text:
# Lagrangian Method- 1st form? Zwangskraefte?

1. Jun 10, 2004

### clumsy9irl

Lagrangian Method-- 1st form? Zwangskraefte?

Mechanics will be the death of me. For some reason, I can do E&M, and particle stuff, but give me a pendulum, and I'm dumbfounded.

THat said, I'm working on some problems using Lagrangians, and the only book I have available to me is Goldstein (which is a bit above where I'm at, but helpful, once it clicks). The course I'm taking will count as my uppper level, undergrad mechanics course.

Anyway, I've got a problem. Supoose you have a particle that's moving frictionless in a gravitational field on the inside of a paraboloid (which is described in cylindrical coordinates as r= az.

How do I come up with the Lagrangian for this? If given one, I can usually figure out more on how to solve this, but it's stumping me. And another thing, I can't figure out what the english term for 'zwangskraft' would be, forced force? I'm supposed to find the size and direction of these forces, but I'm not really sure what it means. Eeep.

There's a note saying that I'm supposed to make it a requirement that energy and rotational momentum are conserved.

THat isn't heling me at all. Suggestions? ANy other suggestions for texts (that aren't too expensive), that might help me? My exam is in 3 weeks, and I'm lost.

2. Jun 10, 2004

### turin

This seems to hint that the problem has a reduced number of dimensions (from 3). In other words, you aparently have a particle confined on some surface. The frictionless stipulation tells you that the Euler Lagrange expression is equation to zero (which gives you the Euler Lagrange equation).

This seems to be a description of the submanifold onto which the particle is constrained to move. However, that equation doesn't look like a paraboloid to me? In fact, I am even more confused about the bold vs. non-bold.

It's hard to say without a better understanding of the physical situation (because Lagrangian are Taylor-made). I will try to give you a general approach:

If this paraboloid is a 2-D surface, then devise some 2-D coordinate system that is capable of unambiguously describing the position of the particle on this surface. It is usually a good idea to do this so that either the kinetic energy or the potential energy only depends on one of the coordinates and not the other. Then, you need to express the two energies in terms of these new coordinates: T = (1/2)mv2 -> ?, V = mgh -> ?. Finally, of course, L = T - V.

If this is a 1-D parabolic curve, then the same as above, except that you don't worry so much about the coordinate dependence of T and V. You still need to express both in terms of the 1 coordinate that you chose. For instance, suppose the parabola is y = x2. Then
V = mgy = mg(x2)
and
T = (1/2)m{vx2 + vy2}
= (1/2)m{vx2 + 2xvx2}
= (1/2)m{1 + 2x}vx2

I'm not clear on what your submanifold is, though.

From the sound of it, I would guess it means "constraint force." This is the same basic idea as the normal force you deal with in 1st semester physics. You don't approach the problem knowing it. In fact, the value doesn't matter to you except that it necessarily counteracts (usually) the weight (and any inertial forces) to balance the force and keep the particle on your submanifold. (In fact, this is just about the most important idea behind the Lagrangian formalism.)

You might try
www.dictionary.com/translator[/URL]

[QUOTE=clumsy9irl]There's a note saying that I'm supposed to make it a requirement that energy and rotational momentum are conserved.[/QUOTE]Well, you have problably have two mechanical energies: kinetic and potential. Since there's no friction, these should be conserved. The rotational momentum is conserved because you're dealing with forces that have a (scalar) potential energy associated with them.

Last edited by a moderator: Apr 20, 2017
3. Jun 10, 2004

### Gokul43201

Staff Emeritus
Must be $$r^2=az$$

Unless it's a conical surface.

4. Jun 10, 2004

### arildno

German has such a delightfully(?) dictatorial ring to it..
"Zwangskräfte" means, (in non-physics), "forces of subjugation"..

5. Jun 10, 2004

### clumsy9irl

Yes, r^2= az. Typo that happened when bolding. Sorry :(

But thank you very much for your response. THat's exactly what I need, more of a step by step thought process on how to solve a problem like this. It all seems a bit clearer now :)


text:
# 3.11. LSTM - Long Short Term Memory¶

http://data.is/1bKs2mG

International airline passengers: monthly totals in thousands. Jan 49 – Dec 60

In [1]:

from conx import Network, Layer, LSTMLayer, plot, frange

Using Theano backend.
conx, version 3.5.5

For this experiment, we will use the monthly counts of international airline passengers on tickets between 1949 and 1961, in thousands:

In [2]:

data = [112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 104, 118, 115,
126, 141, 135, 125, 149, 170, 170, 158, 133, 114, 140, 145, 150,
178, 163, 172, 178, 199, 199, 184, 162, 146, 166, 171, 180, 193,
181, 183, 218, 230, 242, 209, 191, 172, 194, 196, 196, 236, 235,
229, 243, 264, 272, 237, 211, 180, 201, 204, 188, 235, 227, 234,
264, 302, 293, 259, 229, 203, 229, 242, 233, 267, 269, 270, 315,
364, 347, 312, 274, 237, 278, 284, 277, 317, 313, 318, 374, 413,
405, 355, 306, 271, 306, 315, 301, 356, 348, 355, 422, 465, 467,
404, 347, 305, 336, 340, 318, 362, 348, 363, 435, 491, 505, 404,
359, 310, 337, 360, 342, 406, 396, 420, 472, 548, 559, 463, 407,
362, 405, 417, 391, 419, 461, 472, 535, 622, 606, 508, 461, 390,
432]

Plotting the data shows a regular, but varying, cyclic pattern:

In [3]:

plot(["", data],
title="International airline passengers: monthly totals in thousands. Jan 49 – Dec 61",
xlabel="year",
ylabel="counts (thousands)",
xs=frange(1949, 1961, 1/12))

Let’s scale the counts into the range 0 - 1:

In [4]:

def scale(data):
"""
Scale data to between 0 and 1
"""
minv = min(data)
maxv = max(data)
span = maxv - minv
return [(v - minv)/span for v in data]

In [5]:

scaled_data = scale(data)

In [6]:

plot(["Scaled Data", scaled_data])

For our dataset, we will contruct a history sequence. First, we need to put each scaled value into a list. This is the list of features. In our case, we just have the one feature:

In [7]:

sequence = [[datum] for datum in scaled_data]

We wish that the inputs -> targets are constructed as follows:

1. [S0] -> S1
2. [S1] -> S2

where Sn is a list of features in the sequence.

We need to inform the network of the shape of the sequence. We need the:

• time_steps - the length of the history
• batch_size - how many vectors are the inputs composed of?
• features - the length of each input bank vector
In [8]:

time_steps = 1  # history
batch_size = 1  # how many to load at once
features = 1    # features (length of input vector)

In [9]:

def create_dataset(sequence, time_steps):
dataset = []
for i in range(len(sequence)-time_steps-1):
dataset.append([sequence[i:(i+time_steps)],
sequence[i + time_steps]])
return dataset

In [10]:

dataset = create_dataset(sequence, time_steps)

In [11]:

print(dataset[0])
print(dataset[1])

[[[0.015444015444015444]], [0.02702702702702703]]
[[[0.02702702702702703]], [0.05405405405405406]]

Now we construct the network giving the batch_shape in terms of (look_back, banks, width):

In [12]:

net = Network("LSTM")
net.connect()

In [13]:

net.dataset.load(dataset)

In [14]:

net.dashboard()

In [15]:

net.dataset.split(.33)

In [16]:

net.propagate([[.02]])

Out[16]:

[-0.000337743986165151]

In [17]:

outputs = [net.propagate(i) for i in net.dataset.inputs]
plot([["Network", outputs], ["Training data", net.dataset.targets]])

In [18]:

if net.saved():
net.plot_loss_acc()
else:
net.train(100, batch_size=batch_size, shuffle=False, plot=True, save=True)

In [20]:

outputs = [net.propagate(i) for i in net.dataset.inputs]
plot([["Network output", outputs], ["Training data", net.dataset.targets]])

NOTE: Even though the above plot of the Network output appears to closely track the Training data, don’t be fooled! As can be seen in the accuracy plot after training, the trained network has about 70% accuracy. Why does it look so good in the plot? Take a moment to consider how it can look so good, and yet be so bad.

## 3.11.1. LSTM with Window¶

In [21]:

time_steps = 3

In [22]:

dataset = create_dataset(sequence, time_steps)

In [23]:

print(dataset[0])
print(dataset[1])

[[[0.015444015444015444], [0.02702702702702703], [0.05405405405405406]], [0.04826254826254826]]
[[[0.02702702702702703], [0.05405405405405406], [0.04826254826254826]], [0.032818532818532815]]

In [24]:

net2 = Network("LSTM with Window")
net2.connect()

In [25]:

net2.dataset.load(dataset)
net2.dataset.split(.33)

In [26]:

net2

Out[26]:

In [27]:

net2.propagate([[0.1], [0.2], [0.3]])

Out[27]:

[0.010807438753545284]

In [29]:

outputs = [net2.propagate(i) for i in net2.dataset.inputs]
plot([["Network output", outputs], ["Training data", net2.dataset.targets]])

In [30]:

if net2.saved():
net2.plot_loss_acc()
else:
net2.train(100, batch_size=batch_size, shuffle=False, plot=True, save=True)

In [31]:

net2.train(100, batch_size=batch_size, shuffle=False, plot=True, save=True)

========================================================================
|  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
#  200 |   0.00173 |   0.98925 |   0.00713 |   0.68085
Saving network... Saved!

