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# What's the motivation of Runge-Kutta method?

I'm recently taking a ode course and learning Runge-Kutta method on solving ordinary differential equations. To be specific, the 4th order Runge-Kutta method on solving initial value problems of ordinary equations.

My instructor and the textbook told me the formula but didn't say about the thoughts behind the method. I wrote some code and found that Runge-Kutta method does perform better than Euler method. But I can't understand why.

Is anyone willing to give me a hand on how to get the formula of Runge-Kutta? Thanks!

• It 'does perform better than Euler method' because the RK4 method uses 4 stages to get an approximation as opposed to the Euler method which uses 1 stage. See here for a good description of the method. You can find the derivation in most numerical analysis books. – mattos Nov 19 '17 at 10:47
• math.stackexchange.com/questions/528856/… This question is duplicated! See this question. – Chongxu Ren Nov 19 '17 at 10:59

## On the history

See Butcher: A History of the Runge-Kutta method

In summary, people (Nystroem, Runge, Heun, Kutta,...) at the end of the 19th century experimented with success in generalizing the methods of numerical integration of functions in one variable $$\int_a^bf(x)dx,$$ like the Gauss, trapezoidal, midpoint and Simpson methods, to the solution of differential equations, which have an integral form $$y(x)=y_0+\int_{x_0}^x f(s,y(s))\,ds.$$

Carl Runge in 1895[1] came up with ("by some curious inductive process" - "auf einem eigentümlich induktiven Wege" wrote Heun 5 years later) the 4-stage 3rd order method \begin{align} k_1&=f(x,y)Δx,\\ k_2&=f(x+\tfrac12Δx,y+\tfrac12k_1)Δx\\ k_3&=f(x+Δx,y+k_1)Δx\\ k_4&=f(x+Δx,y+k_3)Δx\\ y_{+1}&=y+\tfrac16(k_1+4k_2+k_4) \end{align}

[1] "Über die numerische Auflösung von Differentialgleichungen", Math. Ann. 46, p. 167-178

Inspired by this Karl Heun in 1900[2] explored methods of the type \left.\begin{aligned}k^i_m &= f(x+ε^i_m,y+ε^i_mk^{i+1}_m)Δx,~~ i=1,..,s,\\ k^{s+1}_m&=f(x,y)Δx\end{aligned}\right\},~~ m=1,..,n\\ y_{+1}=y+\sum_{m=1}^n\alpha_mf(x+ε^0_mΔx,y+ε^0_mk^1_m)Δx He computed the order conditions by Taylor expansion and constructed methods of this type up to order 4, however the only today recognizable Runge-Kutta methods among them are the order-2 Heun-trapezoidal method and the order 3 Heun method.

[2] "Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen", Z. f. Math. u. Phys. 45, p. 23-38

Wilhelm Kutta in his publication one year later in 1901[3] considered the scheme of Heun wasteful in the number of function evaluations and introduced what is today known as explicit Runge-Kutta methods, where each new function evaluation potentially contains all previous values in the $$y$$ update. \begin{align} k_1&=f(x,y)Δx,\\ k_m&=f(x+c_mΔx, y+a_{m,1}k_1+...+a_{m,s-1}k_{s-1})Δx,&& m=2,...,s\\[0.5em] y_{+1}&=y+b_1k_1+...+b_sk_s \end{align} He computed order conditions and presented methods up to order $$5$$ in parametrization and examples. He especially noted the 3/8 method for its symmetry and small error term and the "classical" RK4 method for its simplicity in using always only the last function value in the $$y$$ updates.

[3] "Beitrag zur näherungsweisen Lösung totaler Differentialgleichungen", Z. f. Math. u. Phys. 46, p. 435-453

## On the order dependence of the performance

The Euler method has global error order 1. Which means that to get an error level of $$10^{-8}$$ (on well-behaved example problems) you will need a step size of $$h=10^{-8}$$. Over the interval $$[0,1]$$ this requires $$10^8$$ steps with $$10^8$$ function evaluations.

The classical RK4 method has error order 4. To get an error level of $$10^{-8}$$ you will thus need a step size of $$h=0.01$$. Over the interval $$[0,1]$$ this requires $$100$$ steps with $$400$$ function evaluations.

If you decrease the step by a factor of $$10$$ to $$h=0.001$$, the RK4 method will need $$1000$$ steps with $$4000$$ function evaluations to get an error level of $$10^{-12}$$. This is still much less effort than used in the Euler example above with a much better result.

Using double precision floating point numbers you will not get a much better result with any method using a fixed step size, as smaller step sizes result in an accumulating floating point noise that dominates the error of the method.


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# Visualizing the Fourier Transfrom in Python

Last updated: August 26th, 2020

# Fourier Transform¶

If you are only slightly interested in sound, I can only recommend getting to be friends with the Fourier transform. The Fourier transform is used in a variety of cases, such as better understanding the planet Saturn's rings and our own planet's ocean waves. Tell me that is not cool.

Thinking of these use cases, the Fourier transform feels more like a super power than a math equation. With it we can zoom to other planets and stars to see what they're made out of, which direction they're going etc. Well, I am delighted to say that this same super power giving tool is very necessary in the field of acoustics.

If you are not yet familiar, what this lovely equation does, is it takes the digitized form of a wave, which shows the amplitude or energy in the wave in time, and transforms it show which frequency or frequencies exist in the wave (and also the energy level of those frequencies). For a more in-depth introduction, I discovered this jewel of an explanation. If you're interested in a calming YouTube video with stellar twirly visuals, check out this one from 3 Blue 1 Brown.

On to our own visuals... of sound!

## Important Note:¶

### Only .wav files with bitdepth of 16 or 32 are acceptable.¶

In [1]:
# import libraries for playing and loading sound
import IPython.display as ipd
import numpy as np
import matplotlib.pyplot as plt

## Load the wavfile: digital sound wave¶

I am using scipy's wavfile module to read the wavfile I want to work with. As of now, it does not support wavfiles with 24-bit data. (Something to keep in mind if you want to use your own wavfiles in this notebook.)

The module's function read returns two items, which I named sr and samps.

sr refers to the sampling rate of the audio. This just indicates the number of times per second the amplitude of the sound wave is measured. High definition sound has an 'sr' of around 48000 and telephone lines are usually capped to a sampling rate of only 8000.

A higher sampling rate means more data to deal with which usually requires more computational resources and likely more time to process. In real time applications, for example telephoning, it is most important that sound be processed quickly. We all know how frustrating conversations with a time delay are. Non-real time applications, such as listening to music or watching movies, it is more important that the sound be of great quality. Because the processing doesn't have to be real time, a higher sampling rate can be used, resulting in a crisper and more enjoyable sound.

The other variable name, samps, refers to the measurements of amplitude of the sound signal. We will graph them below so you can see what they look like.

In [2]:
wav = 'python.wav'
ipd.Audio(samps, rate=sr)

Out[2]:
In [3]:
# creating variables for clarification
samples_per_second = sr
total_samples = len(samps)
time_seconds = total_samples / samples_per_second
print("Sampling rate: ", samples_per_second)
print("Total number of samples: ", total_samples)
print("Total time in seconds: ", time_seconds)

Sampling rate:  44100
Total number of samples:  38093
Total time in seconds:  0.8637868480725623

In [4]:
time_vector = np.linspace(0, time_seconds, total_samples)

In [5]:
plt.plot(time_vector, samps)
plt.title("'Python' in the time domain")
plt.xlabel('time (sec)')
plt.ylabel('amplitude')

Out[5]:
Text(0,0.5,'amplitude')

### Normalizing the values¶

If you look at the amplitude values, they are al over the place. Let's bring them between -1 and 1.

In [6]:
samps_normed = np.interp(samps,(samps.min(), samps.max()),(-1, 1))

In [7]:
plt.plot(time_vector, samps_normed)
plt.title("'Python' in the time domain")
plt.xlabel('time (sec)')
plt.ylabel('amplitude')

Out[7]:
Text(0,0.5,'amplitude')

### Breaking down the time domain¶

I studied clinical linguistics, so I feel I have to show you some cool linguistics stuff before moving on.

In the waveform above, you can see some changes happening in the signal.

Think about the individual sounds that make up the word 'python'.

One thing you can see very well in the time domain is the difference between voiced and voiceless sounds. The letters 'p' and, in this case, 'th' are voiceless, as no vocal cords are activated when pronouncing them. The letters 'y', 'o' and 'n' are voiced, because one needs to vibrate the vocal cords in order to produce them.

Looking at the signal above, you can probably piece out which sections belong to which letter.

For fun let's zoom into the letter 'p' and its transition to 'y'.

In [8]:
plt.plot(time_vector[6500:8000], samps[6500:8000])
plt.title("'py' in the time domain")
plt.xlabel('time (sec)')
plt.ylabel('amplitude')

Out[8]:
Text(0,0.5,'amplitude')

As you can see, there is almost no energy in the signal at the beginning, and around 0.158 seconds, an explosion of energy comes.

There is a lot to understand about the signal looking at it in the time domain like this. As a linguist, it is helpful when annotating speech: one can see quite clearly when specific sounds are being made.

However, it is difficult to know more specifics about the sound or the speaker. For example, it would be hard to know if the speaker was actually saying 'Python' or rather 'Tafem'. Each of those letters have similar characteristics:

• 'p' and 't' are plosives and voiceless

• 'y' and 'a' are vowels and voiced

• 'th' and 'f' are fricatives and voiceless

• 'o' and 'e' are vowels and voiced

• 'n' and 'm' are nasals and voiced

This is where the Fourier transform, or the FFT, comes in to help us.

### FFT¶

Several libraries allow you to apply the fft, such as Librosa's STFT function, which stands for short-time fourier transform.

Note: all STFT means is that the FFT is conducted on consecutive segments of 'short-time' periods. For example, the FFT could be conducted on a very long signal - not consecutive segments, or windows - if the signal is not constantly changing. But if you have a signal, for example, speech, or even background noise, where the signal changes according to sounds produced ('l' vs 'r') or what is happening in the room, you need to apply the FFT in small windows, aka, you need to apply the STFT to a changing signal. By applying the STFT, or the FFT in successive, overlapping windows of the sound, you can identify the frequencies as they change. If you were to apply the fft to the entire signal, in one big window, all the complexities of that signal would be lost.

### Visualization¶

For visualization purposes, I will be using scipy.signal.spectrogram along with matplotlib.pyplot to calculate the FFT on varying sound types.

In [9]:
import matplotlib.pyplot as plt
from scipy import signal

### FFT: Visualizing a simple signal¶

In [22]:
#create a simple sine wave
def create_signal(freq=500, amplitude=0.4, sampling_rate=8000, dur_sec=0.25):
#The variable time holds the expected number of measurements taken:
#duration in seconds multiplied by the sampling rate.
time = np.linspace(0, dur_sec, np.floor(dur_sec*sampling_rate))
full_circle = 2 * np.pi
sinewave_samples = amplitude * np.sin((freq*full_circle)*time)
return sinewave_samples, sampling_rate, time

In [23]:
sin_samps, sin_sr, sin_time = create_signal() #I'm keeping the defaults

/usr/local/lib/python3.6/site-packages/ipykernel_launcher.py:5: DeprecationWarning: object of type <class 'numpy.float64'> cannot be safely interpreted as an integer.
"""

Which sounds like this:

In [24]:
ipd.Audio(sin_samps, rate=sin_sr)

Out[24]:

### scipy.signal.spectrogram¶

I suggest clicking on the link above, to the documentation, and keep it open on a separate tab. This will help you understand the parameters and function in general, at least better than what I've pasted below, which are the parameters and their defaults.

I also suggest to play around with the settings and see how they influence the picture. If you lose track of the original settings, just come back to my original notebook if you have forked this.

scipy.signal.spectrogram(x, fs=1.0, window=('tukey', 0.25), nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, mode='psd')

In [25]:
f, t, Sxx = signal.spectrogram(x = sin_samps, fs=sin_sr)
plt.pcolormesh(t, f, Sxx)
plt.ylabel('Frequency [Hz]')
plt.xlabel('Time [sec]')
plt.show()

### Break this down¶

Do the numbers of this graph match the sound wave we created?

