text:
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# Why do we fuse Odom, IMU, GPS and other pose messages?

I'm working with navigation stack lately, and I have a indoor location system to get the absolute pose of the robot.

As we all know, the odometry is only right for a short time, so I want to correct the pose with my indoor location system.

I know there is a package robot_pose_ekf can fuse /odom, /imu_data and /vo. But my question is Why do we have to fuse these pose messages? Why don't we just make the /odom correct with indoor location system or GPS or something else?

## EDIT

Hi mig,

Actually, I tried to adapt the tf map->odom before. But I found it hard to deal with the orientaion. You know, to adapt the position (x,y,z), you just need to do some additions and subtractions, however, when it comes to the orientation, if you change it (Quaternion: x, y, z, w), the position of base_link relative to map will change as well. And I also tried to do some 'cos' and 'sin' calculation, but I didn't get the right tf result.

So, is there any packages I can use or refer for this tf map->odom? I know amcl did such thing, but its source code is hard to understand.

Thank you.

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The difference between those is how/what they measure.

Odometry, IMU and Visual Odometry (I guess this what you mean with vo) just measure the internal state of the robot, and thus just deliver relative measurements towards a starting pose and cannot correct for long-term drifts. However, you can fuse those using the robot_pose_ekf to achieve a more stable "fused" odometry guess.

Then, you need a localization that is providing measurements with respect to "world". This can be GPS, IPS, cameras with stored, localized features or laserscanners with a given map. With those, you can correct the drift of the "fused" odometry.

There are several packages providing this or parts of this functionality, e.g. robot_localization or amcl, to name just two.

EDIT

You are right, I did not think of adding a GPS sensor like this. Seems like I misunderstood how they define visual odometry here. However, a world fixed frame does not mean that this is fixed over multiple runs. Typically, any odometry starts of from the pose of the robot where it is turned on. In contrast there are fixed frames (like map coordinates) that are the same whether you turn this on or not.

Thus, vo provides measurements to the vo frame which can different any time you launch the robot, depending on what you use for input.

EDIT 2

Typically, when you add a sensor providing "global corrections", you don't correct the odom frame. The tf odom->base_link is what is typically provided by internal sensors, i.e. wheel encoders, IMU and visual odometry.

If you have another sensor (GPS, laserscanner, ...) I would prefer to adapt the tf map->odom such that the tree map->odom->base_link is correct. This is how it is typically done for mobile robots in ROS, thus I'd prefer this solution.

EDIT 3

This is where the magic happens in amcl.

You can use the TransformPose function of TF to get map->odom (called odom_to_map therein) from the map->base_link that you estimate, and broadcast this (after you bring it into the correct format...)

more

Thanks for your answer. But I'm still confused. I thought the robot_pose_ekf not just fuses relative measurements, because it now supports GPS too. And I thought /vo is also the global pose relative to "world"

( 2016-08-18 04:45:19 -0600 )edit

Hi mig, I know amcl can get the base_link position relative to map, and it corrects the drift by publishing the tf between map and odom.

In my project, I subscribe the /odom and /ips_pose(which is from my indoor position system), correct /odom according to /ips_pose, and then publish tf between odom and base_link.

I wanna ask which solution is better?

more

( 2016-08-18 05:53:26 -0600 )edit

OK, I'm sorry, I'm new to here.

( 2016-08-18 08:30:37 -0600 )edit


text:
# Spin-parity for excited O-15 states

1. Feb 11, 2008

### Warr

1. The problem statement, all variables and given/known data

Use the shell model to find reasonable configurations for the first 5 excited states of O-15.

2. Relevant equations

The spin-parity and energy above the ground state for the first five excited states are:

ground state: (1/2)-

excited states:
5.18 MeV, (1/2)+
5.24 MeV, (5/2)+
6.18 MeV, (3/2)-
6.79 MeV, (3/2)+
6.86 MeV, (5/2)+

Parity of an unpaired nucleon is given by (-1)^l, with l_s = 0, l_p=1, etc.

The order of shells is: $$1s_{1/2}, 1p_{3/2}, 1p_{1/2}, 1d_{5/2}, 2s_{1/2}, 1d_{3/2}, 1f_{7/2}$$

3. The attempt at a solution

First off, O-15 has 8 protons and 7 neutrons, so the ground state config is given by

protons: $$(1s_\frac{1}{2})^2(1p_\frac{3}{2})^4(1p_\frac{1}{2})^2$$
neutrons: $$(1s_\frac{1}{2})^2(1p_\frac{3}{2})^4(1p_\frac{1}{2})^1$$

so for the neutrons we have $$(1p_\frac{1}{2})^{-1}$$. This has a spin of 1/2 and l_p = 1, so the parity is -1, hence the state (1/2)-, confirming the ground state.

For the first excited state, we need the parity to be (1/2)+. Here is where my first confusion is. Can the $$1p_{1/2}$$ in the protons be promoted? If so, how high can it be promoted. For example, lets say it is promoted to the next subshell, $$1d_{5/2}$$

protons: $$(1s_\frac{1}{2})^2(1p_\frac{3}{2})^4(1p_\frac{1}{2})^2$$
neutrons: $$(1s_\frac{1}{2})^2(1p_\frac{3}{2})^4(1d_\frac{5}{2})^1$$

Does this make the spin-parity (5/2)+? Or do we have to take something into account for the fact that there is now no neutrons in the $$1p_{1/2}$$ subshell (thus a gap between the protons in the $$1p_{3/2}$$ and the proton in the excited subshell $$1d_{5/2}$$), yet there are 2 in the $$1p_{1/2}$$ for the neutrons.

are other valid 'excitations' to have a neutron in the $$1s_{1/2}$$ or $$1p_{3/2}$$ be promoted to the $$1p_{1/2}$$ shell, thus creating a spin parity of (1/2)+ and (3/2)- respectively? If these are right, I think I have the first 3 states.

2. Feb 11, 2008

### malawi_glenn

"Does this make the spin-parity (5/2)+? "

Yes, so you only look where you have particles and "holes", and then couple them. In this case you have only one particle outside closed shells and paired nucleons, so its quantum numbers decides the total spin and parity of the nucleus.

And yes, you my break a pair and move that nucleon to higher shell and couple the angular momenta and parities. But remember that it requires energy to break pairs (approx 1MeV).

format: plain


text:
# Subset sum problem for permutations

Given permutations $$g_1,\,\ldots, g_m \in S_n$$ of size $$n$$ and target permutation $$g \in S_n$$, decide if there exists a subset of $$\{g_1,\, \ldots, g_m\}$$, which composition in some order (or, alternatively, as a variant of this problem, in the same order) is equal to $$g$$, i.e. $$g_{i_1} \circ g_{i_2} \circ \cdots \circ g_{i_k} = g$$.

This is actually a subset sum problem, but for symmetric group $$(S_n, \circ)$$ instead of $$(\mathbb{Z}, +)$$.

The questions are:

1. Is there a known solution in polynomial time?
2. Otherwise, is this problem known as NP-complete?

I've found a paper on knapsack problems in groups, however, these results seems to be not applicable for symmetric group.

• To be 100% clear: just like in Subset Sum, you aren't allowed to reuse the same permutation more than once, right? If you're allowed to reuse permuations from the input set then it's a problem about generators rather than a Subset Sum analogue. – Aaron Rotenberg Dec 28 '19 at 4:48
• @AaronRotenberg You can also consider a variant of this problem in which we are allowed to reuse the same permutation more than once, if you think this variant would be easier. – Nastya Koroleva Dec 29 '19 at 14:42
• The variant where you are allowed to reuse permutations is called permutation group membership testing and can be solved in polynomial time by a classic algorithm based on subgroup chains. See Furst, Hopcroft, and Luks. "Polynomial Time Algoirthms for Permutation Groups." (Or any modern source on algorithms for finite groups, I'm guessing.) – Aaron Rotenberg Dec 29 '19 at 19:50
• Permutation group membership testing is not really analogous to a knapsack problem because every element of a finite group has a finite order: if $x$ is available, so is $x^{-1}$, by just stringing together $\operatorname{ord}(x) - 1$ copies of $x$. – Aaron Rotenberg Dec 29 '19 at 19:57

This problem, that I will call Subset-Perm-Sum, is NP-complete. Membership is easy: guess the subset non-deterministically and then check.

For hardness one can reduce from 3CNF-SAT in a very similar way to the standard proof of hardness for Subset-Sum.

Let $$\varphi$$ be an input formula with $$v$$ variables and $$c$$ clauses. We will build an instance of Subset-Perm-Sum over $$S_{2v+4c}$$. For every variable we build $$2$$ permutations (one that will represent the variable, and one that will represent its negation), and for every clause we will build $$2$$ permutations as well.

To each clause $$C_j, 1\leq j \leq c$$ we associate $$4$$ elements: $$2v+4j-k$$ for $$0 \leq k \leq 3$$. To build the $$2$$ permutations associated to a clause we simply do a cycle on its associated elements. That is, $$p(C_j) = (2v+4j-3, 2v+4j-2, 2v+4j-1, 2v+4j)$$ (we will add $$2$$ instance of $$p(C_j)$$ to the set we want to find a subset-sum later.)

Consider the $$i$$-th variable, $$x_i$$, and associate with it the elements $$2i-1$$ and $$2i$$ of $$S_{2v+4c}$$. To build the permutation $$p(x_i)$$ of variable $$x_i$$ simply swap its associated elements ($$2i-1$$ and $$2i$$), and multiply by the permutation of each clause it is contained in. Then, in cycle notation you can write: $$p(x_i) = (2i-1, 2i) \prod_{j | x_i \in C_j}p(C_j)$$

Consider now the multiset $$M$$ that is the union of the $$p(x_i)$$ and $$p(\bar{x_i})$$ and two times each $$p(C_j)$$.

We define the target permutation $$t$$ as:

$$t = \prod_{i=1}^v (2i-1, 2i) \cdot \prod_{j=1}^c p(C_j)^3$$

Claim: There is a subset $$X$$ of $$M$$ that composes to the target permutation $$t$$ (let me write $$p(X)=t$$) if and only if $$\varphi$$ is satisfiable.

