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## Measurement of photon–jet transverse momentum correlations in 5.02 TeV Pb + Pb and $pp$ collisions with ATLAS

Research output: Contribution to journalArticle

Open

### Documents

Original language English Aaboud:2018anc 167-190 Physics Letters B B789 10.1016/j.physletb.2018.12.023 Published - 10 Feb 2019

### Abstract

Jets created in association with a photon can be used as a calibrated probe to study energy loss in the medium created in nuclear collisions. Measurements of the transverse momentum balance between isolated photons and inclusive jets are presented using integrated luminosities of 0.49 nb$^{-1}$ of Pb+Pb collision data at $\sqrt{s_\mathrm{NN}}=5.02$ TeV and 25 pb$^{-1}$ of $pp$ collision data at $\sqrt{s}=5.02$ TeV recorded with the ATLAS detector at the LHC. Photons with transverse momentum $63.1 < p_\mathrm{T}^{\gamma} < 200$ GeV and $\left|\eta^{\gamma}\right| < 2.37$ are paired inclusively with all jets in the event that have $p_\mathrm{T}^\mathrm{jet} > 31.6$ GeV and pseudorapidity $\left|\eta^\mathrm{jet}\right| < 2.8$. The transverse momentum balance given by the jet-to-photon $p_\mathrm{T}$ ratio, $x_\mathrm{J\gamma}$, is measured for pairs with azimuthal opening angle $\Delta\phi > 7\pi/8$. Distributions of the per-photon jet yield as a function of $x_\mathrm{J\gamma}$, $(1/N_\gamma)(\mathrm{d}N/\mathrm{d}x_\mathrm{J\gamma})$, are corrected for detector effects via a two-dimensional unfolding procedure and reported at the particle level. In $pp$ collisions, the distributions are well described by Monte Carlo event generators. In Pb+Pb collisions, the $x_\mathrm{J\gamma}$ distribution is modified from that observed in $pp$ collisions with increasing centrality, consistent with the picture of parton energy loss in the hot nuclear medium. The data are compared with a suite of energy-loss models and calculations.

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# Many Capricorn with Cancer moon/rising

QuestionsCategory: QuestionsMany Capricorn with Cancer moon/rising

I am interested in someone with Sun/Venus/Saturn/Uranrus/Neptune in Capricorn & Moon/Rising in Cancer. She looks so so cool & cold when we first met and worked together for a month, seldom speaks, then I sense her humor and gentleness, sometime she smiles at me when we have eye contacts… she seems to be a bit mysterious but sometimes make me warm…

how do I get closer with her and know if she is interested in me by any behaviours or words?

Only way to warm up a Capricorn like that is time and effort and don’t be fickle. Like nice one day then ignore her the next. She’ll pick up on that.


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# Numbers related to the Riemann hypothesis

Are there numbers $k > 1$ and $c > 1$ such that:

1 ) $\theta(c) \geq c \left( 1-\frac{1}{5 \ln^2(c)} \right)$

2 ) $\frac{c}{1+\frac{1}{\ln^4(c)}} \leq p(\pi(c))$ where $p(n)$ is the $n$-th prime number and $\pi(n)$ is the prime counting function

3 ) $\pi(c) \leq \frac{c}{\ln (c)} (1+\frac{1}{\ln (c)}+\frac{2}{\ln^2 (c)}+\frac{6}{\ln^3 (c)} + \frac{25}{\ln^4 (c)} )$

4 ) $\pi(c) \geq \frac{c}{\ln (c)} (1+\frac{1}{\ln (c)}+\frac{2}{\ln^2 (c)}+\frac{6}{\ln^3 (c)} + \frac{24}{\ln^4 (c)} )$

5 ) $\prod \limits_ {p \leq c} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq c^k} (1+\frac{1}{p}) \leq e^\gamma \ln (c ( 1-\frac{1}{5 \ln^2(c)})+c^k ( 1-\frac{1}{5 k^2 \ln^2(c)}))$?

• I forgot to mention that $k$ have to be bigger than $1$ Dec 25 '16 at 22:04
• Looks like homework. Dec 25 '16 at 22:48
• not a homework, i think that such numbers could help prove RH partially Dec 25 '16 at 22:52
• Also, the typical error bound coming out of the proof of the PNT is way better than $O(\ln(x)^{-N})$, yet the PNT is still a light-year from the RH. Dec 25 '16 at 22:59

## 1 Answer

Proving such results falls into three parts. First you take an effective version of the prime number theorem, which implies all your desired bound for sufficient. Second you write a computer program (or use computations done by somebody else) to check your inequalities for small $x$. Finaly you bridge the gap between the two ranges by improving the general bounds for large $x$ with more specialized arguments using numerical information on zeros of the Riemann $\zeta$-function.

The first part is easy, since you can just cite the error terms of the effective PNT. As far as I know, the bet results are due to Trudgian ( https://arxiv.org/pdf/1401.2689.pdf ). The difficult part is making the different ranges overlap. Note that as your inequalities are rather weak, and $\pi(x)$ can be computed in sublinear time, you should compute $\pi(x_0)$, use an upper bound sieve to compute some $x_1$, such that your inequality holds in $[x_0, x_1]$. Since $x_1$ will be of magnitude $x_0+\frac{x_0}{\log^C x_0}$, the savings are substantial.


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# There Are Cancellations?

Algebra Level 3

$\large a_n=\sqrt{ 1+\left(1+\dfrac{1}{n} \right)^2 }+\sqrt{ 1+\left(1-\dfrac{1}{n} \right)^2 }$

Consider a sequence $a_n$, $n\geq1$ as defined above. Compute $\dfrac{1}{a_1}+\dfrac{1}{a_2}+\cdots+\dfrac{1}{a_{20}}$.

×


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# Iterative Policy Evaluation for Estimating Value Function

#### Introduction

In this tutorial, I am going to code the iterative policy evaluation algotithm from the book “Reinforcement Learning: An Introduction by <NAME> and <NAME>”. I am going to take psuedo code, image and examples from this text. The example I am taking for this tutorial is the gird world maze from Chapter 4 (page 76, example 4.1).

The gird world is a ×4 matrix with the first and last cells are being terminal states. Assuming a random policy, which weights every action equaly, and the only possible actions are up, down, left, or right. For every action except the terminal states the reward will be . The graphical representation of the rules of the game are shown below.

The psuedo code for the algorithm from the text books is shown below.

According to this iterative policy evaluation algorithm, if we iterate over the Bellman value function for many times, it will converge to its optimum value. For each states the algorithm looks for the possible actions, next states and rewards and updates the state value at the present state. Since this is an episodic taks (i.e. having terminal states), the discounting is not required. I have written about the intuition (when and why) behind the usage of the discount factor. Please see the post if you are interested in knowing that.

Following is the code for the environment, agent, and algorithm.

##### Code
import numpy as np
import matplotlib.pyplot as plt

class Env(object):
def __init__(self):
self.states = np.zeros((4,4))
self.fstates = np.zeros((4,4))

def showEnv(self, aS):
fig, ax = plt.subplots()
ax.matshow(self.states, cmap='winter')
for (i, j), z in np.ndenumerate(self.fstates):
ax.text(j, i, round(z,5), ha='center', va='center')
ax.text(aS[1]-0.25, aS[0], ':-)', ha='center', va='center')
plt.axis('off')
plt.show()

class Agent(object):
def __init__(self, start=[0,0]):
self.start = start
self.cstate = start
self.reward = 0.
self.validActions = ['U','D','L','R']
self.policy = np.ones((4,4,4))*0.25

def checkNextState(self, action):
if action not in self.validActions:
print('Not a valid action')
elif action == 'U':
nstate = [self.cstate[0] - 1, self.cstate[1]]
self.reward = -1.
elif action == 'D':
nstate = [self.cstate[0] + 1, self.cstate[1]]
self.reward = -1.
elif action == 'L':
nstate = [self.cstate[0], self.cstate[1] - 1]
self.reward = -1.
elif action == 'R':
nstate = [self.cstate[0], self.cstate[1] + 1]
self.reward = -1.
if (nstate[0] < 0) or (nstate[0] >3) or (nstate[1] < 0) or (nstate[1] >3):
self.reward = -1.
nstate = self.cstate
if nstate == [0, 0]:
self.reward = 0.
if nstate == [3, 3]:
self.reward = 0.
return nstate

def takeAction(self, action):
self.cstate = self.checkNextState(action)

def gpolicy(self, state, a):
a = ['U','D','L','R'].index(a)
return self.policy[state[0], state[1], a]

# Iterative Policy Evaluation (Ex. 4.1, p.76 Sutton & Barto)
theta = 2
myEnv = Env()
myAgent = Agent([0,0])
commands = ['U','D','L','R']
delta = 0
k = 0

while True:
print('Iteration: ', k)
k+=1
for (i, j), z in np.ndenumerate(myEnv.states):
sums = 0
myAgent.start = [i,j]
myAgent.cstate = [i,j]
# print('Agent at ', i, j)
for act in commands:
# print('Checking Action: ', act)
polval = myAgent.gpolicy(myAgent.start, act)
# print('Policy Prob: ', polval)
if myAgent.start ==[0,0]:
step = [0,0]
myAgent.reward = 0.
break
elif myAgent.start == [3,3]:
step = [3,3]
myAgent.reward = 0.
break
else:
step = myAgent.checkNextState(act)
# print('Next Step is: ', step)
# print('Possible Reward is: ', myAgent.reward)
sums+=polval * (myAgent.reward+(myEnv.states[step[0],step[1]]))
# print('Updated Value: ', sums)
myEnv.fstates[i,j] = sums
delta = max(delta, abs(z-sums))
# print(delta)
# myEnv.showEnv(myAgent.cstate)
myEnv.states = myEnv.fstates
print(myEnv.fstates)
if delta>theta:
break
if k > 1000:
break
##### After each iteration, the value function is printed. We can see it converges towards the following value.
Iteration:  1000
[[  0. -13. -19. -21.]
[-13. -17. -19. -19.]
[-19. -19. -17. -13.]
[-21. -19. -13.   0.]]