The time goes to 0.25 seconds. Check.

The yellow frequency bar looks like it is set to 500 Hz. Check.

There is only one frequency in the signal, and there is only one yellow bar. Check.

If you are wondering why the Frequencies go up to 4000, the short answer is: it is half the sampling rate 8000, which we set when we created the signal. To dive into why that is, I explore that in another notebook.

Nice.

What happens with a more complex signal?

### Make something simple slightly more complicated¶

In [26]:
def create_noise(num_samples, amplitude=0.025, random_seed=None):
if random_seed:
np.random.seed(random_seed)
noise = amplitude * np.random.randn(num_samples)
return noise

### Increase number of frequencies¶

In [27]:
# duration and sampling rate need to be the same in order to combine them later
sr_comp = 8000
duration = 1

samps1, sr1, time1 = create_signal(freq=500, amplitude=0.7, sampling_rate=sr_comp, dur_sec=duration)
samps2, sr2, time2 = create_signal(freq=2000, amplitude=0.5, sampling_rate=sr_comp, dur_sec=duration)
samps3, sr3, time3 = create_signal(freq=1111, amplitude = 0.3, sampling_rate=sr_comp, dur_sec=duration)

/usr/local/lib/python3.6/site-packages/ipykernel_launcher.py:5: DeprecationWarning: object of type <class 'numpy.float64'> cannot be safely interpreted as an integer.
"""

In [28]:
sin_mult = samps1+samps2+samps3

In [29]:
noise2add = create_noise(num_samples = len(sin_mult))

### Let's listen to it.¶

I feel like this won't be pleasant.

In [30]:
ipd.Audio(sin_noisy, rate=sr_comp)

Out[30]:

Can you hear the different frequencies as well as the noise?

### FFT: Visualizing a slightly more complex signal¶

In [31]:
f, t, Sxx = signal.spectrogram(x = sin_noisy, fs=sr_comp)
plt.pcolormesh(t, f, Sxx)
plt.ylabel('Frequency [Hz]')
plt.xlabel('Time [sec]')
plt.show()

Can you see all three frequencies? I can, but I'll be honest, I can't see the noise.

The loudest frequency is very easy to see, the yellow one at 500 Hz. As the amplitudes of the frequencies decrease, the harder it is to see them. We can still see the green one at 2000 Hz, but I can hardly see the quietest frequency at 1111 Hz.

When you compare this with what you see in the graph below, I can't say this is too bad! Below you see the same signal, but in the time domain, which we discussed earlier.

Looking back at the FFT visual above, notice that the frequencies stay straight and do not change, across the entire graph. No windowing would be necessary for this. However, when a signal changes, for example in speech, we can't lump the entire signal into one window.

In [32]:
plt.plot(time1,sin_noisy)
plt.xlabel('time (sec)')
plt.ylabel('amplitude')
plt.show()

### FFT: Visualizing speech¶

In [33]:
f, t, Sxx = signal.spectrogram(x = samps_normed, fs=sr)
plt.pcolormesh(t, f, Sxx)
plt.title("'Python': Spectrogram")
plt.ylabel('Frequency [Hz]')
plt.xlabel('Time [sec]')
plt.show()

Hmmmmmmph. Can't see much. I'll work on improving this visual, perhaps with their help.

In [34]:
f, t, Sxx = signal.spectrogram(x = samps_normed, fs=sr)
mag = np.abs(Sxx)
mag_log = np.log(mag)
plt.pcolormesh(mag_log)
plt.title("'Python': Log Spectrogram")
plt.show()

Looks better now that I've taken the natural log of the spectrogram.. but I'm still working on the x and y label scales.

Take a look at this picture that is also at the top of this 'tutorial':

That is basically the same thing I'm trying to graph here, but much more vain: it's me saying my name! The brighter spots 'further in the distance'? Those are the 's' letters in Aislyn Rose! Notice the columns going from left to right? Those are the windows where the FFT was applied, each window lasting appx. 20 ms! Imagine all those colors mixed together, averaged out. That would be what it would look like if the FFT were applied to the signal all at once. Tragic idea.

Aaaanyways, know that this graph above (the mostly purple one where you can hardly see anything) is a spectrogram, or visual representation of the FFT over time, of me saying the word 'Python'. I think you can guess where the sound 'py' is...

Just taking a look at the loudest part of the signal, 'py', there are curvatures there that reveal the complexities within the signal that aren't seen when we look at the signal in the time domain. These curvatures help reveal what type of sounds are actually being made, as in where in the mouth they are made, not only if the sound is voiced or not.

## Closing Remarks¶

It is with images like this that we can train neural networks on speech recognition or speaker identification. We all have different bodies and therefore instruments with which we create sound. This means that the frequencies we each generate are different, and neural networks can learn these differences. Perhaps that is a creepy note to end on, but let's face it, sound analysis can be very very creepy. But also beautiful. We are all musical instruments.

I hope to have a notebook prepared soon, where you can explore how this image changes depending on the features extracted.

format: plain


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# Orders of elements of Dihedral Group D8

Printable View

• October 12th 2011, 02:32 AM
Bernhard
Orders of elements of Dihedral Group D8
R = $\left(\begin{array}{cccc}1&2&3&4 \\ 2&3&4&1\end{array}\right)$

Peter

Moderator edit: Latex fixed. @OP: Please add the rest of the question by replying to this post.


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# Satori

## Introducing OG Image Generation: Fast, dynamic social card images at the Edge - Vercel

### Problems

• Difficult: This solution required launching Chromium in a Serverless Function and taking a screenshot of the given HTML page with Puppeteer. Setting up these tools was hard to implement and often led to errors
• Slow: Chromium needs to be compressed to fit inside a Serverless Function and then decompressed on a cold boot; it's very slow (~4 seconds on average). This can result in slow or broken social card images
• Expensive: Spinning up an entire browser to take a screenshot was not efficient. This led to large Function sizes, which could be expensive and waste computing

### @vercel/og

• Easy: No headless browser is needed. Using Vercel OG, you can define your images using HTML and CSS and automatically generate dynamic images from the generated SVGs
• Affordable: Vercel Edge Functions are ~160x cheaper than running Chromium in a Serverless Function. Further, generated images can be cached and stored at the Edge
• Fast: Vercel OG (500KB) is 100x more lightweight than Chromium + Puppeteer (50MB). 5x faster in P99 TTFB (4.96s → 0.99s) and 5.3x faster in P90 (4s → 0.75s).

### Features

• Basic CSS layout, styling, and typography.
• Works with any framework or frontend application
• Font and Emojis Subsets from Google Fonts and other CDNs
• Tailwind CSS with the tw prop
• Handles Cache automatically

### Engine

• The core engine, Satori, can be used in modern browsers, Node.js, and Web Workers. Building on top of the core engine, Vercel OG can be used inside Edge environments through WebAssembly to create social card images easily.


text:
# Thread: evaluate the definite integral

1. ## evaluate the definite integral

can i take the the 2 so it looks like this

= √2(1/√x)dx??

2. ## Re: evaluate the definite integral

Originally Posted by asilvester635
can i take the the 2 so it looks like this
= √2(1/√x)dx??

What is the derivative of $\sqrt{2x}~?$

3. ## Re: evaluate the definite integral

it will be x^-1/2

4. ## Re: evaluate the definite integral

Originally Posted by asilvester635
it will be x^-1/2
No it is not. If you cannot do derivatives, the integrals are impossible for you.

5. ## Re: evaluate the definite integral

do i rewrite this?

6. ## Re: evaluate the definite integral

Originally Posted by asilvester635
do i rewrite this?
Look at this webpage.

7. ## Re: evaluate the definite integral

The answer to the original question is yes, $\int_1^8\sqrt\frac{2}{x}\ dx = \sqrt{2}\int_1^8\sqrt\frac{1}{x}\ dx$.

Plato has a point - you need some proficiency with derivatives before you try to do integrals.

- Hollywood

8. ## Re: evaluate the definite integral

Hello, asilvester635!

$18.\;\int^8_1\!\sqrt{\frac{2}{x}}\,dx$
Can i take the the 2 so it looks like this? . $\sqrt{2}\left(\frac{1}{\sqrt{x}}\right)$ . Yes!

$\sqrt{\frac{2}{x}} \;=\;\frac{\sqrt{2}}{\sqrt{x}} \;=\;\sqrt{2}\cdot\frac{1}{\sqrt{x}} \;=\;2x^{-\frac{1}{2}}$

Therefore: . $\int^8_1\!\sqrt{\frac{2}{x}}\,dx \;=\;\sqrt{2}\!\int^8_1 x^{-\frac{1}{2}}\,dx$

Got it?

9. ## Re: evaluate the definite integral

Originally Posted by asilvester635
can i take the the 2 so it looks like this

= √2(1/√x)dx??
You can find the answer to this definite integral like this:

\begin{align*}\int^8_1\!\sqrt{\frac{2}{x}}\,dx =& 2\sqrt{2}\sqrt{x}\Big{]}^{8}_{1} \\ =& 2\sqrt{2} \sqrt{8} - 2\sqrt{2}\sqrt{1} \\ =& 8 - 2\sqrt{2}\end{align*}

$\therefore \int^8_1\!\sqrt{\frac{2}{x}}\,dx = 8 - 2\sqrt{2}$

You can check the answer at:

int (2)^(1/2) 1/(x)^(1/2) dx x=1 to 8 - Wolfram|Alpha


text:
# Why is the mass dimension of anticommuting coordinates $[Mass]^{-1/2}$

+ 3 like - 0 dislike
295 views

I am reading a review ( http://arxiv.org/abs/hep-ph/9709356 ) about supersymmetry. On page 29 I have read that the mass dimension of the Grassmann anticommuting coordinates is $-1/2$. Why this? Why don't they have the same mass dimensions as the bosonic coordinates?

asked May 13, 2015
edited May 13, 2015

You can work out the dimension of any field from the fact that the free action must be dimensionless (in units where $\hbar=1$).

+ 5 like - 0 dislike

By definition supersymmetry transformations square to spacetime translations. In a superspace formalism the supersymmetry operator is constructed from the vector field $\partial_\theta$ with respect to the odd coordinates $\theta$. As this operator has to square to the vector field $\partial_x$ with respect to the even coordinates $x$, which is of dimension $1$, the vector field with respect to the odd coordinate has to be of dimension $1/2$ and so the odd coordinate as to be of dimension $-1/2$.

Equivalently, a typical superfield is of the form

$\phi + \theta \psi +...$

where $\phi$ is a scalar and $\psi$ a spinor. In $d$ spacetime dimensions, a scalar is of dimension $(d-2)/2$, a spinor is of dimension $(d-1)/2$ and so $\theta$ has to be of dimension $-1/2$.

answered May 14, 2015 by (5,140 points)
edited May 15, 2015 by 40227

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.


text:
What Is The Best Way To Calculate The Distance Between Two Proteins?
3
5
Entering edit mode
11.9 years ago
Sirus ▴ 820

Hello, My question may be is simple but I am confused, I have a protein-protein interaction network and I want to express the distance between the proteins, you think which is the best way to express the distance or at least what are the usually used way in these circumstances. and does a package or a website that offer this functionality exist? Thank you in advance.

ppi • 7.1k views
6
Entering edit mode
11.9 years ago
Neilfws 49k

By distance, I assume that you mean "shortest path".

There are a lot of software solutions to this problem. For any kind of network analysis, I recommend the igraph package. It can be used from R, Python, Ruby or C. As an example, here is how you would read a network into R:

library(igraph)
# assume network is a file in GML format
g <- read.graph("myfile.gml", format = "gml")

Then consult the R documentation for functions related to "path". For example:

shortest.paths(g)

will return a matrix of N x N (where N is the number of vertices), containing the path lengths.

