text:
Calculating sum of primes using the CPU and GPU

This is a little baffling to me as to why the CUDA code runs about twice as slow as the CPU version. I am just counting all the primes from 0 to (512 * 512 * 512). The CPU version executed in about 97 seconds whereas the GPU version took 182 seconds.

• CPU: Intel Core i7 @ 4 GHz
• GPU: NVIDIA GTX 960

Any ideas why?

#include <cuda.h>
#include <iostream>
#include <cstdint>
#include <stdio.h>
#include <ctime>
#include <vector>
#include <cstdlib>
#include <climits>

using namespace std;

__host__ __device__  bool is_prime(uint32_t n)
{
if(n == 2)
return true;
if(n % 2 == 0)
return false;
uint32_t sr = sqrtf(n);

for(uint32_t i = 3; i <= sr; i += 2)
if(n % i == 0)
return false;
return true;
}

__global__ void prime_sum(unsigned int* count)
{
uint32_t n = (blockIdx.y * gridDim.y + blockIdx.x) * blockDim.x + threadIdx.x;
if(is_prime(n))
}

CPU version

int main()
{
time_t start = time(0);
int pcount = 0;
for(uint32_t i = 0; i < (512 * 512 * 512); i++)
{
if(is_prime(i)) pcount++;
}
start = time(0) - start;
std::cout << pcount << "\t" << start << std::endl;

return 0;
}

CUDA version

int main()
{
time_t start = time(0);
unsigned int* sum_d;
cudaMalloc(&sum_d, sizeof(unsigned int));
cudaMemset(sum_d, 0, sizeof(unsigned int));

prime_sum<<< dim3(512, 512), 512 >>>(sum_d);

unsigned int sum = 0;
cudaMemcpy(&sum, sum_d, sizeof(unsigned int), cudaMemcpyDeviceToHost);
start = time(0) - start;
std::cout << sum << "\t" << start << std::endl;
cudaFree(sum_d);

return 0;
}

• This is lacking in curly braces and could be bad for maintenance:

if(n == 2)
return true;
if(n % 2 == 0)
return false;
uint32_t sr = sqrtf(n);

for(uint32_t i = 3; i <= sr; i += 2)
if(n % i == 0)
return false;
return true;

It's a good idea to still use them for single-line statements, and you should especially use them with for loops. Some extra whitespace in places could be useful as well.

Here's what it should look like:

if (n == 2)
{
return true;
}
if (n % 2 == 0)
{
return false;
}

uint32_t sr = sqrtf(n);

for (uint32_t i = 3; i <= sr; i += 2)
{
if (n % i == 0)
{
return false;
}
}

return true;

• Consider finding an alternative to sqrtf() as it can be slow. It may not make a huge difference in overall runtime, though. Fortunately, it appears to be thread-safe.

• One source of slower performance may be the if statement in prime_sum(). Doing so may cause thread divergence, if threads end up executing different instructions. You'll have to analyze your code to determine this for sure.

• This line seems confusing:

start = time(0) - start;

It appears that the lvalue should instead be named end since you already calculate a starting time beforehand.

It's not surprising that your serial code is faster than your GPU code. You are not taking advantage of the places where GPUs perform exceptionally well, and you are using the GPU in exactly the places where GPUs don't perform well at all.

Regarding the former, GPUs are geared toward floating point rather than integer operations, and on arrays where the same computation is performed on every element in the array. You are using integers rather than floating point numbers, and don't have any arrays at all (but you could if you used a sieve).

Regarding the latter, two things that GPUs don't do well at at all are integer division/modulus, and atomic operations. You are using both. There are over 7.6 million primes less than 5123, so you are calling AtomicAdd on a global variable over 7.6 million times. This alone is a huge bottleneck.


text:
Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 Title Keywords Abstract Author All
Search Results: 1 - 10 of 100 matches for " "
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 中国物理快报 , 2009, Abstract: We present Monte Carlo studies on the singly tagged D mesons, which are crucial in the absolute measurements of D meson decays, based on a full Monte Carlo simulation for the BES-III detector, with the BES-III Offline Software System. The expected detection efficiencies and mass resolutions of the tagged D mesons are well estimated.
 Hai-Bo Li Physics , 2009, DOI: 10.1088/0954-3899/36/8/085009 Abstract: Decays of both $\eta$ and $\eta^\prime$ provide very useful information in our understanding of low-energy QCD, and experimental signatures for these decays would be extremely helpful at BES-III. The rare decays of the $\eta$ and $\eta^\prime$ mesons could serve as a low-energy test of the Standard Model and its beyond. The sensitivities of the measurements of $\eta$ and $\eta^\prime$ decays are discussed at BES-III, in which the $\eta$ and $\eta^\prime$ mesons are produced in the $\psi$ decays.
 High Energy Physics - Phenomenology , 2008, Abstract: This physics book provides detailed discussions on important topics in $\tau$-charm physics that will be explored during the next few years at \bes3 . Both theoretical and experimental issues are covered, including extensive reviews of recent theoretical developments and experimental techniques. Among the subjects covered are: innovations in Partial Wave Analysis (PWA), theoretical and experimental techniques for Dalitz-plot analyses, analysis tools to extract absolute branching fractions and measurements of decay constants, form factors, and CP-violation and \DzDzb-oscillation parameters. Programs of QCD studies and near-threshold tau-lepton physics measurements are also discussed.
 Physics , 2008, Abstract: This physics book provides detailed discussions on important topics in $\tau$-charm physics that will be explored during the next few years at \bes3 . Both theoretical and experimental issues are covered, including extensive reviews of recent theoretical developments and experimental techniques. Among the subjects covered are: innovations in Partial Wave Analysis (PWA), theoretical and experimental techniques for Dalitz-plot analyses, analysis tools to extract absolute branching fractions and measurements of decay constants, form factors, and CP-violation and \DzDzb-oscillation parameters. Programs of QCD studies and near-threshold tau-lepton physics measurements are also discussed.
 Yi-Fang Wang Physics , 2007, Abstract: Interesting results from BES-II and other experiments raised actually many new questions which shall be answered by its upgrade program, BEPCII and BES-III. The design and current status of BEPCII and BES-III are reported.
 Physics , 2009, DOI: 10.1016/j.physletb.2010.02.059 Abstract: We study an experimental search of a GeV scale vector boson at BES-III/BEPCII. It is responsible for mediating a new U(1)$_d$ interaction, as recently exploited in the context of weakly interacting massive particle dark matter. At low energy $e^+ e^-$ colliders this dark state can be efficiently probed. We discuss the direct productions of this light vector $U$ boson and the decay of this state with BES-III data and its foreseen larger data. In particular, we show that Higgs' strahlung in the dark sector can lead to multilepton signatures, which probe the physics range for kinetic mixing parameter $\epsilon \sim 10^{-4} -10^{-3}$ over a large portion of the parameter space.
 Physics , 2010, Abstract: We discuss the possible searching for the oscillation by coherent $\Lambda\bar{\Lambda}$ production in $J/\psi \rightarrow \Lambda \bar{\Lambda}$ decay process. The sensitivity of measurement of $\Lambda - \bar{\Lambda}$ oscillation in the external field at BES-III experiment is considered. These considerations indicate an alternative way to probe the $\Delta B =2$ amplitude in addition to neutron oscillation experiments. Both coherent and time-dependent information can be used to extract $\Lambda -\bar{\Lambda}$ oscillation parameter. With one year's luminosity at BES-III, we can set an upper limit of $\delta m < 10^{-15}$ MeV at 90% confidence level, corresponding to about $10^{-6}$ s of $\Lam-\Lamb$ oscillation time.
 World Journal of Nuclear Science and Technology (WJNST) , 2016, DOI: 10.4236/wjnst.2016.64025 Abstract: The beam energy measurement system is significant and profit for both BES-III detector and BEPC-II accelerator. The detection of the high energy scattering photons is realized by virtue of the Compton backscattering principle. Many advanced techniques and precise instruments are employed to acquire the highly accurate measurement of positron/electron beam energy. During five years’ running period, in order to satisfy the requirement of data taking and enhance the capacity of measurement itself, the upgradation of system is continued, which involves the components reformation of laser and optics subsystem, replacement of view-port of the laser to vacuum insertion subsystem, the usage of electric cooling system for high purity germanium detector, and the refinement of data acquisition and processing subsystem. The upgrading of system guarantees the smooth and effective measuring of beam energy at BEPC-II and accommodates the accurate offline energy values for further physics analysis at BES-III.
 中国物理 B , 2009, Abstract: The $CP$ violation in the $D$ system is predicted to be an unobserved level in the Standard Model. In this paper, we describe the method of searching for $CP$ violation decay processes with the coherently produced $D^0\bar D^0$ mesons from the $\psi(3770)$ decay. The $CP$ violation decay processes can be searched for at the BES-III experiment. The experimental sensitivity for searching for the $CP$ violation can reach about a $10^{-4}$ level with a $\psi(3770)$ data sample of about $20$ fb$^{-1}$.
 Beijiang Liu Physics , 2010, Abstract: Recent BES results on light scalars are reported in this talk, including the observation of a charged kappa decaying to K pi0 with 5.8 * 10^7 J/psi data at BES II and the direct measurements of a0(980)-f0(980) mixing in the processes J/psi --> phi f0(980) --> phi a0(980) --> phi eta pi0 and chi_c1 --> pi0 a0(980) --> pi0 f0(980) --> pi0 pi+ pi- with 2.26 * 10^8 J/psi data and 1.06 * 10^8 psi' data at BES III.
 Page 1 /100 Display every page 5 10 20 Item

format: plain


text:
# Where is the origin?

Nov 11, 2014

The origin $\left(0 , 0\right)$ is located where the $x$-axis and the $y$-axis intersect.

I hope that this was helpful.


text:
# Nice Polynomials of degree 3

Here follows an example inspired by some great mathematical work of Jean-François Burnol.

