
Explanation:
1 $$\frac{7}{9}$$ + 2 $$\frac{5}{12}$$
1 + 2 = 3
$$\frac{7}{9}$$ + $$\frac{5}{12}$$
LCD is 36
$$\frac{28}{36}$$ + $$\frac{15}{36}$$ = $$\frac{43}{36}$$
$$\frac{43}{36}$$ = 1 $$\frac{7}{36}$$
3 + 1 + $$\frac{7}{36}$$ = 4 $$\frac{7}{36}$$
4 $$\frac{7}{36}$$ + 3 $$\frac{5}{9}$$
4 + $$\frac{7}{36}$$ + 3 + $$\frac{5}{9}$$
4 + 3 = 7
$$\frac{7}{36}$$ + $$\frac{5}{9}$$
= $$\frac{7}{36}$$ + $$\frac{20}{36}$$ = $$\frac{27}{36}$$ = $$\frac{3}{4}$$
7 + $$\frac{3}{4}$$ = 7 $$\frac{3}{4}$$
### Chapter Review/Test – Page No. 286
Question 14.
Ursula mixed 3 $$\frac{1}{8}$$ cups of dry ingredients with 1 $$\frac{2}{5}$$ cups of liquid ingredients. Which answer represents the best estimate of the total amount of ingredients Ursula mixed?
Options:
b. about 4 $$\frac{1}{2}$$ cups
d. about 5 $$\frac{1}{2}$$ cups
Answer: about 4 $$\frac{1}{2}$$ cups
Explanation:
Ursula mixed 3 $$\frac{1}{8}$$ cups of dry ingredients with 1 $$\frac{2}{5}$$ cups of liquid ingredients.
3 + 1 = 4
$$\frac{1}{8}$$ is closer to 0.
$$\frac{2}{5}$$ is closer to $$\frac{1}{2}$$
4 + $$\frac{1}{2}$$ = 4 $$\frac{1}{2}$$
Thus the correct answer is option B.
Question 15.
Samuel walks in the Labor Day parade. He walks 3 $$\frac{1}{4}$$ miles along the parade route and 2 $$\frac{5}{6}$$ miles home. How many miles does Samuel walk?
Options:
a. $$\frac{5}{10}$$ mile
b. 5 $$\frac{1}{12}$$ miles
c. 5 $$\frac{11}{12}$$ miles
d. 6 $$\frac{1}{12}$$ miles
Answer: 6 $$\frac{1}{12}$$ miles
Explanation:
Samuel walks in the Labor Day parade.
He walks 3 $$\frac{1}{4}$$ miles along the parade route and 2 $$\frac{5}{6}$$ miles home.
3 + $$\frac{1}{4}$$ + 2 + $$\frac{5}{6}$$
3 + 2 =5
$$\frac{5}{6}$$ + $$\frac{1}{4}$$ = $$\frac{10}{12}$$ + $$\frac{3}{12}$$ = $$\frac{13}{12}$$
$$\frac{13}{12}$$ = 6 $$\frac{1}{12}$$ miles
Thus the correct answer is option D.
Question 16.
A gardener has a container with 6 $$\frac{1}{5}$$ ounces of liquid plant fertilizer. On Sunday, the gardener uses 2 $$\frac{1}{2}$$ ounces on a flower garden. How many ounces of liquid plant fertilizer are left?
Options:
a. 3 $$\frac{7}{10}$$ ounces
b. 5 $$\frac{7}{10}$$ ounces
c. 6 $$\frac{7}{10}$$ ounces
d. 9 $$\frac{7}{10}$$ ounces
Answer: 9 $$\frac{7}{10}$$ ounces
Explanation:
A gardener has a container with 6 $$\frac{1}{5}$$ ounces of liquid plant fertilizer.
On Sunday, the gardener uses 2 $$\frac{1}{2}$$ ounces on a flower garden.
6 + $$\frac{1}{5}$$ + 2 + $$\frac{1}{2}$$
6 + 2 = 8
$$\frac{1}{5}$$ + $$\frac{1}{2}$$
LCD = 10
$$\frac{2}{10}$$ + $$\frac{5}{10}$$ = $$\frac{7}{10}$$
8 $$\frac{7}{10}$$
Question 17.
Aaron is practicing for a triathlon. On Sunday, he bikes 12 $$\frac{5}{8}$$ miles and swims 5 $$\frac{2}{3}$$ miles. On Monday, he runs 6 $$\frac{3}{8}$$ miles. How many total miles does Aaron cover on the two days?
Options:
a. 23 $$\frac{1}{6}$$ miles
b. 24 $$\frac{7}{12}$$ miles
c. 24 $$\frac{2}{3}$$ miles
d. 25 $$\frac{7}{12}$$ miles
Answer: 24 $$\frac{2}{3}$$ miles
Explanation:
Aaron is practicing for a triathlon.
On Sunday, he bikes 12 $$\frac{5}{8}$$ miles and swims 5 $$\frac{2}{3}$$ miles.
On Monday, he runs 6 $$\frac{3}{8}$$ miles.
5 $$\frac{2}{3}$$ + 6 $$\frac{3}{8}$$ = 12 $$\frac{1}{24}$$
12 $$\frac{1}{24}$$ + 12 $$\frac{5}{8}$$ miles
12 + $$\frac{1}{24}$$ + 12 + $$\frac{5}{8}$$
12 + 12 = 24
$$\frac{1}{24}$$ + $$\frac{5}{8}$$ = $$\frac{1}{24}$$ + $$\frac{15}{24}$$ = $$\frac{16}{24}$$ = $$\frac{2}{3}$$
24 + $$\frac{2}{3}$$ = 24 $$\frac{2}{3}$$ mile
The correct answer is option D.
### Chapter Review/Test – Page No. 287
Question 18.
Mrs. Friedmon baked a walnut cake for her class. The pictures below show how much cake she brought to school and how much she had left at the end of the day.
Which fraction represents the difference between the amounts of cake Mrs. Friedmon had before school and after school?
Options:
a. $$\frac{5}{8}$$
b. 1 $$\frac{1}{2}$$
c. 1 $$\frac{5}{8}$$
d. 2 $$\frac{1}{2}$$
Answer: 1 $$\frac{5}{8}$$
Explanation:
The fraction for the above figure is 1 $$\frac{7}{8}$$
The fraction for the second figure is $$\frac{1}{4}$$
1 + $$\frac{7}{8}$$ – $$\frac{1}{4}$$
$$\frac{7}{8}$$ – $$\frac{1}{4}$$ = $$\frac{7}{8}$$ – $$\frac{2}{8}$$
$$\frac{7}{8}$$ – $$\frac{2}{8}$$ = $$\frac{5}{8}$$
1 + $$\frac{5}{8}$$ = 1 $$\frac{5}{8}$$
The correct answer is option C.
Question 19.
Cody is designing a pattern for a wood floor. The length of the pieces of wood are 1 $$\frac{1}{2}$$ inches, 1 $$\frac{13}{16}$$ inches, and 2 $$\frac{1}{8}$$ inches. What is the length of the 5th piece of wood if the pattern continues?
Options:
a. 2 $$\frac{7}{6}$$ inches
b. 2 $$\frac{3}{4}$$ inches
c. 3 $$\frac{1}{2}$$ inches
d. 4 inches
Answer: 2 $$\frac{3}{4}$$ inches
Explanation:
The length of the pieces of wood are 1 $$\frac{1}{2}$$ inches, 1 $$\frac{13}{16}$$ inches, and 2 $$\frac{1}{8}$$ inches
1 $$\frac{1}{2}$$ = $$\frac{3}{2}$$
1 $$\frac{13}{16}$$ inches = $$\frac{29}{16}$$
$$\frac{29}{16}$$ – $$\frac{3}{2}$$ = latex]\frac{5}{16}[/latex]
5th piece = $$\frac{3}{2}$$ + latex]\frac{5}{16}[/latex] (5 – 1)
= $$\frac{3}{2}$$ + latex]\frac{5}{16}[/latex] 4
= $$\frac{3}{2}$$ + latex]\frac{20}{16}[/latex]
= $$\frac{3}{2}$$ × latex]\frac{8}{8}[/latex] + latex]\frac{20}{16}[/latex]
= latex]\frac{44}{16}[/latex] = 2 latex]\frac{3}{4}[/latex]
Thus the correct answer is option B.
Question 20.
Julie spends $$\frac{3}{4}$$ hour studying on Monday and $$\frac{1}{6}$$ hour studying on Tuesday. How many hours does Julie study on those two days?
Options:
a. $$\frac{1}{3}$$ hour
b. $$\frac{2}{5}$$ hour
c. $$\frac{5}{6}$$ hour
d. $$\frac{11}{12}$$ hour
Answer: $$\frac{11}{12}$$ hour
Explanation:
Julie spends $$\frac{3}{4}$$ hour studying on Monday and $$\frac{1}{6}$$ hour studying on Tuesday.
$$\frac{3}{4}$$ + $$\frac{1}{6}$$
LCD = 12
$$\frac{9}{12}$$ + $$\frac{2}{12}$$ = $$\frac{11}{12}$$ hour
So, the correct answer is option D.
### Chapter Review/Test – Page No. 288
Constructed Response
Question 21.
A class uses 8 $$\frac{5}{6}$$ sheets of white paper and 3 $$\frac{1}{12}$$ sheets of red paper for a project. How much more white paper is used than red paper? Show your work using words, pictures, or numbers. Explain how you know your answer is reasonable.
______ $$\frac{□}{□}$$ sheet of white paper
Answer: 5 $$\frac{3}{4}$$ sheet of white paper
Explanation:
A class uses 8 $$\frac{5}{6}$$ sheets of white paper and 3 $$\frac{1}{12}$$ sheets of red paper for a project.
8 $$\frac{5}{6}$$ – 3 $$\frac{1}{12}$$
8 + $$\frac{5}{6}$$ – 3 – $$\frac{1}{12}$$
8 – 3 = 5
$$\frac{5}{6}$$ – $$\frac{1}{12}$$
$$\frac{10}{12}$$ – $$\frac{1}{12}$$ = $$\frac{9}{12}$$
$$\frac{9}{12}$$ = $$\frac{3}{4}$$
5 + $$\frac{3}{4}$$ = 5 $$\frac{3}{4}$$
Question 22.
For a family gathering, Marcos uses the recipe below to make a lemon-lime punch.
A). How would you decide the size of a container you need for one batch of the Lemon-Lime Punch?
Type below:
________
Answer: He may use $$\frac{1}{4}$$ gallon lime juice for one batch of the lemon-lime punch.
Question 22.
B). If Marcos needs to make two batches of the recipe, how much of each ingredient will he need? How many gallons of punch will he have? Show your math solution and explain your thinking when you solve both questions.
Type below:
________
Answer: $$\frac{2}{3}$$ gallon lime juice
Question 22.
C). Marcos had 1 $$\frac{1}{3}$$ gallons of punch left over. He poured all of it into several containers for family members to take home. Use fractional parts of a gallon to suggest a way he could have shared the punch in three different-sized containers.
Type below:
________
Answer: 1 $$\frac{1}{12}$$
Conclusion:
Real-time learning is very important for students. By following the concepts given in the Go Math Grade 5 Chapter 6 Solution Key the students can solve the questions easily in the exam. If you understand the concept you can solve any type of question. Try to solve the questions given at the end of the chapter to test your knowledge. Get Chapter-wise Solutions in our Go Math Answer Key.
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Viewpoint: Second-harmonic generation in microresonators through natural phase matching
Physics 3, 32
Researchers have found a way to naturally double the frequency of laser light with an optical microresonator made from lithium niobate that supports “whispering gallery” modes.
Resonators are ubiquitous in physics, and recently much effort has gone into developing optical microresonators that offer strong spatial and long temporal confinement. The basic principle relies on the fact that light can be trapped inside a microresonator via continuous total internal reflection. The modes formed in this manner are also known as “whispering gallery modes” (WGM), since they are an optical analogy to the acoustical effect that was observed by Lord Rayleigh in Saint Paul’s cathedral in which sounds travel efficiently along a curved wall [1]. Now, in a paper published in Physical Review Letters, Josef Fürst, Dmitry Strekalov, Dominique Elser, Mikael Lassen, Ulrik Andersen, Christoph Marquardt, and Gerd Leuchs from the Max Planck Institute for the Science of Light (Erlangen), the University of Erlangen-Nuremberg, both in Germany, the Jet Propulsion Laboratory in Pasadena, US, and the Technical University of Denmark report a way to easily generate the second harmonic of laser light using such resonators [2] (see Fig. 1).
Optical WGM resonators have a natural “figure of merit”: their optical quality factor $Q$, which expresses their ability to confine light for long amounts of time. While it has long been known that dielectric particles exhibit whispering gallery mode resonances, it was the observation of ultrahigh $Q$ modes that led to optical microresonators being widely employed [3]. Ultrahigh $Q$ modes at optical wavelengths were first demonstrated in 1989 by Braginsky and Gorodetsky [4] in silica microspheres, and remarkably high $Q$ values exceeding ten billion (${10}^{9}$) have been observed. In the last decade, several other varieties of ultrahigh $Q$ microresonators have emerged, which lead to a wide range of new microresonator applications in cavity quantum electrodynamics [5, 6], nonlinear optics [7, 8], cavity optomechanical studies [9], or cavity enhanced sensing schemes. On-chip silica microtoroids have been discovered [10] that allowed bringing this $Q$ factor into a chip scale. While microtoroids and microspheres employ fused silica as the material of the dielectric, a versatile method to extend ultrahigh $Q$ to other materials has been developed by the Jet Propulsion Laboratory in Pasadena, US. By simply polishing a cylinder blank [11] (such as pure ${\text{CaF}}_{2}$ crystals), $Q$ factors exceeding ten billion have been demonstrated, only limited by residual surface roughness.
In contrast to fused silica, crystalline materials have the distinct advantage of being extraordinarily transparent to frequencies from ultraviolet to the mid-infrared. In addition, these materials exhibit low mechanical dissipation [12], which make them appealing for optomechanical studies. The highest optical finesse (the power enhancement factor) demonstrated to date is ${10}^{7}$ [13], which implies that only $1\phantom{\rule{0.333em}{0ex}}\mu \text{W}$ of power gives rise to $10\phantom{\rule{0.333em}{0ex}}\text{W}$ of circulating power inside the crystalline microresonator. Not surprisingly, at such power levels (and due to the small transverse confinement high intensity), a rich set of nonlinear phenomena have been observed in microresonators, for instance, Raman lasing [7], Brillouin scattering [14], parametric oscillations [15, 16] and optical frequency comb generation [17, 18]. A common feature of these nonlinear oscillations in microresonators is their exceptionally low threshold; threshold levels for nonlinear oscillation are regularly in the sub-$\mu \text{W}$ regime, values that are traditionally associated with linear optics. In addition, radiation pressure interactions have been observed in both silica microresonators [19, 20] as well as crystalline resonators [21], whereby the mechanical modes of the microresonators interact with the optical modes. So far, however, most of the nonlinear optical studies have relied on silica or crystalline resonators made out of materials that lack inversion symmetry and therefore only contain a third-order nonlinearity.
A natural extension has been the development of resonators with a second-order nonlinearity, a necessity for processes such as frequency doubling of laser light. Second-order nonlinear materials are particularly interesting from the perspective of quantum optics. The second-order process annihilates two pump photons and creates a photon with twice the frequency and hence twice the energy, which can give rise to optical squeezing [22]. A natural step would hence be to combine the ultralow-loss optical microresonators with a second-order nonlinearity to achieve second-harmonic generation. The paper by Fürst et al.—following pioneering work at the Jet Propulsion Laboratory [11]—the work makes a further important step in this direction. The challenge of making a resonator out of a second-order nonlinear material was relaxed with the advent of micropolishing techniques—by polishing a circular blank (with mm-scale size) of ${\text{LiNbO}}_{3}$, high $Q$ resonators have been fabricated whose quality factor ($Q$ of the order of ${10}^{7}$) is only limited by the material quality [23]. In order to generate a second-harmonic field, however, phase matching has to be obeyed.
For phases to be matched, the wave vectors for the two frequencies have to obey the law of conservation of momentum. The WGM resemble mathematically the solutions for the orbitals of a hydrogen atom, characterized by an azimuthal, angular, and radial mode number (as well as a polarization). Just as in the case of atomic transitions, selection rules apply to, and govern, the second-order nonlinear interaction. As the optical modes are angular momentum eigenstates, the second-order nonlinear interaction requires conservation of the angular momentum mode number. Hence for a given pump frequency, this fixes the angular mode number of the mode in which the second harmonic is converted. This condition means that the second-harmonic generation mode must carry twice as much angular momentum as the pump mode, implying that the mode number of the scattered mode is $2l$, when the pump mode has angular momentum $l$. This condition can always be satisfied. However, the frequency of the second harmonic needs to also satisfy energy conservation—if the resonator is pumped at frequency $\omega$, the optical mode with twice the angular momentum ($2l$) must exist precisely at frequency $2\omega$ in order for second-harmonic generation to be able to take place. Due to the small size of the resonators and the high $Q$, the latter is enormously challenging. In the past, phase matching in microresonators had been achieved using periodic poling of the ${\text{LiNbO}}_{3}$, however, the method relied on accidental frequency matching [23].
The Letter by Fürst et al. takes a much more practical and reliable route to achieve phase matching. Using the fact that the crystal has two refractive indices associated with the ordinary and extraordinary direction of propagation, which, importantly, exhibit different temperature dependence, one can achieve a differential tuning of the pump mode with respect to the second harmonic. This thereby allows continuous tuning of the relative frequency between the pump and the second harmonic until the two satisfy the law of conservation of energy. In this manner, the authors observe efficient second-harmonic generation. This method therefore opens up to study these phenomena in a reliable manner.
The implications of this work are manifold. First, the resonators can serve as efficient doubling cavities, in particular when combined with efficient tapered optical fiber coupling. More fascinating, however, is the ability to generate squeezed states of light. Squeezing of the pump laser as well as the generated second-harmonic generation should be possible, which makes compact sources of squeezed light a tantalizing possibility. While it is yet not clear if other obstacles such as thermorefractive noise [24] may impede such studies, certainly one aspect is clear: the quest for ultrahigh $Q$ in microresonators continues to bring new advances in a wide range of fields, making them even more indispensible in the future.
References
1. L. Rayleigh, Scientific Papers (Cambridge University Press, Cambridge, 1912)
2. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, Phys. Rev. Lett. 104, 153901 (2010)
3. K. J. Vahala, Nature 424, 839 (2003)
4. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989)
5. T. Aoki et al., Nature 443, 671 (2006)
6. D. W. Vernooy et al., Phys. Rev. A 57, R2293 (1998)
7. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, Nature 415, 621 (2002)
8. A. A. Savchenkov et al., Phys. Rev. Lett. 93, 243905 (2004)
9. T. J. Kippenberg and K. J. Vahala, Opt. Express 15, 17172 (2007)
10. D. K. Armani et al., Nature 421, 925 (2003)
11. V. S. Ilchenko et al., Phys. Rev. Lett. 92, 043904 (2004)
12. V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (University of Chicago Press, Chicago, 1985)[Amazon][WorldCat]
13. A. A. Savchenkov et al., Opt. Express 15, 6768 (2007)
14. I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009)
15. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905 (2004)
16. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004)
17. P. Del’Haye et al., Nature 450, 1214 (2007)
18. A. A. Savchenkov et al., Phys. Rev. Lett. 101, 093902 (2008)
19. T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008)
20. Florian Marquardt and Steve Girvin, Physics 2, 40 (2009)
21. J. Hofer, A. Schliesser, and T. J. Kippenberg, arXiv:0911.1178v2
22. S. F. Pereira, Min Xiao, H. J. Kimble, and J. L. Hall, Phys. Rev. A 38, 4931 (1988)
23. V. S. Ilchenko et al., Phys. Rev. Lett. 92, 043903 (2004)
24. M. L. Gorodetsky and I. S. Grudinin, J. Opt. Soc. Am. B, 21, 697 (2004)
About the Author
Tobias J. Kippenberg is Associate Professor of Physics and Electrical Engineering at EPFL and leads the Laboratory of Photonics and Quantum Measurement. He obtained his B.A. at the RWTH Aachen, and M.A. and Ph.D. at the California Institute of Technology (Caltech in Pasadena, US). From 2005 to 2009 he led an Independent Research Group at the MPI of Quantum Optics. He is an alumnus of the Studienstiftung des Deutschen Volkes and winner of the 8th EU Contest for Young Scientists (1996) for his invention of an infrared-microwave radiation ice condition sensor for cars. For his invention of chip-scale frequency combs, he is co-recipient of the Helmholtz Price for Metrology (2009). Moreover, he is recipient of the EFTF Young Investigator Award (2010) and the EPS Fresnel Prize (2009).
Optics
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# On infinite abelian p-group of bounded order
Definition. If $p$ is a prime, then a p-group is a group in which every element has order a power of $p$. Remark: An additively writen group is called $bounded$ if its elements have boundedly finite orders. Of course multiplicative groups with this property are said to have finite exponent but this term is inappropriate in the context of additive groups.
Let $G$ be an infinite abelian p-group of bounded order, then prove that $G\cong \mathbb{Z}_{p^{n}}\oplus\mathbb{Z}_{p^{n}}\oplus H$, for some natural number $n$ and for some abelian group $H$.
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Can you tell me what is the meaning of "infinite abelian p-group of bounded order"? Let me make it clear: I am not telling you, you're wrong. It is just that I don't understand and I am willing to learn about it! – user21436 Feb 5 '12 at 14:21
The "bounded order" you mention is usually called the exponent of the group. So you are asking about an infinite abelian p-group of bounded exponent. Try using the following theorem of Prufer:
Theorem: If $A$ is an abelian p-group of bounded exponent, then $A$ is a direct sum of cyclic groups.
Sketch of Proof: Let $p^n$ be the exponent, and induct on $n$. The base case $n=1$ is a corollary of the fact $A$ is then a vector space over $\mathbb{Z}/p\mathbb{Z}$.
For the inductive step, consider $pA$, which is a direct sum of cyclic groups. Let $pA=\oplus A_i$, with the $A_i$ cyclic and generated by $a_i$. Let $pb_i=a_i$. Then show $B$ (generated by the $b_i$) is a direct sum of cyclic groups.
Finally, pick a subgroup $C$ of $A$ maximal with respect to satisfying $C\cap B=\lbrace0\rbrace$. Show $A=B\oplus C$.
To get the statement you gave, write $A=\oplus X_i$ as the direct sum of cyclic subgroups. Note that each $X_i\cong \mathbb{Z}/p^m\mathbb{Z}$ for some $m\le n$; if there was only (at most) one copy of each, $A$ would be finite.
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# Gerhard Jäger – From fixed points in weak set theories to some open problems
Research seminar, Kurt Gödel Research Center – October 9th
Abstract: Least fixed points of monotone operators are well-studied objects in many
areas of mathematical logic. Typically, they are characterized as the
intersection of all sets closed under the respective operator or as the
result of its iteration from below.
In this talk I will start off from specific $\Sigma_1$ operators in a
Kripke-Platek environment and relate fixed point assertions to alternative
set existence principles. By doing that, we are also led to some
“largeness axioms” and to several open problems.
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# Charging one plate of capacitor and grounding the other plate of capacitor
I want to learn about this way of charging the capacitor. At my university, we charge capacitor with power supply. Its negative power supply. Power supply is grounded (earthed). A conductor from power supply is attached to one plate of capacitor and other plate of capacitor is grounded (earthed) separately. Both earthed points are different (physically). I want to learn how this capacitor is getting charged?
Both earthed points are different (physically). I want to learn how this capacitor is getting charged
The fact that the power supply and one plate of the capacitor are earth grounded at different locations simply potentially introduces additional resistance through which charging occurs. That resistance increases the charging time constant (t=RC) slowing down the rate of charging the capacitor. How slow for a given capacitance C depends on how much resistance exists between the earth connection of the power supply and the earth connection of the one plate, which in turn depends on the conductivity of the soil and the distance between the connections.
Perhaps the diagram below illustrates your university experiment. It assumes there is no other resistor connected between the power supply and the capacitor, i.e., the only resistance is the soil between the earth connections and the resistance of the connecting wires is negligible. Under these conditions if the capacitor is initially uncharged, and the switch shown closes at time $$t=0$$, the theoretical voltage across the capacitor as a function of time, $$V_{C}(t)$$, would be
$$V_{C}(t)= V(1-e^{-t/RC})$$
If your experiment involved another resistor connected between the power supply and capacitor, you would need to add it to the value of $$R$$ in the above equation.
I want to add a further question. What would happen if we remove the supply and connect a ground stick (connected to another separate ground) at the positive terminal of the capacitor shown in the figure above. How would a charged capacitor behave in this scenario?
The capacitor will discharge through the ground. In this case the voltage across the capacitor as a function of time will be $$V_{C}(t)=V_{o}\epsilon ^{-t/RC}$$ where $$V_{o}$$ is the initial voltage across the capacitor.
Hope this helps.
• Thanks alot Bob D. I want to add a further question. What would happen if we remove the supply and connect a ground stick (connected to another separate ground) at the positive terminal of the capacitor shown in the figure above. How would a charged capacitor behave in this scenario? Oct 22 '21 at 19:44
• Are you asking what would happen if you started with a charged capacitor and connected both the positive and negative plates to earth with separate grounding electrodes? Oct 22 '21 at 20:02
• Exactly. I am asking the same. Oct 23 '21 at 7:09
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# MPLAB X, PWM, Compile error
#### Ctenom
Joined Nov 1, 2010
59
Hi i have spent the last few hour's trying to get PWM to work on my pic18f16k22. Im using the C18 compiler in MPLABX V6.1 The probolem im having is it compiles with this error
Error - could not find definition of symbol 'SetDCPWM1' in file './build/default/production/_ext/916313547/PWM.o'.
Errors : 1
The PWM.h file definatly contains the definition and is included in my config.
Rich (BB code):
void SetDCPWM1 ( unsigned int duty_cycle);
I started off wrighting my own code then found several examples on the web and thay all compile with this error
My last code looks like this using some code from a C18 cheat sheet by AtomSoftTech.