For a more visual approach, either Cytoscape or Gephi may be useful. In Gephi, for example, you can import a network file, select the "shortest path" tool from the menu and then click on 2 vertices to see the path length.

0
Entering edit mode

Good answer, well I have seen that there is two kind of distances, topological and biological distance, for the topological one I think this one may suite, but is there any metric based on esome biological information that can be used, if there are some pre-existing tools, it would be fantastic :)

0
Entering edit mode

Well, "biological distance" is a rather nebulous term that could mean many things. I guess you are thinking about clustering the network into "functional modules". I'd suggest starting with a web or PubMed search for precisely those terms: "protein-protein interaction cluster functional modules".

0
Entering edit mode

Yeah, in fact I am trying to implement a clustering algorithm for the protein-protein interaction network :). Thank you for your help, really appreciate it

3
Entering edit mode
0
Entering edit mode

Thank you, those papers seems interesting I will take a look, the links are also good

0
Entering edit mode
8.8 years ago
jackuser1979 ▴ 890

You can use cytoscape PPI network analysis

http://www.cytoscape.org/

Here is short tutorial on Protein interaction network analysis using Cytoscape

http://www.doc.ic.ac.uk/~natasha/course/docs/cytoscapeTutorial.pdf

For implementing cluster algorithm, you can install clusterMaker plugin in Cytoscape

http://www.cgl.ucsf.edu/cytoscape/cluster/clusterMaker.html


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# what is squeeze theorem pls explain

7 years ago

The squeeze theorem is formally stated as follows.

Let I be an interval having the point a as a limit point. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:

g(x) \leq f(x) \leq h(x) \,

and also suppose that:

\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L. \,

Then \lim_{x \to a} f(x) = L.


text:
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## Section2.4Complex Expressions

In this section we look at some equations and inequalities that will come up throughout the text.

###### Example2.4.1.Line equations.

The standard form for the equation of a line in the $xy$-plane is $ax + by + d = 0\text{.}$ This line may be expressed via the complex variable $z = x + yi\text{.}$ For an arbitrary complex number $\beta = s + ti\text{,}$ note that

\begin{align*} \beta z + \overline{\beta z} \amp = \big[(sx - ty)+(sy+tx)i\big] + \big[(sx - ty) - (sy+tx)i\big]\\ \amp = 2sx - 2ty\text{.} \end{align*}

It follows that the line $ax + by + d = 0$ can be represented by the equation

\begin{gather*} \alpha z + \overline{\alpha z} + d = 0 \tag{equation of a line} \end{gather*}

where $\alpha = \frac{1}{2}(a - bi)$ is a complex constant and $d$ is a real number.

Conversely, for any complex number $\alpha$ and real number $d\text{,}$ the equation

\begin{equation*} \alpha z + \overline{\alpha z} + d = 0 \end{equation*}

determines a line in $\mathbb{C}\text{.}$

We may also view any line in $\mathbb{C}$ as the collection of points equidistant from two given points.

Given two points $\gamma$ and $\beta$ in $\mathbb{C}\text{,}$ $z$ is equidistant from both if and only if $|z - \gamma|^2 = |z - \beta |^2\text{.}$ Expanding this equation, we obtain

\begin{align*} (z - \gamma)(\overline{z - \gamma}) \amp = (z - \beta)(\overline{z - \beta})\\ |z|^2 - \overline{\gamma}z - \gamma\overline{z} + |\gamma|^2 \amp = |z|^2 - \overline{\beta}z - \beta\overline{z} + |\beta|^2\\ \overline{(\beta-\gamma)}z + (\beta-\gamma)\overline{z} + (|\gamma|^2 - |\beta|^2) \amp = 0\text{.} \end{align*}

This last equation has the form of a line, letting $\alpha = \overline{(\beta - \gamma)}$ and $d = |\gamma|^2 - |\beta|^2\text{.}$

Conversely, starting with a line we can find complex numbers $\gamma$ and $\beta$ that do the trick. In particular, if the given line is the perpendicular bisector of the segment $\gamma\beta\text{,}$ then $|z - \gamma| = |z - \beta|$ describes the line. We leave the details to the reader.

###### Example2.4.3.Quadratic equations.

Suppose $z_0$ is a complex constant and consider the equation $z^2 = z_0\text{.}$ A complex number $z$ that satisfies this equation will be called a square root of $z_0$, and will be written as $\sqrt{z_0}\text{.}$

If we view $z_0 = r_0e^{i\theta_0}$ in polar form with $r_0 \geq 0\text{,}$ then a complex number $z = re^{i\theta}$ satisfies the equation $z^2 = z_0$ if and only if

\begin{equation*} re^{i\theta}\cdot re^{i\theta} =r_0e^{i\theta_0}\text{.} \end{equation*}

In other words, $z$ satisfies the equation if and only if $r^2 = r_0$ and $2\theta = \theta_0$ (modulo $2\pi$).

As long as $r_0$ is greater than zero, we have two solutions to the equation, so that $z_0$ has two square roots:

\begin{equation*} \pm \sqrt{r_0}e^{i\theta_0/2}\text{.} \end{equation*}

For instance, $z^2 = i$ has two solutions. Since $i =1 e^{i\pi/2}\text{,}$ $\sqrt{i} = \pm e^{i\pi/4}\text{.}$ In Cartesian form, $\sqrt{i} = \pm (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i)\text{.}$

More generally, the complex quadratic equation $\alpha z^2 + \beta z + \gamma = 0$ where $\alpha\text{,}$ $\beta\text{,}$ and $\gamma$ are complex constants, will have one or two solutions. This marks an important difference from the real case, where a quadratic equation might not have any real solutions. In both cases we may use the quadratic formula to hunt for roots, and in the complex case we have solutions

\begin{equation*} z = \frac{-\beta \pm \sqrt{\beta^2 - 4\alpha\gamma}}{2\alpha}\text{.} \end{equation*}

For instance, $z^2 + 2z + 4 = 0$ has two solutions:

\begin{equation*} z = \frac{-2 \pm \sqrt{-12}}{2} = -1 \pm \sqrt{3}i \end{equation*}

since $\sqrt{-1} = i\text{.}$

###### Example2.4.4.Solving a quadratic equation.

Consider the equation $z^2 - (3+3i)z = 2-3i\text{.}$ To solve this equation for $z$ we first rewrite it as

\begin{equation*} z^2-(3+3i)z-(2-3i)=0\text{.} \end{equation*}

We use the quadratic formula with $\alpha = 1\text{,}$ $\beta = -(3+3i)\text{,}$ and $\gamma = -(2-3i)\text{,}$ to obtain the solution(s)

\begin{align*} z \amp = \frac{3+3i \pm \sqrt{(3+3i)^2+4(2-3i)}}{2}\\ z \amp = \frac{3+3i\pm \sqrt{8+6i}}{2}\text{.} \end{align*}

To determine the solutions in Cartesian form, we need to evaluate $\sqrt{8+6i}\text{.}$ We offer two approaches. The first approach considers the following task: Set $x + yi = \sqrt{8+6i}$ and solve for $x$ and $y$ directly by squaring both sides to obtain a system of equations.

\begin{align*} x+yi \amp = \sqrt{8+6i}\\ (x+yi)^2 \amp = 8+6i\\ x^2-y^2+2xy i \amp = 8 + 6i\text{.} \end{align*}

Thus, we have two equations and two unknowns:

\begin{align*} x^2-y^2 \amp = 8 \tag{1}\\ 2xy \amp = 6 \tag{2}\text{.} \end{align*}

In fact, we also know that $x^2+y^2 = |x+yi|^2 = |(x+yi)^2| =|8+6i| = 10\text{,}$ giving us a third equation

\begin{align*} x^2+y^2 \amp = 10 \tag{3}\text{.} \end{align*}

Adding equations (1) and (3) yields $x^2 = 9$ so $x = \pm 3\text{.}$ Substituting $x = 3$ into equation (2) yields $y = 1\text{;}$ substituting $x = -3$ into (2) yields $y = -1\text{.}$ Thus we have two solutions:

\begin{equation*} \sqrt{8+6i} = \pm (3+i)\text{.} \end{equation*}

We may also use the polar form to determine $\sqrt{8+6i}\text{.}$ Consider the right triangle determined by the point $8+6i = 10e^{i\theta}$ pictured in the following diagram.

We know $\sqrt{8+6i} = \pm \sqrt{10}e^{i\theta/2}\text{,}$ so we want to find $\theta/2\text{.}$ Well, we can determine $\tan(\theta/2)$ easily enough using the half-angle formula

\begin{equation*} \tan(\theta/2) = \frac{\sin(\theta)}{1+\cos(\theta)}\text{.} \end{equation*}

The right triangle in the diagram shows us that $\sin(\theta) = 3/5$ and $\cos(\theta)=4/5\text{,}$ so $\tan(\theta/2) = 1/3\text{.}$ This means that any point $re^{i\theta/2}$ lives on the line through the origin having slope 1/3, and can be described by $k(3+i)$ for some scalar $k\text{.}$ Since $\sqrt{8+6i}$ has this form, it follows that $\sqrt{8+6i} = k(3+i)$ for some $k\text{.}$ Since $|\sqrt{8+6i}| = \sqrt{10}\text{,}$ it follows that $|k(3+i)| = \sqrt{10}\text{,}$ so $k = \pm 1\text{.}$ In other words, $\sqrt{8+6i} = \pm (3+i)\text{.}$

Now let's return to the solution of the original quadratic equation in this example:

\begin{align*} z \amp = \frac{3+3i\pm \sqrt{8+6i}}{2}\\ z \amp = \frac{3+3i\pm (3+i)}{2}\text{.} \end{align*}

Thus, $z = 3+2i$ or $z = i\text{.}$

###### Example2.4.5.Circle equations.

If we let $z = x + yi$ and $z_0 = h + ki\text{,}$ then the complex equation

\begin{gather*} |z - z_0| = r \tag{equation of a circle} \end{gather*}

describes the circle in the plane centered at $z_0$ with radius $r> 0\text{.}$

To see this, note that

\begin{align*} |z - z_0| \amp = |(x-h)+(y-k)i|\\ \amp =\sqrt{(x-h)^2 + (y-k)^2}\text{.} \end{align*}

So $|z - z_0| = r$ is equivalent to the equation $(x-h)^2 + (y-k)^2 = r^2$ of the circle centered at $z_0$ with radius $r\text{.}$

For instance, $|z - 3-2i| = 3$ describes the set of all points that are 3 units away from $3+2i\text{.}$ All such $z$ form a circle of radius 3 in the plane, centered at the point $(3,2)\text{.}$

###### Example2.4.6.Complex expressions as regions.

Describe each complex expression below as a region in the plane.

1. $|1/z| \gt 2\text{.}$

Taking the reciprocal of both sides, we have $|z| \lt 1/2\text{,}$ which is the interior of the circle centered at 0 with radius $1/2\text{.}$

2. Im$(z)\lt$ Re$(z)\text{.}$

Set $z = x + yi$ in which case the inequality becomes $y \lt x\text{.}$ This inequality describes all points in the plane under the line $y = x\text{,}$ as pictured below.

3. Im$(z) = |z|\text{.}$

Setting $z = x + yi\text{,}$ this equation is equivalent to $y = \sqrt{y^2 + x^2}\text{.}$ Squaring both sides we obtain $0 = x^2\text{,}$ so that $x = 0\text{.}$ It follows that $y = \sqrt{y^2} = |y|$ so the equation describes the points $(0,y)$ with $y \geq 0\text{.}$ These points determine a ray on the positive imaginary axis.

Moving forward, lines and circles will be especially important objects for us, so we end the section with a summary of their descriptions in the complex plane.