In [33]:

outputs = [net2.propagate(i) for i in net2.dataset.inputs]
plot([["Network output", outputs], ["Training data", net2.dataset.targets]])

## 3.11.2. LSTM with State¶

In [34]:

net3 = Network("LSTM with Window and State")
net3.connect()

In [35]:

net3.dataset.load(dataset)
net3.dataset.split(.33)

In [36]:

net2

Out[36]:

In [37]:

if net3.saved():
net3.plot_loss_acc()
else:
net3.train(100, batch_size=batch_size, shuffle=False, plot=True, save=True)

In [39]:

outputs = [net3.propagate(i) for i in net3.dataset.inputs]
plot([["Network output", outputs], ["Training data", net3.dataset.targets]])

## 3.11.3. LSTM - Stacked¶

In [40]:

net4 = Network("LSTM with Window and State and Stacked")
net4.connect()

In [41]:

net4.dataset.load(dataset)
net4.dataset.split(.33)

In [42]:

net4

Out[42]:

In [43]:

net4.propagate([[0.1], [-0.2], [0.8]])

Out[43]:

[0.0064370655454695225]

In [44]:

if net4.saved():

In [45]:

outputs = [net4.propagate(i) for i in net4.dataset.inputs]

format: plain


text:
# $\lambda _{{{\boldsymbol Z}}}$ INSPIRE search

This coupling is $\mathit CP$-conserving ($\mathit C$- and $\mathit P$- separately conserving).
VALUE EVTS DOCUMENT ID TECN  COMMENT
$-0.088$ ${}^{+0.060}_{-0.057}$ $\pm0.023$ 7171 1
 2004 D
L3 ${\it{}E}^{\it{}ee}_{\rm{}cm}$ = $189 - 209$ GeV
• • • We do not use the following data for averages, fits, limits, etc. • • •
2
 2019 CL
CMS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 13 TeV
3
 2018 BZ
CMS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 13 TeV
4
 2017 S
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 7+8 TeV
5
 2017 U
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
6
 2017 O
CMS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
7
 2017 X
CMS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
8
 2016 AR
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
9
 2016 P
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
10
 2014 Y
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 8 TeV
11
 2013 AL
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 7 TeV
12
 2013 BF
CMS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 7 TeV
13
 2012 CD
ATLS ${\it{}E}^{\it{}pp}_{\rm{}cm}$ = 7 TeV
14
 2012 AC
CDF ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
34 15
 2011
D0 ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
334 16
 2010 K
CDF ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
13 17
 2007 Z
D0 ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
17 18
 2006 H
D0 ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
2.3 19
 2005 S
D0 ${\it{}E}^{\it{}p\overline{\it{}p}}_{\rm{}cm}$ = 1.96 TeV
1  ACHARD 2004D study $WW$ $-$pair production, single$-{{\mathit W}}$ production and single$-$photon production with missing energy from 189 to 209 GeV. The result quoted here is obtained using the $WW$ $-$pair production sample. Each parameter is determined from a single$-$parameter fit in which the other parameters assume their Standard Model values.
2  SIRUNYAN 2019CL study ${{\mathit W}}{{\mathit W}}$ and ${{\mathit W}}{{\mathit Z}}$ production in lepton + jet events, with one ${{\mathit W}}$ boson decaying leptonically (electron or muon), and another ${{\mathit W}}$ or ${{\mathit Z}}$ boson decaying hadronically, reconstructed as a single massive large-radius jet. In the electron channel 2,456 (2,235) events are selected in the ${{\mathit W}}{{\mathit W}}$ ( ${{\mathit W}}{{\mathit Z}}$ ) category, while in the muon channel 3,996 (3572) events are selected in the ${{\mathit W}}{{\mathit W}}$ ( ${{\mathit W}}{{\mathit Z}}$ ) category. Analysing the di-boson invariant mass distribution, the following 95$\%$ C.L. limit is obtained: $-0.0065$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ 0.0066.
3  SIRUNYAN 2018BZ study ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit Z}}$ $~\mathit jet~jet$ events at 13 TeV where ${{\mathit Z}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ $/$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ . Isolated electrons and muons are selected with $p_T$ of the leading/sub-leading lepton $>$ 30/20 GeV and $\vert {{\mathit \eta}}\vert$ $<$ 2.4, with the di-lepton invariant mass within 15 GeV of the ${{\mathit Z}}$ mass. The two highest $p_T$ jets are selected with $p_T$ of the leading/sub-leading jet $>$ 50/30 GeV respectively and dijet invariant mass $>$ 200 GeV. Templates in the transverse momentum of the ${{\mathit Z}}$ are utilized to set limits on the triple gauge couplings in the EFT and the LEP parametrizations. The following 95$\%$ C.L. limit is obtained $-0.010$ $<{{\mathit \lambda}_{{Z}}}<$ $0.010$.
4  AABOUD 2017S analyze electroweak production of a ${{\mathit W}}$ boson in association with two jets at high dijet invariant mass, with the ${{\mathit W}}$ boson decaying to electron or muon plus neutrino. In the signal region of dijet mass larger than 1 TeV and leading-jet transverse momentum larger than 600 GeV, 30 events are observed in the data with $39$ $\pm4$ events expected in the Standard Model, yielding the following limit at 95$\%$ CL for the form factor cut-off scale ${{\mathit \Lambda}_{{FF}}}\rightarrow\infty{}$: $-0.053$ $<{{\mathit \lambda}_{{Z}}}<$ $0.042$.
5  AABOUD 2017U analyze production of ${{\mathit W}}{{\mathit W}}$ or ${{\mathit W}}{{\mathit Z}}$ boson pairs with one ${{\mathit W}}$ boson decaying to electron or muon plus neutrino, and the other ${{\mathit W}}$ or ${{\mathit Z}}$ boson decaying hadronically. The hadronic decay system is reconstructed as either a resolved two-jet system or as a single large jet. Analysing the transverse momentum distribution of the hadronic system above 100 GeV yields the following limit at 95$\%$ CL for the form factor cut-off scale ${{\mathit \Lambda}_{{FF}}}\rightarrow\infty{}$: $-0.013$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ $0.013$.
6  KHACHATRYAN 2017O analyse ${{\mathit W}}{{\mathit Z}}$ production where each boson decays into electrons or muons. Events are required to have a tri-lepton invariant mass larger than 100 GeV, with one of the lepton pairs having an invariant mass within 20 GeV of the ${{\mathit Z}}$ boson mass. The ${{\mathit Z}}$ transverse momentum spectrum is analyzed to set a 95$\%$ C.L. limit of: $-0.018$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ $0.016$.
7  SIRUNYAN 2017X study ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$ / ${{\mathit W}}$ ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\mathit q}}{{\overline{\mathit q}}}$ production at 8 TeV where ${{\mathit \ell}}$ is an electron or muon with $p_T$ $>$ 30 or 25 GeV respectively. Suitable cuts are put on the $p_T$ of the dijet system and the missing $\mathit E_{T}$ of the event yielding a total of 285 and 204 ${{\mathit W}}{{\mathit V}}$ events observed in the electron and muon channels. The following 95$\%$ C.L. limit is obtained: $-0.011$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ 0.011.
8  AAD 2016AR study WW production in ${{\mathit p}}{{\mathit p}}$ collisions and select 6636 ${{\mathit W}}{{\mathit W}}$ candidates in decay modes with electrons or muons with an expected background of $1546$ $\pm157$ events. Assuming the LEP formulation and setting the form-factor $\Lambda$ to infinity, a fit to the transverse momentum distribution of the leading charged lepton, leads to a 95$\%$ C.L. range of $-0.019$ $<{{\mathit \lambda}_{{Z}}}<$ 0.019.
9  AAD 2016P study ${{\mathit W}}{{\mathit Z}}$ production in ${{\mathit p}}{{\mathit p}}$ collisions and select 2091 ${{\mathit W}}{{\mathit Z}}$ candidates in 4 decay modes with electrons and muons, with an expected background of $1825$ $\pm7$ events. Analyzing the ${{\mathit W}}{{\mathit Z}}$ transverse momentum distribution, the resulting 95$\%$ C.L. limit is: $-0.016$ $<{{\mathit \lambda}_{{Z}}}<$ 0.016.
10  AAD 2014Y determine the electroweak ${{\mathit Z}}$-dijet cross section in 8 TeV ${{\mathit p}}{{\mathit p}}$ collisions. ${{\mathit Z}}$ $\rightarrow$ ${{\mathit e}}{{\mathit e}}$ and ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \mu}}{{\mathit \mu}}$ decays are selected with the di-lepton $p_T>$ 20 GeV and mass in the $81 - 101$ GeV range. Minimum two jets are required with $p_T>$ 55 and 45 GeV and no additional jets with $p_T>$ 25 GeV in the rapidity interval between them. The normalized $p_T$ balance between the ${{\mathit Z}}$ and the two jets is required to be $<$ 0.15. This leads to a selection of 900 events with dijet mass $>$ 1 TeV. The number of signal and background events expected is 261 and 592 respectively. A Poisson likelihood method is used on an event by event basis to obtain the 95$\%$ CL limit $-0.15$ $<{{\mathit \lambda}_{{Z}}}$ $<$ 0.13 for a form factor value $\Lambda$ = $\infty{}$.
11  AAD 2013AL study ${{\mathit W}}{{\mathit W}}$ production in ${{\mathit p}}{{\mathit p}}$ collisions and select 1325 ${{\mathit W}}{{\mathit W}}$ candidates in decay modes with electrons or muons with an expected background of $369$ $\pm61$ events. Assuming the LEP formulation and setting the form-factor ${{\mathit \Lambda}}$ = infinity, a fit to the transverse momentum distribution of the leading charged lepton, leads to a 95$\%$ C.L. range of $-0.062$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ 0.059. Supersedes AAD 2012AC.
12  CHATRCHYAN 2013BF determine the ${{\mathit W}^{+}}{{\mathit W}^{-}}$ production cross section using unlike sign di-lepton (${{\mathit e}}$ or ${{\mathit \mu}}$) events with high $\not\!\!p_T$. The leptons have $p_T>$ 20 GeV/c and are isolated. 1134 candidate events are observed with an expected SM background of $247$ $\pm34$. The $p_T$ distribution of the leading lepton is fitted to obtain 95$\%$ C.L. limits of $-0.048$ ${}\leq{}{{\mathit \lambda}_{{Z}}}{}\leq{}$ 0.048.
13  AAD 2012CD study ${{\mathit W}}{{\mathit Z}}$ production in ${{\mathit p}}{{\mathit p}}$ collisions and select 317 ${{\mathit W}}{{\mathit Z}}$ candidates in three ${{\mathit \ell}}{{\mathit \nu}}$ decay modes with an expected background of $68.0$ $\pm10.0$ events. The resulting 95$\%$ C.L. range is: $-0.046$ $<$ ${{\mathit \lambda}_{{Z}}}<$ 0.047. Supersedes AAD 2012V.
14  AALTONEN 2012AC study ${{\mathit W}}{{\mathit Z}}$ production in ${{\mathit p}}{{\overline{\mathit p}}}$ collisions and select 63 ${{\mathit W}}{{\mathit Z}}$ candidates in three ${{\mathit \ell}}{{\mathit \nu}}$ decay modes with an expected background of $7.9$ $\pm1.0$ events. Based on the cross section and shape of the ${{\mathit Z}}$ transverse momentum spectrum, the following 95$\%$ C.L. range is reported: $-0.08$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ 0.10 for a form factor of $\Lambda$ = 2 TeV.
15  ABAZOV 2011 study the ${{\mathit p}}$ ${{\overline{\mathit p}}}$ $\rightarrow$ 3 ${{\mathit \ell}}{{\mathit \nu}}$ process arising in ${{\mathit W}}{{\mathit Z}}$ production. They observe 34 ${{\mathit W}}{{\mathit Z}}$ candidates with an estimated background of 6 events. An analysis of the $p_T$ spectrum of the ${{\mathit Z}}$ boson leads to a 95$\%$ C.L. limit of $-0.077<$ ${{\mathit \lambda}_{{Z}}}<$ 0.093, for a form factor $\Lambda$ = 2 TeV.
16  AALTONEN 10K study ${{\mathit p}}$ ${{\overline{\mathit p}}}$ $\rightarrow$ ${{\mathit W}^{+}}{{\mathit W}^{-}}$ with ${{\mathit W}}$ $\rightarrow$ ${{\mathit e}}$ / ${{\mathit \mu}}{{\mathit \nu}}$ . The $p_T$ of the leading (second) lepton is required to be $>$ 20 (10) GeV. The final number of events selected is 654 of which $320$ $\pm47$ are estimated to be background. The 95$\%$ C.L. interval is $-0.16$ $<$ ${{\mathit \lambda}_{{Z}}}<$ 0.16 for $\Lambda$ = 1.5 TeV and $-0.14$ $<$ ${{\mathit \lambda}_{{Z}}}<$ 0.15 for $\Lambda$ = 2 TeV.
17  ABAZOV 2007Z set limits on anomalous TGCs using the measured cross section and $p_T({{\mathit Z}}$) distribution in ${{\mathit W}}{{\mathit Z}}$ production with both the ${{\mathit W}}$ and the ${{\mathit Z}}$ decaying leptonically into electrons and muons. Setting the other couplings to their standard model values, the 95$\%$ C.L. limit for a form factor scale $\Lambda$ = 2$~$TeV is $-0.17$ $<{{\mathit \lambda}_{{Z}}}<$ 0.21.
18  ABAZOV 2006H study ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit W}}{{\mathit W}}$ production with a subsequent decay ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit \nu}_{{e}}}{{\mathit e}^{-}}{{\overline{\mathit \nu}}_{{e}}}$ , ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit e}^{\pm}}{{\mathit \nu}_{{e}}}{{\mathit \mu}^{\mp}}{{\mathit \nu}_{{\mu}}}$ or ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \nu}_{{\mu}}}{{\mathit \mu}^{-}}{{\overline{\mathit \nu}}_{{\mu}}}$ . The 95$\%$ C.L. limit for a form factor scale $\Lambda$ = 2 TeV is $-0.39$ $<\lambda _{Z}<$ 0.39, fixing $\kappa _{Z}$=1. With the assumption that the ${{\mathit W}}{{\mathit W}}{{\mathit \gamma}}$ and ${{\mathit W}}{{\mathit W}}{{\mathit Z}}$ couplings are equal the 95$\%$ C.L. one-dimensional limit ($\Lambda$ = 2 TeV) is $-0.29$ $<\lambda <$ 0.30.
19  ABAZOV 2005S study ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit W}}{{\mathit Z}}$ production with a subsequent trilepton decay to ${{\mathit \ell}}{{\mathit \nu}}{{\mathit \ell}^{\,'}}{{\overline{\mathit \ell}}^{\,'}}$ (${{\mathit \ell}}$ and ${{\mathit \ell}^{\,'}}$ = ${{\mathit e}}$ or ${{\mathit \mu}}$). Three events (estimated background $0.71$ $\pm0.08$ events) with $WZ$ decay characteristics are observed from which they derive limits on the anomalous couplings. The 95$\%$ CL limit for a form factor scale $\Lambda$ = 1.5 TeV is $-0.48$ $<$ ${{\mathit \lambda}_{{Z}}}$ $<$ 0.48, fixing $\mathit g{}^{{{\mathit Z}}}_{1}$ and ${{\mathit \kappa}_{{Z}}}$ to their Standard Model values.
References:
 SIRUNYAN 2019CL
JHEP 1912 062 Search for anomalous triple gauge couplings in WW and WZ production in lepton + jet events in proton-proton collisions at $\sqrt{s} =$ 13 TeV
 SIRUNYAN 2018BZ
EPJ C78 589 Electroweak production of two jets in association with a Z boson in proton?proton collisions at $\sqrt{s}=$ 13 $\,\text {TeV}$
 AABOUD 2017S
EPJ C77 474 Measurements of Electroweak $\mathit Wjj$ Production and Constraints on Anomalous Gauge Couplings with the ATLAS Detector
 AABOUD 2017U
EPJ C77 563 Measurement of ${{\mathit W}}{{\mathit W}}/{{\mathit W}}{{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\mathit q}}{{\mathit q}^{\,'}}$ Production with the Hadronically Decaying Boson Reconstructed as One or Two Jets in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with ATLAS, and Constraints on Anomalous Gauge Couplings
 KHACHATRYAN 2017O
EPJ C77 236 Measurement of the ${{\mathit W}}{{\mathit Z}}$ Production Cross Section in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 and 8 TeV and Search for Anomalous Triple Gauge Couplings at $\sqrt {s }$ = 8 TeV
 SIRUNYAN 2017X
PL B772 21 Search for Anomalous Couplings in Boosted ${{\mathit W}}{{\mathit W}}/{{\mathit W}}$ ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\mathit q}}{{\overline{\mathit q}}}$ Production in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
JHEP 1609 029 Measurement of Total and Differential ${{\mathit W}^{+}}{{\mathit W}^{-}}$ Production Cross Sections in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector and Limits on Anomalous Triple-Gauge-Boson Couplings
PR D93 092004 Measurements of ${{\mathit W}^{\pm}}{{\mathit Z}}$ Production Cross Sections in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector and Limits on Anomalous Gauge Boson Self-Couplings
JHEP 1404 031 Measurement of the Electroweak Production of Dijets in Association with a $\mathit Z$-Boson and Distributions Sensitive to Vector Boson Fusion in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV using the ATLAS Detector
PR D87 112001 Measurement of ${{\mathit W}^{+}}{{\mathit W}^{-}}$ Production in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV with the ATLAS Detector and Limits on Anomalous ${{\mathit W}}{{\mathit W}}{{\mathit Z}}$ and ${{\mathit W}}{{\mathit W}}{{\mathit \gamma}}$ Couplings
 CHATRCHYAN 2013BF
EPJ C73 2610 Measurement of the ${{\mathit W}^{+}}{{\mathit W}^{-}}$ Cross Section in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 7 TeV and Limits on Anomalous ${{\mathit W}}{{\mathit W}}{{\mathit \gamma}}$ and ${{\mathit W}}{{\mathit W}}{{\mathit Z}}$ Couplings
EPJ C72 2173 Measurement of ${{\mathit W}^{\pm}}{{\mathit Z}}$ Production in Proton$−$Proton Collisions at $\sqrt {s }$ = 7 TeV with the ATLAS Detector
 AALTONEN 2012AC
PR D86 031104 Measurement of the ${{\mathit W}}{{\mathit Z}}$ Cross Section and Triple Gauge Couplings in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
 ABAZOV 2011
PL B695 67 Measurement of the ${{\mathit W}}$ ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\mathit \ell}}{{\mathit \ell}}$ Cross Section and Limits on Anomalous Triple Gauge Couplings in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 1.96 TeV
 AALTONEN 2010K
PRL 104 201801 Measurement of the ${{\mathit W}^{+}}{{\mathit W}^{-}}$ Production Cross Section and Search for Anomalous ${{\mathit W}}{{\mathit W}}{{\mathit \gamma}}$ and ${{\mathit W}}{{\mathit W}}{{\mathit Z}}$ Couplings in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96$~$TeV
 ABAZOV 2007Z
PR D76 111104 Measurement of the ${{\mathit p}}$ ${{\overline{\mathit p}}}$ $\rightarrow$ WZ+X Cross Section at $\sqrt {s }$ = 1.96 TeV and Limits on WWZ Trilinear Gauge Couplings
 ABAZOV 2006H
PR D74 057101 Limits on Anomalous Trilinear Gauge Couplings from ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit e}^{\pm}}{{\mathit \mu}^{\mp}}$ , and ${{\mathit W}}$ ${{\mathit W}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ Events from ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ = 1.96 TeV
 ABAZOV 2005S
PRL 95 141802 Production of ${{\mathit W}}{{\mathit Z}}$ Events in ${{\mathit p}}{{\overline{\mathit p}}}$ Collisions at $\sqrt {s }$ =1.96 TeV and Limits on Anomalous $\mathit WWZ$ Couplings
 ACHARD 2004D
PL B586 151 Measurement of Triple Gauge Boson Couplings of the ${{\mathit W}}$ Boson at LEP