Rich (BB code):
#include <p18Cxxx.h>
#include <pwm.h>
#include <delays.h>
#pragma config FOSC = INTIO67 // Oscillator Selection
#pragma config WDTEN = OFF // Watchdog Timer
#pragma config MCLRE = INTMCLR // MCLR Pin Enable bit
#pragma config DEBUG = ON // Background Debug
#define PWM_Tris TRISCbits.TRISC2 //Define a nice name for the PWM TRIS bit.
char direction; //direction holder for out loop
void main (void)
{
char DutyCyc = 0x66; //Start Duty Cycle
OSCCON = 0x72; //8MHz clock
while(!OSCCONbits.IOFS); //Wait for OSC to become stable
PWM_Tris = 0; //Set the pin a OUTPUT
SetOutputPWM1 (SINGLE_OUT, PWM_MODE_1); //Set as single output pwm and mode 1 (P1A,P1B,P1C,P1D active-high)
OpenPWM1(DutyCyc); //Set the duty cycle and start the PWM module
while(1){
if(DutyCyc == 0x12) //Check if we reached our minimum. If so set direction (up)
direction = 1;
if(DutyCyc == 0x66) //Check if we reached our maximum. If so clear direction (down)
direction = 0;
SetDCPWM1(DutyCyc); //Set the new duty cycle
if(direction == 1) //based on direction if 1 (up) add 2 each time to the duty cycle
DutyCyc+=2;
else // if 0 (down) minus 2 each time to the duty cycle
DutyCyc-=2;
Delay10KTCYx(3); //small delay (15mS)
}
}
Thank you for taking time to read my post.
#### Ctenom
Joined Nov 1, 2010
59
I installed the new version after completely deleting the old one.
And got a slightly different error but still the same problem.
Error - could not find definition of symbol 'SetDCPWM1' in file './build/default/production/PWM.o'.
Errors : 1
And i was expecting that to work as well
#### nsaspook
Joined Aug 27, 2009
10,897
For some of the chips the routines may have been renamed or renumbered. Check the
"PIC18F Peripheral Library Help Document" in the C18 doc folder.
#### Ctenom
Joined Nov 1, 2010
59
Just tried updating my C18 from 3.39 to 3.40 still no joy.
#### Ctenom
Joined Nov 1, 2010
59
Thanks for your help, i belive it might just be teething problems with MPLABX as i have tried just about everything now. I finally got it working using HI-TECH C heres my test code (makes a pretty fading led).
Rich (BB code):
#include <htc.h>
__CONFIG (1,FOSC_INTIO67);
__CONFIG (2,WDTEN_OFF);
__CONFIG (3,MCLRE_INTMCLR);
__CONFIG (4,STVREN_OFF & DEBUG_ON & LVP_OFF);
void main()
{
unsigned char dc ;
TRISC = 0 ;
PORTC = 0 ;
PR2 = 0b01111100 ;
T2CON = 0b00000101 ;
CCP1CON = 0b00001100 ;
CCP2CON = 0b00111100 ;
for(;;)
{
for(dc = 5 ; dc < 128 ; dc++)
{
CCPR1L = dc ;
CCPR2L = 128 - dc ;
_delay(6000);
}
for(dc = 127 ; dc > 5 ; dc--)
{
CCPR1L = dc ;
CCPR2L = 128 - dc ;
_delay(4000);
}
}
}
Now the proper coding can begin.
#### stahta01
Joined Jun 9, 2011
133
The PWM.h file definatly contains the definition and is included in my config.
Rich (BB code):
void SetDCPWM1 ( unsigned int duty_cycle);
That code is the function prototype and is the declaration NOT the definition of the function. If this was normal C (instead of embedded C) I would say you are missing a Library that needs linked into the project.
Since it is a embedded C, could be a missing(or wrong) Compiler or Linker Options; but, I would guess that a object file needs linked that provides the function.
Quote from http://www.informit.com/guides/content.aspx?g=cplusplus&seqNum=188
a definition causes the compiler to allocate storage for an object, whereas a declaration merely associates an object with a certain type, without allocating storage for it.
Tim S.
#### ErnieM
Joined Apr 24, 2011
8,353
Ctenom: When you were using the C18 did you set the project library include search path? I don't use MPLABX (I use the non-X) but that path and a few others needs to be set for every new project. MPLAB will show the correct path by default, but you still have to select it and OK it.
Tim: What would be the difference between embedded and this "normal" C? My PIC C compilers are pretty much ANSI standard, with some minor defined exceptions.
#### stahta01
Joined Jun 9, 2011
133
Tim: What would be the difference between embedded and this "normal" C? My PIC C compilers are pretty much ANSI standard, with some minor defined exceptions.
I have learned that embedded C have a lot of non standard extensions and sometimes you need to link to an system object file instead of with a system library file. Note: Borland also had this linking with system object file; so, it is not just a embedded C issue.
Note: My main experience in the embedded world is the Dynamic C [like] Compiler from Rabbit (now owned by Digi). And old experience (20+ years ago) with OS9 from Microware. And, PIC Assembly/C as a Student Assistant Teacher. Note: Rabbit Dynamic C IS NOT a ANSI C Language; but getting closer every release cycle.
Edit: The Microware C Compiler required linking to system object files to get a good build like the Borland C Compiler did.
Tim S.
Last edited:
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{}
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Application Domains
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
PDF e-Pub
## Section: New Results
### Algorithmic aspects of topological and geometric data analysis
#### DTM-based filtrations
Participants : Frédéric Chazal, Marc Glisse, Raphaël Tinarrage.
In collaboration with H. Anai, Y. Ike, H. Inakoshi and Y. Umeda of Fujitsu.
Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Čech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper [33], we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.
#### Persistent Homology with Dimensionality Reduction: $k$-Distance vs Gaussian Kernels
Participants : Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta.
We investigate the effectiveness of dimensionality reduction for computing the persistent homology for both $k$-distance and kernel distance [34]. For $k$-distance, we show that the standard Johnson-Lindenstrauss reduction preserves the $k$-distance, which preserves the persistent homology upto a ${\left(1-\epsilon \right)}^{-1}$ factor with target dimension $O\left(klogn/{\epsilon }^{2}\right)$. We also prove a concentration inequality for sums of dependent chi-squared random variables, which, under some conditions, allows the persistent homology to be preserved in $O\left(logn/{\epsilon }^{2}\right)$ dimensions. This answers an open question of Sheehy. For Gaussian kernels, we show that the standard Johnson-Lindenstrauss reduction preserves the persistent homology up to an $4{\left(1-ϵ\right)}^{-1}$ factor.
#### Computing Persistent Homology of Flag Complexes via Strong Collapses
Participants : Jean-Daniel Boissonnat, Siddharth Pritam.
In collaboration with Divyansh Pareek (Indian Institute of Technology Bombay, India)
#### Strong Collapse for Persistence
Participants : Jean-Daniel Boissonnat, Siddharth Pritam.
In this paper, we build on the initial success of and show that further decisive progress can be obtained if one restricts the family of simplicial complexes to flag complexes. Flag complexes are fully characterized by their graph (or 1-skeleton), the other faces being obtained by computing the cliques of the graph. Hence, a flag complex can be represented by its graph, which is a very compact representation. Flag complexes are very popular and, in particular, Vietoris-Rips complexes are by far the most widely simplicial complexes used in Topological Data Analysis. It has been shown in that the persistent homology of Vietoris-Rips filtrations can be computed very efficiently using strong collapses. However, most of the time was devoted to computing the maximal cliques of the complex prior to their strong collapse. In this paper [37], we observe that the reduced complex obtained by strong collapsing a flag complex is itself a flag complex. Moreover, this reduced complex can be computed using only the 1-skeleton (or graph) of the complex, not the set of its maximal cliques. Finally, we show how to compute the equivalent filtration of the sequence of reduced flag simplicial complexes using again only 1-skeletons. x On the theory side, we show that strong collapses of flag complexes can be computed in time $O\left({v}^{2}{k}^{2}\right)$ where $v$ is the number of vertices of the complex and $k$ the maximal degree of its graph. The algorithm described in this paper has been implemented and the code will be soon released in the Gudhi library. Numerous experiments show that our method outperforms previous methods, e.g. Ripser.
#### Triangulating submanifolds: An elementary and quantified version of Whitney's method
Participants : Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken.
#### Randomized incremental construction of Delaunay triangulations of nice point sets
Participants : Jean-Daniel Boissonnat, Kunal Dutta, Marc Glisse.
In collaboration with Olivier Devillers (Inria, CNRS, Loria, Université de Lorraine).
Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice.
Randomized incremental constructions are most of the time space and time optimal in the worst-case, as exemplified by the construction of convex hulls, Delaunay triangulations and arrangements of line segments.
However, the worst-case scenario occurs rarely in practice and we would like to understand how RIC behaves when the input is nice in the sense that the associated output is significantly smaller than in the worst-case. For example, it is known that the Delaunay triangulations of nicely distributed points in ${ℝ}^{d}$ or on polyhedral surfaces in ${ℝ}^{3}$ has linear complexity, as opposed to a worst-case complexity of $\Theta \left({n}^{⌊d/2⌋}\right)$ in the first case and quadratic in the second. The standard analysis does not provide accurate bounds on the complexity of such cases and we aim at establishing such bounds in this paper [35]. More precisely, we will show that, in the two cases above and variants of them, the complexity of the usual RIC is $O\left(nlogn\right)$, which is optimal. In other words, without any modification, RIC nicely adapts to good cases of practical value.
Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.
#### Approximate Polytope Membership Queries
Participant : Guilherme Da Fonseca.
In collaboration with Sunil Arya (Hong Kong University of Science and Technology) and David Mount (University of Maryland).
#### Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums
Participant : Guilherme Da Fonseca.
In collaboration with Sunil Arya (Hong Kong University of Science and Technology) and David Mount (University of Maryland).
#### Approximating the Spectrum of a Graph
Participant : David Cohen-Steiner.
In collaboration with Weihao Kong (Stanford University), Christian Sohler (TU Dortmund) and Gregory Valiant (Stanford University).
#### Spectral Properties of Radial Kernels and Clustering in High Dimensions
Participants : David Cohen-Steiner, Alba Chiara de Vitis.
In this paper [40], we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in ${ℝ}^{n}$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but different covariance matrices. This algorithm will succeed if the angle between any two covariance matrices in the mixture (seen as vectors in ${ℝ}^{{n}^{2}}$) is larger than $\Omega \left({n}^{-1/6}{log}^{5/3}n\right)$. In particular, the required angular separation tends to 0 as the dimension tends to infinity. To the best of our knowledge, this is the first polynomial time algorithm for clustering such mixtures beyond the Gaussian case.
#### Exact computation of the matching distance on 2-parameter persistence modules
Participant : Steve Oudot.
In collaboration with Michael Kerber (T.U. Graz) and Michael Lesnick (SUNY).
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In [44] we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.
#### A Comparison Framework for Interleaved Persistence Modules
Participant : Miroslav Kramár.
In collaboration with Rachel Levanger (UPenn), Shaun Harker and Konstantin Mischaikow (Rutgers).
In [43], we present a generalization of the induced matching theorem of [1] and use it to prove a generalization of the algebraic stability theorem for R-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.
#### Discrete Morse Theory for Computing Zigzag Persistence
Participant : Clément Maria.
In collaboration with Hannah Schreiber (Graz University of Technology, Austria)
|
{}
|
#### Howdy, Stranger!
It looks like you're new here. If you want to get involved, click one of these buttons!
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# Azimuth blog overview
Just in case someone finds this useful, out of my frustration to be unable to scroll down the blog easily while looking for older articles (I can use Google though if I know a keyword) I've made this list on Azimuth Blog Overview. In any case, it didn't take me much time and even less concentration, so I don't think the effort is wasted, since I can use it myself.
@John: I don't expect that it's worth any thumbs up ;-)
• Options
1.
edited March 2013
Renamed to Azimuth blog overview.
Hoping that John will add links to his series lists, so that we can gradually build this page up into a full and meaningful index of the blog.
Once it’s in reasonable shape, it would be nice to add a link to it in the homepage of the wordpress blog (if they allow you to do such customizations).
Comment Source:Renamed to [[Azimuth blog overview]]. Organized the links by series. Hoping that John will add links to his series lists, so that we can gradually build this page up into a full and meaningful index of the blog. Once it’s in reasonable shape, it would be nice to add a link to it in the homepage of the wordpress blog (if they allow you to do such customizations).
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2.
Thanks a lot, guys!
I won't give you an official thumbs up, Frederik... you still owe me all those articles! But unofficially, I think this is great.
The blog has limited customizability but there should be some way to link to this information there.
I'm a bit tired, so I only added links to the Game Theory series (which I happen to have prepared today, in a somewhat different format). I have nice annotated lists of posts for the Network Theory and Information Geometry series, which I can include later.
The main series that seems to be completely missing is The Mathematics of Biodiversity, which has, I believe, 7 parts.
Comment Source:Thanks a lot, guys! I won't give you an _official_ thumbs up, Frederik... you still owe me all those articles! But unofficially, I think this is great. The blog has limited customizability but there should be some way to link to this information there. I'm a bit tired, so I only added links to the Game Theory series (which I happen to have prepared today, in a somewhat different format). I have nice annotated lists of posts for the Network Theory and Information Geometry series, which I can include later. The main series that seems to be completely missing is The Mathematics of Biodiversity, which has, I believe, 7 parts.
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3.
The last update I made is from Nov 5, 2011 so unless David added more recent entries everything from 2012 is missing.
If you have a student suffering from procrastination you could urge them to complete the list ;-)
Comment Source:The last update I made is from Nov 5, 2011 so unless David added more recent entries everything from 2012 is missing. If you have a student suffering from procrastination you could urge them to complete the list ;-)
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4.
Right, what I did was to organize the links that you had put there. Then I added a few more.
I just updated the initial statement on the page, to:
This page is an index into the Azimuth blog articles on Wordpress. The sections are for series, or other topic categories, and are listed alphabetically. For other articles, there is a section called Authors, with one subsection per author.
Of course, if you have an improvement to the organization, then go for it, and document it in the header.
Suggestion, can anyone -- whoever is up for it -- who has articles in the subsection called Uncategorized create a subsection for themselves, and move the links into it. Once you have fulfilled this mission, your next challenge will be to add links to your articles that were written after Nov 5, 2011.
Frederik, did you start from the beginning of the blog until Nov 5, 2011?
Comment Source:Right, what I did was to organize the links that you had put there. Then I added a few more. I just updated the initial statement on the page, to: > This page is an index into the Azimuth blog articles on Wordpress. The sections are for series, or other topic categories, and are listed alphabetically. For other articles, there is a section called Authors, with one subsection per author. Of course, if you have an improvement to the organization, then go for it, and document it in the header. Suggestion, can anyone -- whoever is up for it -- who has articles in the subsection called Uncategorized create a subsection for themselves, and move the links into it. Once you have fulfilled this mission, your next challenge will be to add links to your articles that were written after Nov 5, 2011. Frederik, did you start from the beginning of the blog until Nov 5, 2011?
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5.
When I added the articles in Petri net programming, I added the months as well. I think the chronology gives a nice extra touch.
Comment Source:When I added the articles in Petri net programming, I added the months as well. I think the chronology gives a nice extra touch.
• Options
6.
edited March 2013
I added annotated descriptions of each post in the Information Geometry and Network Theory series - I had these descriptions on my website already.
If you have a student suffering from procrastination you could urge them to complete the list ;-)
If we could use the whole world's procrastination, our power would be enormous. It would be a bit like "SETI At Home" and those other projects that use spare computing power to solve scientific problems.
Comment Source:I added annotated descriptions of each post in the Information Geometry and Network Theory series - I had these descriptions on my website already. > If you have a student suffering from procrastination you could urge them to complete the list ;-) If we could use the whole world's procrastination, our power would be enormous. It would be a bit like "SETI At Home" and those other projects that use spare computing power to solve scientific problems.
• Options
7.
edited March 2013
@David Tanzer:
Frederik, did you start from the beginning of the blog until Nov 5, 2011?
I started at the beginning, so I am convinced I did.
When I added the articles in Petri net programming, I added the months as well. I think the chronology gives a nice extra touch.
yes, good idea! Originally I made the links chronologically, but grouping in topics has advantages. (I think you moved Moore's law below Welcome to Azimuth, which was the first entry)
Comment Source:@David Tanzer: > Frederik, did you start from the beginning of the blog until Nov 5, 2011? I started at the beginning, so I am convinced I did. > When I added the articles in Petri net programming, I added the months as well. I think the chronology gives a nice extra touch. yes, good idea! Originally I made the links chronologically, but grouping in topics has advantages. (I think you moved Moore's law below Welcome to Azimuth, which was the first entry)
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8.
edited March 2013
I just:
• created a link to this page from the word ‘Azimuth Blog' on the homepage of the Azimuth Wiki. I think this overview is a better way to get started than simply diving into the blog.
• added a section called ‘Azimuth news', right on top - this is for articles on the blog that are about Azimuth.
• added the last parts of the Game Theory course, which is done. (Yay!)
• moved one more of Tim's articles into the section ‘Tim van Beek'.
Comment Source:I just: * created a link to [this page](http://www.azimuthproject.org/azimuth/show/Azimuth+blog+overview) from the word ‘Azimuth Blog' on the homepage of the Azimuth Wiki. I think this overview is a better way to get started than simply diving into the blog. * created a link from this page to the Azimuth Blog, right at the top. * added a section called ‘Azimuth news', right on top - this is for articles on the blog that are about Azimuth. * added the last parts of the Game Theory course, which is done. (Yay!) * moved one more of Tim's articles into the section ‘Tim van Beek'.
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9.
Cool, these are good ideas.
By the way, the links to the "web versions" in the Information Geometry section don't work -- they are file names, not full URLs.
Comment Source:Cool, these are good ideas. By the way, the links to the "web versions" in the Information Geometry section don't work -- they are file names, not full URLs.
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10.
I just put all of the uncategorized articles into author subsections.
Our next challenge is to get everything from Nov 5, 2011 up to the present -- need to do some fishing on the wordpress blog.
Comment Source:I just put all of the uncategorized articles into author subsections. Our next challenge is to get everything from Nov 5, 2011 up to the present -- need to do some fishing on the wordpress blog.
• Options
11.
I have brought this index of the blog articles up to date. Here we have 379 articles, 27 series and 17 authors.
Comment Source:I have brought this index of the blog articles up to date. Here we have 379 articles, 27 series and 17 authors.
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12.
Thanks a million, David! I will put a link from the Azimuth Blog to this.
By the way, the links to the “web versions” in the Information Geometry section don’t work – they are file names, not full URLs.
Whoops - I fixed that, by adding http://math.ucr.edu/home/baez/information/
Comment Source:Thanks a million, David! I will put a link from the Azimuth Blog to this. > By the way, the links to the “web versions” in the Information Geometry section don’t work – they are file names, not full URLs. Whoops - I fixed that, by adding http://math.ucr.edu/home/baez/information/
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13.
Hi! This is a great idea. Is it OK to add the quantum network theory series?
Comment Source:Hi! This is a great idea. Is it OK to add the quantum network theory series?
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14.
Sure, of course. Just follow the format. But if you want to be really really nice, do all the Azimuth articles after 7-25-2013, and then change that date up at the top of the page. There are only about 8.
Comment Source:Sure, of course. Just follow the format. But if you want to be _really really_ nice, do all the Azimuth articles after 7-25-2013, and then change that date up at the top of the page. There are only about 8.
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15.
I just brought the index up to date. Merry Christmas.
This index is a good way of keeping track of the blog articles, but not the whole stream of comments. The notification mechanism from wordpress periodically forgets my subscription. Any recommendations for a good RSS reader would be welcome.
Comment Source:I just brought the index up to date. Merry Christmas. This index is a good way of keeping track of the blog articles, but not the whole stream of comments. The notification mechanism from wordpress periodically forgets my subscription. Any recommendations for a good RSS reader would be welcome.
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16.
edited December 2013
I just brought the index up to date.
Yay, thanks!
I added a link to Blog articles in progress, and I updated the list to cover my 26 December 2013 post.
By the way, we have an official rule for writing dates, which goes like "26 December 2013". Your index doesn't follow that rule. I don't think this will cause earthquakes and floods, so I won't fix it, since it would take me a long time. I just want people to know we have this policy: day month year, with month spelled out so nobody can get confused.
I'm afraid someone else will have to help you with that... I've sort of given up using an RSS reader, for some reason.
Comment Source:> I just brought the index up to date. Yay, thanks! <img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/> I added a link to [[Blog articles in progress]], and I updated the list to cover my 26 December 2013 post. By the way, we have an official rule for writing dates, which goes like "26 December 2013". Your index doesn't follow that rule. I don't think this will cause earthquakes and floods, so I won't fix it, since it would take me a long time. I just want people to know we have this policy: day month year, with month spelled out so nobody can get confused. > Any recommendations for a good RSS reader would be welcome. I'm afraid someone else will have to help you with that... I've sort of given up using an RSS reader, for some reason.
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17.
edited May 2014
I brought the index up to date -- from January to the present.
We have a number of new authors, which is Great: Marc Harper, Leonard Adleman, Alastair Jamieson-Lane, Vanessa Schweizer, Eugene Lerman, Steve Easterbrook, Ville Bergholm.
Bookkeeping note: I started distinguishing between multi-part blog articles and series. The former consists of articles which are only split up because they don't naturally fit into one article, whereas the latter represent themes which could be explored indefinitely. Examples of non-series multi-part article are Steve's 8-part article on the new IPCC report, and my two-part article on the stochastic resonance program.
Comment Source:I brought the index up to date -- from January to the present. We have a number of new authors, which is Great: Marc Harper, Leonard Adleman, Alastair Jamieson-Lane, Vanessa Schweizer, Eugene Lerman, Steve Easterbrook, Ville Bergholm. Bookkeeping note: I started distinguishing between multi-part blog articles and series. The former consists of articles which are only split up because they don't naturally fit into one article, whereas the latter represent themes which could be explored indefinitely. Examples of non-series multi-part article are Steve's 8-part article on the new IPCC report, and my two-part article on the stochastic resonance program. * [The Pentagram of Venus](http://johncarlosbaez.wordpress.com/2014/01/04/the-pentagram-of-venus/), John Baez, 4 January 2014 * [Lyapunov functions for complex-balanced systems](http://johncarlosbaez.wordpress.com/2014/01/07/lyapunov-functions-for-complex-balanced-systems/), Manoj Gopalkrishnan, 7 January 2014 * [Geometry Puzzles](http://johncarlosbaez.wordpress.com/2014/01/12/geometry-puzzles/), John Baez, 12 January 2014 * [Unreliable biomedical research](http://johncarlosbaez.wordpress.com/2014/01/13/unreliable-biomedical-research/), John Baez, 13 January 2014 * [Wormholes and entanglement](http://johncarlosbaez.wordpress.com/2014/01/20/wormholes-and-entanglement/), 20 January 2014 * (new author) [Relative entropy in evolutionary dynamics](http://johncarlosbaez.wordpress.com/2014/01/22/relative-entropy-in-evolutionary-dynamics/), Marc Harper, 22 January 2014 * (new author) [The rarest things in the universe](http://johncarlosbaez.wordpress.com/2014/01/27/the-rarest-things-in-the-universe/), Leonard Adleman, 27 January 2014 * [Bio-inspired information theory](http://johncarlosbaez.wordpress.com/2014/01/31/bio-inspired-information-theory/), John Baez, 31 January 2014 * [Category theory for better spreadsheets](http://johncarlosbaez.wordpress.com/2014/02/05/category-theory-for-better-spreadsheets/), John Baez, 5 February 2014 * [Categories in control](http://johncarlosbaez.wordpress.com/2014/02/06/categories-in-control/), John Baez, 6 February 2014 * [Network theory talks at Oxford](http://johncarlosbaez.wordpress.com/2014/02/07/network-theory-talks-at-oxford/), John Baez, 7 February 2014 * [Triangular numbers](http://johncarlosbaez.wordpress.com/2014/02/12/triangular-numbers/), John Baez, 12 February 2014 * [Relative entropy, Part 4](http://johncarlosbaez.wordpress.com/2014/02/16/relative-entropy-part-4/), John Baez, 16 February 2014 * [Finding and solving problems](http://johncarlosbaez.wordpress.com/2014/02/18/finding-and-solving-problems/), John Baez, 18 February 2014 * [Network theory overview](http://johncarlosbaez.wordpress.com/2014/02/22/network-theory-overview/), John Baez, 22 February 2014 * (new author) [Markov models of social change (part 1)](http://johncarlosbaez.wordpress.com/2014/02/24/markov-models-of-social-change-part-1/), Alastair Jamieson-Lane, 24 February 2014 * [Network theory I](http://johncarlosbaez.wordpress.com/2014/03/02/network-theory-i/), John Baez, March 2014 * [Network theory II](http://johncarlosbaez.wordpress.com/2014/03/12/network-theory-ii/), John Baez, March 2014 * [Network theory III](http://johncarlosbaez.wordpress.com/2014/03/16/network-theory-iii/), March 2014 * (new author) [Markov models of social change (part 2)](http://johncarlosbaez.wordpress.com/2014/03/05/markov-models-of-social-change-part-2/), Vanessa Schweizer, 5 March 2014 * (new author) [Networks of dynamical systems](http://johncarlosbaez.wordpress.com/2014/03/18/networks-of-dynamical-systems/), Eugene Lerman, 18 March 2014 * [Programming with chemical reaction networks](http://johncarlosbaez.wordpress.com/2014/03/23/programming-with-chemical-reaction-networks/), John Baez, 23 March 2014 * [Civilizational collapse (part 1)](http://johncarlosbaez.wordpress.com/2014/03/25/civilizational-collapse-part-1/), John Baez, 25 March 2014 * (new author) [New IPCC Report (part 1)](http://johncarlosbaez.wordpress.com/2014/04/07/what-does-the-new-ipcc-report-say-about-climate-change-part-1/), Steve Easterbrook, April 2014 * [New IPCC Report (part 2)](http://johncarlosbaez.wordpress.com/2014/04/09/what-does-the-new-ipcc-report-say-about-climate-change-part-2/), April 2014 * [New IPCC Report (part 3)](http://johncarlosbaez.wordpress.com/2014/04/10/what-does-the-new-ipcc-report-say-about-climate-change-part-3/), April 2014 * [New IPCC Report (part 4)](http://johncarlosbaez.wordpress.com/2014/04/11/what-does-the-new-ipcc-report-say-about-climate-change-part-4/), April 2014 * [New IPCC Report (part 5)](http://johncarlosbaez.wordpress.com/2014/04/14/what-does-the-new-ipcc-report-say-about-climate-change-part-5/), April 2014 * [New IPCC Report (part 6)](http://johncarlosbaez.wordpress.com/2014/04/16/what-does-the-new-ipcc-report-say-about-climate-change-part-6/), April 2014 * [New IPCC Report (part 7)](http://johncarlosbaez.wordpress.com/2014/04/18/what-does-the-new-ipcc-report-say-about-climate-change-part-7/), April 2014 * [New IPCC Report (part 8)](http://johncarlosbaez.wordpress.com/2014/04/22/what-does-the-new-ipcc-report-say-about-climate-change-part-8/), April 2014 * (new author) [Noether's Theorem: quantum vs stochastic](http://johncarlosbaez.wordpress.com/2014/05/03/noethers-theorem-quantum-vs-stochastic/), Ville Bergholm, 3 May 2014 * [Quantum frontiers in network science](http://johncarlosbaez.wordpress.com/2014/05/06/quantum-frontiers-in-network-science/), Jacob Biamonte, 6 May 2014 * [The stochastic resonance program (part 1)](http://johncarlosbaez.wordpress.com/2014/05/10/the-stochastic-resonance-program-part-1/), David Tanzer, 10 May 2014 * [West Antarctic ice sheet news](http://johncarlosbaez.wordpress.com/2014/05/16/west-antarctic-ice-sheet-news/), John Baez, 16 May 2014
• Options
18.
edited May 2014
Great!