###### Lines and circles in $\mathbb{C}$.

Lines and circles in the plane can be expressed with a complex variable $z = x + yi\text{.}$

• The line $ax + by + d = 0$ in the plane can be represented by the equation

\begin{equation*} \alpha z + \overline{\alpha z} + d = 0 \end{equation*}

where $\alpha = \frac{1}{2}(a - bi)$ is a complex constant and $d$ is a real number.

• The circle in the plane centered at $z_0$ with radius $r \gt 0$ can be represented by the equation

\begin{equation*} |z - z_0| = r\text{.} \end{equation*}

### ExercisesExercises

###### 1.

Use a complex variable to describe the equation of the line $y = mx + b\text{.}$ Assume $m \neq 0\text{.}$ In particular, show that this line is described by the equation

\begin{equation*} (m+i)z + (m-i)\overline{z} + 2b = 0\text{.} \end{equation*}
###### 2.

In each case, sketch the set of complex numbers $z$ satisfying the given condition.

1. $|z + i| = 3\text{.}$
2. $|z+i|=|z-i|\text{.}$
3. Re$(z) = 1\text{.}$
4. $|z/10 + 1 - i| \lt 5\text{.}$
5. Im$(z) >$ Re$(z)\text{.}$
6. Re$(z) = | z - 2 |\text{.}$
Hint

It may be helpful to set $z = x + yi$ and rewrite the expression in terms of $x$ and $y\text{.}$

###### 3.

Suppose $u, v, w$ are three complex numbers not all on the same line. Prove that any point $z$ in $\mathbb{C}$ is uniquely determined by its distances from these three points.

Hint

Suppose $\beta$ and $\gamma$ are complex numbers such that $|u - \beta| = |u - \gamma|\text{,}$ $|v - \beta| = |v - \gamma|$ and $|w - \beta| = |w - \gamma|\text{.}$ Argue that $\beta$ and $\gamma$ must in fact be equal complex numbers.

###### 4.

Find all solutions to the quadratic equation $z^2 + iz - (2+6i) = 0.$


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My Math Forum 2 person die roll 2 times

 Probability and Statistics Basic Probability and Statistics Math Forum

 May 14th, 2014, 08:07 AM #1 Member   Joined: Apr 2014 From: Missouri Posts: 39 Thanks: 1 2 person die roll 2 times If A and B both rolled two standard dice two times then what is the probability that B gets the same number as A but not necessarily in the same order? I don't understand the question. Can you please solve it? Thanks in advance and Regards
May 14th, 2014, 01:25 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 407

Hello, Shen!

Quote:
 If A and B both rolled two standard dice two times then what is the probability that B gets the same number as A but not necessarily in the same order? I don't understand the question. $\displaystyle \;\;$Neither do I!

Because of the phrase "but not necessarily in the same order",
I believe it asked for the probability that B gets the same two numbers as A.

I listed the outcomes and counted.

$\displaystyle \quad \begin{array}{cccccc} 11 & 12 & 13 & 14 & 15 & 16 \\ 21 &22 & 23 & 24 & 25 & 26 \\ 31&32&33&34&35&36\\ 41&42&43&44&45&46 \\ 51&52&53&54&55&56\\ 61&62&63&64&65&66 \end{array}$

Suppose A rolls a "double".
This happens $\displaystyle \tfrac{6}{36} = \tfrac{1}{6}$ of the time.
Then B must match it; probability: $\displaystyle \tfrac{1}{36}$
$\displaystyle P(\text{Double and match}) \:=\:\tfrac{1}{6}\cdot\tfrac{1}{36} \:=\:\tfrac{1}{216}$

Suppose A rolls some "Other".
This happens $\displaystyle \tfrac{30}{36} = \tfrac{5}{6}$ of the time.
Then B must match it; probability: $\displaystyle \tfrac{2}{36}$
$\displaystyle P(\text{Other and match}) \:=\:\tfrac{5}{6}\cdot\tfrac{2}{36} \:=\:\tfrac{10}{216}$

Therefore: $\displaystyle \:P(\text{Match}) \;=\;\tfrac{1}{216} + \tfrac{10}{216} \;=\;\tfrac{11}{216}$

 May 14th, 2014, 02:09 PM #3 Member   Joined: Apr 2014 From: Missouri Posts: 39 Thanks: 1 Hi soroban Thanks. This is the answer. I was stuck as i did not divide the double rolls and single rolls. Now i understand. Thanks again

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text:
_ContextMethodMixin.set_materialize_grads(value)[source]
This should be called only from inside the forward() method
If true, undefined output grad tensors will be expanded to tensors full of zeros prior to calling the backward() method.


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# Can Blender's bundled "bfont" be used for commercially?

Blender, which I use to create my video intros, comes bundled with a bfont by default so the question is:

Is this font is free for commercial use and what type of license does this "bfont" font have?

While the other answers are about distributing the font file itself, the use of a font to display text in your final image is not the same as sharing the font file and is not affected by the licensing of the font file.

As explained further in this wikipedia article (that you should read)

In the United States, the shapes of typefaces are not officially copyrightable, though they may be protected de facto by copyrights on computer code and trademarks on names.

which means the use of a font to draw text contained within an image does not cause copyright issues (some countries could have differing laws). While some fonts released under commercial licenses may restrict their use to people who paid for the font, the fonts included with blender are freely distributable and you are using them legally.

While there are other things to consider, like recreating some text that resembles a company logo can get you into trademark territory and some phrases have been trademarked etc, the general displaying of text in an image or video is not a copyright issue.

The bfont is actually a modified DejaVu (which in turn is a modified Bitstream Vera).

You are free to use it commercially but:

• If you modify it, it has to be under a new name (it can't be called "bfont")
• You can't sell it by itself
• You have to include the copyright notice (found in release/datafiles/LICENSE-bfont-ttf.txt)

• no i do not have any plan to sell the font seprately i m talking about if i use bfont in my video intro which i made in blender and then use the intro video for commercial purposes Feb 26 '16 at 16:01

I am not a lawyer. This answer is not a substitute for reading the actual license file.


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# mixing action

Let $X$ be a topological space and let $G$ be a semigroup. An action $\phi=\{\phi_{g}\}_{g\in G}$ of $G$ on $X$ is (topologically) mixing if, given any two open subsets $U$, $V$ of $X$, the intersection $U\cap\phi_{g}(V)$ is nonempty for all $g\in G$ except at most finitely many.

Example 1. Let $F:X\to X$ be a continuous function. Then $F$ is topologically mixing if and only if the action of the monoid $\mathbb{N}$ on $X$ defined by $\phi_{n}(x)=F^{n}(x)$ is mixing according to the definition given above.

Example 2. Suppose $X$ is a discrete nonempty set and $G$ is a group; endow $X^{G}$ with the product topology. The action of $G$ on $X^{G}$ defined by

 $\phi_{g}(c)(z)=c(g\cdot z)\;\;\forall c:G\to X$

is mixing.

To prove this fact, we may suppose without loss of generality that $U$ and $V$ are two cylindric sets of the form:

 $\displaystyle U$ $\displaystyle=$ $\displaystyle\{c\in X^{G}\mid\left.{c}\right|_{E}=\left.{u}\right|_{E}\}$ $\displaystyle V$ $\displaystyle=$ $\displaystyle\{c\in X^{G}\mid\left.{c}\right|_{F}=\left.{v}\right|_{F}\}$

for suitable finite subsets $E,F\subseteq G$ and functions $u,v:G\to X$. Then the only chance for $U\cap\phi_{g}(V)$ to be empty, is that $e=gf$ for some $e\in E$, $f\in F$ such that $u(e)\neq v(f)$: but then, $g\in EF^{-1}$, which is finite.

Title mixing action MixingAction 2013-03-22 19:19:32 2013-03-22 19:19:32 Ziosilvio (18733) Ziosilvio (18733) 5 Ziosilvio (18733) Definition msc 37B05


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# LOG#150. Bohr and Doctor Who: A=mc³.

The year 2013 is coming to its end…And I have a final gift for you. An impossible post!

This year was the Bohr model 100th anniversary. I have talked about this subject already, herehere and here. The hydrogen spectrum is very important in Astronomy, Astrophysics and even cosmology, chemistry, and quantum mechanics. Quantum mechanics provides indeed the same results for the hydrogen atom than that of the original Bohr model, but it also includes novel effects: the Stark effect, the Zeeman effects- normal and anomalous, and many others like the hyperfine structure of the atom! However, one of the magical things of the hydrogen atom is that it yields the right results if you neglect purely quantum effects as spin, and other subtle effects like the finite mass effect (this effect can be even obtained with the aid of classical mechanics and the notion or reduced mass but it does not matter to the needs of this post!). The energy levels of the hydrogen atom can be summarized in the next pictures:

Is it cool?

Other useful scheme for you:

The Bohr model was revolutionary…

Recently, this year happened the Doctor Who 50th anniversary as well!!!!!And it has been a revolutionary year for whovians too!

Therefore, I thought (some weeks ago): I am a geek,  I am a whovian, I am a physicist, I am a theoretical physicist, I am a physmatician/physchematician as well. Thus, I have to merge all this crazy stuff together…Then, this special 150th post will try to do it! The Physics, Chemistry, Mathematics of a revolutionary theory and a wonderful (inmortal?) TV show…Impossible? Impossible post? Let me know after you read the final result…

Let me begin with a short review of the popular and well known Bohr model (for hydrogenlike atoms) formulae:

1) Quantization of angular momentum (quantization of “rotational features”).

$\boxed{L(n)=L_n=mv_nr_n=\hbar n=\dfrac{hn}{2\pi}\;\;\;\; n=1,2,\ldots,\infty}$

2) Quantization of radius (quantization of allowed orbits, or quantization of length/space!) and area.

$\boxed{R(n)=R_n=a_Bn^2=a_0n^2=\dfrac{1}{Z\alpha}\dfrac{\hbar}{mc}n^2=\dfrac{1}{Z}\dfrac{\hbar^2}{mK_Ce^2}n^2\;\;\;\; n=1,2,\ldots,\infty}$

The length traveled by an electron in the hydrogen atom is also quantized by the rule $l_n=2\pi R_n$ as well.

Area of circular orbits are given by $S=\pi R_n^2$, and thus

$\boxed{S(n)=S_n=\pi (a_Bn^2)^2=\pi a_B^2n^4=\pi\dfrac{1}{Z^2\alpha^2}\dfrac{\hbar^2n^4}{m^2c^2}=\pi\dfrac{1}{Z^2}\dfrac{\hbar^4}{m^2K_C^2e^4}n^4\;\;\;\; n=1,2,3,\ldots,\infty}$

they are also quantized!

3) Quantization of velocities.

The rate of change in position with respect to time is also quantized through the rule

$\boxed{v(n)=v_n=\dfrac{Z\alpha c}{n}=\dfrac{ZK_Ce^2 }{\hbar n}\;\;\;\; n=1,2,\ldots,\infty}$

4) Quantization of linear momentum.

As a consequence of quantization of velocities (or space quantization as well), the linear momentum is also quantized

$\boxed{p(n)=p_n=mv_n=\dfrac{n\hbar}{R_n}=\dfrac{Z\alpha mc}{n}=\dfrac{ZK_Ce^2 m}{\hbar n}\;\;\;\; n=1,2,\ldots,\infty}$

5) Quantization of acceleration. The centripetal acceleration is quantized as well

$\boxed{a(n)=a_n=\dfrac{v_n^2}{R_n}=\dfrac{Z^3\alpha^3mc^3}{\hbar n^4}= \dfrac{Z^3}{\hbar n^4}mc^3\dfrac{K_C^3e^6}{\hbar^3c^3}=\dfrac{Z^3K_C^3e^6}{\hbar^4n^4}m\;\;\;\;n=1,2,3,\ldots,\infty}$

This equation will be very useful and vital in this post. Keep it in your mind! (In fact, whovians KNOW that $mc^3$ in the time vortex but they are wrong in what it means!).