A visual representation of the same might make more sense.

myEnv.showEnv(myAgent.cstate)

#### Conclusion

As we can see here the converged value function after iterative policy evaluation is symmetric and resonates the reward rule. States next to terminal states are having same values. The agent will be learning to reach a terminal as soon as possible. Hence the value function around the center of the gird are low and relatively high near the terminal states.

The converged values are just one value different than the result shown in the text book. I think this is because we are following an iterative approximation method rather than solving linear equations directly. We can see that our solution gets very close to the analytical one shown in the text book.


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Viewpoint

# A Precise Parity-Violation Measurement in Light Nuclei

Physics 13, 149
A new measurement from the ${n}^{3}$He Collaboration advances understanding of parity violation in few-nucleon systems.

When studying parity violation in nuclei, it is a truth universally acknowledged that what is experimentally easy to measure is theoretically difficult to calculate, and what is theoretically easy to calculate is experimentally difficult to measure. Now, <NAME> of the University of Manitoba in Canada and colleagues have helped address this challenge by precisely measuring the rate of parity violation for a nuclear reaction that is within reach of theoretical understanding—neutron capture by helium-3 nuclei [1]. This measurement helps bring us closer to an understanding of how nucleons interact.

What is parity violation and why does it matter? A parity transformation replaces a three-dimensional coordinate system with one in which all directions are switched to their negatives. For example, a vector $\stackrel{\to }{r}=\left(1,1,1\right)$ in Cartesian coordinates transforms under a parity operation into $-\stackrel{\to }{r}=\left(-1,-1,-1\right)$. Because a vector $\stackrel{\to }{r}$ transforms in this way, we say that it has negative parity. Angular momentum, on the other hand, does not change sign under a parity operation because $\stackrel{\to }{L}=\stackrel{\to }{r}×\stackrel{\to }{p}$ . Both $\stackrel{\to }{r}$ and $\stackrel{\to }{p}$ have negative parity and change sign under a parity transformation, so their cross product $\stackrel{\to }{L}$ has positive parity.

Once nuclear physics was understood to be governed by a few fundamental forces of nature, scientists assumed that those forces remained invariant under parity transformations because no parity violation had been observed in past experiments. However, in 1956, <NAME> and <NAME> pointed out that parity invariance had not been experimentally tested for the weak interactions [2]. Within months, <NAME> and colleagues observed parity violation in cobalt-60 nuclei [3] (see Focus: Landmarks—Breaking the Mirror). Today, researchers continue to study parity violation not only to refine our understanding of nuclear phenomena but to potentially probe for new physics.

To look at parity violation in a simpler system, consider the neutrino. Assuming the neutrino is massless, which is sufficient for the purposes of this discussion, its spin always points in the direction opposite to its momentum’s direction. As discussed above, momentum $\stackrel{\to }{p}$ changes sign under a parity transformation, but angular momentum—including spin—does not. If a neutrino is traveling in the $+\stackrel{̂}{z}$ direction with spin pointed in the $-\stackrel{̂}{z}$ direction, a parity transformation would take it to a neutrino traveling in the $-\stackrel{̂}{z}$ direction with spin in the $-\stackrel{̂}{z}$ direction. But such a particle does not exist! We say that parity is maximally violated by neutrinos. (Taking into account the tiny finite neutrino mass just makes this parity violation slightly less than maximal.)

Things are not so simple, however, in nuclear systems, because parity violation in nuclei arises from an interplay between strong and weak forces. The nucleons—neutrons and protons—that form nuclei consist of quarks and gluons. Both the binding of nucleons into nuclei and the behavior of quarks and gluons inside nucleons are governed by the strong force. Unfortunately, quantum chromodynamics (QCD), the theory of strong interactions, is complicated to calculate at nuclear energies. This makes it very difficult to understand how QCD is manifest in nuclei. QCD itself is invariant under parity, but quarks are also subject to weak interactions, which violate parity. Studying parity violation in nuclei with high precision can serve as a potent window into the inner workings of nucleons.

But weak interactions have their name for a reason—the parity-violating component of the nucleon-nucleon interactions is 7 orders of magnitude smaller than the parity-conserving component. On the experimental side, this makes the signal of parity violation very tiny compared to background effects. So the general strategy is to find observables that would vanish if parity were conserved. Instead of trying to find a tiny effect on top of something much larger, the goal is to find a deviation from zero.

In some special cases, parity-violating signals in nuclei can be enhanced by several orders of magnitude to make measurements easier, but these enhancements can only happen in large nuclei. Unfortunately, a detailed theoretical description of large nuclei using QCD-based calculations is currently out of reach. The fewer the nucleons involved, the easier it is to connect the calculations to QCD. This is one reason why the measurement of a parity-violating asymmetry in the capture of cold neutrons by helium-3 is such an important development.

In the helium-3 experiment [1], Gericke and colleagues of the ${n}^{3}$He Collaboration used the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory to shoot a beam of neutrons along the $\stackrel{̂}{z}$ direction. The neutrons, with spins aligned perpendicularly to the direction of their propagation ( $±ŷ$ ), enter the helium-3 target chamber, which also serves as the detector. In the reaction of interest, a neutron collides with a helium-3 nucleus ( ${}^{3}\text{He}$), yielding an outgoing proton and a tritium nucleus ( ${}^{3}\text{H}$). Measuring how often this process occurs as a function of the proton’s propagation direction and then correlating it with the neutron spin direction determines the parity-violating asymmetry ${A}_{\text{PV}}$. If parity were conserved, the detector would see an identical distribution of protons in its upper and lower halves, and ${A}_{\text{PV}}$ would be zero. But parity was violated in this reaction, so the protons were not evenly distributed. The measured asymmetry was ${A}_{\text{PV}}=1.55×1{0}^{-8}$ with a statistical uncertainty of $0.97×1{0}^{-8}$ and a systematic uncertainty of $0.24×1{0}^{-8}$.

The reported precision is better than that of any previously measured few-nucleon system. To give an example of the level of care the researchers took to achieve this result: The orientation of one of the elements in their detector (a wire frame) had to be measured to within 3 milliradians.

This precision of the ${A}_{\text{PV}}$ measurement was also made possible by the very low background in the helium-3 target chamber, improvements in controlling the spin-flipping of the incoming neutrons, and analysis advancements in understanding SNS beam fluctuations and detector acceptance.

The result from the ${n}^{3}$He Collaboration is a spectacular achievement. Its impact is best seen in the larger landscape of parity-violating nuclear forces and interactions. Parity violation is so complex in nuclei that no single experiment provides sufficient information; a suite of observables must be analyzed to understand the phenomena. There are very few nuclei that are both well understood theoretically and well measured experimentally. It is when the results of the ${n}^{3}$He and other experiments—the NPDGamma experiment [4], the polarized proton-proton scattering experiment [5], and future few-nucleon experiments—are combined with a robust, QCD-based, theoretical understanding [6] that we can start to disentangle the interplay of the fundamental forces in the nucleons. Theory advances—in the use of effective field theories that encode the symmetries of QCD in the language of nucleons, and in numerical simulations of QCD—will continue to complement the experimental efforts. Eventually, understanding QCD and its interplay with other forces in nuclear systems will enable us to use nuclei to search for hints of new particles and new forces that are beyond the standard model.

Correction (5 October 2020): We felt it was important to clarify that the physicist Chien-Shiung Wu was behind the observation of parity violation in cobalt-60.

