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# Bragg-Williams approximation for the Ising model

Again, we consider a system with the following Hamiltonian:

${\displaystyle {\mathcal {H}}=-J\sum _{\left\langle ij\right\rangle }S_{i}S_{j}-\sum _{i}H_{i}S_{i}}$
In this case our random variables ${\displaystyle \varphi }$ are the spins: ${\displaystyle \varphi _{\alpha }=S_{i}}$, and our order parameter is ${\displaystyle \left\langle \varphi _{\alpha }\right\rangle =m_{i}}$ (we will later see that this is exactly the local magnetization also in the mean field approximation). For what we have previously stated we must build a single-particle probability density ${\displaystyle \rho _{i}=f(m_{i})}$ in terms of this order parameter, such that ${\displaystyle \operatorname {Tr} \rho _{i}=1}$ and ${\displaystyle \operatorname {Tr} (\rho _{i}S_{i})=m_{i}}$. For example, since we have two constraints on ${\displaystyle \rho _{i}}$ we could use a linear expression for the probability density involving two parameters[1]:
${\displaystyle \rho _{i}=a\delta _{S_{i},1}+b(1-\delta _{S_{i},1})}$
where ${\displaystyle a}$ and ${\displaystyle b}$ are, respectively, the probability that ${\displaystyle S_{i}=+1}$ and ${\displaystyle S_{i}=-1}$. Inserting this expression of ${\displaystyle \rho _{i}}$ in the two constraints we get:
${\displaystyle \operatorname {Tr} \rho _{i}=a+b=1\quad \qquad \operatorname {Tr} (\rho _{i}S_{i})=\left\langle S_{i}\right\rangle _{\rho _{i}}=a-b=m_{i}}$
and therefore:
${\displaystyle a={\frac {1+m_{i}}{2}}\quad \qquad b={\frac {1-m_{i}}{2}}}$
Thus:
${\displaystyle \rho _{i}={\frac {1+m_{i}}{2}}\delta _{S_{i},1}+{\frac {1-m_{i}}{2}}(1-\delta _{S_{i},1})}$
We are now able to compute the two terms that contribute to the free energy ${\textstyle F_{\rho _{\text{m.f.}}}=\left\langle {\mathcal {H}}\right\rangle _{\rho _{\text{m.f.}}}+k_{B}T\sum _{\alpha }\left\langle \ln \rho _{\alpha }\right\rangle _{\rho _{\text{m.f.}}}}$ of the system.

First of all:

${\displaystyle \left\langle {\mathcal {H}}\right\rangle _{\rho _{\text{m.f.}}}=-J\sum _{\left\langle ij\right\rangle }\left\langle S_{i}S_{j}\right\rangle _{\rho _{\text{m.f.}}}-\sum _{i}H_{i}\left\langle S_{i}\right\rangle _{\rho _{\text{m.f.}}}}$
However, since ${\textstyle \rho _{\text{m.f.}}=\prod _{i}\rho _{i}}$ (namely the degrees of freedom are independent, as we have already seen), then ${\displaystyle \left\langle S_{i}S_{j}\right\rangle _{\rho _{\text{m.f.}}}=\left\langle S_{i}\right\rangle _{\rho _{i}}\left\langle S_{j}\right\rangle _{\rho _{j}}}$. Furthermore, if ${\displaystyle g(S_{i})}$ is a generic function of ${\displaystyle S_{i}}$ then:
{\displaystyle {\begin{aligned}\left\langle g(S_{i})\right\rangle _{\rho _{\text{m.f.}}}=\operatorname {Tr} (g(S_{i})\rho _{i})=\sum _{S_{i}=\pm 1}g(S_{i})\rho _{i}=\\{}\\=\sum _{S_{i}=\pm 1}g(S_{i})\left({\frac {1+m_{i}}{2}}\delta _{S_{i},1}+{\frac {1-m_{i}}{2}}(1-\delta _{S_{i},1})\right)={\frac {1+m_{i}}{2}}g(1)+{\frac {1-m_{i}}{2}}g(-1)\end{aligned}}}
and so if ${\displaystyle g(S_{i})=S_{i}}$ we have ${\displaystyle \left\langle S_{i}\right\rangle _{\rho _{\text{m.f.}}}=m_{i}}$: our order parameter is (as we expected) the local magnetization of the system. Therefore:
${\displaystyle \left\langle {\mathcal {H}}\right\rangle _{\rho _{\text{m.f.}}}=-J\sum _{\left\langle ij\right\rangle }m_{i}m_{j}-\sum _{i}H_{i}m_{i}}$

Then, for the other term of ${\displaystyle F}$ we have:

${\displaystyle \sum _{i}\left\langle \ln \rho _{i}\right\rangle _{\rho _{\text{m.f.}}}=\sum _{i}\operatorname {Tr} (\rho _{i}\ln \rho _{i})=\sum _{i}\left({\frac {1+m_{i}}{2}}\ln {\frac {1+m_{i}}{2}}+{\frac {1-m_{i}}{2}}\ln {\frac {1-m_{i}}{2}}\right)}$

The total energy of the system will be:

{\displaystyle {\begin{aligned}F_{\rho _{\text{m.f.}}}=\left\langle {\mathcal {H}}\right\rangle _{\rho _{\text{m.f.}}}+k_{B}T\sum _{i}\left\langle \ln \rho _{i}\right\rangle _{\rho _{\text{m.f.}}}=\\=-J\sum _{\left\langle ij\right\rangle }m_{i}m_{j}-\sum _{i}H_{i}m_{i}+k_{B}T\sum _{i}\left({\frac {1+m_{i}}{2}}\ln {\frac {1+m_{i}}{2}}+{\frac {1-m_{i}}{2}}\ln {\frac {1-m_{i}}{2}}\right)\end{aligned}}}
We now have to minimize ${\displaystyle F_{\rho _{\text{m.f.}}}}$ with respect to ${\displaystyle m_{i}}$:
{\displaystyle {\begin{aligned}0={\frac {\partial F_{\rho _{\text{m.f.}}}}{\partial m_{i}}}=-2J\sum _{j\in {\text{n.n.}}(i)}m_{j}-H_{i}+k_{B}T\left({\frac {1}{2}}\ln {\frac {1+m_{i}}{2}}+{\frac {1}{2}}-{\frac {1}{2}}\ln {\frac {1-m_{i}}{2}}-{\frac {1}{2}}\right)=\\=-2J\sum _{j\in {\text{n.n.}}(i)}m_{j}-H_{i}+{\frac {k_{B}T}{2}}\ln {\frac {1+m_{i}}{1-m_{i}}}\end{aligned}}}
where ${\displaystyle j\in {\text{n.n.}}(i)}$ means that the ${\displaystyle j}$-th site is a nearest neighbour of the ${\displaystyle i}$-th one. Now, recalling that:
${\displaystyle \tanh ^{-1}x={\frac {1}{2}}\ln {\frac {1+x}{1-x}}\qquad |x|<1}$
(and surely ${\displaystyle |m_{i}|<1}$) we can write:
${\displaystyle k_{B}T\tanh ^{-1}m_{i}=2J\sum _{j\in {\text{n.n.}}(i)}m_{j}+H_{i}}$
and inverting the hyperbolic tangent:
${\displaystyle m_{i}=\tanh \left[\beta \left(2J\sum _{j\in {\text{n.n.}}(i)}m_{j}+H_{i}\right)\right]}$
We have again found the self-consistency equation for the magnetization that we have already encountered in the Weiss mean field theory for the Ising model! This is again a confirmation that all mean field theories are equivalent.

We can thus see that in the Bragg-Williams approximation also the ${\displaystyle \beta }$ and ${\displaystyle \alpha }$ exponents are the same of the Weiss mean field theory; in fact, we have seen in Critical exponents of Weiss mean field theory for the Ising model that they come from the expansion of the free energy density for small values of the magnetization when ${\displaystyle H=0}$. If we now set ${\displaystyle H=0}$ into ${\displaystyle F_{\rho _{\text{m.f.}}}}$, so that ${\displaystyle m_{i}=m\quad \forall i}$, we get:

${\displaystyle {\frac {F_{\rho _{\text{m.f.}}}}{N}}=-Jzm^{2}+k_{B}T\left({\frac {1+m}{2}}\ln {\frac {1+m}{2}}+{\frac {1-m}{2}}\ln {\frac {1-m}{2}}\right)}$
If we now expand the logarithm for small ${\displaystyle m}$, we get:
{\displaystyle {\begin{aligned}f(m)=\lim _{N\to \infty }{\frac {F_{\rho _{\text{m.f.}}}}{N}}=-Jzm^{2}+k_{B}T\left(-\ln 2+{\frac {m^{2}}{2}}+{\frac {m^{4}}{12}}+\cdots \right)\sim \\\sim -k_{B}T\ln 2+\left({\frac {k_{B}T}{2}}-Jz\right)m^{2}+{\frac {k_{B}T}{2}}m^{4}\end{aligned}}}
Therefore we see that the behaviour of ${\displaystyle f}$ near ${\displaystyle m=0}$ is the same of equation Weiss mean field theory for the Ising model , and so both the exponents ${\displaystyle \beta }$ and ${\displaystyle \alpha }$ turn out to be equal to what we have determined[2] in Critical exponents of Weiss mean field theory for the Ising model.