I'd noticed this page was getting a bit out of date... thanks for rejuvenating it.
Comment Source:Great! I'd noticed this page was getting a bit out of date... thanks for rejuvenating it.
• Options
19.
Comment Source:Sure, glad to do it.
• Options
20.
edited October 2014
I just updated the Azimuth Blog Overview with all the articles from 16 May 2014 to the present:
Comment Source:I just updated the [[Azimuth Blog Overview]] with all the articles from 16 May 2014 to the present: * (new author) [Warming slowdown? (part 1)](http://johncarlosbaez.wordpress.com/2014/05/29/warming-slowdown-2/), Jan Galkowski, 29 May 2014 * [Warming slowdown? (part 2)](http://johncarlosbaez.wordpress.com/2014/06/05/warming-slowdown-part-2/), Jan Galkowski, 5 Jun 2014 * [The computational power of chemical reaction networks](http://johncarlosbaez.wordpress.com/2014/06/10/the-computational-power-of-chemical-reaction-networks/), John Baez, 10 Jun 2014 * [Wind power and the smart grid](http://johncarlosbaez.wordpress.com/2014/06/18/wind-power-and-the-smart-grid/), John Baez, 18 Jun 2014 * [El Niño Project (Part 1)](http://johncarlosbaez.wordpress.com/2014/06/20/el-nino-project-part-1/), John Baez, 20 Jun 2014 * [El Niño Project (Part 2)](http://johncarlosbaez.wordpress.com/2014/06/24/el-nino-project-part-2/), John Baez, 24 Jun 2014 * [El Niño Project (Part 3)](http://johncarlosbaez.wordpress.com/2014/07/01/el-nino-project-part-3/), John Baez, 1 Jul 2014 * [El Niño Project (Part 4)](http://johncarlosbaez.wordpress.com/2014/07/08/el-nino-project-part-4/), John Baez, 8 Jul 2014 * [El Niño Project (Part 5)](http://johncarlosbaez.wordpress.com/2014/07/12/el-nino-project-part-5/), John Baez, 12 Jul 2104 * (new author) [El Niño Project (Part6)](http://johncarlosbaez.wordpress.com/2014/07/23/el-nino-project-part-6/), Steven Wenner, 23 Jul 2014 * [El Niño Project (Part 7)](http://johncarlosbaez.wordpress.com/2014/08/18/el-nino-project-part-7/), John Baez, 18 Aug 2014 * [Chemical reaction network talks](http://johncarlosbaez.wordpress.com/2014/06/26/chemical-reaction-networks/), John Baez, 26 Jun 2014 * [Entropy and information in biological systems (part 2)](http://johncarlosbaez.wordpress.com/2014/07/04/entropy-and-information-in-biological-systems-part-2/), John Baez, 4 Jul 2014 * [The harmonograph](http://johncarlosbaez.wordpress.com/2014/07/18/the-harmonograph/), John Baez, 18 Jul 2014 * (new author) [Exploring climate data (Part 1)](http://johncarlosbaez.wordpress.com/2014/08/01/exploring-climate-data-part-1/), John Baez and Dara O Shayda, 1 Aug 2014 * (new author) [Exploring climate data (Part 2)](http://johncarlosbaez.wordpress.com/2014/09/16/exploring-climate-data-part-2/), Blake Pollard, 16 Sep 2014 * [Information aversion](http://johncarlosbaez.wordpress.com/2014/08/22/information-aversion/), John Baez, 22 Aug 2014 * [The stochastic resonance program (part 2)](http://johncarlosbaez.wordpress.com/2014/08/28/the-stochastic-resonance-program-part-2/), David Tanzer, 28 Aug 2014 * [Science, models and machine learning](http://johncarlosbaez.wordpress.com/2014/09/03/science-models-and-machine-learning/), David Tweed, 3 Sep 2014 * [The logic of real and complex numbers](http://johncarlosbaez.wordpress.com/2014/09/08/the-logic-of-real-and-complex-numbers/), John Baez, 8 Sep 2014 * [Network theory news](http://johncarlosbaez.wordpress.com/2014/09/28/network-theory-news/), John Baez, 28 Sep 2014
• Options
21.
Comment Source:<img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/>
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# Heyacrazy: Crosses
This is a Heyacrazy puzzle.
Heyacrazy is an original genre, inspired by the "two border" rule of Heyawake and Heyawacky.
Rules of Heyacrazy:
• Shade some cells of the grid.
• Shaded cells cannot be orthogonally adjacent; unshaded cells must be orthogonally connected.
• When the puzzle is solved, you may not be able to draw a line segment that passes through two borders, and does not pass through any shaded cells or grid vertices.
Example:
Here, the top solution is valid. The first three wrong solutions all break the third rule with the given red segments, and the other two break the two parts of the second rule in the indicated areas.
The puzzle:
• One grid, or 4 smaller grids? – tmpearce Aug 21 '19 at 2:22
• @tmpearce One grid. This is a single puzzle. – Deusovi Aug 21 '19 at 2:33
• Can the "line" go outside the outermost square border? – Omega Krypton Aug 21 '19 at 4:10
• @OmegaKrypton No, lines that go outside the puzzle are not considered. (Otherwise, the example solution would not work - you could draw a horizontal line at the left side of row 3.) – Deusovi Aug 21 '19 at 4:11
• Are cells separated by a border considered orthogonally adjacent? How does that work with angled borders? – Birjolaxew Aug 21 '19 at 7:16
Solution:
Reasoning:
There are a couple of key observations that help us solve this:
1. All $$+$$'s and $$\times$$'s must be shaded diagonally in one of the two possible ways ($$▚$$ or $$▞$$), otherwise they would internally violate rule #3
2. Once a $$+$$ has been shaded, its internal borders are irrelevant since no line can pass through them any more.
3. This means that we only have to be careful about rule #3 for the $$\times$$'s and central borders
With those observations out of the way, we can start solving.
The top-right cell must be shaded (otherwise it would be an enclosed unshaded cell). Being careful to obey rule #2, we can deduce half the board from just that one cell:
The lower-right quarter must have its right side connected to its left side (otherwise the right side is an unconnected group of unshaded cells, since it's already closed off in the upper-right quarter). If we shade the $$+$$ in the $$▚$$ direction, a connection is impossible - therefore we must shade the $$+$$ in the $$▞$$ direction.
From here it's pretty straight forward to solve the lower-left quarter, and get our final solution.
### Old answer (before puzzle was updated)
There are multiple solutions.
Reasoning:
The top-right cell must be shaded (otherwise it would be an enclosed unshaded cell). This gives us most of the top-right square.
All of the +'s can be shaded diagonally in one of two ways. This includes the central +. If we shade the central + in the \ direction, we end up with a situation like this:
However if we shade the + marked "1" in the \ direction, we close off the unshaded cells in each half, which violates rule #2. If we instead mark it in the / direction, then we always end up being able to connect the central borders which violates rule #3.
For this reason we know that the central + must be shaded in the / direction. Knowing that, we can make it this far:
From here it's guesswork. We can shade the + in the lower right square in either of the two directions. I am not sure that the solutions I've given in this answer are the only ones possible.
• Gah, looks like I uploaded the broken version of this one... really sorry about that. I'll edit in the fixed version instead (adding horizontal bars in columns 4 and 5, between rows 3 and 4). That should make the logical path actually work out. – Deusovi Aug 21 '19 at 8:39
• Oof I was late for several minutes.. also got multiple solutions and just realized there was an update haha. Anw, nice puzzle and solving! XD – athin Aug 21 '19 at 8:59
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# The Length of a Vector is Zero if and only if the Vector is the Zero Vector
## Problem 639
Let $\mathbf{v}$ be an $n \times 1$ column vector.
Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.
## Proof.
Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$.
Then we have
$\mathbf{v}^\trans \mathbf{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \sum_{i=1}^n \mathbf{v}_i^2 .$ Because each $v_i^2$ is non-negative, this sum is $0$ if and only if $v_i = 0$ for each $i$. In this case, $\mathbf{v}$ is the zero vector.
## Comment.
Recall that the the length of the vector $\mathbf{v}\in \R^n$ is defined to be
$\|\mathbf{v}\| :=\sqrt{\mathbf{v}^{\trans} \mathbf{v}}.$
The problem implies that the length of a vector is $0$ if and only if the vector is the zero vector.
### More from my site
#### You may also like...
##### A Relation between the Dot Product and the Trace
Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. Prove that \$\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans...
Close
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# Functions of Pancreas
The pancreas is located in the abdominal cavity behind and slightly inferior to the stomach, between the duodenum and spleen. The functions of the pancreas are both exocrine and endocrine glands. The endocrine portion of the gland consists of groups of tiny cells scattered in the exocrine portion. These cells are known as islets of Langerhans. Four types of such cells have been identified secreting different hormones.
1. Alpha cells secrete the hormone glucagon.
2. Beta cells secrete the hormone insulin.
3. Delta cells secrete the hormone Somatostatin or Growth Hormone Inhibiting Hormone (GHIH) and
4. F-cells secrete pancreatic polypeptide. The exact origin of these endocrine cells is not known. The pancreatic endocrine cells have a rich blood supply and are innervated by both sympathetic and parasympathetic nerves.
## Pancreatic Hormones and Functions of Pancreas
Insulin
The beta cells of islets of Langerhans produce insulin. Insulin facilitates the transport of glucose into the cells (Glucose uptake). It enhances glycogenesis (conversion of glucose into glycogen). It accelerates the conversion of glucose into fatty acids (Lipogenesis) and reduces gluconeogenesis (formation of glucose from non-carbohydrate sources). All these effects lead to a decrease in the glucose level of the blood. (Hypoglycemia).
Glucagon
The alpha cells of islets of Langerhans secrete glucagon. It has opposite effects to that of insulin. It increases the blood glucose level. In the liver, it accelerates the conversion of glycogen into glucose (Glycogenolysis). It promotes gluconeogenesis and enhances the release of glucose into the blood.
Somatostatin
The hormone secreted by delta cells of Islets of Langerhans is Somatostatin. It acts as a peregrine to inhibit the secretion of alpha and beta cells of Islets.
The pancreatic polypeptide secreted by F-cells regulates the release of pancreatic digestive enzymes.
Release of Pancreatic Hormones
The release of Pancreatic hormones is controlled by chemical, hormonal and neural stimuli. The blood glucose level appears to be the major factor that governs the release of both insulin and glucagon. A higher blood glucose level stimulates insulin release while a lower blood glucose level stimulates glucagon release. The hormones secreted by cells of the gastrointestinal tract such as secretin, gastrin, and cholecystokinin promote insulin release. Somatostatin inhibits the secretion of both insulin and glucagon. Parasympathetic activity releasing acetylcholine stimulates insulin release while sympathetic transmitters epinephrine and nor-epinephrine inhibit insulin release.
## Pancreatic Disorders
The commonest pancreatic endocrine disorder is a deficiency of insulin leading to diabetes mellitus. There is an increase in blood glucose levels (hyperglycemia). The condition is characterized by Polyuria (excessive urine production), Polydipsia (excessive thirst), and Polyphagia (excessive eating). The glucose is passed in urine (glucosuria). There are two main types of diabetes mellitus: Insulin-Dependent Diabetes Mellitus (IDDM) and Non-Insulin Dependent Diabetes Mellitus (NIDDM). In the first type, there is an absolute deficiency of insulin and the patient needs regular administration of insulin by injection. It appears to be an autoimmune disorder where a person’s immune system destroys the beta cells of Islets. The inability of body cells to uptake and utilize glucose leads to the production of ATP from fatty acids. This causes ketoacidosis. Catabolism of proteins and stored fats leads to weight loss. Mobilization of fat from fat depots and its transport to cells may result in the deposition of fat in the wall of blood vessels. This leads to a variety of cardiovascular problems such as atherosclerosis, ischemic heart disease, peripheral vascular disease, and gangrene. Loss of vision due to cataracts (excessive glucose bound to lens proteins) or damage to blood vessels of the retina is another important complication of diabetes.
Non-insulin-dependent diabetes mellitus (NIDDM) is more common than IDDM. It occurs in persons above 40 years of age. The symptoms are comparatively mild and high blood glucose can be controlled by diet, exercise, and weight loss. An antidiabetic drug that stimulates beta cells to secrete insulin is also useful to control the condition. In this type, there is no deficiency of insulin but cells become less sensitive to insulin and hence cannot uptake and utilize glucose.
Make sure you also check our other amazing Article on: Adrenal Glands
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There's a (someone else's) multi-file addon with a complex structure of libraries and classes with a bug in a particular class and I'd like to try to fix it. However I never made an addon with several separated files and I don't understand how to reload it properly. At the moment I edit the original .py file and restart Blender which seems like a terrible workflow.
• I really am not sure but you can try to reload the module with the 'importlib' module. Use "import importlib" and then "importlib.reload(module)" – Gorgious Mar 4 '20 at 10:00
• Does F3 > Reload Scripts help? You can also bind the operator to any hotkey... – brockmann Mar 4 '20 at 10:57
• @Gorgious I'll try that, thank you – Sergey Kritskiy Mar 4 '20 at 11:50
• @brockmann this doesn't seem to update the code – Sergey Kritskiy Mar 4 '20 at 11:51
• Does it for me. Is the addon enabled? Also see: blender.stackexchange.com/a/28505/31447 – brockmann Mar 4 '20 at 12:04
Not sure why Reload Scripts didn't work at first for me, maybe I didn't save the correct file or whatnot, anyway, that's a possible way. My issues were however with this method is that
• I have to edit files in Blender addons folder, not in my project folder;
• this reloads all the scripts, on my machine with a lot of addons installed it takes about four seconds and plus after that some addons reset their settings;
In the end I followed this article for creating and debugging multifile addons. Author suggests using a specific setup for the __init__.py file and then provides a Blender script in the second part of the article to update only a specific addon. I've assigned this script to a hotkey and it works much faster than reloading all the scripts.
Here are the final scripts from the article. __init__.py:
bl_info = {
'category': 'All',
'version': (0, 0, 1),
'blender': (2, 80, 0)
}
import sys
import importlib
modulesFullNames = {}
for currentModuleName in modulesNames:
if 'DEBUG_MODE' in sys.argv:
modulesFullNames[currentModuleName] = ('{}'.format(currentModuleName))
else:
modulesFullNames[currentModuleName] = ('{}.{}'.format(__name__, currentModuleName))
for currentModuleFullName in modulesFullNames.values():
if currentModuleFullName in sys.modules:
else:
globals()[currentModuleFullName] = importlib.import_module(currentModuleFullName)
setattr(globals()[currentModuleFullName], 'modulesNames', modulesFullNames)
def register():
for currentModuleName in modulesFullNames.values():
if currentModuleName in sys.modules:
if hasattr(sys.modules[currentModuleName], 'register'):
sys.modules[currentModuleName].register()
def unregister():
for currentModuleName in modulesFullNames.values():
if currentModuleName in sys.modules:
if hasattr(sys.modules[currentModuleName], 'unregister'):
sys.modules[currentModuleName].unregister()
if __name__ == "__main__":
register()
import os
import sys
filesDir = "d:/Python/TestMultifile"
initFile = "__init__.py"
if filesDir not in sys.path:
sys.path.append(filesDir)
file = os.path.join(filesDir, initFile)
if 'DEBUG_MODE' not in sys.argv:
sys.argv.append('DEBUG_MODE')
if 'DEBUG_MODE' in sys.argv:
sys.argv.remove('DEBUG_MODE')
I have found an easiest method. In this example the addon has two modules - main and operator
__init__.py:
bl_info = {
# Your regular bl_info goes here.
}
# This is essential part
if "bpy" in locals():
import importlib
else:
import bpy
from . import main
from . import operator
classes = (
• To keep code "modular" I would recommend each module with classes to register having a register method as above.. eg modules = [main, operator] then in the register method loop the modules and mod.register() IMO making adding another module easier to test and integrate into addon. – batFINGER Jun 9 '20 at 1:42
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# A Divisible Subset Sum
Here is a well-known result.
Theorem. Consider a list of $n$ not-necessarily distinct integers $a_1, a_2, \ldots, a_n$. Then there always exists a subset of these numbers with sum divisible by $n$.
Let’s do a quick example for $n = 4$ of what this result means. First, choose any four integers you like. I’m going to go with:
1, 2, 7 and 17.
Then, it follows that there must be a subset of these integers with sum divisible by 4.
Indeed, the subset $\{7, 17\}$ has a sum of 24 which is divisible by 4.
But how would we prove this result? Actually, better question: how would you prove this result? Have a go of this first before reading through the textbook proof below.
Proof. Let $a_1, a_2, \ldots a_n$ be integers. We consider the partial sums of these:
$s_1 = a_1$
$s_2 = a_1 + a_2$
$s_3 = a_1 + a_2 + a_3$
and so on up to
$s_n = a_1 +a_2 + a_3 + \cdots + a_n.$
Now, if any of the partial sums $s_i$ are divisible by $n$ then we are done. If none of them are, then consider the remainders $s_i \mod n$.
As we have discounted zero from being a remainder, it follows that there are only $n-1$ possibilities for these $n$ remainders. By the pigeonhole principle, there must be two partial sums that are equal $\mod n$.
Let’s call these $s_i$ and $s_j$ where $i < j$. Then it follows that $s_j - s_i$ must be divisible by $n$.
Importantly, $s_j - s_i$ is the difference of two partial sums, and so it has the form
$s_j - s_i = a_{i+1} + a_{i+2} + \cdots + a_{j-1} + a_j.$
This is the sum of the subset $\{a_{i+1}, a_{i+2}, \ldots, a_{j-1}, a_j\}$ and so the proof is complete.
We actually proved something slightly stronger than the theorem in the above proof. Can you work out what it is?
Here’s another thought on the proof. There is some standard mathematical backwardsness delivered with a “just trust me” smokiness when the idea of partial sums is conjured up. This does not feel natural until later in the proof when we realise why it was set up that way. Can you come up with a proof that feels more natural and proceeds more linearly?
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“We have 10 minutes left. Do you have any questions from us?” At the end of our technical interview session, I asked one of the interviewers. “yes, I do,” he replied. “I saw your company’s blog post and see that you guys write code in Scala instead of other languages. What is the reason for that?”
I thought about it and said, “We use Scala because Scala has the Functional programming features that we need to scale our business.”
He waited for a while, processing his thought. At that moment, I thought I didn’t clearly explain that functional programming is a huge practice in our team. Then, he asks, “What is good about functional programming?”
I waited before answering this time, thinking through in my head, and give a generic explanation that Function Programming (FP) feature such as immutability and no-side effects helps us reason our code better in a concurrent environment. Then, I explained the basic terminology of FP. However, during that time, which is a year ago, I didn’t understand what it means to write code in a functional style and how functional writing code helps my work be more productive.
More developers use languages with an explicit bias towards FP, such as Scala and Haskell. In contrast, OOP languages and their communities adapt the features and practices (Think about React and Typescript in the Javascript community). However, when I first wrote functional code, limiting myself to immutability in the code neither helps with my productivity in producing quality code nor creating a readable codebase (think about needing to do a recursive function to loop through a list of elements). Besides, developers have a hard time learning all the FP abstract features while having a project deadline.
Immutability and side effects are the first thing that I exposed when learning about FP. However, that is not the reason real reason why the team adopted FP.
When I started down the FP route, this was my view, but now I understand why they enforce these features to developers - for local reasoning and composition. Ironically, writing FP code makes me more productive and makes my job easier, and the language features and programming style are just in service of these goals. In this post, I attempt to explain why functional writing code makes your job easier.
Although it is a debate that static typing increases productivity, having type safety is one of the best things to mitigate errors and surprises during run time. I didn’t understand how helpful type safety was until recently when I tried to debug one of my colleague’s codebase. He asked me how he keeps getting runtime exceptions such as null pointer in one of the microservices. Our services are written in Scala. Therefore, getting a runtime exception is not as common as writing code in Java.
It has various error types that the developer needs to handle. I keep scanning through the entire codebase multiple times, running the same failed test cases, and trace through the bug. Three hours later, I realized that he didn’t handle any exception when getting one of the AttributeValue in DynamoDB.
That one single case causes three hours of resources in the team. The bug is often encountered when we transform Java code into Scala. You cannot see that the return value of function definition may return null, and a developer will need to constantly remind themselves to check for null cases. A type-safe way of writing is to return an Option. Declaring an Option type in your return function indicates to the caller that the returned value may be empty. Therefore, the caller needs to handle any empty case during compile-time, avoiding any surprises such as NullPointerException in the run time.
Let’s take an example of getting items from a database and then parse the underlying item in the database.
An experienced developer will know right away that there are at least two types of errors, network errors, and parsing errors. Network errors may cause when the database is down, and the parse exception may cause by the model mismatch. However, these error is hidden to the caller - we need to look into its implementation and “guess” what sort of error it has.
Error handling in this scenario is usually a try and catch error, and we need to think about what sort of error is possible in the catch block.
It is not required to handle all errors. Thus, you cannot see how complex the function above is and may miss important errors that will crash your program at night.
An aha moment for this is that exceptions are a form of hidden complexity. We can to explicitly tell these two exceptions to the caller as an “effect.” Let’s see what it will look like in the below code snippet:
I won’t go into the implementation since we can determine what error will occur in this function by looking at the function definition itself. The first outer value is IO, which means that this function will have some side-effect. Having IO gives a cautious to the caller that the function may return an error, which is a network error. Secondly, we have an Either of ParsingError or the Json value that we want. Either is a data type in Scala or other functional languages representing that the result can either be a Left or a Right. In convention, a Left is the error case, and the Right is the caller’s result.
You see, right now, the caller is “forced” to handle these scenarios. They cannot retrieve Json without handling the Left case. They will also know what types of errors occur during the “compile” time to handle those errors based on their application logic.
More importantly, type safety helps you forced to handle corner cases. It makes a simple function more complex because it made everything explicit rather than hiding it.
## Reason our Program Flow logic in Compile Time
Have you ever wished to write your program all at once without compiling, and everything works when you compile your program? Writing functional code makes you do that.
One of the sins of writing functional code is to have a side effect. The two mantras that I have to remember when I learn functional programming are that side-effects and mutability are a sin. If functional programmers spot a mutable reference in a Scala code base, their palm starts to sweat. They started to have shortness of breath, and they start scanning through all the files multiple times to figure out where the surprises of the code may be.
However, after two years of learning functional programming, I realized that it is not that functional programming discouraging side effects or mutability. Still, it is to declare the intent of your program as much as possible in the function definition. There are various effects in functional programs. (If you don’t understand what I mean, I wrote an article about what effects mean in functional programming here.
Returning an IO in your function tells the developer that the function is doing an IO call. Thus, some side effects will occur during the run time - a high possibility of “unsafe” code.
Every functional code you wrote is a description, and you will need to initiate that call at your main function. We called it the end of the world. This notion of “description” tells the developer what the program expects to behave in the compile-time and guarantees the developers that it will behave as intended.
One example is IO vs. Future. Future is eager evaluated. Therefore, instantiating a Future will automatically start an execution context and fire another thread. On the other hand, IO is lazily evaluated. Thus, creating an IO will not automatically run the function inside until you call unsafeRun. A function with different behavior may have the same result with Future, but not with IO. Let’s see the code snippet below:
What does the above code do? We want to fetch the database twice. However, when we evaluate with Future, it only resulted in fetching the database once. On the other hand, instantiating with IO only describe what the program will behave. Then, the unsafeRun will execute that program.
The above code will fetch the DB twice. Writing a program as a description instead of a function execution helps us figure out how the program runs in compile time.
## Decreasing the number of lines of code in The Program with Abstraction
We did this by thinking about the “laws” of what we want our type to behave. Usually, this is where all the type category systems such as Monad or Functor came.
You will not need to implement a different variation of your function repeatedly and use the most minimal primitive for all your program logic. You start writing function in terms of what kind of “laws” or behavior you want that function to be. For instance, you can make a function that enforces the passing argument to be associative. It means that we can be sure when running combine(a,b) will be equal to running combine(b,a).
With constructing your function based on laws, you don’t have to be worried that the caller of that function will supply types that are not associative. Thus, you can implement the function in any order you want, and it is very helpful when running any concurrent programs or map-reduce kind of job.
This associative law in category theory is called Semigroup. Therefore, you can have a function that the arguments will need to abide by the Semigroup law.
The more you write functional code, the more you will look at your code in building blocks. You often see how functional programmers are obsessed with creating the most primitive possible. What I mean by primitive is the most generalized form of a function. For instance, if you want to write a program that checks if a certain boolean value is true, you must do a certain action. In general, this can be done with an “if-else” statement, like this:
However, this function is pretty limited. What if I want to have multiple branches of action based on certain conditions instead of two ways? If we have this function, it will be much powerful than the function with a boolean statement. Therefore, thinking about it, we can use a List where the index will replace the checkStatement. Then, based on that index, we can execute some values:
Now, we can use chooseN to implement choose:
chooseN is not the most primitive possible function. You can make it more generalized and realized that you could use map or flatMap to implement these functions. However, I’ll leave that portion to you to derive chooseN into flatMap.
## Conclusion
Writing functional code requires a steep learning curve. Therefore, a high-growth company is hesitant to enforce functional programming languages because it takes time for developers to learn a new way of thinking.
You may need a longer development time because you need to account for all these errors. However, you will also sleep well at night because you can be sure that the code behaves as you intend. Once you get over the hump, you will realize how productive and confident you feel when you push your code. You’ll notice all the errors that may happen in the program before the program runs. You don’t need to hit the run function simultaneously to see how your function will behave.
You will write single responsible functions and decrease the amount of repeated logic in your program. Once you learn to write functional programming code for a while and get back into writing imperative code, you become a much stronger programmer that can see various runtime exceptions and bugs within the program.
Most importantly, you are confident that the feature you build is robust and will create a great user experience for your users.
## Reference
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No 1. Nested Asynchronous Function
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Will I lose my Job To the Younger Generation if I am not Passionate about Technology
##### Use This Mantra to Decide whether You Want to go to Big Tech or a Startup
If you want to Go Deep, Go for Big Tech. If you want to Go Wide, Go for Startup.