All graphics are setup with PSTricks. All animations are generated with the animate package of Alexander Grahn and finally converted to svg format. The calculations within the animations are mostly done with xintexpr (part of the mighty xint bundle) of Jean-François Burnol.

## Short mathematical background

Consider a cubic equation with real coefficients $P=0$ with three distinct roots. These are the abscissas of the vertices of an equilateral triangle whose radius is equal to the horizontal distance between the local extrema of the graph of the given cubic polynomial. The animation shows this fact in varying the second term of the equation $P = u$, which corresponds to the intersection of the graph with the horizontal line $y=u$.

## The dance of the rational roots of a polynomial of degree 3

A much more complicated thing to find only rational roots solved and animated by Jean-François Burnol. All calculations within this animation are done with xintexpr of Jean-François Burnol.

Note: This animation can't be fluently due to only rational roots.

## The dance of the integer roots of a polynomial of degree 3

Even much more complicated thing to find only integer roots solved by Jean-François Burnol.

As an example we chose the cubic: $P(X)=(X-10)(X+1)(X+9)=X^3-91X-90$

The roots of the first derivative are not integers.

Note: This animation can't be fluently due to only integer roots.

## The dance of the integer roots of a $\ZZ$-nice polynomial of degree 3

Even much much more complicated thing to find only integer roots and integer roots of the first derivative solved by Jean-François Burnol.

As an example we chose the $\ZZ$-nice cubic: $P(X)=X(X-9)(X-24)=X^3-33X^2+216X$

This polynomial gives the minimal distance of the abscissas of its local extrema which is 14.

Note: This animation can't be fluently due to only integer roots.

## Documentation

Nice cubic polynomials: symmetry and arithmetic of the Lagrange resolvent
by Jean-François Burnol and Jürgen Gilg (in English)

La voie-sans-effort™ vers le cubisme
by Jean-François Burnol (in French)

Polynômes cubiques plaisants
Excerpts of the documentation by Manuel Luque (in French)

Polynômes cubiques plaisants (2)
Excerpts of the documentation by Manuel Luque (in French)

The $\TeX$ source files can be accessed at

Polynômes plaisants (figures et animation avec PSTricks, calculs avec xint, en particulier les fractions)
Many thanks to Jean-Michel Sarlat


text:
# Chapter 10 Attractive Output

Once you’ve gotten your data into the shape you want to be ready for analysis, you’re going to want to output the results. This chapter won’t walk you through how to do the analysis, it’ll just focus on the presentation of those results. You can use the default formatting that R uses, but that can be a little plain and difficult on the eyes. For instance, let’s take a look at summary statistics and a regression table for data on California schools.

CASchools <- read.csv("https://raw.githubusercontent.com/ejvanholm/DataProjects/master/CASchools.csv")
CASchools2 <- subset(CASchools, select=c("math", "income", "students", "english"))
summary(CASchools2)
##       math           income          students          english
##  Min.   :605.4   Min.   : 5.335   Min.   :   81.0   Min.   : 0.000
##  1st Qu.:639.4   1st Qu.:10.639   1st Qu.:  379.0   1st Qu.: 1.941
##  Median :652.5   Median :13.728   Median :  950.5   Median : 8.778
##  Mean   :653.3   Mean   :15.317   Mean   : 2628.8   Mean   :15.768
##  3rd Qu.:665.9   3rd Qu.:17.629   3rd Qu.: 3008.0   3rd Qu.:22.970
##  Max.   :709.5   Max.   :55.328   Max.   :27176.0   Max.   :85.540
summary(lm(math~income+students+english, data=CASchools))
##
## Call:
## lm(formula = math ~ income + students + english, data = CASchools)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -41.804  -7.577   0.421   7.453  30.415
##
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  6.366e+02  1.590e+00 400.498   <2e-16 ***
## income       1.498e+00  8.262e-02  18.137   <2e-16 ***
## students     6.333e-05  1.553e-04   0.408    0.684
## english     -4.060e-01  3.491e-02 -11.632   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.49 on 416 degrees of freedom
## Multiple R-squared:  0.6274, Adjusted R-squared:  0.6248
## F-statistic: 233.5 on 3 and 416 DF,  p-value: < 2.2e-16

Not the most beautiful thing I’ve ever seen, but it’s okay and you’re probably getting used to it. It’s a lot of numbers, and a lot of information. Some of which you will probably use, other parts of which will be ignored in general. This chapter will walk you through producing that same general output using two packages in R that are designed to make results and output be a bit more presentable: pander and stargazer.

## 10.1 Pander

You’ll need to install the package pander and load it to use it below.

library(pander)

Pander automatically reformats your output when you wrap it around another command. It’s always being used in addition to whatever command you were already running.

Here is how it looks for summary statistics.

pander(summary(CASchools2))
math income students english
Min. :605.4 Min. : 5.335 Min. : 81.0 Min. : 0.000
1st Qu.:639.4 1st Qu.:10.639 1st Qu.: 379.0 1st Qu.: 1.941
Median :652.5 Median :13.728 Median : 950.5 Median : 8.778
Mean :653.3 Mean :15.317 Mean : 2628.8 Mean :15.768
3rd Qu.:665.9 3rd Qu.:17.629 3rd Qu.: 3008.0 3rd Qu.:22.970
Max. :709.5 Max. :55.328 Max. :27176.0 Max. :85.540

That adds space to make the numbers a bit clearer. We can also save whatever we’re doing as an object before using pander.

sum.stats <- summary(CASchools2)
pander(sum.stats)
math income students english
Min. :605.4 Min. : 5.335 Min. : 81.0 Min. : 0.000
1st Qu.:639.4 1st Qu.:10.639 1st Qu.: 379.0 1st Qu.: 1.941
Median :652.5 Median :13.728 Median : 950.5 Median : 8.778
Mean :653.3 Mean :15.317 Mean : 2628.8 Mean :15.768
3rd Qu.:665.9 3rd Qu.:17.629 3rd Qu.: 3008.0 3rd Qu.:22.970
Max. :709.5 Max. :55.328 Max. :27176.0 Max. :85.540

Here it is for a regression.

pander(summary(lm(math~income+students+english, data=CASchools)))
Estimate Std. Error t value Pr(>|t|)
(Intercept) 636.6 1.59 400.5 0
income 1.498 0.08262 18.14 1.374e-54
students 6.333e-05 0.0001553 0.4079 0.6836
english -0.406 0.03491 -11.63 2.849e-27
Fitting linear model: math ~ income + students + english
Observations Residual Std. Error $$R^2$$ Adjusted $$R^2$$
420 11.49 0.6274 0.6248

It removes some of the metrics that were in the default output, but includes most of the key numbers we’d use in interpreting our model.

One benefit of pander is its simplicity. You just need to load it and give it your command or saved output to produce more attractive tables and figures. That’s particularly advantageous when you’re just including a quick look at your table, such as if you’re inserting a table to show what values are in your data.

table(CASchools$grades) ## ## KK-06 KK-08 ## 61 359 pander(table(CASchools$grades))
KK-06 KK-08
61 359

But that simplicity means there is a lack of options to customize the output.

## 10.2 Stargazer

On the other hand, stargazer is loaded with options for customization. You’ll need to install and load it as a package as well.

library(stargazer)

You can find the reference manual for the stargazer package here, and we’ll walk through some of the more common customization options below.

The first thing you’ll want to do is set type=“text”. That’s not the default, so you’ll want to do it each time. There are other options for type which you might use if you’re creating the output to go on a website, but for using stargazer to copy something into a paper or a markdown text is the simplest option.

If you’re producing summary statistics, you can just give stargazer a data frame with all the variables you want included, like we did with CASchools.

stargazer(CASchools2, type="text")
##
## ================================================================
## Statistic  N    Mean    St. Dev.   Min  Pctl(25) Pctl(75)  Max
## ----------------------------------------------------------------
## math      420  653.343   18.754    605   639.4    665.8    710
## income    420  15.317     7.226   5.335  10.639   17.629  55.328
## students  420 2,628.793 3,913.105  81     379     3,008   27,176
## english   420  15.768    18.286     0     1.9      23.0     86
## ----------------------------------------------------------------

But we can also use stargazer to select the subset of variables we want included using the keep= option. We could also remove one or more variables with omit=, if that’s a quicker option.

stargazer(CASchools, type="text",
keep=c("math", "income", "students", "english"))
##
## ================================================================
## Statistic  N    Mean    St. Dev.   Min  Pctl(25) Pctl(75)  Max
## ----------------------------------------------------------------
## students  420 2,628.793 3,913.105  81     379     3,008   27,176
## income    420  15.317     7.226   5.335  10.639   17.629  55.328
## english   420  15.768    18.286     0     1.9      23.0     86
## math      420  653.343   18.754    605   639.4    665.8    710
## ----------------------------------------------------------------

Stargazer adds summary statistics we don’t typically get, and we can select which ones to include as well. There are options for keep. summary .stat or omit.summary.stat depending on how we want to structure our list. I’ll remove the Min and Max below.

stargazer(CASchools, type="text",
keep=c("math", "income", "students", "english"),
omit.summary.stat = c("min", "max"))
##
## ===================================================
## Statistic  N    Mean    St. Dev.  Pctl(25) Pctl(75)
## ---------------------------------------------------
## students  420 2,628.793 3,913.105   379     3,008
## income    420  15.317     7.226    10.639   17.629
## english   420  15.768    18.286     1.9      23.0
## math      420  653.343   18.754    639.4    665.8
## ---------------------------------------------------

Of course, you’ll need to know what each stat is called since stargazer will only recognize certain items. The list is copied from the help guide below.