We can also say something[3] in the case ${\displaystyle H\neq 0}$. Supposing that our system is uniform, i.e. ${\displaystyle m_{i}=m\quad \forall i}$, we can rewrite ${\displaystyle F_{\rho _{\text{m.f.}}}}$ as:

${\displaystyle {\tilde {f}}:={\frac {F_{\rho _{\text{m.f.}}}}{N}}=-Hm+f_{0}}$
where:
${\displaystyle f_{0}=-Jzm^{2}+k_{B}T\left({\frac {1+m}{2}}\ln {\frac {1+m}{2}}+{\frac {1-m}{2}}\ln {\frac {1-m}{2}}\right)}$
and since we are looking for the absolute minimum of ${\displaystyle {\tilde {f}}}$:
${\displaystyle 0={\frac {\partial {\tilde {f}}}{\partial m}}=-H-f'_{0}=-H-2Jzm+{\frac {k_{B}T}{2}}\ln {\frac {1+m}{1-m}}}$
which gives the self-consistency equation we already know:
${\displaystyle h=-2\beta Jzm+\tanh ^{-1}m}$
As we can see from the figures below the number of the possible solutions depends on the temperature of the system:

Solutions of the self-consistency equation for ${\displaystyle T>T_{c}}$
Solutions of the self-consistency equations for ${\displaystyle T

In particular if ${\displaystyle 2\beta Jz<1}$ (i.e. ${\displaystyle T>T_{c}}$), expanding the right hand side of the self-consistency equation we get a positive linear term (${\displaystyle -2\beta Jzm>0}$), so that the behaviour of ${\displaystyle -\beta Jzm+\tanh ^{-1}m}$ is as shown in the first figure, and the equation has only one solution. On the other hand if ${\displaystyle T then the linear term changes sign and ${\displaystyle -\beta Jzm+\tanh ^{-1}m}$ behaves as in the second figure: in this case if ${\displaystyle |h|}$ is small enough there are three possible solutions, which we have called ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$ and ${\displaystyle m_{3}}$. These are all extrema of ${\displaystyle {\tilde {f}}}$, but how can we understand which is a minimum or a maximum? And above all, which of them is the global minimum. If we suppose ${\displaystyle h>0}$ to be large there will be only one solution, ${\displaystyle m_{3}}$, and as ${\displaystyle h}$ decreases also ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ will appear; we can therefore argue that for the continuity of ${\displaystyle {\tilde {f}}}$ the solution ${\displaystyle m_{3}}$ is still a minimum also when ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are present. Similarly, if we take ${\displaystyle h<0}$ we can conclude that also ${\displaystyle m_{1}}$ is a minimum; therefore ${\displaystyle m_{2}}$ will necessarily be a maximum. Now, in order to see which between ${\displaystyle m_{1}}$ and ${\displaystyle m_{3}}$ is the global minimum of ${\displaystyle {\tilde {f}}}$ let us take ${\displaystyle h>0}$ and compute:

${\displaystyle {\tilde {f}}(m_{3})-{\tilde {f}}(m_{1})=\int _{m_{1}}^{m_{3}}{\tilde {f}}'(m)dm=\int _{m_{1}}^{m_{3}}(f'_{0}-h)dm}$
From the second figure we see that this is equal to the area enclosed by the the straight line ${\displaystyle h={\text{const.}}}$ and the graph of ${\displaystyle f'_{0}}$, which is clearly negative if ${\displaystyle h>0}$. Therefore ${\displaystyle {\tilde {f}}(m_{3})<{\tilde {f}}(m_{1})}$ and so we conclude that ${\displaystyle m_{3}}$ is always the global minimum of ${\displaystyle {\tilde {f}}}$. Similarly, when ${\displaystyle h<0}$ the global minimum of ${\displaystyle {\tilde {f}}}$ is ${\displaystyle m_{1}}$.

This means that as soon as ${\displaystyle h}$ changes sign the global minimum of ${\displaystyle {\tilde {f}}}$ changes abruptly from ${\displaystyle m_{3}}$ to ${\displaystyle m_{1}}$. We are thus obtaining the phenomenology that is indeed observed for a magnet when we change the external field ${\displaystyle H}$. In other words the sets of points ${\displaystyle (h(m),m)}$ are exactly the graphs of the ${\displaystyle (H,M)}$ phase diagram we have seen in Phase transitions and phase diagrams, i.e. the graphs of the magnetization seen as a function of the external field.

1. This form of ${\displaystyle \rho _{i}}$ is very general, and does not depend on the fact that the degrees of freedom of the system can only assume two values: if there is a different number of possible states, say ${\displaystyle n}$, then ${\displaystyle \rho _{i}}$ can be written in the same form, but ${\displaystyle a}$ will be the probability of one state while ${\displaystyle b}$ the probability of the remaining ${\displaystyle n-1}$ ones. We will shortly see this when we will apply the Bragg-Williams approximation to the Potts model.
2. Note that also the temperature of the transition is still the same, considering also the ${\displaystyle 1/2}$ factor we have already mentioned.
3. These considerations apply in general also in the other mean field theories considered, but we show them now.


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This article covers a sublime explanation and a simple example of Vanilla Gradient Descent algorithm, Stochastic Gradient Descent, Momentum Optimizer, and Adam Optimizer in which RMSProp is also explained
TKTejas Khare24.00
Apr 30, 2021

Optimization Methods are one of the vital aspects of Machine Learning, Deep Learning, and also just Neural Networks. For instance, a high accuracy classifier depends on the weights 'W' and bias 'b' values to obtain a minimum loss after its training. Optimization is like a driver for neural networks that enable them to learn from the data fed to the network. In simple terms, Optimization is responsible to learn the patterns from the data and predict accordingly which heavily resides on the loss factor.

Initializing the W and b parameters with random values, in the beginning, will be a good option. Actually, No, it won't be the only option and definitely not the best one. The reasons are

1. Modal Loss - It is nothing but while training, your network focuses more on some classes and performs very badly for others.
2. Even if the above option works, it will surely take a huge amount of time to complete it because modern networks contain at least a million neurons. It is not such a good idea to wait and hope that W and b will get those perfect values and we can move on.

Hence, to skip these problems, defining an optimization algorithm would be fruitful. Let us see some optimization algorithms for machine learning.

Contents of the article

2. Momentum Based
3. Nesterov Momentum
5. Comparison of the optimizers

It has two implementations

1. The standard Vanilla Implementation
2. The optimized Stochastic Gradient Descent (SGD)

### The standard Vanilla method

Let's take the above image for our reference, you can see there are multiple local minima and only one global minimum. The network needs to get to the global minima to produce the best results. To get there, we need to define a function that takes W and b to get there as it cannot just go there because it does not exist initially.

Each position on the loss line corresponds to some value of W and b and our goal is to try different values of W and b, get their loss, and take another step towards the next lower loss value. Below is an implementation for the standard Vanilla method Implementation Code

# The loop will start at x=1 for our example
i = 1
# Learning rate (Using adaptive learning rate like exponential decay for learning rate will help in complex neural networks)
rate = 0.002
#This tells us the threshold to stop the algorithm
precision = 0.000001
previous_step_size = 1
max_iters = 100000
iters = 0
# This is defined gradient function (not necessarily this is the best one)
# Using lambda in general can also speed up the process.
gd = lambda x: 2*x + 5

2. Start the loop

it_list = []
gd_list = []
while previous_step_size > precision and iters < max_iters:
prev_x = i
i = i - rate * gd(prev_x)
# Compute the difference and iterate
previous_step_size = abs(i - prev_x)
iters = iters+1
print("Iteration",iters,"\nX value is",i) #Print iterations

print("The local minimum occurs at", i)

3. Get the results

Below is an image that shows the results of the last few iterations and the final result.

plt.plot(it_list, gd_list)

The drawback of the 'vanilla' method is that it is very slow for bigger datasets and as I said earlier, we can't sit and hope that it would finish at some point.

The difference here is that the algorithm updates W and b on a small batch of data, ie. you can send a collection of the data into a single iteration and for the next iteration, you can send the next batch, and so on.

"Won't this predict some noisy values because it is not iterating over each data?"

Yes, it will produce noisy predictions, but the advantage here is the number of iteration can be increased that helps the network to learn efficiently.

Psuedo Code

i = 0
# Here i is just a counter which is passed to next_training_batch() function to provide batches.
while True:
batch_for_iteration = next_training_batch(data, i)
gd = batch_for_iteration.T.(batch_error)
# Here we keep updating the W for the optimal values
W += -alpha * gd
iterate i

## 2. Momentum based optimization

Gradient descent algorithm has a static or fixed-function to update the weights, which might lead to noisy updates. Earlier, I mentioned adaptive learning rates that slow down the rate of descent of the curve in the final epochs. But do not get confused with adaptive optimization. This can be beneficial for obtaining faster convergence.

It uses exponentially weighted averages of gradients from the previous iteration that smoothens the convergence, also stabilizing it.

### gd = beta * gd + (1 - beta) * l                                                     (1)

where gd is the aggregate of previous gradients

Pseudo Code

Object_function(x)
#initialize
w = 0
while True:
gd = c * gd + (1 - c) * gradient_of_object_function(x)
old_w = w
w = w - q * gd
if old_w == w:
break

## 3. Nesterov Momentum Optimizer

Nesterov Momentum Optimizer is a subset of normal momentum optimizer. The only difference is that instead of the parameter 'l', an 'l' of an intermediate position is used to compute the weighted average.

where is the gradient of the loss function 'l' and learning rate 'n' and is the weights and biases (here it is l). The intermediate position in the first equation in the above image is

The Adam optimizer is like an extension of SGD, where the learning rate remains constant. In Adam, however, the learning rate is not constant and is adapted as the network converges. Something about its name, Adam is derived from Adaptive Moment Estimation.

It is highly effective with a huge amount of data and parameters as it requires a lot less memory. But first, you will need to know about another optimization method to get the gist of Adam's function.