6) Quantization of centripetal force. The last quantization rule implies a quantization rule for the centripetal force

$\boxed{F(n)=F_n=ma_n=\dfrac{Z^3\alpha^3m^2c^3}{\hbar n^4}=\dfrac{Z^3}{\hbar n^4}m^2c^3\dfrac{K_C^3e^6}{\hbar^3c^3}=\dfrac{Z^3K_C^3e^6}{\hbar^4n^4}m^2\;\;\;\;n=1,2,3,\ldots,\infty}$

7) Quantization of energies.

$\boxed{E(n)=E_n=\dfrac{1}{2}ma_nR_n=\dfrac{Ry}{n^2}=\dfrac{Z^2\alpha^2mc^2}{2n^2}=\dfrac{Z^2(K_Ce^2)^2m}{2\hbar^2n^2}\;\;\;\;n=1,2,\ldots,\infty}$

In summary, radius, length, area, velocity, linear momentum, angular momentum, acceleration, force and energy is quantized in the Bohr model. Awesome! Are you Bohred? LoL. Now, the whovian part. According to the trivia (and whovian “culture”), it is said that (wikipedia):

“(…)In the science fiction television series Doctor Who, the Time Vortex is the medium that TARDISes and other time machines travel through. The “howlaround” tunnel in most versions of the series’ title sequence is supposed to be a representation of the Time Vortex, although it is sometimes also shown as nothingness.

The Vortex is outside normal spacetime, and therefore normal rules of physics to not apply. For instance, in the Vortex the equation for the relationship between energy and matter is E=mc³ (The Time Monster).

The Vortex is an extremely hostile environment. In the serial Warriors’ Gate opening the TARDIS in flight exposes the interior to the Time Winds, which age whatever they come into contact with.

Beings that dwell in the Vortex include the Chronovores (The Time Monster), the Vortex Wraiths (the Eighth Doctor Adventures novels The Slow Empire and Timeless) and the vortisaurs(the Big Finish audio play Storm Warning).

In the Eighth Doctor Adventures, Sabbath‘s employers set up their headquarters in the Vortex, casting many of the natives out into the linear universe.(…)”

And we also find the following comments in the world wide web:

“(…)E=MC³ in the extra temporal physics of the time vortex. ‘Being without becoming, an ontological absurdity!’ The Doctor makes a ‘time flow analogue’ from a Moroccan burgundy bottle, spoons, forks, corks, keyrings, tea leaves and a mug. ‘The relationship between the different molecular bonds and the actual shapes form a crystalline structure of ratios.’ [Since the Doctor and the Master made these at school (the Academy) to spoil each other’s time experiments, it is only the shape of the things that are important.] The Master works out his landing coordinates with map and compass. A lot of polarities get reversed.(…)”

The point I want to remark is that of the physical interpretation of $E=mc^3$ is wrong. Indeed, that has no dimensions of energy. Energy has (physical, not numerical) dimensions of $ML^2T^{-2}$ irrespectively the number of spacetime dimensions! That is how Physics works. But, you know, The Doctor lies (rule number One!). In fact, the right interpretacion of $mc^3$ is a quantum interpretation. If you have kept your attention to the Bohr model formulae, you should have realized something that is “almost” $mc^3$. Plug $Z=\alpha=n=1$, then you get that accelerations in the Bohr model are quantized and are less than the maximal acceleration

$\boxed{a_{max}=A=\dfrac{mc^3}{\hbar}}$

And if you take units in which the (rationalized) Planck constant is taken to be equal to the unit ($\hbar=1$), you get the “time vortex” equation for acceleration (or gravitational field if you take into account the equivalence principle, being “naive”):

$\boxed{A=mc^3}$

Here you are!!!!!! The geekiest cool explanation of the right physical meaning of A=mc³ (Not E=mc³!!!!!!!). In fact, there is more interesting cool stuff to discuss. Common special relativity is related to the existence of one maximal speed (velocity): the speed of light. Of course, in quantum theory this becomes a bit fuzzy because of the Heisenberg uncertainty principle and even general relativity and some Beyond Standard Model theories (like the Magueijo-Moffat theories of varying speed of light) can change it, but, in the end, “classically” there seems to exist a maximal speed in 3+1 spacetime. It is the speed of light. Similarly, with the aid of the Bohr model (note that the Bohr model as stated and studied here is NOT a model consistent with special relativity! Exercise: state why!;)) you can guess a new “extended” relativity principle: the principle of maximal acceleration! From the Bohr model you obtain quantized accelerations decreasing as $n^{-4}$. $A=mc^3/\hbar$ is the maximal acceleration. Not surprisingly, there has been some speculations about the existence of a “new extended relativity principle” related to that maximal acceleration principle (MAP). The MAP was pioneered by Born in his reciprocal relativity, Caianiello, Nesterenko, Scarpetta, and others. Nowadays, the principle of maximal acceleration (MAP) is seriously considered by some scientists like Brandt, C. Castro, and some quantum gravity theorists like Rovelli himself! It is not crackpottery. MAP is out there. Even more, we can extended MAP further…Is there a maximal jerk? A maximal length? A maximal n-th derivative of position? A maximal -nth (negative or integral) derivative of position? A maximal fractional n-th derivative? What about minimal bounds? Quantum gravity is generally assumed to provide area (and length) quantization, and it is generally assummed that quantum gravity provides a minimal length on very general assumptions (irrespectively you use superstring theory or not!). Thne next “fact” was one of the main lectures obtained from the physics of Black Holes (whovian comment: Time Lords get its power source from a Black Hole!): the Bekenstein conjecture states that the Black Hole entropy, proportional to the area, is quantized. So, area and length are quantized (remarkably similar to some of the features of the Bohr atom!). Thus, momentum itself will be quantized, also angular momentum, velocity and acceleration as well! What about the derivative of acceleration? And the derivative of derivative of acceleration? We can speculate if any derivative (even integrals of position or fractional differentegrals) is quantized! Position, velocity, acceleration, jerk,… I have discussed the naming and physical interpretations of higher order velocities (n-th derivatives) here.

Indeed, a simple and naive (physical) dimensional analysis allow us to recover the main relations of Quantum Mechanics and Special Relativity, and to build new interesting relationships I have discussed in the blog post mentioned above. Let me put this statement into equation a bit further. We define negative derivatives (integrals!) of position as the kinematical variables:

$\displaystyle{\boxed{\mathcal{A}=\int xdt=\mbox{Absement}}}$

$\displaystyle{\boxed{\tilde{\mathcal{A}}=\int x d^2t=\mbox{Absity}}}$

$\displaystyle{\boxed{\tilde{\mathcal{A'}}=\int xd^3t=\mbox{Abseleration}}}$

$\ldots$

Physical dimensions of some classical magnitudes (giving up purely numerical prefactors):

$E=\mbox{Energy}=(\mbox{Mass})(\mbox{Velocity})^2=(\mbox{Force})(\mbox{Space})=(\mbox{Force})(\mbox{Displacement})$

$\mbox{Linear momentum}=(\mbox{Mass})(\mbox{Velocity})$

$\mbox{Angular momentum}=(\mbox{Mass})(\mbox{Velocity})(\mbox{Displacement})$

Some “new” kinematical and dynamical variables:

$\mbox{Yank}=\left(\dfrac{\mbox{Force}}{\mbox{Time}}\right)=(\mbox{Mass})(\mbox{Jerk})$

$(\mbox{Absement})=(\mbox{Displacement})(\mbox{Time})$

And there are many others I will not rewrite here again. In terms of physical dimensions, we also know that:

$E=ML^2T^{-2}=(\mbox{Absity})(\mbox{Tug})=(\mbox{Abseleration})(\mbox{Snatch})$

Can it be generalized further? Yes, it can! Note that:

$E=(LT^2)(MLT^{-4})=(LT^3)(MLT^{-5})$

so we can write

$(1)\boxed{E=\left(D^{-n}_t (x)\right)\left( D_t^{n+2} (x)\right)(\mbox{Mass})}$

for $n=0,1,\ldots,...$. Indeed,  we can also write (writing x=LX, X adimensional)

$(2)\boxed{E=ML^2\left(D^{-\alpha}_t X\right)\left( D_t^{\alpha+4} X\right),\;\;\forall\alpha\in\mathbb{Z}}$

In fact, you can symmetrize (1) as follows:

$(3)\boxed{E=(\mbox{Mass})\left(D^{-n+1}_t (x)\right)\left( D_t^{n+1} (x)\right)}\;\;\;\forall n\in\mathbb{Z}$

and where we define the negative differentiation (integration!) as

$I=\left(\dfrac{d}{dt}\right)^{-n}=D_t^{-n}=\left[T^n\right]$

and where the last term being the physical dimensions. The positive derivatives are defined as usual

$\mathcal{D}=D_t^n=\left(\dfrac{d}{dt}\right)^n=\left[T^{-n}\right]$

A completely symmetric form is also available, from (3):

$(4)\boxed{E=(\mbox{M})\left(D^{\frac{2(-n+1)}{2}}_t (x)\right)\left( D_t^{\frac{2(n+1)}{2}} (x)\right)}\;\;\forall n\in\mathbb{Z}$

or, at formal level of “fractional derivatives”

$(5)\boxed{E=(\mbox{M})\left(\sqrt{D_t^{2(-n+1)}}(x)\right)\left(\sqrt{D_t^{2(n+1)}} (x)\right)}\;\;\forall n\in\mathbb{Z}$

From any of these last equations (3)-(4)-(5) (or (1), (2)), you can derive a complete set of kinematical and dynamical relations for integer values of $n, \alpha$. You can recover classical relationships like

$E=xD_t^2(x)M=Fx=(\mbox{Force})(\mbox{Displacement})=(\mbox{Mass})(\mbox{Acceleration})(\mbox{Displacement})$

or

$E=\mbox{Energy}=MaX=(\mbox{Mass})(\mbox{Velocity})^2$

and you can also build up new ones, like

$E=\mathcal{A}\mathcal{Y}=\mbox{(Absement)}\mbox{(Yank)}=(\mbox{Absement})(\mbox{Mass})(\mbox{Jerk})=(\mbox{Mass})(\mbox{Presement})^{-2}$

This (fractional) operator (differentegral calculus!) formalism is very useful to determine and quickly remember some old results and produce new interesting variables. Furthermore, if you implement a generalized principle of maximal velocity, and maximal (linear) momentum, you recover the usual relations of special relativity! I mean, take the formal expressions

$E=pc$

$E=mc^2$

which are obtained from the relativistic dispersion relationship

$E^2=p^2c^2+m^2c^4\leftrightarrow E=\sqrt{p^2c^2+m^2c^4}$

in the limits of $m=0, p\neq 0$ and $m\neq 0, p=0$. These formal relationships can be obtained from our general expressions above:

$(\mbox{Energy})=(\mbox{Linear momentum})(\mbox{Velocity})=pD_t(X)$

$(\mbox{Energy})=(\mbox{Mass})(\mbox{Velocity})^2=MD_t^2(X)$

from maximal velocity $D_t(x)\leq c$! One should be wonder if similar things do exist for higher order derivatives (likely yes, but nobody knows for sure today, and it is speculative). In fact, from Newton’s 2nd law:

$\displaystyle{F=\mbox{Force}=D_tp=D_t(\mbox{Linear Momentum})\leftrightarrow p=D_t^{-1}F=\int F dt=D_t^{-1}\mbox{(Force)}}$

Is force (acceleration) bounded from maximal force (accelerations)? What about mass, length or absement? What about any n-th order derivative? And fractional derivatives? Minimal action is common in the variational approach of mechanics (lagrangian and hamiltonian dynamics), but we lack a Maximal Action Principle as we lack a Maximal Acceleration Principle in phycics at current time. Physics are based on critical points of action functionals

$\displaystyle{S(q,\dot{q};t)=\int L(q,\partial_t q;t)dt}$

or its higher order generalizations (sometimes referred as higher order lagrangian theory)

