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# The ratio 2 to 1/3 is equal to the ratio

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The ratio 2 to 1/3 is equal to the ratio [#permalink]

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20 Aug 2012, 03:03
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The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

Practice Questions
Question: 22
Page: 155
Difficulty: 550
[Reveal] Spoiler: OA

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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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20 Aug 2012, 03:04
SOLUTION

The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

$$\frac{2}{(\frac{1}{3})}=2*\frac{3}{1}=\frac{6}{1}$$.

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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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20 Aug 2012, 10:00
2
KUDOS
Solved by back substitution

2 of the answer choices have a "1" for the second part of the ratio. Only way this is possible is if both parts are multiplied by 3.

Thus 6 : 1

[Reveal] Spoiler:
A

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Math Expert
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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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24 Aug 2012, 02:17
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Expert's post
SOLUTION

The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

$$\frac{2}{(\frac{1}{3})}=2*\frac{3}{1}=\frac{6}{1}$$.

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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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29 Jun 2014, 06:15
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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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10 Sep 2014, 00:05
Bunuel wrote:
The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

Practice Questions
Question: 22
Page: 155
Difficulty: 550

2:(1/3)

multiply 3 both side -=== 6:1

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02 Feb 2016, 07:39
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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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24 May 2016, 10:48
Bunuel wrote:
The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

When setting up ratios, we can set up a ratio in fraction form. For example, if we were told the ratio of a to b is 1 to 2, we could say: a/b = ½.

Since we are given the ratio of 2 to 1/3, we can write this as:

2/(1/3)

We simplify this complex fraction by multiplying both the numerator and denominator by 3, obtaining:

(2 x 3)/1 = 6/1

Thus, the equivalent ratio is 6 to 1.

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The ratio 2 to 1/3 is equal to the ratio [#permalink]

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12 Jun 2016, 04:30
Toughest Question Ever

2 : 1/3 = 6:1
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Last edited by rishi02 on 12 Jun 2016, 04:38, edited 1 time in total.

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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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12 Jun 2016, 04:34
1
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Bunuel wrote:
The ratio 2 to 1/3 is equal to the ratio

(A) 6 to 1
(B) 5 to 1
(e) 3 to 2
(D) 2 to 3
(E) 1 to 6

Practice Questions
Question: 22
Page: 155
Difficulty: 550

$$2$$ : $$\frac{1}{3}$$

Multiply both sides by 3

$$2*3$$ : $$\frac{1}{3}*3$$

=> 6 : 1

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27 Jun 2017, 04:26
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Re: The ratio 2 to 1/3 is equal to the ratio [#permalink]

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27 Jun 2017, 14:10
The ratio 2 to 1/3 is equal to the ratio

$$\frac{2}{(\frac{1}{3})}=2*\frac{3}{1}=\frac{6}{1}$$

Or Ratio of 6 to 1

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Re: The ratio 2 to 1/3 is equal to the ratio   [#permalink] 27 Jun 2017, 14:10
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# Math Help - Rare events

1. ## Rare events

The ethnicity of about 93.5 percent of the population of Gemany is Caucasian. Suppose you will take a random sample of 473 Germans.

What proportions of Caucasians would be rare events?

how can i solve this one?

i tried msum+1.96*sigma SD, but its all wrong

2. Originally Posted by makaveli89
The ethnicity of about 93.5 percent of the population of Gemany is Caucasian. Suppose you will take a random sample of 473 Germans.

What proportions of Caucasians would be rare events?

how can i solve this one?

i tried msum+1.96*sigma SD, but its all wrong
What do you mean by "rare event"? If it is "happening with probably less that 5%", as I guess, then it is a consequence of the central limit theorem and of usual independence hypotheses that a rare event would be a proportion that differs from the mean ( $0.935$) by more than around $\frac{1.96 \sigma}{\sqrt{n}}$ where $n=473$ and $\sigma=\sqrt{0.935\times (1-0.935)}\simeq 0.247$, that is to say a proportion of Caucasians larger than about 0.958 or less than about 0.912. This correspond to about $442\pm 11$ people among 473.

I'm really surprised that this question is being asked in Germany because I could not imagine it happening in France, where some people would find this politically incorrect to distinguish people according to their "ethnicity". By the way, the law here forbids to do polls on what you call "ethnicity", or origin, and for that reason no official proportion can be given.

format: plain


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# Why is this MOSFET showing this behaviour?

I have solar charge controller PCB in which I am using MOSFET IRF9540. I am using ATmega328P as microcontroller to read the voltages of battery and solar and to enable/disable the charging process. Below is the schematic:

Now in the above schematic, I am giving solar voltage at X1 connector. R1 and R2 are used as a voltage divider and its output is going to an analog pin of the microcontroller. CHARGEPIN is a GPIO of the microcontroller. So when this GPIO is high, it turns on the transistor, which then turn on the MOSFET and solar is then connected to the battery to start the charging process. When the GPIO is low, solar and battery are disconnected. I am also using a serial port to monitor the voltages.