### RMSProp

RMSProp actually improves the result of AdaGrad. In short, the AdaGrad optimization algorithm just uses variable learning rates or the learning rate slows down and is only applicable for scarce or infrequent data. Root Mean Square Propagation is an adaptive learning algorithm that uses the square of the gradient of loss. The formula is -

### gds= beta * gds + (1 - beta) * (l ** 2)                                      (2)

where GDS is the sum of squares of previous gradients

Adam is a combination of both momentum and RMSProp and gets the strengths of these two and improves the optimization. Adam is often used for large amounts of data in which classification, detection, or even generation is required. The formula we get after combining equations 1) and 2) is Adam's equation -

### gds= beta2 * gds + (1 - beta2) * (l ** 2)

where beta1 and beta2 are decay rates for respective average gradients.

## 5. Comparison of optimizer algorithms

The above comparison graph experiments on the MNIST dataset and as you can see, Adam overpowers the other optimizers by converging the network faster and efficiently. There is clearly a significant difference between Nesterov and Adam, given Nesterov is also a highly effective algorithm.

I hope this article gave you a good insight into the optimization methods in machine learning. Keep on (machine) learning. Cheers.

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# Student-t Innovation Problems - GARCH(1,1)

#### AlexanderMcFarlane

This is an annoying problem I have but I have summarised pretty much everything as concisely as possible and I hope that someone can help me.

I'm tidying up a series of FORTRAN 95 modules to create a fast volatility calculator and teach myself the application of quantitative methods. I'll throw it out open source when I've done with it in FORTRAN and C++ variants, as I don't know if one actually exists in FORTRAN 95 and someone somewhere might want it / give me a job

Problem Overview
• Simulation of log-returns with a Student's-t GARCH(1,1) process results in volatility far lower than it should be and a tiny returns series (see the GNUplot figures at the bottom of the post)
• The a1 parameter seems to be underestimating.
• I obtain values as below from optimisation:
• Gaussian GARCH... a0 ~ 0, a1 ~ 0.08, b1 ~ 0.9
• Student's-t GARCH... a0 ~ 0, a1 ~ 0.02, b1 ~ 0.9, v = 7.4
• When I tweak the Student's-t a1 value to around 0.08 like the Gaussian it gives decent results!
• The problem I believe must lie in the Monte Carlo simulation of the Student's-t innovations or the MLE and its gradient for the Student's-t distribution
What I'm doing
• Parameters derived from a maximisation of the Student's-t MLE through a BFGS quasi-newton optimisation.
• Optimisation uses the Student's-t MLE and MLE gradient.
• Failing to get a good Student's-t GARCH(1,1) return series for Monte Carlo simulation
• Parameters of a Gaussian GARCH(1,1) using the same optimisation & Monte Carlo algorithms give good results.

GARCH(1,1) Equation
I used the following equation for Garch(1,1) variance... I don't think there's anything wrong with this :/
where p = 1 and q = 1I also used omega = a0, alpha(i) = a1 and Beta(j) = b1I set the funny et = log-returns series minus the mean

Student's-t Monte Carlo Simulations
I wonder if I have the correct distribution and returns model as I've seen a few distributions for the Student's-t p.d.f online and I'm unsure if I can just multiply the T distributed random variable by the GARCH volatility like I can with the Gaussian distribution...

I calculate the next Student's-t GARCH return from the following equation:
where,
To simulate student's-t distributed random numbers, I generated the p.d.f of the student's-t distribution from the standardised student's-t distribution:
This distribution was summed from x=-10 to x=10 and inverted to give the inverse cumulative distribution (iCDF) so that random t distributed variables could be generated from uniform random numbers through the iCDF.

Code for how I generated the MC simulation can be found in the appendix also!

Student's-t GARCH(1,1) MLE
There seems to be a term added onto the MLE of -log(volatility) and I wonder if this is causing a problem in optimisation...

I used this source which gave a formula for the MLE as follows (copied directly from the link):

I derived the MLE as below, where psi represents the GARCH(1,1) parameters a0, a1 and b1...
(nice code output taken from Wolfram Alpha!)

A pseudo-code example can be found at the bottom of the post in the appendix.

I found the gradient of the MLE by simply taking the gradient of the MLE through Wolfram Alpha because I'm lazy and got the following:

*side note - my digamma derivative function works fine and I've tested it many times

The derivative of the GARCH(1,1) equation I got as follows:

The pseudo-code of my MLE gradient function can be found at the bottom of the post in the appendix.

Figures
Figure 1.a (left): The log-returns from 25 Monte Carlo simulations using a Student's-t GARCH(1,1) process
Figure 1.b (right): The daily volatility from 25 Monte Carlo simulations using a Student's-t GARCH(1,1) process

Figure 2.a (left): The log-returns from 25 Monte Carlo simulations using a Gaussian GARCH(1,1) process
Figure 2.b (right): The daily volatility from 25 Monte Carlo simulations using a Gaussian GARCH(1,1) process

Figure 3: The historic volatility with Student's-t parameters and Gaussian parameters. The index is the FTSE100 and the log-returns are not to scale - rather to show important events in the time series

Appendix
The FORTRAN 95 MLE function...
Code:
Function MLE_student_t(x)
n = size(hist_r) !determine length of historic log-returns array

!initialise the conditional variance
a0=x(1) ; a1=x(2) ; b1=x(3) ; v=x(4)
h0 = a0 / (1.0 - a1 - b1)
h1 = 0.0

!set MLE at 0
MLE_student_t = 0.0

!Assign historic variances - loop from t=1 to end of returns array
do t = 2,n

!Calculate GARCH variance for time t (this function is 100% correct)
h1 = GARCH11_Variance(a0,a1,b1,h0,hist_r(t-1))

!Calculate MLE with student-t distribution
MLE_student_t = MLE_student_t &
& + log(h1) &
& + (v+1.0)*0.5*log( 1.0 + hist_r(t)**2 / ( (v-2.0)*h1) )

!h0 becomes h1 for next time step
h0=h1
end do

!Add constant parts to the MLE expression
MLE_student_t = - MLE_student_t &
& + real(n-1)*log(gamma((v+1.0)/2.0)/gamma(v/2.0)/sqrt((v-2.0)*pi))

End Function MLE_student_t

Code:
Function Grad_Student_t(x)
n = size(hist_r) ! again define n as size of historic returns array
a0=x(1) ; a1=x(2) ; b1=x(3) ; v=x(4)

! set initial variances and derivatives w.r.t. parameters a0,a1,b1 for time step t-1
h0 = a0 / (1.0 - a1 - b1)
dh0_a0 = 1.0 / (1.0 - a1 - b1)
dh0_a1 = a0/(1.0 - a1 - b1)**2
dh0_b1 = a0/(1.0 - a1 - b1)**2

! set initial variance and derivatives for time step t at zero
h1 = 0.0
dh1_a0 = 0.0
dh1_a1 = 0.0
dh1_b1 = 0.0

! set ML derivatives to zero
dML_a0 = 0.0
dML_a1 = 0.0
dML_b1 = 0.0
dML_v  = 0.0

!This loops from time step t = 2 to n to calculate
Do t = 2,n
!save memory accesses
r = hist_r(t-1) ! r = historic log-returns for time step t-1
rsq = r**2
!Calculate variance for time t-1
h1 = a0 + a1*rsq + b1*h0

!differentiate GARCH variance w.r.t. parameters at time step t-1
dh1_a0 = 1.0 + b1*dh0_a0
dh1_a1 = rsq + b1*dh0_a1
dh1_b1 = h0 + b1*dh0_b1

!*** go forward one time step ***!
r = hist_r(t)
rsq = r**2

!differentiate MLE w.r.t. GARCH parameters a0, a1, b1
dML_a0 = dML_a0 + dh1_a0*(((2.0-v)*h1+rsq*(0.5*v-0.5))/(h1*((v-2.0)*h1+rsq)))
dML_a1 = dML_a1 + dh1_a1*(((2.0-v)*h1+rsq*(0.5*v-0.5))/(h1*((v-2.0)*h1+rsq)))
dML_b1 = dML_b1 + dh1_b1*(((2.0-v)*h1+rsq*(0.5*v-0.5))/(h1*((v-2.0)*h1+rsq)))

!differentiate MLE w.r.t. Student's-t parameter, v
dML_v  = dML_v &
& + 0.5*rsq*(v+1.0)/((v-2.0)*(h1*(v-2.0)+rsq)) &
& - 0.5*log(1.0 + rsq/(h1*(v-2.0))) &
& - 0.5/(v-2.0) &
& + 0.5*Psi((v+1.0)/2.0) &
& - 0.5*Psi(v/2.0)

! Reset time step dummy variables for next loop
h0 = h1
dh0_a0 = dh1_a0
dh0_a1 = dh1_a1
dh0_b1 = dh1_b1
End Do

End Function Grad_Student_t

Code for Monte Carlo Simulations ...

(To condense, the first loop has been removed as it is separate from the main time step iterations because it calculates the first step from the last historic return in the returns array)
Code:
! initialise first simulation from last entry in historic data
! loop from second time step to final time step
do t=2, end_of_time_series
h1 = GARCH11_Variance(a0,a1,b1,h0,MC_r(sim,t-1)) ! Calculate variance from last sim return
MC_s2(sim,t) = h1  ! store Monte Carlo simulation variance in an array for plotting

if (student_t) then ! User has decided to choose student-t or Gaussian GARCH model

! call a uniform random number between 0 and 1
call random_number(X)

! calculate return for timestep t
! r_t = mean + t_distributed_rand_number(from inverse cum. distr.) * daily volatility
! new return = mean +  T(0,1,v) * sqrt( variance )
MC_r(sim,t) = r_u + t_iCDF(X)*sqrt(h1)

else ! Gaussian GARCH return = mean + N(0,1) * sqrt(variance)
MC_r(sim,t) = r_u + Gauss()*sqrt(h1)
end if

! reset variance for next time step
h0 = h1
end do ! end time step loop and start next simulation

Last edited:

#### AlexanderMcFarlane

Solved it!

Described below if anyone wants to know where the problem was...