Assume such an $$X$$ exists, then we know $$X$$ contains exactly one out of $$p(x_i)$$ and $$p(\bar{x}_i)$$, as it is the only way for the composition of $$X$$ to include the cycle $$(2i-1, 2i)$$. Furthermore, we can see that for each clause $$C_j \in \varphi$$, at least one if variables satisfies it, as to have $$p(C_j)^3$$ in $$p(X)$$ we need to have included a $$p(\ell_i)$$ such that literal $$\ell_i$$ is in $$C_j$$. Note that taking the two instances of $$p(C_j)$$ in $$X$$ is not enough. Therefore, there is an assignment of variables that satisfies every clause. For the backward direction, let $$\sigma$$ be a satisfying assignment of $$\varphi$$. If $$\sigma(x_i) = 1$$ then we let $$X$$ contain $$p(x_i)$$ and otherwise we let it contain $$p(\bar{x}_i)$$. For each clause $$C_j$$, we know there are between $$0 \leq r_j \leq 2$$ variables on it that are not satisfied by $$\sigma$$, and we add $$r_j$$ occurrences of $$p(C_j)$$ to $$X$$, so this way $$p(X)$$ includes $$p(C_j)^3$$. It is clear that the composition of $$X$$ equals $$t$$.

The subset sum problem is even harder(NP-hard) for special groups which you can embed into $$S_n$$. See the paper you have linked in the problem description.

$$\textbf{Observation:}$$ $$g$$ can be writtens as combination of elements $$\{g_1,\ldots,g_k\}$$ if and only if $$g \in \langle g_1,\ldots,g_k\rangle$$. Under the assumption that each of $$g_i$$ can appears any number of times.

The complexity of the problem depends upon the input representation. There are two two most commonly used ways, Cayley table and generating set. For Cayley table see this paper for results. Read the CGM(Cayley membership problem) Link

$$\textbf{CGM}$$

$$\textbf{Input :}$$ A group $$G$$ by its Cayley table, $$X \subseteq G$$ and $$t \in G$$.

$$\textbf{Output :}$$ Does $$t$$ belong to the subgroup $$\langle X \rangle$$ generated by $$X$$?

In general problem is in Symmetric log space.

$$\langle A \rangle$$ means subgroup generated by $$A$$

• I'm not sure if this is exactly right, because I suspect in the original problem no group element can be used more than once. You can have $g \in \langle g_1,\dots,g_k \rangle$ but there is no way to obtain $g$ by a composition of a subset of the $g_i$'s if you are restricted to using each only once. Therefore, I'm not convinced the problem is equivalent to CGM. – D.W. Dec 28 '19 at 6:02
• @D.W Ok Let me edit the answer. – user35837 Dec 28 '19 at 6:18
• It seems the SSP problem defined in that paper is not allowed to change the order of the given elements, which is different from OP's problem. – xskxzr Dec 28 '19 at 15:04
• @aaag Well, the size of Cayley table of $S_n$ is $O((n!)^2)$, which is actually too big to build, therefore I think this doesn't answers if there exists a polynomial time algorithm for original problem, nor provides an evidence of NP-completeness. Am I right? – Nastya Koroleva Dec 29 '19 at 14:23
• @aaag "for special groups which you can embed into $S_n$" — for which groups? It seems that every special group mentioned in the paper is infinite, however all $S_n$'s are finite. – Nastya Koroleva Jan 3 '20 at 19:22


text:
July 5, 2022

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A:

Your question is in error because the output of a UPS is not a balanced sine wave even though the power source is balanced.
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This means that the supply is un-unipolar and is the reason for the voltage imbalances.

After a successful 2013 campaign in which the young midfielder was voted the Fans’ Choice MLS Rookie of the Year, Mavinga was selected in the 2014 MLS Expansion Draft by expansion side New York City FC.

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_p)(p,a))$$for all \varphi \in C_0(G) and \psi \in {{\mathcal{H}}}\subseteq C_0(G) and a \in G. By Lemma $lemma.prop.f\_l$, the map f satisfies the following property. $lemma.prop.fg\_l$ For any \varphi \in C_0(G), f_l(\varphi \cdot f_r)=\varphi \cdot f_l and f_l(\varphi \cdot f_l)=\varphi \cdot f_l. Now we define two maps on C_0(G) as follows. Let \pi:C_0(G) \to {{\mathcal{H}}} be the map defined by$$\pi(\varphi):=f_r(\varphi \cdot f_l)-f_l(\varphi \cdot f_l) \cdot f_r$$for \varphi \in C_0(G). Then \pi is a homomorphism with respect to the operations on C_0(G) as follows. $lemma.pi.coho$ The homomorphism \pi:C_0(G) \to {{\mathcal{H}}} satisfies \pi(f_l(\varphi \cdot f_r))=\varphi \cdot f_l and \pi(f_r(\varphi \cdot f_r))=\varphi \cdot f_l for \varphi \in C_0(G). Let \varphi \in C_0(G). Since f_l(\varphi \cdot f_r)=\varphi \cdot f_l by Lemma $lemma.prop.f\_l$, the lemma follows. On the other hand, we define a map \bar{\pi}:C_0(G) \to {{\mathcal{H}}} to be$$\bar{\pi}(\varphi):=f_r(\varphi \cdot f_l)-f_

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text:
## ThinkDSP¶

This notebook contains code examples from Chapter 1: Sounds and Signals

## Think DSP module¶

thinkdsp is a module that accompanies Think DSP and provides classes and functions for working with signals.

In [1]:
# Get thinkdsp.py

import os

if not os.path.exists('thinkdsp.py'):
!wget https://github.com/AllenDowney/ThinkDSP/raw/master/code/thinkdsp.py

## Signals¶

Instantiate cosine and sine signals.

In [2]:
from thinkdsp import CosSignal, SinSignal

cos_sig = CosSignal(freq=440, amp=1.0, offset=0)
sin_sig = SinSignal(freq=880, amp=0.5, offset=0)

Plot the sine and cosine signals. By default, plot plots three periods.

In [3]:
from thinkdsp import decorate

cos_sig.plot()
decorate(xlabel='Time (s)')

Here's the sine signal.

In [4]:
sin_sig.plot()
decorate(xlabel='Time (s)')

Notice that the frequency of the sine signal is doubled, so the period is halved.

The sum of two signals is a SumSignal.

In [5]:
mix = sin_sig + cos_sig
mix

Out[5]:
<thinkdsp.SumSignal at 0x7fc6b8436160>

Here's what it looks like.

In [6]:
mix.plot()
decorate(xlabel='Time (s)')

## Waves¶

A Signal represents a mathematical function defined for all values of time. If you evaluate a signal at a sequence of equally-spaced times, the result is a Wave. framerate is the number of samples per second.

In [7]:
wave = mix.make_wave(duration=0.5, start=0, framerate=11025)
wave

Out[7]:
<thinkdsp.Wave at 0x7fc6b83912b0>

IPython provides an Audio widget that can play a wave.

In [8]:
from IPython.display import Audio
audio = Audio(data=wave.ys, rate=wave.framerate)
audio

Out[8]:

Wave also provides make_audio(), which does the same thing:

In [9]:
wave.make_audio()

Out[9]:

The ys attribute is a NumPy array that contains the values from the signal. The interval between samples is the inverse of the framerate.

In [10]:
print('Number of samples', len(wave.ys))
print('Timestep in ms', 1 / wave.framerate * 1000)

Number of samples 5512
Timestep in ms 0.09070294784580499

Signal objects that represent periodic signals have a period attribute.

Wave provides segment, which creates a new wave. So we can pull out a 3 period segment of this wave.

In [11]:
period = mix.period
segment = wave.segment(start=0, duration=period*3)
period

Out[11]:
0.0022727272727272726

Wave provides plot

In [12]:
segment.plot()
decorate(xlabel='Time (s)')

normalize scales a wave so the range doesn't exceed -1 to 1.

apodize tapers the beginning and end of the wave so it doesn't click when you play it.

In [13]:
wave.normalize()
wave.apodize()
wave.plot()
decorate(xlabel='Time (s)')

You can write a wave to a WAV file.

In [14]:
wave.write('temp.wav')

Writing temp.wav

wave.write writes the wave to a file so it can be used by an exernal player.

In [15]:
from thinkdsp import play_wave

play_wave(filename='temp.wav', player='aplay')

read_wave reads WAV files. The WAV examples in the book are from freesound.org. In the contributors section of the book, I list and thank the people who uploaded the sounds I use.

In [16]:
from thinkdsp import read_wave

In [17]:
wave.make_audio()

Out[17]:

I pulled out a segment of this recording where the pitch is constant. When we plot the segment, we can't see the waveform clearly, but we can see the "envelope", which tracks the change in amplitude during the segment.

In [18]:
start = 1.2
duration = 0.6
segment = wave.segment(start, duration)
segment.plot()
decorate(xlabel='Time (s)')

## Spectrums¶

Wave provides make_spectrum, which computes the spectrum of the wave.

In [19]:
spectrum = segment.make_spectrum()

Spectrum provides plot

In [20]:
spectrum.plot()
decorate(xlabel='Frequency (Hz)')

The frequency components above 10 kHz are small. We can see the lower frequencies more clearly by providing an upper bound:

In [21]:
spectrum.plot(high=10000)
decorate(xlabel='Frequency (Hz)')

Spectrum provides low_pass, which applies a low pass filter; that is, it attenuates all frequency components above a cutoff frequency.

In [22]:
spectrum.low_pass(3000)

The result is a spectrum with fewer components.

In [23]:
spectrum.plot(high=10000)
decorate(xlabel='Frequency (Hz)')

We can convert the filtered spectrum back to a wave:

In [24]:
filtered = spectrum.make_wave()

And then normalize it to the range -1 to 1.

In [25]:
filtered.normalize()

Before playing it back, I'll apodize it (to avoid clicks).

In [26]:
filtered.apodize()
filtered.plot()
decorate(xlabel='Time (s)')

And I'll do the same with the original segment.

In [27]:
segment.normalize()
segment.apodize()
segment.plot()
decorate(xlabel='Time (s)')

Finally, we can listen to the original segment and the filtered version.

In [28]:
segment.make_audio()

Out[28]:
In [29]:
filtered.make_audio()

Out[29]:

The original sounds more complex, with some high-frequency components that sound buzzy. The filtered version sounds more like a pure tone, with a more muffled quality. The cutoff frequency I chose, 3000 Hz, is similar to the quality of a telephone line, so this example simulates the sound of a violin recording played over a telephone.

## Interaction¶

The following example shows how to use interactive IPython widgets.

In [30]:
import matplotlib.pyplot as plt
from IPython.display import display

def filter_wave(wave, start, duration, cutoff):
"""Selects a segment from the wave and filters it.

