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# Sandbox for drafts of long, complex answers

This sandbox is intended for saving drafts of long, complex answers, especially answers whose composition takes a long time. It serves to localize to one thread the front-page "bumps" caused by edits to drafts of such answers, so that they may be easily ignored. Also, it helps to guard against losing longly-composed answers due to system crashes.

When you are happy with your draft here, you may simply copy the code and paste it to the desired location.

Proper Use of the Sandbox

1. Do not post a new answer! We wish all the answers on this page to be owned by the Community user (so that only a non-sentient bot is informed of edits to these answers). Posting a new answer will make you the owner, meaning that you will be notified whenever another user makes an edit to that answer.
2. Do not delete answers! Deleting seems like a reasonable option, but there are no "hard deletions" on Stack Exchange, and users will sufficient privileges will still see your supposedly deleted postings.
3. Do look for an answer which indicates that it is free and then edit it to your heart's content. If none looks available, take over the one that has been left unchanged the longest (which will appear at the bottom of the page if you order answers by "activity").
4. Do clear your draft when you are finished. This includes removing all $\LaTeX$ from your answers. Replacing all code with a simple statement like
This answer is free for anyone to use
is sufficient. Periodically users may go through and free up answer slots that have not been edited in, say, over one month. But you can aid in the smooth running of this sandbox by clearing away your drafts when you are finished with them.
5. Do not create new such sandboxes. The point of having a unique such sandbox is that it minimizes the noise on the front page when the sandbox is edited. If there were multiple sandboxes they will frequently occupy numerous front page slots, pushing other topics off the front page, and increasing noise.
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I have added a [sandbox] tag to allow people ignore it more easily (via software support of ignoring tags), and since it seems that we have two sandboxes now, a tag may seem a bit more in place here. –  Asaf Karagila Jul 18 '12 at 8:35
+1, great idea! It might be preferable if moderators delete "free for use" posts from time to time. You don't want two users writing their extensive, carefully crafted answers as edits of the same post and overwrite each other. Also, Chrome crashes from time to time. For some reason, I don't lose what I've written in an original post, but lose it when editing. So if I would use this thread, I would still prefer to create a new post. –  Michael Greinecker Jul 18 '12 at 11:06
@MichaelGreinecker, since deleted answers are still visible for 10k+ users, they would just accumulate in a useless state if they were deleted rather than reused. Also, if you prefer working in a pristine post rather than saving-and-then-editing (such as due to the autosave), then this sandbox is probably not for you anyway. You could do that just as easily at the eventual location of your post. –  Henning Makholm Jul 18 '12 at 11:13
(+1) For thinking outside the (sand)box. –  cardinal Jul 18 '12 at 19:40
My own practice for sandboxes is to post a stub question and then delete it immediately. If you don't leave the page, you can continue to edit the deleted question, and then undelete it when you are finished. –  MJD Jul 23 '12 at 18:14
I have discovered a drawback of the method I described in the previous comment: moderators might choose to undelete your sandbox without consulting you. If you do use this method, it is probably a good idea to attach a header that says draft in large letters. –  MJD Jul 24 '12 at 13:40
I must say I am somewhat curious to why there are up/down-votes on the sandbox answers... –  Willie Wong Jul 24 '12 at 13:44
@Mark Please don't delete any answers in this sandbox. Deleted answers are displayed to users with 10k+ rep. If every answer was deleted when done, then the thread would eventually have hundreds of deleted answers, which would make page navigation and MathJax rendering very complex for 10k+ users. That is why I advocated reuse of answers. –  anon Jul 24 '12 at 14:14
@Willie My guess would be that they are used to keep the answers sorted (so that they are not permuted randomly and user who edited one of the answers finds it in the same place when it returns). Perhaps it is slightly more comfortable that way. –  Martin Sleziak Aug 6 '12 at 8:37
I find the sandbox is too slow to edit a long answer. –  Makoto Kato Aug 26 '12 at 0:37
@Makoto MathJax can get very slow when rendering long answers. I don't think the sandbox plays any role in what you see, since there is no other TeX code on the page. –  anon Aug 26 '12 at 2:15
@MakotoKato, unfortunately, you have chosen the worst possible browser to use. IE8 is slower than all other browsers (e.g., IE9 and IE7 are twice as fast, chrome is 5 times as fast for MathJax). Switching browsers would give you an immediate performance improvement. Short of that, the incremental preview that I link to above would help you a lot as well. I will look into it more, but IE8 has such poor performance, I doubt I can do much to help. Do you shut down your browser between uses, or is it running all the time? –  Davide Cervone Aug 29 '12 at 10:33
I just did some timing tests, and here are the results. Editing Makoto's post (with over 400 equations), to retypeset the entire page: in IE8, 2 minutes 25 seconds, in IE9, 12 seconds, in Safari 5, 10 seconds, in Firefox 15, 13 seconds. So you see that IE8 is really a bad choice or this type of work. The reason it is so bad is that any time MathJax asks for the size of some piece of mathematics (so it can place the superscripts properly, etc), IE reflows the entire page, and that slows things down considerably... –  Davide Cervone Aug 29 '12 at 11:18
At the suggestion of the moderators, I have gone and changed the associated owners of all the answers here to the Community user. This way, the original owners will not receive excess pings for each time another user uses the draft space for their work. Enjoy! –  Grace Note Oct 5 '12 at 14:45
To prevent crashes I've found the "Bookmarks to disable/enable MathJax", provided in here, pretty useful. –  leo Dec 17 '12 at 18:03

This answer is $\;\bbox[3pt,color:red;border-radius:4px;box-shadow:2px 2px 4px firebrick]{\verb/Free/}\;$ for everyone to use

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This answer is free for everyone.

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available.....................

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$\begin{array}{c|ccc} ··&··X··&··Y··&··Z··\\ \hline &c&s&0\\ \to&x&{{\sqrt{10}y+2z}\over\sqrt{14}}&{{-2y+\sqrt{10}z}\over{\sqrt{14}}}\\ &c&\sqrt{35}s\over7&-{\sqrt{14}s\over7}\\ \to&{x-3z}\over\sqrt{10}&y&{3x+z}\over\sqrt{10}\\ \end{array}$

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This answer is free for anyone to use

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This answer is free for everyone to use.

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Free for anyone at all to use.

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This answer is free for everyone.

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This answer is free for everyone to use.

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In progress for question 879618.

The substitution $x=\tan \phi$ converts this integral to \begin{align} \int_{0}^{\pi/2}\frac{\cos \phi\,d\phi}{\sqrt{1+\frac{1}{3}\sin^22\phi+\sqrt{1+\frac{1}{3}\sin^22\phi}}} \end{align} Note that the transformation $\phi\mapsto \pi/2-\phi$ simply replaces $\cos\phi\mapsto\sin\phi$ in the numerator. Consequently we symmeterize these two expressions and write $$\int_{0}^{\pi/4}\frac{d\phi \,(\sin\phi+ \cos \phi)}{\sqrt{1+\frac{1}{3}\sin^22\phi+\sqrt{1+\frac{1}{3}\sin^22\phi}}}$$ where we have used the symmetry to cut the bounds of integration in half. Since $\cos\phi+\sin\phi =\sqrt{2}\cos(\pi/4-\phi),$ we take $\phi\mapsto\pi/4-\phi$ to obtain

$$\sqrt{2}\int_{0}^{\pi/4}\frac{d\phi\cos\phi}{\sqrt{1+\frac{1}{3}\cos^2 2\phi+\sqrt{1+\frac{1}{3}\cos^22\phi}}}$$

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I think your last integral is equivalent to what I got. –  Olivier Oloa yesterday

This answer is free for anyone to use.

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A bit prettier, perhaps: If $P=3ac-b^2$ and $Q=27a^2d-9abc+2b^3$, then $x={{{\sqrt[3]{4\left(-Q+\sqrt{4P^3+Q^2}\right)}}+{\sqrt[3]{4\left(-Q-\sqrt{4P^3+Q^2}\right)}}-{2ab}}\over{6a}}$

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This answer is free for everyone to use.

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This answer is free for anyone to use.

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Task: Develop a fraction equivalent to $1\over{\sum_{i=0}^{n-1}{c_i\sqrt[n]{n^i}}}$ in which the denominator is rational.

Step 1: Develop the denominator. The preliminary answer you seek will be a fraction in which the numerator and denominator are two nearly-similar determinants which differ only in their top rows. The initial form of the denominator is

$\begin{vmatrix} c_0&c_{n-1}n&c_{n-2}n&\cdots&c_3n&c_2n&c_1n\\ c_1&c_0&c_{n-1}n&\cdots&c_4n&c_3n&c_2n\\ c_2&c_1&c_0&\cdots&c_5n&c_4n&c_3n\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ c_{n-3}&c_{n-4}&c_{n-5}&\cdots&c_0&c_{n-1}n&c_{n-2}n\\ c_{n-2}&c_{n-3}&c_{n-4}&\cdots&c_1&c_0&c_{n-1}n\\ c_{n-1}&c_{n-2}&c_{n-3}&\cdots&c_2&c_1&c_0\\ \end{vmatrix}$.

Step 2: The numerator is merely the same determinant as the denominator, except that its top row is replaced by

$\begin{matrix} 1&\sqrt[n]n&\sqrt[n]{n^2}&\cdots&\sqrt[n]{n^{n-3}}&\sqrt[n]{n^{n-2}}&\sqrt[n]{n^{n-1}}\\ \end{matrix}$.

This gives rise, for example, to the following:

${1\over{A+B\sqrt n}}= {{\begin{vmatrix} 1&\sqrt n\\ B&A\\ \end{vmatrix}}\over {\begin{vmatrix} A&Bn\\ B&A\\ \end{vmatrix}}}={{A-B\sqrt n}\over{A^2-B^2 n}}$ and
${1\over{A+B\sqrt[3] n+\sqrt[3]{n^2}}}= {{\begin{vmatrix} 1&\sqrt[3]n&\sqrt[3]{n^2}\\ B&A&Cn\\ C&B&A\\ \end{vmatrix}}\over{ \begin{vmatrix} A&Cn&Bn\\ B&A&Cn\\ C&B&A\\ \end{vmatrix}}}={{A^2-BCn+(C^2n-AB)\sqrt[3]n+C\sqrt[3]{n^2}}\over{A^3+B^3 n+C^3 n^2}}$.