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# Stepsize and Linesearch
Most iterative algorithms determine a direction along which the algorithm will proceed and determine a step size to find the next iterate. How advanced the step size computation can be implemented depends (among others) on the properties the corresponding problem provides.
Within Manopt.jl, the step size determination is implemented as a functor which is a subtype of [Stepsize](@refbased on
Manopt.StepsizeType
Stepsize
An abstract type for the functors representing step sizes, i.e. they are callable structures. The naming scheme is TypeOfStepSize, e.g. ConstantStepsize.
Every Stepsize has to provide a constructor and its function has to have the interface (p,o,i) where a AbstractManoptProblem as well as AbstractManoptSolverState and the current number of iterations are the arguments and returns a number, namely the stepsize to use.
Linesearch
source
Usually, a constructor should take the manifold M as its first argument, for consistency, to allow general step size functors to be set up based on default values that might depend on the manifold currently under consideration.
Currently, the following step sizes are available
Manopt.ArmijoLinesearchType
ArmijoLinesearch <: Linesearch
A functor representing Armijo line search including the last runs state, i.e. a last step size.
Fields
• initial_stepsize – (1.0) and initial step size
• retraction_method – (default_retraction_method(M)) the rectraction to use
• contraction_factor – (0.95) exponent for line search reduction
• sufficient_decrease – (0.1) gain within Armijo's rule
• last_stepsize – (initialstepsize) the last step size we start the search with
• linesearch_stopsize - (0.0) a safeguard when to stop the line search before the step is numerically zero. This should be combined with StopWhenStepsizeLess
• initial_guess ((p,o,i,l) -> l) based on a AbstractManoptProblem p, AbstractManoptSolverState o and a current iterate i and a last step size l, this returns an initial guess. The default uses the last obtained stepsize
Constructor
ArmijoLineSearch(M)
with the Fields above as keyword arguments and the retraction is set to the default retraction on M.
The constructors return the functor to perform Armijo line search, where two interfaces are available:
• based on a tuple (amp, ams, i) of a AbstractManoptProblem amp, AbstractManoptSolverState ams and a current iterate i.
• with (M, x, F, gradFx[,η=-gradFx]) -> s where Manifold M, a current point x a function F, that maps from the manifold to the reals, its gradient (a tangent vector) gradFx$=\operatorname{grad}F(x)$ at x and an optional search direction tangent vector η=-gradFx are the arguments.
source
Manopt.ConstantStepsizeType
ConstantStepsize <: Stepsize
A functor that always returns a fixed step size.
Fields
• length – constant value for the step size.
Constructors
ConstantStepsize(s::Real)
initialize the stepsize to a constant s.
ConstantStepsize(M::AbstractManifold=DefaultManifold(2); stepsize=injectivity_radius(M)/2)
initialize the stepsize to a constant stepsize, which by default is half the injectivity radius, unless the radius is infinity, then the default step size is 1.
source
Manopt.DecreasingStepsizeType
DecreasingStepsize()
A functor that represents several decreasing step sizes
Fields
• length – (1) the initial step size $l$.
• factor – (1) a value $f$ to multiply the initial step size with every iteration
• subtrahend – (0) a value $a$ that is subtracted every iteration
• exponent – (1) a value $e$ the current iteration numbers $e$th exponential is taken of
• shift – (0) shift the denominator iterator $i$ by $s$.
In total the complete formulae reads for the $i$th iterate as
$$$s_i = \frac{(l - i a)f^i}{(i+s)^e}$$$
and hence the default simplifies to just $s_i = \frac{l}{i}$
Constructor
DecreasingStepsize(l=1,f=1,a=0,e=1,s=0)
Alternatively one can also use the following keyword.
DecreasingStepsize(
M::AbstractManifold=DefaultManifold(3);
length=injectivity_radius(M)/2, multiplier=1.0, subtrahend=0.0, exponent=1.0, shift=0)
initialiszes all fields above, where none of them is mandatory and the length is set to half and to $1$ if the injectivity radius is infinite.
source
Manopt.LinesearchType
Linesearch <: Stepsize
An abstract functor to represent line search type step size deteminations, see Stepsize for details. One example is the ArmijoLinesearch functor.
Compared to simple step sizes, the linesearch functors provide an interface of the form (p,o,i,η) -> s with an additional (but optional) fourth parameter to provide a search direction; this should default to something reasonable, e.g. the negative gradient.
source
Manopt.NonmonotoneLinesearchType
NonmonotoneLinesearch <: Linesearch
A functor representing a nonmonotone line search using the Barzilai-Borwein step size[Iannazzo2018]. Together with a gradient descent algorithm this line search represents the Riemannian Barzilai-Borwein with nonmonotone line-search (RBBNMLS) algorithm. We shifted the order of the algorithm steps from the paper by Iannazzo and Porcelli so that in each iteration we first find
$$$y_{k} = \operatorname{grad}F(x_{k}) - \operatorname{T}_{x_{k-1} → x_k}(\operatorname{grad}F(x_{k-1}))$$$
and
$$$s_{k} = - α_{k-1} * \operatorname{T}_{x_{k-1} → x_k}(\operatorname{grad}F(x_{k-1})),$$$
where $α_{k-1}$ is the step size computed in the last iteration and $\operatorname{T}$ is a vector transport. We then find the Barzilai–Borwein step size
$$$α_k^{\text{BB}} = \begin{cases} \min(α_{\text{max}}, \max(α_{\text{min}}, τ_{k})), & \text{if } ⟨s_{k}, y_{k}⟩_{x_k} > 0,\\ α_{\text{max}}, & \text{else,} \end{cases}$$$
where
$$$τ_{k} = \frac{⟨s_{k}, s_{k}⟩_{x_k}}{⟨s_{k}, y_{k}⟩_{x_k}},$$$
if the direct strategy is chosen,
$$$τ_{k} = \frac{⟨s_{k}, y_{k}⟩_{x_k}}{⟨y_{k}, y_{k}⟩_{x_k}},$$$
in case of the inverse strategy and an alternation between the two in case of the alternating strategy. Then we find the smallest $h = 0, 1, 2, …$ such that
$$$F(\operatorname{retr}_{x_k}(- σ^h α_k^{\text{BB}} \operatorname{grad}F(x_k))) \leq \max_{1 ≤ j ≤ \min(k+1,m)} F(x_{k+1-j}) - γ σ^h α_k^{\text{BB}} ⟨\operatorname{grad}F(x_k), \operatorname{grad}F(x_k)⟩_{x_k},$$$
where $σ$ is a step length reduction factor $∈ (0,1)$, $m$ is the number of iterations after which the function value has to be lower than the current one and $γ$ is the sufficient decrease parameter $∈(0,1)$. We can then find the new stepsize by
$$$α_k = σ^h α_k^{\text{BB}}.$$$
Fields
• initial_stepsize – (1.0) the step size we start the search with
• linesearch_stopsize - (0.0) a safeguard when to stop the line search before the step is numerically zero. This should be combined with StopWhenStepsizeLess
• memory_size – (10) number of iterations after which the cost value needs to be lower than the current one
• min_stepsize – (1e-3) lower bound for the Barzilai-Borwein step size greater than zero
• max_stepsize – (1e3) upper bound for the Barzilai-Borwein step size greater than min_stepsize
• retraction_method – (ExponentialRetraction()) the rectraction to use
• strategy – (direct) defines if the new step size is computed using the direct, indirect or alternating strategy
• storage – (x, gradient) a StoreStateAction to store old_x and old_gradient, the x-value and corresponding gradient of the previous iteration
• stepsize_reduction – (0.5) step size reduction factor contained in the interval (0,1)
• sufficient_decrease – (1e-4) sufficient decrease parameter contained in the interval (0,1)
• vector_transport_method – (ParallelTransport()) the vector transport method to use
Constructor
NonmonotoneLinesearch()
with the Fields above in their order as optional arguments (deprecated).
NonmonotoneLinesearch(M)
with the Fields above in their order as keyword arguments and where the retraction and vector transport are set to the default ones on M, repsectively.
The constructors return the functor to perform nonmonotone line search.
source
Manopt.WolfePowellBinaryLinesearchType
WolfePowellBinaryLinesearch <: Linesearch
A Linesearch method that determines a step size t fulfilling the Wolfe conditions
based on a binary chop. Let $η$ be a search direction and $c1,c_2>0$ be two constants. Then with
$$$A(t) = f(x_+) ≤ c1 t ⟨\operatorname{grad}f(x), η⟩_{x} \quad\text{and}\quad W(t) = ⟨\operatorname{grad}f(x_+), \text{V}_{x_+\gets x}η⟩_{x_+} ≥ c_2 ⟨η, \operatorname{grad}f(x)⟩_x,$$$
where $x_+ = \operatorname{retr}_x(tη)$ is the current trial point, and $\text{V}$ is a vector transport, we perform the following Algorithm similar to Algorithm 7 from [Huang2014]
1. set $α=0$, $β=∞$ and $t=1$.
2. While either $A(t)$ does not hold or $W(t)$ does not hold do steps 3-5.
3. If $A(t)$ fails, set $β=t$.
4. If $A(t)$ holds but $W(t)$ fails, set $α=t$.
5. If $β<∞$ set $t=\frac{α+β}{2}$, otherwise set $t=2α$.
Constructors
There exist two constructors, where, when prodivind the manifold M as a first (optional) parameter, its default retraction and vector transport are the default. In this case the retraction and the vector transport are also keyword arguments for ease of use. The other constructor is kept for backward compatibility. Note that the linesearch_stopsize to stop for too small stepsizes is only available in the new signature including M, for the first it is set to the old default of 1e-9.
WolfePowellBinaryLinesearch(
retr::AbstractRetractionMethod=ExponentialRetraction(),
vtr::AbstractVectorTransportMethod=ParallelTransport(),
c1::Float64=10^(-4),
c2::Float64=0.999
)
WolfePowellLinesearch(
M,
c1::Float64=10^(-4),
c2::Float64=0.999;
retraction_method = default_retraction_method(M),
vector_transport_method = default_vector_transport(M),
linesearch_stopsize = 0.0
)
source
Manopt.WolfePowellLinesearchType
WolfePowellLinesearch <: Linesearch
Do a backtracking linesearch to find a step size $α$ that fulfils the Wolfe conditions along a search direction $η$ starting from $x$, i.e.
$$$f\bigl( \operatorname{retr}_x(αη) \bigr) ≤ f(x_k) + c_1 α_k ⟨\operatorname{grad}f(x), η⟩_x \quad\text{and}\quad \frac{\mathrm{d}}{\mathrm{d}t} f\bigr(\operatorname{retr}_x(tη)\bigr) \Big\vert_{t=α} ≥ c_2 \frac{\mathrm{d}}{\mathrm{d}t} f\bigl(\operatorname{retr}_x(tη)\bigr)\Big\vert_{t=0}.$$$
Constructors
There exist two constructors, where, when prodivind the manifold M as a first (optional) parameter, its default retraction and vector transport are the default. In this case the retraction and the vector transport are also keyword arguments for ease of use. The other constructor is kept for backward compatibility. Note that the linesearch_stopsize to stop for too small stepsizes is only available in the new signature including M. For the old (deprecated) signature the linesearch_stopsize is set to the old hard-coded default of 1e-12
WolfePowellLinesearch(
retr::AbstractRetractionMethod=ExponentialRetraction(),
vtr::AbstractVectorTransportMethod=ParallelTransport(),
c1::Float64=10^(-4),
c2::Float64=0.999
)
WolfePowellLinesearch(
M,
c1::Float64=10^(-4),
c2::Float64=0.999;
retraction_method = default_retraction_method(M),
vector_transport_method = default_vector_transport(M),
linesearch_stopsize = 0.0
)
source
Manopt.linesearch_backtrackMethod
linesearch_backtrack(M, F, x, gradFx, s, decrease, contract, retr, η = -gradFx, f0 = F(x); stop_step=0.)
perform a linesearch for
• a manifold M
• a cost function F,
• an iterate x
• the gradient $\operatorname{grad}F(x)$
• an initial stepsize s usually called $γ$
• a sufficient decrease
• a contraction factor $σ$
• a retraction, which defaults to the ExponentialRetraction()
• a search direction $η = -\operatorname{grad}F(x)$
• an offset, $f_0 = F(x)$
• a keyword stop_step as a minimal step size when to stop
source
Manopt.max_stepsizeMethod
max_stepsize(M::AbstractManifold, p)
Get the maximum stepsize at point p on manifold M`. It should be used to limit the distance an algorithm is trying to move in a single step.
source
• Iannazzo2018
B. Iannazzo, M. Porcelli, The Riemannian Barzilai–Borwein Method with Nonmonotone Line Search and the Matrix Geometric Mean Computation, In: IMA Journal of Numerical Analysis. Volume 38, Issue 1, January 2018, Pages 495–517, doi 10.1093/imanum/drx015
• Huang2014
Huang, W.: Optimization algorithms on Riemannian manifolds with applications, Dissertation, Flordia State University, 2014. pdf
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The horizontal pressure difference generates a pressure gradient force that accelerates the air from the region of higher pressure (the warm air) towards the lower pressure (the cold air). Only the nondimensional profiles are available for this experiment. The Pressure Gradient Force weakens at the same time the Coriolis has not had time to adjust and decrease. - Conditions downstream of exit plane impact flow in domain. Effects of adverse and favorable pressure gradients on entropy generation in a transitional boundary layer under the influence of freestream turbulence. The normal hydrostatic pressure gradient for freshwater is 0. Adverse-Pressure-Gradient Effects on Turbulent Boundary Layers: Statistics and Flow-Field Organization Carlos Sanmiguel Vila 0 1 Ramis O¨ rlu¨ 0 1 Ricardo Vinuesa 0 1 Philipp Schlatter 0 1 Andrea Ianiro 0 1 Stefano Discetti 0 1 0 Linne ́ FLOW Centre, KTH Mechanics , SE-100 44 Stockholm , Sweden 1 Aerospace Engineering Group, Universidad Carlos III de Madrid , Legane ́s , Spain This. The streamwise vortices were created by three pairs of half-delta wing vortex generators, while impulsively initiated opposite-wall suction created a strong adverse pressure gradient. This limits its use to the non-separated flows only. alveolar/arterial gradient synonyms, alveolar/arterial gradient pronunciation, alveolar/arterial gradient translation, English dictionary definition of alveolar/arterial gradient. A common adverse pressure-gradient flow for tests of turbulence models is the experiment of Samuel & Joubert 15. the variation of a reynolds analogy parameter, which indicates the ratio of skin friction to heat transfer, is from zero to 7. In purely theoretical terms, this physical measuring principle is the most appropriate one because it only provides a signal during dynamic pressure curves, but with a very good signal-to-noise interval and for high frequencies. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. Pressure gradients are usually characterized by the Clauser pressure gradient parameter 6 6. com with free online thesaurus, antonyms, and definitions. This is true at all values of x. The power of improved heating technology can completely change the way the food and beverage industry does business. Strong gradients = strong winds Weak gradients = low wind speeds * That means, moving in any horizontal direction away from the "Low" will result in an increase in pressure. Once a mass of fluid is moving, it keeps moving without any pressure gradient. The thermal wind occurs above the. 86; namely:. For research of the characteristics of the steady and unsteady flows and heat transfer at high-free-stream turbulence intensity the turbulence two-parametric models have been presented. (c) Determine The Pressure Gradient. Mack Faculty ofEngineering,University ofTechnologySydney, PO Box123,Broadway,NSW 2007,Australia Abstract: The flow field experimentally realized by the wind tunnel for the investigation of the control of. The last term is −τ o/ρ since τ must be close to zero near the main stream. In the stream-function formulation the pressure is eliminated via the incompressibility condition, however occasionally it is useful to examine it in order to gain some insight. Kennedy once observed that the word “crisis” in Chinese is composed of two characters—one representing danger, the other opportunity. The assessment of mitral stenosis relies on measurement of the pressure gradient and on calculation of the valve area. Now we will go ahead to understand the basic concept of velocity potential function and stream function, in the field of fluid mechanics, with the help of this post. He may not have been entirely correct on the linguistics, but the sentiment is true enough: a crisis presents a choice. If a concentration gradient exists, the molecules (which are constantly moving) will eventually become evenly distributed (a state of equilibrium). The easiest way to calculate pressure from depth is to use the pressure gradient of the given fluid. COHES and ELI RESHOTKO SUMMARY Stewartson's ti ansformation is applied to the laminar cotn- pressiblp bou ntiary-layer equation. 460 psi/ft or 10. May 31, 2020 - Relation between shear stress, pressure gradient and velocity distribution JEE Notes | EduRev is made by best teachers of JEE. These temperature differences produce a horizontal pressure gradient that drives geostrophic and gradient winds. The streamwise vortices were created by three pairs of half-delta wing vortex generators, while impulsively initiated opposite-wall suction created a strong adverse pressure gradient. Simple two-layer model for temperature distribution in the inner part of turbulent boundary layers in adverse pressure gradients International Journal of Heat and Mass Transfer, Vol. ’-’’ This article reviews noninvasive quantifica- tion of intracardiac pressure and assessment of cardiac function. The Significance Of The Hydraulic Gradient Normally when a pipe is laid, attempts are made to keep the pipe at or below the hydraulic gradient. Stream ecosystem function along a land-use gradient. where U1 is the local free stream velocity and Ut is the friction velocity. 4Re p-1/3 + O(Re p-2/3), where Re p is the Reynolds number based on the longitudinal pressure gradient. Thus, in discussing a boundary layer, we may speak interchangeably about a stream- wise pressure gradient or an external-flow velocity gradient. , (1984) and. Lecture 13: Stream Function Velocity: Part 10: Example- Velocity Potential. Simplified method for calculation of compressible laminar boundary with arbitrary free-stream pressure gradient, report. where pi is the surface pressure measured at location i on the surface, p∞ is the pressure in the free stream, ρ is air density, and U∞ is the free-stream velocity given by () ρ ∞ ∞ − = p p U stagnation 2 (2) where pstagnation is the stagnation pressure measured at the tip of the Pitot tube. The Gradient (also called Slope) of a straight line shows how steep a straight line is. An excellent test case, and case to familiarize yourself with some of the turbulence models available in OpenFOAM is a 2D flat plate with zero pressure gradient. With the influence of adverse-pressure-gradient and shock/turbulent-boundary-layer interactions, the flow exhibits extensive separation at the bump trailing-edge. Integrate Eqn. The intensity of the pressure gradient increases up to the troposphere. Print Send Add Share. 8 kPa/m] in a well with a true vertical depth of 8000 ft [2440 m] would predict a fracturing pressure of 5600 psi [38. That’s just like a rigid mass moving in space. 1000 f(x) = 7. Differences in air pressure and the pressure gradient force are caused by the unequal heating of the Earth's surface when incoming solar radiation concentrates at the equator. Operational Draft Regional Guidebook for the Functional Assessment of High-gradient Ephemeral and Intermittent Headwater Streams in Western West Virginia and Eastern Kentucky. The wind around a surface high pressure center in the Northern Hemisphere blows: a. 123–151, 1951. The portosysterific pressure gradient dropped from 24 mmHg before TIPSS to 11 mmHg and remained stable after shunt occlusion. The hydraulic gradient is a vector gradient between two or more hydraulic head measurements over the length of the flow path. For example, a fracture gradient of 0. The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression {eq}\psi{/eq} = -19(x - y) where the stream function has the units of ft{eq}^2{/eq}/s. This happens at each value of x. | Meaning, pronunciation, translations and examples. Selected Answer: geostrophic winds Correct Answer: geostrophic winds Question 6 Match the term with the correct description. A two-dimensional, incompressible fluid flow is described by the stream function Ψ = x(y^3) m2/s on the Cartesian x-y plane. Integrate Eqn. Computing the pressure as a function of depth in a homogeneous crust is a straightforward calculation. The term was originally coined by the drillers, who need to pump mud into the well to counteract pressures. This pressure gradient consists in two parts, baroclinic and barotropic term: () ∂ ∂ − + ∂ ∂ = ⋅ ∂ ∂ x z g x g x p η η ρ ρ 1 1 (1. com with free online thesaurus, antonyms, and definitions. An extensive data set describing separated flows with different Reynolds numbers (Re), free-stream turbulence intensities (Tu) and adverse pressure gradients (APG) is used for the statistical characterization of laminar separation bubbles (LSBs). The boundary conditions are: u (0) = 0 (lower plate velocity), that varies as a function of the pressure gradient and the upper plate. 15 R = 1 Results Based on Flowing Pressure Gradient Plot: DEPTH (M) 400-800. '-'' This article reviews noninvasive quantifica- tion of intracardiac pressure and assessment of cardiac function. (a) Is the continuity equation satisfied? (h) Is the flow field irrotational? If so, determine the corresponding velocity potential. For your convenience, you can input the pressures required in either mmHg or kPa. Lecture 18: Pipe Pressure Drop/Minor Losses: Part 1. The letter g stands for the acceleration due to gravity and h is the fluid's elevation. - The atmosphere above the Jet Stream becomes more stable with height due to the Stratosphere’s property of an increase in temperature with height. The boundary layer thickness is reduced due to the presence of pressure gradient on the velocity profile. Mild aortic stenosis: Valve area is between 1. Thomas,a) and Robert C. When the pressure gradient force is balanced by the Coriolis force, high altitude _____ move parallel to isobars. Pressure Gradient Force (PGF) - causes horizontal pressure differences and winds 2. () = pressure as a function of depth = force of gravity = depth = density as a function of depth. 465 psi/ft for water with 100,000 ppm total dissolved solids (a typical Gulf Coast water), or 10. Fluids – Lecture 12 Notes 1. considered, will be removed by the hydrostatically consistent calculation of the pressure gradient term of the pressure gradient force, but the influence of the other, of † Tj+1/2,k+1, will not. To investigate the. Pressure gradient: the change in pressure with distance. IMPROVED NEW HIGH QUALITY GARDEN HOSE- Leak-Resistant Connection / Anti-Rust Metal Fittings / On Off Valve Setting / Nature Latex Pipe. The transport of sugar in plants takes place from the roots to the leaves. 8 In contrast, commonly used. Update in intracranial pressure evaluation methods and translaminar pressure gradient role in glaucoma Lina Siaudvytyte,1 Ingrida Januleviciene,1 Arminas Ragauskas,2 Laimonas Bartusis,1,2 Brent Siesky3 and Alon Harris1,3 1Eye Clinic, Lithuanian University of Health Sciences, Kaunas, Lithuania, 2Health Telematics Science Centre of Kaunas University. Stream Function. Strong gradients = strong winds Weak gradients = low wind speeds * That means, moving in any horizontal direction away from the "Low" will result in an increase in pressure. Air pressure is not uniform across the planet, however. P is the wake parameter, b = (d∗~t0)(dp/dx) is the Clauser pressure gradient parameter where d∗ is the displace-ment thickness, p is the free stream static pressure, t0 is the wall shear stress, x is the stream wise distance and the non equi-. As the difference in pressures rises, filtration increases from the area of high pressure to the area of low pressure. Your understanding of the concept is crucial. These non-linear effects have been modelled for flexible surfaces in a flow that has zero mean-flow pressure gradient (Newman & Goland, 1982). Convergence at the jet stream level forces air to sink because the highly stable tropopause prevents air from rising. The horizontal pressure difference generates a pressure gradient force that accelerates the air from the region of higher pressure (the warm air) towards the lower pressure (the cold air). , (1984) and. A Crustal Geostatic Gradient Pressure increases with depth in the earth due to the increasing mass of the rock overburden. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. Bars spaced far apart represent a gradual pressure gradient and light winds. Figure 2, (a) the stream function and its contour lines. The gradient strength at the center of ROI is 2. Oxygen and carbon dioxide move into and out of our blood by diffusion. 5) Figure 5. Army Corps of Engineers Environmental Laboratory : July 2010. As with Darcy’s law, the dependent variable always is pressure p. The gradient encompassed the full range of sediment types and organic carbon concentrations of the southern North Sea. This horizontal Pressure Gradient can be compared with the 2-D vector that arose due to the components drawn in a local horizontal plane. This study evaluates which factor between them plays more significant role for the determination of the gradient in association with continence function. The last term is −τ o/ρ since τ must be close to zero near the main stream. 4 for a surface of temperature twice the free-stream. zero-pressure gradient is applied to complex ows: the ow over a ramp, ow over the RAE2822 airfoil and ONERA M6 wing. Compute Fn = (Vn) x. During the day in summer when the land heats up more quickly than water, heat-related low pressure causes rising air over land which moves over the water and cools, then returns to land as a cooling "sea breeze". In other words, ΔP is increased by either an increase in flow or resistance. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = -0. Let define the free stream velocity. This is particularly true today. in particular that the pressure P is only defined up to a constant, which is fine, since only the gradient of P enters the momentum equation. Antonyms for concentration gradient. pressure gradient force is put into action. Temperature gradients between water and land can also cause local atmospheric circulations which affect winds. The produce large circular currents in all. The pressure on the convex surface by SST model does not. He found it paradoxical that the ejected blood from the ventricle continues into the aorta despite the positive pressure gradient. If the pressure is different in two parts of the atmosphere next to each other, there will be a greater force on one side. Sound waves parallel to the plane of the diaphragm produces no pressure differential, and so pressure-gradient microphones have figure-eight directional characteristics. Because of the capability to partly account for the effects of pressure gradients and departure from equilibrium, the non-equilibrium wall functions are recommended for use in complex flows involving separation, reattachment, and impingement where the mean flow and turbulence are subjected to severe pressure gradients and change rapidly. C, the constant, lets you know that the sum of a fluid's static pressure and dynamic pressure, multiplied by the fluid's velocity squared, is constant at all points along the flow. The difference in pressure between two areas is called the pressure gradient, and it is this gradient that plays a role in wind. ratio of the free stream velocity U to the friction velocity up. In atmospheric science, the pressure gradient is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The experimental and computed velocity profiles at two downstream locations (Samuel & Joubert's station 9 and station 12) are shown in Figure 1. It is shown that optimally excited algebraic mechanisms are. This boundary condition sets the pressure gradient to the provided value such that the flux on the boundary is that specified by the velocity boundary condition. Gradient definition: A gradient is a slope, or the degree to which the ground slopes. attached boundary layer subjected to adverse pressure gradient, contours of constant two-point spatial correlation of wall-pressure fluctuations are more elongated in the spanwise direction. The Pressure Gradient Force flows from a southerly to northerly direction. Isobars at a sufficiently large time in a circular cavity, which correspond to those in a steady flow (Re = 10) IMPROVEMENT OF A PRESSURE GRADIENT METHOD can be compared for sufficiently small values of Re, using an asymptotic analytical solution for Re -+ 0, where the stream function t+b and the pressure p can be expressed as t+b= -\$(r2-1. A common adverse pressure-gradient flow for tests of turbulence models is the experiment of Samuel & Joubert 15. The streamwise pressure gradient is furthermore assumed to be constant in the wall-normal direc-tion. However, data on the effects of improvements in LV function in humans and the relationship to IVPGs. 1) If stream is unaltered or no obstructions to natural water flow, and there is no excessive ponding within the channel, the score for this variable is 1. The associated workflow provides a systematic integration of the predictive saturation gradient and saturation pressure models to estimate the degree of undersaturation, and hence, potential GOC depth in a reservoir characterized by. Partial Pressure Gradients. 