We can also change the names of the variables. Variable names are the default, but the are often written to be short and identify the basics of the data, but sometimes we want to offer a clearer explanation for our readers. We can change the labels with covariate.labels and a list.

stargazer(CASchools, type="text",
keep=c("math", "income", "students", "english"),
omit.summary.stat = c("min", "max"),
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers", "Math Test Scores"))
##
## ==========================================================================
## Statistic                         N    Mean    St. Dev.  Pctl(25) Pctl(75)
## --------------------------------------------------------------------------
## Number of Students at School     420 2,628.793 3,913.105   379     3,008
## Median Income of Parents         420  15.317     7.226    10.639   17.629
## % of Non-Native English Speakers 420  15.768    18.286     1.9      23.0
## Math Test Scores                 420  653.343   18.754    639.4    665.8
## --------------------------------------------------------------------------

We can do a lot of the same things in building regression results using stargazer. However, we’ll need to give stargazer our saved output to generate a table. It’s important that you don’t wrap the regression command in summary before saving the object, in this case you just want to leave the lm() (or glm()) command alone.

school.reg <- lm(math~income+students+english, data=CASchools)
stargazer(school.reg, type="text")
##
## ===============================================
##                         Dependent variable:
##                     ---------------------------
##                                math
## -----------------------------------------------
## income                       1.498***
##                               (0.083)
##
## students                      0.0001
##                              (0.0002)
##
## english                      -0.406***
##                               (0.035)
##
## Constant                    636.628***
##                               (1.590)
##
## -----------------------------------------------
## Observations                    420
## R2                             0.627
## Residual Std. Error      11.488 (df = 416)
## F Statistic          233.540*** (df = 3; 416)
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

Looks good. A bit more space, with the coefficients and model metrics a little more organized. Let’s change the name of the variables first. We can do that with the same option as we did above.

stargazer(school.reg, type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers") )
##
## ============================================================
##                                      Dependent variable:
##                                  ---------------------------
##                                             math
## ------------------------------------------------------------
## Number of Students at School              1.498***
##                                            (0.083)
##
## Median Income of Parents                   0.0001
##                                           (0.0002)
##
## % of Non-Native English Speakers          -0.406***
##                                            (0.035)
##
## Constant                                 636.628***
##                                            (1.590)
##
## ------------------------------------------------------------
## Observations                                 420
## R2                                          0.627
## Residual Std. Error                   11.488 (df = 416)
## F Statistic                       233.540*** (df = 3; 416)
## ============================================================
## Note:                            *p<0.1; **p<0.05; ***p<0.01

And let’s change the metrics we show at the bottom of the table with keep.stat (notice that it isn’t keep.summary.stat here), which would line up well with the discussion hereabout different metrics we can include.

stargazer(school.reg, type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers"),
keep.stat=c("n", "adj.rsq", "aic") )
##
## ============================================================
##                                      Dependent variable:
##                                  ---------------------------
##                                             math
## ------------------------------------------------------------
## Number of Students at School              1.498***
##                                            (0.083)
##
## Median Income of Parents                   0.0001
##                                           (0.0002)
##
## % of Non-Native English Speakers          -0.406***
##                                            (0.035)
##
## Constant                                 636.628***
##                                            (1.590)
##
## ------------------------------------------------------------
## Observations                                 420
## ============================================================
## Note:                            *p<0.1; **p<0.05; ***p<0.01

I asked for the AIC to be included, but it wasn’t. That’s because it wasn’t automatically calculated for R for this model (some models it is, others not.) In that case I can add it manually by adding a line with add.lines=. It’s a bit more complicated, because with add.lines I want to add multiple entries across the columns, so I need to tell R I’m making a list, and then set up each set of entries as a separate group with c().

It’s also made tougher because I need to tell R what to calculate to include, so I’m first adding a label for the row “AIC”, then telling it to calculate the AIC to insert. I also have to tell it to round that number using round() so that it doesn’t show a 9-digit decimal. The 2 at the end tells R to put two decimal points at most.

stargazer(school.reg, type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers"),
add.lines = list(c("AIC", round(AIC(school.reg),2))) )
##
## ============================================================
##                                      Dependent variable:
##                                  ---------------------------
##                                             math
## ------------------------------------------------------------
## Number of Students at School              1.498***
##                                            (0.083)
##
## Median Income of Parents                   0.0001
##                                           (0.0002)
##
## % of Non-Native English Speakers          -0.406***
##                                            (0.035)
##
## Constant                                 636.628***
##                                            (1.590)
##
## ------------------------------------------------------------
## AIC                                        3248.6
## Observations                                 420
## ============================================================
## Note:                            *p<0.1; **p<0.05; ***p<0.01

I’ll add a second note to try and make that a little clearer.

stargazer(school.reg, type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers"),
c("Regression", "Yes"))
)
##
## ============================================================
##                                      Dependent variable:
##                                  ---------------------------
##                                             math
## ------------------------------------------------------------
## Number of Students at School              1.498***
##                                            (0.083)
##
## Median Income of Parents                   0.0001
##                                           (0.0002)
##
## % of Non-Native English Speakers          -0.406***
##                                            (0.035)
##
## Constant                                 636.628***
##                                            (1.590)
##
## ------------------------------------------------------------
## AIC                                        3248.6
## Regression                                   Yes
## Observations                                 420
## ============================================================
## Note:                            *p<0.1; **p<0.05; ***p<0.01

You can see all the statistics related to models that are available below.

One of the big advantageous of using stargazer is the ability to include multiple regressions in a single table. Say, for instance, you’re looking at the effect of including difference variables in your model. We can provide stargazer multiple regression objects.

school.reg1 <- lm(math~income+students+english, data=CASchools)
school.reg2 <- lm(math~income+students+english+expenditure, data=CASchools)
school.reg3 <- lm(math~income+students+english+expenditure+calworks, data=CASchools)

stargazer(school.reg1, school.reg2, school.reg3, type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers", "Total School Spending", "Percent Qualifying for CalWorks "),
add.lines = list(c("AIC", round(AIC(school.reg1),2), round(AIC(school.reg2),2), round(AIC(school.reg3),2))))
##
## =================================================================
##                                        Dependent variable:
##                                  --------------------------------
##                                                math
##                                     (1)        (2)        (3)
## -----------------------------------------------------------------
## Number of Students at School      1.498***   1.553***   1.146***
##                                   (0.083)    (0.087)    (0.095)
##
## Median Income of Parents           0.0001    0.00002     0.0001
##                                   (0.0002)   (0.0002)   (0.0001)
##
## % of Non-Native English Speakers -0.406***  -0.400***  -0.361***
##                                   (0.035)    (0.035)    (0.033)
##
## Total School Spending                        -0.002*     0.0004
##                                              (0.001)    (0.001)
##
## Percent Qualifying for CalWorks                        -0.462***
##                                                         (0.056)
##
## Constant                         636.628*** 645.349*** 645.144***
##                                   (1.590)    (4.842)    (4.497)
##
## -----------------------------------------------------------------
## AIC                                3248.6    3246.94    3185.84
## Observations                        420        420        420
## Adjusted R2                        0.625      0.627      0.678
## =================================================================
## Note:                                 *p<0.1; **p<0.05; ***p<0.01

You can also do that if you have multiple regressions with different dependent variables. You can change the labels at the top of the column to identify what the dependent variable is in each regression.

school.reg1 <- lm(math~income+students+english, data=CASchools)

stargazer(school.reg1, school.reg2,  type="text",
covariate.labels = c("Number of Students at School", "Median Income of Parents", "% of Non-Native English Speakers", "Total School Spending", "Percent Qualifying for CalWorks "),
add.lines = list(c("AIC", round(AIC(school.reg1),2), round(AIC(school.reg2),2) )) )
##
## =============================================================
##                                      Dependent variable:
##                                  ----------------------------
##                                       (1)            (2)
## -------------------------------------------------------------
## Number of Students at School        1.498***      1.501***
##                                     (0.083)        (0.074)
##
## Median Income of Parents             0.0001        -0.0001
##                                     (0.0002)      (0.0001)
##
## % of Non-Native English Speakers   -0.406***      -0.569***
##                                     (0.035)        (0.031)
##
## Total School Spending              636.628***    641.224***
##                                     (1.590)        (1.431)
##
## -------------------------------------------------------------
## AIC                                  3248.6        3160.46
## Observations                          420            420
## Note:                             *p<0.1; **p<0.05; ***p<0.01


text:
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#### Description:

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is $1 \frac12$ miles away....

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#### CCSS.Math.Content.6.NS.A.1: Common Core State Standards for Mathematics

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
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# Math Help - Solving trigonometric equations

1. ## Solving trigonometric equations

Solve over Real numbers:
a)5sin^2t+cos^2t-2=0 b)sec^2t+tantt=7 c)root of sint=2sint-1 d)tant-sintcost=0

Solve over [0, 2pi[ :

e)sin^2x-cos^2x=0 f)2tan^2x+3tanx-5=0 g)2sin^2x-sinx-1=0 h)6cos^2x+5cosx=0

solve:
2(cos(3x-pi)+2=1 "x" element of reals

if "x" element [0 degrees,180degrees] solve 4sin2(x-5)=1

Find the exact value:
cosx=-1/3 pi/2<0<pi

evaluate
sin2x tan2x

2. Hello, etheen!

I'll do a few of them . . .