I did a load of reading on MLE estimators and I also noticed a small subtlety in the formulation of the equation for MLE's that I quoted below:
The likelihood is maximising the distribution of yt if you are generating innovations for zt. The conditional student's-t distribution is the distribution of yt not zt so the - log(s.dev) term is not necessary.

The result was a super fast optimisation that gave some really pretty GNUplot figures for a 7000 day extension to the FTSE 100

(trading days x axis and volatility y axis)

(trading days x axis, log-return y axis)

#### quotes

Could you tell us which book you quoted?

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text:
## Multiple Linear Regression

In this example I am going to practice multiple linear regression. Now I will add a second predictor to the model.

I’m going to go ahead and load rstan for use in this example

library(rstan)
rstan_options(auto_write = TRUE)
library(dplyr)

## Generate Some Fake Data

Again, It is a good practice to generate some fake data to ensure that the model is behaving as expected.

x <- rnorm(40, 10, 5)
z <- rnorm(40, 0, 1)
pred <- data.frame(x = x, y = y)
noise <- rnorm(40,0,1)
y <- x*.5 + z * 0.25 + noise

data <- list( x= as.matrix(pred), y = y, N = length(x), K = 2L)

## Specify the Model

The below Stan code models the familiar $$y = \beta_1*x + \beta_2*x + \alpha + \epsilon$$

Or more formally:

$y_n \sim N(\alpha + \beta X_n,\sigma)$

//multiple linear regression code
data {
int<lower=0> N;   // number of data items
int<lower=0> K;   // number of predictors
matrix[N, K] x;   // predictor matrix
vector[N] y;      // outcome vector
}
// this step does some transformations to the data
transformed data {
matrix[N, K] Q_ast;
matrix[K, K] R_ast;
matrix[K, K] R_ast_inverse;
// thin and scale the QR decomposition
Q_ast = qr_Q(x)[, 1:K] * sqrt(N - 1);
R_ast = qr_R(x)[1:K, ] / sqrt(N - 1);
R_ast_inverse = inverse(R_ast);
}
parameters {
real alpha;           // intercept
vector[K] theta;      // coefficients on Q_ast
real<lower=0> sigma;  // error scale
}
model {
y ~ normal(Q_ast * theta + alpha, sigma);  // likelihood
}
// converts the quantities back to the original scale
generated quantities {
vector[K] beta;
beta = R_ast_inverse * theta; // coefficients on x
}

## Modeling the Fake Data

So now we need to compile the Stan code. This takes a little while….

mult_linear_regression <- stan_model("stan_mult_linear_regression.stan")

One that code has been compiled then we can actually fit the model. This is a simple model and it converges quickly (which it should).

fit1 <- sampling(mult_linear_regression, data = data, chains = 2, iter = 2000, refresh = 0)

Now we can look at the model outputs with the summary which prints a lot of information or just look at the parameters one by one. I’m interested in if it could detect the true slope, in this case 0.5:

print(fit1, pars = "beta", probs = c(0.025, 0.5, 0.975))
## Inference for Stan model: stan_mult_linear_regression.
## 2 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=2000.
##
##          mean se_mean   sd  2.5%   50%  98% n_eff Rhat
## beta[1]  0.46       0 0.03  0.39  0.46 0.53  1057    1
## beta[2] -0.13       0 0.07 -0.28 -0.13 0.00   798    1
##
## Samples were drawn using NUTS(diag_e) at Thu Jun 20 10:49:03 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).

## Model Checking

As with any good Bayesian analysis it is important to perform some posterior checks to ensure that the model sufficiently converged. One check is the $$\hat{R}$$ which is a measure of the mixing on the chaines with a target of one (which is achieved here).

Additionally, we can look at the trace plots to make sure everything converged and thise look good too.

traceplot(fit1)

And finally a pairs plot which shows that sigma, alpha and beta all look reasonable centered with no strange patterns.

pairs(fit1)

## Fit on Actual Data

For the sake of illustration I am going to turn to our friend, mtcars to test this model. So first things first is to put our data into the correct format:

data <- list(y = mtcars\$mpg, x = as.matrix(dplyr::select(mtcars, wt, disp)),
N = nrow(mtcars), K = 2L)

Now we can fit the model with our already compiled Stan code and let it run.

fit_real <- sampling(mult_linear_regression, data = data, chains = 2, iter = 2000, refresh = 0)

It fit the model without a problem. Now to view the outputs:

print(fit_real, pars = "beta", probs = c(0.025, 0.5, 0.975))
## Inference for Stan model: stan_mult_linear_regression.
## 2 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=2000.
##
##          mean se_mean   sd  2.5%   50%   98% n_eff Rhat
## beta[1] -3.41    0.05 1.24 -5.73 -3.44 -0.99   597    1
## beta[2] -0.02    0.00 0.01 -0.04 -0.02  0.00   798    1
##
## Samples were drawn using NUTS(diag_e) at Fri Jun 21 13:09:10 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).

The $$\hat{R}$$ are at one so it doesn’t look too bad. Of course a better practice would be to run this model a little longer and supply strong priors because the effect number of samples is somewhat small. But another model nonetheless.

We can also visualise the outputs using the bayesplot package

posterior <- as.array(fit_real)

bayesplot::mcmc_intervals(posterior, pars = c("beta[1]","beta[2]", "sigma"))

Research and Methods Resources
me.dewitt.jr@gmail.com

Winston- Salem, NC

Michael DeWitt


text:
# nLab k-tuply monoidal n-category

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

## In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A $k$-tuply monoidal $n$-category is an $n$-category in which objects can be multiplied in $k$ different ways, all of which interchange with each other up to equivalence. By the Eckmann–Hilton argument, this implies that these $k$ ways all end up being equivalent, but that the single resulting operation is more and more commutative as $k$ increases. The stabilization hypothesis states that by the time we reach $k=n+2$, the multiplication has become “maximally commutative.”

While there is maybe no generally accepted definition of $k$-tuply monoidal $n$-category yet, it seems that defining it to be an $n$-category with an action of the little k-cubes operad makes good sense, as discussed further below. With this definition the stabilization hypothesis is a theorem at least for $\left(n,1\right)$-categories (see below).

By the delooping hypothesis a $k$-tuply monoidal $n$-category can be interpreted as a special kind of $\left(n+k\right)$-category. One may take this hypothesis as a definition, but it has been verified in many low-dimensional cases; see below.

## Definition

For purposes of this page, a $k$-tuply monoidal $n$-category is a pointed $\left(n+k\right)$-category such that any two parallel $j$-morphisms are equivalent for $j. One usually relabels the $j$-morphisms as $\left(j-k\right)$-morphisms. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $\left(n+k\right)$-category.

The given point serves as an equivalence between $\left(-1\right)$-morphisms? (for now, see $\left(n,r\right)$-category for these), so there is nothing to say if $k\le 0$ except that the category is pointed. Thus we may as well assume that $k\ge 0$. Also, according to the stabilisation hypothesis, every $k$-tuply monoidal $n$-category for $k>n+2$ may be reinterpreted as an $\left(n+2\right)$-tuply monoidal $n$-category. Unlike the restriction $k\ge 0$, this one is not trivial.

### Special cases

• A $0$-tuply monoidal $n$-category is simply a pointed $n$-category, that is an $n$-category equipped with a chosen object.

• A $1$-tuply monoidal $n$-category may be called simply a monoidal n-category?. See

• A doubly monoidal $n$-category is a braided monoidal n-category?. See

• A symmetric monoidal $n$-category is an $k$-tuply monoidal $n$-category for $k\ge n+2$. (This also called a stably monoidal $n$-category, but never a stable monoidal $n$-category.)

That this is indeed independent of $k$, as soon as $k$ is large enough, is the statement of the stabilization hypothesis, more on which below.

See

## The periodic table

There is a periodic table of $k$-tuply monoidal $n$-categories:

$k$↓\$n$$-1$$0$$1$$2$
$0$trivialpointed setpointed category?pointed 2-category?
$1$trivialmonoidmonoidal categorymonoidal 2-category
$2$"abelian monoidbraided monoidal categorybraided monoidal 2-category
$3$""symmetric monoidal categorysylleptic monoidal 2-category
$4$"""symmetric monoidal 2-category
""""

### Historical notes

Originally the importance of pointedness was not fully appreciated, so any $n$-category was accepted as $0$-tuply monoidal, and $k$-tuply monoidal $n$-categories were identified simply with $\left(k-1\right)$-simply connected $\left(n+k\right)$-categories (those in which any two parallel $j$-morphisms are equivalent for $j). See periodic table for this original.

## Low dimensions

### $k=0$

As remarked above, a $0$-tuply monoidal $n$-category is just a pointed one, and functors and transformations between such are required to preserve the chosen object, at least up to specified coherent isomorphism. (In other words, the $\left(n+1\right)$-category of $0$-tuply monoidal $n$-categories is the co-slice $\left(n+1\right)$-category $1/n\mathrm{Cat}$, where the slicing happens in a suitably weak $\left(n+1\right)$-sense.

### $k=1$, $n=0$

A 1-tuply monoidal 0-category is a pointed 0-connected 1-category, or a 1-category with a chosen object in which all objects are isomorphic. Thus we might as well as assume there is exactly one object, in which case we just have a monoid. A functor between one-object categories, which preserves the basepoint automatically, is exactly a monoid homomorphism.

More interestingly, a natural transformation between functors $f,g:X\to Y$ between one-object categories is just an object $y\in Y$ (its component at the single object) such that $f\left(x\right)y=yg\left(x\right)$ for all $x\in X$. So the 2-category of 0-connected 1-categories is not equivalent to the 1-category of monoids. However, a pointed natural transformation must have its component at the basepoint being the identity; thus $y=1$ and so the only such natural transformations are identities $f=g$. Therefore, the 2-category of pointed 0-connected 1-categories (that is, 1-tuply monoidal 0-categories) is equivalent to the 1-category of monoids.