20-25% of all calories consumed by the human body are used to maintain this vital concentration gradient! Function of Concentration Gradients. NB: Specific Gravity is always relative to pure water. If you want the gradient at a specific point, for example, at (1, 2, 3), enter it as x,y,z=1,2,3, or simply 1,2,3 if you want the order of variables to be detected automatically. The Gradient (also called Slope) of a straight line shows how steep a straight line is. The threshold pressure gradient, which is associated with non-Darcy flow in low permeability reservoirs, is defined as the level of pressure gradient that must be attained to enable the fluid to overcome the viscous forces and start to flow. different upstream histories leading to a particular pressure gradient condition. The data used in the analysis come from the experiment per-formed for the pressure gradient conditions represen-. Nov 2010, Berlin, Deutschland. , pressure gradient) across the vessel length or across the valve (P 1-P 2 in the figure to the right). The value of the strength (or norm) of the pressure gradient in the troposphere is typically of the order 9 Pa/m (or 90 hPa/km). Dye line may refer either to a streakline: dye released gradually from a fixed location during time; or it may refer to a timeline: a line of dye applied instantaneously at a certain moment in time, and observed at a later instant. Therefore, the jet stream flows from the west to east. In this lesson, we explore wind velocity and air pressure on weather maps and examine how we can effectively determine wind velocity as a result of the illustrated air pressure gradient. This is the second part of Theorem 3. pressure gradient and Prandtl number on velocity and temperature profiles on convective heat transfer in boundary is the similarity function and is the stream function, and simply by replacing and components of velocity by a single function. Pressure gradient seems difficult, but it is simply using the density of the fluid and converting units: The density of pure water is 1000 kg/m3. Simple two-layer model for temperature distribution in the inner part of turbulent boundary layers in adverse pressure gradients International Journal of Heat and Mass Transfer, Vol. 10 Pressure gradient in radial flow. in particular that the pressure P is only defined up to a constant, which is fine, since only the gradient of P enters the momentum equation. Pressure gradient force (PGF) The pressure gradient force starts the horizontal movement of air over Earth’s surface. The SCL-10AVP conveniently linksall VP Series modules via fiber optic interfacing for easy "plug-and. The individual gas bubble moves with unique velocities as a function of its diameter8. [3] examined mean velocity pro les of. Pressure gradients for incompressible fluids have units of pressure/depth. DEPT H (ft) f(x) = 4. Osmosis occurs until the concentration gradient of water goes to zero or until the hydrostatic pressure of the water balances the osmotic pressure. If LV function improves with dobutamine infusion and if the gradient increases, aortic stenosis is severe and is associated with secondary LV dysfunction. A blood pressure gradient refers to a difference in the blood pressure between two points in the vasculature. In fact, the same pressure change observed in the lowest 30 m (98 ft) of the troposphere may not be equaled over a horizontal distance of 200 km (124 mile) at sea level. Lecture 18: Pipe Pressure Drop/Minor Losses: Part 1. Deviations from normal pressure are described as high or low pressure. Actually, HVPG usually used in liver disease diagnosis. Pressure gradient force (PGF) The pressure gradient force starts the horizontal movement of air over Earth’s surface. (6) Now, from (4), knowing w o = 0 w δ. The velocity components are given by udPdxyhy=−()12µ()(2) and v =0, where µ is the fluid's viscosity. Wavenumber spectra are converted to frequency spectra, and compared with experiments. A surface region of low atmospheric pressure around which winds spiral inwards H. So the fluid can only flow when the pressure gradient acting on the fluid is bigger than a critical value, which is known as the threshold pressure gradient (TPG). A normal aortic valve area is greater than or equal to 2. Favorable pressure gradient (accelerating flow) Adverse pressure gradient (decelerating flow). 4 indicates that the pressure is a function of x and y. Objectives The aim was to assess the effects of interventional treatment on external and internal heart power (EHP, IHP) in patients with. - Conditions downstream of exit plane impact flow in domain. It is referred to as transepidermal water loss (TEWL). The stream flows from 500 feet to 300 feet in a distance of 40. "The meridional temperature gradient between the equator and poles that gives rise to the jet stream also produces secondary atmospheric circulations, or eddies. Pressure and depth: "Pressure and Depth" is the FUNDAMENTAL relationship in the oil industry. A special case occurs when the barotropic pressure gradient is equal to but of the opposite sign of the baroclinic pressure. 86; namely:. This document is highly rated by JEE students and has been viewed 5482 times. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. Density and sea surface height are used to calculate weight of the water column above a given point. The associated workflow provides a systematic integration of the predictive saturation gradient and saturation pressure models to estimate the degree of undersaturation, and hence, potential GOC depth in a reservoir characterized by. Outlet pressure for multiphase pipe flow by Gray correlation, [psia] It is commonly used for gas wells that are also producing liquid. The streamwise vortices were created by three pairs of half-delta wing vortex generators, while impulsively initiated opposite-wall suction created a strong adverse pressure gradient. 2D Boundary Layer Modelling The analysis of an aerofoil's boundary layer can be done as an extension of the results for a simple flat plate. The power of improved heating technology can completely change the way the food and beverage industry does business. Th s a gi en t ti l di t implies the same ESS227 Prof. zero-pressure gradient is applied to complex ows: the ow over a ramp, ow over the RAE2822 airfoil and ONERA M6 wing. Britain will also make existing retail customer disclosure rules known as PRIIPs function better, he said. (a) Determine the pressure gradient along this streamline. The Gradient Wind Approximation is a useful construct but to understand it properly it is necessary to first review the conventions concerning such things as the radius of curvature of motion. EASY TO OPERATE: Single handle design integrates to control water temperature and flow volume easily. The simplified Bernoulli equation: PG = 4 (V)' allows calculation of the pressure gradient from the blood velocity. Pressure and Velocity Distribution Near the Convex Surface. , (1984) and. • Near the top of the ball the local external pressure. So not only is the propagation pressure not equal to the frac gradient but, like breakdown pressure, the propagation pressure is not solely a function of formation stress (in the case of breakdown. Ali Keshavarz. Assuming the density of sea water to be 1025 kg/m³ (in fact it is slightly variable), pressure increases by 1 atm with each 10 m of depth. The best part: It’s all voluntary. The following lines are a guide to how you should use this A-a gradient calculator: - O2 Arterial pressure – PaO2 is the partial arterial pressure of the oxygen in the arteries; its range is between 75-100 mmHg. The gradient is a vector operation which operates on a scalar function to produce a vector whose magnitude is the maximum rate of change of the function at the point of the gradient and which is pointed in the direction of that maximum rate of change. • Adverse pressure gradient boundary layer very common in aerodynamics Project Motivation and Aims DLR. 高等流力 之剛式教學 3 stream function, velocity potential, Laplace equation, linearity by skwang 義守大學 王曉剛 高等流力 之剛式教學 13 pressure, pressure gradient by. This is similar to. Instability waves and transition in adverse-pressure-gradient boundary layers Abstract Transition to turbulence in incompressible adverse-pressure-gradient (APG) boundary layers is investigated by direct numerical simulations. The gradient is called OBG or overburden gradient. Water will move as horizontal currents from regions of high pressure to regions of lower pressure. For this type of an application, it is necessary to study how pressure gradients affect the analogy functions. The expression for pressure rise is computed numerically by evaluating the numerical integration. The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression {eq}\psi{/eq} = -19(x - y) where the stream function has the units of ft{eq}^2{/eq}/s. In purely theoretical terms, this physical measuring principle is the most appropriate one because it only provides a signal during dynamic pressure curves, but with a very good signal-to-noise interval and for high frequencies. Consider a volume of air, dV=dxdydz. Assuming the density of sea water to be 1025 kg/m³ (in fact it is slightly variable), pressure increases by 1 atm with each 10 m of depth. determining horizontal pressure gradients to understand the movement of waters through the ocean. An integration is required to get a function of height vs. recall that the strength of the geostrophic wind is proportional to the magnitude of the pressure gradient force. It is shown that there exist bodies such that in two-dimensional steady inviscid incompressible flow the pressure gradient is favourable over the entire surface of the body, and the lift is non-zero, if the body is immersed in a uniform stream and there are also two trapped point vortices. Pressure gradients (PG) are calculated by measuring the velocity (V) across the valve. You may have to register before you can post: click the register link above to proceed. organism level (function together) eleven organ systems. Velocity Potentials and Stream Functions As we have seen, We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace's equation. The difference in pressure between two areas is called the pressure gradient, and it is this gradient that plays a role in wind. Differences in air pressure and the pressure gradient force are caused by the unequal heating of the Earth's surface when incoming solar radiation concentrates at the equator. Integrate Eqn. Oedema will occur if ISF oncotic pressure is not kept low. these walls due to the pressure gradient that exists between the blood in the capillaries and the fluid in the Bowman's capsule. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. In the linear model, the transpulmonary pressure gradient (TPG) is only a function of flow rate (Q) as shown in a), and is not affected by pulmonary capillary wedge pressure (Ppcw), as shown in b), whatever the pulmonary vascular resistance. IMPROVED NEW HIGH QUALITY GARDEN HOSE- Leak-Resistant Connection / Anti-Rust Metal Fittings / On Off Valve Setting / Nature Latex Pipe. Thanks for the reply. To calculate water pressure gradient (P grad), use the following formula: where: ρ w = water density For example, given ave. Moreoer,v the complex interaction between separa-tion induced by adverse pressure gradients and the ensuing transition process can also haev a. As the difference in pressures rises, filtration increases from the area of high pressure to the area of low pressure. Thus the pressure gradient force is balanced by friction and Coriolis force. Flashcards. Stream Function. Compute the atmospheric gradient Richardson number and, optionally, the Brunt-Vaisala, buoyancy and shear. NB: Specific Gravity is always relative to pure water. Normal pressure gradients depend only on the density of the fluid in the pores, integrated from surface to the depth of interest. 3 to yield Also note that the flow is negative, i. As a function of geopotential height, the pressure gradient acceleration is the acceleration due to gravity (g) times the gradient of the geopotential height (see Wallace & Hobbs Ch. 2 Average gradient (EMBGN) We notice that the gradient of a curve changes at every point on the curve, therefore we need to work with the average gradient. 0 cm2 with a pressure gradient of less than 25 mmHg. Methods: Forty-eight patients with HOCM participated (mean age:58±12 years). Consider a volume of air, dV=dxdydz. edu Port 80. Jet stream lab. Pressure increases predictably with depth in areas of normal pressure. Backflow prevention for high pressure gradient systems: 2008-07-08: Andrews et al. Pore pressures can be expressed as a gradient, either psi/ft or Pa/m. 3 m/s and mean pressure gradient (PG) of 23. (c) Determine The Pressure Gradient. It is shown that there exist bodies such that in two-dimensional steady inviscid incompressible flow the pressure gradient is favourable over the entire surface of the body, and the lift is non-zero, if the body is immersed in a uniform stream and there are also two trapped point vortices. For smooth-wall flow,. For a horizontal-radial system, the pressure gradient is positive (see Figure 1. The other component of gas exchange is the delivery of oxygen from the atmosphere, through the lungs, and into the blood. The calculation of pressure gradient in reference to the subsea, RKB, DF can leads to a serious misleading calculation. Let us consider that V is the resultant velocity of a fluid particle at a point in a flow filed. Pressure gradients for incompressible fluids have units of pressure/depth. The partial pressure of oxygen (PO 2) of venous blood is 40 mm Hg; The PO 2 in the alveoli is ~100 mm H g; Steep gradient allows PO 2 gradients to rapidly reach equilibrium (0. 2: 6679274: Clean-in-place method for cleaning solution delivery systemes/lines: 2004-01-20: Gruszczynski et al. Adverse pressure gradient decreases the dimensionless velocity of the boundary layer. This is an unprecedented time. The horizontal pressure difference generates a pressure gradient force that accelerates the air from the region of higher pressure (the warm air) towards the lower pressure (the cold air). 1) If stream is unaltered or no obstructions to natural water flow, and there is no excessive ponding within the channel, the score for this variable is 1. Problem 1: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by Psi = -2(x-y) where the stream function has the units of ft^2/s with x and y in feet. pressure gradient and Prandtl number on velocity and temperature profiles on convective heat transfer in boundary is the similarity function and is the stream function, and simply by replacing and components of velocity by a single function. Last summer's hot-and-dry weather in the Southwest was a function of the so-called "Four Corners High" — a high-pressure system that typically sets up over the region — failing to slide east. The stream function equation is discretized using the standard central difference, and can be solved using an iterative elliptic solver, such as Jacobi or Gauss-Seidel. In this calculator, you have three input values: the distance (in km) of the two locations, or centers of high and low pressures the pressure (in kPa) at the first location (the area of low pressure) the pressure (in kPa) at the second location. The Coriolis force eventually balances the horizontal pressure gradient force resulting in a strong stream of air that flows from the west towards the east. 7 PressureGradientHarBrown: Pressure gradient for multiphase pipe flow by Hagedorn and Brown correlation with Griffith modification , [psi/ft]. Number of equations. Simplified method for calculation of compressible laminar boundary with arbitrary free-stream pressure gradient Auxiliary functions usea in equations (31) and (32). Example: Find the pressure on a scuba diver when she is 12 meters below the surface of the ocean. If the motion is irrotational, curl v = 0, then we know that v can be derived from a potential function φ, v = -grad φ. Gravity ball can help the nozzle automatically retracted to its original position after use. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. For example, psi/ft, bar/m. Hepatic venous pressure gradient commonly called as HVPG and it has function to diagnose the condition of a patient. - In unsteady flows with variable density. The simplified Bernoulli equation: PG = 4 (V)' allows calculation of the pressure gradient from the blood velocity. 26 R = 1 2000. Offshore, the pressure of the reservoirs and seals of these upper unconsolidated sediments has the same gradient (+/- 0. The calculated AVA is showing 0. pressure gradient region usually develops just down-stream of the blunt leading edge. 12 For each of the following stream functions, with units of m2/s, determine the magnitude and the angle the velocity (Sts. However, upon implantation in vivo, a major challenge for clinically relevant large‐size grafts is the maintenance of cell. (b) If the 6. Aortic stenosis or aortic valve stenosis (AS) is defined as the presence of an increase in pressure across the aortic valve (AV). Since a closely spaced gradient implies. DEPT H (ft) f(x) = 4. This document is highly rated by JEE students and has been viewed 1005 times. The normal vectors to the level contours of a function equal the normalized gradient of the function: Create an interactive contour plot that displays the normal at a point: View expressions for the gradient of a scalar function in different coordinate systems:. 8 kPa/m] in a well with a true vertical depth of 8000 ft [2440 m] would predict a fracturing pressure of 5600 psi [38. That combination will give us non-pressure vorti-city transport equation which in non-steady form can be written as follows: ∂ζ ∂t +u ∂ζ. The pressure gradient often has a small but critical horizontal component, which is largely responsible for the wind circulation. Temperature gradients between water and land can also cause local atmospheric circulations which affect winds. (6) Now, from (4), knowing w o = 0 w δ. The three functions of the lymphatic system are: Return of protein and fluid from the ISF to the circulation to maintain a low interstitial fluid protein concentration and maintain the oncotic pressure gradient across the capillary membrane. , periodic in x and y) we can easily calculate the pressure (defined up to a constant), using the boundary conditions that gradients in the pressure are subject to periodic BCs: ∇ p (x, y) = ∇ p (x + L, y) etc. To be clear, this pressure is an auxiliary field, and the accuracy of the simulation isn't dependent on it. On the left side, the scheme will fail to take account of the information that the continuous pressure gradient force needs, that of † Tj-1/ 2,k-1. (6) Now, from (4), knowing w o = 0 w δ. equation for the stream function. to find the pressures and stream function, but omitted the local acceleration terms. So not only is the propagation pressure not equal to the frac gradient but, like breakdown pressure, the propagation pressure is not solely a function of formation stress (in the case of breakdown. Quantifying Aortic Stenosis. Transmitral pressure gradients. Stream Gradient Calculations The formula for gradient is: the difference in elevation between two points on the stream distance along the stream or the RISE RUN Using this formula, calculate the gradients in the problems below. , periodic in x and y) we can easily calculate the pressure (defined up to a constant), using the boundary conditions that gradients in the pressure are subject to periodic BCs: ∇ p (x, y) = ∇ p (x + L, y) etc. The Significance Of The Hydraulic Gradient Normally when a pipe is laid, attempts are made to keep the pipe at or below the hydraulic gradient. where pi is the surface pressure measured at location i on the surface, p∞ is the pressure in the free stream, ρ is air density, and U∞ is the free-stream velocity given by () ρ ∞ ∞ − = p p U stagnation 2 (2) where pstagnation is the stagnation pressure measured at the tip of the Pitot tube. 5) Figure 5. "The meridional temperature gradient between the equator and poles that gives rise to the jet stream also produces secondary atmospheric circulations, or eddies. Laminar and Turbulent Flow : Eqn. A Crustal Geostatic Gradient Pressure increases with depth in the earth due to the increasing mass of the rock overburden. Consider a volume of air, dV=dxdydz. tissue level (2 or more cell types) common function 4. Synonyms for concentration gradient in Free Thesaurus. The gradient of a river is defined as grade measured in by the ratio of drop in elevation of a stream per unit of horizontal distance (in other words, the "steepness" of a river). Question Correct Match Selected Match Cyclone H. 460 psi/ft or 10. Solution: The density of sea water is 1. 465 psi/ft in GoM) and is a function of depth and sea water density. (b) If the 6. Report presenting a simplification of the Karman-Polhausen integral method as applied to compressible laminar boundary layers. to find the pressures and stream function, but omitted the local acceleration terms. On the left side, the scheme will fail to take account of the information that the continuous pressure gradient force needs, that of † Tj-1/ 2,k-1. The units of pressure gradient are Pa/m in SI units. Mathematically, it is obtained by applying the del operator to a pressure function of position. "Concentration" refers to how much of a solute there is compared to solvent. 86; namely:. For research of the characteristics of the steady and unsteady flows and heat transfer at high-free-stream turbulence intensity the turbulence two-parametric models have been presented. The average gradient between any two points on a curve is the gradient of the straight line passing through the two points. Since the Earth is rotating, however, the air does not flow directly from high to low pressure, but it is deflected to the right (in the Northern Hemisphere; to the left in the Southern Hemisphere), so that the wind flows mostly around the high and low pressure areas. Compute dew point temperature as a function of actual vapor pressure as described in FAO 56. This project report presents a study on measurements and prediction of laminar-turbulent transition at high free-stream turbulence in boundary layers of the airfoil-like geometries with presence of the external pressure gradient changeover. The tricuspid regurgitation pressure gradient (TRPG) strongly correlates with the pulmonary artery systolic pressure (PASP). The other component of gas exchange is the delivery of oxygen from the atmosphere, through the lungs, and into the blood. – Gradients in flow direction are significant. Blood Pressure Gradients are ultimately responsible for the flow of blood within the vasculature. Yang and I. Pressure gradient determination in production and injection wells is essential to obtain information concerning the effects of flow-rate, pipe size, and pressure on each other. Instability waves and transition in adverse-pressure-gradient boundary layers Abstract Transition to turbulence in incompressible adverse-pressure-gradient (APG) boundary layers is investigated by direct numerical simulations. (c) Determine The Pressure Gradient. It is often true that the horizontal pressure gradient, and hence the geostrophic velocity, decreases with depth in the ocean. REPORT 1293 SIMILAR SOLUTIONS FOR THE COMPRESSIBLE LAMINAR BOUNDARY LAYER WITH HEAT TRANSFER AND PRESSURE GRADIENT l. 433 psi/ft, or 9. These non-linear effects have been modelled for flexible surfaces in a flow that has zero mean-flow pressure gradient (Newman & Goland, 1982). a3 10 O 10 a5 20 George M. Some patients, however, manifest a restricted AVA but have a paradoxically low pressure gradient with a normal ejection fraction (EF). 8 Baroclinic Instability pressure gradient on geometric height surfaces. Nelson Hessert Center for Aerospace Research, The University of Notre Dame, Notre Dame, Indiana 46556. pressure gradient, the remaining factors have been taken into account very completely by Crocco (ref. The pressure gradient ∂p ∂x is assumed not to be a function of z so the term becomes simply the gradient multiplied by δ. Strong gradients = strong winds Weak gradients = low wind speeds * That means, moving in any horizontal direction away from the "Low" will result in an increase in pressure. The matrices x, y, u, and v must all be the same size and. The Pressure Gradient Force weakens at the same time the Coriolis has not had time to adjust and decrease. Brookfield Submitted to the Department of Aeronautics and Astronautics on May 17, 1993 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An analytical and computational study was performed to examine the effects of adverse. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = -0. The results show the existence of an inertial subrange for a separation range of about one decade. The simplified Bernoulli equation: PG = 4 (V)’ allows calculation of the pressure gradient from the blood velocity. Kline et al. Answer: We have seen that flow between parallel plates (fixed) for incompressible, steady state flow is: =−(𝜕 𝜕 ) ℎ2 2𝜇 [1−. This project report presents a study on measurements and prediction of laminar-turbulent transition at high free-stream turbulence in boundary layers of the airfoil-like geometries with presence of the external pressure gradient changeover. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. In fact, the same pressure change observed in the lowest 30 m (98 ft) of the troposphere may not be equaled over a horizontal distance of 200 km (124 mile) at sea level. The paper presents the attempt to modify the scal-ing proposed by Alfredsson et al. The associated workflow provides a systematic integration of the predictive saturation gradient and saturation pressure models to estimate the degree of undersaturation, and hence, potential GOC depth in a reservoir characterized by. function is formulated for the low-wavenumber range. If the pressure is different in two parts of the atmosphere next to each other, there will be a greater force on one side. For flow between two parallel fixed plates due to the pressure gradient, compute: (a) The wall shear stress, (b) The stream function, (c) The vorticity, (d) The velocity potential, (e) The average velocity. 37) a 5+49/14 hah. R824777 (Final) not available: Dissertation/Thesis (1) Journal Article : Particle transport and transient storage along a stream-size gradient in the Hubbard Brook Experimental Forest. Density and sea surface height are used to calculate weight of the water column above a given point. 19 At a given point on a horizontal streamline in flowing air, the static pressure is −2. On the other hand, onshore, where topography has a great impact on the hydrology of this zone, lateral piezometric pressure gradient applies. The normal vectors to the level contours of a function equal the normalized gradient of the function: Create an interactive contour plot that displays the normal at a point: View expressions for the gradient of a scalar function in different coordinate systems:. Pressure gradient is just the difference in pressure between high- and low-pressure areas. Complementing this function, is the identification of a robust empirical model to predict saturation pressure. 10) and Darcy’s equation can be expressed in the following generalized radial form: v = q r A r = k µ ∂p ∂r [1. The Coriolis force eventually balances the horizontal pressure gradient force resulting in a strong stream of air that flows from the west towards the east. With only horizontal components to the flow, the Navier-Stokes equations for incompressible flow are. Now, if you're sensible, you are wondering why I dragged you through that big long calculation after all, although it is nice to know what the alveolar partial pressure of oxygen is, it probably doesn't seem very useful right now. 138 patients had radial-femoral artery pressure gradient after cardiopulmonary bypass (group P) but 263 were not (group N). An extensive data set describing separated flows with different Reynolds numbers (Re), free-stream turbulence intensities (Tu) and adverse pressure gradients (APG) is used for the statistical characterization of laminar separation bubbles (LSBs). References P. In particular, the x-momentum equation, Eq. pressure gradient force is put into action. Problem 1: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by Psi = -2(x-y) where the stream function has the units of ft^2/s with x and y in feet. The results of the theory are in good agreement with experimental data. Digital artists can use a variety of art tools in software to create images, including. the frictional force 22. counterclockwise and inward toward the center. It represents 1 newton/m 2. Differences in air pressure and the pressure gradient force are caused by the unequal heating of the Earth's surface when incoming solar radiation concentrates at the equator. Britain will also make existing retail customer disclosure rules known as PRIIPs function better, he said. tissue level (2 or more cell types) common function 4. In pure mitral stenosis, the blood flow from the left atrium into the left ventricle is impaired, resulting in a pressure gradient between the two chambers during diastole. The effects of gas and water flow rate on pressure gradient. The pressure gradient is a way to describe the difference in atmospheric pressure from one location to another. This limits its use to the non-separated flows only. Similarly, the average channel profile based on link slope can be related to link magnitude or discharge in the form of a power equation. organ level (2 or more tissue types) specialized task 5. So, the situation of an equator to pole temper- The background uniform zonal wind shear means the basic state stream function can be written as ψ= −Λyz. Jet Stream Formation - Polar Jet - pressure gradient aloft. If you step into the reference frame of the rocket, it would be nice to have the rocket aft-end have the same velocity as the free-stream air approaching the rocket from the front. NS equations reduce to a infinite set of equations. Formation fluid pressure gradient = 0. For flow between two parallel fixed plates due to the pressure gradient, compute: (a) The wall shear stress, (b) The stream function, (c) The vorticity, (d) The velocity potential, (e) The average velocity. Points in the direction of greatest increase of a function (intuition on why)Is zero at a local maximum or local minimum (because there is no single direction of increase). A Coupled Modeling System to Predict Morphology Changes and a Comparison of Pressure Gradient Forces to Shear Stresses i. 210/656: 6780315: Backflow prevention for high pressure gradient systems: 2004-08-24: Richardson et al. Governing Equations: Continuity: Number of unknowns. It is pressure gauge. 7 PressureGradientHarBrown: Pressure gradient for multiphase pipe flow by Hagedorn and Brown correlation with Griffith modification , [psi/ft]. • Density no longer appears explicitly in the pressure gradient force; this is a distinct advantage of the isobaric system. 2 pounds; 1 m = 39. The boundary layer and its interaction with the local pressure gradient plays a major role in affecting the flow over a cylinder. The Coriolis force eventually balances the horizontal pressure gradient force resulting in a strong stream of air that flows from the west towards the east. are shown for the various methods in Figure 1, while the pressure gradient function P is plotted in Figure 2. 