Solve over Real numbers:

$a)\;5\sin^2\!t+\cos^2\!t-2\:=\:0$
Replace $\cos^2\!t$ with $1 - \sin^2\!t$

$5\sin^2\!t + (1 - \sin^2\!t) - 2 \:=\:0 \quad\Rightarrow\quad 4\sin^2\!t - 1 \:=\:0 \quad\Rightarrow\quad 4\sin^2\!t \:=\:1$

. . $\sin^2\!t \:=\:\tfrac{1}{4} \quad\Rightarrow\quad \sin t \:=\:\pm\tfrac{1}{2} \quad\Rightarrow\quad t \;=\;\begin{Bmatrix}\frac{\pi}{6} + 2\pi n \\ \\[-3mm] \frac{5\pi}{6} + 2\pi n \end{Bmatrix}$

$b)\;\sec^2\!t+\tan t\:=\:7$
Replace $\sec^2\!t$ with $\tan^2\!t + 1$

$(\tan^2\!t + 1) + \tan t \:=\:7 \quad\Rightarrow\quad \tan^2\!t + \tan t - 6 \:=\:0 \quad\Rightarrow\quad (\tan t - 2)(\tan t + 3) \:=\:0$

We have: . $\begin{array}{cccccccccc}
\tan t - 2 \:=\:0 & \Rightarrow & \tan t \:=\:2 & \Rightarrow & t \:=\:\arctan(2) + \pi n \\
\tan t + 3 \:=\:0 & \Rightarrow & \tan t \:=\:\text{-}3 & \Rightarrow & t \:=\:\arctan(\text{-}3) + \pi n \end{array}$

$c)\;\sqrt{\sin t}\:=\:2\sin t-1$

Square both sides: . $\left(\sqrt{\sin t}\right)^2 \:=\:(2\sin t - 1)^2 \quad\Rightarrow\quad \sin t \:=\:4\sin^2\!t - 4\sin t + 1$

. . $4\sin^2\!t - 5\sin t + 1 \:=\:0 \quad\Rightarrow\quad (\sin t - 1)(4\sin t - 1) \:=\:0$

We have: . $\begin{array}{ccccccc}
\sin t - 1 \:=\:0 & \Rightarrow & \sin t \:=\:1 & \Rightarrow & t \:=\:\frac{\pi}{2} + 2\pi n \\ \\[-3mm]
4\sin t - 1 \:=\:0 & \Rightarrow & \sin t \:=\:\frac{1}{4} & \Rightarrow& t \:=\:\arcsin\left(\frac{1}{4}\right) + 2\pi n \end{array}$

$d)\;\tan t-\sin t\cos t\:=\:0$

We have: . $\frac{\sin t}{\cos t} - \frac{\sin t \cos^2\!t}{\cos t} \:=\:0 \quad\Rightarrow\quad \frac{\sin t - \sin t\cos t}{\cos t} \:=\:0$

. . $\frac{\sin t(1 - \cos^2\!t)}{\cos t} \:=\:0 \quad\Rightarrow\quad \frac{\sin t\cdot\sin^2\!t}{\cos t} \:=\:0 \quad\Rightarrow\quad \frac{\sin^3\!t}{\cos t} \:=\:0$

. . $\sin^3\!t \:=\:0 \quad\Rightarrow\quad \sin t \:=\:0 \quad\Rightarrow\quad t \:=\:\pi n$

3. hm..i see, I see. These are the harder ones. I'm going to do a few more and ask if I am stuck...Thank you so much :}


text:
My Math Forum Show a Function is not Uniformly Continuous
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 Real Analysis Real Analysis Math Forum

 July 1st, 2015, 02:14 PM #1 Newbie   Joined: Mar 2014 Posts: 16 Thanks: 2 Show a Function is not Uniformly Continuous Show that the function L(x, y) = xy is not uniformly continuous.
 July 2nd, 2015, 05:17 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 On what set?
 July 2nd, 2015, 08:31 AM #3 Newbie   Joined: Mar 2014 Posts: 16 Thanks: 2 On R^2
 July 2nd, 2015, 10:22 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 "Continuous" on a R^2 means that, given a point (x_0, y_0) in R^2, given any $\displaystyle \epsilon> 0$, there exist $\displaystyle delta> 0$ such that if $\displaystyle \sqrt{(x- x_0)^2+ (y- y_0)^2}< \delta$ then $\displaystyle |xy- x_0y_0|< epsilon$. "Uniformly continuous" means that, for a given $\epsilon$ a value of $\delta$ can be chosen that works for all points in R^2. Could you prove that xy is continuous on R^2 at, say, (0, 0)? Where would that proof break down if you used a general point rather than (0, 0)? Thanks from dpsmith
 July 3rd, 2015, 03:44 PM #5 Newbie   Joined: Mar 2014 Posts: 16 Thanks: 2 Actually, I think I've solved the problem. Here's what you have to do. Find an epsilon (I don't know how enter the math symbol) so that, given any delta > 0, you can find two points in the domain within delta of each other, BUT their function values differ by greater than epsilon. To that end, for L(x, y) = xy, let epsilon = 1. Let delta > 0. Choose N so that sqrt(2)/N < delta. Let M > N/2. Look at the two points (M, M) and (M + 1/N, M + 1/N). They differ by sqrt(2)/N < delta. Yet, | L(M + 1/N, M + 1/N) - L(M, M) | = (2M)/N + 1/N^2 > 1 = epsilon. Voila. Last edited by dpsmith; July 3rd, 2015 at 03:48 PM.

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1)The allowable axial stress in the plate material,σ = 24 ksi, and all the allowable shear...

Question:

1)The allowable axial stress in the plate material,σ = 24 ksi, and all the allowable shear stress for the bolt material,τall = 14 ksi.

a)Determine the maximum value of T for complete assembly. (hint: First find the maximum T for each component.

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# Union grouping in bipartite graphs?

I'm trying to figure out a good (and fast) solution to the following problem:

I have two models I'm working with, let's call them players and teams. A player can be on multiple teams and a team can have multiple players). I'm working on creating a UI element on a form that allows a user to select multiple teams (checkboxes). As the user is selecting (or deselecting) teams, I'd like to display the teams grouped by the players.

So for examples:

1. If the selected teams have no players that intersect, each team would have its own section.

2. If the user selects two teams and they have the same players, there would be one section containing the names of the two teams and all the players.

3. If TEAM_A has players [1, 2, 4, 5] and TEAM_B has players [1, 3, 5, 6]. There would be the following sections: SECTION_X = [TEAM_A, TEAM_B, 1, 5], SECTION_Y = [TEAM_A, 2, 3], SECTION _Z = [TEAM_B, 3, 5]

I hope that's clear. Essentially, I want to find the teams that players have in common and group by that. I was thinking maybe there is a way to do this by navigating a bipartite graph? Not exactly sure how though and I might be overthinking it. I was hoping to do this by creating some type of data structure on the server and using it on the client. I would love to hear your suggestions and I appreciate any help you can give!

• Are you sure you can't code that logic using a many-to-many relationship between your models. Hopefully, when you start writing your server-side code, your favorite framework will allow you to declare models with some fields. Therefore, you will be able to instruct that the model Team should a have a field called players that requires a many-to-many relationship to your other model Player. This allows you to retrieve which players are in each team and which teams are associated to each player. – Robert Smith Jan 23 '15 at 3:47
• Have you chosen any particular technology? This could be done with Neo4J and Cypher. – sheldonkreger Mar 4 '15 at 20:22

Have you heard about the intersection graph? You can try to draw players as points, connections (team mates) as edges and teams as transparent coloured blobs on top.

As for your original question, I cannot understand your goal. I think your formulation is not well defined / incomplete. Suppose you have teams A [1,2] B [2,3] and C [1,3]. What do you want to display? Do you want to list the parts of the Venn diagram? I think that for more than 3 sets this can become more cumbersome than the bipartite graph itself = simple listing of team compositions.

Store the edges (relations) in your server:

(TeamID, playerID)

When you want to find common elements just filter all edges where:

TeamID="TeamA" or TeamID="TeamB"

(You could use indexes to speed is up, etc)

Then group by playerID and check how many items are in each group. The groups with two items belong to both teams and are shared.


text:
# 10.20: Estimate the Difference between Population Proportions (1 of 3)

### Learning Objectives

• Recognize when to use a hypothesis test or a confidence interval to compare two population proportions or to investigate a treatment effect for a categorical variable.
• Construct a confidence interval to estimate the difference between two population proportions (or the size of a treatment effect) when conditions are met. Interpret the confidence interval in context.

In “Distributions of Differences in Sample Proportions,” we used simulation to observe the behavior of the differences in sample proportions when we randomly select many, many samples. From the simulation, we developed a normal probability model to describe the sampling distribution of sample differences. With this model, we are now ready to do inference about a difference in population proportions (or about a treatment effect.)

When our goal is to estimate a difference between two population proportions (or the size of a treatment effect), we select two independent random samples and use the difference in sample proportions as an estimate. Of course, random samples vary, so we want to include a statement about the amount of error that may be present. Because the differences in sample proportions vary in a predictable way, we can also make a probability statement about how confident we are in the process that we used to estimate the difference between the population proportions. You may recognize that what we are describing is a confidence interval.

In Inference for One Proportion, we calculated confidence intervals to estimate a single population proportion. In this section, “Estimate the Difference between Population Proportions,” we learn to calculate a confidence interval to estimate the difference between two population proportions. If the data comes from an experiment, we estimate the size of the treatment effect.

### Learn By Doing

https://assessments.lumenlearning.co...sessments/3927

### Confidence Interval for a Difference in Two Population Proportions: the Basics

Every confidence interval has this form:

$\mathrm{statistic}\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}$

To estimate a difference in population proportions (or a treatment effect), the statistic is a difference in sample proportions, so the confidence interval is

$(\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{sample}\text{}\mathrm{proportions})\text{}±\text{}\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error}$

When we select two random samples and calculate the difference in the sample proportions, we do not know the exact amount of error for this particular pair of samples. We therefore use the standard error as a typical amount of error and calculate the margin of error from the standard error, as we did in Inference for One Proportion.

If a normal model is a good fit for the sampling distribution, we can use it to make probability statements that describe our confidence in the interval. More specifically, we use the normal model to describe our confidence that the difference in population proportions lies within a given margin of error of the difference in sample proportions. For example, we can state that we are 95% confident that the difference in population proportions is contained in the following interval:

$(\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{sample}\text{}\mathrm{proportions})\text{}±\text{}2(\mathrm{standard}\text{}\mathrm{error})$

## Nuclear Power

The following problem is based on a report, “Opposition to Nuclear Power Rises amid Japanese Crisis,” by the Pew Research Center (March 21, 2011).

After the nuclear reactor accidents in Japan during the spring of 2011, there was a shift in public support for expanded use of nuclear power in the United States. A few months before the accident, 47% of a random sample of 1,004 U.S. adults supported expanded use of nuclear power. After the nuclear accident in Japan, 39% of a different random sample of 1,004 U.S. adults favored expanded use.

https://assessments.lumenlearning.co...sessments/3928

https://assessments.lumenlearning.co...sessments/3929

### What Does 95% Confident Really Mean?

95% confident comes from a normal model of the sampling distribution.