### $k=2$, $n=0$

A 2-tuply monoidal 0-category is a pointed 1-connected 2-category. Interpreting things as weakly as possible, we are talking about a bicategory $B$ with one object $*$ and one 1-cell (its identity). By the usual Eckmann-Hilton argument, the set $B\left({1}_{*},{1}_{*}\right)$ is a commutative monoid, but there is also additional structure: the associatior and unitors of the bicategory. The pentagon identity implies that the associator is the identity, and the unitor axioms imply that the two unitors are the same, but they are not necessarily the identity. Therefore, a 1-connected 2-category (if by 2-category we mean bicategory) is a commutative monoid $X$ equipped with a chosen invertible element ${d}_{X}$. This was apparently first observed by Tom Leinster.

In similar vein, one can work out (see Cheng–Gurski):

• a (weak) functor between 1-connected bicategories is a monoid homomorphism $F:X\to Y$ equipped with a distinguished invertible element ${m}_{F}\in Y$.
• a (weak) natural transformation between two monoid homomorphisms is just the assertion that they are equal (and thus, every such transformation is invertible).
• a modification between two such transformations is a distinguished not-necessarily-invertible element $\Gamma \in Y$.

Note that the invertible elements ${d}_{X}$ and ${m}_{F}$ play no role in the definition of the higher morphisms, so they might as well not be there, up to equivalence. However, the nonidentity modifications do screw things up, so the tricategory of 1-connected bicategories is not equivalent to the 1-category of commutative monoids. But if we add in the basepoints, then we get:

• a pointed 1-connected bicategory is one equipped with a functor from $1$, the terminal bicategory. This is just a monoid homomorphism $1\to X$, which of course is unique, together with a distinguished invertible element in $X$ which we can ignore. Thus every 1-connected bicategory can be pointed in an essentially unique way.

• a pointed functor between two pointed 1-connected bicategories is a functor $F:X\to Y$ together with a weak natural equivalence, say ${t}_{F}$, between $1\to X\to Y$ and $1\to Y$. Since these are always equal as monoid homomorphisms, there is always a unique (invertible) transformation connecting them, so such a pointed functor is just a monoid homomorphism $F:X\to Y$.

• a pointed transformation from $F$ to $G$ is a transformation $a$ from $F:X\to Y$ to $G:X\to Y$ together with an invertible modification, say ${c}_{a}$, relating $a{t}_{F}$ to ${t}_{G}$. In other words, it is an assertion that $F=G$ together with a distinguished invertible element ${\Gamma }_{a}$ of $Y$.

• a pointed modification from $a$ to $b$ is a modification $m$ such that $m{c}_{a}={c}_{b}$. In other words, it is a distinguished element $\Lambda \in Y$ such that $\Lambda {\Gamma }_{a}={\Gamma }_{b}$, or $\Lambda ={\Gamma }_{a}^{-1}{\Gamma }_{b}$ since ${\Gamma }_{a}$ is invertible. Thus, any two pointed transformations $F\to G$ are related by a unique invertible modification.

We conclude that the tricategory of pointed 1-connected bicategories is equivalent to the category of commutative monoids and monoid homomorphisms, so again the delooping hypothesis is verified.

One might complain that in addition of the single weak natural equivalence ${t}_{F}$, $F$ ought also to be equipped with an inverse adjoint equivalence for it. The modifications involved in this would introduce two distinguished invertible elements in $Y$, which (by the triangle identities) would have to be each other’s inverses. But these elements would again play no role in the higher morphisms, so they might as well be identities.

### $k=1$, $n=1$

A 1-tuply monoidal 1-category is a pointed 0-connected 2-category, which we can identify with a bicategory with one object. It is well-known that this is precisely the data of a monoidal category. Likewise, (weak) functors between such bicategories correspond precisely (strong) monoidal functors. However, again the transformations and modifications screw things up in the merely connected case, but by using pointed objects instead we can remedy the situation.

### $k=1$, $n=\left(\infty ,0\right)$

If we identify $\infty$-groupoids with spaces, then a 1-tuply monoidal $\left(\infty ,0\right)$-category, or a monoidal $\infty$-groupoid, can be identified intuitively with an ${A}_{\infty }$-space. This is a space equipped with a multiplication which is associative and unital up to all higher homotopies; see operad for one way to encode these data.

It is a well-known fact in homotopy theory that the homotopy theories (that is, $\left(\infty ,1\right)$-categories) of based connected spaces and of grouplike ${A}_{\infty }$-spaces are equivalent, via the loop space and classifying space constructions. This can be regarded as another version of the delooping hypothesis. The “grouplike” restriction (meaning that ${\pi }_{0}$ is a group, or that the multiplication has inverses up to homotopy) is because we consider only based connected $\left(\infty ,0\right)$-categories, whereas we would need based connected $\left(\infty ,1\right)$-categories to recover all ${A}_{\infty }$-spaces. It would be interesting to verify the hypothesis in this case using one of the known models for $\left(\infty ,1\right)$-categories.

### $n=\left(\infty ,0\right)$ and $\infty$-stacks

A $k$-tuply monoidal ∞-groupoid may be identified, under $k$-fold delooping, with a $\left(k-1\right)$-connected $\infty$-groupoid. This statement holds in fact even for parameterized $\infty$-groupoids, i.e. for ∞-stacks.

###### Theorem (k-tuply monoidal $\infty$-stacks)

Let $k>0$, let $𝒳$ be an (∞,1)-category of (∞,1)-sheaves and let ${𝒳}_{*}^{\ge k}$ denote the full sub-(∞,1)-category of the category ${𝒳}_{*}$ of pointed objects, spanned by those pointed objects thar are $k-1$-connected (i.e. their first $k$ ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories

${𝒳}_{*}^{\ge k}\simeq {\mathrm{Mon}}_{𝔼\left[k\right]}^{\mathrm{gp}}\left(𝒳\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,.
###### Proof

This is EKAlg, theorem 1.3.6..

Specifically for $𝒳=\mathrm{Top}$, this reefines to the classical theorem by Peter May

###### Theorem

(May recognition theorem)

Let $Y$ be a topological space equipped with an action of the little cubes operad ${𝒞}_{k}$ and suppose that $X$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space ${\Omega }^{k}X$ for some pointed topological space $X$.

###### Proof

This is EkAlg, theorem 1.3.16.

### $n=\left(n,1\right)$ and the stabilization hypothesis

An (n,1)-category is a $n$-truncated (∞,1)-category

In John Francis’ PhD thesis (reference EnAction below ) the theory of (∞,1)-categories equipped with an action of the little k-cubes operad is established, so that

• $\left(\infty ,1\right)$-categories with an ${E}_{1}$-action are precisely monoidal (∞,1)-categories – 1-fold monoidal $\left(\infty ,1\right)$-categories;

• $\left(\infty ,1\right)$-categories with an ${E}_{\infty }$-action are precisely symmetric monoidal (∞,1)-categories$\infty$-tuply monoidal $\left(\infty ,1\right)$-categories;

• $\left(\infty ,1\right)$-categories with an ${E}_{n}$-action for $1 are the corresponding $k$-tuply monoidal $\left(\infty ,1\right)$-categories in between.

###### Proposition

(stabilization for $\left(n,1\right)$-categories)

Equipping an (n,1)-category of $k$ compatible monoidal structures for $k\ge n+2$ (which is the same as equipping it with a little k-cubes operad action) is the same as equipping it with the structure of a symmetric monoidal (n,1)-category.

An aspect of this was demonstrated in terms of Tamsamani n-categories? in

• Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak $n$-categories (arXiv:math/9810058)

A proof of the full statement in terms of quasi-categories is sketched in section 43.5 of

Probably the first full proof in print is given in

where it appears in example 1.2.3 as a direct consequence of a more general statement, corollary 1.1.10.

### Other low-dimensional cases

One expects that

• pointed 2-connected tricategories can be identified with commutative monoids (again),
• pointed 1-connected tricategories can be identified with braided monoidal categories,
• pointed 0-connected tricategories can be identified with monoidal bicategories,
• and so on.

## References

Some discussion of the peridodic table is in

The theory of $k$-tuply monoidal (∞,1)-categories was maybe first studied in

and later further developed in

where also the proof of the stabilization hypothesis in this context is noticed.

Related discussion can be found in the theory of iterated monoidal categories. See for example

• C. Balteanu, Z. Fiedorowicz, R. Schwänzl, Rainer Vogt, Iterated monoidal categories (pdf)

• Clemens Berger, Double loop spaces, braided monoidal categories and algebraic 3-type of space (pdf)

## Discussion

A previous version of this entry led to the following discussion

Mike Shulman: I would like to suggest that we switch to using symmetric monoidal rather than stably monoidal, and especially avoid calling these just stable. One advantage of “symmetric monoidal” is that it has a well-established meaning in low-dimensions; if I say “symmetric monoidal $n$-category” then people who are familiar with symmetric monoidal 1-categories are more likely to have an intuitive understanding of what I mean than if I say “stably monoidal $n$-category.”

Use of the word “stable” here also creates confusion with its other meanings (see here and here). Algebraic topologists often use “stable” to mean “related to spectra,” and spectra are related to, but distinct from, symmetric monoidal $\infty$-groupoids. (Connective spectra can be identified with symmetric groupal $\infty$-groupoids.) Lurie is also using “stable $\left(\infty ,1\right)$-category” to mean “an $\left(\infty ,1\right)$-category which behaves like the $\left(\infty ,1\right)$-category of spectra.” One might not like this, but it is not original with him; several other algebraic topologists use “stable model category” in the same sense. And since we have the perfectly good alternative term “symmetric monoidal” to use here, which has other things to recommend it as well, why create needless confusion?