Plots the spectrum and displays an Audio widget.

wave: Wave object
start: time in s
duration: time in s
cutoff: frequency in Hz
"""
segment = wave.segment(start, duration)
spectrum = segment.make_spectrum()

spectrum.plot(color='0.7')
spectrum.low_pass(cutoff)
spectrum.plot(color='#045a8d')
decorate(xlabel='Frequency (Hz)')
plt.show()

audio = spectrum.make_wave().make_audio()
display(audio)

Adjust the sliders to control the start and duration of the segment and the cutoff frequency applied to the spectrum.

In [31]:
from ipywidgets import interact, fixed


text:
1. ## Estimation ad confidence Interval Q2

These questions are on behalf of Bill who needs urgent help but cant start threads/post for some reasons

Question 1 : Suppose that

$X_1 , X_2,...X_n \color{red}--\color{black}^{iid} N(\mu , \sigma^2)$

Consider the following estimators for
$\sigma^2$.

$S^{*2} =\sum_{i=1}^{n}(\frac{(X_i - \overline X)^2}{n})$ and

$S^2 = \sum_{i=1}^{n}\frac{(X_i - \overline X)^2}{n-1}$

Calculate MSE of those estimators and discuss which one is closer to

$\sigma^2$.

HINT: If $Z_1 , Z_2,...Z_n \color{red}--\color{black}^{iid} N(0 , 1)$

than

$\sum_{i=1}^{n} (Z_i-\overline{Z})^2 \color{red}--\color{black} {\chi}^2_{n-1}$

Note: MSE - Mean squared Error

iid = independent and identically distributed

--
= ~ and iid is above it

2. ## Estimation ad confidence Interval Q2

The question 2 of the attachment, Can't type now , Sorry, again this is also for Bill

3. ${(n-1)S^2\over \sigma^2}$ is a $\chi^2$ random variable with n-1 degrees of freedom.

Hence $S^2$ is unbiased for $\sigma^2$.

The variance of a $\chi^2$ random variable with n-1 degrees of freedom is 2(n-1).

4. ## HELP with POSTED assignment (Both questions by Bill)

Please help with both questions, I have the same problem set, and it looks Japanese to me!!!
The Gamma one is very hard.

5. Originally Posted by polyluka15
Please help with both questions, I have the same problem set, and it looks Japanese to me!!!
The Gamma one is very hard.
So where are you stuck? What part of matheagle's reply do you need clarification on? What have you tried?

6. There is question #2 about the Gamma distribution.
a) its a F-distribution where
Z=(X/k) over (Y/m), but in this question I dunno what k and m are??? and even if Z is correct

b) NO CLUE

c) I found this:

Estimate + - (critical value)*(standard error)

where the critical value comes from the relevant distribution e.g. Normal, t, chi-square, F, Gamma, and so on.

However, the standard error will vary depending on statistic being estimated.

Of which I don't know what is what.

Am I out of it? I have little idea of what is going on


text:
0
36kviews
Show that every square matrix can be uniquely expressed as a sum of symmetric and skew symmetric matrix.

Not understanding plz explain detailed

0
796views

Let, A be any square matrix.

Now, $A = \dfrac{1}{2} (A+A') + \dfrac{1}{2} (A-A') \; = \; say, \; P+Q \; where \\ \; \\ P=\dfrac{1}{2} (A+A') \; \; and \; \; Q=\dfrac{1}{2} (A-A') \\ \; \\ To \; prove: \; P \; is \; symmetric, \; i.e. \; P' \; = P \; and \; Q \; is \; skew - symmetric, \; i.e. \; Q' \; = \; -Q \\ \; \\ \; \\ \therefore P'=\dfrac{1}{2} \Big(A+A'\Big)' \; = \; \dfrac{1}{2} \bigg(A'+(A')'\bigg) \; = \; \dfrac{1}{2} (A'+A) \; = \; P \\ \; \\ \therefore P \; is \; symmetric \\ \; \\ \; \\ \therefore Q'=\dfrac{1}{2} \Big(A-A'\Big)' \; = \; \dfrac{1}{2} \bigg(A'-(A')'\bigg) \; = \; \dfrac{1}{2} (A'-A) \; = \; -Q \\ \; \\ \therefore Q \; is \; skew-symmetric \\ \; \\ \; \\ \; \\ To \; prove \; uniqueness,\; let \; A=R+S \; where \; R \; is \;symmetric\; and\; S\; is\; skew- symmetric \\ \; \\ \; \\ \therefore A' \;=\; (R+S)' \; = \; R'+S' \; = \; R-S \\ \; \\ \{ \because R'=R \; and \; S'=-S \; by \; Definition \; of \; symmetric \; and \; skew-symmetric \; matrices \} \\ \; \\ \; \\ \therefore \dfrac{1}{2} (A+A') \; = \; \dfrac{1}{2} (R+S+R-S) \; = \; R\; = \; P \\ \; \\ \dfrac{1}{2} (A-A') \; = \; \dfrac{1}{2} (R+S-R+S) \; = \; S\; = \; Q$

Hence, the representation $A=P+Q$ is unique. Hence, it is proved that every square matrix can be uniquely expressed as a sum of symmetric and skew-symmetric matrix.


text:
# Does S^6 have the structure of an algebraic variety?

More accurately, the question should be: Is it known that $S^6,$ the 6-dimensional sphere, is $not$ a (proper) complex algebraic variety, or algebraic space? And is there a reference? It's easy to see that it cannot be projective (as $H^2=0$), but I don't see how it can violate the usual general properties of algebraic varieties [e.g. by Chow's lemma one can get a projective (and smooth, by resolution of singularities) variety lying over and birational to it, but the cohomology groups can get larger when we go up...]. Maybe one needs some theory of classification of 3-folds (if there is one such theory).

Knowing that it cannot be algebraic will somehow make the analogous question of complex structure much sharper (by the way, it does have an almost complex structure).

Thank you.

-
See mathoverflow.net/questions/1973/… for the discussion of the existence of a complex structure on $S^6$, as well as remarks on its non-algebraicity. –  ACL Nov 20 '11 at 10:55

Suppose that $X$ is a smooth complete positive-dimensional algebraic space over $\mathbb C$. Then $\mathrm H^2(X, \mathbb Q)$ can not be 0. In fact, every algebraic space contains an open dense subscheme. Let $U \subseteq X$ be an open affine subscheme; by a well known-result, the complement $C$ of $U$ has pure codimension 1; consider $C$ as a divisor, with its reduced scheme structure. I claim that the cohomology class of $C$ can not be 0. Let $Y \to X$ a birational morphism, where $Y$ is a smooth projective variety; this exists, by Chow's lemma for algebraic spaces, and resolution of singularities. The pullback of $C$ is a non-zero effective divisor on $Y$, so its cohomology class is not 0. Since formation of cohomology classes of divisors is compatible with pullbacks, this implies the result.

This says very little about the problem of existence of a complex structure. Complete algebraic spaces, or, if you prefer, Moishezon manifolds, are not so distant from projective varieties; it seems clear to me that if $S^6$ has a complex structure, this will be a very exotic animal, very far away from the world of algebraic geometry.

: why does $C$ have codimension 1? This is Corollaire 21.12.7 of EGAIV, when $X$ is a scheme. In general you can reduce to the case of a scheme by considering an étale morphism $Y \to X$, where $Y$ is an affine scheme. This map is affine, because $X$ is separated, so the inverse image of $U$ in $Y$ is affine.

-
Thanks, Angelo. What is this "well known-result" that $C\subset X$ has pure codim. 1 (and any reference)? And I suppose that the cohom. class of an effective divisor in a projective variety is non-zero because one can find another (smooth) effective alg. cycle with non-empty intersection (a finite set of points) with $C,$ right? –  shenghao Nov 27 '11 at 4:48
For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.
[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.
There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.
[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.
The nearly Kahler structure on $S^6$ is definitely not integrable, this is well known. But it certainly does not follow from that fact that there could be no other integrable almost complex structure on $S^6$. What is known (due to LeBrun) is that there can be no integrable complex structure which is orthogonal with respect to the standard round metric on $S^6$. As far as I know, this question of existence of some integrable complex structure on $S^6$ is still wide-open. –  Spiro Karigiannis Nov 20 '11 at 12:56


text:
### Impossible Differential Cryptanalysis on Deoxys-BC-256

##### Abstract

Deoxys is a third-round candidate of the CAESAR competition. This paper presents the first impossible differential cryptanalysis of Deoxys-BC-256 which is used in Deoxys as an internal tweakable block cipher. First, we find a 4.5-round ID characteristic by utilizing a miss-in-the-middle-approach. We then present several cryptanalyses based upon the 4.5 rounds distinguisher against round-reduced Deoxys-BC-256 in both single-key and related-key settings. Our contributions include impossible differential attacks on up to 8-rounds Deoxys-BC-256 in the tweak-key model which is, to the best of our knowledge, the first independent investigation of the security of Deoxys-BC-256 in the single-key model. Our attack reaches 9 rounds in the related-key related-tweak model which has a slightly higher data complexity than the best previous results obtained by a rectangle attack presented at FSE 2018 but requires a lower memory complexity with an equal time complexity.

Available format(s)
Category
Secret-key cryptography
Publication info
Published elsewhere. The ISC International Journal of Information Security
Keywords
authenticated encryptionblock cipherDeoxys-BCrelated- tweakrelated-keyimpossible differential cryptanalysis.
Contact author(s)
a mehrdad @ mail sbu ac ir
History
2018-08-08: revised
See all versions
Short URL
https://ia.cr/2018/048

CC BY

BibTeX

@misc{cryptoeprint:2018/048,


text:
1. ## Trip identity

$csc^4(x) - cot^4(x)$

Could someone walk me though simplifying this? Thanks!

2. Originally Posted by absvalue
$csc^4(x) - cot^4(x)$

Could someone walk me though simplifying this? Thanks!
It factors to $(\csc^2x+\cot^2x)({\color{blue}\csc^2x-\cot^2x})=({\color{red}\csc^2x}+\cot^2x)({\color{b lue}1})={\color{red}\cot^2x+1}+\cot^2x$ $=2\cot^2x+1$

Does this make sense?

3. Thanks for your helpful post. I was stuck on the second step... I understand it now. Thanks again!


text:
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# Ethicist- Studies have documented the capacity of placebos

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Ethicist- Studies have documented the capacity of placebos [#permalink]  26 Aug 2006, 15:16
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Ethicist- Studies have documented the capacity of placebos to reduce pain in patients who belive that they are receiving beneficial drugs. Some doctors say that they administer placebos because medically ineffective treatment reinforced by the placebo effect sometimes helps patients recover faster than good treatment alone . But administering placebos is nonetheles ethically questionable, for even if a placebo benefits a patient, a doctor might, for example, have prescribed it just to give the patient satisfaction that something was being done.