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Solved papers for BCECE Medical BCECE Medical Solved Papers-2015

done BCECE Medical Solved Papers-2015

• question_answer1) A ball is thrown from the ground to clear a wall 3 m high at a distance of 6 m and falls 18 m away from the wall, the angle of projection of ball is

A) ${{\tan }^{-1}}\left( \frac{3}{2} \right)$

B) ${{\tan }^{-1}}\left( \frac{2}{3} \right)$

C) ${{\tan }^{-1}}\left( \frac{1}{2} \right)$

D) ${{\tan }^{-1}}\left( \frac{3}{4} \right)$

• question_answer2) Consider if the electric flux entering and leaving an enclosed surface respectively are ${{\phi }_{1}}$ and ${{\phi }_{2}}$, the electric charge inside the surface will be

A) ${{\varepsilon }_{0}}({{\phi }_{1}}+{{\phi }_{2}})$

B) $\frac{{{\phi }_{2}}+{{\phi }_{1}}}{{{\varepsilon }_{0}}}$

C) $\frac{{{\phi }_{1}}-{{\phi }_{2}}}{{{\varepsilon }_{0}}}$

D) $\frac{{{\phi }_{1}}+{{\phi }_{2}}}{{{\varepsilon }_{0}}}$

• question_answer3) If the escape speed of a projectile on Earths surface is 11.2 $km{{s}^{-1}}$ and a body is projected out with thrice this speed, then determine the speed of the body far away from the Earth.

A) 56.63 $km{{s}^{-1}}$

B) 33 $km{{s}^{-1}}$

C) 39 $km{{s}^{-1}}$

D) 31.7 $km{{s}^{-1}}$

• question_answer4) Determine the potential drop between the points A and C in the following circuit. Resistances $1\,\,\Omega$ and $2\,\,\Omega$ are representing the internal resistances of the respective cells.

A) $\frac{4}{5}V$

B) 1.75 V

C) 2.25 V

D) 5 V

• question_answer5) A tank is filled with a liquid upto a height H. A small hole is made at the bottom of this tank. Consider ${{t}_{1}}$ be the time taken to empty first half of the tank and ${{t}_{2}}$ be the time taken to empty rest half of the tank. Then, determine the ratio $\frac{{{t}_{1}}}{{{t}_{2}}}$.

A) 1.33

B) 1.5

C) 2

D) 0.414

• question_answer6) An electron and a proton enter a magnetic field perpendicularly both have same kinetic energy. Which of the following statements are true?

A) Trajectory of proton is less curved

B) Trajectory of electron is less curved

C) Both trajectories are equally curved

D) Both move on straight line path

• question_answer7) The coefficient of volume expansion of glycerine is $49\times {{10}^{-5}}{{K}^{-1}}$. What is the fractional change in its density for a ${{30}^{o}}C$ rise in temperature?

A) $1.5\times {{10}^{-2}}$

B) $2\times {{10}^{-4}}$

C) $3.5\times {{10}^{-3}}$

D) $2.5\times {{10}^{-2}}$

• question_answer8) If a conducting rod of length 4 $l$ is rotated about at point O in a uniform magnetzic field B directed into the paper and $DE=l,\,EA=3l$, then

A) ${{v}_{A}}-{{v}_{E}}=\frac{9}{2}B\omega \,{{i}^{2}}$

B) ${{v}_{E}}-{{v}_{A}}=\frac{9}{2}B\omega \,{{i}^{2}}$

C) ${{v}_{D}}-{{v}_{E}}=\frac{B\omega \,{{i}^{2}}}{2}$

D) ${{v}_{A}}-{{v}_{E}}=4B\omega \,{{i}^{2}}$

• question_answer9) An open organ pipe has a fundamental frequency of 300 Hz. The first overtone of a closed organ pipe has the same frequency as the first overtone of organ pipe. How long is each pipe?

A) 41.25 cm

B) 42.3 cm

C) 49.5 cm

D) 40.5 cm

• question_answer10) In L-C-R circuit, $f=\frac{50}{\pi }\,Hz$, V = 50 V, R = 300 $\Omega$ If L = 1 H and C = 20 $\mu C$, then the voltage across capacitor is

A) 50V

B) 20V

C) zero

D) 30 V

• question_answer11) In a car race, car A takes a time t less than car B at the finish and passes the finishing point with speed v more than that of the car B. Assuming that both the cars starts from rest and travel with constant accelerations ${{a}_{1}}$ and ${{a}_{2}}$ respectively. So, the value of v will be

A) $(\sqrt{{{a}_{1}}/{{a}_{2}}}\,)t$

B) $(\sqrt{{{a}_{2}}/{{a}_{1}}})\,t$

C) $({{a}_{1}}\sqrt{{{a}^{2}}})\,t$

D) $(\sqrt{{{a}_{1}}{{a}_{2}}})\,t$

• question_answer12) If an equiconvex lens is cut into two halves along (i) XOX and (ii) YOY as shown in the figure. Let $f,f,f$ be the focal lengths of complete lens of each, in case (i) and of each half in case (ii) respectively. Choose the correct statement from the following?

A) $f=f,\,f=f$

B) $f=2f,\,f=2f$

C) $f=f,\,f=2f$

D) $f=2f,\,f=f$

• question_answer13) The bob of a pendulum is released from < horizontal position A as shown in the figure If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lower most point B, given that it dissipated 5% of its initial energy against air resistance?

A) 5 m/s

B) 5-5 m/s

C) 5.3 m/s

D) 4.4 m/s

• question_answer14) In a photoemissive cell with executing wavelength $\lambda$, the fastest electron has speed v. If the exciting wavelength is changed to $\frac{3\lambda }{4}$the speed of fastest emitted electron will be

A) greater than$v{{\left( \frac{4}{3} \right)}^{1/2}}$

B) $v{{\left( \frac{3}{4} \right)}^{1/2}}$

C) $v{{\left( \frac{4}{3} \right)}^{1/2}}$

D) less than $v{{\left( \frac{4}{3} \right)}^{1/2}}$

• question_answer15) A ball of mass m moving at a speed v makes a head on collision with an identical ball at rest. The kinetic energy at the balls after the collision is 3/4th of the original. What is the coefficient of restitution?

A) $1/\sqrt{3}$

B) $1/\sqrt{2}$

C) $\sqrt{2}$

D) $\sqrt{3}$

• question_answer16) A radioactive material decays by simultaneous emission of two particles with half-lives 1620 yr and 810 yr respectively. The time in years after which one-fourth of material remains, is

A) 1080 yr

B) 2340 yr

C) 4860 yr

D) 3240 yr

• question_answer17) A cylinder rolls up an inclined plane at an angle of ${{30}^{o}}$. At the bottom of the inclined plane, the centre of mass of the cylinder has speed of 5 m/s. How long will it take to return to the bottom?

A) 2s

B) 3 s

C) 1.5 s

D) 4 s

• question_answer18) The current gain in the common emitter amplifier mode of a transistor is 10. The input impedance is 20 $k\Omega$ and load of resistance is 100 $k\Omega$. The power gain is

A) 300

B) 500

C) 200

D) 100

• question_answer19) What will be the displacement equation of the simple harmonic motion obtained by combining the motions? ${{x}_{1}}=2\,\sin \omega \,t$, ${{x}_{2}}=4\,\sin \left( \omega \,t+\frac{\pi }{6} \right)$ and${{x}_{3}}=6\,\sin \left( \omega \,t+\frac{\pi }{3} \right)$

A) $x=10.25\sin \,(\omega \,t+\phi )$

B) $x=10.25\sin \,(\omega \,t-\phi )$

C) $x=11.25\sin \,(\omega \,t+\phi )$

D) $x=11.25\sin \,(\omega \,t-\phi )$

• question_answer20) An ammeter A reads 2 A and the voltmeter V reads 20 V. The value of resistance R is (assuming finite resistaneces of ammeter and voltmeter)

A) exactly 10 $\Omega$.

B) less than 10 $\Omega$.

C) more than 10 $\Omega$.

D) we cannot definitely say

• question_answer21) A stream of water flowing horizontally with the speed of 15 $m{{s}^{-1}}$ gushes out of a tube of cross-sectional area ${{10}^{-2}}{{m}^{-2}}$ and hits a vertical wall nearby. What is the force exerted on the wall by the impact or water? Assuming it does not rebound.

A) 2250 N

B) 2000 N

C) 1500N

D) 1000N

• question_answer22) Two charges $+6\,\,\mu C$ and $-4\,\,\mu C$ are placed 15 cm apart as shown. At what distances from A to its right, the electrostatic potential is zero?

A) 4, 9, 60

B) 9. 15, 45

C) 20, 30, 40

D) 9, 45, infinity

• question_answer23) A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is 80 $m{{s}^{-1}}$. What is the recoil speed of the gun?

A) 1.6 $cm{{s}^{-1}}$

B) 0.5 $cm{{s}^{-1}}$

C) 2 $cm{{s}^{-1}}$

D) 3 $cm{{s}^{-1}}$

• question_answer24) If $E=100\sin (100\,t)$ volt and $I=100\sin \left( 100\,t+\frac{\pi }{3} \right)mA$ are the instantaneous voltage and current, then the rms values of voltage and current are respectively;

A) 70.7 V, 70.7 mA

B) 60.9 V, 69.3 mA

C) 90.6 V,-141.4 mA

D) 60 V, 70 mA

• question_answer25) If torques of equal magnitudes are applied to a hollow cylinder and a solid sphere both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

A) ${{\omega }_{1}}>{{\omega }_{2}}$

B) ${{\omega }_{1}}={{\omega }_{2}}$

C) ${{\omega }_{2}}>{{\omega }_{1}}$

D) None of these

• question_answer26) . The threshold frequency for certain metal is$3.3\times {{10}^{14}}Hz$. If light of $8.2\times {{10}^{14}}Hz$ frequency is incident on the metal, then the cut-off voltage of the photoelectric current will be

A) 3V

B) 4V

C) 2V

D) 2.5V

• question_answer27) If two simple pendulums first of bob mass ${{M}_{1}}$and length ${{l}_{1}}$ second of bob mass ${{M}_{2}}$ and length ${{l}_{2}}$. Given ${{M}_{1}}={{M}_{2}}$ and ${{l}_{1}}=2{{l}_{2}}$. If the vibrational energies of both are same, then which of the following is correct?

A) Amplitude of B is smaller than A

B) Amplitude of B is greater than B

C) Amplitude will be same

D) None of the above

• question_answer28) A thin film of soap solution $({{\mu }_{s}}=1.4)$ lies on the top of a glass plate $({{\mu }_{g}}=1.5)$. When visible light is incident almost normal to the plate, two adjacent reflection maxima are observed at two wavelengths 420 nm and 630 nm. The minimum thickness of the soap solution are

A) 420 nm

B) 500 nm

C) 450 nm

D) 490 nm

• question_answer29) An air bubble stats rising from the bottom of a lake. Its diameter is 3.6 mm at the bottom and 4 mm at the surface. The depth of the lake is 250 cm and the temperature at surface is${{40}^{o}}C$. What is the temperature at the bottom of the lake? Given atmpospheric pressure = 76 cm of Hg and $g=980\,\,cm/{{s}^{2}}$.