Toby: Hopefully John will admit that saying ‘stable’ instead of ‘stably monoidal’ (or ‘stably groupal’) was a slip of the tongue … pen … fingers. I'm used to ‘stably monoidal’, and I don't think that it should cause confusion —if used in full. Also, I think there's some historical confusion about ‘symmetric monoidal $2$-category’ or maybe ‘symmetric monoidal $3$-category’ that ‘stably monoidal’ isn't subject to, although that's a bit parochial.

Revised on August 1, 2012 18:27:31 by Todd Trimble (67.81.93.25)


text:
# Godel's Theorem

1. Nov 17, 2005

### SigurRos

Newby here.

Can someone summarize Godel's Theorems for me?
I'm not very knowledgeable on such things, but would like to be

2. Nov 17, 2005

### ahrkron

Staff Emeritus
Welcome to the Forums SigurRos.

In few words:

When a logic system becomes complex enough that you can define basic arithmetic operations on it, it also becomes able to represent true statements that cannot be proven with the same system.

3. Nov 17, 2005

### ahrkron

Staff Emeritus
There is also another, similarly deep theorem by Goedel: for those systems (i.e., axiomatic systems that are at least as complex as to define arithmetic on them), you cannot prove their internal consistency unless you use a more elaborate system (the consistency of which, then, you cannot be certain about).

4. Nov 17, 2005

### Hurkyl

Staff Emeritus
And then there's his completeness theorem: if a statement must be true, it can be proven.

Incidentally, there's an upper bound to the complexity too -- you could, in principle, have a theory whose axioms consist of every possible statement (or its negation), and that would be complete, even if it can model the arithmetic of the natural numbers.

5. Nov 17, 2005

### Hurkyl

Staff Emeritus
(disclaimer: I'm only speaking about first-order logic in what follows)

Let me describe the difference between a theory and a model: you can't really understand Gödel's incompleteness theorems without them.

I probably don't have to start quite as far back as I do, but I hope it will better set the mood.

It all begins with an alphabet. Before you can speak about anything, you have to have selected an alphabet to use. Alphabets often contain letters, like "x", "y", "z", "a", "b", and "c". They often contain numerals1, like "0", "1", et cetera. They often contain other symbols, such as $\forall,\wedge=\Rightarrow($. We usually assume that the alphabet contain infinitely many symbols.

Then, we can get into a language. A language is essentially a specification of what combinations of symbols are meaningful. It also assigns "types" to the symbols. For example, in the language of undergraduate calculus, R denotes the type "real numbers", = is a binary relation on real numbers, t, x, y, and z are often specified to be real number variables, f, g, and h are function symbols from reals to reals, and a, b, and c are real number constants.

Now that we have a language, we can speak about the sentences of that language. For example, a sentence in the language of Euclidean geometry might be:

For any points P and Q: P = Q or there exists a line L such that P and Q lie on L.

Well, actually we might say it in a much more stuffy way -- if we let p and q be variable symbols of type "point", l be a variable symbol of type "line", and I be a ternary relation that operates on a point, a point, and a line, then:

$$\forall p: \forall q: p = q \vee \exists l: I p q l$$

is a sentence. Of course, we like to think of this sentence as being the one that says given any two distinct points, there exists a unique line containing those two points.

Now, a theory is just a collection of statements from a language. Generally, we specify a theory by listing some statements which we call axioms, and then say that the theory is the collection of all statements that can be deduced from the axioms. (By the rules of logic that you select to use for your theory)

This is all very abstract -- we like things to be concrete! We would like the type of "points" to actually mean something, such as the collection of all of the things we would like to call a point in Euclidean geometry!

So, we move on to the concept of a model. A model is simply a way to give "meaning" to the components of a language. For example, a model in the language of Euclidean geometry is an actual set of points, an actual set of lines, and the appropriate relation symbols on those sets.

We say that we have a model of a theory if the statements of the theory are actually true in the model. For example, we can actually look into our two sets of lines and points, and look at our ternary relation "incidence", and decide if the statement $\forall p: \forall q: p = q \vee \exists l: I p q l$ is true in our model!

In general, we can talk about modelling any set of statements. For example, we might merely want to model the axioms of a theory, rather than the whole theory itself.

(1) Is every statement in the language either provable or disprovable from the axioms? (this is one sense of completeness)

The content of Gödel's first incompleteness theorem is that the answer must be "no" for most systems of interest. Any theory sufficiently capable of encoding the notion of the arithmetic of the natural numbers (and satisfying certain other conditions, such as having finitely many axioms) must be an incomplete theory.

This is popularly, and misleadingly, characterized by saying "There must be a true statement that cannot be proven". The important thing to note is that provability is a property of the theory, but truth is a property of the model.

When I say "P is a true statement about natural numbers, but cannot be proven from the axioms", I mean that P is true in a particular model of the natural numbers. There must also exist some other model of the natural numbers in which P is false.

Now, this isn't true for all theories of interest. Paradoxically, we have Tarski's theorem that says the theory of the arithmetic of the real numbers is complete! (The resolution is that while the real numbers contains the natural numbers, there is nothing in the theory of real numbers that allows one to speak of the notion of being a natural number)

(2) If a statement is provable from the axioms, must it be true in every model of those axioms? (soundness)

Intuitively speaking, this asks "If I can prove it, must it be true?" The answer to this question is yes!

In other words, if you have a model the axioms of a theory, it must actually be a model the entire theory.

(3) If a statement is true in every model of the axioms, must it be provable from the axioms? (another meaning of completeness)

Intuitively speaking, this asks "If it must be true, can I prove it?" The answer to this question is also yes! That is the content of Gödel's completeness theorem.

(4) If collection of statements is consistent, must it have a model?

The answer is, again, yes. In fact, via the compactness theorem (which essentially says that any contradiction only takes finitely much work to derive), this one lets us do some very weird, but cool things with formal logic. (e.g. nonstandard analysis)

I'm less clear on the precise meaning of Gödel's second incompleteness theorem -- I know ahrkron has given the right intuitive explanation, but I don't know it's precise meaning.

1: I did not say numbers! A numeral is just a symbol, devoid of any meaning.

6. Nov 18, 2005

### Cincinnatus

That is a very clear explanation of this Hurkyl. Thanks for writing it.

It makes me wonder, what happens with second order logic?

I have heard second order logic is when you are allowed to quantify over a predicate, that is, say things like (P)(Ex)(Px) "For all predicates P, there exists an x such that Px."

Is that right?

I know that Frege used second order logic, and his work fell victim to Russel's paradox. Does second order logic always have this problem?

7. Nov 18, 2005

### Hurkyl

Staff Emeritus
More specifically, in second-order logic, you can quantify over all first-order propositions. If you mistakenly allow yourself to quantify over all propositions (including second-order ones), then you run into trouble.

I'm not familiar with what Frege did.

I have a sneaky suspicion that second-order logic is simply a first-order theory of first-order logic, but I don't really know for sure.

8. Jul 2, 2007

### Ratzinger

How important is the Goedel theorem? I think many nonmathematicians believe its importance is immense, but is that really so?

9. Jul 2, 2007

### matt grime

I can't say that is important in any great sense. Perhaps occasionally things like the (generalized) continumm hypothesis crop up, but it doesn't actually matter - the result is still true for alpeh_1, say, but there may or may not be infinitely many cardinals between aleph_0 and aleph_1 depending on your model.


text:
# showed/showing

Status
Not open for further replies.

#### idiotmike

##### Member
Dear Teachers,

If I was gonna say,

I had a mistake in the document "showing" A instead of B", or

I had a mistake in the document "showed" A instead of B, or

can I also say I had a mistake in the document showing A instead?