## References

1. <NAME> et al. (n3He Collaboration), “First precision measurement of the parity-violating asymmetry in cold neutron capture on ${}^{3}\text{He}$,” Phys. Rev. Lett. 125, 131803 (2020).
2. <NAME> and <NAME>, “Question of parity conservation in weak interactions,” Phys. Rev. 104, 254 (1956).
3. <NAME> et al., “Experimental test of parity conservation in beta decay,” Phys. Rev. 105, 1413 (1957).
4. <NAME> et al., “First observation of $P$-odd $𝛾$ asymmetry in polarized neutron capture on hydrogen,” Phys. Rev. Lett. 121, 242002 (2018).
5. <NAME> et al., “Parity violation in proton-proton scattering at 13.6 MeV,” Phys. Lett. B 256, 11 (1991).
6. <NAME> et al., “Chiral effective field theory analysis of hadronic parity violation in few-nucleon systems,” Phys. Rev. C 89, 064004 (2014).

<NAME> is an associate professor at the University of South Carolina. He obtained his Ph.D. from the Johannes Gutenberg University Mainz, Germany, and held postdoctoral positions at Ohio University and the George Washington University, Washington D.C. His research focuses on the use of effective field theories and symmetries in light nuclei.

<NAME> is a professor of physics at Duke University, North Carolina.  She obtained her Ph.D. from the California Institute of Technology and is a fellow of the American Physical Society. She has expertise in effective field theories of quantum chromodynamics and weak interactions.

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Astrophysics

### Counting All the Antistars in the Sky

Analyzing gamma-ray sources leads to an upper limit on how many antimatter stars could exist in the Milky Way. Read More »


text: π The Schrödinger coherent state for the 2D isotropic harmonic oscillator is a product of two infinite series. Opto-electronic oscillator. 2 I have a wave function which is the ground state of a harmonic oscillator (potential centered at x=0)... but shifted by a constant along the position axis (ie. H= 6. ω For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude. Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. are constants that depends on the initial conditions. and The corresponding energy eigenvalues are labeled by a single quantum number n, where h is Planck's constant and ν depends on t… Harmonic Shift Oscillator. Legal. By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation: the latter being Newton's second law of motion. {\displaystyle \omega } , the time for a single oscillation or its frequency g This is the Schro¨dinger equation for the one-dimensional harmonic oscillator, whose energy eigenvalues and eigenfunctions are well known. Phase-shift oscillator. s How does this decompose into eigenfunctions?!?! For $$D >1$$, the strong coupling regime, the transition with the maximum intensity is found for peak at $$n \approx D$$. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is called the viscous damping coefficient. Pierce oscillator. See the harmonic stride, and harmonic level modulation available, even a single HSO can produce extremely complex, evolving soundscapes with no other input. 5. The harmonic oscillator and the systems it models have a single degree of freedom. Harmonic Oscillator In many physical systems, kinetic energy is continuously traded off with potential energy. If the system has a ﬁnite energy E, the motion is bound 2 by two values ±x0, such that V(x0) = E. The equation of motion is given by mdx2 dx2 = −kxand the kinetic energy is of course T= 1mx˙2 = p 2 2 2m. (x+xo)?/2, where to = mc2 and (mw/h)Ż. The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions: For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients. \begin{array} {l} {U _ {g}^{\dagger} a U _ {g} = e^{i n \omega _ {0} t} a e^{- i n \omega _ {0} t} = a e^{i ( n - 1 ) \omega _ {0} t} e^{- i n \omega _ {0} t} = a e^{- i \omega _ {0} t}} \\ {U _ {g}^{\dagger} a^{\dagger} U _ {g} = a^{\dagger} e^{+ i \omega _ {0} t}} \end{array} \right. Robinson oscillator. , the number of cycles per unit time. New Systems Instruments - Harmonic Shift Oscillator & VCA. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. . The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. ω {\displaystyle x_{0}} The resonances coincide with the corresponding resonances of the unshifted impact oscillator after adding the displacement shift. Robinson oscillator. 0 Illustration of how the strength of coupling $$D$$ influences the absorption lineshape $$\sigma$$ (Equation \ref{12.38}) and dipole correlation function $$C _ {\mu \mu}$$ (Equation \ref{12.32}). It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. Q Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. It should be possible by using a coherent state I guess, because a coherent state can be seen as kind of a 'shifted' number state. sin Using as initial conditions Wien bridge oscillator. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard. ) is maximal. i Note that physically the dephasing function describes the time-dependent overlap of the nuclear wavefunction on the ground state with the time-evolution of the same wavepacket initially projected onto the excited state, $F (t) = \left\langle \varphi _ {g} (t) | \varphi _ {e} (t) \right\rangle \label{12.11}$. The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2. The differential equation that describes the motion of the of a damped driven oscillator is, Here m is the mass, b is the damping constant, k is the spring constant, and F 0 cos(ωt) is the driving force. This effect is different from regular resonance because it exhibits the instability phenomenon. θ Ñêmw. The rain and the cold have worn at the petals but the beauty is eternal regardless of season. 0 The driving force creating resonances is also harmonic and with a shift. When a spring is stretched or compressed by a mass, the spring develops a restoring force. A new {SU}(1,1) position-dependent effective mass coherent states (PDEM CS) related to the shifted harmonic oscillator (SHO) are deduced. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period \label{12.21}\], $\hat {p} (t) = i \sqrt {\frac {m \hbar \omega _ {0}} {2}} \left( a^{\dagger} e^{i \omega _ {0} t} - a e^{- i \omega _ {0} t} \right) \label{12.22}$, So for the dephasing function we now have, $F (t) = \left\langle \exp \left[ d \left( a^{\dagger} e^{i \omega _ {0} t} - a e^{- i \omega _ {0} t} \right) \right] \exp \left[ - d \left( a^{\dagger} - a \right) \right] \right\rangle \label{12.23}$, where we have defined a dimensionless displacement variable, $\underset{\sim}{d} = d \sqrt {\frac {m \omega _ {0}} {2 \hbar}} \label{12.24}$, Since $$a^{\dagger}$$ and $$a$$ do not commute ($$\left[ a^{\dagger} , a \right] = - 1$$), we split the exponential operators using the identity, $e^{\hat {A} + \hat {B}} = e^{\hat {A}} e^{\hat {B}} e^{- \frac {1} {2} [ \hat {A} , \hat {B} ]} \label{12.25}$. For the motion of a classical 2D isotropic harmonic oscillator, the angular momentum about the . General theory. 2.3].) θ {\displaystyle \zeta } = It provides similar capabilities to FM synthesis, but with a more direct relationship between the parameters and the resulting spectrum. Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. The Low-Pass Variant . Here the oscillations at the electronic energy gap are separated from the nuclear dynamics in the final factor, the dephasing function: \begin{align} F (t) & = \left\langle e^{i H _ {g} t / \hbar} e^{- i H _ {c} t / h} \right\rangle \\[4pt] & = \left\langle U _ {g}^{\dagger} U _ {e} \right\rangle \label{12.10} \end{align}, The average $$\langle \ldots \rangle$$ in Equations \ref{12.9} and \ref{12.10} is only over the vibrational states $$| n _ {g} \rangle$$. 0 The absorption lineshape is obtained by Fourier transforming Equation \ref{12.32}, \begin{align} \sigma _ {a b s} ( \omega ) & = \int _ {- \infty}^{+ \infty} d t \,e^{i \omega t} C _ {\mu \mu} (t) \\[4pt] & = \left| \mu _ {e g} \right|^{2} e^{- D} \int _ {- \infty}^{+ \infty} d t\, e^{i \omega t} e^{- i \omega _ {e s} t} \exp \left[ D e^{- i \omega _ {0} t} \right] \label{12.36} \end{align}, $\exp \left[ D \mathrm {e}^{- i \omega _ {0} t} \right] = \sum _ {n = 0}^{\infty} \frac {1} {n !} Other analogous systems include electrical harmonic oscillators such as RLC circuits. This is a vibrational progression accompanying the electronic transition. Any vibration with a restoring force equal to Hooke’s law is generally caused by a simple harmonic oscillator. The energy is 2μ1-1 =1, in units Ñwê2. Remembering $$a^{\dagger} a = n$$, we find, \[\left. The Damped Harmonic Oscillator. r = 0 to remain spinning, classically. This is a perfectly general expression that does not depend on the particular form of the potential. model A classical h.o. angular momentum of a classical particle is a vector quantity, Angular momentum is the property of a system that describes the tendency of an object spinning about the point . the force always acts towards the zero position), and so prevents the mass from flying off to infinity. is described by a potential energy V = 1kx2. We start by writing a Hamiltonian that contains two terms for the potential energy surfaces of the electronically excited state $$| E \rangle$$ and ground state $$| G \rangle$$, \[H _ {0} = H _ {G} + H _ {E} \label{12.1}$, These terms describe the dependence of the electronic energy on the displacement of a nuclear coordinate $$q$$. 0 The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator. Notice that in this approximation the period {\displaystyle \omega _{s},\omega _{i}} What is so significant about SHM? Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. In the case of a sinusoidal driving force: where ω φ The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity. 0 {\displaystyle \theta _{0}} When this assumption is not valid then one must account for the much more complex possibility of emission during the course of the relaxation process. {\displaystyle \theta _{0}} has translated the center of the harmonic oscillator and shifted the spectrum by a constant energy. must be zero, so the linear term drops out: The constant term V(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved: Thus, given an arbitrary potential-energy function is described by a potential energy V = 1kx2. Further, one can establish that, \left.\begin{aligned} \sigma _ {a b s} ( \omega ) & = \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} \right) t + g (t)} \\ \sigma _ {f l} ( \omega ) & = \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} \right) t + g^{*} (t)} \\ g (t) & = D \left( e^{- i \omega _ {0} t} - 1 \right) \end{aligned} \right. Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. Due to frictional force, the velocity decreases in proportion to the acting frictional force. If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the position x of the mass and a constant k. Balance of forces (Newton's second law) for the system is Quantum Harmonic Oscillator Think of a sliding block, constrained to move along one direction on an idealized frictionless surface, attached to an idealized spring. If we approximate the oscillatory term in the lineshape function as, \[\exp \left( - i \omega _ {0} t \right) \approx 1 - i \omega _ {0} t - \frac {1} {2} \omega _ {0}^{2} t^{2} \label{12.40}, \begin{align} \sigma _ {e n v} ( \omega ) & = \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \omega t} e^{- i \omega _ {e g} t} e^{D \left( \exp \left( - i \omega _ {0} t \right) - 1 \right)} \\ & \approx \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} t \right)} e^{D \left[ - i \omega _ {0} t - \frac {1} {2} \omega _ {0}^{2} t^{2} \right]} \\ & = \left| \mu _ {e g} \right|^{2} \int _ {- \infty}^{+ \infty} d t e^{i \left( \omega - \omega _ {e g} - D \omega _ {0} \right) t} e^{- \frac {1} {2} D \omega _ {0}^{2} t^{2}} \label{12.41} \end{align}, This can be solved by completing the square, giving, $\sigma _ {e n v} ( \omega ) = \left| \mu _ {e g} \right|^{2} \sqrt {\frac {2 \pi} {D \omega _ {0}^{2}}} \exp \left[ - \frac {\left( \omega - \omega _ {e g} - D \omega _ {0} \right)^{2}} {2 D \omega _ {0}^{2}} \right] \label{12.42}$, The envelope has a Gaussian profile which is centered at Franck–Condon vertical transition, $\omega = \omega _ {e g} + D \omega _ {0} \label{12.43}$, Thus we can equate $$D$$ with the mean number of vibrational quanta excited in $$| E \rangle$$ on absorption from the ground state. is the mass on the end of the spring. 1 For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency ( Sinusoidal oscillator with low total harmonic distortion (THD) is widely used in many applications, such as built-in-self-testing and ADC characterization. F The classical varactor parametric oscillator oscillates when the diode's capacitance is varied periodically. Bright, like a moon beam on a clear night in June. This is the value of $$H_e$$ at $$q=0$$, which reflects the excess vibrational excitation on the excited state that occurs on a vertical transition from the ground state. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators – their output waveform, resonant frequency, damping factor, etc. is small. The Hamiltonian for each surface contains an electronic energy in the absence of vibrational excitation, and a vibronic Hamiltonian that describes the change in energy with nuclear displacement. They are the source of virtually all sinusoidal vibrations and waves. The transient solutions are the same as the unforced ( RC&Phase Shift Oscillator. The shifted harmonic oscillator is obtained by adding a relatively bounded per-turbation of the harmonic oscillator P 0, which implies that the resolvent of P a is compact. is independent of the amplitude This allows us to work with the spectral decomposition of P adespite the fact that P ais not a normal operator. How can one solve this differential equation? Here we will discuss the displaced harmonic oscillator (DHO), a widely used model that describes the coupling of nuclear motions to electronic states. The solution to this differential equation contains two parts: the "transient" and the "steady-state". In electrical engineering, a multiple of τ is called the settling time, i.e. z Displacement r from equilibrium is in units è!!!!! The potential-energy function of a harmonic oscillator is. Harmonic Oscillator Assuming there are no other forces acting on the system we have what is known as a Harmonic Oscillator or also known as the Spring-Mass-Dashpot. That is, we want to solve the equation M d2x(t) dt2 +γ dx(t) dt +κx(t)=F(t). Mathematically, the notion of triangular partial sums … A damped oscillation refers to an oscillation that degrades over a … It is common to use complex numbers to solve this problem. x It is therefore the energy that must be dissipated by vibrational relaxation on the excited state surface as the system re-equilibrates following absorption. 0 The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude $$F(t)$$ quantifies the overlap of vibrational wavepackets on ground and excited state, which peaks once every vibrational period. The vertical lines mark the classical turning points, that is, the displacements for which the harmonic potential equals the energy. 3. F How do we know that we found all solutions of a differential equation? The shift