433 psi/foot or 9. Yang, “ Numerical study of transition process in a separated boundary layer on a flat plate with two different leading edges,” WSEAS Trans. In addition to the solution steps, we have the visualization step, in which the stream function Qn is computed. Problem 1: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by Psi = -2(x-y) where the stream function has the units of ft^2/s with x and y in feet. So, the situation of an equator to pole temper- The background uniform zonal wind shear means the basic state stream function can be written as ψ= −Λyz. This boundary condition sets the pressure gradient to the provided value such that the flux on the boundary is that specified by the velocity boundary condition. Subsequently the total pressure gradient in the lower layer is directed up stream. Pressure increases predictably with depth in areas of normal pressure. It is not an easy matter to find velocity fields that satisfy the equation of continuity. A Crustal Geostatic Gradient Pressure increases with depth in the earth due to the increasing mass of the rock overburden. R824777 (Final) not available: Dissertation/Thesis (1) Journal Article : Particle transport and transient storage along a stream-size gradient in the Hubbard Brook Experimental Forest. The normal hydrostatic pressure gradient for freshwater is 0. The Significance Of The Hydraulic Gradient Normally when a pipe is laid, attempts are made to keep the pipe at or below the hydraulic gradient. Stream Function. 1b] where: q r = volumetric flow. , (1984) and. The Pressure Gradient When part of the atmosphere has a lower pressure than the surrounding area, a pressure gradient exists. Brookfield Submitted to the Department of Aeronautics and Astronautics on May 17, 1993 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An analytical and computational study was performed to examine the effects of adverse. 2) and Chapman and Rubesin (ref. termine the pressure gradient, ðp/ðx, (as a func- tion of x) needed to produce this flow. The term gradient is often applied in cardiology when describing pressure variation in the vicinity of a stenotic heart valve (for example). Water movement up a xylem is a function of (mark all correct answers): (A) positive pressure potential in turgid leaf cells (B) negative (suction) pressure potential further up the plant (C) negative osmotic potential in roots (D) negative water potential in leaves due to photosynthesis and the evaporation of water. Basic Concepts in Physical Oceanography: Approximations Approximations. The line integration of the pressure gradient field. Static pressure and pressure gradient value do not vary across boundary layer thickness. When expressed scientifically, pressure change over a unit distance is called pressure gradient force, and the greater this force the faster the winds will blow. 01 x 10 5 N/m 2. Initially, in the entrance region, the pressure falls rapidly with distance but far from the entrance the pressure falls linearly with distance, (b) The linear variation of the pressure gradient (Ap/L) in the region remote from the entrance as a function o{ the volume flow-rate Q through the tube. Last summer's hot-and-dry weather in the Southwest was a function of the so-called "Four Corners High" — a high-pressure system that typically sets up over the region — failing to slide east. Fresh water with zero salinity will generate a pressure gradient of 0. Given that the strength of the jet stream is influenced by the magnitude of the temperature-gradient, it follows that warming of the Arctic could lead to a weakening of the jet stream and a greater tendency to meander as it slows down. The normal hydrostatic pressure gradient for freshwater is 0. The results concerning the Reynolds-number dependence of the coefficients of the wall-region scaling. In Drake and Calantoni (2001), the near bed free stream velocity ~ubis driven by the pressure gradient F(t), and F(t)follows F(t) / X4 n=0 1 2n sin (n+1) 2ˇ T t+n˚ (14) and ˚ is dened as the flwaveform parameterfl. (b) If the upstream pressure is p0, integrate the pressure gradient to obtain the pressure p(x) for − ∞ ≤ x ≤ − a. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = -0. Stream Gradient Calculations The formula for gradient is: the difference in elevation between two points on the stream distance along the stream or the RISE RUN Using this formula, calculate the gradients in the problems below. The maximum catheter pressure gradient was calculated as the difference between the proximal and the minimum pressure measurements (P prox −P. After air spraying, the nascent. The horizontal Pressure Gradient is the horizontal component of the Pressure Gradient. 433 psi/ft, or 9. The negative gradient of pressure is known as the force density. The analysis was conducted under the assumptions of a Prandtl number of 1, zero heat transfer, and a linear viscosity-temperature relation. Closure Gradient = Closure Pressure / Formation Depth; Net Fracture Pressure (Δp net) - Net fracture pressure is the additional pressure within the frac above the pressure required to keep the fracture open. – Gradients in flow direction are significant. Osmosis occurs until the concentration gradient of water goes to zero or until the hydrostatic pressure of the water balances the osmotic pressure. Pressure Gradient Force (PGF) - causes horizontal pressure differences and winds 2. Synonyms for gradient at Thesaurus. If the pressure is different in two parts of the atmosphere next to each other, there will be a greater force on one side. In SI units, pressure (Pascals) is the force (Newtons) per unit area (meters2) such that 1 Pa = 1 N/m2. An integration is required to get a function of height vs. 26) In other words, the contours of the velocity potential and the stream function cross at right. G-Function Analysis. Jet Stream Formation - Polar Jet - pressure gradient aloft. Despite differences in mako and tuna gill morphology, mouth gape and basal swimming speeds, measurements of mako O2 utilization at the gills (53. Formation fluid pressure gradient = 0. Flashcards. (ANS: 2(y2 x2 ) C) (ANS: 64 R 27 r1 3) 7708d_review_R1-R22 8/27/01 2:25 PM Page R-14 mac79 Mac 79:1st shift:. 86; namely:. The gradient encompassed the full range of sediment types and organic carbon concentrations of the southern North Sea. A processing of recent experimental data by Nagib and Hites [Nagib, H. Coppola‐Owen E-mail address: [email protected] A special case occurs when the barotropic pressure gradient is equal to but of the opposite sign of the baroclinic pressure. Report presenting a simplification of the Karman-Polhausen integral method as applied to compressible laminar boundary layers. a part of a railway, road, etc. zero-pressure gradient is applied to complex ows: the ow over a ramp, ow over the RAE2822 airfoil and ONERA M6 wing. A Pascal is the unit of pressure in the metric system. Favorable pressure gradient have the opposite effect for the dimensionless velocity profile. Jin-Yi Yu • Thus, a given geopotential gradient implies the same geostrophic wind at any height, whereas a given horizontal pressure gradient implies different values. Pressure Gradient Force operates from the high pressure area to a low pressure area and causes wind movement. Jun 09, 2020 - Unit-5: Relation between shear stress, pressure gradient and velocity distribution - Lecture Notes JEE Notes | EduRev is made by best teachers of JEE. For smalf pressure gradients, Low (ref. When the pressure gradient force is balanced by the Coriolis force, high altitude _____ move parallel to isobars. 12 For each of the following stream functions, with units of m2/s, determine the magnitude and the angle the velocity (Sts. It is a unit vector in the direction of the motion. LEMDIASOV,1 REINHOLD LUDWIG2 1 Insight Neuroimaging Systems,39Salisbury treet,Worcester,Massachusetts 01609, USA 2 ECE Department ,WorcesterPolytechnic I nstitute, 100 I Road ,Massachusetts 01609, USA ABSTRACT: A new design approach for the construction of gradient coils for magnetic resonance imaging is presented. The result is expressed in feet per mile (ft. Comparison of Methods for Estimating Stream Channel Gradient Using GIS David Nagel, John Buffington, and Daniel Isaak USDA Forest Service, Rocky Mountain Research Station Boise Aquatic Sciences Lab Boise, ID September 14, 2006 Special thanks to Sharon Parkes…. Consider a volume of air, dV=dxdydz. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): unsteady flow, pressure gradient. Brookfield Submitted to the Department of Aeronautics and Astronautics on May 17, 1993 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An analytical and computational study was performed to examine the effects of adverse. The rate of diffusion is determined by partial pressure gradients across the respiratory membrane in our lungs. Description. Last summer's hot-and-dry weather in the Southwest was a function of the so-called "Four Corners High" — a high-pressure system that typically sets up over the region — failing to slide east. Complementing this function, is the identification of a robust empirical model to predict saturation pressure. Quantifying Aortic Stenosis. Mathematically, it is obtained by applying the del operator to a pressure function of position. A special case occurs when the barotropic pressure gradient is equal to but of the opposite sign of the baroclinic pressure. where the stream function has the units of ft 2 /s with x and y in feet. Despite differences in mako and tuna gill morphology, mouth gape and basal swimming speeds, measurements of mako O2 utilization at the gills (53. To calculate the Gradient:. pressure gradient, the remaining factors have been taken into account very completely by Crocco (ref. IMPROVED NEW HIGH QUALITY GARDEN HOSE- Leak-Resistant Connection / Anti-Rust Metal Fittings / On Off Valve Setting / Nature Latex Pipe. (a) Is The Continuity Equation Satisfied? (h) Is The Flow Field Irrotational? If So, Determine The Corresponding Velocity Potential. Flowing Pressure Gradient Plot PRESSURE (psi) 600 0. This region pro-motes early transition and thereby substantially re-duces the percentage of laminar ow ervo the wing. The average gradient between any two points on a curve is the gradient of the straight line passing through the two points. Bars spaced far apart represent a gradual pressure gradient and light winds. Overlying theme: wind is the result of a horizontal difference in pressure- Wind always blows initially from high to low pressure (in the absence of all other forces) and this is due to the pressure gradient force (PGF)- Once the parcel starts to move (as a result of the PGF) the Coriolis force begins to act to the right of the wind (in the northern hemisphere), balancing the. 210/656: 6780315: Backflow prevention for high pressure gradient systems: 2004-08-24: Richardson et al. At any given pressure gradient (ΔP), the flow rate is determined by the resistance (R) to that flow. In the stream-function formulation the pressure is eliminated via the incompressibility condition, however occasionally it is useful to examine it in order to gain some insight. counterclockwise and inward toward the center. Brookfield Submitted to the Department of Aeronautics and Astronautics on May 17, 1993 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An analytical and computational study was performed to examine the effects of adverse. Compute Fn = (Vn) x. 7 The TRPG has recently been proposed as a responsive metric of worsening heart failure, with a lower PASP having been associated with less clinically severe heart decompensation during AHF. The Pressure Gradient When part of the atmosphere has a lower pressure than the surrounding area, a pressure gradient exists. The greater the horizontal temperature difference, the stronger the jet stream. An extensive data set describing separated flows with different Reynolds numbers (Re), free-stream turbulence intensities (Tu) and adverse pressure gradients (APG) is used for the statistical characterization of laminar separation bubbles (LSBs). The pressure gradient parameter suppresses the boundary layer growth, and when the pressure gradient parameter is, this is on a two-dimensional stagnation flow. a measure of such a slope, esp the ratio of the vertical distance between two. 2%) and the pressure gradient driving. 4 KPa/meter. When the pressure gradient force is balanced by the Coriolis force, high altitude _____ move parallel to isobars. ’-’’ This article reviews noninvasive quantifica- tion of intracardiac pressure and assessment of cardiac function. It has been shown that these effects are a function of the free-stream turbulence intensity, the turbulence length scale, and the boundary layer momentum thickness Reynolds number. The streamwise vortices were created by three pairs of half-delta wing vortex generators, while impulsively initiated opposite-wall suction created a strong adverse pressure gradient. (a) Is the continuity equation satisfied? (b) Is the flow field irrotational? If, so, determine the corresponding velocity potential. The gradient of a river is defined as grade measured in by the ratio of drop in elevation of a stream per unit of horizontal distance (in other words, the "steepness" of a river). Hense space is fake. The symbol used is (psi). The present study aimed to assess the effects of verdpamil on pno-hepatic pressure gradient and on hepatic. Therefore, the normal pressure gradient along y = n is ∂p = −ρV2κ ∂n This is the normal-momentum equation, sometimes also called the centrifugal formula. IMPROVED NEW HIGH QUALITY GARDEN HOSE- Leak-Resistant Connection / Anti-Rust Metal Fittings / On Off Valve Setting / Nature Latex Pipe. The pressure on the convex surface by SST model does not. The velocity at the entrance section is assumed constant. In Drake and Calantoni (2001), the near bed free stream velocity ~ubis driven by the pressure gradient F(t), and F(t)follows F(t) / X4 n=0 1 2n sin (n+1) 2ˇ T t+n˚ (14) and ˚ is dened as the flwaveform parameterfl. Gravity will tend to pull the water down the "hill" or pile of water against the pressure gradient. The horizontal Pressure Gradient is the horizontal component of the Pressure Gradient. In this lesson, we explore wind velocity and air pressure on weather maps and examine how we can effectively determine wind velocity as a result of the illustrated air pressure gradient. The heart causes blood to flow by creating a mean arterial pressure that is greater than the pressure in the veins. It's a vector (a direction to move) that. Since there is no flow rate normal to a stream line, then it follows that the stream function is the same between O and any point P, P' or P'' on the same stream line. Thomas,a) and Robert C. If the fluid is incompressible, then the equation of continuity div v = 0 becomes div grad φ = 0. 13, the equation works as follows: Table of water pressure gradients. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a Swing bowling (2,543 words) [view diff] no match in snippet view article find links to article. for the case of favorable pressure gradients with heated walls, the velocity within a portion of the boundary layer is shown to exceed the local external velocity. The gradient of a river is defined as grade measured in by the ratio of drop in elevation of a stream per unit of horizontal distance (in other words, the "steepness" of a river). But the Coriolis Force intervenes and cause the water to move to the right (in the northern hemisphere) around the mound of water. A partial condenser functions as. The partial pressure of oxygen (PO 2) of venous blood is 40 mm Hg; The PO 2 in the alveoli is ~100 mm H g; Steep gradient allows PO 2 gradients to rapidly reach equilibrium (0. This experiment studied the effect of streamwise vortices on a turbulent boundary-layer exposed to an unsteady adverse pressure gradient in a water tunnel at a momentum thickness Reynolds number of 1840. alveolar/arterial gradient synonyms, alveolar/arterial gradient pronunciation, alveolar/arterial gradient translation, English dictionary definition of alveolar/arterial gradient. For smalf pressure gradients, Low (ref. to the left, for a positive pressure gradient, dp/dx. 4) has, by a perturbation analYSiS, treated the general prob lem of the isothermal surface. Internal Pressure Gradient Errors in ˙-coordinate Ocean Models: The Finite Volume and Weighted Approaches Master thesis in Applied and Computational Mathematics Helene Hisken Pedersen Department of Mathematics University of Bergen April 27, 2010. a coupled modeling system to predict morphology changes and a comparison of pressure gradient forces to shear stresses in the nearshore by william l. influence of a free stream pressure gradient. Hence, a pressure gradient drives the blood from the arteries to the veins. Alveolar expansion during spontaneous inspiration (equivalent to the change in volume) is proportional to the difference between the alveolar and pleural pressures at end inspiration. equation for the stream function. Derivation of formula for pressure gradient (fluid mechanics) Ask Question Asked 7 years, 9 months ago. It is a unit vector in the direction of the motion. Laminar and Turbulent Flow : Eqn. Elevated interstitial fluid pressure, a hallmark of solid tumors, can compromise the delivery of therapeutics to tumors. The normal range of the Earth's air pressure is from 970 MB to 1,050 MB. cmH 2 O -1. The three principal functions it utilizes: It breaks the stream into smaller streams. Based on nitrate penetration depth and concentration gradient in the porewater we estimated benthic nitrate consumption rates assuming either diffusive transport in cohesive sediments or advective transport in permeable sediments. To convert to gradient: 1 kg = 2. Question: Problem 1: The Stream Function For A Two-dimensional, Nonviscous, Incompressible Flow Field Is Given By Psi = -2(x-y) Where The Stream Function Has The Units Of Ft^2/s With X And Y In Feet. That comes first. So, the situation of an equator to pole temper- The background uniform zonal wind shear means the basic state stream function can be written as ψ= −Λyz. Coppola‐Owen E-mail address: [email protected] This document is highly rated by JEE students and has been viewed 1005 times. An integration is required to get a function of height vs. 2%) and the pressure gradient driving. , (1984) and. 2: 6679274: Clean-in-place method for cleaning solution delivery systemes/lines: 2004-01-20: Gruszczynski et al. The Gradient. The normal range of the Earth's air pressure is from 970 MB to 1,050 MB. Outlet pressure for multiphase pipe flow by Gray correlation, [psia] It is commonly used for gas wells that are also producing liquid. In rectangular coordinates its components are the respective partial derivatives. amplification is obtained by stream-wise oriented vortices, in agreement with previous results for the Blasius boundary layer. The optimization of the blade surface velocity distribution is promising a reduction of turbine cascade losses.
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# “why is current the derivative of charge and not integral of charge?”
Current is defined as the amount of charge passing a given point per unit time. The word amount throws me off. sorry if this question seems dumb, but
why can current not be equal to integral of charge from time t=t1 to t2?
since we want to know the amount of charge passing a given point, we can add up charge from time t1 until time t2
• Compare this to kinematics: "$\Delta x$ is defined as the amount of change in position. So $\Delta x = \int x \, dt$, since we want to know the amount of change in position, we add up the position." – knzhou Oct 8 '18 at 15:59
• You defined current as for 'per unit time' . Hope you have got enough hint.. – Jnan Oct 8 '18 at 16:00
Current is defined as the amount of charge passing a given point per unit time.
In fact, electric current is defined as the flow of electric charge. From the Wikipedia article Electric current
An electric current is a flow of electric charge.
From the Britannica article Electric current
Electric current is a measure of the flow of charge
A flow is a rate, i.e., an amount over an elapsed time. If an amount of electric charge $$\Delta Q$$ flows into a region in some time $$\Delta t$$, then there is an electric current into the region (with an average value of)
$$\bar{I} = \frac{\Delta Q}{\Delta t}$$
In the electric circuit context, we have a circuit law (Kirchhoff's Current Law) that requires the current into a region equal the current out of the region and so we can think of the current through the region, e.g., the body of a resistor.
This is because current is strictly defined as the rate at which charge passes through a given cross sectional area of a conductor. Here, we are required to measure "how many" charges cross the perpendicular cross section in a given time, therefore derivative is used here as a 'rate measurer'.
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# Question #68aaf
Jan 28, 2017
Conversion factor miles to km
$1 \text{mile} = 1.60934 k m$
So $15000 \text{miles} = 15000 \times 1.60934 k m = 24140.1 k m$
In 1 year motor cycle is driven $24140.1 k m$
So in 5 year motor cycle is driven $24140.1 \times 5 k m = 120700.5 k m$
As $9.5 g$ CO is produced for 1km journey,
So in 5 year motor cycle will produce $120700.5 \times 9.5 g$ CO.
Conversion factor gram to pound is
$1 g = 0.00220462 \text{ pound}$
So the amount of CO generated in 5 years is
$120700.5 \times 9.5 \times 0.00220462 \text{ pound"~~2527.94" pound}$
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# Mixed/dual cycle
(Redirected from Mixed/Dual Cycle)
The dual combustion cycle (also known as the mixed cycle, Trinkler cycle, Seiliger cycle or Sabathe cycle) is a thermal cycle that is a combination of the Otto cycle and the Diesel cycle, first introduced by Russian-German engineer Gustav Trinkler. Heat is added partly at constant volume and partly at constant pressure, the advantage of which is that more time is available for the fuel to completely combust. Because of lagging characteristics of fuel this cycle is invariably used for Diesel and hot spot ignition engines. It consists of two adiabatic and two constant volume and one constant pressure processes. Efficiency lies between Otto and Diesel cycle.
Pressure-Volume diagram of Sabathe cycle
Temperature-Entropy diagram of Sabathe cycle
The dual cycle consists of following operations:
• Process 1-2: Isentropic compression
• Process 2-3: Addition of heat at constant volume.
• Process 3-4: Addition of heat at constant pressure.
• Process 4-5: Isentropic expansion.
• Process 5-1: Rejection of heat at constant volume.
The cycle is applicable for automobile sector.
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# 2 entangled electrons in QFT
In field theory, by quantizing a dirac field, we can obtain a creation operator for a single electron of definite momentum, of definite spin up or down, these respectively are: $$a^\dagger_{+}(p)|0\rangle, {a^\dagger}_{-}(p)|0\rangle$$ Where we've defined the former to create a spin +1/2 electron, the latter to create a spin -1/2 electron. By addition and repeated-application of these creation operators we can write down a state of any number of particles, each having any superposition of spins. : $$\int dp f(p)\prod_{i=0}^n(\alpha_ia^\dagger_{i+}(p)+\beta_i{a^\dagger}_{i-}(p))|0 \rangle$$ Where of course $i$ labels the particle, and $a,b,f$ are some distributions.
Question: For a given field theory, how does one write down a creation operator for a pair of entangled particles? (say electrons in a Dirac theory of spinors)
In quantum mechanics, an entangled state is one which lives in a tensor product Hilbert space, but does not have a tensor product decomposition. Since Fock space is essentially built up with a bunch of tensor products of Hilbert spaces, it doesn't seem unreasonable to demand that it contains such entangled states. But how does one explicitly write down such a state?
• I would like to note that this is not a duplicate of any previous questions on measurement in QFT or entanglement in QFT, for I am asking for something specific and explicit that has not been answered by any previous questions. – zzz Jul 25 '14 at 17:36
• 1007.1569 appears to be a relevant reference, maybe I will read through it and write an answer. – zzz Jul 25 '14 at 17:51
• this previous question gives an example of what's not an answer. – zzz Jul 25 '14 at 17:55
A state like $\frac{1}{\sqrt{2}}(a^\dagger_+(\vec p)a^\dagger_+(-\vec p) + a^\dagger_-(\vec p)a^\dagger_-(-\vec p))|0\rangle$ would be an example. It is both entangled in spin and entangled in momenta.
• It is an entangled state. If you measure the spin of one of the $2$ particles relatively to an axis $Oz$ and find $+1$, then a measure of the spin of the other particle relatively to this axis $Oz$ will give you $+$ too. The measurement will project the initial state, into the state $a^\dagger_+(\vec p)a^\dagger_+(-\vec p)$ – Trimok Jul 27 '14 at 11:41
Let $f(x,y)\in L^2(\mathbb{R}^{2d})$ and $\Omega$ the vacuum of the symmetric Fock space $\Gamma_s(L^2(\mathbb{R}^d))$. Suppose there is no $f_1,f_2\in L^2(\mathbb{R}^d)$ such that $f(x,y)=f_1(x)f_2(y)$: then $f_s$ (the symmetrized of $f$) is an "entangled" two particle state of $\Gamma_s(L^2(\mathbb{R}^d))$. This is created by $$\frac{1}{\sqrt{2}}\int f(x,y) a^*(x)a^*(y)dxdy\Omega\; .$$ For antisymmetric particles and/or more degrees of freedom the reasoning is the same.
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# A new Approach for CMOS Op-Amp Synthesis
Mandal, Pradip and Visvanathan, V (1999) A new Approach for CMOS Op-Amp Synthesis. In: Twelfth International Conference On VLSI Design, 1999, 7-10 January, Goa,India, 189 -194.
Preview
PDF
a_new.pdf
A new approach for CMOS op-amp circuit synthesis has proposed here. The approach is based on the observation that the first order behavior of an MOS transistor in the saturation region is such that the cost and the constraint functions for this optimization problem can be modeled as convex functions. Second order effects are then handled by formulating the problem as one of solving a sequence of convex programs. Numerical experiments show that the solutions to the sequence of convex programs converges to the same design point for widely varying initial guesses. This strongly suggests that the approach is capable of determining the globally optimal solution to the problem. Performance of the synthesized op-amps has been verified against detailed SPICE simulation for a $1.6\mu$ CMOS process.
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# Reverse GAN, changing the distribution of the data to normal noise
I have a dataset which contains the data from 10 classes. The classes look well-separated that the accuracy rate of different classifiers is more than 95%.
The goal is to change the data distribution in a way that they can’t be not easily classified. Some sort of noise should be added to the data that changes the distribution of the data but I am only allowed to change the input data with these actions: 1- increase the positive values 2- decrease the negative values 3- or no change to the feature
For example, if the input is [ +1, -4, +6, -8], one possible noise is [+2, -1, +1, -2], which the final input after adding this noise is: [+3, -5, +7, -10].
If we consider d as a random noise between [0,1], then the modified input is: X’ = X + sign(X) * d
The goal is to add the noise to the data that changes the data distribution to normal distribution that they are not well separated and can't be easily classified. BTW, the amount of noise added to the input should not be too much.
In order to learn such a noise, I am going to use some idea like GAN (Generative Adversarial Network). The generator learns the noise and the discriminator measures how the data is separable. It seems like a reverse GAN.
In GAN, we learn to change the noise distribution to the data distribution but here we change the data distribution to the normal distribution (I mean change the data in a way that they can't be classified).
The above figure shows the architecture. The discriminator tries to measure the separability of the data after adding noise, and generator learns how to add noises to the data to confuse the discriminator.
If X is the input with label y: y is categorical.
noise = generator(X)
X_prime = X + tf.sign(X) * noise
output = discriminator(X_prime)
real = tf.reduce_sum(y * output, 1) # the probability of the true class label
other = tf.reduce_max((1 - y) * output - y * 10000,1) # the maximum probability, among all other classes than true class
The discriminator tries to minimize this loss function
d_loss = tf.reduce_sum(tf.maximum(0.0, (other - real)))
And generator is minimizing two factors: 1- the L2 norm of X and X_prime 2- it tries to minimize the negative of the discriminator loss function.
l2dist = tf.reduce_sum(tf.square(X - X_prime))
G_loss = tf.reduce_sum(tf.maximum(0.0, (real - other)))) +C * l2dist
C is a tuning factor that does not allow one factor to outpower the other one.
I implemented this scenario but the generator does not learn the noise distribution and the discriminator can classify the noisy input with high accuracy. I even tried to train the generator much more often than discriminator not to allow the discriminator outpower the generator but it still does not work.
The main problem is that I first train the discriminator, then I train the generator. The generator learns how to add noise to the data to fool the previously trained discriminator but in the next round when I update (train) the discriminator with the noise data from generator they are separable. I mean the generator learns how to fool the discriminator but does not learn how to make the data inseparable.
The question is whether this model can learn the noise distribution added to the input to make the data distribution inseparable or not. Is my model is right or am I missing something in the model?
• I don't see a question mark here. Assuming you're asking "how can I make this model work?": Probably either (a) the problem is just too hard to turn into your target noise with these restrictions, or (b) you're running into optimization issues with your GAN, which are rampant in the literature. We don't really know enough from your question to know which one is the case, and it's probably too broad of a question for this format in any case. – Dougal Apr 18 '18 at 1:32
• @Dougal thanks I edited the question, The question is whether this model can learn the noise distribution added to the input to make the data distribution inseparable or not. Is my model is right or am I missing something in the model? – MOH Apr 18 '18 at 6:57
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{}
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# TWOSTRS - Editorial
Author: Viktar Sharepa
Tester: Danya Smelskiy
Editorialist: Viktar Sharepa
Medium
# PROBLEM:
We are given an array of strings S, the beauty of each string t in S is given by a positive integer b_t.