A) ${{11}^{o}}C$

B) ${{12.36}^{o}}C$

C) ${{13}^{o}}C$

D) ${{10.37}^{o}}C$

• question_answer30) If four atoms of hydrogen combine to form an 2 He atom, then the amount of energy released is

A) 26.7 MeV

B) 25.3 MeV

C) 22 MeV

D) 24.5 MeV

• question_answer31) A simple harmonic wave of amplitude 8 unit travels along positive x-axis. At any given instant of time, for a particle at a distance of 10 cm from the origin, the displacement is + 6 unit and for a particle at a distance of 25 cm from the origin, the displacement is + 4 unit. Calculate the wavelength,

A) 200 cm

B) 230 cm

C) 210 cm

D) 250 cm

• question_answer32) When a battery of emf 8 V with internal resistance $0.5\,\,\Omega$ is charged by a 120 V DC supply using a series resistance of $15.5\,\,\Omega$, then the terminal voltage of the battery is

A) 11.5 V

B) 20 V

C) 21.5 V

D) 12,3 V

• question_answer33) A rigid bar of mass 15 kg is supported symmetrically by three wires each 2 m long. Those at each end are of copper and the middle one is of iron. Determine the ratio of their diameters if each is to have the tension.

A) 12.6 : 2

B) 1.31 : 1

C) 4.65 : 3

D) 2.69 : 4

• question_answer34) Figure given below shows a network of four capacitors of capacitance equal to ${{C}_{1}}=C$, ${{C}_{2}}=2C,\,\,{{C}_{3}}=3C,\,\,{{C}_{4}}=4C$ are connected to a battery. The ratio of charges on ${{C}_{2}}$ and ${{C}_{4}}$ is

A) 7/4

B) 7/22

C) 3/22

D) 4/7

• question_answer35) Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 $km{{h}^{-1}}$m the same direction, with A head of speed B. The driver of B decides to overtake A and accelerates by $1\,m/{{s}^{-2}}$. If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them?

A) 100 m

B) 1150 m

C) 1300 m

D) 1250 m

• question_answer36) If a fully charged capacitor C with initial charge ${{q}_{0}}$ is connected to a coil of self inductance L at$t=0$. The time at which the energy is stored equally between the electric field and magnetic field is

A) $\pi \sqrt{LC}$

B) $\frac{\pi }{4}\sqrt{LC}$

C) $\frac{\pi }{2}\sqrt{LC}$

D) $\frac{\pi }{6}\sqrt{LC}$

• question_answer37) A rocket is fired from the Earth towards the Sun. At what distance from the Earths centre, the gravitational force on the rocket is zero? Mass of the Sun $=2\times {{10}^{30}}kg$ and mass of the Earth $=6\times {{10}^{24}}kg$.

A) $2.6\times {{10}^{8}}m$

B) $3.2\times {{10}^{8}}m$

C) $3.9\times {{10}^{9}}m$

D) $2.3\times {{10}^{9}}m$

• question_answer38) In the figure given below, for an angle of incidence ${{45}^{o}}$ at the top surface, what is the minimum refractive index need- to total internal reflection at vertical face?

A) $\frac{\sqrt{2}+1}{2}$

B) $\sqrt{\frac{1}{2}}$

C) $\sqrt{\frac{3}{2}}$

D) $\sqrt{2}+1$

• question_answer39) A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad ${{s}^{-1}}$. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of cylinder? What is the magnitude of angular momentum of cylinder about its axis?

A) 62.5 T-s

B) 70.4 T-s

C) 79.6 T-s

D) 60.5 T-s

• question_answer40) A change of 8.0 mA in the emitter current 1} rings a change of 7.9 mA in the correct collector current. The values of a and p are

A) 0.99, 90

B) 0.96, 79

C) 0.97, 99

D) 0.99, 79

• question_answer41) A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with speed 40 rev/min in a horizontal plane. What is the tension in the string and what is the maximum speed with which the stone can be whirled around, if the string can withstand a maximum tension of 200 N?

A) 6.6 N, 35 $m{{s}^{-1}}$

B) 6 N, 37 $m{{s}^{-1}}$

C) 7.5 N, 46 $m{{s}^{-1}}$

D) 8 N, 38 $m{{s}^{-1}}$

• question_answer42) A straight wire of mass 200 g and length 1.5 m carries a current of 2 A. It is suspended in mid air by a uniform horizontal magnetic field B. The magnitude of B (in tesla) is

A) 0.65

B) 0.55

C) 0.75

D) 0.45

• question_answer43) A U -shaped wire is dipped in a soap solution and removed. The thin soap film formed between the wire and light slider supports a weight of $1.5\times {{10}^{2}}N$. The length of the slider is 30 cm. What is the surface tension of the film?

A) $3\times {{10}^{-3}}N{{m}^{-1}}$

B) $2\times {{10}^{-5}}N{{m}^{-1}}$

C) $4\times {{10}^{-4}}N{{m}^{-1}}$

D) $2.5\times {{10}^{-2}}N{{m}^{-1}}$

• question_answer44) At constant pressure, the ratio of increase in volume of an ideal gas per degree rise in kelvin temperature to its original volume is

A) ${{7}^{2}}$

B) $\frac{1}{T}$

C) ${{T}^{3}}$

D) T

• question_answer45) The length of a magnet is large compared to its width and breadth. The time period of its oscillation in vibration magnetometer is 2 s. The magnet is cut along its length into three equal parts and three parts are then palced on each other with their like poles together. The lime period of this combination will be

A) $\frac{2}{3}s$

B) $\sqrt{\frac{2}{3}}s$

C) $\frac{3}{2}s$

D) $\sqrt{\frac{3}{2}}s$

• question_answer46) A body of mass 1 kg is moving in a vertical. circular path of radius 1 m. The difference between the kinetic energies at its highest and lowest positions is

A) $4\sqrt{5}\,\,J$

B) 20 J

C) 10 J

D) 30 J

• question_answer47) If electrical force between two charges is 200 N and we increase 10% charge on one of the charges and decrease 10% charge on the other, then electrical force between them for the same distance becomes

A) 200 N

B) 202 N

C) 198 N

D) 199 N

• question_answer48) The acceleration due to gravity on the planet is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2 m on the surface of A. What is the height of jump by the same on the planet B7

A) 6 m

B) 2/3 m

C) 2/9 m

D) 18 m

• question_answer49) In hydrogen atom spectrum, frequency of $2.7\times {{10}^{15}}Hz$ of EM wave is emitted when transmission takes place from 2 to 1. If it- moves from 3 to 1, the frequency emitted will be

A) $3.2\times {{10}^{15}}Hz$

B) $32\times {{10}^{15}}Hz$

C) $1.6\times {{10}^{15}}Hz$

D) $16\times {{10}^{15}}Hz$

• question_answer50) If a ray of light in a denser medium strikes a rarer medium at an angle of incidence 2, the angles of reflection and refraction are respectively r and r. If the reflected and refracted rays are at right angles to each other, the critical angle for the given pair of media is

A) ${{\sin }^{-1}}(\tan r)$

B) ${{\sin }^{-1}}(\tan r)$

C) ${{\tan }^{-1}}(\sin i)$

D) ${{\cot }^{-1}}(\tan i)$

• question_answer51) Consider the following reaction sequence of alkene $C{{H}_{3}}CH=CHC{{H}_{3}}\xrightarrow{{{O}_{3}}}A\xrightarrow[Zn]{{{H}_{2}}O}B$ The product B is

A) $C{{H}_{2}}C{{H}_{2}}COC{{H}_{3}}$

B) $C{{H}_{3}}COC{{H}_{3}}$

C) $C{{H}_{3}}CHO$

D) $C{{H}_{3}}C{{H}_{2}}CHO$

• question_answer52) Choose the correct statement about the industrial preparation of hydrogen from water gas $(CO+{{H}_{2}})$,

A) ${{H}_{2}}$ is removed through occlusion with Pd

B) CO and ${{H}_{2}}$ are fractionally separated using differences in their densities

C) CO is removed by absorption in aq. $C{{u}_{2}}C{{l}_{2}}$solution

D) CO is oxidised to $C{{O}_{2}}$ with steam in the presence of a catalyst followed by absorption of $C{{O}_{2}}$ in alkali

• question_answer53) The energy required to excite the electron in the atom from $n=1$ to $n=2$, when the ionization enthalpy of hydrogen atom is $1.312\times {{10}^{6}}J/mol$ will be (in the unit of ${{10}^{5}}$).

A) 8.53

B) 7.51

C) 8.92

D) 9.86

• question_answer54) An ideal mono-atomic gas follows the path ABCD. Then, the work done during the complete cycle is

A) zero

B) $-\frac{1}{2}\,pV$

C) $-pV$

D) $-2\,\,pV$

• question_answer55) From the following which pairs give the faster ${{S}_{N}}2$ reaction?

A)

B)

C)

D)

• question_answer56) The reactivity order of halogenation of alkanes is ${{F}_{2}}>C{{l}_{2}}>B{{r}_{2}}>{{I}_{2}}$. Choose the correct statements.

 I. Lower the activation energy for the chain initiation step, more reactive is the halogen. II. Lower the activation energy for the first chain propagation step, more reactive is the halogen. III. Lower the activation energy for the, second chain propagation step, more reactive is the halogen. IV. More negative is the overall heat of the reaction ($\Delta H_{r}^{o}$ of halogenation of alkane) more reactive is the halogen.

A) I and II

B) I, II and III

C) II and IV

D) II. Ill and IV

• question_answer57) What is the bond angle between $Cl-O-Cl$in$C{{l}_{2}}{{O}_{7}}$?

A) ${{109}^{o}}$

B) ${{119}^{o}}$

C) ${{108}^{o}}25$

D) ${{120}^{o}}$

• question_answer58) The solubility of fluorides of alkali metals in water is

A) LiF > NaF > KF > RbF > CsF

B) CsF > RbF > KF > NaF > LiF

C) LiF > KF > NaF > CsF > RbF

D) CsF > KF > NaF > RbF > LiF

• question_answer59) Which of the following relation is correct for gaseous and reversible reactions?