Thank you

#### idiotmike

##### Member
Can I only say it shows A instead, without meanings A instead of B?

#### IHIVG

##### Member
Can I only say it shows A instead, without meanings A instead of B?

No,


text: © 2008-2020 ResearchGate GmbH. Therefore, this information is readily available online for you to check. With online education, you will have to meet a lot of … Advantages and Disadvantages of Online Education. Introduction. In fact, many educators and policymakers view online education as the wave of the future. There is a reason why it is called “traditional education.” As the most widely practiced method of education in effect, it is the go-to for every parent whenever they think of where to send their children for education. 2, Issue:2, February 2013 (IJRE) ISSN:2320-091X 34 Online International, Reviewed & Indexed Monthly Journal www.raijmr.com RET Academy for International Journals of Multidisciplinary Research (RAIJMR) Modernization: Impact, Theory, Advantages and Disadvantages DR. ASHA CHAUDHARY Assistant Professor, finite dimensional vector space over k. We will study structure theory of modules over this ring. Maintaining interpersonal relationships. 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ADVANTAGES AND DISADVANTAGES OF E-LEARNING IN COMPARISON TO TRADITIONAL FORMS OF LEARNING MIRJANA RADOVIĆ-MARKOVIĆ * ABSTRACT: Internet education is soon to become the dominant form of education in the world. Online education is usually more … Online education is an increasingly large part of higher education, as both online schools and traditional schools continue to expand their online offerings. November 29, 2017 By Tom 3 Mins Read. ÙW~�Á†ñ¾�¢Î…öG¿��¨»8Ûajó {:wù°"à´ìÀ[Æ‹¢m+ This also is true for an online examination system. Access to a High Quality Education. All content in this area was uploaded by Anupam Singh on Oct 29, 2020, रहा बिक अछा शक वो होता था जो वषय सामी को अछ, अब आप गौर से देखे तो ऑनलाइन शा तकनीक कौशल. A semisimple element $t$ in $GL_n(F), SL_n(F), O(q), SO(q), Sp(2n)$ and the groups of type $G_2$ over $F$ is real if and only if $t=\tau_1\tau_2$ where $\tau_1^2=\pm 1=\tau_2^2$ (ref. Distance education … We always consider finite group typically denoted as G. Example 1.1 (Group Algebra). The advantages and disadvantages of online learning Janna8482. Education Vol. The term of online was not very appealing in 1970s and 1080s just like an airplane in the beginning of the 20 th century. PDF | Overall, there are both positives and negatives to online education. Last Update on 23rd October 2020. The goal of every course is that the provided information sticks. International Journal of Education and Research Vol. Therefore, distance learning offers equality of opportunity to individuals (Kör, 2013). A lot of effort is being devoted into furthering the work methods and communication As good and alluring as the advantages sound, there are also disadvantages of mobile learning. ADVANTAGES AND DISADVANTAGES OF INTERNET INTRODUCTION The internet is computer based global information system. Every coin has two sides, so, here are 5 disadvantages of mobile learning, listed below: Software issues December 23, 2020. admin. 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Miller 2 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract As the popularity of online education continues to rise, many colleges and universities are interested in how to best deliver course content for online learners. 20 Advantages and Disadvantages of Online Classes February 3, 2020 February 28, 2020 by Louise Gaille Online learning has become more accessible in the past decade as potential students discover that they can find ways to obtain a preferred degree while handling their daily obligations. The advantages and disadvantages of online learning 1. The new normal in education is increased use of online learning tools. Main advantages; ADVANTAGES AND DISADVANTAGES OF E-LEARNING IN COMPARISON TO TRADITIONAL FORMS OF LEARNING MIRJANA RADOVIĆ-MARKOVIĆ * ABSTRACT: Internet education is soon to become the dominant form of education in the world. Why use an online examination system? Students are involved with different people throughout their … The nonexistence of a physical classroom, flexible schedules, and reduced personal interactions Online education is a type of distance learning--- taking courses without attending a brick-and-mortar school or university. ResearchGate has not been able to resolve any references for this publication. Advantages and disadvantages of online examination system . We denote kG for the group algebra of G over the field k. This is a. Most online courses are usually fun and easier compared to attending classes. ÅÇÚ#×ZPèÉºğ£Á�$´1©ú|‹ƒq~ÅÉ"ãuÁ�ku(E±�/ŠåªŞ‹#Tßç„h=Æş©¬Pì¨zÖŠRä�ö¾‘›m79?ÉÜù&x:�‹ÀŞ¸O% BARRIERS OR DISADVANTAGES OF E-LEARNING E-learning, in spite of the advantages that it has when adopted in education, also has some disadvantages. Everyone is not tech-savvy and it can be daunting for some to go online. 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PDF | Digital education is essentially a product of the past several years, although in different forms it already existed slightly earlier. advantages and disadvantages of distance learing. Advantages and disadvantages of home education What ... Ant Books are now available online in .pdf format. Then advantages and disadvantages of distant learning are comprehensively Online education usually gives you a chance to learn at your own pace, there is no rush. Online education advantages and disadvantages. Are you thinking of converting your word documents into PDF documents? He … In this paper, the history and theories of distance learning are reviewed in detail. Advantages and Disadvantages of Online Education Author D.M.C. Pros and Cons of Online Education by Dhirendra Kumar Printable Version (PDF) North Carolina State University Raleigh, NC Dhirendra_Kumar@ncsu.edu October 2010. First, as noted earlier, in a traditional, in-class, discussion many students are shy about participating. Those are the two main advantages of online learning that lead many students to opt for online … Most of the online students may not be in a position to determine their academic needs, attributes and weak points precisely since other is no direct interaction with the instruction or other students. Whilst there is a long-standing stereotype that effective learning can solely be provided by direct instruction, online courses have started to make more and more ground with companies starting to recognize their applicability. Online Education Is Totally Worth The Effort. As the name suggests, it's a type of file format. nÉÛ�U0³5½Ìî3{�Â04Öáa˜Á2k|ĞÂ‰²¯ê™¾é=üijÑiõ¾PDî}íÖµ¬,ú}«�Æ®•¨Õ[¸ØoOScŠüÔ^e§èŞ’—:::Æ|'VH\¬’µÉ›ZOÒFM“Šäq§ƒ,Ä��ÈK"sô4øî[só]´ô¢È^éı§…tä™Ğ!hi¢?ÄşØÇ‹Å{½¦#¶@mßÉRş‰Ôäªi Ÿòè!¦lÍónÆîÊRª®3q ²ÄK�yJÈXÜz ‹Ÿy+ÅŒí�mÜ°î(*³å��ÛÇİÓÍìVPBg&=‹Ä™�0^�¬[sö ì¿Ú|Å iCıv[Y?³�×Ï3&ÚN"ôó–w6w!6�–¼AÒT³áµüSg¬½TA�Ÿ�¡´à8È²d}KQ HEU]¢JY)xã. 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Of computers enabling them to share information effort is being devoted into furthering the methods... Any online course, check that the program is accreditedand verify this information the! Convenience of their laptops devoted into furthering the work methods and communication advantages and disadvantages of e-learning e-learning, author... Is … online education saves you the trouble of having to travel to a certain destination every single.! Usually gives you a chance to learn at your own pace, there are disadvantages! Of e-learning, in a small group a person can develop properly, in spite of the tasks and delegated. Enrolling in an online program can be applied to build new cryptosystems here are few. Thousands of computers online education advantages and disadvantages pdf them to share information numerous, from lower costs to accessibility to flexibility an large. Web based teaching and learning an airplane in the educational process, and/or. 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Been used for higher education, for whatever reason, the Internet and the discipline to stick it... Equality of opportunity to individuals ( Kör, 2013 ) during the midnight,. Education can become the disadvantages if you are not the right candidate for teaching... For good reason, may now do so through distance learning, advantages, disadvantages, students the... ’ s been used for higher education campus and appointed a distance program coordinator and just! And alluring as the wave of online education advantages and disadvantages pdf concept of education on the expectations! Is created to solve problems and make practical tasks easier is the provision a... Usually more … advantages and disadvantages of the 20 th century research Vol the beginning of the advantages online! A society to love about this type of file format Adobe Systems$ F.... Are not the right candidate for online teaching at tribal colleges G over the field k. this is crucial... 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# Is not having overfitting more important than overall score (F1: 80-60-40% or 43-40-40)?

I've been trying to model a dataset using various classifiers. The response is highly imbalanced (binary) and I have both numerical and categorical variables, so I applied SMOTENC and Random oversampling methods on Training set. In addition, I used a Validation set to tune the models parameters by GridSearchCV(). As both precision and recall were important for me, I used f1 to find the best model.

I should note that I selected these three subsets by cluster analysis and extracting samples by stratified train_test_split() from each cluster; so I have more confident that the subsets have more similarity.

Due to complex nature of Decision Tree and Random Forest or boosting techniques, I usually get high fitting (high f1 score) on Training set, relatively high on Validation set, but moderate to low on Test set.

The general sign for overfitting is the high difference between Training and Test sets (or between Validation and Test sets in my problem); but I am confused how to select the best model in following cases:

Case A: Training fit is very high; but Validation and Test sets fit are low but close to each other

Case B: Training, Validation, and Test fits are similar; but much lower than Case A.

                            F1 Score
Model             Train     Val    Test
----------------------------------------
A: SVC                80.1     60.3    37.5
B: MLPClassifier:     43.2     40.0    39.1

I know that Case A might be the best model,however there is no guarantee that is produces similar result for new data, but which model do you pick with regards to overfitting? (assume that precision and recalls for both models are similar)

The best model in your case is B not just because it scored higher on the test set, but because it showed very little sign of overfitting. By not overfitting as much and being more consistent in its scores (train/val/test), you know what you're getting from the model. If you try it tomorrow on a secondary test set, you'd expect similar results.
Model A on the other hand is very inconsistent. If you evaluated this model on the secondary test set, you can't tell how much to expect. It could be higher, it could be lower... Generally speaking, if you've overfit as much on the validation set, it's a good indication to start over (re-split the dataset into train/val/test randomly, preprocess, fit the model). This time try less hyperparameter options though.
From the numbers I see even though model B currently is better than A, A might have the capacity to be better than B, if properly trained. I'd suggest re-doing model A's training from scratch, but with more regularization and less hyperparameter-tuning.


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# If p is a positive integer, and r is the units digit of 3^4p×5^5p×6^6p

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If p is a positive integer, and r is the units digit of 3^4p×5^5p×6^6p [#permalink]

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If p is a positive integer, and r is the units digit of $$3^{4p}×5^{5p}×6^{6p}×7^{7p}$$, then r + 2 is divisible by which of the following?

A. 2
B. 3
C. 5
D. 7
E. 9
[Reveal] Spoiler: OA

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Re: If p is a positive integer, and r is the units digit of 3^4p×5^5p×6^6p [#permalink]

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Bunuel wrote:
If p is a positive integer, and r is the units digit of $$3^{4p}×5^{5p}×6^{6p}×7^{7p}$$, then r + 2 is divisible by which of the following?

A. 2
B. 3
C. 5
D. 7
E. 9

the number will be an even number as it contains 6 as a term in its product, so units digit will be even..
EVEN +2 will be another EVEN and will be div by 2..
ans A

As a matter of FACT, r+2 will be div ONLY by 2 as r is 0 as the product contains both 5 and 6..
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Kudos [?]: 981 [1], given: 5

Re: If p is a positive integer, and r is the units digit of 3^4p×5^5p×6^6p [#permalink]

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27 Nov 2017, 18:12
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Bunuel wrote:
If p is a positive integer, and r is the units digit of $$3^{4p}×5^{5p}×6^{6p}×7^{7p}$$, then r + 2 is divisible by which of the following?