$\displaystyle{S(q,\dot{q},\ddot{q},\ldots;t)=\int L(q,\partial_t,\partial_{tt}q,\ldots;t)}dt$

and, for this particle actions, their field theory analogues are, respectively

$\displaystyle{S(\phi,\partial\phi;x)=\int \mathcal{L}(\phi,\partial\phi;x)d^Dx}$

$\displaystyle{S(\phi,\partial \phi,\partial^2\phi,\ldots;x)=\int \mathcal{L}(\phi,\partial\phi,\partial^2 \phi,\ldots;x)d^Dx}$

in D spacetime dimensions, with

$x=(x^\mu)$

$\partial=D_\mu=\partial_{\mu}$

$\partial^2=\partial_{\mu_1\mu_2}$

and so on! Why am I commenting this? Well, recently, the so-called “double field theory” became popular in several branches of theoretical physics: string theory/M-theory, QFT, dualities and related stuff. Well, it seems pretty interesting to me that the doubling of fields can be suggested by this operator formalism (both in the particle and field pictures) by the following map:

$q\rightarrow 1/q$

$\partial_t\rightarrow \partial_t^{-1}=\dfrac{1}{\partial_t}$

$\phi\rightarrow 1/\phi$

$\partial_\mu\rightarrow\partial_\mu^{-1}$

$\partial\rightarrow 1/\partial=\partial^{-1}$

This T-duality (S-duality if you go to the realm of coupling constants as well) is very suggestive…It extends the classical actions and (at least) enlarge (duplicates,multiplies) the degrees of freedom (double field theory is contained in this operational approach, but it gets generalized!):

$\displaystyle{S=\int\mathcal{L}(q,\partial_tq;t)dt\rightarrow S=\int\mathcal{L}\left(q,\dfrac{1}{q}=q^{-1},\partial_tq,\partial^{-1}_tq,\partial^{-1}_tq^{-1};t\right)dt}$

$\displaystyle{S=\int\mathcal{L}(q,\partial_tq,\partial^2_tq,\ldots;t)dt\rightarrow S=\int\mathcal{L}\left(q,\dfrac{1}{q}=q^{-1},\partial_tq,\partial^{-1}_tq,\partial^{-1}_tq^{-1},\partial^2_tq,\partial^2_tq^{-1},\partial^{-2}_tq,\partial^{-2}_tq^{-1},\ldots;t\right)dt}$

$\displaystyle{S=\int\mathcal{L}(\phi,\partial\phi;x)d^Dx\rightarrow S=\int\mathcal{L}\left(\phi,\phi^{-1},\partial \phi,\partial^{-1}\phi,\partial^{-1}\phi^{-1};x\right)d^Dx}$

$\displaystyle{S=\int\mathcal{L}(\phi,\partial\phi,\partial^2\phi,\ldots;x)d^Dx\rightarrow S=\int\mathcal{L}\left(\phi,\phi^{-1},\partial\phi,\partial^{-1}\phi,\partial^{-1}\phi^{-1},\partial^2\phi,\partial^2\phi^{-1},\partial^{-2}\phi,\partial^{-2}\phi^{-1},\ldots;x\right)d^Dx}$

Does such a generalized MAP/generalized special relativity/Maximal Length Principle exist? I think so. Even more, minimal action, minimal length, minimal velocity, minimal acceleration,…principles should exist as well to explain the quantum realm in some generalized enlarged “pseudoclassical” extended theory of relativity. I can tell you more: I work on it! The existence of such a principle, I think, is the true door to quantum gravity and the true Theory Of Everything (TOE). Indeed, classical mechanics has a principle of minimal action (minimal velocity?) and it also holds in Quantum Mechanics (due to the Heisenberg Uncertainty Principle, HUP). Generalized Uncertainty Principles (GUP) are also discussed in the phenomenology of Quantum Gravity. Of course, all this stuff speculative. I have also asked this question in physics stack exchange:

http://physics.stackexchange.com/questions/61522/extended-born-relativity-nambu-3-form-and-ternary-n-ary-symmetry

Indeed, I wrote a message to a young graduate student who wrote a preprint time AFTER I posted the above question. The preprint is this one:

http://arxiv.org/abs/1308.4044

Its title (and I urge you to read the whole paper) says it all: Notes on several phenomenological laws of quantum gravity.

It is a great paper! Elementary but fascinating! It serves as a good starting point for newbies into the fascinating topic of extended theories of relativity! A subject in which I am actively involved, you have really no idea of how! :).

Only to say a little bit, quantum physics is (currently) believed to quantize (or be based on the quantization of) angular momentum and (specially) energy (neglecting continuous spectra obtained in the large n limit). Quanta of energy are sometimes called quanta, but you could call them “energons” (I am sorry, I am a fan of the transformers as well, so let me borrow the term at the moment and call them quanta in your daily life if you want). Quanta of angular momentum is related to spin, so we could call “spinons” to those elementary quanta. Quanta of charge could be called “chargons”. And you could guess similar quanta of physical magnitudes, even for those new (generally unknown) quantities we call absement, presement, absity, abseleration, tug,…But what about a quantum of space or time? You can call them choraons and chronons (chronons have been discussed in the past by Finkelstein, Caldirola and some other physicists), and these quanta are related to the quanta of linear momentum or energy in a simple way. Loop Quantum Gravity using spin networks derive a similar concept. Indeed, mass itself is related to the reation of quanta of energy, time and space by simple physical dimensions! Remember: $E=ML^2T^{-2}$. If you quantize energy, length and time, mass itself MUST be quantized (to be more precise, it should be “fractionalized” or it will be a rational multiple of some integer/rational number! It is strongly similar to the phenomenon of fractional charge quantization -fractionalization of charge- now popular in condensed matter physics!).

In summary: There is no choice there! Everything is a quantum/some quanta of something! Every magnitude should be quantized in physics. There should be a minimal/maximal or MIN/MAX X-principle acting in fundamental physics over any fundamental system of magnitudes. Position or length, velocity, acceleration, energy, mass, angular momentum or action, linear momentum, even absement, jerk and any n-th derivative (positive, integer or fractional) should reach some minimal/maximal values. Surprised? You should not! It is pure (atomic like) logic!

By the other hand, the role of generalized Nambu mechanics in physics (and its quantization) is yet unclear. There has been some recent advances I follow closely, and some cool new applications of Nambu Mechanics in hydrodynamics, atmospheric dynamics (essentially, the Lorenz system can be written as a Nambu system) or pure electromagnetism and electrodynamics. However, this Nambu stuff will be told in the near future, in a forthcoming TSOR post!

And all these lines were my impossible 150th post!!!!!! I hope you liked it! See you in a new amazing TSOR post! See you in 2014! Stay tuned!

PS: Happy new year 2014 everyone and everywhere out there! Best wishes for all of you from my inner soul, heart and (crazy) mind! Hahahahahaha… Take care and enjoy your mortal existence!

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# Are Our TDFs Massively Underweight Inflation-Fighting Assets?

July 2018
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Key Points
• An industry-wide underallocation to inflation-fighting assets in TDFs leaves investors vulnerable to macroeconomic and market risks, including high-surprise-inflation conditions, bear markets, and volatile markets.

• We urge investors to proactively put inflation-fighting assets in place when the need does not seem self-evident—not after the markets have already inflicted damage and the need for protection is obvious.

• A diversified exposure across multiple inflation-fighting asset classes via a contrarian multi-asset approach can serve as powerful insurance for retirement portfolios.

We both recently moved and bought homeowners’ insurance, once again. Like most Americans,1 for our most valuable assets—our homes typically being the most valuable asset—we willingly pay for protection against a variety of natural disasters such as earthquakes, hurricanes, tornados, and floods.2 Even though the possibility that any of these individual risks materializes is remote, we don’t bat an eye when it comes to buying insurance coverage. As long as risks specific to our geographic location are fully covered by our yearly premium, we “sleep easy.”

Protecting our retirement assets should be no different. Inflationary waves, market downturns, and jolts in volatility are all legitimate threats that can shrink our retirement nest eggs, compromising the spending power of our eventual withdrawals. Unfortunately, traditional retirement portfolios offer minimal protection against these risks, even though the cost of insuring against them is cheap and the likelihood of their materializing is rising.

In this article, we delve into 1) why an industry-wide underallocation to real, or inflation-fighting, assets in target-date funds (TDFs) leaves investors vulnerable to macroeconomic and market risks that can undermine their financial security in retirement; 2) how adding real assets to retirement portfolios may lead to improved outcomes; and 3) the potential challenges investors face in gaining the protection they desire for their retirement portfolio