To review this idea in the context of differences in sample proportions, let’s start with a picture to help us visualize a confidence interval and its relationship to the sampling distribution. The following normal model represents the sampling distribution.

In the diagram, notice the sample difference. In the sampling distribution, we can see that the error in this sample difference is less than the margin of error. We know this because the distance between the sample difference and the population difference is shorter than the length of the margin of error (abbreviated MOE in the figure). When we create a confidence interval with this sample difference, we mark a distance equal to a margin of error on either side of the sample difference. Notice that this interval contains the population difference, which makes sense because the distance between the population difference and the sample difference has not changed.

So where does the “95%” come from? If the normal model is a good fit for the sampling distribution, then the empirical rule applies. The empirical rule says that 95% of the values in a normal distribution fall within 2 standard deviations of the mean. So 95% of the sample differences are within 2 standard errors of the mean difference. Remember that the mean difference is the difference in population proportions. Now consider the confidence interval centered at a sample difference. From the empirical rule, it follows that 95% of the confidence intervals, with a margin of error equal to 2 standard errors, will contain the population difference.

Following is another illustration of 95% confidence, a concept that is often misinterpreted. This diagram helps us remember the correct interpretation. If we construct confidence intervals with a margin of error equal to 2 standard errors, then 95% confidence means that in the long run, 95% of these confidence intervals will contain the population difference, and 5% of the time, the interval we calculate will not contain it. We show one of these less common intervals with a red dot at the sample difference.

Of course, in reality, we don’t know the difference in population proportions. (This is the why we want to estimate it with a confidence interval!) So, in reality, we will not be able to determine if a specific confidence interval does or does not contain the true difference in population proportions. This is why we state a level of confidence. For a specific interval, we say we are 95% confident that the interval contains the true difference in population proportions.

## Correct and Incorrect Interpretations of 95% Confidence

Recall our earlier example about the change in public opinion after the nuclear accident in Japan. We concluded, “We are 95% confident that there was a 4% to 12% drop in support for the expanded use of nuclear power in the United States after the nuclear accident in Japan.”

Here are some accurate ways to describe the phrase “95% confident” for this confidence interval:

• There is a 95% chance that poll results from two random samples will give a confidence interval that contains the true change in public support for the expanded use of nuclear power in the United States after the nuclear accident in Japan
• 95% of the time, this method produces an interval that covers the true difference in the proportions of the U.S. adult population supporting expanded use of nuclear power in the United States before and after the nuclear accident in Japan.

Here are some incorrect interpretations:

• There is a 95% chance that the true difference in public opinion (before and after the nuclear accident in Japan) is between 4% and 12%.
• There is a 95% chance that there was a 4% to 12% drop in support for the expanded use of nuclear power in the United States after the nuclear accident in Japan.

What do you notice? In the correct interpretations, the 95% is a probability statement about the random event of sampling. So statements like “95% chance that two random samples give a confidence interval” and “95% of the time, this method produces an interval” describe the chance that confidence intervals, in the long run, contain the difference in population proportions.

In the wrong interpretations, the phrase “95% chance” is a probability statement about the specific interval 4% to 12%. Since we already found the confidence interval, there is no random event in this description, so we cannot make a probability statement about a single specific interval. For this reason, we use the phrase “95% confident” when we are describing a single interval from a specific study.

### Learn By Doing

https://assessments.lumenlearning.co...sessments/3930


text:
# A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has been proved under many different regularity assumptions, but it still remains an open problem in its original formulation. I realized that the solution I proposed in the linked thread contains the main ideas of a solution of the $C^2$ case of the square peg problem: for any $B\in\gamma$, by considering a sequence of circles centered at $\gamma$ with slowly increasing radii we get a solution of $A,C\in\gamma$, $AB=BC$ and $AB\perp BC$. Given an isosceles and right triangle $ABC$, we may define $B'$ as the symmetric of $B$ with respect to the midpoint of $AC$. If for some $B\in\gamma$ we have $B'\in\gamma$ the square peg problem is solved, and solving the square peg problem with suitable smoothness assumptions boils down to showing that two curves intersect (like in Gauss' original proof of the fundamental Theorem of Algebra).

$\hspace{1in}$

The green curve and the ellipse intersect, hence the square peg problem can be solved in an ellipse.

Context's over. Now I propose the following variation: let $\gamma$ be a (simple,) closed curve of class $C^2$ without any inflection point. Given some $P\in\gamma$, we define $P'$ as the symmetric of $P$ with respect to the center of curvature of $P$. We denote as $\gamma'$ the set $\{P':P\in\gamma\}$.

Q1. Is it true that $\gamma$ and $\gamma'$ always have an intersection?

Q2. Has this problem already been studied in the literature?


text:
# DSPACE

In computational complexity theory, DSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine. It represents the total amount of memory space that a "normal" physical computer would need to solve a given computational problem with a given algorithm. It is one of the most well-studied complexity measures, because it corresponds so closely to an important real-world resource: the amount of physical computer memory needed to run a given program.

## Complexity classes

The measure DSPACE is used to define complexity classes, sets of all of the decision problems which can be solved using a certain amount of memory space. For each function f(n), there is a complexity class SPACE(f(n)), the set of decision problems which can be solved by a deterministic Turing machine using space O(f(n)). There is no restriction on the amount of computation time which can be used, though there may be restrictions on some other complexity measures (like alternation).

Several important complexity classes are defined in terms of DSPACE. These include:

• REG = DSPACE(O(1)), where REG is the class of regular languages. In fact, REG = DSPACE(o(log log n)) (that is, Ω(log log n) space is required to recognize any nonregular language).
• L = DSPACE(O(log n))
• PSPACE = $\bigcup_{k\in\mathbb{N}} \mbox{DSPACE}(n^k)$
• EXPSPACE = $\bigcup_{k\in\mathbb{N}} \mbox{DSPACE}(2^{n^k})$

## Machine models

DSPACE is traditionally measured on a deterministic Turing machine. Several important space complexity classes are sublinear, that is, smaller than the size of the input. Thus, "charging" the algorithm for the size of the input, or for the size of the output, would not truly capture the memory space used. This is solved by defining the multi-string Turing machine with input and output, which is a standard multi-tape Turing machine, except that the input tape may never be written-to, and the output tape may never be read from. This allows smaller space classes, such as L (logarithmic space), to be defined in terms of the amount of space used by all of the work tapes (excluding the special input and output tapes).

Since many symbols might be packed into one by taking a suitable power of the alphabet, for all c ≥ 1 and f such that f(n) ≥ 1, the class of languages recognizable in c f(n) space is the same as the class of languages recognizable in f(n) space. This justifies usage of big O notation in the definition.

## Hierarchy Theorem

The space hierarchy theorem shows that, for every space-constructible function $f: \mathbb{N} \to \mathbb{N}$, there exists some language L which is decidable in space $O(f(n))$ but not in space $o(f(n))$. See space hierarchy theorem for details.

## Relation with other complexity classes

DSPACE is the deterministic counterpart of NSPACE, the class of memory space on a nondeterministic Turing machine. By Savitch's theorem,[1] we have that

$\mbox{DSPACE}[s(n)] \subseteq \mbox{NSPACE}[s(n)] \subseteq \mbox{DSPACE}[(s(n))^2].$

NTIME is related to DSPACE in the following way. For any time constructible function t(n), we have

$\mbox{NTIME}(t(n)) \subseteq \mbox{DSPACE}(t(n))$.

## References

1. ^ Arora & Barak (2009) p.86


text:
# 3.(b)  Replace $\square$ in each of the following by the correct number :  (b)  $\frac{5}{8}=\frac{10}{\square }$

$\frac{5}{8}\times \frac{2}{2}=\frac{10}{16}$


text:
# Manipulating matrices for 3D drawing

I've written a short library for manipulating matrices for 3D drawing. I've tried to strike a balance between speed and readability. Anything to improve?

%!
%mat.ps
%Matrix and Vector math routines

/ident { 1 dict begin /n exch def
[
1 1 n { % [ i
[ exch % [ [ i
1 1 n { % [ [ i j
1 index eq { 1 }{ 0 } ifelse % [ [ i b
exch % [ [ b i
} for % [ [ b+ i
pop ] % [ [ b+ ]
} for % [ [b+]+ ]
]
end } def

/rotx { 1 dict begin /t exch def
[ [ 1  0      0         ]
[ 0  t cos  t sin neg ]
[ 0  t sin  t cos     ] ]
end } def

/roty { 1 dict begin /t exch def
[ [ t cos      0  t sin ]
[ 0          1  0     ]
[ t sin neg  0  t cos ] ]
end } def

/rotz { 1 dict begin /t exch def
[ [ t cos  t sin neg  0 ]
[ t sin  t cos      0 ]
[ 0      0          1 ] ]
end } def

/.error where { pop /signalerror { .error } def } if

/dot { % u v
2 copy length exch length ne {
/dot cvx /undefinedresult signalerror } if
% u v
0 % u v sum
0 1 3 index length 1 sub { % u v sum i
3 index 1 index get exch % u v sum u_i i
3 index exch get % u v sum u_i v_i
mul add % u v sum
} for % u v sum
3 1 roll pop pop % sum
} bind def

% [ x1 x2 x3 ] [ y1 y2 y3 ]  cross  [ x2*y3-y2*x3 x3*y1-x1*y3 x1*y2-x2*y1 ]
/cross { % u v
dup length 3 ne 2 index length 3 ne or {
/cross cvx /undefinedresult signalerror } if
% u v
exch aload pop 4 3 roll aload pop % x1 x2 x3 y1 y2 y3
[
5 index 2 index mul % ... [ x2*y3
3 index 6 index mul sub % ... [ x2*y3-y2*x3
5 index 5 index mul % ... [ x2*y3-y2*x3 x3*y1
8 index 4 index mul sub % ... [ x2*y3-y2*x3 x3*y1-x1*y3
8 index 5 index mul % ... [ x2*y3-y2*x3 x3*y1-x1*y3 x1*y2
8 index 7 index mul sub % ... [ x2*y3-y2*x3 x3*y1-x1*y3 x1*y2-x2*y1
]
7 1 roll 6 { pop } repeat
} bind def