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# If A : B = C : D, then the value of $$\begin{array}{*{20}{c}} {{A^2}}& + &{{B^2}}\\ {{C^2}}& + &{{D^2}} \end{array}$$ is

This question was previously asked in
CMAT 2021 Slot 1 Official Paper
View all CMAT Exam Papers >
1. $$\frac{1}{2}$$
2. $$\begin{array}{*{20}{c}} {{A^{}}}& + &{{B^{}}}\\ C& + &{{D^{}}} \end{array}$$
3. $$\begin{array}{*{20}{c}} {{A^{}}}& - &{{B^{}}}\\ C& - &{{D^{}}} \end{array}$$
4. $$\begin{array}{*{20}{c}} {{A^{}}}& \times &{{B^{}}}\\ C& \times &{{D^{}}} \end{array}$$

Option 4 : $$\begin{array}{*{20}{c}} {{A^{}}}& \times &{{B^{}}}\\ C& \times &{{D^{}}} \end{array}$$
Free
MAH MBA CET 2018 Quantitative Aptitude Previous Year Paper
1993
50 Questions 50 Marks 40 Mins

## Detailed Solution

These questions are always best solved by using the options and assuming values of the variables. And options will be provided in questions like these for sure.

Let us assume A = 1; B = 2; C = 3; D = 6

Now going through the options one by one, we find that only the 4th option holds true here.

Important Points

Values must be assumed carefully otherwise the calculations can become tedious.


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## Linux Bash Notes - Things I use often, a nice place to keep a reminder...

Linux Bash Notes - Things I use often, a nice place to keep a reminder...

# BASH

I use a lot of Bash and Powershell scripts, I will update this page mainly as a aide-memoire for usefull scripts and snippets…

### 1. Time the execution time of your command or script

time yourscript.sh

--- or ---

start=date +%s
stuff
end=date +%s

runtime=$((end-start)) --- or --- SECONDS=0 myScript.sh echo$SECONDS


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# Eigenfrequencies of an elastic bar with viscous damping

I want to know the mode shapes and associated eigenfrequencies and damping ratios of a bar that vibrates in a longitudinal direction and has linear elastic as well as viscous material properties. Consider the bar shown below which has length $$L$$ and is clamped on the left hand side and free on the right. Its motion is desribed by the longitudinal displacements $$u(x,\,t)$$. The material is assumed to behave according to the viscoelastic Kelvin–Voigt model, which relates stress and strain at any point of the bar by

$$\sigma = E\,\varepsilon + \eta\,\dot{\varepsilon}$$

where $$\varepsilon = \frac{\partial\,u}{\partial\,x}$$ is the longitudinal strain, $$E$$ the elastic modulus of the material and $$\eta$$ the viscosity. The first thing I did was to try and obtain the equation of motion for this system. The relationship between stress and acceleration is

$$\rho\,\frac{\partial^2\,u}{\partial\,t^2} = \frac{\partial\,\sigma}{\partial\,x}$$

where $$\rho$$ is the bar's density. Combining this with the material model and multiplying with the cross section area $$A$$ results in the equation of motion

$$\rho A\,\frac{\partial^2\,u}{\partial\,t^2} = EA\,\frac{\partial^2\,u}{\partial\,x^2} + \eta A\,\frac{\partial^3\,u}{\partial\,x^2\,\partial\,t}$$

This is where I'm stuck because I don't know how to solve this. What I'm interested in is the eigenfrequency $$\omega$$ and damping ratio $$\zeta$$ of the first eigenmode (if that's even applicable). Without the damping term the problem reduces to the 1D wave equation which can be solved e.g. by separation of variables, but that doesn't seem to work in this case.

• Probably assuming the form $u(x,t) =X(x) \exp(-i \omega t)$ will simplify the problem. – <NAME> Nov 20 '19 at 16:01

There was a mistake in the governing equation of motion, a missing derivative in the damping term. I edited the question. The problem can now be solved by separation of variables.