text:
Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $$\Psi^*\Psi$$ is probability density function.

Consider the two statements below

1. Probability will change with time. If probability of finding a particle at one place os increased, then the probability of finding this particle in some other place is decreased. So,we can define a probability flow, which gave the continue equation in QM.
2. If wave function is normalised once to one. It can be derived that the wave function is normalised for ever. Means, it's independent of time.

Question:

As per 2nd statement probability is independent of time and as first, probability is dependent of time. What is actually correct? Or is that implies that we can't apply continue equation, once it's normalised?

• Normalization is a global property. There is no contradiction in a normalized function which locally varies with time. This si something which has nothing to do with QM. – GiorgioP Feb 2 at 9:30

The probability density is dependent on time. This is different from what does means for the wavefunction to be normalized forever. We cannot conclude from $$\frac{d}{dt}\int_{-\infty}^{\infty}|\Psi(x, t)|^2dx = 0$$ that $$\frac{\partial}{\partial t} |\Psi(x, t)|^2 = 0$$. As an example, momentum and energy being both conserved doesn't tell us much about how the distribution of momentum and energy change with time: It just says that the total, wherever it is, stays the same. In the same way, the probability density may change with time, so long as the total is conserved.


text:
# All Questions

5 views

### Input equation to find Fixed Point

I have the following equation: Subscript[x, n+1]=Subscript[x, n]^2+c with Subscript[x, 0]=0. this equation is restricted to R and c is Real. How do I input this into Mathematica to find the fixed ...
8 views

### How to transform abstract finite group to permutation group?

It seems that mma only has group function for permute group? Then there is a step to transform the abstract finite one to permute one. As an example, consider the following problem Suppose ...
9 views

### errors STL export, contains rotate or GeometricTransformation

I failed to export a STL file containing a Graphics3D with a Rotate or GeometricTransformation. ...
41 views

### Using Java log4j in Mathematica

Someone has a code example of how log4j can be used to create logs in Mathematica? RLink uses it, as can be checked in ...
22 views

### Streamplot in one direction

R gives explicitly the seeding points and I want the stream lines to be drawn in one direction only, outside the circle ...
29 views

### Lyapunov exponent of a Henon map?

Can anyone help me with calculating Lyapunov exponent of 2D map, for example Henon map? If there is simple code, that would be great!
26 views

### “Reduce” works fine for a non-linear system with 9 equations, but cannot solve it if 10 equations. Any ways to improve the code?