The pleasure of a string C is defined by {\sum\limits_{t \in S}\text{(number of occurrences of }t\text{ in C)}\cdot b_t}.
Given two strings A, B, find the string with maximum pleasure that can be constructed by taking one substring of A and adding one substring of B to its end.
# QUICK EXPLANATION:
• The string with maximum pleasure can be constructed by taking a non-empty prefix of A and a non-empty suffix of B.
• Let’s calculate the answer for all the {|A| \cdot |B|} possibilities. Suppose we choose {A[0..i]} and {B[j..|B|-1]}. There are three types of string occurrences:
1. Occurrences that are completely in {A[0..i]}. We can precompute these sums for each i using the Aho-Corasick algorithm.
2. Occurrences that start in {A[0..i]} and end in {B[j..|B|-1]}. All such occurrences end in {B[j..j+24]} so we can take 25 steps from trie node corresponding to {A[0..i]} in the Aho-Corasick automaton. We can precompute such nodes along with the sums of the first case.
3. Occurrences that are completely in {B[j..|B|-1]}. Note that we have already counted the ones that are completely in {B[j..j+24]}. All that is left to count are occurrences that end in {B[j+25..|B|-1]}. We can do it similarly to the first case
# EXPLANATION:
Given a string C = {A[k..i]} + {B[j..l]} we can construct another string D = {A[0..i]} + {B[j..|B|-1]}. Note that any substring of C is also a substring of D, since all the beauties are positive, the pleasure of D is not smaller than the pleasure of C. That means that the string with maximum pleasure always can be constructed by taking a non-empty prefix of A and a non-empty suffix of B.
There are {|A| \cdot |B|} ways of taking a prefix of A and a suffix of B. To quickly calculate the pleasure of each resulting string, we’ll use the following data structures:
1. Let’s build an automaton on all the strings from S using the Aho–Corasick algorithm. For each node u let sum[u] be the sum of beauties of all strings which nodes are ancestors of u in the suffix link tree.
2. Let prefASum[i] be the sum of beauties of all occurrences of strings from S in the prefix {A[0..i]} and prefANode[i] be the automaton node corresponding to the prefix {A[0..i]}. The arrays prefASum and prefANode can be calculated iterating over A character by character and moving on the automaton.
C++ code
vector<long long> prefASum(a.size());
vector<int> prefANode(a.size());
for (int i = 0; i < a.size(); i++) {
ahoCorasick.move(a[i]);
prefASum[i] = ahoCorasick.getCurrentNodeSum();
if (i != 0) {
prefASum[i] += prefASum[i - 1];
}
prefANode[i] = ahoCorasick.currentNode;
}
1. Similarly to the previous step let suffBSum[i] be the sum of beauties of all occurrences of strings from S in B that end in the suffix {B[i..|B|-1]}.
C++ code
vector<long long> suffBSum(b.size());
for (int i = 0; i < b.size(); i++) {
ahoCorasick.move(b[i]);
suffBSum[i] = ahoCorasick.getCurrentNodeSum();
}
for (int i = (int) b.size() - 2; i >= 0; i--) {
suffBSum[i] += suffBSum[i + 1];
}
Consider the string C = {A[0..i]} + {B[j..|B|-1]}. To calculate the pleasure of C, let’s analyze the three types of string occurrences in it:
1. Occurrences that are completely in {A[0..i]}.
2. Occurrences that start in {A[0..i]} and end in {B[j..|B|-1]}.
3. Occurrences that are completely in {B[j..|B|-1]}.
• The sum of occurrences of the first type is equal to prefASum[i].
• Let’s start at prefANode[i] and move in the automaton character by character of the substring {B[j..j+24]}, keep adding the sum[node] of each node that we visit. In this way we get the sum of beauties of all occurrences that lie in the concatenation of {A[0..i]} and end at {B[j..j+24]}, remember that the strings in S have a maximum lenght of 26. This will allow us to calculate the sum of all the occurrences of the second type and some occurrences of the third type.
• All is left to add to current sum are occurrences of the third type that end somewhere in {B[j+25...|B|-1]}. Their sum is equal to suffBSum[j + 25].
Time complexity per test case: build an automaton in {\mathcal{O}(\sum\limits_{t \in S}|t|)} + calculate prefASum, prefANode, suffBSum in {\mathcal{O}(|A|+|B|)} + calculate the pleasure for each prefix and suffix pair in {\mathcal{O}(|A| \cdot |B| \cdot \max\limits_{t \in S}|t|)} = {\mathcal{O}(|A| \cdot |B| \cdot \max\limits_{t \in S}|t| + \sum\limits_{t \in S}|t|)}.
# SOLUTIONS:
Setter's Solution
#include <bits/stdc++.h>
using namespace std;
template<int ALPHABET_SIZE, unsigned char MINIMAL_CHAR>
struct AhoCorasick {
static constexpr int NON_EXISTENT_NODE_ID = -1;
static constexpr int FAKE_NODE_ID = 0;
static constexpr int ROOT_ID = 1;
int currentNode;
vector<array<int, ALPHABET_SIZE>> edges;
vector<long long> sum;
explicit AhoCorasick(const vector<pair<string, int>> &a) : currentNode(ROOT_ID), edges(2),
suffixLink(2, FAKE_NODE_ID), sum(2, 0) {
edges[FAKE_NODE_ID].fill(ROOT_ID);
edges[ROOT_ID].fill(NON_EXISTENT_NODE_ID);
for (const auto &p : a) {
int node = ROOT_ID;
for (unsigned char c : p.first) {
c -= MINIMAL_CHAR;
if (edges[node][c] == -1) {
edges[node][c] = edges.size();
edges.emplace_back();
edges.back().fill(NON_EXISTENT_NODE_ID);
sum.push_back(0);
}
node = edges[node][c];
}
sum[node] += p.second;
}
queue<int> q;
q.push(ROOT_ID);
while (!q.empty()) {
int node = q.front();
if (suffixLink[node] != NON_EXISTENT_NODE_ID) {
}
q.pop();
for (int i = 0; i < ALPHABET_SIZE; i++) {
int child = edges[node][i];
if (child == NON_EXISTENT_NODE_ID) {
} else {
q.push(child);
}
}
}
}
void setNode(int node) {
currentNode = node;
}
void resetNode() {
setNode(ROOT_ID);
}
long long getCurrentNodeSum() {
return sum[currentNode];
}
void move(unsigned char c) {
c -= MINIMAL_CHAR;
currentNode = edges[currentNode][c];
}
};
void solve() {
string a, b;
cin >> a >> b;
int n;
cin >> n;
vector<pair<string, int>> s(n);
for (int i = 0; i < n; i++) {
cin >> s[i].first >> s[i].second;
}
AhoCorasick<26, 'a'> ahoCorasick(s);
vector<long long> prefASum(a.size());
vector<int> prefANode(a.size());
for (int i = 0; i < a.size(); i++) {
ahoCorasick.move(a[i]);
prefASum[i] = ahoCorasick.getCurrentNodeSum();
if (i != 0) {
prefASum[i] += prefASum[i - 1];
}
prefANode[i] = ahoCorasick.currentNode;
}
ahoCorasick.resetNode();
vector<long long> suffBSum(b.size());
for (int i = 0; i < b.size(); i++) {
ahoCorasick.move(b[i]);
suffBSum[i] = ahoCorasick.getCurrentNodeSum();
}
for (int i = (int) b.size() - 2; i >= 0; i--) {
suffBSum[i] += suffBSum[i + 1];
}
long long ans = 0;
for (int i = 0; i < a.size(); i++) {
for (int j = 0; j < b.size(); j++) {
long long cur = prefASum[i];
ahoCorasick.setNode(prefANode[i]);
for (int k = j; k <= j + 24 && k < b.size(); k++) {
ahoCorasick.move(b[k]);
cur += ahoCorasick.getCurrentNodeSum();
}
if (j + 25 < b.size()) {
cur += suffBSum[j + 25];
}
ans = max(ans, cur);
}
}
cout << ans << '\n';
}
int main() {
ios_base::sync_with_stdio(false);
cin.tie(nullptr);
cout.tie(nullptr);
#ifdef HOME
freopen("in.txt", "r", stdin);
#endif
int tests;
cin >> tests;
while (tests--) {
solve();
}
}
Tester's Solution
#include <iostream>
#include <string>
#include <cassert>
#include <cstdio>
#include <cstring>
#include <vector>
using namespace std;
int Go(int node_id, int next_char);
const int N = 10'000 + 5;
const int NODES_CNT = N * 26 + 5;
string s[N];
int price[N];
int nodes = 0;
int nxt[NODES_CNT][26];
int go[NODES_CNT][26];
int node_value[NODES_CNT];
int anc_c[NODES_CNT];
int anc[NODES_CNT];
bool processed[NODES_CNT];
int node_price[NODES_CNT];
long long dp[1005];
void clear_node(int id) {
memset(nxt[id], 0, sizeof(nxt[id]));
memset(go[id], 255, sizeof(go[id]));
node_value[id] = 0;
}
void init() {
nodes = 0;
clear_node(0);
anc[0] = -1;
}
void add_new_string(const string& w, int cost) {
int last = 0;
for (int i = 0; i < (int)w.size(); ++i) {
int c = w[i] - 'a';
if (!nxt[last][c]) {
nxt[last][c] = ++nodes;
anc_c[nodes] = c;
anc[nodes] = last;
clear_node(nodes);
}
last = nxt[last][c];
}
node_value[last] += cost;
return;
}
int Go(int id, int c) {
if (go[id][c] != -1)
return go[id][c];
if (nxt[id][c])
return go[id][c] = nxt[id][c];
if (id == 0)
return go[id][c] = 0;
return go[id][c] = Go(get_link(id), c);
}
int get_link(int id) {
if (link[id] != -1)
if (id == 0 || anc[id] == 0)
return link[id] = 0;
}
int process_node(int id) {
if (processed[id])
return node_price[id];
processed[id] = true;
node_price[id] = node_value[id];
return node_price[id];
}
void build() {
for (int i = 0; i <= nodes; ++i) {
processed[i] = false;
node_price[i] = 0;
}
for (int i = 0; i <= nodes; ++i) {
process_node(i);
}
}
void calc_dp(const string& b) {
int m = (int)b.size();
int last = 0;
for (int i = 1; i <= m; ++i) {
dp[i] = 0;
int c = b[i - 1] - 'a';
last = Go(last, c);
dp[i] = node_price[last];
}
dp[m + 1] = 0;
for (int i = m; i > 0; --i) {
dp[i] += dp[i + 1];
}
return;
}
void solve(int test_id) {
init();
string a, b;
cin >> a >> b;
int n, m;
cin >> n;
m = (int)b.size();
for (int i = 1; i <= n; ++i) {
cin >> s[i] >> price[i];
assert(s[i].size() <= 26);
}
build();
calc_dp(b);
int last = 0;
long long beauty = 0;
long long result = 0;
for (int i = 1; i <= (int)a.size(); ++i) {
int c = a[i - 1] - 'a';
last = Go(last, c);
beauty += node_price[last];
for (int j = 1; j <= m; ++j) {
int last_c = last;
long long beauty_c = beauty;
for (int k = 0; k < 26; ++k) {
int r = j + k;
if (r > m) break;
int cc = b[r - 1] - 'a';
last_c = Go(last_c, cc);
beauty_c += node_price[last_c];
}
if (j + 26 <= m)
beauty_c += dp[j + 26];
if (beauty_c > result) {
result = beauty_c;
}
}
}
cout << result << '\n';
return;
}
int main(int argc, const char * argv[]) {
ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0);
int tests;
cin >> tests;
assert(1 <= tests && tests <= 10);
for (int i = 0; i < tests; ++i) {
solve(i);
}
return 0;
}
10 Likes
I solved it exactly the same way!!!
5 Likes
Can also be solved using just tries without Aho-Corasick. Though the time complexity becomes O(|A|.|B|.|max_length_of_Si|^2) but it can be optimized to pass in the given time constraint. I am guessing a lot of people solved it that way.
I didn’t know about Aho-Corasick. It’s pretty sick. Definitely worth learning.
2 Likes
@harisisbatman Could you please explain how you optimize this |A| \cdot |B| \cdot 26^2 solution?
@harisisbatman Hey can you plz tell me your approach for this one because I find the editorial approach a little more complex . I am thinking the way you said but not able to solve it.thankyou.
@harisisbatman I can tell you what I do.Basically I calculate the answer for A and then B and then combine every possibility of A with B store in vector.Then for every possiblity I just calculate the part which have a substring in array S. If I calculate my time complexity then it should approx (A * B *(26 * 26)) . But it should time out . I didnt use trie because didnt get where I use to optimize that . I am not getting how you are able to optimized it with trie as I am tried so hard but not able to come up with the solution.
something wrong with the timecomplexity of geeksforgeeks implementation of aho-corasick
1 Like
For occurrences that are completely in A and completely in B, we can do it with just tires with and extra added factor of 26 in the time complexity but that is no big deal of course. (Let me know if you have confusion on how to do this)
The main part is checking all the substrings and finding the strings starting in A and ending in B.
Naively, each of it can be done in 26*26 time. Lets say we are taking the string A till index i and string B from index j.
First we find all strings in S which start at A[i] and end before B[j+25]
Then we find all strings in S which start at A[i-1] and end before B[j+24] and so on until A[i-24] to B[j]
Now this can be optimized in two ways-
1. Instead of taking all the different strings in A again and again, we take every different string only once and get its location in trie and then instead of traversing the trie from the top, we traverse it from this location and then for all required strings in B. We do this for strings of length 25, 24, 23 upto 13 in A.
2. When the length of strings in A become less than 13, it is no longer the most efficient thing to do so now we reverse our strategy and we instead start checking the strings in the reverse order from B to A. So now we take the different strings in B of length 14, 15, . . . 25 and check all required strings of A.
The process is a little overwhelming to code. We need tries of strings in S as well as reversed strings in S and we need to traverse the strings A and B both forwards and backwards.
Heres the code. Its not the neatest code though.
https://www.codechef.com/viewsolution/32986222
The main part of the code is-
B_trie is the trie of normal strings of S.
A_trie is the trie of strings of S in reversed order.
long long bridge_answers[1002][1002];
long long check_all_substrings(){
long long ans = -1, tmp_ans;
int i, j, k;
for(i=0; i<A.length(); i++) for(j=0; j<B.length(); j++) bridge_answers[i][j] = 0;
for(k=25; k>=13; k--){
for(i=0+k-1; i<A.length(); i++){
struct Node *node = B_trie.get_node(A, i-k+1, k, 1);
for(j=0; j<(int)B.length()-k+1; j++){
bridge_answers[i][j] += B_trie.get_value(node, B, j, 26-k, 1);
}
}
}
for(k=1; k<=25; k++){
for(i=0; i<=(int)B.length()-k; i++){
struct Node *node = A_trie.get_node(B, i+k-1, k, -1);
for(j=0; j<A.length(); j++){
bridge_answers[j][i] += A_trie.get_value(node, A, j, min(26-k, 12), -1);
}
}
}
for(i=0; i<A.length(); i++){
for(j=0; j<B.length(); j++){
ans = max(ans, A_pleasure[i] + B_pleasure[j] + bridge_answers[i][j]);
}
}
return ans;
}
With this, the time-complexity becomes O(|A|.|B|.13.13)
3 Likes
what is the time complexity of the solution??
Seems like it works right under the TL in the worst case (smth about 0.8s) but passes.
The testcases for the first two subtasks seem to be weak. My wrong logic https://www.codechef.com/viewsolution/32961034 fetched me 50 points passing subtask 1 and 2 and only failed on subtask 3, when the only difference between the subtasks is with respect to the time complexity.
1 Like
Yes… I had to make all other optimizations as well. Like I couldn’t afford to copy strings and then reverse them to serve to the trie which would have made the code a little easier. Had to do everything by reference to the original A and B strings.
I bet the solution in the editorial looks easier after you read mine
How much time will it take to calculate ratings anyone please?
It is much easier if you know and understand Aho-Corasick. Thank you for sharing your idea and optimizations.
1 Like
How much time will it take to calculate ratings please sharepo?
It is difficult to create a test for each existing incorrect solution. By the way, the only test you got WA on was a random one. You were lucky and unlucky at the same time.
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{}
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## Boards:25 Issue
#### 5 polymerization setting
Dear all
Hello, I'm trying to run GRRM 17 program to examine polymerization reactions including Butadiene.
At first, I have tried to calculate the very simple reaction as below. However, any TS structure has NEVER been generated although GAMMA parameter was very large, SC-AFIR was changed to SC-AFIR2, or the different functionals were used.
To react these molecules, could anyone teach me what options should be tested, or how to set the input file.
Thanks.
S. Mieda
----Input file sample-----------
# SC-AFIR/B3LYP/6-31G
-1 1
C -6.04851 1.08683 -0.12952
C -4.65656 0.67303 -0.47198
C -3.56045 0.80445 0.37107
C -2.26684 0.40808 0.06896
H -6.39564 1.86874 -0.83809
H -4.49975 0.21350 -1.45756
H -3.73818 1.26439 1.35332
H -2.03118 -0.05733 -0.89259
C -7.06260 -0.07494 -0.17775
H -6.60275 -1.00092 0.30256
H -6.06390 1.54725 0.87543
H -1.44681 0.54598 0.77726
H -7.32889 -0.30211 -1.26268
C 3.12328 4.11991 -0.42148
C 1.91711 4.54010 0.00136
C 0.68735 4.38233 -0.75186
C -0.52022 4.79402 -0.32435
H 4.02317 4.26027 0.18117
H 3.24530 3.62228 -1.38871
H 1.82997 5.03655 0.97605
H 0.77488 3.88934 -1.72815
H -0.64143 5.28864 0.64465
H -1.42164 4.64671 -0.92286
H -8.00546 0.21979 0.39125
Options
GAMMA = 1000
Fragm.1 = 1-13,24
Fragm.2 = 14-23
1 2
END
GauMem = 2000
GauProc = 4
• #### Re: polymerization setting
I have tested a SC-AFIR calculation using your sample input; the artificial force was not applied between any fragments because the two moieties were too far away (the input structure (EQ0) had judged as a dissociation channel (DC) structure by the default parameter DownDC=8, and you would be able to see the string "Not to be applied" in the file jobname_EQ_test.rrm.). Therefore, no reaction paths had been computed and no TS structures had also been generated.
Does the intermolecular distance have a meaning (e.g. experimentally solved structure)? If no, it might be worth to try a MC-AFIR calculation. In addition, I think "Stable=Opt" option is recommended in your case.
----- Sample Input for MC-AFIR -----
# MC-AFIR/B3LYP/6-31G
-1 1
C -6.04851 1.08683 -0.12952 1
C -4.65656 0.67303 -0.47198 1
C -3.56045 0.80445 0.37107 1
C -2.26684 0.40808 0.06896 1
H -6.39564 1.86874 -0.83809 1
H -4.49975 0.21350 -1.45756 1
H -3.73818 1.26439 1.35332 1
H -2.03118 -0.05733 -0.89259 1
C -7.06260 -0.07494 -0.17775 1
H -6.60275 -1.00092 0.30256 1
H -6.06390 1.54725 0.87543 1
H -1.44681 0.54598 0.77726 1
H -7.32889 -0.30211 -1.26268 1
C 3.12328 4.11991 -0.42148 2
C 1.91711 4.54010 0.00136 2
C 0.68735 4.38233 -0.75186 2
C -0.52022 4.79402 -0.32435 2
H 4.02317 4.26027 0.18117 2
H 3.24530 3.62228 -1.38871 2
H 1.82997 5.03655 0.97605 2
H 0.77488 3.88934 -1.72815 2
H -0.64143 5.28864 0.64465 2
H -1.42164 4.64671 -0.92286 2
H -8.00546 0.21979 0.39125 1
Options
NFault = 50
GAMMA = 1000
Fragm.1 = 2-4
Fragm.2 = 14-17
1 2
END
Stable = Opt
GauMem = 2000
GauProc = 4
--------------------
This is just my comment, and this might not be a solution to your problem...
• #### Thank you
Saita-Sensei
Thank you for the great advice!
I'm trying to shorten the distance between two fragments, to use the options, and to generate multi-structires by using MC-AFIR.
BTW... could you teach me why the fragment ranges were changed? Did the wide ranges have bad influence on the reaction?
Best regards,
S. Mieda
• #### Re: Thank you
In the above sample input for MC-AFIR, only the sp2 carbon atoms are selected for the fragment. It would be a natural choice in terms of the chemical knowledge, but it was not a strong suggestion. In order to make a global search, the wider fragment range would be required, but it would take a long computational time. So, I chose only the sp2 carbon atoms to the fragments as a sample job.
NOTE: The terms "part" and "fragment" are used in different meanings in GRRM17 program:
· "Fragment" means the fragment to which the artificial force is applied by the AFIR function.
· "Part" means a moiety of the system (or a monomer in the molecular cluster). The defined parts to be used for random geometry generation (all parts to be randomly distributed).
The fragment is not necessarily identical to the part. For details, please see the manual page.
In the SC-AFIR calculation, the fragments to be automatically defined, so that you do not have to put the lines "Fragm.1 = 1-13,24" and "Fragm.2 = 14-23" into the SC-AFIR input file.
• #### Re: polymerization setting
(For your information) this might be an example input file for SC-AFIR local search:
----- Sample Input for SC-AFIR -----
# SC-AFIR/B3LYP/6-31G
-1 1
C -6.04851 1.08683 -0.12952 1
C -4.65656 0.67303 -0.47198 1
C -3.56045 0.80445 0.37107 1
C -2.26684 0.40808 0.06896 1
H -6.39564 1.86874 -0.83809 1
H -4.49975 0.21350 -1.45756 1
H -3.73818 1.26439 1.35332 1
H -2.03118 -0.05733 -0.89259 1
C -7.06260 -0.07494 -0.17775 1
H -6.60275 -1.00092 0.30256 1
H -6.06390 1.54725 0.87543 1
H -1.44681 0.54598 0.77726 1
H -7.32889 -0.30211 -1.26268 1
C 3.12328 4.11991 -0.42148 2
C 1.91711 4.54010 0.00136 2
C 0.68735 4.38233 -0.75186 2
C -0.52022 4.79402 -0.32435 2
H 4.02317 4.26027 0.18117 2
H 3.24530 3.62228 -1.38871 2
H 1.82997 5.03655 0.97605 2
H 0.77488 3.88934 -1.72815 2
H -0.64143 5.28864 0.64465 2
H -1.42164 4.64671 -0.92286 2
H -8.00546 0.21979 0.39125 1
Options
NRUN = 1
Target = 2-4,14-17
GAMMA = 200
END
SC = InterOnly
Stable = Opt
GauMem = 2000
GauProc = 4
--------------------
NRUN is used in order to generate a random initial structure.
SC = InterOnly limits the number of paths to be computed (artificial force to be applied between different parts only).
This is just an example. The SC-AFIR has various options, so you may wish to experiment to decide which one fits your needs best.
• #### Re: Re: polymerization setting
Saita-sensei
Thank you very much for the useful comments.
I understand the difference between Frangments and Parts (and Targets).
Especially, SC = InterOnly option seems to be very useful for my calculation. I'm trying to use the other options for SC-AFIR, too.
If the reaction is carried out, I write the input file here to share the information.
Best regards,
S. Mieda
#### 3 GRRM17 together with SIESTA
Hello,
I'm trying to run GRRM17 together with SIESTA, so I've followed the instructions in the Manual web page (How to run GRRM with SIESTA).
I have correctly set all environment variables and created two sample files (h2o_min.com and h2o_min.inp).
But if I execute:
GRRM17p h2o_min
I get an error:
ioctl failed
What could be the cause?
• #### Re: GRRM17 together with SIESTA
In your environment (on your workstation), can you execute SIESTA as a standalone program (without GRRM17) without such errors?
Please confirm your SIESTA program has been installed succesfully and you have set the environment variables "subsiesta" and "submpi" properly.
• #### GRRM17 together with SIESTA
Yes, the siesta program works Ok and I have set the environment variables "subsiesta" and "submpi" properly.
• #### Re: GRRM17 together with SIESTA
Did you put the pseudopotential files (*.psf) in the same directory? In the sample case, two psf files O.psf and H.psf are required.
#### 3 IRC Calculations
Good afternoon,
I am working with GRRM17 to run IRC from transition states calculated in Gaussian.
When I use Gaussian to calculate IRC, I run 2 jobs, “reverse” and “forward”. Then, I can check that with “reverse” log file I get the previous structure of the transition state and with “forward” file I can see the movement from the transition state to the product. When I run the IRC calculation in GRRM, I can see only the forward movement (from the transition state to the product).
Is it possible also to check the reverse movement from transition state to previous stratucture (or stating structure)?
Thank you very much
Best regards
• #### Re: IRC Calculations
What options do you specify in your GRRM17 input file?
In GRRM17, the IRC calculation always computes forward and backward IRC paths (unless the "Meta-IRC” option is specified).
Of course, the forward and the backword paths can be computed from a saddle point (transition state). In other words, from a non-stationary point, only a mass-weighted steepest-descent path can be computed.
Did you see the string "IRC FOLLOWING (FORWARD) STARTING FROM FIRST-ORDER SADDLE” in your GRRM17 log file? If you got "STEEPEST-DESCENT PATH FOLLOWING STARTING FROM NON-STATIONARY POINT”, it meant your input structure was not recognized as a saddle point.
Even if your input structure was recognized as a 1st-order saddle point, it might correspond to the transition state structure between two conformers of the product.
When you use the pre-optimized transition state structure by Gaussian (without GRRM17), such a problem sometimes appears. The SCF cycles might be converged in a different electronic structure. Try using the “MO Guess = filename.chk” option in order to read the Gaussian Checkpoint file as a proper initial guess.
Or try the SADDLE calculation before the IRC calculation. The IRC paths can be automatically computed by specifying “Saddle+IRC” option.
The DS-AFIR calculation would be able to help you to get the IRC paths between the reactant and the product. (Like the QST2 method in Gaussian.)
• #### IRC Calculations
In my GRRM17 input file, I specify the option IRC. I see in my file.log "STEEPEST-DESCENT PATH FOLLOWING STARTING FROM NON-STATIONARY POINT" but at the end of the file also I see: "IRC following along both forward and backward directions were finished"
I converted the file.log in file.mol to see both movements in gaussview and, at this point, I can see only the forward movement. But as it is indicated in the file.log (IRC following along both forward and backward directions were finished), should doI see also the backward movement?
Best regards
• #### Re: IRC Calculations
No. The string "IRC following along both forward and backward directions were finished" just means the IRC calculation was normally terminated (termination message). Your GRRM17 output file (file.log) says that your input structure (pre-optimized TS structure obtained by Gaussian) did not converge in a saddle point in GRRM17. From a non-stationary point, only a mass-weighted steepest-descent path can be computed (the "backward" path cannot be theoretically defined).