The ethicist's argument depends on which one of the following assumptions?

(A) A patient's psychological satisfaction is not a consideration in administering medical treatment
(B) The motivation for administering a placebo can be relevant to the ethical justification for doing so
(C) Medical treatment that relies on the placebo effect alone is ethically indefensible
(D) The pain relief produced by the placebo effect justifies the deception involved in administering a placebo
(E) Administering a placebo is not ethically justified if that treatment is not prescribed by a Dr.

Source- Next 10 LSAT

Please explain( if possible using the negation technique)

Heman
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Re: CR- Placebo Ethicist [#permalink]  26 Aug 2006, 22:48
heman wrote:
Ethicist- Studies have documented the capacity of placebos to reduce pain in patients who belive that they are receiving beneficial drugs. Some doctors say that they administer placebos because medically ineffective treatment reinforced by the placebo effect sometimes helps patients recover faster than good treatment alone . But administering placebos is nonetheles ethically questionable, for even if a placebo benefits a patient, a doctor might, for example, have prescribed it just to give the patient satisfaction that something was being done.

The ethicist's argument depends on which one of the following assumptions?

(A) A patient's psychological satisfaction is not a consideration in administering medical treatment ---> 180 NOPE
(B) The motivation for administering a placebo can be relevant to the ethical justification for doing so ---> Conservatively phrased and within context, Yep!
(C) Medical treatment that relies on the placebo effect alone is ethically indefensibl---> Extreme NOPE
(D) The pain relief produced by the placebo effect justifies the deception involved in administering a placebo ---> Off topic NOPE
(E) Administering a placebo is not ethically justified if that treatment is not prescribed by a Dr. ---> HAHA!! Ridiculous answer choice NOPE

Source- Next 10 LSAT

Please explain( if possible using the negation technique)

Heman

I'm on (B) here in under 90 seconds.

Bring on the LSAT CRs Heman, we love em!!
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Re: CR- Placebo Ethicist [#permalink]  27 Aug 2006, 00:12
heman wrote:
Ethicist- Studies have documented the capacity of placebos to reduce pain in patients who belive that they are receiving beneficial drugs. Some doctors say that they administer placebos because medically ineffective treatment reinforced by the placebo effect sometimes helps patients recover faster than good treatment alone . But administering placebos is nonetheles ethically questionable, for even if a placebo benefits a patient, a doctor might, for example, have prescribed it just to give the patient satisfaction that something was being done.

The ethicist's argument depends on which one of the following assumptions?

(A) A patient's psychological satisfaction is not a consideration in administering medical treatment
(B) The motivation for administering a placebo can be relevant to the ethical justification for doing so
(C) Medical treatment that relies on the placebo effect alone is ethically indefensible
(D) The pain relief produced by the placebo effect justifies the deception involved in administering a placebo
(E) Administering a placebo is not ethically justified if that treatment is not prescribed by a Dr.

Source- Next 10 LSAT

Please explain( if possible using the negation technique)

Heman

A - \$
B - Right, the motive of a doctor is always to cure his patient. So are placebos to replace medicines? Out.
C - This simply restates the argument. Out
D - Contrary to the argument.
E - Out of scope. Discussion is entirely around prescriptions by doctors.
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[#permalink]  27 Aug 2006, 02:37
Hey .. shouldn't it be A ..
The assumption of ethicist is that the psychological satisfaction of patient should not be a consideration in prescribing him/her the medicine.
Based on this assumption only, he finds the prescription against ehics.

OA??
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--I never think of future, it comes soon enough!!

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[#permalink]  27 Aug 2006, 05:09
(A) A patient's psychological satisfaction is not a consideration in administering medical treatment
My choice.
(B) The motivation for administering a placebo can be relevant to the ethical justification for doing so
That is what the argument claims. It is not an assumption.
(C) Medical treatment that relies on the placebo effect alone is ethically indefensible
Out of scope/irrelevant
(D) The pain relief produced by the placebo effect justifies the deception involved in administering a placebo
Out of scope/irrelevant
(E) Administering a placebo is not ethically justified if that treatment is not prescribed by a Dr.
Out of scope/irrelevant
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[#permalink]  27 Aug 2006, 11:02
anandsebastin wrote:
(A) A patient's psychological satisfaction is not a consideration in administering medical treatment
My choice.
(B) The motivation for administering a placebo can be relevant to the ethical justification for doing so
That is what the argument claims. It is not an assumption.
(C) Medical treatment that relies on the placebo effect alone is ethically indefensible
Out of scope/irrelevant
(D) The pain relief produced by the placebo effect justifies the deception involved in administering a placebo
Out of scope/irrelevant
(E) Administering a placebo is not ethically justified if that treatment is not prescribed by a Dr.
Out of scope/irrelevant

would agree with anand. A is the answer.
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[#permalink]  28 Aug 2006, 19:33
I select B..

OA?
_________________

A.P.

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[#permalink]  28 Aug 2006, 19:43
(A) A patient's psychological satisfaction is not a consideration in administering medical treatment

Conclusion: Prescribing medication should not be based on the satisfaction of the patient. In that sense, it is unethical...

Confusing one for me...
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[#permalink]  28 Aug 2006, 19:58
OA-B
No OE available.

I myself picked A. Gmat73 can you elaborate between A & B.

Thanx

Heman
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[#permalink]  29 Aug 2006, 10:59
Late But B. I'm happy that I picked the right one.
[#permalink] 29 Aug 2006, 10:59
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# Ethicist- Studies have documented the capacity of placebos

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# #4 Morgan Tankersley Announces Verbal Commitment to Stanford

Anne Lepesant
##### by Anne Lepesant 76

October 08th, 2017

Tampa, Florida’s Morgan Tankersley, our #4 recruit in the high school class of 2018, has made it official: she intends to swim for Stanford University in the fall. In an embarrassment of riches, the Cardinal now boasts 4 of the SwimSwam Top 10 and 6 of our Top 20. Verbal commits to the class of 2022 are: #2 Taylor Ruck, #3 Zoe Bartel, #4 Tankersley, #9 Allie Raab, #15 Amalie Fackenthal, #16 Lucie Nordmann, and Anya Goeders.

Tankersley swims for H.B. Plant High School and Greater Tampa Swimming Association, and is the top full-range freestyler in the class. She can compete with the best of them in the low and high ends of the range, but is at her best in the 200-500 freestyles where her current times would already score at NCAAs.

At 2016 FHSAA Class 4A Championship last November, Tankersley broke the Florida high school state records in the 200 free (1:44.31), 500 free (4:37.60), and 100 free (48.69) while leading off the Plant 400 free relay in prelims. In club swimming this spring and summer she swam the 1650/1500 free for the first time in years, and dropped 45 seconds in each. Perhaps this portends well for a Day Three event at NCAAS…

Best Times:

• 500 free – 4:37.60
• 200 free – 1:44.31
• 100 free – 48.69
• 50 free – 22.60
• 1650 free – 16:27.61

If you have a commitment to report, please send an email with a photo (landscape, or horizontal, looks best) and a quote to [email protected].

### In This Story

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CHEEZ
5 years ago

Are the Stanford “endowments” for ALL students that have parental income of $125k or less, or are they competitive funds awarded to applicants based on their academic achievement? I find it odd that 68% of the current Stanford squad is “Undeclared”. If the endowments are based on academic achievement, then something doesn’t add up. Jback 5 years ago Stanford swimming was not doing so well before Greg and Tracy arrived. They are obviously some of the best recruiters in the nation and have masterfully leveraged the assets Stanford has to their advantage. swammer 5 years ago Setting the question of scholarship slots vs. financial aid vs. endowments aside, can someone please explain to me how 7 of the top swimmers in the nation for their class just happens to be among the smartest in the country? Also 6 of the best HS swimmers in Stanford’s last class, and 5 in 2016. We are talking about one of the toughest colleges in the US to get into academically. One that is said to have a 4.6% acceptance rate. Not a comment on Tankersley’s academics BTW. Her grades are known to be exemplary. Stanford is lucky to get someone of her academic & swim stature over Cal & several other top programs! final_word 5 years ago This article was about Morgan Tankersly. Hope she has a long and successful swimming career, gets a great education and has fun. Swimswam should do an article (or series of articles) on the state of the sport at all levels. Then everyone could voice their opinions without taking anything away from any athlete or school. It would probably be the most read series ever. The acceptance rate of incoming freshman for Stanford is 4.6%, below Harvard,(5.4%), Yale (6.3%), Cal (17.5%) and far below everyone else. I hope everyone enjoys going to NCAA’s in March knowing today and for years to come, what team is going to win. How many people would go to the Superbowl, the Rose Bowl, the… Read more » Azfan400 5 years ago Stanford isn’t all it’s cracked up to be. Almost any accredited institution can provided a motivated individual a GREAT education. Next time you go to the doctor look at where they graduated from. It’s probably somewhere obscure. The women’s swiimming stuff is all about using endowment$ to get around the scholarship limits. It was reported on swimswam that even the head coaches compensation is endowed. Elitetism and entitlement are two words that can describe this use of endowment money. How about Stanford stick to the scholarship limits and use their endowment dollars to really help those in need. As for the “rigorous academic standards”? I guarantee you the only academic standard that was used when Chelsea Clinton was admitted… Read more »

CraigH
5 years ago

As I stated above the Stanford endowment money IS used to help all students, not just swimmers or athletes. Any person whose family falls in the middle-class or below gets free tuition using funds from endowed, need-based scholarships at Stanford. Frankly it would be nice if more universities were able to provide this.

Swamfanl
5 years ago

I agree it really doesn’t matter where you go to school. If you’re motivated you can get a great education anywhere. Regarding Chelsea Clinton’s admission to Stanford, she was a straight-A student in high school and now has two masters degrees, a ph.d from oxford, and is a professor at Columbia Uniiversity. You can disagree with her and her family’s politics but there is no question she is incredibly bright.