I am trying to solve a class of problems, which are basically about a solution to a system of non-linear equations subject to the constraint that some subset of solutions must be non-decreasing. I ...
62 views

Why does this work: Graphics[{Circle[{0, 0}, -Pi/4]}] But this doesn't? Graphics[{Circle[{0, 0}, N[-Pi/4]]}] Telling me: ...
29 views

### Plotting a parameterized vector in the x-y plane

I am trying to plot the trajectory of a parachutist descending at a constant vertical rate of 8 m/s while a south wind also blows at a rate of 8 m/s. He initially jumps from a height of 300 m. I am ...
37 views

### Comparing unit normal definition in calculus with FrenetSerretSystem

I'm trying to compare the unit normal definition in calculus texts (i.e., $\vec N=\vec T'/||\vec T'||$) where $\vec T$ is the unit tangent vector, with the unit normal vector returned by the ...
61 views

### DSolve documention on web differs greatly from distributed MMA 10.2.0 version

In the course of improving my answer to 95361, I noticed that the DSolve documentation on the web is much better written than that distributed with Mathematica ...
112 views

### Generate multiple integer variables with constant sum

I have asked similar question three day ago[The old question]. it was incorrectly marked as duplicated. This time I make this question more general and closer to reality. I want to build a model of N ...
39 views

### Lloyd relaxation on VoronoiMesh

How can we apply Lloyd's relaxation algorithm to a VoronoiMesh? Thanks. UPDATE Thanks a lot KennyColnago for your answer. Based on Simon Woods sugestion of using ...
111 views

### Trying to model a chemical reaction - NDSolve::litarg error

I am trying to model a chemical reaction with NDSolve. I have specified the known differential equations and parameters, ...
50 views

### Purchasing a laptop to fit Mathematica system requirements [on hold]

I need to buy a new laptop and I intend to run the latest student edition of Mathematica (I am not sure if that is equivalent to Mathematica 10). What I don't understand are the specifications needed ...
68 views

### Plot Text Data in Mathematica

I am trying to analyze some text data for my thesis! How can I plot this data in Mathematica in a barplot based on the frequency of the words and combinations of them? On the Abscissa (...
45 views

### A bug in Rasterize?

I plot a straight diagonal line and then simply rasterize it. Consider what happens (Mathematica 10.2 on Windows 7 x64): ...
54 views

### How can I illustrate the solution space of these equations in 3D?

I would like to plot the solution space $\{\theta,\varphi,\Omega\}$ of this vector equation, \begin{pmatrix} \sin (\theta ) \sin (\varphi ) \sin (2 \pi \Omega )-\sin (2 \theta ) \cos (\varphi ) ...
98 views

### PixelValues along a straight Line

UPDATE to the question I have reformulated the question in order to make it distinct from the one suggested as a duplicate: "Line Intensity Profile From Image." The key difference is that I'm ...
38 views

### Program for fractional-order Lorenz system

Here, I am writing the Mathematica program for fractional-order Lorenz system by using Euler method. Please check the following code and correct it. Program: ...
23 views

### Solve the maximum point of B-spline basis when its mathematical expression is unknown

Description Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denotes a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$. and the $i$-th B-spline basis function of ...
27 views

### Filling option failing when used with ListLogLinearPlot

As mentioned in this question the following filling seems to work well. ...
15 views

### Using Solution of NSolve when Passing a Function Inside [on hold]

I apologize if any of my questions are super easy, I am new to Mathematica. If I do something like this: f[x_] := 1/2 (3 x - x^3) a = NSolve[f == x, x] I will ...
39 views

### Solve 4 equations using Solve [on hold]

I'm trying to solve $4$ systems of equations with $4$ unknowns. There are some trig involved and I keep getting error "Solve::ifun: inverse functions are being used by Solve." Please help me to ...
39 views

### HoldForm -> InputForm without processing the contents

I have programmatically generated a large number of simple equations like 1+1/2. I need them to be 'unprocessed', so they've been run through ...
34 views

### Why do notebooks reset themselves now and then? [on hold]

Now and then I am working on a notebook and suddenly it reset itself. What I mean is that I am entering commands and when I type enter, the notebook command number goes to 1 and I have lost all ...
35 views

### Fourth derivative of a Bessel function [migrated]

I am given with a relation $$\frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x).$$ Using the above equation, I get ...
75 views

### Newton's Method Converges in Mathematica for non-convergent functions

In mathematica I set f[x_] := Sign[x] Sqrt[Abs[x]] and then I solved it in a root finding for Newton's method by doing ...
32 views

### ListPlot3D and plotting a 3 dimensional array [on hold]

I have a 3D array of the form: {{{a1,a2,a3},{b1,b2,b3}...}}} and ListPlot3D only accepts data of form: ...
56 views

### Correlation suspiciously high [on hold]

I used the Correlation function and got a value that seemed suspiciously high. The two series look like this when plotted: Correlation[fri1E, fri2E] // N ...
28 views

### How can i interpret these Clayton copula (parameter 0.723) graph in hydrology? [on hold]

PDF, CDF and Contour plot please explain these three graphs...........
71 views

### Find subsequence A in sequence B A007908 - OEIS

I have sequence A as A007908 - OEIS and pattern B which is a sub-sequence of that sequence A. I need to find index of first occurrence of the given pattern. For example pattern ..1010... I've looked ...
59 views

### Turning table into association for Classify

I have the following sample data for 20 users. I know their sex, and the response for three yes/no questions. I am trying to building a classifier to guess their response to the third question, based ...
85 views

### Inverse of parametrized ellipse area

If you parametrize an ellipse (with the center, not a focus, at the origin) as: {a Cos[t], b Sin[t]} the area traced out from the focus (not origin) for a given ...
65 views

### Can I map a function over a sequence of arguments without making the arguments into a list first?

I would like to set evenFunction[f_][a_, b_, c_, ...] = f[Abs[a], Abs[b], Abs[c], ...] I have come up with two ways to do this so far. Use pure functions ...
81 views

### DSolve gives an empty solution set to a solvable ODE

I really hope this isn't a duplicate, today I was answering this question and was to lazy to solve the quadratic equation on my own and so just asked mathematica to give me the solution of the ...
38 views

### Fail to Create a package [on hold]

This is rather basic. In an empty notebook I type in a single cell ...
41 views

### Call an executable

I have one data file, one executable file and one *.txt with the results from running the executable, all within the same folder. The folder has the path C : \ Users \ Dimitris \ Desktop \C32EXE ...
29 views

### How to prevent space trimming when importing HTML with ImportString?

I am importing a HTML-formatted string with ImportString[string, "HTML"] to properly parse all HTML-escaped characters like ...
154 views

### Consecutive identical integer members of a list

Can you help me to find and count sub-sequences of consecutive, identical integer members of a long list (at least 1000 members)? By sub-sequences I mean runs like ...
25 views

### Launch Front End from .NET MathKernel and return result to .NET

Is it possible to launch a front end and open a notebook from a .NETLink kernel and return a result from the notebook to .NET? In the code below "x" appears in the .NET textbox after 10 seconds but ...
168 views

### A tricky algebraic simplification

This code produces a complicated output: ...
46 views

### Ctrl + Shift + 6 produces some weird symbol instead of power

I'm trying to configure the Ctrl+^ shortcut to produce Superscript instead of Power as described here, but for some reason I ...
51 views

### Mathematica computes too slowly at the start of a session [on hold]

Everyone, I'm a Mathematica new user. I find it a problem that, when I start a new Mathematica session, it executes too slowly for the first command (say, "1+1", it may take a few seconds before it ...
240 views

### Plotting a Bifurcation diagram

I have the following system equation v'(t)=2*G*J1[v(t-τ)]cos(w*τ)-v(t) How do you plot the bifurcation diagram, τ in the x ...
432 views

### Defining indefinitely many functions

Is it possible to define a class of indefinitely many functions f1, f2, f3, ... such that ...
40 views

### How to generate a Log BarLegend for a DensityHistogram

I have a Log count histogram: DensityHistogram[myData, 100, {"Log", "Count"}, ColorFunction -> "Rainbow", ChartLegends -> Automatic] But ...
49 views

### Factoring out a common term

Very simple question, but I can't find a simple answer here or in the documentation. I have an expression: qq = a x + b x^2 + c x^3 and wish to factor out a ...
New to the site and Mathematica. I'm trying to use the finite element method to get a temperature profile. So if $q_0=f(T_0,T_1)$, $q_n=f(T_{n-1},T_n)$ and we also know that ...


text:
# All Questions

371 views

### How can I suppress Print in parallel evaluation?

When using a function that includes Print statements, one can suppress the output by ...
283 views

### n-fold symbolic integral in Mathematica

I am trying to compute symbolically a n-fold integral (n is a parameter of a function) over, say, the cube [0,a]^n. My code looks like this ...
2k views

### Plot Indefinite integral

I'm having a hard time to find a solution to a numerical limitation on mathematica: ...
855 views

### Defining an integer-valued function on an integer domain

This should be an easy question! I want to define a function with domain = the 12 integers {1,12}, with the values f[1]=31, ...
4k views

### Product of elements within a list

Please Consider : list={{5,2,6},{2,8,3}} I need to get the product of all the element within list : As of now I do this : ...
41 views

### How can I download a tweeted image given the tweet id?

ServiceExecute[] supports sending an image via "ImageTweet" i.e. posting an image in a tweet. But how can I achieve the reverse of this and download the image given ...
117 views

### Calculate pixel intensity in two regions

I am just beginning to learn about image processing. I would like to know 1) How to show an original image with a visible mask of an arbitrary shape? 2) how to calculate the pixel intensities in the ...
85 views

### How to prevent Round with hided fractions

I found a strange behavior in Round. If we try: ToString[Round[4.811, 0.01], InputForm] we get: 4.8100000000000005 When I expected 4.81 In order to ...
191 views

### Solving a system of nonlinear equations 2

Regarding this question: Solving a system of nonlinear equations When I try to solve the same system (with the code proposed in Daniel Lichtblau's answer) with different Right Hand Side values, ...
150 views

### How to use the same PlotRange and the same PlotLabel size for a sequence of Graphics?

I wish to use ListAnimate to show a sequence of plots with different plot ranges and differently sized plot lables, all unknown in advance, in a way that keeps the ...
84 views

### Problem with NumberForm when exporting

I have a two-part question regarding NumberForm First question In my attempt at answering this i stumbled upon a strange behavior by NumberForm when the final result is exported. Lets say I ...
222 views

### How do I feed data points into an equation to solve NUMERICALLY?

I start with this equation and solve it numerically for $z(x,y)$ in the range $1 < x < 5$ and $1 < y < 5$: $$\frac{3}{xyz} - 2x - 3y - 5z = 0$$ Then using the data points of $z$ above, ...
150 views

### Finding the minimum of a function

I asked this question Mark Minimum Value in DensityPlot. Another question has come up, or better said, a task: Find the minimum of a function. I haven't been able to write the needed code for that, ...
118 views

### ListPlot with conditions

I have a data file containing the position of about $10^5$ stars of a spiral galaxy. Here is the plot and this is the corresponding code ...
206 views

### How do I exclude the vertical line plotted at a discontinuity?

Condsider function = Tan[1 x] + Tan[2 x] Plot[function, {x, 0, 1.5}] It has a strong discontinuity around 0.8. The problem is , if I want to find ...
189 views