A) $\frac{{{K}_{C}}}{{{K}_{P}}}={{(RT)}^{\Delta {{n}_{g}}}}$

B) $\frac{{{K}_{p}}}{{{K}_{x}}}={{(p)}^{\Delta {{n}_{g}}}}$

C) $\frac{{{K}_{C}}}{{{K}_{X}}}={{(p)}^{-\Delta {{n}_{g}}}}$

D) $\frac{{{K}_{C}}}{{{K}_{X}}}={{\left( \frac{p}{RT} \right)}^{\Delta {{n}_{g}}}}$

• question_answer60) Consider the following sequence of reactions,

A)

B)

C)

D)

• question_answer61) If the dipole moment of HBr is $2.60\times {{10}^{-30}}Cm$and the interatomic spacing is 1.41 $\overset{o}{\mathop{A}}\,$, then the per cent ionic character of HBr is

A) 16.23%

B) 13.21%

C) 11.50%

D) 15.81%

• question_answer62) Consider the following reaction, $CC{{l}_{3}}CH=C{{H}_{2}}\xrightarrow[{{H}_{2}}O/HOCl]{NaB{{O}_{4}}}$ Product (P) is

A) $CC{{l}_{3}}\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,HC{{H}_{2}}Cl$

B) $CC{{l}_{3}}\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,H\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,{{H}_{2}}$

C) $CC{{l}_{3}}\underset{\begin{smallmatrix} | \\ Cl \end{smallmatrix}}{\mathop{C}}\,H-C{{H}_{2}}OH$

D) $CC{{l}_{3}}\underset{\begin{smallmatrix} | \\ Cl \end{smallmatrix}}{\mathop{C}}\,H-\underset{\begin{smallmatrix} | \\ Cl \end{smallmatrix}}{\mathop{C}}\,{{H}_{2}}$

• question_answer63) Among ${{[Ni{{(CN)}_{4}}]}^{2-}},\,{{[NiC{{l}_{4}}]}^{2-}}$ and $[Ni\,\,{{(CO)}_{4}}]$

A) ${{[NiC{{l}_{4}}]}^{2-}}$ is square planar and ${{[Ni{{(CN)}_{4}}]}^{2-}},\,Ni{{(CO)}_{4}}$ are tetrahedral

B) $Ni\,{{(CO)}_{4}}$ is square planar and ${{[Ni\,{{(CN)}_{4}}]}^{2-}},\,{{[NiC{{l}_{4}}]}^{2-}}$ are tetrahedral

C) ${{[Ni\,{{(CN)}_{4}}]}^{2-}}$ is square planar and $[Ni\,{{(C{{l}_{4}}]}^{2-}},Ni{{(CO)}_{4}}$ are tetrahedral

D) None of the above

• question_answer64) Match the following Column I with Column II.

 Column - I Column - II [a] Acetic acid and methyl formate 1. Ring chain tautomerism [b] Maleic and fumaric acid 2. Functional isomers [c] 3. Geometrical isomers [d] 4. Metamerism

A) A- 2 B-3 C-1 D-4

B) A-2 B-1 C-3 D-4

C) A- 4 B-2 C-1 D- 3

D) A-4 B-1 C-2 D-3

• question_answer65) Beryllium gives a compound X with the following percentage composition: Be-6.1%, N-37.8%, Cl-48%, H-8.1%. Molecular weight of X is $148\,\,g\,mo{{l}^{-1}}$ and that of Be is $9\,\,g\,mo{{l}^{-1}}$. The molecular formula of the compound is

A) $Be{{N}_{4}}C{{l}_{2}}{{H}_{12}}$

B) $Be{{N}_{2}}Cl{{H}_{6}}$

C) $Be{{N}_{4}}C{{l}_{2}}{{H}_{6}}$

D) $Be{{N}_{4}}Cl{{H}_{8}}$

• question_answer66) The increasing dipole moments of the following compounds is

A) I < III < IV < II

B) II = IV < III < I

C) II = I < IV < III

D)  III < I < II < IV

• question_answer67) Compound A undergoes following reactions to form B and C Black coloured solid [A] $\xrightarrow{KN{{O}_{3}}+KOH}$ Green coloured solution [B] Green coloured $\xrightarrow{C{{O}_{2}}}$ Pink solution[C] + [A] Solution [B] [C] decolourised by $F{{e}^{3+}}$ Identify A, B and C respectively

A) $Mn{{O}_{2}},\,KMn{{O}_{4}}$ and ${{K}_{2}}Mn{{O}_{4}}$

B) $Mn{{O}_{2}},\,{{K}_{2}}Mn{{O}_{4}}$ and $KMn{{O}_{4}}$

C) $KMn{{O}_{4}},\,Mn{{O}_{2}}$ and ${{K}_{2}}Mn{{O}_{4}}$

D) $KMn{{O}_{4}},\,{{K}_{2}}Mn{{O}_{4}}$ and $Mn{{O}_{2}}$

• question_answer68) The reaction between nitric oxide (NO) and oxygen to form nitrogen dioxide $(N{{O}_{2}})$ is a key step in photochemical smog formation $2NO(g)+{{O}_{2}}(g)\xrightarrow{{}}2N{{O}_{2}}(g)$ The grams of $N{{O}_{2}}$ are formed by complete by reaction of 1.20 g of NO is

A) 7.2g

B) 2:3 g

C) 1.8g

D) 4.1 g

• question_answer69) $2HC\equiv CH\,\,\xrightarrow[+N{{H}_{4}}Cl]{C{{H}_{2}}C{{l}_{2}}}\,A\,\,\xrightarrow[HCl]{1\,\,mol}\,\,B$ Identify the compounds A and B.

A)

B)

C)

D)

• question_answer70) At ${{80}^{o}}C$, the vapour pressure of pure liquid A is 520 mm Hg and that of pure liquid B is 1000 mm Hg. If a mixture solution of A and B boils at ${{80}^{o}}C$and 1 atm pressure, the amount of A in the mixture is

A) 50 mol per cent

B) 54 mol per cent

C) 32 mol per cent

D) 44 mol per cent

• question_answer71) The correct increasing order of solubility of the sulphates in water is

A) $BeS{{O}_{4}}>MgS{{O}_{4}}>CaS{{O}_{4}}>SrS{{O}_{4}}>BaS{{O}_{4}}$

B) $BeS{{O}_{4}}>SrS{{O}_{4}}>CaS{{O}_{4}}>MgS{{O}_{4}}>BeS{{O}_{4}}$

C) $BeS{{O}_{4}}>CaS{{O}_{4}}>SrS{{O}_{4}}>BeS{{O}_{4}}>MgS{{O}_{4}}$

D) $BaS{{O}_{4}}>SrS{{O}_{4}}>MgS{{O}_{4}}>CaS{{O}_{4}}>BeS{{O}_{4}}$

• question_answer72) Consider the following reaction, $ZnC{{l}_{2}}+NaOH\xrightarrow{{}}\underset{White\text{ }ppt.}{\mathop{B}}\,\xrightarrow{NaOH}\underset{Soluble}{\mathop{C}}\,$ Then, the final product C is

A) $N{{a}_{2}}Zn{{O}_{2}}$

B) $Zn{{(OH)}_{2}}$

C) ZnO

D) $ZnS{{O}_{4}}$

• question_answer73) The vapour pressure of pure water is 23.5 mm Hg. Then, the vapour pressure of an aqueous solution which contains 5 mass percent of urea is (Molar mass of urea is 60).

A) 23 mm Hg

B) 18 mm Hg

C) 31 mm Hg

D) 35 mm Hg

• question_answer74) Which of the following organic chloro compound shows stereochemical inversion during a ${{S}_{N}}2$ reaction?

A) ${{(C{{H}_{3}})}_{3}}CCl$

B) ${{(C{{H}_{3}})}_{2}}CHCl$

C) ${{({{C}_{2}}{{H}_{5}})}_{2}}CHCl$

D) $C{{H}_{3}}Cl$

• question_answer75) Consider the following reactions, $C\,(s)+{{O}_{2}}(g)\xrightarrow{{}}C{{O}_{2}}(g),\,\Delta H=-94\,kcal$ $2CO\,(g)+{{O}_{2}}\xrightarrow{{}}2C{{O}_{2}}(g),\,\Delta H=-135.2\,kcal$ Then, the heat of formation of CO (g) is

A) 26.4 kcal

B) -26.4 kcal

C) 41.2 kcal

D) -41.2 kcal

• question_answer76) If the $p{{K}_{a}}$ of a weak acid HA is 4.80 and the$p{{K}_{b}}$ , of a weak base BOH is 4.78. Then, the pH of an aqueous solution of the corresponding salt, BA will be

A) 8.23

B) 9.41

C) 7.01

D) 5.91

• question_answer77) Which one of the following is the correct statement?

A) ${{B}_{3}}{{N}_{3}}{{H}_{6}}$ is known as inorganic benzene

B) Chlorides of both beryllium and aluminium have bridged chloride structure in gas phase

C) Boric acid is a protonic acid

D) Beryllium exhibits coordination number of six

• question_answer78) A nitrogenous substance A is treated with$HN{{O}_{2}}$ and the product formed is further treated with $NaOH$ solution which produces blue colouration. Then, A will be

A) $C{{H}_{3}}C{{H}_{2}}N{{H}_{2}}$

B) ${{(C{{H}_{3}})}_{2}}CHN{{O}_{2}}$

C) $C{{H}_{3}}C{{H}_{2}}N{{O}_{2}}$

D) $C{{H}_{3}}C{{H}_{2}}ONO$

• question_answer79) Argon crystallises in fee arrangement and the density of solid and liquid Ar is $1.59\,\,g/c{{m}^{3}}$ and $1.42\,\,g/c{{m}^{3}}$, respectively. The percentage of empty space in liquid Ar is

A) 34.84%

B) 43.8%

C) 23.4%

D) 21.6%

• question_answer80) Which one of the following is an exothermic reaction?

A) ${{N}_{2}}(g)+{{O}_{2}}(g)+180.8\,kJ\xrightarrow{{}}2NO\,(g)$

B) $C\,(graphite)+2S\,(s)\xrightarrow{{}}C{{S}_{2}}(l)-91.9\,kJ$

C) ${{N}_{2}}\,(g)+3{{H}_{2}}(g)-92\,kJ\xrightarrow{{}}2N{{H}_{3}}(g)$

D) $C\,(g)+{{H}_{2}}O(g)\xrightarrow{{}}CO\,(g)+{{H}_{2}}(g)-131.1\,kJ$

• question_answer81) When 4A of current is passed through a 1.0 L, 0.10 M $F{{e}^{3+}}(aq)$ solution for 1 h, it is partly reduced to Fe(s) and partly of $F{{e}^{2+}}(aq)$. Identify the incorrect statement.

A) 0.10 mole of electrons are required to convert all $F{{e}^{3+}}$ to $F{{e}^{2+}}$

B) 0.025 mol of Fe(s) will be deposited

C) 0.075 mol of iron remains as $F{{e}^{2+}}$

D) 0.050 mol of iron remains as $F{{e}^{2+}}$

• question_answer82) Which of the following statement are correct?