G.I.N.A
5 years ago

Agree . Did those pesky Serbs realise they were shooting at a pre – Stanford Quaker educated A student as she bravely ran from a hail of sniper bullets? A grade I tell you .

dave
5 years ago

Lets not forget that most of the Stanford recruits are Academic All Americans, as well as ranked as a top 20 swimmers. Stanford U, is one of the most highly ranked academic universities in the world, and therefore also a top choice for Academic All American students. The Stanford swim team had a team average GPA of I believe 3.65, so an academic all American might also find this a plus. So when you are parent and have to send your child to college, don’t you think you will select the college with the best academics Plus one of the best swim programs? Sorry, that there isn’t more swim competition in the PAC12 dual meets and the NCAA, but for… Read more »

NCSwimFan
5 years ago

Well yeah, of course. That’s not a debate, and the facts prove it. The debate comes when people come into the comments section and proclaim that Stanford is the ONLY top academic school and top swimming school. That’s just not true, and it makes no sense to argue that case.

4444
5 years ago

Regarding the diversity comment, be careful not to judge a book by its cover. I am pleased that the team is of various ethnicities. One of the seven 2018 Stanford Women recruits is of mixed race(Black&White).

G.I.N.A
5 years ago

What % of each counts as mixed race these days?

Azfan400
5 years ago

As i have commented many times in the past ncaa d1 women’s swimming is not a competition it is an exhibition. The outcome of most events is not a question. The concentration of elite athletes at Stanford makes it difficult for other schools to recruit. The obvious use of endowment money to give athletes “academic aid” to get around scholarship limits is a joke. Why should other pac12 teams waste their time swimming the dual meets with Stanford. They should forfeit the meets and put the time to better use. I am sure USA swimming likes having their entire Olympic squad in one place. Lack of competition and a no win situation in women’s swimming will cause many schools to… Read more »


text:
# How do you graph f(x)=3/x+1 using holes, vertical and horizontal asymptotes, x and y intercepts?

gaby131o 2022-09-18 Answered
How do you graph $f\left(x\right)=\frac{3}{x}+1$ using holes, vertical and horizontal asymptotes, x and y intercepts?
You can still ask an expert for help

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## Answers (1)

Wischarm1q
Answered 2022-09-19 Author has 5 answers
y axis intercepts occur when x=0. x=0 is undefined for this function, so no y intercept.
x axis intercepts occur when y=0:
$\frac{3}{x}+1=0⇒x=-3$
Coordinate: (−3,0)
as $x\to \infty$, $888\frac{3}{x}+1\to 1$
as $x\to -\infty$ , $888\frac{3}{x}+1\to 1$
The line y=1 is a horizontal asymptote.
as $x\to {0}^{+}$ , $888\frac{3}{x}+1\to \infty$
as $x\to {0}^{-}$, $888\frac{3}{x}+1\to -\infty$
The line x=0 is a vertical asymptote.
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Research Short Communication| Volume 102, ISSUE 3, P2051-2053, March 01, 2019

# Short communication: Effect of carbon emission trading on European Union butter prices

Open ArchivePublished:January 10, 2019

## ABSTRACT

Since its inception, the European Union (EU) carbon emission market has been vastly successful in reducing greenhouse gas emissions. Accordingly, the usage of environmentally friendly fuels (e.g., ethanol, biodiesel) has increased significantly over the last few years. Given that EU biodiesel is mainly produced from rapeseed oil and soybean oil, higher carbon taxes are likely to increase the demand of these important vegetable oils, which further affects the prices of its close substitute such as butter. Nevertheless, the association between the EU emission trading scheme and butter prices remains understudied. In this paper, we aim to fill this vacuum in the existing literature. Applying the autoregressive distributed lag bound testing procedure, we show that emission market seems to have a long-term effect on EU butter prices, implying that changes in the levels of carbon taxes will lead to changes in the price level of butter. These results are of vital importance to policymakers and investors.

## Short Communication

Since its inception, the European Union (EU) emission trading scheme (ETS) has been remarkably efficient in the reduction of greenhouse gas (GHG) emissions. In 2015, for instance, GHG emissions in the EU-28 countries were down by 22% compared with 1990 levels, representing an absolute reduction of 1,265 million tonnes of CO2-equivalents, putting the EU on track to surpass its 2020 target, which is to reduce GHG emissions by 20% by 2020 and by 40% by 2030 compared with 1990 (data from https://ec.europa.eu/eurostat/statistics-explained/pdfscache/1180.pdf).
The success of the EU allowance (EUA) market has promoted the use of environmentally friendly fuels all over the world. For instance, biodiesel has emerged as the leading biofuel used in the EU transport sector. A recent report published by the European Commission indicates that biodiesel production capacity has increased to about 26.3 billion liters, with an annual production of about 10.5 billion liters (8.3 million tonnes of oil equivalents) or 40% of total capacity. Such significant growths in biodiesel could be due to the concerns about energy security, GHG emissions, and oil market uncertainty (
• Chiu F.P.
• Hsu C.S.
• Ho A.
• Chen C.C.
Modeling the price relationships between crude oil, energy crops and biofuels.
;
• Dutta A.
Cointegration and nonlinear causality amongst ethanol-related prices: Evidence from Brazil.
).
In the EU, biodiesel is mainly produced from rapeseed oil and soybean oil. At present, rapeseed oil accounts for about 80% of the biodiesel feedstock. Given that vegetable oil is the main feedstock for producing biodiesel in the EU, it is likely that higher carbon permit prices would raise the demand for vegetable oil, which further affects the prices of a close substitute such as butter.
• Crago C.L.
• Khanna M.
Carbon abatement in the fuel market with biofuels: Implications for second best policies.
also argue that as biofuel production competes for land with agricultural production, higher carbon tax could increase land rents, which would then raise feedstock prices. Nevertheless, investigation into the linkage between carbon emission and butter prices is understudied in the existing literature. This study aims to fill this void. As understanding the dynamics of butter prices plays a pivotal role in designing the EU dairy policy, our research could be of vital importance to policymakers and investors.
This paper uses monthly prices of the EU-ETS and butter covering the period from July 2009 to December 2017. The sample consists of 102 observations. The data were retrieved from Thomson Reuters DataStream (https://eikon.thomsonreuters.com/index.html). For the EU emission market, the data for 3 different periods are available. Phase I covers the period from January 2005 to December 2007, and phase II covers January 2008 to December 2012. At present, the EU-ETS is completing the third phase. The data for phase I are not included because the allowance prices were close to zero toward the end of phase I as a consequence of interphase EUA banking restrictions (
• Tian Y.
• Akimov A.
• Roca E.
• Wang V.
Does the carbon market help or hurt the stock price of electricity companies? Further evidence from the European context.
).
Table 1 reports the descriptive statistics for EUA and butter price indexes. These numbers suggest that food prices are more volatile than the emission market as evidenced by the corresponding standard deviations. Moreover, both indexes are negatively skewed and kurtosis was >3 for each series, implying that the data follow leptokurtic distributions. The Jarque-Bera test suggests that the data are normally distributed only for the butter price index.
Table 1Main descriptive statistics (values in parentheses denote P-values) of the 2 indices under study
ΔEUA and ΔButter = first-order differences in European Union allowance and butter prices, respectively. .
IndexMeanSDSkewnessKurtosisJarque-Bera test
ΔEUA−0.05301.0631−0.41344.239.29 (0.00)
Significant at the 1% level.
ΔButter1.44358.0547−0.03613.481.01 (0.60)
1 ΔEUA and ΔButter = first-order differences in European Union allowance and butter prices, respectively. .
** Significant at the 1% level.
Table 2 shows the unit root test results for the indexes used. The results of both augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests indicate that although the data follow a nonstationary process at levels (i.e., raw data), the stationary condition is satisfied when first differences are considered. The ADF and PP tests are used to examine whether the data contain any trend. Nonstationary processes usually have trends, whereas stationary processes do not contain such trends.
Table 2Results of unit root tests
EUA = European Union allowance. The values in parentheses denote P-values.
IndexAugmented Dickey-Fuller testPhillips-Perron test
LevelFirst differenceLevelFirst difference
EUA−1.65 (0.45)−12.29 (0.00)***−1.55 (0.45)−12.28 (0.00)**
Butter−1.44 (0.56)−12.63 (0.00)***−1.31 (0.62)−12.45 (0.00)**
Significant at the **1% or ***0.1% level.
1 EUA = European Union allowance. The values in parentheses denote P-values.
Methodologically, we used the autoregressive distributed lag (ARDL) model to examine the long-term association between carbon emission trading and EU butter prices. The ARDL bound test has several attractive features. First, all the testing equations are allowed to have different lags. Second, it can be applied regardless of whether the underlying variables are stationary, that is, I(0); integrated of order 1, that is I(1); or fractionally integrated (
• Bouri E.
• Jain A.
• Biswal P.C.
• Roubaud D.
Cointegration and nonlinear causality amongst gold, oil, and the Indian stock market: Evidence from implied volatility indices.
). Finally, the test is free from the spurious regression problem. Note that this test has a prerequisite that the variables under study should not be integrated of order 2 or higher.
Now, we construct the following unrestricted ARDL regressions without any time trend component:
$ΔEUAt=ω1+∑i=1nα1,iΔEUAt−i+∑i=1nβ1,iΔButtert−i+a1EUAt−1+b1Buttert−1+ɛ1t,$
[1]