Step 1: Separation of variables

Assume that $$u(x,\,t) = X(x)\,T(t)$$. Insert that into the equation of motion and separate terms that depend on $$t$$ from those that depend on $$x$$,

\begin{align} \rho A\,X(x)\,\ddot{T}(t) &= EA\,X''(x)\,T(t) + \eta A\,X''(x)\,\dot{T}(t) \\ &\ \ \vdots \\ \frac{X''(x)}{X(x)} &= \frac{\rho A\,\ddot{T}(t)}{EA\,T(t) + \eta A\,\dot{T}(t)} =: -\alpha^2 \end{align}

Apply the usual argument: As one side of the equation only depends on position and the other only on time they have to be constant in order to be equal. We choose that constant to be $$-\alpha^2$$ and obtain two ordinary differential equations in $$x$$ and $$t$$ respectively.

Step 2: Solving the ODE for X

The first equation $$X''(x) + \alpha^2 X(x) = 0$$ has the general solution $$X(x) = C\cdot\sin(\alpha\,x - \varphi).$$ The constants $$\alpha$$ and $$\varphi$$ are determined by the boundary conditions of the bar. The left hand side is fixed, therefore

\begin{align} u(0,\,t) = 0: \quad\quad\quad X(0)\,T(t) &= 0 \\ C\cdot\sin(-\varphi) &= 0 \\ \varphi &= 0 \end{align}

The case $$T(t) = 0$$ was ignored here because it is of no interest. The right hand side is free, so the boundary condition is

\begin{align} \sigma(L,\,t) = 0: \quad\quad E\,\varepsilon(L,\,t) + \eta\,\dot{\varepsilon}(L,\,t) &= 0 \\ &\ \ \vdots \\ (EA\,T(t) + \eta A\,\dot{T}(t))\,X'(L) &= 0 \\ X'(L) = C\alpha\cdot\cos(\alpha\,L) &= 0 \\ \\ \alpha = \frac{\pi}{L}k - \frac{\pi}{2L}&,\quad k \in \mathbb{Z} \end{align}

So there is an infinte number of solutions for $$\alpha$$, each corresponding to a different mode shape $$X(x)$$. The case $$EA\,T(t) + \eta A\,\dot{T}(t) = 0$$ was ignored because it implies $$\ddot{T}(t) = 0$$.

Step 3: Analyzing the ODE for T

The differential equation for $$T(t)$$ resembles that of a damped harmonic oscillator. The undamped natural frequency $$\omega_0$$ and the damping ratio $$\zeta$$ can be obtained by simply comparing the coefficients,

$$\ddot{T}(t) + \underbrace{\alpha^2\frac{\eta A}{\rho A}}_{\Large{2\,\zeta\,\omega_0}}\,\dot{T}(t) + \underbrace{\alpha^2\frac{EA}{\rho A}}_{\Large{\omega_0^2}}\,T(t) = 0$$

Using the values for $$\alpha$$ from above the final answer to the initial question is

$$\omega_0 = \frac{\pi(2k - 1)}{2L}\cdot\sqrt{\frac{EA}{\rho A}},\quad \zeta = \frac{\pi(2k - 1)}{4L}\cdot\frac{\eta A}{\sqrt{\rho A \cdot EA}},\quad k \in \mathbb{Z}$$

Sanity check 1: The undamped eigenfrequencies are the same as found in the literature for the more common case of a bar without damping.

Sanity check 2: The damping ratio increases with $$\eta A$$ and $$k$$ and decreases with $$\rho A$$, $$EA$$ and $$L$$, which makes sense.

Sanity check 3: The units seem to be correct.


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# NCG

## Seminar

### Topological boundary maps in physics, general theory and applications

<NAME> (<NAME> 1)
Tuesday 19 December 2006, 16:30-17:00

Seminar Room 1, Newton Institute

#### Abstract

$K$-theory boundary maps and their duals in cyclic cohomology are used to relate topological invariants of bulk and boundary in physical systems. We explain in particular an application to potential scattering thus providing a topological explanation of Levinson's theorem.


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# DOCPLEX: tips to manipulate data input (binary parameter)

I am coding a MIP problem in docplex and I would like to ask for some tips.

I would like to input a binary parameter that states a compatibility relation:

I have a set of characteristics $$H$$ to be satisfied in a product. The product is made by a set of components $$C$$ and each component can have many variants $$V$$. For stating the compatibility between variant of component and required characteristics, I stated the following binary parameter:

• $$G_h{_{c}}_{v}=1$$ if the characteristic $$c$$ can be accomplished by variant $$v$$ of component $$c$$, and zero otherwise

I was wandering to use a 3D numpy array to state this relation. But I if I'm not wrong, I did not find examples using 3D numpy array. Instead, I saw namedTuple, that I guess was used to state similar relation for string elements.

How can I code binary parameters in docplex keeping the index of each element, by accessing the value of each parameter and also its key ?

If someone could help me, I will be very grateful.

One thing I would personally do is stick to dictionaries as accessing each key is easier and faster (Its complexity is $$O(1)$$). If you have three collections for your keys, namely your $$H$$, $$C$$, and $$V$$, then, as you probably know, you can use binary_var_cube where you can see, from the documentation linked:
Using dictionaries will help you tremendously especially in cases where you have a sparse matrix. e.g. only a handful of variants $$V$$ are present in each component $$C$$. That way, rather than having a 3D numpy array or list of list of list, you can have only those indices for your variables that exist. Of course, you can achieve that with tuple or namedTuple, but as I mentioned, accessing dictionaries is faster (you can read more about some of Python data types here)


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## C# Tutorial

The Queue Collection is used when the data items need to be arranged in a FIFO (First In First Out) manner. When a data item is entered into the collection, it is called enqueue. When the data item is removed from the collection, it is called dequeue.Constructors in Queue CollectionThe different constructors and their description is given as follows:Table: Constructors in Queue Collection in C#ConstructorsDescriptionQueue<T>()This constructor initializes a new instance of the Queue<T> class that is empty and has the default initial capacity.Queue<T>(IEnumerable<T>)This constructor initializes a new instance of the Queue<T> class that contains elements copied from the specified collection and has sufficient capacity to accommodate the number of elements copied.Queue<T>(Int32)This constructor initializes a new instance of the Queue<T> class that is empty and has the specified initial capacity.Properties in Queue CollectionThe different properties and their description is given as follows:Table: Properties in Queue Collection in C#PropertiesDescriptionCountThis property gets the number of elements contained in the Queue<T>.Methods in Queue CollectionThe different methods and their description is given as follows:Table: Methods in Queue Collection in C#Source: MSDNMethodsDescriptionClear()This method removes all objects from the Queue<T>.Contains(T)This method determines whether an element is in the Queue<T> or not.CopyTo(T[], Int32)This method copies the Queue<T> elements to an existing one-dimensional Array, starting at the specified array index.Dequeue()This method removes and returns the object at the beginning of the Queue<T>.Enqueue(T)This method adds an object to the end of the Queue<T>.Equals(Object)This method determines whether the specified object is equal to the current object.GetEnumerator()This method returns an enumerator that iterates through the Queue<T>.GetHashCode()This method serves as the default hash function.GetType()This method gets the Type of the current instance.MemberwiseClone()This method creates a shallow copy of the current Object.Peek()This method returns the object at the beginning of the Queue<T> without removing it.ToArray()This method copies the Queue<T> elements to a new array.ToString()This method returns a string that represents the current object.TrimExcess()This method sets the capacity to the actual number of elements in the Queue<T>, if that number is less than 90 percent of current capacity.Queue OperationsSome of the stack operations are given as follows:EnqueueAn element is added to the end of the queue using the Enqueue() method. The program that demonstrates this is given as follows:Source Code: Program to enqueue elements in Queue in C#using System; using System.Collections; namespace QueueDemo {   class Example   {      static void Main(string[] args)      {         Queue q = new Queue();         q.Enqueue(1);         q.Enqueue(2);         q.Enqueue(3);         q.Enqueue(4);         Console.WriteLine("Queue elements are:");         foreach (int i in q)         {             Console.Write(i + " ");         }      }   } }The output of the above program is as follows:Queue elements are: 1 2 3 4DequeueThe element at the beginning of the queue can be removed using the Dequeue() method. The program that demonstrates this is given as follows:Source Code: Program to dequeue elements in Queue in C#using System; using System.Collections; namespace QueueDemo {   class Example   {      static void Main(string[] args)      {         Queue q = new Queue();         q.Enqueue(1);         q.Enqueue(2);         q.Enqueue(3);         q.Enqueue(4);         Console.Write("Original queue: ");         foreach (int i in q)         {             Console.Write(i + " ");         }         q.Dequeue();         q.Dequeue();         Console.WriteLine();         Console.Write("Queue after Dequeue() operation: ");         foreach (int i in q)         {             Console.Write(i + " ");         }      }   } }The output of the above program is as follows:Original queue: 1 2 3 4 Queue after Dequeue() operation: 3 4PeekThe element at the beginning of the queue can be viewed without removing it using the Peek() method. The program that demonstrates this is given as follows:Source Code: Program to peek beginning element in Queue in C#using System; using System.Collections; namespace QueueDemo {   class Example   {      static void Main(string[] args)      {         Queue q = new Queue();         q.Enqueue(1);         q.Enqueue(2);         q.Enqueue(3);         q.Enqueue(4);         Console.Write("Queue elements are: ");         foreach (int i in q)         {             Console.Write(i + " ");         }         Console.WriteLine();         Console.Write("Element at the beginning of the queue is: {0}", q.Peek());      }   } }The output of the above program is as follows:Queue elements are: 1 2 3 4 Element at the beginning of the queue is: 1Count the number of elements in a QueueTo count the number of elements in a Queue, use the Count property:Source Code: Program to count the number of elements in a Queue in C#using System; using System.Collections; namespace QueueExample {   class Example {      static void Main(string[] args) {         Queue q = new Queue();                 q.Enqueue('A');         q.Enqueue('B');         q.Enqueue('C');         q.Enqueue('D');         q.Enqueue('E');         q.Enqueue('F');         q.Enqueue('G');         q.Enqueue('H');                 Console.WriteLine("Count = {0} ", q.Count);                 Console.ReadKey();      }   } }The output of the above program is as follows:Count = 8

# Queue Collection in C#

The Queue Collection is used when the data items need to be arranged in a FIFO (First In First Out) manner. When a data item is entered into the collection, it is called enqueue. When the data item is removed from the collection, it is called dequeue.