### How to enter chemical equations?

When I want to take chemistry notes and write down a formula such as H_2O or C_2H_5OH it enters "Math mode" which changes the font size and italicizes the letters. Is there an easy way to properly ...
84 views

### Can I change the font of all elements in my barchart with one property?

I have this BarChart: created with this code: ...
125 views

183 views

### Permuting elements within a list

I have s = {2, 3, 5, 7, 1, 4, 6} (* always with diffente elements in {1,2....7}*) l = Length[s] n = 3 and n tells us that we ...
216 views

### Differential Equation in Complex Plane and Parametric Plot

I would like to solve $dx/dt=\sqrt{1 + (I x)^3}$, where x is complex, for some initial condition like $1 - 5 I$ and plot the imaginary part of the solution versus the real part. (A somewhat similar ...
78 views

### ReplaceAll in an expression including large InterpolatingFunction

I have a large InterpolatingFunction that I'm transforming to another basis with a few simple multiplications and additions by (symbolic) constants. The ...
72 views

### How to make the code inside Compile conciser without hurting performance?

Consider the following sample: ...
156 views

### Manipulate for variable number of sliders

I'm trying to use Manipulate for plotting multidimensional data, where higher dimensions are defined as sliders. First, this simple example works: ...
217 views

### Area between Contours in ContourPlot

I feel slightly foolish for asking this because I am so close, but I'm having trouble, so I will anyway. I asked this question two days ago regarding finding the lengths of contours. Now, I'd like to ...
95 views

### How do I constrain movement for lists of Locators using 2nd argument of Dynamic?

I need to display edge labels of a graph in a way that allows the edge labels to be moved. Locator seems the simplest and most obvious function to use. My ...
178 views

### Dynamically Updating Function

I use Mathematica 8 and am working with DynamicModule. I'm trying to enter a function in an InputField then later apply it to two values entered in subsequent Input Fields. The function just does not ...
149 views

### Modular surface of tri-focal cassini curve ContourPlot3D missing feet

I am wondering why the following fails to cover the surface at points near k = 0. ...
123 views

### adding vectors of unequal length [duplicate]

If I have hundreds of vectors of unequal length that I need to sum, what is the cleanest way to do so? So, if I have as input inp={{a,b,c,d},{e,f,g},{h}} I want ...
180 views

### Render dynamic Graphics objects static

As was discussed in the comments of another question (MathLink Graphical Output (Plots) to File), Graphcis objects produced by commands such as ...
139 views

### Plotting lists, exact color values

I read some descriptions of similar problems but I cannot make them work in my example, I don't know why. I have list of 3 dim values to plot: x, ...
282 views

### Enter exact rational numbers easily with decimal notation

Is there a simple way to enter decimal numbers such as 0.123 so that Mathematica interprets it as an exact rational number 123/1000?
238 views

### Replacing values above a threshold with 1 [duplicate]

I have a list with some values and I want to replace a value which is greater than 1 with 1 otherwise keep the value. In this case 10562: ...
381 views

### Estimating a distribution using binned data

I have data from which information has been irretrievably lost through binning and I am trying to find a plausible distribution from which it derives. There is an academic paper suggesting that the ...
281 views

### Can Mathematica identify formulae or sequences of numbers?

Can Mathematica suggest what a formula means? For example if I input $$\sum_{i=0}^n i$$ can it infer that this is a sum of integer values from $0$ to $n$? Or if I enter $0,1,1,2,3,5,8,13$, can ...
86 views

### NotebookWrite Cell with Grid

I'd like to NotebookWrite a Cell in StandardForm with a ...
201 views

### How can I merge multiple sets of morphological components (perhaps selected using different metrics)?

Let's say I use SelectComponents to select morphological components in an image according to some criterion, like "Elongation". Then let's say, I pull out a ...
113 views

### Function to differentiate between date pattern and specific string format

I'd like to write a function that behaves one way when the argument is a Mathematica date, such as {2013,12,15}, or with added hours, minutes and seconds if ...
1k views

### System Modeler can't find g++ in Mac OS X Mavericks

I've just purchased SystemModeler (Mac OS X Mavericks) but can't verify the g++ compiler path. I assume it's in the Xcode package but nothing I've tried seems to work, for example, ...
433 views

### Locator to show coordinates of plot

Old I would like to make the locator follow a plot, giving the coordinates as it moves. I don't really know where to start with this one, but I would like the dynamic plot to look something like ...
144 views

### How to make an efficient difference of sets function?

How to write a nice code that removes all elements from given set A, that occur also in set B? I would like to use DeleteCases[] for this purpose, but have no good ...
301 views

### Finding maximal sublist of linearly independent vectors

Given a list of vectors v = {v1, ..., vn}, which is the fastest way to find a maximal sublist of linearly independent vectors? I could add the vectors one by one to ...
179 views

### Transform Piecewise[] into sum of indicators

I'm interested in transforming a piecewise defined function into a sum of indicator functions, ultimately with the aim to be better able to integrate them. As an example I would like to transform the ...
135 views

### Delay of events detection dependent of “MouseDown” position

Let's say we have a Rectangle with color which depends of CurrentValue["MouseOver"]: ...
145 views

### How to embed graphic-element-specific metadata within a Graphics object?

One of the most crucial requirements for rich, interactive scientific graphics is being able to "annotate" individual graphic elements (e.g. the data points of a scatterplot, the individual curves in ...
84 views

### Constrain movement of Locator

Continuing the discussion started in the thread here, I've tried to recreate the code from scratch. I thus looked to constrain the movement of a locator to the graph defined by ...
109 views

### How to define a function of z[bar], etc

Using Displaying index as subscript on output: e.g. C[i] -> C_i with Notation[...] or Interpretation[..]? I have often ended up wanting to define functions of annotated variables to better match my ...
193 views

### Reducing quality of Graphics3D scene to improve performance?

ProteinData["SERPINA1", "MoleculePlot"] gives the detailed Graphics3D plot below. I'd like to make many small copies of this ...
Why is Mathematica plotting a horizontal line at $y=2$ in the following example: ListPlot[Table[i,{i,1,10}],Frame->True,GridLines->None] How to get rid ...


text: take the object's spectrum, measure the wavelengths of several of the absorption lines in its spectrum, and; use the Doppler shift formula above to calculate its velocity. Radial velocity Stars with planets aren't stationary. Other articles where Radial velocity is discussed: Milky Way Galaxy: Solar motion calculations from radial velocities: For objects beyond the immediate neighbourhood of the Sun, initially it is necessary to choose a standard of rest (the reference frame) from which the solar motion is to be calculated. If we combine our radial velocity measurements with the transits observed by Kepler we are able to obtain the real planet mass (and not just a lower limit). Answer: The centripetal acceleration of a rider can be found using the formula: The tangential velocity formula is given by, Where, r is the radius of the circular path and; ω is the angular velocity; The tangential velocity formula is applied in calculating the tangential velocity of any object moving in a circular path. This method does not measure a rate of change and requires less precision. This gives us a first approximation of what the planet is made of, and paves the the first step along the way of testing for habitability. If so why? Formula ; Online calculator to calculate radial velocity. RESULTS: Maximum radial velocity was successfully measured in 126 healthy fetuses (94%) at gestational ages of 16-40 weeks. (The z in the galaxy context is the measured Δλ/λ, and knowledge of both relativity and cosmology is necessary to translate that correctly into radial velocity.) Velocity gradient ( dv/dx): It is defined as the rate of change of velocity with distance in the direction of increasing distance. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity. Doppler Shift is the change in the frequency of a wave for an observer if the observer is moving relative to the source of the wave. Recap: Your astronomy professor has tasked the class with determining the velocity of Andromeda with respect to the Milky Way. We will discuss it here. The radial velocity of our Sun measured from a point coplanar to the plane of the solar system. Centripetal Acceleration Formula Questions: 1) A certain ride at an amusement park includes a circle of radius 3.00 m, that spins the riders at a velocity of 6.00 m/s. Tangential Velocity Formula Questions. Angular velocity can also be expressed as (angular acceleration = constant): ω = ω o + α t (2c) where. $\begingroup$ @MAFIA36790 I'm trying to see what represents the tangential component and radial component of the linear velocity from the diagram above. One is radial velocity. The Radial Velocity (aka. The observed radial velocity of an astronomical object is subject to several projection effects such as the rotation and the orbital motion of the Earth, the motion of the Sun around the Galactic centre, the motion of our Galaxy within the Local Group, etc. In order to measure stellar radial velocities, we rely on the well-known Doppler effect. Doppler Shift and Radial Velocity. Radial velocity is the component of wind velocity parallel to the direction of the radar beam either toward or away from the radar. Below is the spectrum of M31. The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: {\displaystyle \omega ={\frac {d\phi }{dt}}={\frac {v_{\perp }}{r}}.} V t = ω r . Note that this formula, in which c is the speed of light, is valid only if v << c. This is certainly the case in the present context, though it is not correct for measuring the radial velocities of distant galaxies. Solved Examples. It is given by dv/dx. You thought of three possible ways to do this, one of which will give you the right answer. It is expressed in meter per second (m/s). 1) If the angular velocity of a turning bicycle wheel is 42 rad/s, and the wheel diameter is 68 cm, what is the tangential velocity? Angular Velocity and Acceleration. Method 3: If a star belongs to a known moving group or cluster of stars, then its radial velocity can be deduced from the total velocity of the cluster, the proper motion and the parallax, since the vector sum of the radial velocity and tangential velocity must equal the total velocity. It is the angular portion under the arc’s shadow, between the two lines originating from the center and connected to its ends. But what we will get is just a radial velocity (scalar, move away will resulting to < 0 value and the other hand for the wind that come closer will have a > 0 speed), not wind as a cartesian vector (north-south, east-west, and up-down component). Doppler Spectroscopy) Method relies on measurements of a planet's "wobble" to determine the presence of one or more planets around it. Can anybody also explain me better what is a "shadow" and why do I have these high velocities there? The strong signal with a period of 12 years and a semiamplitude of 12.5 m s −1 is caused by Jupiter, while the longer periodic and smaller variation is the signal caused by Saturn. Radial Velocity Christophe Lovis Universit´e de Gen `eve Debra A. Fischer Yale University The radial velocity technique was utilized to make the ﬁrst exoplanet discoveries and continues to play a major role in the discovery and characterization of exoplanetary systems. First, we calculate the angular displacement, ‘ q ‘, which is the ratio of the length of the arc ‘s’ that an object traces on this circle to its radius ‘r’. Radial velocity methods alone may only reveal a lower bound, since a large planet orbiting at a very high angle to the line of sight will perturb its star radially as much as a much smaller planet with an orbital plane on the line of sight. It is the y-axis velocity (when we think the Earth as (0,0)), and it represents the velocity vector of going backward or forward from the Earth itself. This is a very common technique used to measure the radial component of the velocity of distant astronomical objects. This is usually done by selecting a particular kind of star or… The modern formula tion ofthe principle based on the restricted theory ofrelativity ofAlbert Einstein ... stellar radial velocity especially oflater type stars with many narrow lines. This is most easily pictured by considering just one planet orbiting a star. I want to talk about transverse velocity and radial velocity today. The usual wind shear formula is a vector operation. Systolic and early diastolic maximum velocities increased with gestational age in both ventricles. Example 1. It indicates how velocity changes between two points. $\endgroup$ – Reinhild Van Rosenú Jun 12 '16 at 12:11 The rate of change of the distance between the object and the point is determined as the velocity of an object. Answer: The radius, r = 1/2 diameter of 68 cm = 34 cm = 0.34 m. The angular velocity, ω = 42 rad/s. ω o = angular velocity at time zero (rad/s) The radial velocity is usually measured using the observed Doppler shift of spectral lines, given by the formula Δλ/λ = v/c, where Δλ is the shift in wavelength observed for the object compared to the rest wavelength, λ, and vis the velocity of the object along the line of sight. By balancing radial velocity with transversal/angular velocity can help you pull range or close in on a target, while still being able to survive. What is the centripetal acceleration of a rider? Dear all, it is not clear to me what is meant in Fluent by "radial" and "tangential" velocity .Specifically, in my problem of flow around a cyilinder, radial velocity on cylinder surface ("shadow") is different from zero and of the same order of tangential velocity. It was Edwin Hubble and his assistant Milton Humason who confirmed this principle. The steps are to. There are two components of star velocity when we look from the Sun (or the Earth). Next, we'll look at two expanded examples to further explain these concepts and relate it to Eve. The first thing I wanted to ask was if the tangential component of the linear velocity is represented by the tangent at B, and the second was what represented the radial velocity. Formula of Tangential Velocity. The quantity c is the speed of light (299,792 km/s). This principle is named after Christian Doppler who first proposed the principle in 1842. You've decided to try using the Doppler shift of emission lines from M31's spectrum to find the velocity of M31. Depending on whether the star moves toward us or away from us, its light will be blue- or redshifted, as compared to a nonmoving source. radial velocity (ray -dee-ăl) (line-of-sight velocity) Symbol: v r.The velocity of a star along the line of sight of an observer. However, this isn’t true – in reality, the planets and the Sun orbit their common centre of mass. Formula for Tangential Velocity. A method called radial velocity which is the most effective method for locating extrasolar planets with existing technology. Since velocity is just change of position per unit time (in this case V r = dr/dt) the change in phase between the two pulses gives us the radial velocity. On the basis, the effect of three-dimensional velocity distribution in the flow field within cyclones and other parameters on the grade-efficiency calculation are analyzed and a new equation for grade-efficiency estimation is introduced. It is measured in radians. The formula is not only of advantage theoretically but also fit with experimental results quite well. Radial velocity is an omnipresent concept in astronomy, and a quantity whose precision of determination has im-proved signiﬁcantly in recent years. The tangential velocity of the tire can be calculated as. Visually, angular velocity may be thought of as the arc, or curved line, on a sphere centered at you. This video demonstrates how to calculate radial velocity which in conjunction with transverse velocity can be used to calculate true or real space velocity. v = (π radians/s) ((26 inches) / 2) = 40.8 inches/s. Use the equation for tangential velocity. To be able to interpret the observed radial velocity one must convert it into an appropriate rest frame. Radial velocity definition is - the component of velocity of a particle in the direction of its radius vector. Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? We often picture our Solar System with the Sun in the middle, completely stationary, while all the planets move around it. The radial velocity variations due to the other planets are negligible. The radial velocity of a star can be determined in absolute values, or differentially if only changes of the velocity are of interest. So how we detect or calculate wind shear using radar? In this chapter we describe how the technique works, and the current precision and limitations. Its meaning is gen-erally understood as the object’s motion along the line of sight, a quantity normally deduced from observed spectral-line displacements, interpreted as Doppler shifts.