$ΔButtert=ω2+∑i=1nα2,iΔEUAt−i+∑i=1nβ2,iΔButtert−i+a2EUAt−1+b2Buttert−1+ɛ2t,$
[2]

where Δ indicates the first difference operator: ΔEUA indicates the first-order difference for EUA prices, ΔButter is the first-order difference for EU butter prices, α and β refer to the lag coefficients for ΔEUA and ΔButter, and ε1 and ε2 are error terms. To examine whether cointegrating relationship exists among the series under study, it is sufficient to test the null hypothesis H0: a = b = 0, where a and b refer to the first-order lag coefficients for the EUA and butter prices. The general F-statistics are further calculated and compared with 2 different sets of critical values provided by
• Pesaran M.H.
• Shin Y.
• Smith R.J.
Bounds testing approaches to the analysis of level relationships.
. One of these sets is used as the upper bound for purely I(1) series, and the other is used as the lower bound for purely I(0) series. Cointegration or a long-run relationship is said to be present only if the computed F-statistic surpasses the upper bound critical value.
Table 3 shows the results of the ARDL bound tests. These findings reveal that cointegration is present when butter prices act as the dependent variable. We do not find any significant linkage when emission market is the dependent variable. Such results indicate that causality runs from emission to butter price index, implying that changes in the price level of the EUA market cause significant changes in the butter price level.
Table 3Results of autoregressive distributed lag bound tests
The critical F-statistic at the 5% level for model with all I (1) series is 5.73. See Table CI(iii) with k = 1 (where k = number of explanatory variables) on page 300 of Pesaran et al. (2001). ΔEUA and ΔButter = first-order differences in European Union allowance and butter prices, respectively.
Dependent variableTest statisticsDecision
ΔEUA1.70No cointegration
ΔButter5.85
Significant at the 5% level.
Cointegration exists
1 The critical F-statistic at the 5% level for model with all I (1) series is 5.73. See Table CI(iii) with k = 1 (where k = number of explanatory variables) on page 300 of
• Pesaran M.H.
• Shin Y.
• Smith R.J.
Bounds testing approaches to the analysis of level relationships.
. ΔEUA and ΔButter = first-order differences in European Union allowance and butter prices, respectively.
* Significant at the 5% level.
In addition to the ARDL bound tests, we also employ the Toda-Yamamoto (TY) version of Granger causality test to investigate whether any short-term linkage exists between these 2 sectors. See
• Toda H.Y.
• Yamamoto T.
Statistical inference in vector autoregressions with possibly integrated processes.
for more details about the TY test. The results of the TY test, displayed in Table 4, reveal that the EUA market and the butter industry do not move together in the short run.
Table 4Results of the Toda-Yamamoto test
The lag used in the Toda-Yamamoto test is 1. The Akaike information criterion and Bayesian information criterion are used to determine the required number of lags. EUA = European Union allowance.
CausalityTest statisticsP-value
EUA to butter1.320.25
Butter to EUA0.880.35
1 The lag used in the Toda-Yamamoto test is 1. The Akaike information criterion and Bayesian information criterion are used to determine the required number of lags. EUA = European Union allowance.
Overall, these results suggest that the EUA market seems to have a long-term effect on the butter price index. Such findings are not surprising given that higher carbon taxes promote the production of biodiesel and hence the demand for vegetable oil, which is the main feedstock for the EU biodiesel industry, would increase considerably. Because butter and vegetable oil appear to be substitute commodities, the growing demand for vegetable oil leads to a significant rise in the butter price index. Thus, the carbon emission trading has an indirect, but prominent effect on EU butter prices.
The outcomes of this empirical research could be important for policymakers in formulating appropriate hedging strategies to avoid the contagious risk emanating from the EU emission trading scheme. In addition, the information on the EUA prices could be useful in predicting the price variations of the EU butter industry as we find a significant long-run linkage between these 2 markets. Moreover, in the near future, the production of biodiesel could undergo incredible growth, which could, in turn, cause uncertainties in the butter industry. It is important therefore to develop and improve the futures market for butter to precisely measure the risk associated with this important food commodity. Such monitoring system could play a crucial role in planning EU food policy.

## REFERENCES

• Bouri E.
• Jain A.
• Biswal P.C.
• Roubaud D.
Cointegration and nonlinear causality amongst gold, oil, and the Indian stock market: Evidence from implied volatility indices.
Resour. Policy. 2017; 52: 201-206
• Chiu F.P.
• Hsu C.S.
• Ho A.
• Chen C.C.
Modeling the price relationships between crude oil, energy crops and biofuels.
Energy. 2016; 109: 845-857
• Crago C.L.
• Khanna M.
Carbon abatement in the fuel market with biofuels: Implications for second best policies.
J. Environ. Econ. Manage. 2014; 67: 89-103
• Dutta A.
Cointegration and nonlinear causality amongst ethanol-related prices: Evidence from Brazil.
Glob. Change Biol. Bioenergy. 2018; 10: 335-342
• Pesaran M.H.
• Shin Y.
• Smith R.J.
Bounds testing approaches to the analysis of level relationships.
J. Appl. Econ. 2001; 16: 289-326
• Tian Y.
• Akimov A.
• Roca E.
• Wang V.
Does the carbon market help or hurt the stock price of electricity companies? Further evidence from the European context.
J. Clean. Prod. 2016; 112: 1-8
• Toda H.Y.
• Yamamoto T.
Statistical inference in vector autoregressions with possibly integrated processes.
J. Econom. 1995; 66: 225-250


text:
Problem in number theory using logarithms.

Question:-

If n is a natural number such than $n=p_1^{a_1}.p_2^{a_2}.p_3^{a_3}.......p_k^{a_k}$ and $p_1,p_2,...p_k$ are distinct primes, then show that $log(n)>=klog(2)$.

What I think is p1 is greater than or equal to 2. So is p2 and all the way to pn. So their product should be greater than and equal to $2^{k}$. Is this proof sufficient?

Source: IIT JEE 1984

• Yes, it is correct. – TBTD Apr 23 '17 at 19:04
• Yes you're right. – Iti Shree Apr 23 '17 at 19:20
• Yes.It's right... If you write p_1 then iwith dollar signs you get $p_1$ which is better than $p1$. ... Click on Help at the top bar, choose the Help Center, choose the Q "How can I format mathematics here?" – DanielWainfleet Apr 23 '17 at 23:04
• @user254665 Thank you. i did check the post, but it had very few examples. – Reeshabh Ranjan Apr 24 '17 at 3:09
• Did you go to the bottom line and click on the MathJax quick reference? – DanielWainfleet Apr 25 '17 at 2:28


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# Causality, Correlation, and Brownian Motion

A recent question on Quora asked if machine learning could learn something from the Black Scholes model of Finance

http://www.quora.com/Machine-Learning/Can-we-learn-any-lessons-from-the-Black-Scholes-solution-to-pricing-risk-to-machine-learning-algorithms-for-personalization-recommendation-algorithms

I have been curious about this myself, but from a slightly different perspective, which I share here:

#### Introduction

There is a deep relation between statistical mechanics (stat mech) and machine learning (ML). From Exponential Families to Maximum Entropy to Deep Learning, stat mech is all over ML. Here, we are interested in distinguishing between causality and correlation, and the hope is we can learn from stat mech how to do this. We are motivated by a recent neuroscience paper [3] that suggests a new approach to Granger Causality via non-equilibirum statistical mechanics, specifically the Mori-Zwanzig Projection Operator formalism (see [6], my favorite reference, or just Google it).  Also, very recent results [10] demonstrate that one can apply  the Mori-Zwanzig formalism to a set of coupled stochastic variables like we see in Granger Causality  and other time series models.

This is, however, pretty esoteric stuff, even if you have a PhD in Theoretical Chemistry like me. So before we dive in, in this post, we first look at a something ubiquitous in science: Brownian Motion

#### Causality in Statistical Mechanics

What is causality? Let’s pose it as a problem in time series analysis. Say we want to describe the dynamics of a system where one set of variables $X(t)$ dominate, but are driven, seemingly stochastically, by some other set of variables $Y(t)$.

For example, $X$ might represent a particle floating in a viscous fluid $Y$. As the particle moves, the fluid relaxes around it, leaving a slowly decaying trail.

So even though the particle $X$ appears to move randomly, its position at time $t+\delta t$ depends on both it’s entire history $X(t)$ and the history of the local surroundings $Y(t)$. The more viscous, or stiff, the fluid is, the more the surroundings matter.

When can we say that $Y(t)$ causes $X(t)$ ?

The challenge in statistical mechanics is to decide when the particle is just random (i.e. a Brownian motion) and when the fluid is strongly affecting (i.e causing) a part of the dynamics.

#### Stochastic Problems in Astronomy and Physics

To get started, we need a some basic physics.

There is a classic review of Stochastic Problems in Astronomy and Physics [1] by the late Subrahmanyan ChandrasekharNobel Laureate (Physics, 1983) and former head of the department of physics when I was a student at the University of Chicago.

I am a firm believer that every scientist should study and know the great works of the Nobel Laureates, so I strongly encourage the reader to dig deep [1] on their own–or to at least continue reading.

Chandrasekhar introduces the Langevin equation, and uses it to describe stochastic processes such as Random Walks and Mean-Reverting Walks (which are important for cointegration, but not discussed here)

#### Random Walks

When our particle is behaving randomly, and does not remember its history, we say it is undergoing a Random Walk. This is also known as Brownian Motion. Below we plot the path of several 1-D random walks over time. Notice that they all start at (0,0), but, over time, eventually diverge from each other.

Brownian motion is ubiquitous; it appears everywhere in Science. In Finance and Economics, it is described as a Wiener Process or with the Ito Stochastic Calculus. However, if we go old school, we can also describe the random walk using something familiar from high school calculus/physics. The model is

#### the Langevin equation:

$m\dfrac{d^{2}}{d^{2}t}x(t)=-m\gamma\dfrac{d}{dt}x(t)-R(t)$

where $m$ is the mass, $m\gamma$ is a friction coefficient, and $R(t)$ represents the noise, interpreted as a random force.

What is this? We all know–or have least heard of–Newton’s famous equation

$F=ma$

Force equals mass times acceleration

Here, we flip this around, and have

$ma=F=F_f-F_R$

mass ($m$) times acceleration $\dfrac{d^{2}}{d^{2}t}x(t)$ = Force

= Frictional Force ($F_f$ minus Random Force  ($F_R$)

where

the Frictional Force, $F_f=m\gamma\dfrac{d}{dt}x(t)$,  is a constant times velocity

and

the Random Force $F_R=R(t)$ is defined below

.
The random force represents the interactions of the particle with the solvent, which undergoes the thermal, random fluctuations (shown here)

We can not describe the random force with a dynamical equation, so, instead, we define $R(t)$ through it’s time-time auto-correlation functions

$\langle R(t)R(t)\rangle=0$

$\langle R(t_1)R(t_2)\rangle=2\pi_{R}G\delta(t_1-t_2)$

This correlation function is dot product in a Hilbert space; in fact it is a Kernel.

Here it is a constant; below we define a ‘memory Kernel’ that describes what is causing the random behavior (and also deals with the failures of the theory at short times)

#### Diffusion and the Einstein relation:

The particle $X$ moves around randomly; we say it diffuses. We identify a Diffusion constant, which is just the limit of average mean squared position, taken to very long times

$D=\underset{t\rightarrow\infty}{\lim}\dfrac{1}{2t}\langle (X(t)-X(t_{0}))^{2}\rangle$

It turns out the random force is not just any random force; it is intimately related to the other parameters in the equation. Einstein noted that the friction constant $m\gamma$ is related to the diffusion $D$ constant by

$D= \dfrac{k_{B}T}{m\gamma}$

where $k_{B}$ is Boltzman’s constant, and $T$ is the temperature.

With a little work (not shown here), we can also use the Einstein relation and our definition of the random force to express the Diffusion constant in terms of the time correlation of the velocity $u(t)$

$D=\int_{0}^{\infty}\langle u(t_{0})u(t_{0}+t)\rangle dt$, where

This means that if the velocities are weakly correlated (over time), the surroundings drag on the particle, and it diffuses slower. On the other hand, when the velocities are strongly correlated, the particle does not feel the surroundings, and it diffuses faster.