/transpose { STATICDICT begin
/A exch def
/M A length def
/N A 0 get length def
[
0 1 N 1 sub { /n exch def
[
0 1 M 1 sub { /m exch def
A m get n get
} for
]
} for
]
end } dup 0 6 dict put def

/matmul { STATICDICT begin
/B exch def
B 0 get type /arraytype ne { /B [B] def } if
/A exch def
A 0 get type /arraytype ne { /A [A] def } if
/Q B length def
/R B 0 get length def
/P A length def
Q A 0 get length ne {
/A A transpose def
/P A length def
Q A 0 get length ne {
A B end /matmul cvx /undefinedresult signalerror
} if
} if

[
0 1 P 1 sub { /p exch def % rows of A
[
0 1 R 1 sub { /r exch def % cols of B
0
0 1 Q 1 sub { /q exch def % terms of sum
A p get q get
B q get r get mul
} for
} for
]
} for
]

end } dup 0 10 dict put def

%u v {operator}  vop  u(op)v
%apply a binary operator to corresponding elements
%in two vectors producing a third vector as result
/vop { 1 dict begin
/op exch def
2 copy length exch length ne {
/vop cvx end /undefinedresult signalerror
} if

[ 3 1 roll % [ u v
0 1 2 index length 1 sub { % [ ... u v i
3 copy exch pop get % u v i u_i
3 copy pop get      % u v i u_i v_i
op exch pop         % u v u_i(op)v_i
3 1 roll            % u_i(op)v_i u v
} for % [ ... u v
pop pop ]

end } def

%length of a vector
/mag { 0 exch { dup mul add } forall } def

/elem { % M i j
3 1 roll get exch get % M_i_j
} def

/det {
dup length 1 index 0 get length ne { /det cvx /typecheck signalerror } if
1 dict begin /M exch def
M length 2 eq {
M 0 0 elem
M 1 1 elem mul
M 0 1 elem
M 1 0 elem mul sub
}{
M length 3 eq {
}{ /det cvx /rangecheck signalerror } ifelse
} ifelse
end
} def

-
Current version: github.com/luser-dr00g/xpost/blob/master/data/mat.ps – luser droog Sep 16 '15 at 21:31
This code is used to produce the images in these posts across the SE network: 1 2 3 – luser droog Jan 30 at 7:25
– luser droog Jan 30 at 7:25

As far as readability goes, maybe you'd like to add an arg1 arg2 arg3 -- result1 result2 type of comment to each function? So we know what was on the stack before invoking, and what we expect to find afterward? Almost equivalently, you might show a unit test invoking your function and verifying the result.

-
Good advice. I usually do that and wonder why I didn't here. I'm working up a new interpreter to run it, but it will have these things before entering the repository. :) – luser droog May 2 '13 at 6:42
Oh, I remember! I was hoping the stylized local argument convention would make the parameters apparent on the functions without the usual comment. They either construct the argument directly where the layout naturally illustrates the result, or contain intermediate stack-comments up through the final line, illustrating the result. But, it appears it's too clever which is bad. The entire lack of comments in the determinant function det was an oversight. Rats! matmul and transpose need comments, too. :( – luser droog May 2 '13 at 6:49

I just noticed some issues with the vop procedure. It's not nestable the way it's written. So you can't use it to add two matrices like m1 m2 { {add}vop }vop. I've got a favorite trick using token that'll fix this. But I've got serious versioning issues, because I've post this code everywhere with no identifying marks!! I'm an idiot.

So another big improvement would be, maybe, a copyright notice, um, a release date, a version number?!

I've also got a handful of routines that I've considered adding because I keep needing them in the same programs that use this. But they're not really related to matrices, so I've kept them out to avoid the kitchen-sink problem.

/div { dup 0 eq { pop 10000 }{ div } ifelse } bind def
/normalize{[1 index mag 2 index length 1 sub{dup}repeat]{div}vop}def
/tan { dup sin exch cos div } def

Replacing div with a version that doesn't explode on divide-by-zero has gotten things to run that otherwise wouldn't, often enabling me to locate the problem visually.

normalize probably should be in there, if it weren't for the versioning issue.

And tan is curiously omitted from the PostScript operator set. Not sure if this is the best way, but it's probably the simplest. But this requires the div guard, because tangents go to infinity when the cos is 0. The practical selection of infinity is perhaps best left to the individual program. So, I suppose those two should stay out.

-

Much later, I notice an error in the matmul routine if the right argument is a vector, it should be promoted to a column vector in order to multiply with a left matrix. This code doesn't do that.

Rather than

B 0 get type /arraytype ne { /B [B] def } if

it needs to make unit subarrays of each element of B.

B 0 get type /arraytype ne {
/B [ 0 1 B length 1 sub {
B exch 1 getinterval
} for ] def
} if

Oh, and mag should probably take a sqrt.

And det should signal the undefinedresult error rather than typecheck if the argument is not a square matrix. Perhaps it should signal a typecheck is the argument is not an array or scalar number.

-


text:
# The range of specific speed of a Kaplan or propeller turbine in SI units, is:

1. 30 -51
2. 51 -225
3. 8.5 - 30
4. 250 - 850

Option 4 : 250 - 850

## Reaction Turbine MCQ Question 1 Detailed Solution

Explanation:

Specific speed:

• It is defined as the speed of a similar turbine working under a head of 1 m to produce a power output of 1 kW. The specific speed is useful to compare the performance of the various type of turbines. The specific speed differs for the different type of turbines and is the same for the model and actual turbine.
• $${N_s} = \frac{{N\sqrt P }}{{{H^{\frac{5}{4}}}}}$$

Following are the range of specific speed of different turbines

• The specific speed of Pelton wheel turbine (single jet) is in the range of 10-35
• The specific speed of Pelton wheel turbine (multiple jets) is in the range of 35-60
• The specific speed of Francis turbine is in the range of 60-300.
• The specific speed of Kaplan/propeller turbine is greater than 300.

So the Kaplan turbine has the highest specific speed.

The classification of the turbine based on the basis of operating head

 Type of turbine Operating head (m) Pelton 300 m and above Francis 60 m to 300 m Kaplan 20 m to 60 m Bulb 2 m to 20 m

The specific speed of a turbine is in the range of

 Turbine Ns Pelton wheel 10 - 60 Francis 60 - 300 Kaplan > 300

# Guide angle as per the aero foil theory of Kaplan turbine blade design is defined as the angle between

1. lift and resultant force
2. drag and resultant force
3. lift and tangential force
4. lift and drag

Option 1 : lift and resultant force

## Reaction Turbine MCQ Question 2 Detailed Solution

Explanation:

where α1 and α2 are the guide vane angles, and β1 and β2 are runner blade angle.

Guide angle as per the aerofoil theory of Kaplan turbine blade design is defined as the angle between lift and resultant force.

Since the vertical component of the force on aerofoil is called lift so the guide angle is the angle between the lift and the resultant force.

# Kaplan turbine is a type of which of the following?

1. Reaction turbine
2. None of the above
3. Propeller turbine
4. Francis turbine

Option 1 : Reaction turbine

## Reaction Turbine MCQ Question 3 Detailed Solution

Explanation:

Turbines are hydraulic machines that convert hydraulic energy into mechanical energy.

Classification of turbines based on the energy at inlet:

 Impulse turbines Reaction turbines These have only Kinetic energy at the inlet. They have both Kinetic energy and pressure energy at the inlet. Examples: Pelton wheel, Girard turbine, Banki turbine, etc. Examples: Kaplan turbine, Francis turbine, Propeller turbine, etc

Specific Speed of Turbines:

 Type of turbine Specific Speed (Ns) Kaplan and Propeller 300 - 1000 Pelton 10 - 35 Francis 60 - 300

 Type of turbine Head High head turbines: Pelton turbine Above 250 m Medium head turbines: Francis turbine 60 m - 250 m Low head turbines: Kaplan and Propeller turbine Below 60 m

# Draft tube at the exit of a reaction turbine used for the hydroelectric project is __________.

1. above or below the water surface, depends on unit speed of the turbine
2. always above the water surface
3. always immersed in water
4. may either be below or above the water surface

Option 3 : always immersed in water

## Reaction Turbine MCQ Question 4 Detailed Solution

Explanation:

Draft tube

It is a conduit which connects the runner exit to the tailrace where the water is being finally discharged from the turbine.

Hence, Draft tube at the exit of a reaction turbine used for the hydroelectric project is always immersed in water.

Function

The primary function of the draft tube is to reduce the velocity of the discharged water to minimize the loss of kinetic energy at the outlet.

# The flow ratio in a Francis turbine varies from

1. 0.15 to 0.3
2. 0.4 to 0.5
3. 0.6 to 0.9
4. 1 to 1.5

Option 1 : 0.15 to 0.3

## Reaction Turbine MCQ Question 5 Detailed Solution

Explanation:

Flow ratio

The flow ratio of Francis turbine is defined as the ratio of the velocity of flow at the inlet to the theoretical jet velocity.

$$Flow\;Ratio = \frac{{{V_{f1}}}}{{\sqrt {2gH} }}$$

In the case of Francis turbine,

Flow ratio varies from 0.15 to 0.3

Speed ratio varies from 0.6 to 0.9

# The turbine suitable for low heads and high flow rates is:

1. Pelton wheel
2. Francis
3. Kaplan
4. All of these

Option 3 : Kaplan

## Reaction Turbine MCQ Question 6 Detailed Solution

Concept:

In Francis Turbine the medium head is required generally in the range of (100-500) meter. In Kaplan Turbine very low head is required, generally less than 100-meter

In Francis Turbine medium flow rate is required. In Kaplan Turbine very large flow rate is required

• High head turbine: In this type of turbines, the net head varies from 150m to 2000m or even more, and these turbines require a small quantity of water. Example: Pelton wheel turbine.
• Medium head turbine: The net head varies from 30m to 150m, and also these turbines require a moderate quantity of water. Example: Francis turbine.
• Low head turbine: The net head is less than 30m and also these turbines require a large quantity of water. Example: Kaplan turbine.