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Guidelines for authors Publishing Ethics Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 CMFD: Year: Volume: Issue: Page: Find

 CMFD, 2006, Volume 16, Pages 68–95 (Mi cmfd49)

Topological degree and approximation of solutions for nonregular problems of mechanics: Oscillations of satellites on elliptic orbits

I. I. Kosenko

Abstract: We construct the method of approximate solution for the differential equation of oscillations of a satellite on elliptic orbits subject to the gravity torque and the light pressure torque. Parabolic orbits are included as a limiting case. The metric of a weighted Sobolev space is used as a measure of the vicinity. This allows us to construct a uniform approximation of solution with respect to the eccentricity of the orbit.
To prove such an approximation, we use the Leray–Schauder degree theory and the Krasnosel'skij theorem of Galerkin approximations for compact vector fields adapted to the problem under consideration. To establish the uniform estimate of the convergence of approximate solutions to the solution, we also use a modification of an appropriate Krasnosel'skij theorem.

Full text: PDF file (284 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences, 2008, 149:5, 1539–1566

Bibliographic databases:

UDC: 517.5+519.6+629.195.1

Citation: I. I. Kosenko, “Topological degree and approximation of solutions for nonregular problems of mechanics: Oscillations of satellites on elliptic orbits”, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 2, CMFD, 16, PFUR, M., 2006, 68–95; Journal of Mathematical Sciences, 149:5 (2008), 1539–1566

Citation in format AMSBIB
\Bibitem{Kos06} \by I.~I.~Kosenko \paper Topological degree and approximation of solutions for nonregular problems of mechanics: Oscillations of satellites on elliptic orbits \inbook Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2005). Part~2 \serial CMFD \yr 2006 \vol 16 \pages 68--95 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd49} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336446} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 149 \issue 5 \pages 1539--1566 \crossref{https://doi.org/10.1007/s10958-008-0081-5} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84920834817}

• http://mi.mathnet.ru/eng/cmfd49
• http://mi.mathnet.ru/eng/cmfd/v16/p68

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Burov, I. I. Kosenko, “Dvizhenie sputnika s peremennym raspredeleniem mass v tsentralnom pole sil gravitatsii”, Nelineinaya dinam., 13:4 (2017), 519–531
•  Number of views: This page: 236 Full text: 74 References: 30


text:
# The gas in a closed container has a pressure of 3.00 * 10^2 kPa at 30°C (303 K). What will the pressure be if the temperature is lowered to -172°C (146 K)?

Feb 22, 2016

The final pressure will be 145 kPa.

#### Explanation:

This is an example of Gay-Lussac's law, which states that the pressure of a given gas is directly proportional to its Kelvin temperature, when held at a constant volume. The equation that represents this law is ${P}_{1} / {T}_{1} = {P}_{2} / {T}_{2}$.

Given/Known
${P}_{1} = 3.00 \times {10}^{2} \text{kPa}$
${T}_{1} = \text{303 K}$
${T}_{2} = - {127}^{\circ} \text{C+273.15=146 K}$ **

Unknown
${P}_{2}$

Solution
Rearrange the equation to isolate ${P}_{2}$. Substitute the given/known values into the equation and solve.

${P}_{1} / {T}_{1} = {P}_{2} / {T}_{2}$

${P}_{2} = \frac{{P}_{1} {T}_{2}}{T} _ 1$

P_2=(3.00xx10^2"kPa" xx 146cancel"K")/(303cancel"K")="145 kPa"

**Kelvin temperature changed to 146 K, which is $\left(- {127}^{\circ} \text{C} + 273.15\right)$

Excellent Resource for the gas laws: http://chemistry.bd.psu.edu/jircitano/gases.html


text:
# Pure vs mixed strategy Nash Equilibria

Just learning about Nash Equilibria. The pure strategy one is explained as an outcome where both/all players feel like they couldn't have done better given what the others were doing. Mixed strategy is one where even after announcing your strategy openly, your opponents can make any choice without affecting their expected gains. Is there a relationship between the two ? Is the pure strategy one some sort of special case for the mixed strategy one ? (I doubt it, since there can be multiple pure ones, while the linear equations of the mixed one can only give one or infinite number of results). I'd like to intuitively understand the relationship. Thanks !

If you like, you can think of a pure strategy as a mixed strategy in which a player has a 100% chance of picking a certain strategy.

The equilibrium definition is the same for both pure and mixed strategy equilibria ("even after announcing your strategy openly, your opponents can make any choice without affecting their expected gains"). The difference is that in a mixed equilibrium, you are announcing your probability distribution, not the strategy that it randomly produces.

Example: Rock-Paper-Scissors. There are no pure strategy equilibria: If I announce "I'm going to play definitely Rock!" then clearly my opponent will choose Paper; if I know they're going to play paper then I don't want to play Rock anymore, so this is not stable. However, if I announce "I'm going to secretly roll a die, play Rock if it shows 1-2, Scissors for 3-4, and Paper for 5-6!" then my opponent is equally happy with any choice he makes. If he therefore chooses the same strategy as me, then I am equally happy with any choice I make, so this is a mixed equilibrium.

Also, your statement "the linear equations of the mixed one can only give one or infinite number of results" isn't true - there are many games with an in-between number of equilibria (Chicken, for example, has 3 in a two-player version of the game). As an aside, a randomly-generated game will have a finite, odd number of equilibria with probability 1, but that's about all you can say about the number of equilibria.

• GMB, from what I understand, being in pure equilibrium doesn't mean that I could make any choice; it means that I couldn't have done better, but could have done worse. Take the intersection game, where either both cars cross at the same time and crash, or both stop, or one stops and the other goes. The pure strategy NE is the last, because if A knows that B will stop, then why should he waste time stopping, and if B knows that A will go then B wants to avoid crash. However A announcing that he will go doesn't mean that B can do anything. See where I'm confused ? B can have a worse outcome. – Balazs Rau Apr 6 '14 at 17:12
• I agree with your NE analysis. The "equally happy with anything" bit of my post referred specifically to mixed equilibria, sorry if that was unclear. It turns out that if $A$ plays a mixed-strategy equilibrium where he has nonzero chance of playing $k$ different strategies, then there is some set of $k$ strategies for $B$ that are all equally good (and therefore $B$ can randomize among them however he likes). In the special case of $k=1$ ("pure strategy equilibrium"), that means there is (at least) $1$ strategy for $B$ that stands up to scrutiny, which is not so informative. – GMB Apr 6 '14 at 17:20
• This is off on a bit of a tangent from your original post, though =) the answer to your original question is "a pure strategy is just a mixed strategy over exactly 1 strategy." – GMB Apr 6 '14 at 17:22
• Oh man, I feel a little thick... So how does "equally happy with anything" in the mixed case become "I could not have done better" in the general case ? – Balazs Rau Apr 6 '14 at 18:37
• @BalazsRau : Let's say you're playing a game where the players can pick between $n$ different strategies. In general, there might be mixed equilibria that only mix over $k$ of those $n$ strategies. In this case, there will be some linear combination of my $k$ strategies that will make you like some $k$ of your strategies equally well (and the other $n-k$ strategies less than that). In the pure case ($k=1$), this statement is uninteresting: it just says that you like $1$ of your strategies at least as well as all the others, which is obvious. – GMB Apr 6 '14 at 21:25

As far as I know, a mixed strategy vector is a randomized vector where each player can play some action with some probability. i.e., if you play action $a$ with probability $p_a$, I can play (for example) an action $b$ with probability $p_b$. So the set of actions is randomized. There is a random play! That's why it is called $\mathbf{mixed}$.