## Constructors in Queue Collection

The different constructors and their description is given as follows:

Table: Constructors in Queue Collection in C#

ConstructorsDescription
Queue<T>()This constructor initializes a new instance of the Queue<T> class that is empty and has the default initial capacity.
Queue<T>(IEnumerable<T>)This constructor initializes a new instance of the Queue<T> class that contains elements copied from the specified collection and has sufficient capacity to accommodate the number of elements copied.
Queue<T>(Int32)This constructor initializes a new instance of the Queue<T> class that is empty and has the specified initial capacity.

## Properties in Queue Collection

The different properties and their description is given as follows:

Table: Properties in Queue Collection in C#

PropertiesDescription
CountThis property gets the number of elements contained in the Queue<T>.

## Methods in Queue Collection

The different methods and their description is given as follows:

Table: Methods in Queue Collection in C#

Source: MSDN

MethodsDescription
Clear()This method removes all objects from the Queue<T>.
Contains(T)This method determines whether an element is in the Queue<T> or not.
CopyTo(T[], Int32)This method copies the Queue<T> elements to an existing one-dimensional Array, starting at the specified array index.
Dequeue()This method removes and returns the object at the beginning of the Queue<T>.
Enqueue(T)This method adds an object to the end of the Queue<T>.
Equals(Object)This method determines whether the specified object is equal to the current object.
GetEnumerator()This method returns an enumerator that iterates through the Queue<T>.
GetHashCode()This method serves as the default hash function.
GetType()This method gets the Type of the current instance.
MemberwiseClone()This method creates a shallow copy of the current Object.
Peek()This method returns the object at the beginning of the Queue<T> without removing it.
ToArray()This method copies the Queue<T> elements to a new array.
ToString()This method returns a string that represents the current object.
TrimExcess()This method sets the capacity to the actual number of elements in the Queue<T>, if that number is less than 90 percent of current capacity.

## Queue Operations

Some of the stack operations are given as follows:

### Enqueue

An element is added to the end of the queue using the Enqueue() method. The program that demonstrates this is given as follows:

Source Code: Program to enqueue elements in Queue in C#

using System;
using System.Collections;
namespace QueueDemo
{
class Example
{
static void Main(string[] args)
{
Queue q = new Queue();
q.Enqueue(1);
q.Enqueue(2);
q.Enqueue(3);
q.Enqueue(4);
Console.WriteLine("Queue elements are:");
foreach (int i in q)
{
Console.Write(i + " ");
}
}
}
}

The output of the above program is as follows:

Queue elements are:
1 2 3 4

### Dequeue

The element at the beginning of the queue can be removed using the Dequeue() method. The program that demonstrates this is given as follows:

Source Code: Program to dequeue elements in Queue in C#

using System;
using System.Collections;
namespace QueueDemo
{
class Example
{
static void Main(string[] args)
{
Queue q = new Queue();
q.Enqueue(1);
q.Enqueue(2);
q.Enqueue(3);
q.Enqueue(4);
Console.Write("Original queue: ");
foreach (int i in q)
{
Console.Write(i + " ");
}
q.Dequeue();
q.Dequeue();
Console.WriteLine();
Console.Write("Queue after Dequeue() operation: ");
foreach (int i in q)
{
Console.Write(i + " ");
}
}
}
}

The output of the above program is as follows:

Original queue: 1 2 3 4
Queue after Dequeue() operation: 3 4

### Peek

The element at the beginning of the queue can be viewed without removing it using the Peek() method. The program that demonstrates this is given as follows:

Source Code: Program to peek beginning element in Queue in C#

using System;
using System.Collections;
namespace QueueDemo
{
class Example
{
static void Main(string[] args)
{
Queue q = new Queue();
q.Enqueue(1);
q.Enqueue(2);
q.Enqueue(3);
q.Enqueue(4);
Console.Write("Queue elements are: ");
foreach (int i in q)
{
Console.Write(i + " ");
}
Console.WriteLine();
Console.Write("Element at the beginning of the queue is: {0}", q.Peek());
}
}
}

The output of the above program is as follows:

Queue elements are: 1 2 3 4
Element at the beginning of the queue is: 1

### Count the number of elements in a Queue

To count the number of elements in a Queue, use the Count property:

Source Code: Program to count the number of elements in a Queue in C#

using System;
using System.Collections;

namespace QueueExample {
class Example {
static void Main(string[] args) {
Queue q = new Queue();

q.Enqueue('A');
q.Enqueue('B');
q.Enqueue('C');
q.Enqueue('D');
q.Enqueue('E');
q.Enqueue('F');
q.Enqueue('G');
q.Enqueue('H');

Console.WriteLine("Count = {0} ", q.Count);

Console.ReadKey();
}
}
}

The output of the above program is as follows:

Count = 8

## Leave a Reply

Your email address will not be published. Required fields are marked *

### Comments

#### Kshitiz

Hi Team <NAME>, Thank you for such an informative post like this. I am completely new to this digital marketing field and do not have much idea about this, but your post has become a supportive pillar for me. After reading the blog I would expect to read more about the topic. I wish to get connected with you always to have updates on these sorts of ideas. Regards, Kshitiz

#### Ed

The reason abstraction can be used with this example is because, the triangle, circle. Square etc can be defined as a shape, for example.....shape c = new circle(5,0)...the abstract object c now points at the circle class. Thus hiding implementation

## Swift Tutorial

Introduction to Swift Tutorial
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text:
0 like 0 dislike
Use the distributive property and gather the like terms in the equation below.

0 like 0 dislike
-- - - - - - - - - - -- - - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - - - -

$$\blue\textsf{\textbf{\underline{\underline{Question:-}}}}$$

What is -2(4x+5) equal to?

$$\blue\textsf{\textbf{\underline{\underline{Answer and How to Solve:-}}}}$$

Multiply -2 times 4x:-

-8x

Multiply -2 times 5:-

-10

Put the terms together:-

-8x-10

Good luck.

-- - -- - - - - - - - - - - - - - - - - -- - - - - - - - -- - - - - - - - - - - - - - - - - - - - -
by
0 like 0 dislike
-8x - 10

Step-by-step explanation:

To use the distributive property, distribute what is outside the parenthesis to each term in the parenthesis. When you do this, you multiply each term by what's outside the parenthesis.

-2(4x) + (-2)(5)

= -8x - 10

hope this helps :)
by


text:
# Irreducible polynomial in $\mathbb C[x_1,x_2]$ also irreducible in $\mathbb C[x_1,x_2,…x_r]$? [duplicate]

Let $$f_1(x_1), f_2(x_2)$$ be polynomials in a single variable, of relatively prime degree, with complex coefficients. If $$f_1(x_1)+f_2(x_2)$$ is irreducible in $$\mathbb C[x_1,x_2]$$, then is it irreducible in $$\mathbb C[x_1,x_2,...x_r]$$ for every $$r\ge 3$$ ?