# Hydraulic efficiency of Francis turbine whose vanes are radial at inlet is (α – guide blade angle)

1. $$\frac{{2 + {{\tan }^2}\alpha }}{2}$$
2. $$\frac{{{{\tan }^2}\alpha }}{2}$$
3. $$\frac{2}{{2 + {{\tan }^2}\alpha }}$$
4. $$\frac{2}{{ta{n^2}\alpha }}$$

Option 3 : $$\frac{2}{{2 + {{\tan }^2}\alpha }}$$

## Reaction Turbine MCQ Question 7 Detailed Solution

Explanation:-

Fransis turbine is a radial flow inward reaction turbine.

Inlet triangle:

U1 = velocity of blade at inlet, V1  = absolute velocity of entering water, Vr1 = relative velocity of entering water, Vf1  = velocity of  flow at inlet

Outlet triangle:

U2 = velocity of blade at outlet, Vr2 = relative velocity of leaving  water, Vf2  = V2 = velocity of flow at outlet

We know that,

$$W = \frac{{V_1^2 - V_2^2}}{{2g}} + \frac{{V_{r2\;}^2 - \;V_{r1}^2}}{{2g}} + \frac{{U_1^2 - U_2^2}}{{2g}}$$

From inlet and outlet velocity triangles,

$${V_1} = {U_1}\sec \alpha$$

$${V_2} = {V_{f2}}\; = \;{U_1}\tan \alpha = {U_2}\tan \beta \;\;$$

Using these relations into the expression of W, we have $$W = \frac{1}{{2g}}\;\left[ {U_1^2{{\sec }^2}\alpha \; - \;U_1^2{{\tan }^2}\alpha + \;U_2^2{{\sec }^2}\beta - \;U_1^2{{\tan }^2}\alpha + U_1^2\; - \;U_2^2} \right]$$

$$W = \frac{1}{{2g}}\;\left[ {U_1^2 + U_2^2 + U_2^2{{\tan }^2}\beta \; - U_1^2{{\tan }^2}\alpha + U_1^2\; - \;U_2^2} \right]$$

$$W = \frac{1}{{2g}} \times 2U_1^2 = \frac{{U_1^2}}{g}$$

H = work head + energy rejected from turbine

$$H = \frac{{U_1^2}}{g} + \frac{{V_2^2}}{{2g\;}}$$

$$H = \;\frac{1}{{2g}}\left( {\;2U_1^2 + V_2^2\;} \right)$$

$$H = \;\frac{1}{{2g}}(\;2U_1^2 + U_1^2{\tan ^2}\alpha \;)$$

$$H = \;\frac{{U_1^2}}{{2g}}(\;2 + {\tan ^2}\alpha \;)$$

∴ Hydraulic Efficiency

$${\eta _h} = \frac{{work\;equivalent\;head}}{{available\;head}}$$

$${\eta _h} = \frac{{\frac{{U_1^2}}{g}}}{{\frac{{U_1^2\;\left( {2 + {{\tan }^2}\alpha \;} \right)}}{{2g}}}} = \frac{2}{{2 + {{\tan }^2}\alpha }}$$

# Draft tube is used in reaction turbine to

1. Pass water downstream without eddies formation
2. Convert the kinetic energy into pressure energy by the gradual expansion of the flow cross-section
3. Provide safety to the turbine
4. Prevent water splitting

Option 2 : Convert the kinetic energy into pressure energy by the gradual expansion of the flow cross-section

## Reaction Turbine MCQ Question 8 Detailed Solution

Explanation:

Draft Tube: The draft tube is a diverging tube connects the runner exit to the tailrace. At the exit of a turbine runner, the fluid has appreciable kinetic energy and to recover some of this kinetic energy which would otherwise be wasted the flow enters an expanding area diffuser called a draft tube. It is used with reaction turbines only.

The draft tube has two purposes as follows:

1. It permits a negative or suction head to be established at the runner exit, thus making it possible to install the turbine above the tailrace level without loss head.
2. It is used to convert kinetic energy at the exit of the turbine to the pressure energy.

A draft tube is made divergent so as to reduce the velocity at the outlet to a minimum. Therefore, a draft tube is basically a diffuser and should be designed properly with the angle between the walls of the tube to be limited to about 8 degrees so as to prevent the flow separation from the wall and to reduce accordingly the loss of energy in the tube.

# For a non-dimensional specific speed value of 1, for maximum efficiency, which of the following turbines is preferred?

1. Pelton wheel
2. Francis turbine
3. Kaplan turbine
4. Tyson wheel

Option 2 : Francis turbine

## Reaction Turbine MCQ Question 9 Detailed Solution

Non-dimensional specific speed is given by

$${N_S} = \frac{{N\sqrt P }}{{{{\left( {gH} \right)}^{\frac{5}{4}}} \cdot \sqrt \rho }}$$

The Range for non-dimensional specific speed is given in the table below

 Non-Dimensional specific speed Type of turbine Less than 0.18 Single jet Pelton 0.18 – 0.30 Multi Jet Pelton 0.30 – 1.50 Francis 1.50 – 5.12 Kaplan

# Kaplan water turbine is commonly used when the flow through its runner is

1. axial and the head available is more than 100 m
2. axial and the head available is less than 10 m

Option 2 : axial and the head available is less than 10 m

## Reaction Turbine MCQ Question 10 Detailed Solution

Explanation:

Classification of turbines on the various basis is given in the table below:

 Flow Energy Head Specific speed Example Tangential Impulse High (300 m and above) Low (0 – 60 RPM) Pelton Wheel turbine Radial Reaction Medium (30 m to 300 m) Medium (60 – 300) RPM Francis turbine Axial Reaction Low (less than 30 m) High (300 – 600) RPM Propeller turbine (600 – 1000) RPM Kaplan turbine

# In Kaplan Turbine, the flow velocity:

1. directly proportional to the radius

2. maximum at midpoint of the radius
3. inversely proportional to the radius
4. is constant along the radius

Option 4 : is constant along the radius

## Reaction Turbine MCQ Question 11 Detailed Solution

Explanation:

• If the water flows parallel to the axis of rotation of the shaft, the turbine is known as an axial flow turbine.
• For the axial flow reaction turbine, the shaft of the turbine is vertical.
• The lower end of the shaft is made larger which is known as a hub or boss.
• When the vanes on the hub are adjustable then the turbine is known as the Kaplan turbine and when the vanes are fixed to the hub then the turbine is known as the Propeller turbine.
• Kaplan turbine is a low head axial flow turbine with an adjustable vane.
• In a Kaplan turbine, the velocity of flow through the rotor is constant along the radius.

Classification of turbines on various bases is given in the table below:

 Flow Energy Head Specific speed Example Tangential Impulse High (300 m and above) Low (0 – 60 RPM) Pelton Wheel turbine Radial Reaction Medium (30 m to 300 m) Medium (60 – 300) RPM Francis turbine Axial Reaction Low (less than 30 m) High (300 – 600) RPM Propeller turbine (600 – 1000) RPM Kaplan turbine

# The discharge passing through a turbine, which is working under a unit head, is called as:

1. unit velocity
2. unit speed
3. unit power
4. unit discharge

Option 4 : unit discharge

## Reaction Turbine MCQ Question 12 Detailed Solution

Concept:

In order to predict the behaviour of a turbine working under a varying condition of head, speed output, etc, the results are expressed in terms of quantities that may be obtained when the head on the turbine is reduced to unity.

The following are three important unit quantities.

 Unit speed Speed of turbine working under  a unit head $${N_u} = \frac{N}{{\sqrt H }}$$ Unit discharge Discharge passing through a turbine  which is working under a unit head $${Q_u} = \frac{Q}{{\sqrt H }}$$ Unit power Power developed by a turbine   working under unit head $${P_u} = \frac{P}{{{H^{3/2}}}}$$

# The flow ratio of a Francis turbine, if it is working under a head of 62 m and velocity at inlet 7 m/s (g = 10 m/s2) is

1. 0.3
2. 0.2
3. 0.1
4. 0.4

Option 2 : 0.2

## Reaction Turbine MCQ Question 13 Detailed Solution

Explanation:

Flow ratio

The flow ratio of Francis turbine is defined as the ratio of the velocity of flow at the inlet to the theoretical jet velocity.

$$Flow\;Ratio = \frac{{{V_{f1}}}}{{\sqrt {2gH} }}$$

In the case of Francis turbine,

Flow ratio varies from 0.15 to 0.3

Speed ratio varies from 0.6 to 0.9

Calculation:

$$Flow\;Ratio = \frac{{7}}{{\sqrt {2\times10\times62} }}=0.2$$

# The change in head across a small turbine is 10 m, the flow rate of water is 1 m3/s and the efficiency are 80%. The power developed by the turbine is approximately:

1. 100 kW
2. 78 kW
3. 1 MW
4. 50 kW

Option 2 : 78 kW

## Reaction Turbine MCQ Question 14 Detailed Solution

Concept:

The overall efficiency ηo of turbine = volumetric efficiency (ηv)× hydraulic efficiency (ηh)× mechanical efficiency (ηm)

$${\eta _o} = {\eta _v} \times {\eta _h} \times {\eta _m}$$

$${{\rm{\eta }}_{\rm{v}}} = \frac{{{\rm{volume\;of\;water\;actually\;striking\;the\;runner}}}}{{{\rm{volume\;of\;water\;actually\;supplied\;to\;the\;turbine}}}}$$

$${{\rm{\eta }}_{\rm{h}}} = \frac{{{\rm{Power\;deliverd\;to\;runner}}}}{{{\rm{Power\;supplied\;at\;inlet\;}}}} = \frac{{{\rm{R}}.{\rm{P}}}}{{{\rm{W}}.{\rm{P}}}}$$

$${{\rm{\eta }}_{\rm{m}}} = \frac{{{\rm{Power\;at\;the\;shaft\;of\;the\;turbine}}}}{{{\rm{Power\;delivered\;by\;water\;to\;the\;runner}}}} = \frac{{{\rm{S}}.{\rm{P}}}}{{{\rm{R}}.{\rm{P}}}}$$