Now what's happening is when I have not connected the solar pins and only connected the battery pins. At that time, charging is off (because we do not have a solar voltage), so CHARGEPIN is low thus, the MOSFET will be OFF. So solar and battery should be disconnected. But somehow battery voltage is going to solar voltage because when I read the voltage at the MBR2045 diode, it shows the battery voltage, which should not show because the MOSFET is OFF. Why is the MOSFET acting like this?

• Do measurements of the voltages of all terminals of the FET, then you will see why it does what it does. – PlasmaHH Feb 9 '17 at 12:41
• @PlasmaHH At drain, it is 12.32 and at gate and source it is 11.7. Can you explain more. Thanks – S Andrew Feb 9 '17 at 12:43
• Hm, I just saw you are using a PMOS... Anyways, have you considered the effect of the body diode? – PlasmaHH Feb 9 '17 at 12:47
• Look up "body diode". – WhatRoughBeast Feb 9 '17 at 12:47

MOSFETs have an internal diode that conducts in the reverse direction thus you see battery voltage on the diode node: -

Note the diode above. There is usually a diode junction from both source and drain to the substrate in both P and N channel MOSFETs. Here's the P channel MOSFET: -

What usually happens is that the manufacturer connects the "body" terminal to the source terminal so that one of the diodes can never be forward biased and this leaves the other diode able to be forward biased only when drain rises higher than source (P channel MOSFET).

• This is why I insist that when appropriate, MOSFET symbols get drawn with the body diode present! It's an immediate reminder / clue into behavior of your circuit. – Krunal Desai Feb 9 '17 at 18:59

The P-channel power MOSFET has a body diode that points from drain to source.

The diode is actually the PN junction between the drain contact and the substrate. The substrate needs to be held at a higher voltage than all of the other terminals in order to keep those PN junctions from conducting. In a MOS IC the P-channel substrate is normally connected directly to Vdd. However, in nearly all discrete MOSFETs, the substrate is not brought out as a separate terminal; instead, it is bonded directly to the source terminal, which is expected to be the most positive terminal of the device.

If you want bidirectional isolation, you'll have to use two MOSFETs in a back-to-back arrangement. Driving the gates can be tricky; things are simplified somewhat if you connect the sources together — then the gates share a common reference and can also be connected together.


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# Tensorflow ROCM Performance for Probabilistic Programming

This post investigates the Tensorflow ROCM (AMD) library for econometric problems. We perform various comparisons of operations that might be used for econometric applications and find a mixed bag for ROCM compared to Nvidia Cuda in terms of execution times (at the time of this writing March 2020). In some cases Tensorflow ROCM performs brilliantly but in others not so much. Since I have found no good benchmarking for these types of tensorflow operations on the internet, I hope this post will be useful.

Nvidia Cuda has a near monopoly for scientific computing, machine learning, and other compute heavy applications. The importance of GPU processing for scientific workloads can't be overstated as speedups can be dramatic to the point of making methodologies viable. With Cuda you are getting a development stack that is mature, can be applied to a multitude of problems, and that just works (once the drivers are installed). Above all it is very fast. The downside to Cuda is twofold in my opinion:

1. All of the software stack from drivers to compute libraries is closed source and proprietary
2. The hardware is very expensive particularly if installing an Nvidia card in a data center (where consumer-grade cards aren't allowed due to licensing restrictions)

Practically speaking, the first limitation can lead to a software stack that is unreliable (read: computer won't boot to gui or the Cuda stack could break) on Kernel upgrades and it can be quite difficult to install the Cuda framework depending on the linux distribution you choose. The second limitation means that hardware can be exhorbinantly expensive. But the upside is a robust and fast development environment for running models.