Because the velocity platys such an important role, frequently we express the Langevin equation in terms of the velocity

$m\dfrac{d}{dt}u(t)=-m\gamma u(t)-R(t)$

We can also express the velocity-velocity auto-correlation function as a simple exponential decay

$\langle u(t_{0})u(t_{0}+t)\rangle=\langle u^2\rangle\exp(-\gamma|t|)$

where

$\langle u^2\rangle=\dfrac{D}{\gamma}$

Not that our theory breaks down at short time scales because we expect the the velocity process to be stationary process. That is, we expect

$\langle u(t)\dfrac{d}{dt}u(t) \rangle=0$

which is clearly not satisified.

On long timescales, the Langevin equation describes a mathematical Brownian motion, but on small timescales, the Langevin equation includes inertial effects that are not present in the Brownian description. These inertial affects can be very real, as shown above in the image of the pollen in the viscous fluid, and are corrected for below.

Also, we are considering systems that actually have velocities–this is important since we don’t usually think of pure stochastic processes, such as economic or financial time series, as having a well defined or instantaneous velocity.

Still, we are on the right track. We can relate the random forces directly to the macroscopic diffusion $D_{u}$ (in velocity space) through the correlation function of the random forces

$D_{u}\sim\int_{0}^{\infty}\langle R(t_{0})R(t_{0}+t)\rangle dt$

So we can determine how strongly the environment ’causes’ the dynamics by measuring an auto-correlation function of the random forces

We begin to see the link between correlation functions and causality. This is a direct example of

the Fluctuation-Dissipation theorem

<The fluctuation-dissipation theorem is a general result of statistical mechanics that quantifies the relation between the fluctuations in a system at equilibrium and its response to external perturbations. Basic examples include Brownian motion and Johnson–Nyqvist noise, but this phenomena also arise in non-equilibrium systems, and, perhaps, even in the neocortex.

Here, the same random forces that cause the erratic motion of a particle also causes drag. If something pulls the particle through the fluid, it feels this drag. The particle’s random motions and dissipative frictional forces have the same origin, or cause.

We can see this in 2 different ways:

1. We can apply an external force to the system and monitor the (linear) response
2. We can construct equilibrium velocity distribution and relate this to the self-diffusion

More generally, we can observe any external action $Y$  that “causes” $X$ to move by inferring the ‘equilibrium’ distribution, and  computing the appropriate correlation function of the random forces; this is our link to causality.

What does this have to do with Machine Learning? Notice I said infer. For a general system, we will infer the so-called ‘equilibrium’ , or most likely, (Boltzman) distribution, using a variational method. Then we can directly evaluate the correlation functions (which are the partial derivatives of our calculated partition function). But first, we need a more robust Langevin equation to work with.

#### the Generalized Langevin Equation (GLE)

We can generalize the Langevin equation by redefining the Random Force through a different Kernel. Let

$\langle R(t_1)R(t_2)\rangle=\xi(t_1-t_2)$

where $\xi(t)$ is called the memory Kernel.

This gives rise to a Generalized Langevin Equation (GLE)

$\dfrac{d}{dt}u(t)=-\int_{t_0}^{t}\xi(t-t')u(t')dt+R(t)$

A typical application of the GLE in chemical physics to extend Brownian to describe the dynamics of a particle in a viscoelastic fluid (a fluid that is both thick and deformable); the memory Kernel gives rise to a generalized frictional force.

Some examples:

When memory effects matter, we have causality.  Memory effects an appear in the velocity auto-correlation function (VACF) either through the presence of an oscillatory behavior or by means of slowly decreasing correlations [11].   In liquids, oscillations appear, for example, when particles are trapped, or caged, by their surroundings.  And correlations slowly decrease in cases of anomalous diffusion, such as liquid-like defects [12].  Indeed, these effects arise all over physics, and, we suspect, in other systems as well.   Only very recently, however, has the Mori-Zwanzig formalism been recognized as useful as a general mathematical tool for both optimal prediction [13],  Markov Models [14], and Neuroscience [3]

#### Towards a Generalized Granger Causality

I would like to apply the GLE to problems in general machine learning. It would be necessary tointroduce a more abstract form of the GLE. We will then derive this GLE using the Mori-Zwanzig Projection Operator formalism from statistical mechanics.

It is then necessary how to infer the model parameters using variational techniques. Finally, we will demonstrate the relationship between Granger causality and a new measure that, we hope, will be more suitable in noisy, non-linear regimes. If time permits we will try to get to this

#### References

[1] Engle, Robert F., Granger, Clive W. J. (1987) “Co-integration and error correction: Representation, estimation and testing“, Econometrica, 55(2), 251–276.

[3] D.Hsu and M. Hsu (2009) Zwanzig-Mori projection operators and EEG dynamics: deriving a simple equation of motion

[4] Peter Hänggi, Fabio Marchesonia! Introduction: 100 years of Brownian motion

[5]http://web4.cs.ucl.ac.uk/staff/C.Bracegirdle/BayesianConditionalCointegration.php

[6] J.P. Hansen and I. R. McDonald (1986) , Theory of Simple Liquids

[7] http://www.dam.brown.edu/documents/Stinis_talk1.pdf

[8] Kubo, the Fluctuation-Dissipation Theorem

[10] Jianhua Xing and K. S. Kim , Application of the projection operator formalism to non-Hamiltonian dynamics J. Chem. Phys. 134, 044132 (2011)

[12] C. H. Martin and S. J. Singer, “The behavior of point defects in a model crystal near melting”,Phys. Rev. B44: 477 (1991)

[13] Alexandre J. Chorin*, Ole H. Hald*, and Raz Kupferman, Optimal prediction and the MoriZwanzig representation of irreversible processes, PNAS (1999)

[14a] C. L. Beck, S. Lall§, T. Liang and M. West, Model reduction, optimal prediction, and the Mori-Zwanzig representation of Markov chains
[
14b] Madhusudana Shashanka, A FAST ALGORITHM FOR DISCRETE HMM TRAINING USING OBSERVED TRANSITIONS


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group theory

∞-Lie theory

Examples

$\infty$-Lie algebras

Differential geometry

differential geometry

synthetic differential geometry

Contents

Definition

Given a Lie group $G$, it acts smoothly on the dual $g^*$ of its Lie algebra $g$ by the coadjoint action. The orbits of that action are called coadjoint orbits.

Coadjoint orbits are especially important in the orbit method of representation theory or, more generally, geometric quantization.

Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of Virasoro algebra).

Properties

As symplectic leafs of the Lie-Poisson structure

The dual $g^*$ of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, called the Lie-Poisson structure, namely for any $a\in g^*$,

$\{ f, g\}(a) := \langle [d f_a, d g_a],a\rangle$

The coadjoint orbits are the symplectic leaves of that structure; hence each orbit is a symplectic manifold.

References

• B. C., The Structure of the Space of Coadjoint Orbits of an Exponential Solvable Lie Group, ransactions of the American Mathematical Society Vol. 332, No. 1 (Jul., 1992), pp. 241-269, (JSTOR)

Revised on October 29, 2013 23:46:24 by Urs Schreiber (82.169.114.243)


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# How do you find the Remainder term in Taylor Series?

There are lots of different ways this can be thought about, though probably the simplest and most common way is to use the inequality |R_{k}(x)|\leq \frac{M\cdot r^{k+1}}{(k+1)!}, where the $\left(k + 1\right)$st derivative ${f}^{\left(k + 1\right)} \left(x\right)$ of $f$ satisfies $| {f}^{\left(k + 1\right)} \left(x\right) | \setminus \le q M$ over the interval $\left[c - r , c + r\right]$ (assuming sufficient differentiability/smoothness of $f$ over the interval).
The function ${R}_{k} \left(x\right)$ is the "remainder term" and is defined to be ${R}_{k} \left(x\right) = f \left(x\right) - {P}_{k} \left(x\right)$, where ${P}_{k} \left(x\right)$ is the $k$th degree Taylor polynomial of $f$ centered at $x = a$: P_{k}(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots.
For example, if $f \left(x\right) = {e}^{x}$, $a = 0$, and $k = 4$, we get ${P}_{4} \left(x\right) = 1 + x + \setminus \frac{{x}^{2}}{2} + \setminus \frac{{x}^{3}}{6} + \setminus \frac{{x}^{4}}{24}$. Moreover, if we consider this over the interval $\left[- 2 , 2\right]$ so that $r = 2$, then the inequality above becomes |R_{4}(x)|\leq \frac{e^{2}\cdot 2^{5}}{5!}=\frac{4}{15}e^{2}\approx 1.97. So we can expect ${P}_{4} \left(x\right)$ to approximate ${e}^{x}$ to within this amount over the interval $\left[- 2 , 2\right]$ (actually, the approximation is even better than the error bound gives...the point is that the error bound gives a guarantee ).


text:
# Solution - He Ratio of Rates of Fall of Temperature During the Two Intervals - Temperature and Heat

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#### Question

A metal ball cools from 64 °C to 50 °C in 10 minutes and to 42 °C in next 10 minutes. The ratio of rates of fall of temperature during the two intervals is _______.

(A) 4/7

(B) 7/4

(C) 2

(D) 2.5

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Solution for question: He Ratio of Rates of Fall of Temperature During the Two Intervals concept: Temperature and Heat. For the courses HSC Science (Computer Science), HSC Science (Computer Science), HSC Science (Electronics), HSC Science (Electronics), HSC Science (General) , HSC Science (General)
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# ACT PracticeQuestion of the Day

Work on that higher ACT score every day by building practice into your daily routine.

Build prep into your daily routine!

Get a question delivered to your inbox every day, with a detailed explanation of the correct answer. The next time you see a similar question—you’ll be ready.

# ACT Question for May 19th

### Weaver High School is organizing a Spooky Haunted House event. Tickets to the event are $4.50 for adults and$3 for students. To cover costs, a total of $900 must be collected from ticket sales. One of the following graphs in the standard (x,y) coordinate plane, where x is the number of adult tickets sold and y is the number of student tickets sold, represents all the possible combinations of ticket sales that cover at least$900 in expenses. Which graph is it?

Incorrect. This represents 600 adult tickets and 600 student tickets.

Incorrect. This represents 600 adult tickets and 200 student tickets.

This is the credited response! This graph represents 200 adult tickets and 300 student tickets.