## marked as duplicate by <NAME>, Community♦Feb 11 at 21:01

Yes; if $$f:=f_1(x_1)+f_2(x_2)$$ is irreducible in $$\Bbb{C}[x_1,x_2]$$ then it is prime, so the quotient $$\Bbb{C}[x_1,x_2]/(f)$$ is an integral domain. For every $$r\geq3$$ you have $$\Bbb{C}[x_1,\ldots,x_r]/(f)\cong(\Bbb{C}[x_1,x_2]/(f))[x_3,\ldots,x_r],$$ which is again an integral domain, so $$f$$ is also prime and hence irreducible in $$\Bbb{C}[x_1,\ldots,x_r]$$.
Alternatively, suppose $$f$$ is reducible in $$\Bbb{C}[x_1,\ldots,x_r]$$ for some $$r\geq3$$, and write $$f=gh$$ with $$g,h\in\Bbb{C}[x_1,\ldots,x_r]$$ nonconstant. Because $$f$$ is irreducible in $$\Bbb{C}[x_1,x_2]$$ either $$g$$ or $$h$$ must have a monomial term that is a multiple of some $$x_k$$ for some $$k\geq3$$. That is to say, there exists some $$k\geq3$$ such that either $$\deg_{x_k}g>0$$ or $$\deg_{x_k}h>0$$. Then $$\deg_{x_k}f=\deg_{x_k}gh=\deg_{x_k}g+\deg_{x_k}h>0,$$ a contradiction. So $$f$$ is also irreducible in $$\Bbb{C}[x_1,\ldots,x_r]$$.
If $$D$$ is any domain and if for $$a\in D$$, $$a\neq 0$$, we have $$a=bc$$ for $$b,c\in D[x]$$, then $$b,c\in D$$. To see this, note that $$D[x]$$ is also a domain, so in $$D[x]$$ we have: $$0=\deg(a)=\deg(bc)=\deg (b)+\deg(c)$$, where from $$\deg(b)=\deg(c)=0$$, i.e. $$b,c\in D$$. (We use here that for $$a\in D[x]$$ such that $$a\neq 0$$: $$a\in D$$ iff $$\deg(a)=0$$.)
Now by induction we see that any factorization in $$D[x_1,\ldots,x_n]$$ of an element from $$D$$ is already a factorization in $$D$$. Here, if you have an irreducible element in $$D=\mathbb C[x_1,x_2]$$, it stays irreducible in $$D[x_3,\ldots,x_n]=\mathbb C[x_1,\ldots,x_n]$$.


text:
# I Symmetry on the partition function

1. Jun 4, 2017

### Mr rabbit

We have a partition function

$\displaystyle Z=\frac{1}{N! \: h^{f}} \int dq\: dp \:e^{-\beta H(q,p)}$

And we obtain, for example, the pressure by $\displaystyle p = \frac{1}{\beta} \frac{\partial\: \ln Z}{\partial V}$. So if we do the transformation $Z \rightarrow a Z$ where $a >0$ we obtain the same pressure. This is not a symmetry? It should exist a conserved quantity?

2. Jun 6, 2017

### Twigg

The transformation $Z \rightarrow aZ$ simply takes every Boltzmann factor and multiplies it by $a$, which is equivalent to adding a constant amount of energy $\frac{1}{\beta}ln(a)$ to the energy of each state. Shifting every energy up or down consistently won't change any observables. There's no reason why that would result in any conservation laws. If you're referring to Noether's theorem, the Hamiltonian of the system isn't invariant and the partition function is a dynamical variable. Hope that helps!

3. Jun 6, 2017

Thank you!


text:
Opened 8 years ago

Closed 8 years ago

# Plugin not visible in the Menu selection

Reported by: Owned by: davide.frignati@… <NAME> normal RevtreePlugin normal Menu selection 0.11

### Description

I've installed the plugin and update the trac.it within [components] with revtree=*.enabled and within [trac] with repository_tpye = snv and base_url with the position of the trac project where folders "attachemnts,conf,db,htdocs,log,plugins,....." are

The problem is that I cannot find the Menu selection. Where I've made installation wrong?

Thanks

### comment:1 Changed 8 years ago by <NAME>

There are a LOT of typos in your description. Can you please fix them?

Please copy here the changes (and only them) you made to trac.ini, from the description it seems you mixed and matched unrelated configuration options.

### comment:2 Changed 8 years ago by anonymous

Modification done in trac.ini:

[components] revtree.* = enabled

[trac] repository_type = svn base_url =
sdfg1\Elec_daily\trac\AgrotronSoftware

====================================================== the folder
sdfg1\Elec_daily\trac\AgrotronSoftware contains the subfolder attachemnts,conf,db,htdocs,log,plugins,templates

### comment:3 follow-up:  4 Changed 8 years ago by davide.frignati@…

Modification done in trac.ini:

[components] revtree.* = enabled

[trac] repository_type = svn base_url =
sdfg1\Elec_daily\trac\AgrotronSoftware?

====================================================== the folder
sdfg1\Elec_daily\trac\AgrotronSoftware? contains the subfolder attachemnts,conf,db,htdocs,log,plugins,templates

### comment:4 in reply to:  3 Changed 8 years ago by <NAME>

[trac] repository_type = svn base_url = \\sdfg1\Elec_daily\trac\AgrotronSoftware

Not related to this issue, but \\sdfg1\Elec_daily\trac\AgrotronSoftware is not a URL, it is a Windows share path. Not a web standard, not supported. A valid URL should starts with either http:// or https://

Be sure to enable "DEBUG" trace level in Trac and verify within the debug traces that the Revtree plugin is actually loaded.

BTW: be sure to preview your comment before submitting it: the comment text box expects Wiki-formatted text.

### comment:5 follow-up:  6 Changed 8 years ago by davide.frignati@…

I've enabled the DEBUG trace and in the log file the plugin appears as installed.
On the plugin Menu I cannot see that the plug in is installed. Have I to give some permission?
I've installed the plugin version 0.6.3 since I'm using Trac 0.11
Sorry for my bad Wiki formatting, I'm new in this tool
Concerning windows share path, this is working for the allocation of the svn repository

### comment:6 in reply to:  5 Changed 8 years ago by <NAME>

On the plugin Menu I cannot see that the plug in is installed. Have I to give some permission?

You meant the plugin menu from the Administration panel, right? No, you don't need special permission for the plugin to appear in this panel. However, you need to give the REVTREE_VIEW permission to get the Revtree menu in the main menu bar.

I've installed the plugin version 0.6.3 since I'm using Trac 0.11

Right.

Concerning windows share path, this is working for the allocation of the svn repository

SVN repository needs a PATH, so it is correct. base_url needs a URL.

PATH and URL are not the same kind of information.

### comment:7 follow-ups:  8  9 Changed 8 years ago by anonymous

Looking better at the log I can find :

2009-07-06 17:46:28,390 Trac[loader] ERROR: Skipping "revtree.optimizer = revtree.optimizer": (can't import "No module named revtree.optimizer")

2009-07-06 17:46:28,405 Trac[loader] ERROR: Skipping "revtree.web_ui = revtree.web_ui": (can't import "No module named revtree.web_ui")

2009-07-06 17:46:28,405 Trac[loader] ERROR: Skipping "revtree.enhancer = revtree.enhancer": (can't import "No module named revtree.enhancer")

It seem that the plugin is looking for something in c:\python25 that cannot be found

### comment:8 in reply to:  7 Changed 8 years ago by <NAME>

Looking better at the log I can find

Tip: use block quoting / verbatim blocks to quote traces, it will save you a lot of hassle:
use {{{ ... }}} where ... is the pasted text you want to quote.

### comment:9 in reply to:  7 Changed 8 years ago by <NAME>

Looking better at the log I can find :

It seem that the plugin is looking for something in c:\python25 that cannot be found

It should not, there is an installation issue here but hard to diagnose.

How did you install the plugin exactly:

• how did you retrieve it, how did you install it, how is configured you Trac setup?
• have you defined the EGG cache directory?
• have you installed any other plugin?

### comment:10 follow-up:  11 Changed 8 years ago by davide.frignati@…

built with python
Installed with plugin intrface
EGG cache directory is for me unknow item
Other plugin are installed (excel report plugin, ticket import, etc...)

Wiki formatting is not so easy, WikiFormatting help page should have more examples

### comment:11 in reply to:  10 ; follow-up:  12 Changed 8 years ago by <NAME>

from svn or as a zip file?

EGG cache directory is for me unknow item

Then you should really consider reading Trac installation guide: wiki:TracPlugins#SettingupthePluginCache

### comment:12 in reply to:  11 Changed 8 years ago by william

Hello Emmanuel, thanks for the support!!

There was an error during the build procedure so that we ended up with an empty egg actually! Now the main plugin is working, but we have a problem with the RevtreePlugin/MergeInfoEnhancer plugin.

The plugin builds and installs correctly and it works, unless we select a revision range that includes revision not shown in the (basic) revision graph.

For example we have

• revision 7: deleted Branches/branch_xxx
• revision 8: created Branches/branch_yyy

Both are not shown in the basic revision graph. If we enable the MergeInfoEnhancer plugin and we build the revision graph including this two revision we have an error

No node Branches/branch_xxx at revision 7

Both were empty branches. I have looked inside the code enhancer.py and I have seen the FIXME message related to empty branches. Can it be related?

### comment:13 Changed 8 years ago by william.spinelli@…

Resolution: → fixed new → closed

It seems that the problem with the RevtreePlugin/MergeInfoEnhancer described above was similar to that of ticket #4837.

Applying the patch attachment:ticket:4837:deleted-branches.patch the problem was solved.