A) $C{{O}_{2}}$ and $Si{{O}_{2}}$ both are gases

B) $C{{O}_{2}}$ is gas and $Si{{O}_{2}}$ is a liquid

C) $C{{O}_{2}}$ is a gas but $Si{{O}_{2}}$ is a solid

D) Carbon and Si both form oxides of similar properties

• question_answer83) Identify the product B of the following reactions

A)

B)

C)

D)

• question_answer84) Given that, $E_{F{{e}^{2+}}/Fe}^{o}=-0.44\,\,V$, $E_{F{{e}^{2+}}/F{{e}^{3+}}}^{o}=-0.77\,\,V$. If $F{{e}^{2+}},F{{e}^{3+}}$ and Fe solid are kept together, then

A) $F{{e}^{3+}}$ decreases

B) $F{{e}^{3+}}$ increases

C) $F{{e}^{2+}}$ decreases

D) $F{{e}^{2+}},F{{e}^{3+}}$ remain unchanged

• question_answer85) Which of the following compound in its anionic form is aromatic?

A)

B)

C)

D)

• question_answer86) In terms of Arrhenius equation, $k=A{{e}^{-{{E}_{a}}RT}}$the temperature dependence of rate constant (k) of a chemical reaction is written. Then, the activation energy $({{E}_{a}})$ of the reaction can be calculated by plotting.

A) $\log \,k\,vs\frac{1}{\log \,T}$

B) $\log \,k\,vs\frac{1}{T}$

C) $k\,vs\,\,T$

D) $k\,vs\,\,\frac{1}{\log \,T}$

• question_answer87) Which one of the following electrolytes is most effective for the coagulation of $Fe{{(OH)}_{3}}$?

A) NaCI

B) $N{{a}_{2}}S{{O}_{4}}$

C) $N{{a}_{3}}P{{O}_{4}}$

D) $A{{s}_{2}}{{S}_{3}}$

• question_answer88) $PhC{{H}_{2}}C{{H}_{3}}\xrightarrow[(ii)\,{{H}_{3}}{{O}^{+}}]{(i)\,Cr{{O}_{2}}C{{l}_{2}}/CC{{l}_{4}}}PhC{{H}_{2}}CHO$ The above reaction is an example of which, name reaction?

A) Rossenmund reduction

B) Birch reduction

C) Mendius reduction

D) Etard reduction

• question_answer89) Which of the following statement is applicable for Tyndall effect?

A) The diameter of the dispersed particle is much smaller than the wavelength of the light used

B) The diameter of the dispersed phase is much smaller than the wavelength of the light used

C) The refractive indices of the dispersed phase and the dispersion medium must be same

D) The refractive indices of the dispersed phase and the dispersion medium must differ greatly in magnitude

• question_answer90) Match the following and choose the correct option.

 Column I Column II A. Ribose 1. Ninhydrin B. Biotin 2. Nucleic acid C. Enzyme 3. Vitamin D. Glycine 4. GIobular protein

A) A-1 B-4 C-2 D- 1

B) A-2 B-3 C-4 D-1

C) A-1 B-2 C-3 D-4

D) A-2 B-3 C-1 D-4

• question_answer91) On reduction of glycolic acid with HI, the product formed is

A) acetic acid

B) iodo acetic acid

C) formic acid

D) None of these

• question_answer92) Consider the following reaction, Identify the product B.

A)

B)

C)

D)

• question_answer93) Arrange the following compounds in the increasing order of their reactivity towards HCN.

 I. Acetaldehyde II. Acetone III. Di-tert-butylketone IV. Methyl-tert-butyl ketone

A) III < II < IV < I

B) II <1 < IV < II

C) IV < III < II < I

D) II < IV < I < III

• question_answer94) The enolic form of acetone contains

A) 10 $\sigma$-bonds, $2\pi$ -bonds and 1 lone pair

B) 9 $\sigma$-bonds, $2\pi$-bonds and 2 lone pairs

C) 8 $\sigma$-bonds, $2\pi$-b0nds and 2 lone pairs

D) 9 $\sigma$-bonds, $2\pi$-bond and 2 lone pairs

• question_answer95) The magnetic moment of ${{M}^{x\,+}}$ (atomic number = 25) is $\sqrt{15}\,BM$. Then, the oxidation number $x$ of M is

A) 3

B) 4

C) 2

D) 1

• question_answer96) When 0.1 M NaOH is titrated with 0.1 M, 20 mL HA till the end point, ${{K}_{a}}(HA)=6\times {{10}^{-6}}$ and degree of dissociation of HA is small as compared to unity. What is the pH of the resulting solution at the end point?

A) 6.23

B) 9.22

C) 7.21

D) 8.95

• question_answer97) Arrange the following in the decreasing order of basic character.

 I. p-toluidine II. N, N-dimethyl-p-toluidine III. p-nitroaniline IV. Aniline

A) II > III > I > IV

B) I > II > III > IV

C) II > I > IV > III

D) IV > III > I > II

• question_answer98) What is the enthalpy of the disproportionation of $MgCl$ if the enthalpy of formation of hypothetical $MgCl$ is $-125\,kJ/mol$ and the $MgC{{l}_{2}}$ is $-642\,kJ/mol$?

A) - 767 kJ/mol

B) 767 kJ/mol

C) -392 kJ/mol

D) 392 kJ/mol

• question_answer99) The decreasing order of reactivity towards electrophilic addition of the following is

 I. $CH=CH$ II. $C{{H}_{2}}=C{{H}_{2}}$ III. ${{H}_{2}}C=CH=Cl$ IV. $H\underset{\begin{smallmatrix} | \\ Cl \end{smallmatrix}}{\mathop{C}}\,=\underset{\begin{smallmatrix} | \\ Cl \end{smallmatrix}}{\mathop{C}}\,H$

A) I > II > 111 > IV

B) ll > 1 > 111 > IV

C) IV > III > I > II

D) IV > III > I! > I

• question_answer100) Given the following bond dissociation enthalpies $(kJ\,\,mo{{l}^{-1}})$. $P\equiv P-490,\,P-P=209,\,N\equiv N-946,N-N-160$Compare the enthalpy changes for the process (i) $2{{P}_{2}}(g)\xrightarrow{{}}{{P}_{4}}(g)$ (ii) $2{{N}_{2}}(g)\xrightarrow{{}}{{N}_{4}}(g)$ Choose the correct option.

A) ${{P}_{4}}$ is feasible but ${{N}_{4}}$is hot feasible

B) ${{P}_{4}}$ is not feasible but ${{N}_{4}}$ is feasible

C) Both ${{P}_{4}}$ and ${{N}_{4}}$ are feasible

D) Both ${{P}_{4}}$ and ${{N}_{4}}$are not feasible

• question_answer101) Stelar tissues like xylem and phloem, etc, arise from which of the following

A) protpderm

B) cambium

C) procambium

D) lateral meristem

• question_answer102) Chlamydomonas and Chlorella, earlier placed in algae within plants (both having cell wall) are now placed together in which new kingdom. Choose the correct option.

A) Fungi

B) Animalia

C) Protista

D) Monera

A) Chordates are bilaterally symmetrical.

B) Notochord is present at some stage of life.

C) Non-segmented, false segmented or true metamerically segmented animals.

D) Gill clefts present at some stage of life.

• question_answer104) Out of the following scientific names given below, identify the option written incorrectly.

A) Mangifera.indica

B) Trypanosoma gambiense

C) Apis indica

D) Leo panther

• question_answer105) The most commonly occurring species of an earthworm in India is

A) Pheretima communissima

B) Eutyphacus

C) Pheretima posthuma

D) Lumbricus

• question_answer106) Which type of ribosomal subunit are found in prokaryotic cells?

A) 80 S (60 S + 40 S)

B) 80 S (30 S + 50 S)

C) 70 S (35 S + 35 S)

D) 70 S (50 S + 30 S)

• question_answer107) Identify the correct statement about the fluid mosaic model of cell membrane.

A) It was proposed by J David Robertson in 1972.

B) Model resembles the iceberg (as proteins) floating in the sea (of phospholipids).

C) Hydrophilic phosphate head of phospholipids face inwards.

D) Hydrocarbon tails face outward.

• question_answer108) If more than one nucleoli are seen in the nucleus of cell. What does it corresponds to?

A) Presence of false nucleoli

B) Absence of true nucleoli

C) Breakage in nucleolus

D) None of the above

• question_answer109) Polytene or salivary gland chromosome (giant chromosomes) were first observed by

A) Du Praw

B) Balbiani

C) Hemming

D) Ruckert

• question_answer110) Synaptonemal complex is a structure of

A) cytokinesis

B) terminalisation

C) chromosomal disfunction

D) chromosomal pairing

A) sphingosine

B) sphingomyelins

C) sphingbphospholipids

D) cerebroside

• question_answer112) Which of the following acts as an isomerase enzyme?

A) Phosphoglucomutase

B) Fumerases

C) Transphosphorylases

D) Nitrate reductase

• question_answer113) Addition of a competitive inhibitor to any reaction mixture will

A) decrease km

B) decrease ${{V}_{\max }}$

C) increase km

D) increase ${{V}_{\max }}$

• question_answer114) The value of water potential for any given solute is always

A) insignificant

B) negative

C) slightly negative

D) positive

• question_answer115) Radioactive $C{{O}_{2}}{{(}^{14}}C{{O}_{2}})$ is added to the suspension of a photosynthesising green algae. Which compound of the following will be labelled first with $^{14}C{{O}_{2}}$?

A) Glucose

B) RuBP

C) GP(PGA)

D) Triose phosphate

• question_answer116) ATP is the most important molecule released during all the respiratory cycles. Which one of the following is true about ATP?

A) ATP is prosthetic part of an enzyme

B) ATP is an enzyme

C) ATP is an organic Jon of an enzyme

D) ATP is a coenzyme

• question_answer117) Morphogenesis inducing hormone used in plant tissue culture is

A) abscisic acid

B) gibberellins

C) cytokinins

D) 2, 4-D

• question_answer118) Specialised phagocytic cells of liver present in sinusoidal wall are

A) macrophages

B) nematocytes

C) kupffer cells

D) oxyntic cells

• question_answer119) Bundle of His comprises a network of

A) nerve fibres distributed throughout ventricles

B) nerve fibres present in ventricles

C) muscle fibres present throughout the heart walls

D) muscle fibres distributed in ventricle wall only

• question_answer120) Trigohe in excretory system is

A) a urine filled cavity within the kidney

B) a muscular sphincter at the neck of the urinary bladder

C) a smooth connective tissue region in the urinary bladder

D) a tunic of the ureter

• question_answer121) A fractured coracoid process would involve the

A) clavicle

B) scapula

C) ulna

• question_answer122) A hormone which also acts as a neurotransmitter is

A) vasopressin

B) epinephrine

C) thyroxine

D) insulin

• question_answer123) In which phase of cell division will you observe the phenomenon of congression?