Incorrect. This represents 900 adult tickets and 300 student tickets.

Incorrect. This represents 200 adult tickets and 900 student tickets.

### Let's get started

Kaplan | 750 Third Ave | New York, NY 10017


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### Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Comparing Continued Fractions

Which of these continued fractions is bigger and why?

# Not Continued Fractions

##### Stage: 4 and 5 Challenge Level:
1. The key here is that $x$ has to be the integer part of $N$ because the 'continued fraction' part of the expression gives a value less than one.

As $y$ and $z$ are positive integers (whole numbers), $y + 1/z > 1$ and $1/(y+1/z) < 1$ so we know that this must equal $3/7$ and $x = 1$.

Hence $y + 1/z = 7/3$. Again $y$ has to be the integer part of $7/3$ so $y = 2$ and $z = 3$.

2. As in the first part, if $N = 8/5$, then we must have $x = 1$ and $y + 1/z = 5/3$.

To make $y$ and $z$ positive integers we must have $1/z < 1$ and $y = 1$.

It then follows that $1/z = 2/3$ so it is impossible to find positive integer values for $x$, $y$ and $z$.


text:
Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01mg74qp90d
 Title: Large-scale Multi-output Gaussian Processes for Clinical Decision Support Authors: Cheng, Li-Fang Advisors: Engelhardt, Barbara ELi, Kai Contributors: Electrical Engineering Department Keywords: Electronic health recordsGaussian processreinforcement learningtime series analysistreatment effect estimation Subjects: StatisticsArtificial intelligenceMedicine Issue Date: 2019 Publisher: Princeton, NJ : Princeton University Abstract: In the scenario of real-time monitoring of hospital patients, high-quality inference of patients’ health status using all information available from clinical covariates and lab tests is essential to enable successful medical interventions and improve patient outcomes. Developing a computational framework that can learn from observational large-scale electronic health records (EHRs) and make accurate real-time predictions is a critical step in achieving this goal. However, existing EHRs pose several challenges for conventional methods. For instance, many covariates are sparsely sampled in time across patients. In addition, there are substantial uncertainties in patient condition and disease progression at any time. These properties make inferring the physiological status of a patient or joint analysis of time series across patients challenging. This dissertation first presents MedGP, a statistical framework that provides accurate real-time predictions of physiological states. MedGP is based on multi-output Gaussian processes, a Bayesian nonparametric model, to capture temporal structures between clinical covariates from noisy and irregularly sampled time series data. MedGP has a number of benefits over current methods. First, alignments of time series across patients are not required. Next, MedGP learns robust and interpretable relationships across covariates through regularization. Lastly, to tackle the computational bottlenecks, an approximate inference approach is derived and shown to achieve orders of speedup. The empirical results show significant improvements over baselines in making online predictions on two real-world medical data sets consist of tens of thousands of patients and millions of observations. Finally, two extended frameworks based on MedGP are explored to enhance clinical decision-making processes, including unifying with reinforcement learning algorithms for action recommendation and designing non-stationary kernels to learn the underlying dynamics of clinical treatments. Both methods demonstrate encouraging results on improving clinical practices and advancing towards precision medicine. URI: http://arks.princeton.edu/ark:/88435/dsp01mg74qp90d Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: catalog.princeton.edu Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Electrical Engineering

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# 物理代写|热力学代写Thermodynamics代考|ENES232 What Is the Second Law of Thermodynamics?

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## 物理代写|热力学代写Thermodynamics代考|What Is the Second Law of Thermodynamics?

The concepts of the Second Law of Thermodynamics and of entropy have their roots in the early nineteenth century, when engineers started to develop and to operate heat engines. One of the problems was that not all forms of energy are of equal “utility”. For example, electrical or mechanical work (from a resistance heater or a stirrer) can-without restriction be used to increase the internal energy of a water reservoir. It is not possible, however, to transform all the energy stored within a warm water reservoir into useful work. This statement holds for all processes. In a Clausius-Rankine process, as an example of a heat

engine discussed in Chapter 6 , the heat provided by the combustion of coal can only partially be converted into work. In heat engines, part of the energy provided in the form of heat always leaves the engine as “waste” heat at a low temperature.

This can be seen in Figure 3.1, which depicts the basic principle of a heat engine. $Q$ indicates the heat typically provided by a combustion process at a high temperature $T$, whereas $Q_{0}$ indicates the heat lost at a low temperature $T_{0}$ to the environment. Therefore, only the difference may be obtained as useful mechanical (shaft) work $W_{\mathrm{s}}$.
An energy balance shows
$$Q+W_{\mathrm{s}}+Q_{0}=0 \quad \text { or } \quad\left|W_{\mathrm{s}}\right|=Q-\left|Q_{0}\right| .$$
In Equation 3.1, $W_{\mathrm{s}}$ and $Q_{0}$ have negative values because they are leaving the system. In general, and if not otherwise specified, quantities added to the system have positive values while leaving quantities count negatively (see Questions 1.4.2 and 1.4.3).

The second form of Equation $3.1$ has been written using absolute values for $W_{\mathrm{s}}$ and $Q_{0}$ because we are normally interested in these rather than in the negative values. As a result, $Q$ represents the heat transferred to the system at high temperature, whereas $\left|W_{\mathrm{s}}\right|$ and $\left|Q_{0}\right|$ stand for the shaft work obtained and the heat leaving the system at a low temperature. It follows that a thermal efficiency factor of the machine, defined as the ratio between the net work output, and the heat input may be computed as follows (see Chapter 6)
$$\eta_{\text {th }} \equiv \frac{\left|W_{\mathrm{s}}\right|}{Q}=1-\frac{\left|Q_{0}\right|}{Q} .$$

## 物理代写|热力学代写Thermodynamics代考|How Did the Need for a New State Function Arise?

The big question in the early days of heat machines was how much heat had to be “wasted” as $\left|Q_{0}\right|$ because this would determine the amount of energy that could be retrieved as shaft work and also the efficiency of the machine: The lower this could be kept, the more efficient the process would be.

In a purely mathematical analysis of heat engines Sadi Carnot showed (Carnot 1824) that the fraction of wasted heat with respect to the heat provided must indeed reach a minimum and that the efficiency would reach a maximum value if the heat engine could be operated strictly reversibly. If there was a way to predict this “magical” heat minimum $Q_{0 \text { rev }}$ one could use the above equations to predict at least the theoretical maximum efficiency of heat engines. Unfortunately, this is not possible because heat is not a state function. Carnot has also shown that the efficiency must be exactly the same for any conceivable reversible heat engine operating between the same temperatures, and that they will be the more efficient the larger the temperature difference $\left(T-T_{0}\right)$.

Based on these results, the question arose whether a new state function could be found that would permit one to predict if not $Q_{0}$ then at least $Q_{0 \mathrm{rev}}$ in the case of reversible processes. Since the heat engine is a cyclic process (see Question 6.1), the amount of this state function in the engine cannot change over a complete cycle, because the cycle could only work repeatedly if the system returned to the exact same state after a complete cycle. The amount of this state function taken up would therefore have to be exactly equal to the amount released over a whole cycle.

A crucial element in the search for such a state function was probably the realization by Lord Kelvin around 1850 (see e.g. Levenspiel 1996), and in a preliminary form already by Carnot, that the amounts of heat reversibly exchanged with the environment are proportional to the absolute temperatures $T$ and $T_{0}$ :
$$\frac{\left|Q_{0 \text { rev }}\right|}{Q_{\text {rev }}}=\frac{T_{0}}{T} \quad \text { or } \quad \frac{\left|Q_{0 \text { rev }}\right|}{T_{0}}=\frac{Q_{\text {rev }}}{T} .$$
The last equation demonstrates that the same amount of $Q_{\mathrm{rev}} / T$ is released and taken up during a complete working cycle, so that the amount accumulated inside the engine stays constant. The function $Q_{\mathrm{rev}} / T$ can thus be considered as the desired state function. It can also be shown that the change of such a state function between two states of the system only depends on the two states and not on the path taken to bring the system from one state to the other, as is the case with any state function.

Clausius defined, around 1865, the new state function, which he named entropy, for the case of non-isothermal processes as
$$\mathrm{d} S \equiv \frac{\delta Q_{\mathrm{rev}}}{T} .$$
For isothermal heat transfer, this becomes
$$\Delta S \equiv \frac{Q_{\mathrm{rev}}}{T} .$$
These definitions are not just applicable to heat engines but are absolutely general. In Equations $3.4$ and $3.5$ and in the following equations, $\delta Q_{\text {rev }}$ and $Q_{\text {rev }}$ do not therefore only signify the heat provided (reversibly) to a heat engine at high temperature, but any amount of heat reversibly exchanged between any closed system and its environment.

## 物理代写|热力学代写Thermodynamics代考|What Is the Second Law of Thermodynamics?

$$Q+W_{\mathrm{s}}+Q_{0}=0 \quad \text { or } \quad\left|W_{\mathrm{s}}\right|=Q-\left|Q_{0}\right| .$$

$$\eta_{\mathrm{th}} \equiv \frac{\left|W_{\mathrm{s}}\right|}{Q}=1-\frac{\left|Q_{0}\right|}{Q} .$$

## 物理代写|热力学代写Thermodynamics代考|How Did the Need for a New State Function Arise?

$$\frac{\left|Q_{0 \mathrm{rev}}\right|}{Q_{\mathrm{rev}}}=\frac{T_{0}}{T} \quad \text { or } \quad \frac{\left|Q_{0 \mathrm{rev}}\right|}{T_{0}}=\frac{Q_{\mathrm{rev}}}{T} .$$

$$\mathrm{d} S \equiv \frac{\delta Q_{\mathrm{rev}}}{T} .$$

$$\Delta S \equiv \frac{Q_{\mathrm{rev}}}{T} .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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I am a postdoctoral investigator in the Mahadevan Group at Woods Hole Oceanographic Institution. My work is motivated by the question of how ocean circulation and turbulence affect Earth’s climate – a big question that leads to broad interests. My current research centers on intraseasonal oscillations of the Indian monsoon over the Bay of Bengal.

My PhD work was in string theory. My thesis on M-theory/heterotic duality was advised by Professor David Morrison, and I was a member of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics. I was previously a National Science Foundation Graduate Research Fellow, and before that I completed my BSc in physics and mathematics at Stanford University.

6/8/21: My paper with Bobby Acharya and David R. Morrison on heterotic duals of M-theory on $G_2$ orbifolds is now available as a preprint on arXiv. Click here to view.