Try SADDLE calculation with "Saddle+IRC" and “MO Guess = filename.chk” options. You will get an actual TS structure and the "forward" and "backward" IRC paths.
For example:
0 1
C -0.079213112255 0.000038936045 -0.592031587948
O 0.018800714554 -0.000018308130 0.717081381477
H 0.322739833191 0.909186505065 -0.947006485140
H 0.322739833191 -0.909359858691 -0.946723086445
Options
GauProc = 4
GauMem = 100
MO Guess = filename.chk
#### 1 Microiteration with external programs
Dear Developers,
I am interested in microiteration calculations with external programs (not with Gaussian). Could you please show us what data and format should be written in xxx_OUT4GEN.rrm. It seems that the manual has no descirptions about the case of TASK: MICROITERATION in xxx_INP4GEN.rrm.
I would appreciate if you could help us.
Best wishes,
Hiroko Satoh
• #### Re: Microiteration with external programs
The file xxx_OUT4GEN.rrm should be written in the same format in every case (TASK: "MAKE GUESS", "MICROITERATION", "ENERGY", "ENERGY and GRADIENT", or "ENERGY, GRADIENT, and HESSIAN").
The required format, for example, for 3-atom system is:
RESULTS
CURRENT COORDINATE
atom1 x1 y1 z1
atom2 x2 y2 z2
atom3 x3 y3 z3
ENERGY = E1 E2 E3
= E4 E5 E6
S**2 = <S2>
g(x1)
g(y1)
g(z1)
g(x2)
g(y2)
g(z2)
g(x3)
g(y3)
g(z3)
DIPOLE = μ(x) μ(y) μ(z)
HESSIAN
h(x1,x1)
h(y1,x1) h(y1,y1)
h(z1,x1) h(z1,y1) h(z1,z1)
h(x2,x1) h(x2,y1) h(x2,z1) h(x2,x2)
h(y2,x1) h(y2,y1) h(y2,z1) h(y2,x2) h(y2,y2)
h(z2,x1) h(z2,y1) h(z2,z1) h(z2,x2) h(z2,y2)
h(x3,x1) h(x3,y1) h(x3,z1) h(x3,x2) h(x3,y2)
h(y3,x1) h(y3,y1) h(y3,z1) h(y3,x2) h(y3,y2)
h(z3,x1) h(z3,y1) h(z3,z1) h(z3,x2) h(z3,y2)
h(z2,z2)
h(x3,z2) h(x3,x3)
h(y3,z2) h(y3,x3) h(y3,y3)
h(z3,z2) h(z3,x3) h(z3,y3) h(z3,z3)
DIPOLE DERIVATIVES
(x,x1) (y,x1) (z,x1)
(x,y1) (y,y1) (z,y1)
(x,z1) (y,z1) (z,z1)
(x,x2) (y,x2) (z,x2)
(x,y2) (y,y2) (z,y2)
(x,z2) (y,z2) (z,z2)
(x,x3) (y,x3) (z,x3)
(x,y3) (y,y3) (z,y3)
(x,z3) (y,z3) (z,z3)
POLARIZABILITY
α(x,x)
α(x,y) α(y,y)
α(x,z) α(y,z) α(z,z)
If the file xxx_OUT4GEN.rrm is written in different formats, GRRM17 cannot read xxx_OUT4GEN.rrm. Therefore you need to put zero (0.0) into matrix elements of GRADIENT, DIPOLE, HESSIAN, DIPOLE DERIVATIVES, and/or POLARIZABILITY that cannot be obtained from your "sub link = aaa" program. Note: ENERGY requires 6 elements (E1E6). In usual cases, the energy from your "sub link = aaa" program have to be put into the first element (E1), and put zero into the others (E2E6). The other E2E6 elements are reserved for the multistate calculations or some special usages.
In the case "TASK: MICROITERATION", GRRM17 requests you to perform microiterations by your "sub link = aaa" program and to give xxx_OUT4GEN.rrm which contains the coordinates after microiterations, energy, and gradients of the external atoms (although δV/δQn is almost zero for coordinates of external atoms Qn after microiterations). GRRM17 does not perform microiterations, and this is a specification (this might seem to be strange, but this is designed for general use).
For further details, try a microiteration calculation with Gaussian program, and look at the xxx_LinkJOB.rrm file. You will understand how GRRM17 does a microiteration calculation. Even if you use the general interface, GRRM17 requires the same information.
• #### Microiteration with external programs
Dear Prof. Saita,
Thank you very much for the info and taking your time. We will give it a try.
Best wishes,
Hiroko Satoh
#### 1 IRC Calculation
I am interested to run IRC calculations with GRRM of some transition states that I obtained with Gaussian.
2 log files are generated in the calculation: name.log and name_GauJOB.log.
I run one file getting normal termination but when I open the file name_GauJOB.log in GaussView I cannot see the expected movement of the molecule (reverse or forward).
Maybe I did not indicate in the file the correct parameters.
My file:
# IRC/b3lyp/GenECP
0 1
COORDINATES
Options
GauMem = 200
GauProc = 8
NoBondRearrange
Bond Condition
12 75 < 2.06
Fragm.1=1-40
Fragm.2=41-75
1 2
GAMMA = 200
END
GauInpB
C H N O F Si
6-31g**
****
Au 0
SDD
END
In this example, I indicate as Bond Condition 12 75: I am working with transition states:
Here, do I have to indicate the numbers of the 2 atoms involved in the formation of the new bond, right?
1. When I open the log file of my transition state (obtanied in Gaussian) in gaussview I see the movement of the atoms when I select the option Vibrations. Which is the distance that I have to indicate for the calculation in GRRM?. Is the distance that I have before to select vibrations? The distance that I have to indicate is the difference between the distance of these atoms in the product and the precursor? or between the transition state and the product?
2. Do I indicate > or < based on what?
• #### Re: IRC Calculation
The GRRM17 program uses Gaussian program to only compute single point calculation, and the file name_GauJOB.log is the Gaussian log file of the last step of your calculation. In this case, it would correspond to the freq calculation log file at the minimum point of the backward IRC path, so that you cannot see the IRC path information in the file name_GauJOB.log with GaussView. The IRC information is stored in the file name.log. To visualize the structure in IRC path computed by GRRM17 with GaussView, you need to plot the corresponding structure and energy separately from the file name.log.
In the "Bond Condition ... END" option, "12 75 < 2.06" means "the distance between atoms 12 and 75 to be less than or equal to 2.06 angstroms". This bond condition will be appled to the SC-AFIR search, but not to the IRC path calculation. Do not confuse this option with the IRC=Phase=(N1,N2 [,N3 [,N4]]) option in Gaussian. In GRRM17, an IRC calculation always computes forward and backward IRC paths.
The options "NoBondRearrange", "Bond Condition ... END", and "Add Interaction ... END" have no effect on the IRC calculation, because "IRC" keyword just performs the IRC path calculation. Such options are effective in the "SC-AFIR" calculation. In SC-AFIR, the "Bond Condition ... END" option cannot be used simultaniously with the "NoBondRearrange" option (the latter keyword in the input file is only effective).
You may need to add the basis to the valence part of Au:
GauInpB
C H N O F Si 0
6-31g**
****
Au 0
SDD
****
Au 0
SDD
END
#### AFIR Web unavailable: 31 Aug - 2 Sep 2019
Due to a planned power outage in Hokkaido University, all services on our website will be unavailable during the following date and times:
31 August 2019, 15:00 (GMT+9) - 2 September 2019, 13:00 (GMT+9)
• #### AFIR Web server has been back
All services on our website are running now.
#### Elementary questions
Dear GRRM developers,
Very recently, I became a new user of GRRM17. I managed to perform MC-AFIR calculation for exploring new reaction between different two molecules. I have three question now.
Question 1 : I would like to understand the meaning of "Fragment" and "Part". In "MC-AFIR", "part" means the index of molecules for generating initial configration of geometry. I found same question and answer ( Applying artificial force in SC-AFIR , https://afir.sci.hokudai.ac.jp/documents/manual/32) "Fragment" means the index of atoms that is applied to AFIR external force during reaction searching. If so, if I could guess the atoms of reaction area properly, the whole calculation time could be reduced.
Question 2: What is the usage of "GRRM17.out" ? If it is something for summarizing reaction pathway, I would like to use it.
Question 3: When I could not found the desired or supposed reaction pathway among the extracted reaction pathways, could large value setting for NFault or GAMMA be effective ? In my understanding, NFault just limits the number of reaction patth collection, and GAMMA limits the walkable energy hieght during TS survey. If you have effective option, I would like to know.
Best regards,
Yu Kaneko
Dear GRRM developers,
I'm trying to use GRRM17 combined with MOLPRO2015 but the combination does not work. The GRRM tutorial reads that the version 2006, 2008,2010,2012 were successfully tested with GRRM. Kind helps or information would appreciated to use the combination, GRRM17 plus MOLPRO2015.
Best regards,
Yasuhiro Shigemitsu
• #### GRRM17 linked with Molpro 2015 or later
GRRM17 does not recognize the keyword "%link=molpro2015" or later. Consequently, GRRM17 is officially compatible with Molpro 2006, 2008, 2010, and 2012. However, you can use Molpro 2015 or later at your own risk.
With Molpro 2015:
Whereas we have not fully tested Molpro2015 jobs, the formats of output files of Molpro 2015 seems to be the same as those of Molpro2012, and GRRM17 will work with Molpro2015 by using the link keyword "%link=molpro2012".
With Molpro 2018 or later:
GRRM17 knows a Molpro job has been terminated normally if the string "Variable memory released" is appeared in the Molpro output file (XXX_MolJOB.out), because Molpro2015 or earlier prints the string in the last line of the output. But, Molpro2018 or later uses "Molpro calculation terminated" instead of "Variable memory released". Thus, GRRM17 will never understand the termination of the Molpro2018 (or later) calculation, and then GRRM17 program will stop with an error. This issue will be evaded by using the TEXT card in Molpro input. For example:
Molpro Input
BASIS=6-311+G(d,p)
{ITERATIONS;DO,UNCOUPLE,11,TO,39;}
START,*
ORBITAL,*
Occ,12;
Closed,4;
WF,16,1,0;
STATE,2;}
{RS2,Mix=2,Root=2,SHIFT=0.3,MaxIt=99,MaxIti=999;
ORBITAL,IGNORE_ERROR;
WF,16,1,0;
STATE,2;}
FORCE;
IF(STATUS.LT.0) STOP
$STR='Variable memory' TEXT,$STR released
---
RSPT2 STATE 2.1
END
We are aware of another issue; the format of the punch file (XXX_moljob.pun) is different from that of Molpro2015 or earlier, then GRRM17 cannot read the energies from the punch file. This issue will be evaded by setting "submol" command to the shellscript in which the punch file format to be converted. For example:
setenv submol "./script.csh" (csh/tcsh case)
and prepare "script.csh" like as
#!/bin/csh -f
set PUN=echo $2:r.pun | tr $A-Z$ $a-z$ set TMP='tmp' molpro -W./ -d./$1 $2 grep "Energy"${PUN} > ${TMP} set i = 1 set n = wc -l${TMP} | awk '{print $1}' while ($i <= $n ) set line = head -n$i ${TMP} | tail -n 1 echo$line >> ${PUN} @ i =$i + 1
end
\rm -f ${TMP} unset i, n, line unset PUN, TMP exit Alternatively, you can use the general interface in GRRM17. Please see "General interface with external ab initio programs". We are aware of another issue; the format of the punch file (XXX_moljob.pun) is bit different from that of Molpro2015 or earlier, then GRRM17 cannot read the energies from the punch file. For example: RSPT2 STATE 1.1 Energy -114.21130105 (Molpro2015 or earlier) RSPT2 STATE 1.1 Energy -114.21130105 (Molpro2018 or later) So, make sure whether you provide an appropriate string to the Molpro Input section. For example, Molpro Input IF(STATUS.LT.0) STOP$STR='Variable memory'
TEXT,\$STR released
---
RSPT2 STATE 2.1
END
We have made several tests using Molpro 2015.1 (patch level 33), Molpro 2018.2, and Molpro 2019.1.2, but we cannot give any warranty.
#### ONIOM on MC-AFIR
Dear Prof. Maeda & laboratory people,
I'm trying ONIOM method on MC-AFIR.
But, I don't understand the exact format of the ONIOM layer & MC-AFIR part.
This is my input file with error calculation log.
---------------------------------------------------------------------------------------
# MC-AFIR/ONIOM(uwb97xd/6-31G : upm6) integral(coarsegrid)
0 1
H -5.09198232 1.46148646 0.64415956 H 1
H -5.05103711 1.51846702 -1.80580446 H 1
H -5.08619301 3.61195144 -0.53172902 H 1
H -3.07552771 2.21038282 -0.53072587 H 1
Si -4.57617575 2.20057460 -0.55601978 H 1
Si 0.92262713 -0.42951851 -0.06586408 H 2
H 0.32586471 0.89377867 0.44846657 H 2
H 0.92141623 -0.42839950 -1.60585939 H 2
Si -0.36596427 -2.22060001 0.71381911 H 2
Si 0.54148635 -4.23086015 -0.06762047 H 2
...
Si 7.89452207 2.71361008 -0.06626650 L H 16 2
Si 6.60446476 0.92311796 3.05275285 L H 16 2
Si 3.11867252 -0.64788568 3.05254379 L H 12 2
Si -2.56235026 -2.00324774 -0.06393534 L H 9 2
Si 1.83058292 -2.43960890 3.83181251 L H 19 2
Si 1.45021923 -6.24149307 3.83086153 L H 19 2
Si -0.74565359 -6.02268238 0.71234080 L H 10 2
Si 2.35719613 -8.25146150 3.04887994 L H 23 2
Si 4.55185684 -8.47158271 3.82932265 L H 33 2
External Atoms
Si 7.50985596 -1.08833081 6.95223274 L 2
Si 7.12790465 -4.89067932 6.95082004 L 2
Si 8.03487102 -6.90152356 6.17032595 L 2
Si 8.41640895 -3.09947186 6.17176893 L 2
...
H 6.74535629 -8.69308650 8.48930406 L 2
H 7.50965005 -1.08890581 8.49222140 L 2
H 4.02610987 -2.65912846 8.49169783 L 2
H 3.64455597 -6.46260745 8.48999760 L 2
H 0.54408428 -4.23167248 8.49284240 L 2
Options
MicroIt = (MMOnly)
GauProc = 4
NSample = 20
MinFC = -1
MC = ReactivePathOnly
Fragm.1 = 1-5
Fragm.2 = 6-43
Fragm.3 = 66-145
1 2
1 3 -
GAMMA = 600
END
-------------------------------------------------------------------------------------
I think that the number order of ONIOM layer or MC-AFIR part is not good.
Please teach me the exact number order of ONIOM layer & MC-AFIR part.
Best regards
• #### Re: ONIOM on MC-AFIR
To combine ONIOM and MC-AFIR, please use the following format.
atom x y z part-number layer [link-atom info]
where
atom : atom type
x : X-coordinate
y : Y-coordinate
z : Z-coordinate
part-number : part number for MC-AFIR (see Part designation for random structure generation)
layer : layer assignment (H/M/L) for ONIOM
H -5.09198232 1.46148646 0.64415956 1 H
H -5.05103711 1.51846702 -1.80580446 1 H
H -5.08619301 3.61195144 -0.53172902 1 H
H -3.07552771 2.21038282 -0.53072587 1 H
Si -4.57617575 2.20057460 -0.55601978 1 H
Si 0.92262713 -0.42951851 -0.06586408 2 H
H 0.32586471 0.89377867 0.44846657 2 H
H 0.92141623 -0.42839950 -1.60585939 2 H
Si -0.36596427 -2.22060001 0.71381911 2 H
Si 0.54148635 -4.23086015 -0.06762047 2 H
...
Si 7.89452207 2.71361008 -0.06626650 2 L H 16
Si 6.60446476 0.92311796 3.05275285 2 L H 16
Si 3.11867252 -0.64788568 3.05254379 2 L H 12
Si -2.56235026 -2.00324774 -0.06393534 2 L H 9
Si 1.83058292 -2.43960890 3.83181251 2 L H 19
Si 1.45021923 -6.24149307 3.83086153 2 L H 19
Si -0.74565359 -6.02268238 0.71234080 2 L H 10
Si 2.35719613 -8.25146150 3.04887994 2 L H 23
Si 4.55185684 -8.47158271 3.82932265 2 L H 33
...
#### 2 Metal Element Containing Surface Reaction Analysis
Dear Prof. Maeda & Laboratory People
I'm struggling to analyze metal element containing surface reaction analysis such as { TiCl4 on TiN }, { EtOH on HfO2 }, { SiH4 on Ru } and so on.
I use GRRM17 which is not feasible PBC surface model.
Then, I use cluster surface model on these metal element containing surfaces.
However, it is difficult for me to converge these surfaces { TiN, HfO2, Ru } SCF calculation by Gaussian.
I also use Materials Studio DMol3 which is easy to converge SCF calculation but not suitable for GRRM.
Please teach me how to converge these SCF calculation by Gaussian or some alternative convergent technology.
Sencerely
• #### Re:
Although GRRM17 doesn't contain an interface with DMol3, GRRM17 can be used together with any energy computation code if a user prepare a simple interface script. Please see the following page:
https://afir.sci.hokudai.ac.jp/documents/manual/48
• #### Other QM code trial
I'll consider the GRRM application trial not only on Gaussian but also on the other QM codes.
DMol3 doesn't have any analytical hessian calculation method.
Therefore, I think that DMol3 may be very slow on GRRM.
And I hope to choice more established way because I'm a biginner on these operations.
How is the performance of SIESTA (or Turbomole) on cluster model analysis ?
Will you please teach me SIESTA (or Turbomole) setting manuals ?
|
{}
|
Consecutive Integers
Number Theory Level pending
Two sets $$\{a_n\}$$ and $$\{b_n\}$$ of 4 consecutive positive integers have exactly one integer in common. Let
• $$A$$ denote the sum of the integers in $$\{a_n\}$$, which is the set with the greater numbers, and
• $$B$$ denote the sum of the integers in $$\{b_n\}$$.
Find $$A-B$$.
×
|
{}
|
# The State of LDC on Windows
LDC is one of the three major D compilers. It uses the same frontend as DMD, the reference implementation of the language, but leverages LLVM for optimization and code generation. While it has been stable on Linux and OS X for quite some time, support for the Windows operating system family was virtually non-existent so far. There have been substantial advances recently, and this post gives an overview of the current situation.
Before going on to discuss the present status, though, let me quickly answer the inevitable question: Why did it take so long? It is not that the importance of Windows as a target platform would not have been recognized by the D community (or the LDC contributors in particular). Instead, the reason for the lack on of a working Windows port was caused by the fact that LLVM itself did not support all the required operating system specific features. Notably, exception handling was not implemented at all on Windows for a long time.
This applies to 32-bit variants of Windows (Win32) as well as to the newer 64-bit operating systems (Win64), but interestingly the reasons for this are completely different. In the latter case, the problem was just that nobody took the time to implement the (table-driven) Win64 exception handling scheme in the LLVM backend. This is not so surprising, as most of the big companies sponsoring LLVM development are not using LLVM on Windows, or in an application domain that does not require features such as native exception handling or thread-local storage support.
However, Kai Nacke has tackled this problem recently, among with a number of other LLVM issues blocking development of the Visual Studio-based Win64 port of LDC. A patch fixing the bulk of the bugs in the exception handling implementation is currently under review on the LLVM development mailing list, and Kai has prepared a binary preview version of LDC with all the latest patches. For more information, you can also visit the Building and hacking LDC on Windows using MSVC page on the LDC wiki.
The rest of this post will discuss the situation specifically on Win32/MinGW. Here, the root problem is that Structured Exception Handling (SEH), the default exception handling mechanism on 32-bit Windows, is covered by a Borland-held patent. It will not expire until next year, and while Borland seems to dismiss any related concerns, the GCC and LLVM projects have decided to not include an implementation of SEH in their compiler backends for fear of legal trouble.
Recently, however, support for DWARF 2-style exception handling appeared in GCC/MinGW. Here, the Windows-“native” SEH is forgone for the same table-based exception handling scheme that is also used on Linux. The downside of this approach is obviously that it doesn’t integrate with SEH exceptions raised by the OS or other C libraries. But even if it is theoretically possible to catch those from D, this (DMD) feature isn’t really used widely, and as such virtually all D projects should be oblivious to the exception handling mechanism used under the hood.
## Status Overview
So, what can you expect from LDC on Win32/MinGW today? First, the good parts:
• Exception handling works, and all the related test cases that also pass on the various Posixen also pass on Win32/MinGW. Why this qualification? Just like GDC, LDC unfortunately doesn’t implement all the fine details of D’s exception chaining mechanism on any platform yet.
• Thread-local storage (TLS) support is solid. Seeing this item on the list might surprise you, as TLS is central to each and every D2 application. However, it regularly turns out to be a pain point when porting D to new platforms, as it is typically not so important for other native languages. Thus, the related parts of the toolchains are typically less well tested, and LLVM on MinGW unfortunately was no exception here. At this point, however, my fixes to TLS support have arrived in the upstream versions of both mingw-w64 and LLVM, so no custom patches are required any longer (this is also the reason why LDC requires a very recent version of both).
• The DMD, druntime and Phobos test suites mostly pass, and some smaller applications I tested build and work just fine. This notably includes most functionality associated with 80-bit reals (aka long double), which is notoriously problematic as the Microsoft Visual C/C++ runtime (MSVCRT) does not support this type of floating point numbers at all.
• LDC is sufficiently ABI-compatible to DMD on 32-bit Windows that virtually all of the inline assembly code in druntime and Phobos works without changes. This only covers a surprisingly small part of the total ABI though, so even if DMD emitted COFF object files, it would still be a hopeless endeavor to try and link object files produced by the two compiles together, just as it is on the other operating systems.
Now, for the less pleasant points:
• There are still a few issues related to floating-point math, particularly with complex 80-bit numbers. Single tests in std.complex, std.math, std.mathspecial and std.internal.math.gammafunctionstill fail, and core.stdc.fenv is not implemented properly yet. It seems to be likely that most of these problems are again caused by functions lacking from MSVCRT respectively their MinGW replacements (one specific example is fmodl, which seems to cause interesting ABI issues).
• The core.sys.windows.dll tests do not build, and while this would be easy to work around, DLL creation is entirely untested at this point.
• While MinGW theoretically supports COM, the std.windows.iunknown tests do not link yet because of missing symbols. There is likely an easy fix, but interfacing with COM has not been tested at all.
• There are also still two rather disconcerting test failures in core.time and rt.util.container which have not been tracked down yet.
• LDC currently relies on using the MinGW as for emitting object files, as the LLVM integrated assembler does not correctly support writing the DWARF exception handling tables yet. This is suboptimal, as it causes several issues with non-ASCII characters in symbol names and generally has a negative effect on compiler performance. It currently also causes an issue with building the std.algorithm unit tests in debug mode, where the humongous symbol names (in the tens of kilo(!)bytes) overflow some as-internal data structures.
• And most importantly, LDC/MinGW is still virtually untested on larger real-world applications. There will certainly be a number of bugs which have not been caught by any of the test suites.
## Getting Started
So, how to try out LDC on Windows? The easiest thing would be to just download the latest binary (preview) release. For this, first grab a very recent mingw32-w64 snapshot, such as this one (rubenvb personal build, .7z, ~27 MB) and extract it to an arbitrary location. It is important that you pick one built with Dwarf 2 exception handling enabled; when in doubt, just use the above one.
Then, download and extract the latest LDC binary release for MinGW (.7z, ~8.5 MB). It is a “DMD-style” package that should work from any location without any extra installation steps. Before invoking LDC, you need to make sure that the MinGW bin directory is on your path, though. This is easiest to achieve by starting a shell using mingw32env.cmd in the MinGW root directory, or of course using a MSYS shell altogether.
If you prefer building LDC from source yourself, a guide on building LDC on MinGW x86 is available on the wiki. Any help with LDC/MinGW development would be very much appreciated!
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GKC
Background: When I was the editor of Gilbert Magazine, I was responsible for the “Tremendous Trifles” column. It was occasionally hard to find a sufficient amount of interesting GKC material to fill the page, so John Peterson sent me a file full of Chesterton ancedotes. They were idiosyncratic, historical, and Chestertonian. He gave me permission to use them here. I hope y’all find them as interesting as I have over the years. Most of them have never been published.
Chesterton Short(s)
In his 1979 study of the Oxford Inklings, Humphrey Carpenter concludes that the mind of C.S. Lewis had two aspects, the poet and the debater, and when the debater was in ascendancy, Carpenter says interestingly enough that Lewis showed the mark of Chesterton. [The Inklings, Houghton Mifflin, p. 221]
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odd doctor 已更新
# CRB and Tree
Time Limit
4s
Memory Limit
65536KB
Judge Program
Standard
Ratio(Solve/Submit)
100.00%(1/1)
Description:
CRB has a tree, whose vertices are labeled by 1, 2, …, N. They are connected by N – 1 edges. Each edge has a weight.
For any two vertices u and v(possibly equal), f(u,v) is xor(exclusive-or) sum of weights of all edges on the path from u to v.
CRB’s task is for given s, to calculate the number of unordered pairs (u,v) such that f(u,v) = s. Can you help him?
Input:
There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:
The first line contains an integer N denoting the number of vertices.
Each of the next N - 1 lines contains three space separated integers ab and c denoting an edge between a and b, whose weight is c.
The next line contains an integer Q denoting the number of queries.
Each of the next Q lines contains a single integer s.
1 ≤ T ≤ 25
1 ≤ N ≤ 105
1 ≤ Q ≤ 10
1 ≤ ab ≤ N
0 ≤ cs ≤ 105
It is guaranteed that given edges form a tree.
Output:
For each query, output one line containing the answer.
Sample Input:
1
3
1 2 1
2 3 2
3
2
3
4
Sample Output:
1
1
0
Hint:
For the first query, (2, 3) is the only pair that f(u, v) = 2.For the second query, (1, 3) is the only one.For the third query, there are no pair (u, v) such that f(u, v) = 4.
Source:
2015 Multi-University Training Contest 10
Submit
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1. ## Integration
How do you evaluate integral(x^1/2)sinx dx ?
2. You can use the Taylor expansion of $\displaystyle \sin x$ around $\displaystyle x=0$, then multiply each term of the series by $\displaystyle x^{\frac{1}{2}}$ and finally integrate term by term...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
3. Originally Posted by CandyKanro
How do you evaluate integral(x^1/2)sinx dx ?
An exact answer cannot be found using a finite number of elementary functions. Where has this integral come from? Is an exact answer required?
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