A) Prophase

B) Metaphase

C) Anaphase

D) Telophase

• question_answer124) A person met with an accident and was diagonsed with memory loss. Choose the correct option for the part of brain injured in the accident.

A) Hippocampus

B) Diencephalon

C) Cerebrum

D) Amygdala

• question_answer125) The inactive form of the gastric enzymes is

A) argentaffin

B) zymogens

C) oxyntic

D) None of these

• question_answer126) Native people of plain areas moves to hilly areas, after a few months will have

A) physically unfit to indulge in high energy games

B) have normal blood (RBC) count with Hb having high binding affinity to ${{O}_{2}}$

C) have increased count of RBC, Hb shows decreased affinity to ${{O}_{2}}$

D) suffer from nausea, fatigue and other symptoms of altitude sickness

• question_answer127) Eggs having yolk sac in their centre and cytoplasm in peripheral layer are called

A) homolecithai

B) meioiecithal

C) centrolecithal

D) telolecithal

• question_answer128) Which of the following is a diploid cell?

A) Spermatid

B) Secondary spermatocyte

C) Spermatogonium

D) Spermatozoa

• question_answer129) Given below are few comments regarding Medical Termination of Pregnancy (MTP). Identify the true statement.

A) Government of India legalised MTP in 1970.

B) MTPs, are considered relatively safe up to 12 weeks of pregnancy.

C) Majority of MTPs are legally performed by qualified doctors to an abort healthy child.

D) Amniocentesis demotivates people to carry on MTP.

• question_answer130) Pasteurs experiment to refute the theory of spontaneous generation was successful spontaneous generation did not occur in his set up because

A) he did not boil the flask for long

B) the S-shaped flask used by Pasteur prevented the contamination of broth by air

C) extreme precautions were used by him to kept his flask covered

D) he used fine muslin cloths to cover his experimental set up

A) Proterospongia

B) Neopilina

C) Archaeopteryx

D) Penpatus

• question_answer132) Dihybrid test-cross results in ratio of

A) 9 : 3 : 3 : 1

B) 1 : 2 : 4 : 1

C) 3 : 1 : 1 :3

D) 1 : 2

• question_answer133) The direct ancestor of living modern man is

A) Cro-magnon man

B) Neanderthal .man

C) Heidelberg man

D) Homo habilis

• question_answer134) To identify the genotype of one of the parents, which cross is most appropriate?

A) Test cross

B) Back cross

C) Either [a] or [b]

D) None of these

• question_answer135) The flow of genetic information is correctly indicated by

A) DNA $\to$mRNA $\to$ RNA $\to$ proteins

B) DNA $\to$ mRNA $\to$ proteins

C) RNA $\to$ DNA $\to$ proteins

D) mRNA $\to$ DNA $\to$ proteins

• question_answer136) Who among the following is not involved in the formation of a eukaryotic transcription initiation complex?

A) Transcription factors

B) SnRNA

C) RNA polymerase-11

D) Promoter

• question_answer137) An individual contracted an allergic substance and started wheezing and sneezing. In response the bodys cell secrete some chemicals against allergens, which among those given below are?

A) Melatonin

B) Histamine

C) Sebum

D) Threonine

• question_answer138) Malignant cancer cells can cross the cellular barriers and spread themselves, causing cancer in any part of the body, this property of cancer cells is

A) contact inhibition

B) metastasis

C) malignancy

• question_answer139) LSD, morphine and bhang are derived from

A) Theobroma, Papaver and Cannabis

B) Claviceps, Papaver and Cannabis

C) Cannabis, Erythroxylon and Theobroma

D) Rauwolfia, Cannabis and Papaver

• question_answer140) Splenic fever occurs in catties, goats, camel and sheep etc., and is caused by

A) Corynebactehum pyogenes

B) Cowpox virus

C) Klebsiella pneumoniae

D) Bacillus anthracis

• question_answer141) Triticale the first man made cereal is a cross between

A) wheat and millet

B) wheat and Sorghum

C) wheat and Brassica

D) wheat and rye

• question_answer142) After an organ transplant operation, a doctor prescribed an immuno suppressant drug to the patient. Which of the following would thus be included in for this purpose?

A) Streptokinase

B) Cyclosporin-A

C) Statins

D) Cephalosporins

• question_answer143) The antisense construct of ACC synthase gene (RNA) was used in production of

A) Bt maize

B) Bt cotton

C) Golden rice

D) Flavr savr (tomato)

• question_answer144) T-DNA, extensively used in genetic engineering procedures in extracted from

A) Bacillus thermus aquaticus

B) Caenorhabditis elegans

C) Agrobacterium tumefaciens

D) Bacillus thuringiensis

• question_answer145) Persistent notochord is the feature of which among the following?

A) Chondrichthyes

B) Cyclostomata

C) Loa loa

D) None of these

• question_answer146) The most productive ecosystem is

A) ocean

B) desert

C) grasslands

D) temperate forests

• question_answer147) Critically Endangered (CR) species are

A) whose last individual has died

B) liable to become extinct if not allowed to realise their full biotic potential

C) those facing an extremely high risk of extinction in immediate future

D) low risk species

• question_answer148) Indias first national park was established in which year and known as?

A) 1970. Hailey National Park

B) 1974, Bandipur National Park

C) 1975, Gir National Park

D) 1935. Jim Corbett National Park

• question_answer149) In which zone of the biosphere, no human activity is permitted?

A) Buffer zone

B) Core zone

C) Manipulation zone

D) All of these

• question_answer150) The thinning of egg shell and their premature breaking is causing a decline in bird population in certain areas. The probable reason for this could be

A) biomagnification of DDT

B) eutrophication by fertilisers

C) excessive use of organic manures

D) faulty calcium metabolism in birds

format: plain


text:
I have forgotten

• https://me.yahoo.com
COST (GBP)
4.99
5.10
0

Random Sample

viewed 3301 times and licensed 56 times
Generates random numbers following a standard gamma distribution.
Controller: CodeCogs

C++
Excel

Class RandomSample

The general formula for the probability density function of the gamma distribution is where is the shape parameter, is the location parameter, is the scale parameter, is the gamma function.

The case where and is called the standard gamma distribution. Therefore the density of this particular distribution, which we also use with this random number generator is

Using this class, the diagram below is generated from two distinct sequences of 1000 random numbers. Each pair of numbers are plotted against each other, to illustrate the standard gamma behaviour of this non-uniform random number generator.

Speed:

The average running time for generating 100,000,000 random numbers using this class on a 750MHz microprocessor is 38 seconds.

References:

• NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/
• The Newran03 random number generator library of Robert Davies, http://www.robertnz.net/nr03doc.htm

Example 1

The following example displays 40 random floating point numbers from a standard gamma distribution. It uses two different generators to achieve this. The first generator uses a particular value to initialize the seed, while the second one is using the system timer. Notice that it was necessary to divide the timer with the MERSENNEDIV value in order to keep the seed in the (0, 1) interval. Since the seed of the first generator is never changed, the first 20 numbers will always remain the same. However since the second generator is initialized via the system timer, the next 20 numbers will obviously vary with each execution of the program,
#include <iostream>
#include <time.h>
#include <codecogs/stats/dists/continuous/gamma/randomsample.h>
using namespace std;

int main()
{
Stats::Dists::Continuous::Gamma::RandomSample A(3.71, 0.175);
Stats::Dists::Continuous::Gamma::RandomSample B(4.85, time(0) / MERSENNEDIV);

for (int i = 0; i < 20; ++i)
cout << A.genReal() << endl;
cout << endl;

for (int i = 0; i < 20; ++i)
cout << B.genReal() << endl;
return 0;
}
Below you will find 20 numbers corresponding to the output of the first generator :
6.87529
7.7321
4.06328
5.90752
9.7893
10.2651
3.52183
3.73508
2.43065
0.505826
1.67514
2.52252
4.4189
2.71961
2.33248
4.09904
4.40664
4.07805
1.78729
3.71478

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Unknown

 RandomSample::RandomSampleunknown( double alpha double s )
Constructor that sets up the class variables and initializes the associated random number generator with the given seed.
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Sample

 doublesample( double alpha double seed = 0 )
This function is a simple wrapper around the randomsample class provided in this module. It uses a static to keep a single instance of this class, so that each call to this function returns a new random number. As a result this function is not necessarily thread safe, in the sense that with identical initial seed, the sequence of random numbers may differ on a multitasking or multi threaded system.

The seed is only set on the first call to this function. Thereafter this parameter is ignored. If you do no want to set the seed, then we suggest you use the system clock the first time you call this function, i.e.
#include <time.h>
...
sample(alpha, time(0) / MERSENNEDIV);
If you require more advance behaviour, we strongly recommend that directly use the underlying class randomsample that is provided with this module.

Parameters

 seed sets the initial seed for the random generator. Only used in the first call to this function
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


text:
# Tag Info

## Hot answers tagged matrices

3

Hint Consider the eigenvalues of $A \in SO(3, \Bbb R)$. In particular, (1) their product is $\det A = 1$, (2) they all have modulus $1$, and (3) any nonreal eigenvalues come in complex conjugate pairs. Now, if $A \neq I$, what can you say about the $1$-eigenspace of $A$?

3

If $A$ is nonsingular, then $AA^{-1} = I$, so $$1 = ||I|| = ||AA^{-1}|| \leqslant ||A||\cdot||A^{-1}||.$$ In general, then $1 \leqslant ||A||\cdot||A^{-1}|| \implies ||A||^{-1} \leqslant ||A^{-1}||$. Equality is thus not necessarily guaranteed for arbitrary nonsingular $A$; however, the inequality above implies that equality may occur. Consider an ...

2

A direct argument may be obtained by arguing by induction. You should try the small cases $n = 2, 3, 4$ to see what this means in practice. Fix $k$. Say that a matrix $B$ satisfies condition null-$k$ if $b_{ij} = 0$ if $i \ge j - k$. (Clearly $B = A$ satisfies null-$0$.) Suppose $B$ satisfies null-$k$. Consider the product $C = A B$, and let $c_{ij}$ be ...

2

$$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$

1

The characteristic polynomial of a linear map is defined without any additional structure beyond that of a finite dimensional vector space while the notion of unitarity or normality requires the structure of an inner product so you should be suspicious if someone tells you that the characteristic polynomial tells you something about unitarity or normality. ...