In pure strategy, if you play $a$ (with probability $1$), I can play for example the same action $a$ but with probability $1$. There is no random play! That's why it is called $\mathbf{pure}$.

In fact, mixed strategy is used to guarantee that every game has at least one Nash equilibrium. Randomize the set of strategy to make them convex sets and apply Brouwer fixed point Theorem to prove this result. A well known theorem due to Nash.

• You might be correct, but it's a little over my head. Someone else please tell me if I should accept. – Balazs Rau Apr 7 '14 at 5:12


text:
MeowLover17 one year ago Simplify √5(10-4√2) A,B,C,D ANSWER CHOICES

1. anonymous

14 15 square root of 2 5 square root of 2 end root minus 4 square root of 10 None of the above

2. anonymous

3. anonymous

$10\sqrt{5}-4\sqrt{10}$

4. anonymous

So the answer would be non of the above?

5. anonymous

None*

6. anonymous

yes

7. anonymous

Thanks you got a medal+fan :D

8. anonymous

:) Thanks !

9. anonymous

I'm going to open another question so i can re-medal you if you can help me with it is that okay?

10. anonymous

ok, post it


text: Subtraction is just addition with negation on the second element and division is just multiplication with inverse on the second element. Then, multiplication follows addition from the distributive property of fields: $a(b+c)=ab+ac$. Real numbers with addition and multiplications form a field and one of the properties of fields is the distributive property. If you wanted to switch them around, you would have to change the operations fundamentally. –  fhyve Dec 14 '12 at 9:55


text:
# Unity: Move Collider up stairs/ramps

The bots have a simple Capsule collider which cannot tilt. Movement is applied across the horizontal axes, gravity is enabled on the rigidbody. It moves up stairs okay with steps roughly 0.2 units high. When the bots speed is too slow or he is moving up steep ramps he gets stuck. I've tried tinkering to fix the problem but to no avail.

    float gravity = currentMovementSpeed * 2f;
Physics.gravity = new Vector3(0, -gravity, 0);

Here is the problem: https://youtu.be/ypCavT3_As0

What is a simple way to consistently force a collider up ramps and stairs?

I managed to tinker my way to success. But if someone has a REAL answer to this question, please go ahead and post it.

I set an empty gameobject inside the collider and raycasted two rays, one straight down, one at a 45deg angle forward (The bots only move forwards). I calculate the incline of the slope from the two hitpoints of the rays.

The steeper the slope, the lower gravity acts onto the object.

Trouble is if the distance of the gameobject off the ground is too low it will not recognise a slope because the collider bumps into it much further away. If the distance is too high, the collider will think the slope has flattened before it actually has. To fix the issues I actually raycasted twice, once from very close off the ground, and one from a distance equal to the radius of the collider:

Seems to work: https://youtu.be/6qfwMhLqrfo

Code:

void setGravity()
{
RaycastHit hitInfoDown;
RaycastHit hitInfoAngle;

Ray rayDown = new Ray(slopeReader.transform.position, -Vector3.up);
Ray rayAngle = new Ray(slopeReader.transform.position, transform.forward + -Vector3.up);

if (Physics.Raycast(rayDown, out hitInfoDown, 1000, shootLayerMask)) { }
if (Physics.Raycast(rayAngle, out hitInfoAngle, 1000, shootLayerMask)) { }

float angleHigh = Vector3.Angle(slopeReader.transform.forward, hitInfoAngle.point - hitInfoDown.point); //Angle 0.5m above the ground
if (hitInfoDown.point.y > hitInfoAngle.point.y)
angleHigh = -angleHigh;

rayAngle = new Ray(slopeReader.transform.GetChild(0).transform.position, transform.forward + -Vector3.up);

if (Physics.Raycast(rayDown, out hitInfoDown, 1000, shootLayerMask)) { }
if (Physics.Raycast(rayAngle, out hitInfoAngle, 1000, shootLayerMask)) { }

float angle = Vector3.Angle(slopeReader.transform.GetChild(0).transform.forward, hitInfoAngle.point - hitInfoDown.point); //Angle 0.1m above the ground
if (hitInfoDown.point.y > hitInfoAngle.point.y)
angle = -angle;

if (Mathf.Abs(angleHigh) > Mathf.Abs(angle))
angle = angleHigh;

float distanceToFloor = (hitInfoDown.point - slopeReader.transform.GetChild(0).transform.position).magnitude - 0.1f;
if (angle > 0 && distanceToFloor < 0.4f)
{
angle = Mathf.Clamp(angle, 0, 60);
float grav = Mathf.Pow(1 - (angle / 60), 5); //Increase power for higher slopes / higher speed on steep slopes (5 works fine for 60 degree slopes)
float speedModifier = Mathf.Clamp(currentMovementSpeed / 2, 0, 1);

grav = grav * speedModifier;
grav = grav - 1; //-(1-grav)


text:
My Math Forum not an integer
 User Name Remember Me? Password

 Number Theory Number Theory Math Forum

 August 8th, 2010, 05:59 PM #1 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 not an integer let n be integers in Z with n>1. Show that ?(^n)(_i=1) (1/i) is not an integer. So i am trying to prove that 1/i is not an integer. I put some number in to see if this is correct and it is. however i don't know how to generalize it if n >1 for all n. thanks.
 August 8th, 2010, 06:26 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: not an integer You're trying to show that H_1 = 1 is the only integer harmonic number. The key step is looking at the denominator and seeing how many primes divide it.
 August 8th, 2010, 06:40 PM #3 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: not an integer Hi again tinynerdi. It's pretty clear that for all n > 1, 1/n is not an integer, right? Because no positive integer greater than 1 divides 1. Now, for the sum, you probably want to use induction on n. Clearly, if n = 2, then $\sum_{i=1}^{2}\frac{1}{i} = \frac{3}{2}$ is not an integer. Now, assume that for some n > 2, the statement is true, i.e. suppose that $S_n= \sum_{i=1}^{n}\frac{1}{i}$ is not an integer. Clearly, it is a rational number, though, so let's say $S_n= p/q$ in lowest terms. We want to show that $S_{n+1}$ is not an integer. But of course $S_{n+1}= S_n + \frac{1}{n+1} = \frac{p}{q} + \frac{1}{n+1} = \frac{pn+p+q}{qn+q}$ Now, see if you can prove that the above fraction cannot be an integer. [Hint: Assume the contrary and derive a contradiction.] Best of luck. Addendum. Sorry, I noticed just after I posted that CRGreathouse had already replied. I did not mean to try to "one-up" his advice, which is just as good.
August 8th, 2010, 06:53 PM   #4
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: not an integer

Quote:
 Originally Posted by Ormkärr Addendum. Sorry, I noticed just after I posted that CRGreathouse had already replied. I did not mean to try to "one-up" his advice, which is just as good.
Yours is much more useful; I was in too much of a hurry to give a real answer (but figured I'd at least leave a hint),

 August 8th, 2010, 07:28 PM #5 Senior Member   Joined: Nov 2009 Posts: 129 Thanks: 0 Re: not an integer thanks two both

 Tags integer

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post BenFRayfield Number Theory 11 July 14th, 2013 10:56 AM harrypham Number Theory 1 July 30th, 2012 11:42 PM Denis Algebra 12 May 6th, 2012 06:01 AM najaa Algebra 5 March 11th, 2012 12:40 PM najaa Number Theory 3 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.


text:
# The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases

Mark A. Webster1,2, Benjamin J. Brown1,3, and Stephen D. Bartlett1

1Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia
2Sydney Quantum Academy, Sydney, NSW, Australia
3Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, University of Copenhagen, 2100 Copenhagen, Denmark

### Abstract

We propose an extension to the Pauli stabiliser formalism that includes fractional $2\pi/N$ rotations around the $Z$ axis for some integer $N$. The resulting generalised stabiliser formalism – denoted the XP stabiliser formalism – allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' – a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.

The Pauli stabiliser formalism allows us to efficiently describe certain quantum states. To describe a state, we specify a list of 'stabiliser generators' which are strings of Pauli $X$, $Y$ and $Z$ operators. The $X$, $Y$ and $Z$ operators can be thought of as half rotations around the $X$, $Y$ and $Z$ axes of the Bloch sphere. The stabiliser generators define a quantum code, and we can perform various calculations efficiently by working with the stabiliser generators rather than the state itself.

In our work, we extend the Pauli stabiliser formalism by defining a $P$ operator which is a $1/N$ rotation around the Z axis. We allow stabiliser generators to be made from $X$ and $P$ operators. This allows us to describe a much wider range of states. We generalise many of the algorithms from the Pauli stabiliser formalism. Unlike the Pauli stabiliser formalism, operations on XP codes cannot always be classically simulated – for instance measurement of arbitrary XP operators on codespaces. The XP formalism allows us to identify a wider range of fault-tolerant logical operators on stabiliser codes – including transversal non-Clifford logical operators which are important in realising universal quantum computation.

### ► References

[1] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/PhysRevA.70.052328.
https://doi.org/10.1103/PhysRevA.70.052328

[2] Juan Bermejo-Vega and Maarten Van Den Nest. Classical simulations of abelian-group normalizer circuits with intermediate measurements. Quantum Info. Comput., 14 (3–4): 181–216, Mar 2014. ISSN 1533-7146. 10.26421/QIC14.3-4-1.
https://doi.org/10.26421/QIC14.3-4-1

[3] Juergen Bierbrauer. Introduction to coding theory. Discrete mathematics and its applications. Taylor and Francis, Boca Raton, second edition, Sep 2016. ISBN 9781315371993. 10.1201/9781315371993.
https://doi.org/10.1201/9781315371993

[4] Benjamin J. Brown, Daniel Loss, Jiannis K. Pachos, Chris N. Self, and James R. Wootton. Quantum memories at finite temperature. Rev. Mod. Phys., 88: 045005, Nov 2016. 10.1103/RevModPhys.88.045005.
https://doi.org/10.1103/RevModPhys.88.045005

[5] Johannes Buchmann and Stefan Neis. Algorithms for linear algebra problems over principal ideal rings. Technical Report TI-7/96, TU Darmstadt, Jan 1996. URL http://tubiblio.ulb.tu-darmstadt.de/101111/.