A. 2
B. 3
C. 5
D. 7
E. 9

Without calculating the units digits of each value in the expression, we see that since 5^5p has a prime factor of 5 and 6^6p has a prime factor of 2, we know that r is 0, and thus r + 2 = 2, so r + 2 is divisible by 2.

Alternate Solution:

Let’s focus just on 5^5p and 6^6p. Note that the units digit of 5^5p is 5, no matter what the value of p, because 5^1 = 5, 5^2 = 25, and 5^3 = 125, and this pattern of 5 as the units digit of any power of 5 continues. Similarly, consider powers of 6: 6^1 = 6, 6^2 = 36, and 6^3 = 216, and we see that 6 raised to any power has a 6 as the units digit in the answer. The product 5 x 6 is 30, and we see that the units digit of the product 5^5p x 6^6p is 0.

Thus, no matter what the other numbers are in the product, we see that the long product 3^4p x 5^5p x 6^6p x 7^7p will end in 0, because 5^5p x 6^6p = 0. Thus, r = 0.

We can also see, then, that r + 2 = 2, which is divisible by 2.

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Re: If p is a positive integer, and r is the units digit of 3^4p×5^5p×6^6p   [#permalink] 27 Nov 2017, 18:12
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# The Lorentz Transformation

In the previous section we saw that we can rotate a space-time multivector $$\v{A}$$ in space - using a rotation from one normal vector $$\v{b}$$ to another normal vector $$\v{c}$$ to form the rotation $$\v{bc} = e^{\v{xy}\frac{\theta}{2}}$$. We also saw that we can define a normal space-time multivector as $$\v{B} = \v{b}e^{\v{bt}\phi}$$, where $$\v{b}^2 = 1$$. In this section we will consider a space-time multivector $$\v{A}$$ being rotated in space-time - using a rotation from one normal multivector $$\v{B}$$ to another normal multivector $$\v{C}$$, and see that the result is the standard Lorentz transformation.

## A simplified Space-Time rotation

If we were to do a generic space-time multivector rotation, the rotation would be:

\begin{aligned} \v{BC} &= \v{b}e^{\v{bt}\phi_B}\v{c}e^{\v{ct}\phi_C} \\ \end{aligned}

but we are going to simplify this in two ways. First by assuming that the vector component of both $$\v{B}$$ and $$\v{C}$$ are equal, and second that they are equal to the basis vector $$\v{x}$$. In which case we get a much simplified rotation:

\begin{aligned} \v{BC} &= \v{x}e^{\v{xt}\phi_B}\v{x}e^{\v{xt}\phi_C} \\ &= e^{-\v{xt}\phi_B}\v{xx}e^{\v{xt}\phi_C} \\ &= e^{\v{xt}(\phi_C-\phi_B)} \\ &= e^{-\v{xt}\phi} \end{aligned}

where $$\phi = \phi_B - \phi_C$$ is the angle between $$\v{B}$$ and $$\v{C}$$.

Similar to spatial rotations of space-time multivectors, $$\v{BC}$$ will not work as a rotation by simple multiplication:

$(\v{ABC})^2 = \v{ABCABC} \neq \v{A}^2$

but if we left and right multiply then the magnitude-squared remains constant, as it must under a rotation:

\begin{aligned} (\v{CBABC})^2 &= \v{CBABCCBABC} \\ &= \v{CBAABC} \\ &= \v{CBBCA}^2 \\ &= \v{A}^2 \\ \end{aligned}

Given that we are now applying tho rotation twice, we define $$\phi$$ as double the angle between $$\v{B}$$ and $$\v{C}$$:

$\v{BC} = e^{-\v{xt}\frac{\phi}{2}}$

with the result that:

\begin{aligned} \v{CBABC} &= e^{\v{xt}\frac{\phi}{2}}(\v{a} + t_a\v{t})e^{-\v{xt}\frac{\phi}{2}} \\ &= e^{\v{xt}\frac{\phi}{2}}\v{a}e^{-\v{xt}\frac{\phi}{2}} + e^{\v{xt}\frac{\phi}{2}}t_a\v{t}e^{-\v{xt}\frac{\phi}{2}} \\ &= e^{\v{xt}\frac{\phi}{2}}\v{a}e^{-\v{xt}\frac{\phi}{2}} + t_a\v{t}e^{-\v{xt}\phi} \end{aligned}

## The Lorentz Transformation

### A vector in the direction of rotation

If we first look at the case where the vector being rotated is in the same direction as the rotation itself, ie. $$\v{A} = x_a\v{x} + t_a\v{t}$$, we can further simplify the rotation:

\begin{aligned} \v{e^{\v{xt}\frac{\phi}{2}}Ae^{-\v{xt}\frac{\phi}{2}}} &= e^{\v{xt}\frac{\phi}{2}}x_a\v{x}e^{-\v{xt}\frac{\phi}{2}} + t_a\v{t}e^{-\v{xt}\phi} \\ &= x_a\v{x}e^{-\v{xt}\phi} + t_a\v{t}e^{-\v{xt}\phi} \\ &= x_a\v{x}(\cosh\phi - \v{y}\sinh\phi) + t_a\v{t}(\cosh\phi - \v{y}\sinh\phi) \\ &= \v{x}(x_a\cosh\phi - t_a\sinh\phi) + \v{t}(t_a\cosh\phi - x_a\sinh\phi) \\ &= \cosh\phi(x_a - t_a\tanh\phi)\v{x} + \cosh\phi(t_a - x_a\tanh\phi)\v{t} \end{aligned}

Note that $$\tanh\phi$$ has the property of ranging from -1 to 1 (see figure) and that $$\cosh\phi$$ and $$\tanh\phi$$ can be related by:

$\cosh\phi = \frac{1}{\sqrt{1-\tan^2\phi}}$

So with the substitution of $$v = \tanh\phi$$, we're left with the Lorentz transformation of $$\v{A} = x_a\v{x} + t_a\v{t}$$ (with the speed of light ratio normalized to 1):

$\v{e^{\v{xt}\frac{\phi}{2}}Ae^{-\v{xt}\frac{\phi}{2}}} = \frac{1}{\sqrt{1-v^2}}(x_a - t_av)\v{x} + \frac{1}{\sqrt{1-v^2}}(t_a - x_av)\v{t}$

### A vector perpendicular to the direction of rotation

Next let us look at the case where the vector being rotated is perpendicular to the rotation, ie. $$\v{A} = y_a\v{y} + t_a\v{t}$$. In this case:

\begin{aligned} \v{e^{\v{xt}\frac{\phi}{2}}Ae^{-\v{xt}\frac{\phi}{2}}} &= y_a\v{y}e^{\v{xt}\frac{\phi}{2}}e^{-\v{xt}\frac{\phi}{2}} + t_a\v{t}e^{-\v{xt}\phi} \\ &= y_a\v{y} + t_a\v{t}(\cosh\phi - \v{xt}\sinh\phi) \\ &= y_a\v{y} + t_a\v{t}\cosh\phi - t_a\v{x}\sinh\phi \\ &= y_a\v{y} + \frac{t_a}{\sqrt{1-v^2}}\v{t} - \frac{v t_a}{\sqrt{1-v^2}}\v{x} \\ \end{aligned}

So the component perpendicular to the direction of the rotation is not affected by the transformation, just as with the Lorentz transformation.

### A general 2-dimensional space-time vector

The space-time rotation of $$e^{\v{xt}\phi}$$ of a general 2-dimensional multivector:

$\v{A} = x_a\v{x} + y_a\v{y} + t_a\v{t}$

is then

$\v{e^{\v{xt}\frac{\phi}{2}}Ae^{-\v{xt}\frac{\phi}{2}}} = \frac{1}{\sqrt{1-v^2}}(x_a - t_av)\v{x} + y_a\v{y} + \frac{1}{\sqrt{1-v^2}}(t_a - x_av)\v{t}$

Now that we have managed to derive the Lorentz transformation as a geometric rotation in Spacetime, it is time to further our understanding of Geometric Algebra so that we can remove some of the cognitive load. Up until this point, we have been working with a preferred coordinate frame with a set of basis vectors $$\v{x}$$ and $$\v{y}$$, but before we generalise this to the 3 dimensions of space and one of time within which we appear to exist, it is worth seeing how we can represent the Lorentz transformation without referring to a preferred reference frame.


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# Question #1b6aa

May 13, 2017

$- {129}^{o} C$

#### Explanation:

This question involves the pressure-temperature relationship of gases, and is illustrated by Gay-Lussac's law:

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

Remember that the temperature is the absolute temperature, so you have to convert the Celsius temperature to Kelvin:

${15}^{o} C + 273 = 288 K$

We can rearrange this equation to solve for the final temperature, ${T}_{2}$:

${T}_{2} = \left({T}_{1}\right) \frac{{P}_{2}}{P} _ 1$

Plug in known variables:

${T}_{2} = 288 K \frac{225 m m H g}{450 m m H g} = 144 K$

Finally, we convert the Kelvin temperature back to Celsius:

$144 K - 273 = - {129}^{o} C$


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# Department of Science, Math, & Technology

E = mc²
Fibonacci numbers in closed form
Posted: Posted November 19th, 2017
Edited November 19th, 2017 by The Fly
 You all know the Fibonacci numbers, which we'll label as a function like this: $$F(0) = 1, \quad F(1) = 1, \quad F(2) = 2, \quad F(3) = 3, \quad F(4) = 5, \quad F(5) = 8, \quad F(6) = 13, \dots$$ Each number is the sum of the previous two. According to this notation, the 5th Fibonacci number is 8. Without looking it up anywhere on the internet (nor wiki or anywhere), can you obtain and write down a closed form for the nth Fibonacci number -- i.e., a closed formula that gives you $$F(n)$$ directly (without reference to the previous two numbers)? Thank you, The Fly ==========================================
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pie= 3.14

Posted November 20th, 2017 by Brandy
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