Introduced in 1994, TDFs have been touted as a “one-stop” investment vehicle for retirement savings. They are intended to be a well-diversified, dynamic, multi-asset portfolio set-it-and-forget-it option that de-risks over time by increasing exposure to bonds at the expense of equities.3 Given that they automatically address asset allocation, rebalancing, and portfolio construction considerations relevant to specific age cohorts, TDFs have—not surprisingly—grown rapidly, especially over the last decade. Having assumed the role of default option for the majority of corporate sponsors of defined contribution plans, TDFs have accumulated over $1 trillion in assets as of year-end 2017 (Holt, 2018). Are TDFs really the one-stop solution they claim to be, providing adequate diversification to protect our retirement assets over a variety of market and inflationary regimes? Or are we “sleeping easy” just before our basement, which is notcovered by our insurance policy, is flooded? Let’s look under the hood of the 10 most dominant providers of mutual fund TDFs, which had combined assets of just over$820 billion of the total $880 billion, an all-time high, across all vintages as of December 31, 2016 (Holt, 2017).4 All of these portfolios hold considerable positions in US stocks and bonds. For instance, on an average asset-weighted basis, the 2030 TDF vintage across the 10 largest providers allocates 63% to US assets. Further, we conservatively estimate that over three-fourths of the exposure within the Global Stocks and Bonds category is actually developed-market stocks and bonds,5 which raises the total allocation to mainstream stocks and bonds to over 80%.6 But what about real assets? The average TDF exposure to real assets, those assets typically considered to offer direct protection against rising inflation, was only 4.5% at year-end 2016. These “traditional inflation fighters” include 1) Treasury Inflation Protected Securities (TIPS), whose prices adjust annually with the CPI inflation rate and are guaranteed by the US Treasury; 2) commodity futures, whose prices change to reflect changes in the cost of the respective underlying raw material; and 3) real estate investment trusts (REITs), whose prices change in line with changes in the rents earned by the real estate property the REIT owns.7 Asset classes other than real assets also provide inflation protection, even if they are not commonly viewed as inflation fighters. We at Research Affiliates refer to these asset classes as “stealth inflation fighters,” a category that includes high-yield bonds and bank loans, along with allocations away from the US dollar into emerging markets (EM) currencies, bonds, and equities. (After all, what is inflation, if it’s not US dollar debasement?) These asset classes have demonstrated positive correlation with US inflation, with many offering inflation protection superior to that provided by TIPS.8Across the 2030 TDF vintages offered by the 10 largest providers, these stealth inflation-fighters had an average allocation of 6.4% as of December 31, 2016. Adding this to the 4.5% allocation to real assets, the total allocation to inflation-fighting assets was only a modest 10.9%. For investors getting closer to retirement, the average exposure to inflation-fighting assets only falls from these low allocations as they age. Consider (as of July 2018) the most popular target retirement fund built for investors who are retiring five years hence. These soon-to-be retirees have barely 4.0% in traditional inflation-fighting markets, and their exposure to high-yield bonds, EM bonds, and EM currencies is zilch. Their sole allocation to stealth inflation fighters is 3.5% to EM equities, which puts combinedinflation-fighting exposure at a measly 7.5%. We believe investors should, instead, be increasing portfolio diversification and raising inflation protection as retirement gets closer. ## Underinsured for Market Risks Target-date funds have such scant exposure to inflation-fighting asset classes today because investment experience tends to shape mindsets and behaviors, and for nearly four decades, investors’ experience has been shaped by a disinflationary environment. Since peaking in March 1980, US inflation levels have fallen from 14.6% to 2.8% through June 2018,9 resulting in a substantial tailwind to the returns of developed stocks and bonds. Mainstream stocks and bonds thrive in disinflationary conditions. Bond prices tend to rise as fears fall that future inflation will erode the purchasing power of bonds’ fixed coupon payments. Over the disinflationary period 1980–2018, US aggregate bond yields fell from 14.1% to 3.1%, leading to annualized total returns of 7.8% over that span. Mainstream stocks also tend to experience return tailwinds through rising valuations in disinflationary regimes, because falling or low inflation reduces the rate that market participants use to discount future cash flows. For example, over the same roughly 37-year period, the US CAPE (cyclically adjusted price-to-earnings, or Shiller PE) ratio rose from 8.1x to 32.1x, soaring from the lowest 6th to the highest 97th percentile rank since 1871. As a result, annualized total returns to US stocks came in at an extremely elevated 11.9% over that period. Not only have these asset classes offered impressive historical returns, but they also complement each other well because bonds tend to outperform in disinflationary bear markets, whereas stocks naturally outperform during disinflationary bull markets. Given the continuous disinflationary environment of the last nearly 40 years, combined with the common practice of using the past to forecast the future,10 the fact that the overwhelming majority of TDF assets is allocated to disinflationary-biased stocks and bonds is easy to understand. Recall that TDFs were launched within the last quarter-century, and disinflationary conditions have defined the collective experience of investors active during this time. In fact, some TDF providers are increasing their exposure to mainstream equities now, believing that doing so can improve their odds of funding the retirements of populations with increasing life expectancies.11 At the present time, it seems little attention is being paid to how TDF portfolios could suffer in inflationary conditions. Because expected inflation levels are typically baked into prices in the short term, we focus on how asset classes fare in inflation-surprise environments, or unanticipated moves in inflation levels relative to what was expected at the beginning of each quarter.12 When inflation is lower than anticipated, developed-market stocks and bonds shine. For example, since 1973, in low-inflation-surprise environments, these assets delivered brilliant average annualized returns in excess of the cash return of 12.6% for US stocks and 9.3% for a developed 60/40 mix. An equally weighted basket of inflation-fighting assets13 lagged these conventional assets, but nevertheless delivered a reasonable positive average excess return of 5.1%. When upward moves in inflation catch investors off guard, portfolios of conventional assets suffer. High-inflation shocks lead to losses in developed-market stocks and bonds, the asset classes that compose the vast majority of most retirement portfolios. In the high-inflation-surprise environments experienced since 1973, the average annualized excess return of US stocks was –0.9% and for a developed 60/40 portfolio was −0.3%. And whereas bonds are intended to provide diversification away from equity risk, they fail to do so in inflationary environments. Across these periods of positive inflation surprises since 1973, the average correlation levels between developed-market stocks and bonds rose two-fold to 33%. In other words, the complementary nature of stocks and bonds breaks down in environments when a portfolio complement is needed most! We find that the appropriate portfolio complement in inflation surprise environments is a basket of inflation-fighting assets, which earned positive annualized excess returns averaging 1.4% in such environments. Inflation-fighting assets also provide powerful protection in other conditions of distress for developed-market stocks and bonds—both bear markets and volatile markets. Consider the times over the last 45 years when the S&P 500 Index delivered a negative 12-month rolling return. In these bear markets, mainstream assets gravely suffered, delivering an average rolling 12-month return of −13.7% for US stocks and −5.7% for a developed 60/40 portfolio. In stark contrast, under these same conditions, an equally weighted mix of real-return inflation-fighters provided significant downside protection, providing an average gain of 1.9%! Similarly, in volatile environments, as characterized by an average rolling 12-month US stock volatility exceeding 30%,14 mainstream assets fared poorly. On average, US stocks tumbled by 11.2% and a global 60/40 portfolio fell by 1.0%. Inflation fighters, however, returned an average annual gain of 0.4% during such market turbulence. When to Insure? We all understand the need to buy insurance before we actually need it, but sometimes the cost is hard to swallow when no impending dangers are evident on the horizon. In extended environments of disinflation or prolonged periods of low volatility and positive movement in the equity market, some investors will continue to position their portfolios for a continuation of recent history. Such an approach is behaviorally understandable, but increasingly unlikely to pay off. A “wait and see” mode, that is, seeking the catalyst to a regime shift before repositioning the portfolio for the new regime, is typically too little, too late. As is the case with most natural disasters, a change in economic or market regime rarely pre-announces its arrival. Thus, lacking the clairvoyance needed to accurately and consistently identify regime shifts in advance (as is the case for most of us), investors will likely be repositioning their portfolios after the catalyst. Such an approach is akin to buying insurance after our homes are damaged and premiums have spiked. It’s classic performance chasing. Contrarians, on the other hand, patiently buy insurance before the regime shift and while insurance premiums are low. In other words, they proactively put real-return-producing assets in place when the need doesn’t seem self-evident—not after the markets have already inflicted damage and the need for protection is obvious. Given that market and economic environments, which warrant some portfolio protection, happen more frequently than investors have been conditioned to accept, we believe inflation-fighting assets with their powerful diversification benefits merit substantially higher exposure within investors’ retirement portfolios. The current environment, in fact, may prove timely for investors to add portfolio protection, if for no other reason than a simple truism: current conditions will not persist indefinitely. Consider the following: • Since 1802, the average length in the United States of an equity market rally—maintaining a positive annual return over rolling monthly outcomes—is slightly over three-and-a-half years.15 At nine years and counting, today’s rally is only 18 months away from being the longest bull market of the past two centuries! • The current period of persistently calm markets—characterized by an average rolling volatility less than 20%—has reached nine years, surpassing the six-year average span of historical quiet periods. This near-decade-long period of calm falls within the longest 83rd percentile rank of all tranquil outcomes since 1860. • In the last five years, we have not experienced a singlequarter (that’s not a typo!) of a high-inflation-surprise environment, which has been the case one-third of the time over the last 45 years. With current positive market conditions appearing excessively stretched, adding insurance now seems prudent before it’s too late and the regime shifts.16 Additionally, we believe the long-term return prospects of inflation-fighting, real-return-producing assets are currently attractive relative to the investment opportunity set. Based on a building-block-return approach that focuses on expected yield, growth in yield, and future changes in asset and currency valuations, the Research Affiliates Asset Allocation Interactive (AAI) tool, which is available on our website, provides 10-year return estimates for asset classes and portfolios. Even after a rebound in many inflation-fighting markets since the 2013–2015 bear market, an equally weighted basket of these higher-yielding assets is poised to deliver an annualized nominal return of 5.3% over the coming decade. In other words, the inflation-fighting asset classes today are real bargains—priced to deliver a prospective excess return of 1.4% a year compared to a global 60/40 portfolio and 1.6% a year versus a 2030 TDF proxy over the coming decade. For instance, if we were to start today with$100, 10 years from now we would expect our inflation-fighting basket to rise to $168, surpassing the end values of a 60/40 portfolio and a 2030 TDF proxy,$147 and $144, respectively. Simply increasing our exposure to inflation-fighting real assets is likely to deliver improved returns and diversification benefits beyond what mainstream-centric TDFs can offer. Should our longer-term forecasts materialize or we experience unanticipated upside moves in inflation, renewed bouts of market volatility, and/or bear markets, then increasing exposure to inflation-fighting real assets now would clearly deliver enhanced return and diversification benefits to our otherwise mainstream-centric TDFs. Selecting the Best Strategy for Protection How we add inflation-fighting asset exposure to retirement portfolios is important in achieving the best possible outcome and the best possible portfolio protection. Consider, for example, the volatility characteristics of commodities and REITs. Since January 1973, the annualized standard deviation of their returns has been 16.9% and 17.6%, respectively. Assuming normally distributed returns, an investor who invests in REITs or commodities—as a stand-alone asset class—can expect to lose more than 25% once every 20 years!17 Such painful experiences encourage many investors to succumb to the natural tendency of exiting these regret-maximizing positions precisely when their forward return potential is highest. Study after study (e.g., Kinnel 2005, 2014, 2015, 2016; Hsu, Myers, and Whitby, 2016; and Arnott, Kalesnik, and Wu, 2017) has confirmed that investors consistently buy recently winning exposures while selling recent losers, thereby locking in losses and ensuring lower forward-return prospects. Given these behavioral consistencies, adding stand-alone exposure to these asset classes would almost certainly lead to disappointing results. TIPS may seem a superior option given their lower annualized volatility levels of 9.4% since inception in March 1997, but their current yield of 0.7% is grossly below the 3–4% yield levels the JPMorgan TIPS 1-10 Year Index offered before March 2002. In today’s environment, TIPS’ low long-term nominal expected returns of approximately 3.0% pale in comparison to other inflation-fighting markets, which are offering both higher return prospects and superior inflation-hedging characteristics at this stage in the cycle. Research suggests that broad exposure across several inflation-fighting asset classes via a contrarian multi-asset approach can mitigate the line-item risk of investing in a single inflation-protection asset class. The benefits of diversification are well established (e.g., French and Poterba, 1991; Binstock, Kose, and Mazzoleni, 2017; and Viciera, Wang, and Zhou, 2017), and investors exploiting this “free lunch” and the “buy high, sell low” tendency of their peers through their inflation-protection solution can benefit from consistently higher returns at more tolerable levels of volatility. Such an approach can serve as powerful insurance for retirement portfolios, protecting future retirees in the very conditions that render mainstream assets vulnerable—namely, inflation surprises, market turbulence, and/or bear markets. ## Is Opportunity Knocking Now? The best time to protect our retirement portfolios is when the insurance against the looming possibilities of inflationary waves, market downturns, or volatility jolts is cheap. Now seems an opportune time to increase our retirement portfolios’ protection against damage from unforeseen, but not entirely unexpected, turbulence in financial conditions. FEATURED TAGS Learn More About the Authors ## Endnotes 1. The Insurance Information Institute’s (III) Opinion Research Corporation International poll estimates that 95% of homeowners in the United States have homeowners’ insurance. More information on how the III arrived at the 95% figure is available here. 2. We attribute this metaphor to Irwin (2018). 3. Arnott (2012) provides more information about the potential flaws of this standard de-risking approach. 4. According to the Morningstar report by Holt (2017), these fund companies and their TDF assets as of year-end 2016 include Vanguard ($280 billion), Fidelity Investments ($193 billion), T. Rowe Price ($148 billion), American Funds ($54 billion), JPMorgan ($45 billion), TIAA-CREF Asset Management ($31 billion), Principal Funds ($26 billion), American Century Investments ($17 billion), John Hancock ($16 billion), and BlackRock (\$12 billion).

5. Based on a holdings analysis in FactSet as of May 31, 2018, 91% of the iShares MSCI ACWI Index is in developed equities (54% in US equities and 37% in developed ex US equities).

6. This observation is not unique to the 2030 vintage. Across all vintages, TDFs are heavily allocated to developed-market stock and bond markets, exhibiting a large home-country bias.

7. The pass-through of rents is not 100% as some slippage from maintenance costs and so forth occurs.

8. From March 1997 to March 2018, the quarterly correlations of returns to US inflation were 50% for bank loans, 31% for EM currency, 28% for high-yield bonds, and 21% for EM equities. These levels exceed or are comparable to the correlation of US TIPS return to inflation, which is measured at 22% over the same span.

9. Inflation is measured as the 12-month percentage change in the US CPI City Average Seasonally-Adjusted Index, as provided by the US Bureau of Labor Statistics (BLS).

10. West and Ko (2017), the first of eight articles in a series by Research Affiliates focused on the needs of financial advisors, provides more details about the impact of using historical returns to forecast the future.