### comment:14 Changed 8 years ago by <NAME>

Resolution: fixed closed → reopened

### comment:15 Changed 8 years ago by <NAME>

Resolution: → duplicate reopened → closed

### Modify Ticket

Change Properties


text:
Differences between revisions 11 and 12
 ⇤ ← Revision 11 as of 2011-12-04 14:15:49 → Size: 6259 Editor: OliverSchaefer Comment: ← Revision 12 as of 2019-04-30 12:00:41 → ⇥ Size: 6243 Editor: OliverSchaefer Comment: updated doxygen home page link Deletions are marked like this. Additions are marked like this. Line 1: Line 1: '''Doxygen''' is a documentation system for C++, C, Java, Objective-C, Python, IDL (Corba and Microsoft flavors), Fortran, VHDL, PHP and C#. For details, see http://www.stack.nl/~dimitri/doxygen/. Here only some basic suggestions of how to document your code will be given. '''Doxygen''' is a documentation system for C++, C, Java, Objective-C, Python, IDL (Corba and Microsoft flavors), Fortran, VHDL, PHP and C#. For details, see http://www.doxygen.nl. Here only some basic suggestions of how to document your code will be given.

Doxygen is a documentation system for C++, C, Java, Objective-C, Python, IDL (Corba and Microsoft flavors), Fortran, VHDL, PHP and C#. For details, see http://www.doxygen.nl. Here only some basic suggestions of how to document your code will be given.

In order to generate doxygen based documentation, you need to follow four steps:

• have the doxygen executable installed on your computer (this is already done on our DESY machines)

• create a configuration file
• run doxygen to create the documentation (HTML or LaTeX based).

All these steps will be explained in the following.

## How to document your code

It is highly recommended that you document your code. This makes your life easier not only for potential users of your code, but also for you, if you are going to reuse your code after a long period of time.

• Note that the actual documentation consists in comments you write in the header file

What you should in any case document is:

• who is the author of the class (Marlin processor, etc) and the last modification date
• what is the purpose of your class
• how it can be used (and/or possible caveats)
• what are the methods of the class good for? (Even if it seems trivial for you, don't be lazy, and tell it also other people)
• if you use special algorithms or if your class is based on certain assumptions, document them

Bellow is an example of a dummy class:

#ifndef MYCLASS_HH
#define MYCLASS_HH

/** @brief Dummy class used for illustration purposes. Doing something with it.

Detailed description follows here.
@author <NAME>
@date March 2008
*/
class myClass{
/** Default constructor. Does something.
*/
myClass();
/** My constructor. Initializes the coordinates.
*/
myClass(float x, float y, float z);
/** Default destructor.
*/
~myClass();

public:
/**Print out the private members of the class
*/
void print();
/**Calculate distance from origin.
The distance is calculated with the formula
\f$\sqrt{(x-x0)^2+(y-y0)^2+(z-z0)^2}\f$.
@param x - x coordinate of the point
@param y - y coordinate of the point
@param z - z coordinate of the point
@return distance to point
*/
float getDistance(float x, float y, float z);

private:
float _x0;/**<initial x-coordinate*/
float _y0;/**<initial y-coordinate*/
float _z0;/**<initial z-coordinate*/

};

#endif

Have a look at the doxygen output. It is easy to see which is the effect of the comments.

Basically, C-style comments are used, e.g.

/**
Commenting something...
*/

The doxygen \brief command was used in the class description. It ends at the end of the paragraph. You can try to delete it, to see the difference.

The methods of a class are easily described like this:

/** My function doing something...
@param param1 first parameter
@param param2 second parameter
@return value return value
*/
float myFunction(float param1, float param2){
....
}

Formulas can be inserted as in LaTeX files, but put between a pair of \f$commands: \f$x_0 + y^3\f\$

Members of a class are documented like this:

int var; /**< Detailed description after the member */

where the sign '<' just tells doxygen that documentation follows after the member.

These are only a few very basic and easy steps which you need to follow to create/improve the documentation Of course, this is not the only way in which things can be done. But as you can see, you don't need to be an expert in doxygen to improve significantly the documentation which we all miss...

## How to create a configuration file

Usually, the HCAL software packages already contain a doxygen configuration file, see for example /group/hcal/calice_soft/pro_test/source/calice_userlib/doc/Doxyfile, such that you don't need to create an additional one. However, if you want to do this for private purposes, you have to:

doxygen -g <config_file>

Then you can edit the default configuration file to serve your needs. The options are explained in the file. Some of the important things you need to pay attention to are:

• the name of your project:

PROJECT_NAME = 'your project name'
• the input files (relative to the directory where you run doxygen):

INPUT = ../include ../src
• the directory where to put the documentation (if you leave it empty, then the documentation will be created in the directory where you run doxygen):

OUTPUT_DIRECTORY = 'your output directory'
• the type of documentation you want to generate (HTML, LaTeX and/or something else):

GENERATE_HTML = YES
• If HTML chosen, the following tells doxygen where to put the html documentation relative to OUTPUT_DIRECTORY:

HTML_OUTPUT = html/

and

HTML_FILE_EXTENSION = .html

tells what the extension of the html files should be (.htm or .html).

Here you can find an example of a Doxyfile with which the above example was generated. Have fun with it and feel free to modify it according to your needs.

## How to run doxygen

doxygen <config_file>

This takes a bit, probably you will see some warnings about undocumented methods, etc, which you can ignore in the first stage (you still get the documentation you added).

Suppose you decided yourself for an HTML documentation. In the directory you ran doxygen, you will get a directory called html. There, the file you are interested in is index.html. You can view it in your favourite browser. For example, open: file:///group/hcal/calice_soft/pro_test/source/calice_userlib/doc/html/index.html in your internet browser (on a FLC machine). By clicking, you can find a lot of useful information about the classes and methods present in the package calice_userlib.

How to document your code using doxygen (last edited 2019-04-30 12:00:41 by OliverSchaefer)


text:
# Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

## Author

Listed:
• <NAME>

() (Hunan Institute of Engineering)

• <NAME>

(Hunan Normal University)

• <NAME>

(University of Calgary)

## Abstract

Abstract In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters $$(\beta ^{\top },\theta ^{\top })^{\top }$$ ( β ⊤ , θ ⊤ ) ⊤ is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of $$\beta$$ β is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of $$\beta$$ β . The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.

## Suggested Citation

• <NAME> & <NAME> & <NAME>, 2018. "Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(3), pages 255-281, April.
• Handle: RePEc:spr:metrik:v:81:y:2018:i:3:d:10.1007_s00184-018-0642-7
DOI: 10.1007/s00184-018-0642-7
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## References listed on IDEAS

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1. <NAME> & <NAME>, 2006. "Empirical likelihood confidence regions in a partially linear single‐index model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 549-570, June.
2. <NAME> & Lau, Tai-Shing, 2000. "Empirical Likelihood for Partially Linear Models," Journal of Multivariate Analysis, Elsevier, vol. 72(1), pages 132-148, January.
3. <NAME> & <NAME> & <NAME> & <NAME>, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
4. <NAME> & <NAME> & <NAME>, 2009. "Effects of data dimension on empirical likelihood," Biometrika, Biometrika Trust, vol. 96(3), pages 711-722.
5. <NAME> & <NAME>, 2013. "Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 305-322, March.
6. Xia, Yingcun & Härdle, Wolfgang, 2006. "Semi-parametric estimation of partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1162-1184, May.
7. Xue, Liu-Gen & Zhu, Lixing, 2006. "Empirical likelihood for single-index models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1295-1312, July.
8. <NAME> & <NAME> & <NAME>, 2006. "Efficient semiparametric estimator for heteroscedastic partially linear models," Biometrika, Biometrika Trust, vol. 93(1), pages 75-84, March.
9. <NAME> & <NAME>, 2014. "Semiparametric efficient estimation for partially linear single-index models with responses missing at random," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 33-50.
10. <NAME>, 2002. "Empirical likelihood-based inference in linear errors-in-covariables models with validation data," Biometrika, Biometrika Trust, vol. 89(2), pages 345-358, June.
11. <NAME>, 2009. "Empirical likelihood for heteroscedastic partially linear models," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 387-396, March.
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### Keywords

Empirical likelihood; High-dimensional data; Heteroscedasticity; Partially linear single-index model; Semiparametric efficiency;

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text:
University Calculus: Early Transcendentals (3rd Edition)

$\dfrac{-\ln |x+1|}{2}+\dfrac{3\ln |x+3|}{2} +c$
Integration by parts formula suggests that $\int p'(x) q(x)=p(x) q(x)-\int p(x) q'(x)dx$ $\int \dfrac{x}{x^2+4x+3}=\int \dfrac{x}{(x+1)(x+3)}$ Write the integral into partial fractions as follows: $\int \dfrac{x}{(x+1)(x+3)}=\int [-\dfrac{1}{2(x+1)}+\dfrac{3}{2(x+3)}]dx$ Hence, $\int [-\dfrac{1}{2(x+1)}+\dfrac{3}{2(x+3)}]dx=\dfrac{-\ln |x+1|}{2}+\dfrac{3\ln |x+3|}{2} +c$