[6] Earl T. Campbell. The smallest interesting colour code. Blog Post, Sep 2016. URL https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/.
https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/

[7] Andrew Cross, Graeme Smith, John A. Smolin, and Bei Zeng. Codeword stabilized quantum codes. IEEE Transactions on Information Theory, 55 (1): 433–438, Aug 2009. 10.1109/TIT.2008.2008136.
https://doi.org/10.1109/TIT.2008.2008136

[8] Julio Carlos Magdalena de la Fuente, Nicolas Tarantino, and Jens Eisert. Non-Pauli topological stabilizer codes from twisted quantum doubles. Quantum, 5: 398, Feb 2021. 10.22331/q-2021-02-17-398.
https://doi.org/10.22331/q-2021-02-17-398

[9] David Elieser Deutsch. Quantum computational networks. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, 425 (1868): 73–90, Sep 1989. ISSN 1364-5021. 10.1098/rspa.1989.0099.
https://doi.org/10.1098/rspa.1989.0099

[10] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur E. Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson. Pauli stabilizer models of twisted quantum doubles. arXiv:2112.11394 [quant-ph], Dec 2021. 10.48550/arXiv.2112.11394.
https://doi.org/10.48550/arXiv.2112.11394
arXiv:2112.11394

[11] Vlad Gheorghiu. Standard form of qudit stabilizer groups. Physics Letters A, 378 (5): 505–509, Jan 2014. ISSN 0375-9601. 10.1016/j.physleta.2013.12.009.
https://doi.org/10.1016/j.physleta.2013.12.009

[12] D Gottesman. The Heisenberg representation of quantum computers. arXiv:quant-ph/9807006, Jun 1998.
arXiv:quant-ph/9807006

[13] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia. Measurement-based quantum computation beyond the one-way model. Phys. Rev. A, 76: 052315, Nov 2007. 10.1103/PhysRevA.76.052315.
https://doi.org/10.1103/PhysRevA.76.052315

[14] L Hartmann, J Calsamiglia, W Dür, and H J Briegel. Weighted graph states and applications to spin chains, lattices and gases. Journal of Physics B: Atomic, Molecular and Optical Physics, 40 (9): S1–S44, Apr 2007. 10.1088/0953-4075/40/9/s01.
https://doi.org/10.1088/0953-4075/40/9/s01

[15] John A. Howell. Spans in the module $(\mathbb{Z}_m)^s$. Linear and Multilinear Algebra, 19 (1): 67–77, Jan 1986. 10.1080/03081088608817705.
https://doi.org/10.1080/03081088608817705

[16] A.Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of physics, 303 (1): 2–30, Jan 2003. ISSN 0003-4916. 10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[17] Jacob Miller and Akimasa Miyake. Hierarchy of universal entanglement in 2D measurement-based quantum computation. npj Quantum Information, 2: 16036, Nov 2016. 10.1038/npjqi.2016.36.
https://doi.org/10.1038/npjqi.2016.36

[18] Xiaotong Ni, Oliver Buerschaper, and Maarten Van den Nest. A non-commuting stabilizer formalism. Journal of Mathematical Physics, 56 (5): 052201, May 2015. 10.1063/1.4920923.
https://doi.org/10.1063/1.4920923

[19] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, Dec 2010. 10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667

[20] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117–158, Oct 2014. ISSN 0003-4916. 10.1016/j.aop.2014.06.013.
https://doi.org/10.1016/j.aop.2014.06.013

[21] Eric M. Rains, R. H. Hardin, Peter W. Shor, and N. J. A. Sloane. A nonadditive quantum code. Phys. Rev. Lett., 79: 953–954, Aug 1997. 10.1103/PhysRevLett.79.953.
https://doi.org/10.1103/PhysRevLett.79.953

[22] Narayanan Rengaswamy, Robert Calderbank, Michael Newman, and Henry D. Pfister. On optimality of CSS codes for transversal T. IEEE Journal on Selected Areas in Information Theory, 1 (2): 499–514, Aug 2020. 10.1109/JSAIT.2020.3012914.
https://doi.org/10.1109/JSAIT.2020.3012914

[23] M Rossi, M Huber, D Bruß, and C Macchiavello. Quantum hypergraph states. New Journal of Physics, 15 (11): 113022, Nov 2013. 10.1088/1367-2630/15/11/113022.
https://doi.org/10.1088/1367-2630/15/11/113022

[24] Ady Stern. Anyons and the quantum Hall effect—a pedagogical review. Annals of Physics, 323 (1): 204–249, Jan 2008. ISSN 0003-4916. 10.1016/j.aop.2007.10.008.
https://doi.org/10.1016/j.aop.2007.10.008

[25] Arne Storjohann. Algorithms for Matrix Canonical Forms. PhD thesis, Department of Computer Science, Swiss Federal Institute of Technology – ETH, 2000. URL https://cs.uwaterloo.ca/ astorjoh/diss2up.pdf.
https://cs.uwaterloo.ca/~astorjoh/diss2up.pdf

[26] Maarten Van den Nest, Jeroen Dehaene, and Bart De Moor. Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A, 69: 022316, Feb 2004. 10.1103/PhysRevA.69.022316.
https://doi.org/10.1103/PhysRevA.69.022316

### Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2023-02-07 10:13:12). On SAO/NASA ADS no data on citing works was found (last attempt 2023-02-07 10:13:13).


text:
PRIME

Platonic

Realms

Interactive

Mathematics

Encyclopedia

# arithmetic

The branch of mathematics concerned with addition, subtraction, multiplication, and division of numbers.

All elementary arithmetic is based on counting. Addition is generalized counting (which is why people often add by counting on their fingers) and multiplication is repeated addition. Subtraction and division are inverses of these operations.

In modern mathematics arithmetic is formalized in the Peano axioms.

Contributors

• Smith, B. Sidney: author

Citation Info

• [MLA] Smith, B. Sidney. "arithmetic." Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 21 Feb 2013. Web. 21 Feb 2013. <http://platonicrealms.com/>
• [APA] Smith, B. Sidney (21 Feb 2013). arithmetic. Retrieved 21 Feb 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/arithmetic/


text:
# Quarter Wavelength Matching : Missing Reflections?

With regards to the quarter wavelength impedance matcher, I understand it is used to correct a mismatch and looks as follows where $Z_T$ is the matching line:

I understand that we have set up the problem such that the input impedance of the combined matching line and the load is the same as $Z_0$ meaning that we have no reflections here at all. We find that $Z_T$ must be the geometric mean of $Z_0$ and $Z_L$ through this condition.

However, surely the matching line and the load itself aren't matched and hence we should have some reflections at this point itself?

As in any situation like this, it can be analysed in the time domain, or the frequency domain. Both will agree, but one or the other might be easier to comprehend. Flipping between one and the other has to be done carefully, as things that are simple in one domain are not in the other, which is why either might be helpful. In the time domain, a single step that's wideband in frequency is good, in the frequency domain, a single sinewave is easier to handle.

Time Domain

Let's launch a step along the line, and stay with it as it negotiates the junctions.

It gets to the Z0/ZT junction, and some is reflected back. Of the energy that carries on, some is absorbed in the load, and the rest is reflected back. Some of that reflection passes the ZT/Z0 junction, and some is reflected back towards the load. So you can see that some energy, decreasing every bounce, is trapped on the ZT line, and that there have been two steps reflected back, with more to come. This creates a succession of steps, separated by 2*ZT's length.

As out quarter-wave transformer is only supposed to work at a single frequency, we need to concentrate on the effect at that frequency, so we need to put our frequency domain hat on now.

The successsion of steps separated by t has energy at zero frequency, no energy at 1/2t, energy at 1/t, no energy at 3/2t and so on.

A $\frac{\lambda}{4}$ 'transformer' only works for signals with wavelength $\lambda$. And at that wavelength, we can see that there's no energy in that train of reflected steps.

If instead of a wideband step, we send in a single frequency wave of the right frequency, each reflected pulse will be replaced by a sinewave. The time-shifted sinewaves, with a half-period spacing defined by the length of the ZT line, will add up to give you no nett reflection.

So what's that about no energy at 3/2t? Yes, a $\frac{3\lambda}{4}$, and in fact every odd multiple, works as well.

Frequency Domain

A $\frac{\lambda}{4}$ transformer makes its load 'appear' as if it has an impedance of $\frac{{Z_T}^2}{Z_L}$, when you do the line input impedance sums properly, which accounts for taking the ZT/ZL reflection and phase shifting it to the start of the line. This reflection has therefore already been accounted for, and must not be double counted.

As the ZT line now appears to have an input impedance of Z0, there is no reflection at the Z0/ZL junction either.

Summary

As a $\frac{\lambda}{4}$ transformer works only for certain frequencies, it's most appropriate to analyse it in the frequency domain, where steps and other wideband signals don't exist. Which means if you do try a time domain approach, ie 'there will be reflections because the lines do have different impedances', then you have to follow the analysis all the way through to analysing the results in the frequency domain again, where the 'transformer' 'works'.

• @NightStrider It's also important to bear in mind that quarter-wave matching can mathematically only be done for one single frequency (although in practice the match is acceptable over a small range). The reflections only cancel out at the matched frequency. – Tom Carpenter Apr 19 '17 at 21:57
• Thank you for your response. I thought that the entire purpose of the impedance matching was to eliminate reflections at the first junction because the impedance there matches that of the line perfectly. Is this not the case? – DreamsOfHummus Apr 19 '17 at 22:16
• @NightStrider, yes. It does that by arranging that the reflections from the second junction interfere destructively with the reflections from the first junction. – The Photon Apr 19 '17 at 22:19
• That's a different way to say the same thing. It "sees Z_0" because the reflections from the second junction are interfering with the immediate reflections at the first junction to make it that way. – The Photon Apr 19 '17 at 22:31
• What third junction? Can you edit your diagram and label what you mean by "first junction" and "2nd junction". I thought you meant the junction between the Z0 line and the ZT line when you said "first junction", and the junction between the ZT line and the load when you said "second junction". I don't see any third junction in your diagram. – The Photon Apr 19 '17 at 23:05

If you look at the situation from within the time/reflections framework, then of course there are reflections. Infinite reflections, in fact.

However, the widely used (and more convenient) frequency/impedance framework allows us to "ignore" those multiple reflections. How can this be?

Well, because it's just a steady-state simplification: those reflections are actually there, but they're only relevant for a time/transient analysis.

From Chapter 2.5 in Pozar, D., Microwave Engineering:

[...] the matching property of the quarter-wave transformer comes about by properly selecting the characteristic impedance and length of the matching section so that the superposition of all of the partial reflections adds to zero. Under steady-state conditions, an infinite sum of waves traveling in the same direction with the same phase velocity can be combined into a single traveling wave. Thus, the infinite set of waves traveling in the forward and reverse directions on the matching section can be reduced to two waves traveling in opposite directions.