1

This is obvious using Hamilton-Cayley's theorem: the only eigenvalue of such a matrix is $0$, since its characteristic polynomial is $\chi_A(x)=(-1)^nx^n$. By Hamilton-Cayley, $$\chi_A(A)=0=(-1)^nA^n.$$

1

The rank is 2 (the first two columns span the image) and trace is zero. If there is one non-zero eigenvalue $a$, there is another non-zero eigenvalue $-a$. Hence the image would be spanned by the two eigenvectors of the non-zero eigenvalues and the same would automatically hold for all powers of $A$. But the first column is contained in the kernel and in the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Does this set contain the given integer?

Let $a = 31$. Consider the set of integers $T = \{a, 8a, 8^{2}a, 8^{3}a, \cdots \}$. Does $T$ contain the integer:

$999999999900000000000090909090000000000000000008$?

So far I've deduced that if we work mod $9$ that the set $T$ can be reduced to a reduced residue system modulo $9$ by using $8$ as a root. Additionally, because $(a, 9) = (31, 9) = 1$ we can eliminate $a$ from the elements of $T$.

Continuing, remark that $ord_{9}(2) = \phi(9) = 6$ and $ord_{9}(2^{3}) = \frac{6}{3} = 2$. Taking $8^{k} \mod 9$ for $k \in \mathbb{Z^{+}}$ we see that we get the set of reduced residues $\{8, 1\}$. $8$ is an element of this set so $T$ contains $999\ldots 0008$.

That's where I am so far but I have a feeling that I went wrong early on in this one.

-
Can whoever just downvoted my question explain the downvote? –  user39898 Dec 17 '12 at 19:22
I don't know, but maybe is because it looks too exaggerated to be an actual question and you show no own work on it? This can be solved sharing some of your own efforts, insights and self work in this problem and, perhaps, adding some background of where/how did this question come from. –  DonAntonio Dec 17 '12 at 19:25
I suppose...it was downvoted like 10 seconds after I uploaded it though so whoever downvoted it didn't even bother to think about it. –  user39898 Dec 17 '12 at 19:26

Hint

$$8 \equiv -1 \pmod 9$$ $$31 \equiv 4 \pmod 9$$

Thus

$$8^k a \equiv ??? \pmod 9$$

-
I'm sure you're on the right track but could you explain where you're coming from? I'm not seeing your thought process here. –  user39898 Dec 17 '12 at 19:25
@decave Your number is huge, but the "huge" part contains only 9's and 0's, so it is natural to look at it mod 9... Modulo 9, the question you ask implies "Can $8^ka \equiv 8 \pmod 9$?" –  N. S. Dec 17 '12 at 19:29
@N.S: Amusing! :) What if the number was 999999999900000000000090909090000000000000000004? –  Isomorphism Dec 17 '12 at 19:30
So we know that $9 \mid 99999\ldots00090909\ldots 0000$? I guess you could just factor out a 9....wow...I really need to learn to work better under pressure haha. –  user39898 Dec 17 '12 at 19:31
@Isomorphism That's why I said implies... If the number was what you said, that one needs to find a differenta approach, because that is a DIFFERENT problem ;) –  N. S. Dec 17 '12 at 19:31

The largest power of $8$ that divides our number is $8^1$. Our number is not $(8)(31)$.

Note that the only thing that was used is that our number ends in $008$.

-
Also, the number is not divisible by $31$. It is, in fact, $8 \times 220654023912941 \times 566497713347021146672084609906661$ where the last two factors are primes. –  Robert Israel Dec 17 '12 at 19:49
But $31$ is further than I can comfortably count, even after taking my shoes off. –  André Nicolas Dec 17 '12 at 19:53


text:
Direction.

Study the following graph carefully and answer the questions given below it.
% profit Earned by a company over the years Per Cent Profit = (Income Expenditure) / expenditure x 100.

# The per cent profit earned by the company in the year 2009 was what percent of the per cent profit earned by the company in the year 2005?

200

240

260

255

Please do not use chat terms. Example: avoid using "grt" instead of "great".


text:
# American Institute of Mathematical Sciences

• Previous Article
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• JCD Home
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Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations
June  2019, 6(1): 39-68. doi: 10.3934/jcd.2019002

## Mori-Zwanzig reduced models for uncertainty quantification

 Pacific Northwest National Laboratory, Richland, WA 99354, USA

* Corresponding author: Panos Stinis

Published  February 2019

Fund Project: This research at Pacific Northwest National Laboratory (PNNL) was partially supported by the U.S. Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4), under Award Number DE-SC0009280 and partially by the U.S. DOE ASCR project "Uncertainty Quantification For Complex Systems Described by Stochastic Partial Differential Equations". PNNL is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.

In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions.

Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which is better suited for reduced models with long memory. The reformulation can be used as a starting point for approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.

Citation: Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2019, 6 (1) : 39-68. doi: 10.3934/jcd.2019002
##### References:

show all references

##### References:
Linear ODE: Evolution of the memory kernel $(Le^{tQL}QLu_1,h^{01})$ (see text for details)
Linear ODE: Evolution of the resolved variables $u_0,u_1$ predicted by the full model (black line), the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)
Linear ODE: Relative error with respect to the true solution for the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)
Nonlinear ODE: Evolution of the resolved variables $u_0,u_1$ predicted by the full model (black line), the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)
Nonlinear ODE: Logarithimic scale relative error for $u_0,u_1$ with respect to the true solution for the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)
Burgers equation: Evolution of the mean of the energy of the solution using only the first two Legendre polynomials
Burgers equation: Evolution of the standard deviation of the energy of the solution using only the first two Legendre polynomials
Burgers equation: Evolution of the mean of the squared $l_2$ norm of the gradient of the solution calculated using only the first two Legendre polynomials
Burgers equation: Evolution of the standard deviation of the squared $l_2$ norm of the gradient of the solution calculated using only the first two Legendre polynomials
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Impact Factor:


text:
시간 제한메모리 제한제출정답맞힌 사람정답 비율
1 초 512 MB71373151.667%

문제

Suppose 4 teachers are grading the presentation of a student and all their given marks are integers. The actual score is computed by taking arithmetic average of these 4 marks. Even though the marks are integers, the average can still be a fraction. But in some cases the average can also be an integer and in some rare cases the one of given marks can be equal to the average. Such rare incidents are called matching events. Considering all possible marks given by all teachers, you will have to count the total number of matching events. If the average mark matches with marks given by more than 1 teacher then for each match a matching event should be counted.

입력

There are at most 1001 test cases. The description of each test case is given below:

The input for each test case consists of two integers N (2 ≤ N ≤ 60) and fullmarks (1 ≤ fullmarks ≤ 200). Here N denotes the number of teachers who are grading the student and fullmarks denotes the maximum possible mark that any of the teachers can give. The minimum mark that a teacher can give is always zero.

Input is terminated by a line containing two zeroes.

출력

For each test case produce one line of output. This line contains the value M % 1 000 000 007 where M is the total number of matching events for the given input.

예제 입력 1

4 100
3 1
0 0

예제 출력 1

1373732
6


text:
# Do simple, multi-dimensional generalizations of this continued fraction formula exist?

## Background

Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}.$$ Let $$\frac{p_n}{q_n} = [a_1, a_2, \dots, a_n]$$ be the $n$th convergent, and let $T^n x = [a_{n+1}, a_{n+2}, a_{n+3},\dots]$.

A well-known formula gives $$x- \frac{p_n}{q_n} = \frac{(-1)^n \cdot x \cdot Tx \cdot T^2 x \cdot \dots \cdot T^{n}x}{q_n} = \frac{(-1)^n }{q_n(q_{n+1}+q_nT^{n+1} x)}.$$ This formula holds (with very minor changes, if any) for most kinds of one-dimensional continued fractions, including the various real variants, such as even continued fractions, odd continued fractions, Nakada's $\alpha$-continued fractions, etc., and - I believe - for Hurwitz's complex continued fractions as well.

## The question

On page 132 of his book on Multidimensional Continued Fractions, Schweiger gives a generalization of the formula as a corollary to Perron's Identity; however, this formula is comparatively much more complicated.

Are there specific multi-dimensional continued fraction algorithms for which this formula simplifies considerably to something closer to what is seen for one-dimensional continued fractions? Do there exist other analogs of this formula for multi-dimensional continued fractions?

## Motivation

Anton Lukyanenko and I worked on a paper studying continued fractions on the Heisenberg group (a two-dimensional complex system), and were surprised to see a simple analog of the formula in our work (Theorem 3.23 on page 22). We believe this is unique among multidimensional continued fractions, but we were not sure if there existed other simple analogs that were just not as well known.

## 1 Answer

One of natural generalizations of continued fractions (CF) is 3D Voronoi-Minkowski CF. For Voronoi construction see Delone & Fadeev's The theory of irrationalities of the third degree for Minkowski see Hancock, H. Development of the Minkowski geometry of numbers. Vol. 1 (ch. X). This object is nice from different points of view. But it is essentially 2D object.

We can consider classical CF as a chain or as broken line, while 3D CF is a plane graph. We can walk along this graph in different directions. The answer depends on your problem. You can try to find all minima of a linear form $x+\alpha y+\beta z$ and you'll go in one direction, you can try to minimize simultaneously two forms $x+\alpha y$ and $x+\beta z$, and then you must go in another direction.

There is no universal formula because time is 2-dimesional and there is no universal direction. But one good news. We have universal invariant measure $$dx_2\,dx_3\,dy_1\,dy_3\,dz_1\,dz_2\begin{vmatrix} 1 & x_2 & x_3\\ y_1 & 1 & y_3\\ z_1&z_2&1\\ \end{vmatrix}^{-3}$$ which is an analog of classical Gauss measure $$dx_2\,dy_1\begin{vmatrix} 1 & x_2 \\ y_1 & 1 \\ \end{vmatrix}^{-2}.$$ It arises in different contexts.

(See for example Kontsevich, M. L. & Suhov, Y. M. Statistics of Klein polyhedra and multidimensional continued fractions Pseudoperiodic topology, Amer. Math. Soc., 1999, 197, 9-27,

Karpenkov, O. On invariant Mobius measure and Gauss-Kuzmin face distribution ArXiv Mathematics e-prints, 2006

A. A. Illarionov The average number of relative minima of three-dimensional integer lattices of a given determinant. Izvestiya: Mathematics, 2012, 76:3, 535–562.)


text:
# Wavelet-domain gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be specified by multipliers in wavelet domain. A realization of one such GP is: $$s(t) = M_h m(b,a) W_g \eta(t)\, ,$$ where $\eta(t)$ is white noise, $W_g$ is the continuous wavelet transform with respect to wavelet $g$, $m(b,a)$ is the multiplier (kinda like a Fourier coefficient) with scale $a$ and time $b$, and $M_h$ is the inverse wavelet transform with reconstruction wavelet $h$.