11. Pollock (2018) provides more information on some TDFs’ boosts in stock exposure.

12. Consistent with the analysis by Johnson (2012), inflation surprises reflect the difference between actual inflation at the end of the quarter and expectations for inflation at the start of the quarter, so that a “high (low) inflation surprise” is a quarter that falls in the top (bottom) third of all historical inflation surprises from June 1973 through March 2018.

13. This basket is an equally weighted mix of traditional and stealth inflation-fighting asset classes, which are added into the mix in the first month the returns are available. The asset classes, representative indices, and dates of available data are as follows: 1) High-yield bonds, IA Barclays US HY Corporate Bond Index (Jan 1973–Dec 1992) and BofA ML US High Yield, BB-B Rated, Constrained Index (Jan 1993 onward); 2) Commodities, S&P GSCI TR Index (Jan 1973–Dec 1990) and DJ UBS Commodity TR USD Index (Jan 1991 onward); 3) REITs, FTSE REIT All REITs TR Index (Jan 1973); 4) Bank loans, CSFB Leveraged Loan Index (Jan 1992); 5) EM equities, MSCI EM GR USD TR Index (Jan 1988); 6) EM currencies/local bonds, JPM ELMI Index (Jan 1994–Dec 1993) and JPM GBI EM Global Diversified TR USD (Jan 1994 onward); and 7) US TIPS, Barclays US Treasury US TIPS TR USD Index (Mar 1997–Jun 1998) and Barclays Capital US Treasury Inflation Notes: 10+ Year Index (Jul 1998 onward).

14. We acknowledge that rolling 12-month volatility exceeding 30% is extreme. Since 1973, such bouts of severe turbulence occurred slightly less than 5% of the time, notably in April–May 1975, October 1987–September 1988, and March–September 2009. When we relax the threshold and use 12-month volatility of 20%, we find that highly volatile markets occurred 15% of the time from January 1973 to March 2018. Over this period, the average 12-month rolling return is 1.5% for the S&P 500, 3.2% for a developed 60/40 portfolio, and 4.3% for the inflation-fighting basket. Therefore, at the 12% level, inflation-fighters typically outperformed conventional assets, although the latter did deliver positive, albeit average returns. In the section “When to Insure?” we use the 12% threshold to characterize calm versus volatile markets.

15. This calculation is based on rolling 12-month returns of the US stock market from 1802 through March 2018. The duration of a market rally is the total number of consecutive positive rolling return periods without a gap exceeding two rolling periods. For example, the length of the latest rally is 9.25 years or 99 12-month rolling outcomes (99+12)/12. Within this span, the longest time during which the rolling 12-month return was consecutively negative was two periods (the 12-month periods ending January 2016 and February 2016); that is, the largest gap is two, and therefore this period is captured within the latest rally episode. The sources of data used to compute the US stock market are Ibbotson/Morningstar, Robert Schiller’s Online Data, and William Schwert’s Indexes of US Stock Prices.

16. Aside from these excessively stretched conditions, we believe reversal risks are skewed to the upside for other reasons. First, the United States is emerging from a period of highly irregular monetary policy, which was designed to bring about market and macroeconomic stability and keep interest rates low (i.e., bond prices high). With the removal of such accommodative policy, market and macroeconomic volatility should naturally rise as the Fed “put” is stricken further out of the money. Second, the US Federal Reserve’s Federal Open Market Committee’s dual mandate of full employment and price stability skews inflation risks to the upside. From 2012 to mid-2018, the Fed’s preferred measure of inflation—the Personal Consumption Expenditures (PCE) inflation rate—has fallen below its long-term 2% target nearly 95% of the time (73 of 77 months). In light of this persistent undershooting and the recent emphasis on this target’s symmetrical nature, the Fed will likely tolerate modest overshooting of the PCE inflation rate even after having met the 2.0% PCE objective in June 2018. Third, the Fed isn’t the only entity seeking higher inflation. Because fiscal austerity measures are unpopular, governments are also incentivized to create higher inflation in an effort to reduce the real value of their debt burdens. In the United States, late-cycle fiscal stimulus and trade tariffs only increase inflationary upside along with growing debt and deficits. Finally, decomposing the components of the core inflation basket also points to a continued upside for inflation. US dollar depreciation in recent years may result in flattening (instead of falling) import prices, causing the core goods component of inflation to be less of a drag on realized inflation moving forward. On the core services front, a negative unemployment gap indicates higher upside potential to services prices as wage pressures loom amid an economy with an unemployment rate that, since early 2017, has fallen below NAIRU (the non-accelerating inflation rate of unemployment).

17. Shepherd (2017) provides more information on the volatility characteristics of commodities and REITs.

## References

Arnott, Robert. 2012. “The Glidepath Illusion.” Research Affiliates Fundamentals (September).

Arnott, Robert, Vitali Kalesnik, and Lillian Wu. 2017. “The Folly of Hiring Winners and Firing Losers.” Research Affiliates Publications (September).

Binstock, Jay, Engin Kose, and Michele Mazzoleni. 2017. “Diversification Strikes Again: Evidence from Global Equity Factors.” Research Affiliates Publications (October).

French, Kenneth, and James Poterba. 1991. “Investor Diversification and International Equity Markets.” American Economic Review, vol. 81, no. 2 (May):222–226.

Holt, Jeff. 2017. “2017 Target-Date Fund Landscape: Answers to Frequently Asked Questions.” Morningstar Manager Research (April 20).

———. 2018. “2018 Target-Date Fund Landscape: Sizing Up the Trillion-Dollar, Increasingly Passive Giant.” Morningstar Manager Research (May 7).

Hsu, Jason, Brett Myers, and Ryan Whitby. 2016. “Timing Poorly: A Guide to Generating Poor Returns While Investing in Successful Strategies.” Journal of Portfolio Management, vol. 42, no. 2 (Winter):90–98.

Irwin, Neil. 2018. “Why Markets Have Gotten So Jumpy.” The New York Times (February 23).

Johnson, Nicholas. 2012. “Inflation Regime Shifts: Implications for Asset Allocation.” PIMCO In Depth(October).

Kinnel, Russel. 2005. “Mind the Gap: How Good Funds Can Yield Bad Results.” Morningstar FundInvestor.

———. 2014. “Mind the Gap 2014.” MorningstarAdvisor (February 27).

———. 2015. “Mind the Gap 2015.” MorningstarAdvisor(August 11).

———. 2016. “Mind the Gap 2016.” MorningstarAdvisor (June 14).

Pollock, Michael. 2018. “Target-Date Funds Adjust for a Stock Pullback.” The Wall Street Journal (February 4).

Shepherd, Shane. 2017. “Risk: Preparing Clients for an Uncertain Journey.” Research Affiliates Publications (October).

Viciera, Luis, Zixuan Wang, and John Zhou. 2017. “Global Portfolio Diversification for Long-Horizon Investors.” Harvard Business School Finance Working Paper No. 17−085.

West, John, and Amie Ko. 2017. “The Most Dangerous (and Ubiquitous) Shortcut in Financial Planning.” Research Affiliates Publications (September).


text:
# How to: Force, work & dot product

1. Dec 3, 2005

### Lisa...

A force in the xy plane is given by: F= - (b/r3) (x i^ + y j^) where b is a positive constant and r= sqrt(x2 + y2)

a) Show that the maginitude of the force varies as the inverse of the square of the distance to the origin and that its direction is antiparallel (opposite) to the radius vector r= x i^ + y j^).

Could anybody please verify my work?:

The following needs to be true:
F ~ 1/r2 => F= c /r2, where c is a constant.

F * F= F2 = (b2 x2/r6) + (b2 y2/r6)

F= sqrt((b2 x2/r6) + (b2 y2/r6) )= sqrt((b2/r6) (x2 + y2)) = b/r3 sqrt(x2 + y2)

Because r= sqrt(x2 + y2) substitution gives:

b/(sqrt(x2 + y2))3 sqrt(x2 + y2)= b/ (x2 + y2).

r2= (x2 + y2) so F= c /r2 is true where the constant c= the constant b in this case.
----------------------------------------------------------------------
F= - (b/r3) (x i^ + y j^) needs to be antiparallel to r= x i^ + y j^).

If two vectors are parallel the following needs to be true: AB (dotproduct)= AB (length vectors).

The dotproduct AB gives: -bx2/r3 + -by2/r3
Substitution of r= sqrt(x2 + y2) and adding the two terms gives:

(-bx2 -by2)/(sqrt(x2 + y2))3= (-b (x2 + y2))/(sqrt(x2 + y2))3= -b/(x2 + y2)

The length of F= b/(x2 + y2), calculated in the first part of the question.

The length of r= r * r= r2= x2+ y 2
r= sqrt(x2+ y 2)

Fr (length of the vectors) gives: b (sqrt(x2+ y 2)/(x2 + y2) = b/ (sqrt(x2 + y2))

Therefore it is true that Fr (dotproduct)= Fr (length vectors):
-b/(x2 + y2)= b/ (sqrt(x2 + y2)) with the only restriction that Fr (dotproduct) is negative and therefore F and r are antiparallel.

Ok till now I get the problem, but I'm not sure how to solve b, c and d... so please help me ...

b) If b= 3 Nm2 find the work done by this force on a particle moving along a straight line path between an initial position x= 2 m, y= 0 m and a final position x= 5 m, y= 0 m.

I know W= Fs and s= 3 m but I don't know how to calculate the force with - (3/r3) (x i^ + y j^)...

c) Find the work done by this force on a particle moving once around a circle of radius r= 7 mcentered around the origin.

d) If this force is the only force acting on the particle what's the particle's speed as it moves along this circular path? Assume that the particle's mass is m= 2 kg.

2. Dec 3, 2005

### Galileo

Correct so far (didn't check the rest yet). At this stage I recommend looking back at the solution and try to see the 'big picture' of what you have done. Perhaps you will be able to see the solution in a glance or see how you could have gotten there much quicker.
What you did was subsitute r^2=x^2+y^2 back and forth again and again. When you reached the point F= b/r^3 sqrt(x2 + y2) you used $r=\sqrt{x^2+y^2}$, but subsituted r in favor of x and y (!) instead of the other way around and later you 'undid' that. Writing $\sqrt{x^2+y^2}=r$ in $F= \frac{b}{r^3}\sqrt{x^2 + y^2}$ gives $F=b/r^2$ right away.
The way to know what way to go is partly experience, but in this case you want the answer to be in terms of r, so try to eliminate x and y right away.

If you know vector notation the question becomes much easier and the solution more insightful, since $\vec r=x \hat i+y\hat j$ we have:

$$\vec F= - \frac{b}{r^3}(x \hat i + y \hat j) =-b \frac{\vec r}{r^3}=-b \frac{\hat r}{r^2}$$

Where $\hat r=\vec r/r$ is the unit vector in the direction of $\vec r$. In this case the answers are immediate.

Last edited: Dec 3, 2005
3. Dec 6, 2005

### Lisa...

Using your formula for F I've found:

b) The work is found by integrating your formula for F with respect to r between 2 and 5. W25= 3/5 - 3/2 = -0.9 J

c) Because the force is perpendicular to the path the dotproduct will equal zero. Therefore the work is zero when the particle is doing circular motion.

d) F= -b r^/r2 with r= 7 m and b= 3 N/m.
F= -3/49= -0.0612244898 N= Fres

Fres= m a= mac= m v2/r
v2= Fres*r/m= -0.06122* 7/2= -0.2142857
v= sqrt(0.2142857)= -0.46291 m/s

Is this correct?

4. Dec 6, 2005

### Galileo

For d), remember that the F you should consider in the equation F=mv^2/r is the magnitude of $\vec F$, which is always positive. You should get a warning from the fact that the v^2 you calculated was negative.

The units of b are Nm^2, not N/m. But apart from these things it looks good. Well done.

5. Dec 7, 2005

### Lisa...

Ah of course :) But then it's simple: I only need to make the negative signs positive and I'll get the right answer, correct? So the answer would be 0.46291 m/s and not - 0.46291 m/s?

6. Dec 7, 2005

### Galileo

Yes, that's right