Overall efficiency: $${\eta _o} = \frac{{S.P}}{{W.P}}$$

Water Power = ρ × Q × g × h

Calculation:

Given:

ηo = 0.8, Head h = 10 m, and Q = 1 m3/s.

$${\eta _o} = \frac{{S.P}}{{W.P}} = \frac{{S.P}}{{\rho \times Q \times g \times h}}$$

$$0.8 = \frac{{S.P}}{{1000 \times 1 \times 9.81 \times 10}} \Rightarrow S.P = 78480\;W \approx 78\;kW$$

# In Kaplan turbine runner, the number of blades is generally of the order _____.

1. 2 – 4
2. 4 – 8
3. 8 – 16
4. 16 – 24

Option 2 : 4 – 8

## Reaction Turbine MCQ Question 15 Detailed Solution

Explanation:

Kaplan turbine:

• In Kaplan turbine, water flows parallel to the axis of rotation of the shaft, hence it is called axial flow turbine.
• Head at the inlet of the turbine is the sum of pressure energy and kinetic energy.
• During the flow of water through the runner, a part of pressure energy is converted into kinetic energy.
• It indicates that the Kaplan turbine is a reaction turbine. When the vanes on the hub are adjustable, the turbine is known as the Kaplan turbine.
• When vanes are not adjustable, the turbine is called the propeller turbine. Kaplan turbine components are shown in the figure below.

• Kaplan turbine has adjustable runner blades. Kaplan Turbine has a very small number of blades 3 to 8.

Important Points

• Francis Turbine has a very large number of blades 16 to 24

# Jet ratio (m) is defined as the ratio of

1. Diameter of the jet of water to diameter of the Pelton wheel
2. Velocity of the vane to velocity of the jet of water
3.  Velocity of flow to velocity of the jet of water
4. Diameter of Pelton wheel to the diameter of the jet of water

Option 4 : Diameter of Pelton wheel to the diameter of the jet of water

## Reaction Turbine MCQ Question 16 Detailed Solution

Explanation:

Jet ratio (m):

It is defined as the ratio of the diameter of the pitch circle to the diameter of the jet.

Jet ratio = $$\rm \dfrac{Diameter\;of\;wheel }{Jet\; diameter}$$

m =$$\dfrac{D}{d}$$,

Where, D = Diameter of the pitch circle, d = Diameter of the jet.

The value of m varies from, 11 ≤ m ≤ 15,

Number of vanes (n) = 15 + $$\dfrac{m}{2}$$, Where, m = Jet ratio

 Impulse turbine Reaction turbine All hydraulic energy is converted into kinetic energy. A part of hydraulic energy is converted into kinetic energy and the rest to pressure energy. Pressure throughout remains atmosphere. Pressure at the inlet is higher than the outlet Watertight casing not nessacary The runner must be enclosed within the watertight casing Always installed above the tailrace and no use of draft tube It may be installed above or below the tailrace with the help of a draft tube. Flow regulation is done using the needle valve. Flow regulation is done with the help of a guide vane

# A Francis turbine is an-

1. Inward flow reaction turbine
2. Inward flow impulse turbine
3. Outward flow reaction turbine
4. Outward flow impulse turbine

Option 1 : Inward flow reaction turbine

## Reaction Turbine MCQ Question 17 Detailed Solution

Explanation:

Impulse Turbine: If at the inlet of the turbine, the energy available is only kinetic energy, the turbine is known as impulse turbine. e.g. a Pelton wheel turbine.

Reaction Turbine: If at the inlet of the turbine, the water possesses kinetic energy as well as pressure energy, the turbine is known as a reaction turbine. e.g. e Francis and Kaplan turbine.

Tangential flow turbines: In this type of turbine, the water strikes the runner in the direction of the tangent to the wheel. Example: Pelton wheel turbine

Radial flow turbines: In this type of turbine, the water strikes in the radial direction. accordingly, it is further classified as

• Inward flow turbine: The flow is inward from periphery to the centre (centripetal type); Example: old Francis turbine
• Outward flow turbine: The flow is outward from the centre to periphery (centrifugal type); Example: Fourneyron turbine

Axial flow turbine: The flow of water is in the direction parallel to the axis of the shaft. Example: Kaplan turbine and propeller turbine

∴ Francis turbine is a radial inward flowing reaction turbine.

# A Kaplan turbine is a/an ______ turbine.

1. Axial flow
2. Mixed flow
4. Tangential flow

Option 1 : Axial flow

## Reaction Turbine MCQ Question 18 Detailed Solution

Explanation:

According to the direction of flow of fluid over the blades, turbines can be classified in the following categories:

• Axial flow turbine: The flow of water is in the direction parallel to the axis of the shaft.
• Example: Kaplan turbine and propeller turbine.
• Mixed flow turbine: water flows through the runner in the radial direction but leaves in the direction parallel to the axis of rotation of the runner. In simple terms, Fluid flows both in radial as well as in axial direction.
• Example: Modern Francis Turbine
• Radial flow turbines: In this type of turbines, the water strikes in the radial direction. accordingly, it is further classified as:
• Inward flow turbine: The flow is inward from periphery to the center (centripetal type); Example: old Francis turbine
• Outward flow turbine: The flow is outward from the center to periphery (centrifugal type); Example: Fourneyron turbine
• Tangential flow turbines: In this type of turbines, the water strikes the runner in the direction of the tangent to the wheel. Example: Pelton wheel turbine.

# The degree of reaction for Parson’s reaction turbine is __________.

1. 0.25
2. 0.5
3. 0
4. 0.75

Option 2 : 0.5

## Reaction Turbine MCQ Question 19 Detailed Solution

Explanation:

The above-mentioned figure is a schematic diagram of Parson's reaction turbine

• The velocity triangles for inlet & outlet conditions are symmetrical.
• ∴  $$\left\{ {\begin{array}{*{20}{c}} {{\alpha _1} = {\beta _2}}\\ {{\alpha _2} = {\beta _1}} \end{array}} \right\}\;\;\;\;\;\;\;\& \;\;\;\;\left\{ {\begin{array}{*{20}{c}} {{V_{{r_1}}} = {V_2}}\\ {{V_{{r_2}}} = {V_1}} \end{array}} \right\}$$
• This consists of a symmetrical stator and rotor blades.

• And the degree of reaction is defined as the (ratio of enthalpy drop in the rotor to that of the stator

• the pressure drop is equally shared by the stator and the rotor for a turbine.

• It is a  very widely used reaction turbine also known as a 50% reaction turbine

Degree of reaction $${\rm{R}} = \frac{{{\rm{Static\;enthalpy\;drop\;or\;rise\;in\;rotor}}}}{{{\rm{Stagnationenthalpy\;drop\;or\;rise\;in\;stage}}}}$$

Or, $$R = \frac{{static\;pressure\;drop\;or\;rise\;in\;rotor}}{{stagnation\;pressure\;drop\;or\;rise\;in\;stage}}$$

In the case of impulse turbine, there is no change in the enthalpy in the rotor, so the degree of reaction is zero. R = 0.

And for Parson’s turbine or 50% reaction turbine R = 0.5

And for Hero’s turbine R = 1.

# For high head, suitable turbine is _________

1. Pelton
2. Francis
3.  Kaplan
4.  None of the above

Option 1 : Pelton

## Reaction Turbine MCQ Question 20 Detailed Solution

Explanation:

Classification of the turbine based on the head:

 Type of head Head under which it works and example Low head turbine Operates in the head range of fewer than 45 meters. Kaplan turbine is one of these turbines If the head is less than 3 meters, it is considered an ultra-low head. Medium head turbine Working range for heads of 45 to 250m. Francis turbines generally operate in such conditions High head turbine Turbines works in the head range of higher than 250 m Pelton turbine generally operates in this range


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### Home > INT3 > Chapter 5 > Lesson 5.1.2 > Problem5-25

5-25.

Graph $f(x) = \frac { 1 } { 2 }(x - 1)^3$ and its inverse on the same set of axes. 5-25 HW eTool (Desmos). Homework Help ✎

Identify three or more important points on the graph of $y = f\left(x\right)$ to get started. Then make a table for the inverse by switching the inputs and outputs.

$(3, 4) \rightarrow (4, 3)$

$(1, 0) \rightarrow (0, 1)$

$(−1, −4) \rightarrow\: ?$

Complete the table in the eTool below and view the graphs.
Click the link at right for the full version of the eTool: INT3 5-25 HW eTool


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# What is the domain and range of y=ln(x^2)?

Apr 9, 2017

Domain for $y = \ln \left({x}^{2}\right)$ is $x \in R$ but $x \ne 0$, in other words $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$ and range is $\left(- \infty , \infty\right)$.
We cannot have logarithm of a number less than or equal to zero. As ${x}^{2}$ is always positive, only value not permissible is $0$.
Hence domain for $y = \ln \left({x}^{2}\right)$ is $x \in R$ but $x \ne 0$, in other words $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$
but as $x \to 0$, $\ln \left({x}^{2}\right) \to - \infty$, $y$ can take any value from $- \infty$ ao $\infty$ i.e. range is $\left(- \infty , \infty\right)$.


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# Foundations of Analysis

American Mathematical Society, 2001 - Mathematics - 136 pages
Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous ""Grundlagen der Analysis"" - also available from the AMS - answers these important questions.

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### References to this book

 Foundations of Modern Analysis, Volume 1J. DieudonneNo preview available - 2006
 Introduction to Metric and Topological SpacesLimited preview - 1975
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