One key result of the paper is that if the multipliers $m(b,a)$ only change slowly, then the realization themselves are only "weakly" dependent on the actual choices of $g$ and $h$. Thus $m(b,a)$ specifies the process. They go on to create some significant tests to help infer the wavelet multipliers based on realizations.

Two questions:

1. How do we evaluate the standard GP likelihood which is $p(D) = \mathcal{N}(0,K)$?

I'd guess we're effectively doing a change of coordinates so $K^{-1} = W^T M^{-1} W$ where $W$ are the wavelets and $M$ is the (diagonal?) matrix of wavelet coefficients $m(a,b)$. However, they use a non-orthonormal CWT so I don't know if this is correct.

2. How can this wavelet domain GP be related to a real-space GP? Specifically, can we calculate a real-space (non-stationary) kernel $k$ from $m(a,b)$?

For comparison, the kernel of a stationary Gaussian processes is the Fourier dual of its spectral density (Bochner's theorem, see Rasmussen chapter 4) - which gives an easy way to switch between a real space GP and a frequency space one. Here I'm asking if there's such a relationship in the wavelet domain.

• Did you get anywhere with this. I'm not sure the change of variables is right as that would contradict when they say $K_{g,h}(b−b/a,a/a)=W_{g,h}(b−b/a)$ is called the reproducing kernel? – tdc Nov 10 '17 at 10:30


text:
# Documentation

Std.Tactic.ShowTerm

show_term tac runs tac, then prints the generated term in the form "exact X Y Z" or "refine X ?_ Z" if there are remaining subgoals.

(For some tactics, the printed term will not be human readable.)

Equations
• One or more equations did not get rendered due to their size.

show_term e elaborates e, then prints the generated term.

(For some tactics, the printed term will not be human readable.)

Equations


text:
# Tag Info

12

You can interpolate the solution onto a finer mesh and then plot it: from dolfin import * coarse_mesh = UnitSquareMesh(2, 2) fine_mesh = refine(refine(refine(coarse_mesh))) P2_coarse = FunctionSpace(coarse_mesh, "CG", 2) P1_fine = FunctionSpace(fine_mesh, "CG", 1) f = interpolate(Expression("sin(pi*x[0])*sin(pi*x[1])"), P2_coarse) g = interpolate(f, ...

11

I suggest you start by looking at the FEniCS Navier-Stokes demo which is documented here: http://fenicsproject.org/documentation/dolfin/1.2.0/python/demo/pde/navier-stokes/python/documentation.html For your specific test case, you might want to look at the NSBENCH set of test problem (which are described in Chapter 21 of the FEniCS Book). The code for that ...

10

To implement your problem in FEniCS, you have to replace the integrals in terms of boundaries by integrals in terms of edges. This introduces jumps/averages in the test functions, which you entirely miss in your implementation. Hence, the system is not invertible and your solution does not look right. Equation (3.3) in Arnold et. al. 2002 gives you a tool to ...

10

I've been developing a lightweight finite element library in Python 2.7 harnessing the power of NumPy arrays and SciPy sparse matrices. The general idea is that given a mesh and a finite element, you have more-or-less one-to-one correspondence between the bilinear form and a (sparse) matrix. The user can then use the resulting matrix as he or she sees fit. ...

8

You can use a Nitsche-type method for this. See the following reference: J. Freund, R. Stenberg. On weakly imposed boundary conditions for second order problems. Proceedings of the Ninth Int. Conf. Finite Elements in Fluids, Venice 1995. M. Morandi Cecchi et al., Eds. pp. 327-336. I have implemented this a while ago in some simple FEniCS-Code to deal with ...

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First, a general point: you cannot prescribe arbitrary boundary conditions for a partial differential operator and expect that the partial differential equation (which always includes both operator and boundary conditions) is well-posed, i.e., admits a unique solution that depends continuously on the data -- all of which is a necessary condition for actually ...

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That depends how did you generated the file. For example your mesh generator can produce some cell or facet markers according to its input. Then dolfin-convert script may or may not succeed converting all these markers to XML. They're then stored within XML mesh file or in separate XML. You can then use them as FacetFunction # if stored within mesh ...

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x = V.cell().x then use x[0] and x[1] as $x$ and $y$ respectively. Right-hand side of LinearVariationalProblem needs to be rank 1 form - this is expressed by dependendnce on TestFunction and independence on TrialFunction.

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I've been working a bit on adaptive refinement to do the job (see the code below). The scaling of the error indicator with total mesh size and total variation of the mesh function is not perfect, but you could fit that to your needs. The images below are for testcase #4. The number of cells increases from 200 to about 24,000, which may be a bit over the top, ...

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As one of the library's authors, I would of course love for deal.II to come out on top with this comparison. But I suspect it may not, and the answer lies in a factor you omit from your comparison: how long it actually takes to implement your code. Few people in academia with the skills to implement a FEM code from scratch spend more time solving PDEs than ...

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Use dolfin-convert from a terminal window.

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Part of your problem is your choice of boundary conditions. This paper (which I wrote) develops the criteria for imposing a pressure boundary condition on the incompressible Navier-Stokes equations. The short version is that you need to impose the normal component of the normal traction on the inlet and outlet to be the pressures you want on each surface, ...

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Doing u, p = U.split() gives you a view into U, i.e. u, p and U share degree of freedom data. Doing u, p = U.split(deepcopy=True) gives a u and a p that have their own degree-of-freedom vectors. Use of shallow and deep sub-functions is demonstrated in the DOLFIN demo demo/undocumented/stokes-taylor-hood (from both C++ and Python). The syntax is a bit ...

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Your sample code will not work in parallel because you are creating the same vertices and cells on all processes. If you add mesh entities only on process 0 and then call MeshPartitioning.build_distributed_mesh(mesh) on all processes, your sample code runs fine.

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I've used this exact workflow for solving the Navier-Stokes equations in FEniCS. Almost certainly the hardest part in the process will be implementing the numerical solution method; there is no out-of-the-box support for that in FEniCS (although of course all differential operators you need are available). For reference, check out nsbench, a Navier-Stokes ...

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You can always use an iterator to access the entities connected to a cell, like the edges in your case. Just as you suggest, you first need to create a dolfin::Cell from the ufc::cell. Then do something like this: void eval(Array<double>& v, const Array<double>& x, ...

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Looks like you're using the Python interface, in which case u is both a DOLFIN Function object and a UFL object (it inherits from both). The type information you're seeing is related to UFL. You can evaluate the function u at an arbitrary point: f = u((0.5, 0.5, 0.5)) if DOLFIN is configured with CGAL. I don't know how you could check that the solution ...

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That's actually very easy to do in Dolfin: from dolfin import * # define mesh, function space (piecewise linear) mesh = UnitSquareMesh(64,64) V = FunctionSpace(mesh,'CG',1) # inhomogeneous boundary conditions (otherwise the solution is trivial) bc = DirichletBC(V, Constant(1.0), lambda x,on_boundary: on_boundary) # define bi(non)linear form # note that ...

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This is I think rotating cylinder and Karman vortex street behind it. You can also uncomment SUPG/PSPG stabilization. But I suspect that mesh size h computed as 2*circumradius is not good quantity and prefer cell diameter computed this way https://scicomp.stackexchange.com/a/7181/4254 from dolfin import * import time #set_log_level(PROGRESS) parameters['...

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As the error message suggest the DirichletBC does not hit any mesh entities with corresponding dofs. You need to examine your subdomains. One way to debug these are to apply the DirichletBC to a Function and plot the result: def left_boundary(x, on_boundary): r = math.sqrt(x[0] * x[0] + x[1] * x[1]) return r < 0.5 leftPlate = DirichletBC(V, u_L, ...

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Since the problem is treated in subdomains-poisson for the case, where the mesh is aligned with the interface, I assume in the following that in your case the jump can occur anywhere. I think the first step here is not to think about implementation FEniCS, but rather to find an adequate variational form for this problem, where the interface conditions are ...

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Take a look at the CSG (Constructive Solid Geometry) demos in DOLFIN. You should be able to generate a rectangle with two holes with something like mesh = Mesh(Rectangle(...) - Ellipse(...) - Ellipse(...), resolution)

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Make sure that the mesh is aligned with x = 0. Then mark the facets along that line and compute the flux using an "interior facet integral". You can do this by supplying a facet function markers that (e.g.) marks those facets by 1 and the remaining facets by 0. Then compute the flux by line_segment = AutoSubDomain(lambda x: near(x[0], 0)) markers = ...

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You need to evaluate the Function at the given point. Assuming phi1 is a scalar Function, try: class Bulk(SubDomain): def inside(self, x, on_boundary): return phi1(x) >= 0.1 and similar for the other SubDomains

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The following compiles: from dolfin import * from random import random lmbda = Constant(1.0) mu = Constant(1.0) def sigma(v): return 2.0*mu*sym(grad(v)) + lmbda*tr(sym(grad(v)))*Identity(v.cell().d) mesh = UnitCubeMesh(8,8,8) V = VectorFunctionSpace(mesh, "CG", 1) # Create random displacement vector displacement = Function(V) for i in xrange(V.dim()...

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You can define the Mini element in this alternative manner if you want a mixed enriched element rather than an enriched mixed element: U = FunctionSpace(mesh, "Lagrange", 1) B = FunctionSpace(mesh, "Bubble", 3) M = U + B Mini_h = MixedFunctionSpace([M, M])

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There will be non-negligible overhead when moving between meshes (some current developments will make this faster than it is now). You could first interpolate, then to evaluate vertex values efficiently use the function Function::compute_vertex_values. For spaces that are not continuous it can yield some unexpected results.

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I don't know about these things in FEniCS. For openFOAM there are libraries for multiphase flow. The implementation of population balance equations to track particle size distributions in openFOAM has been done and discussed in this paper. The question 'openFOAM or FEniCS' can maybe decided by deciding on whether you want to use finite volumes or finite ...

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You can't expect that solution to your altered problem would be a solution to Poisson problem because you need to change the problem somehow to make it well-posed. One could guess that possible problem formulation is to minimize F(u, \lambda) = \int_\Omega \frac{1}{2}|\nabla u|^2 \;\mathrm{d}x - \int_\Omega f u \;\mathrm{d}x - \int_{\Gamma_\mathrm{N}} g ...

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Vatiational formulation of Poisson problem with $\varepsilon \in L^\infty$ Both interfacial conditions are incorporated in following variational problem. Given $\Omega$ Lipschitz domain $\partial\Omega = \Gamma_\mathrm{D} \cup \Gamma_\mathrm{N}$ $V=\{v\in H^1(\Omega); v|_{\Gamma_\mathrm{D}}=0 \}$ $\varepsilon\in L^\infty(\Omega)$ \$g \in L^2(\Gamma_\...

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