I was extremely excited to hear about the Radeon Open Compute (ROCM) initiative and AMD drivers in linux. The primary benefit (as of March 2020) is that you can use the AMDGPU driver which is an open source in-tree kernel library for linux (so it is very easy to install and maintain). Additionally, the scientific library is open source. Finally, in 2019 AMD released the Radeon VII with 16GB of GPU memory (which is nearly identical to AMD's data center card, the MI50) and for what you get is very inexpensive ($699 at the time of this writing). I read a bunch of compute benchmarks on reddit, the ROCM Github Issues page, and other sources (and this one). All of these benchmarks show that the Radeon VII on ROCM comes very close to the Nvidia 2080TI Nvidia flagship comsumer card for Tensorflow machine learning applications like resnet, graphics rendering, and gaming. Unfortunately, my applications don't match any of these benchmarks. My work involves heavy usage of tensorflow operations, high memory usage, and the use of the tensorflow_probability library and not image processing. So I bought a Radeon VII and installed it in my machine along with the ROCM stack. The card installed easily, runs without any driver installation required, and is not loud or hot. Since I am using Tensorflow for some models I am running right now, I immediately wanted to try my Cuda working code (on an Nvidia P10016GB or V10016GB Nvidia GPU at the William and Mary Sciclone cluster) on the ROCM tensorflow stack (I term this ROCM from here on) on the Radeon VII installed on my personal workstation. I was able to run code and here is a quick summary of results: • Once I disabled XLA support in my code, everything just worked. The code ran and gave me the same results as if I was on Cuda or the CPU. But disabling XLA can mean compute times are longer. • For basic tensorflow operations, the ROCM tensorflow stack is very mature and comparable to Cuda (with a few exceptions). For these tensorflow functions, usually the Radeon VII on ROCM was every bit as fast (and sometimes faster) as an Nvidia V100 16GB. Since that card costs as much as$8,000 that is impressive.
• For other functions (like tensorflow-probability.mcmc), the performance hit from having to disable XLA can be so large that even CPU with XLA is faster than ROCM on GPU. That is a big disappointment. I've opened an issue at the ROCM tensorflow github site. Hopefully it will be resolved soon. I will be updating when the issues are addressed.

Hardware details and full disclosure: the times presented in this post are not apples to apples comparisons since the Radeon VII card is not installed on the cluster but rather on my own workstation. My workstation has fewer cores (18 vs 64) but a faster processor (Xeon W2155 @3.30GHz vs E52683v4 @2.10 Ghz). Memory is roughly comparable. For these reasons, I recommend IGNORING the CPU comparisons between Rocm and Cuda, except to note how important XLA compilation is for some problems. All ROCM tests are run on Ubuntu 18.04 with the HWE kernel 5.3, whereas Cuda tests are on Redhat 7 with a 3.10 kernel. All cuda tests are run by decorating functions with @tf.function(experimental_compile=True) where possible.

## Tensorflow Operations

Code for this problem can be found here. I have picked a few operations to compare in this test, some are linear algebra operations, some are for reshaping/copying data, and others are reduce or simple mathematical operations. This list isn't meant to be definitive in comparing Cuda and Rocm, but rather are functions that are or might be important to my workflow. In the table below, all times are in milliseconds.

CPU-Cuda GPU-Cuda CPU-ROCM GPU-ROCM
tf.linalg.inv() (ms) 124.49 9.80 88.86 88.23
tf.linalg.matmul() (ms) 2.77 2.02 3.62 1.68
tf.gather_nd() (ms) 2.002 3.01 5.34 4.05
tf.scatter_nd() (ms) 278.75 7.12 161.98 5.56
tf.gather_nd() (ms) 0.54 0.73 2.89 1.19
tf.math.bincount() (ms) 5.19 2.19 4.29 2.98
tf.multiply() (ms) 9.89 0.03 26.39 0.01
tf.reduce_sum() (ms) 4.89 0.06 5.65 0.02
tf.add() (ms) 0.27 0.64 0.43 0.83

Some things stand out in these results:

1. The ROCM GPU stack is always faster than base tensorflow (labeled CPU-Cuda) for every operation considered in the table.
2. Operations using ROCM GPU is faster than Cuda GPU operations in 4 of the 9 tests considered here. When ROCM is faster differences are usually small (less than 25%).
3. In 4 of the 5 occasions where the ROCM GPU operation is slower, differences are usually small (less than 25%).
4. In one instance (matrix inversion), ROCM appears to not implement this on the CPU and it is 9 times slower than its Cuda counterpart.

Apart from the first row in the preceding table (which is a big slow-down), we see that ROCM is a viable alternative to CUDA if your work only requires tensorflow operations. This is fantastic and speaks to the advances made by the ROCM community. But caution is warranted. If your function isn't implemented yet on the GPU, you may not get enough speedup from ROCM to justify switching from CUDA.

## A Toy Problem

Code for this problem can be found here. This problem differs from the preceding test as follows:

1. Tensorflow functions are created from basic tensorflow operations.
2. These functions are timed on their own
3. These functions are used as inputs to tensorflow_probability calls like mcmc (for the NUTS sampling rows) and optimize (for the Maximum Likelihood Estimation rows)
CPU-Cuda GPU-Cuda CPU-ROCM GPU-ROCM
Function and Gradient (ms) 36.40 37.50 34.70 27.60
Maximum Likelihood Estimation (s) 9.63 8.04 4.26 8.93
NUTS Sampling (s) 108.92 1.58 99.50 47.29

The results show that ROCM GPU is actually signficantly faster than GPU-Cuda (by around 30%) for the function and gradient calculations (recall these are pure tensorflow operations)- that's fantastic! When we estimate using maximum likelihood using tensorflow_probability, we see that on GPU, ROCM holds its own compared to CUDA (although the ROCM CPU is the fastest probably due to the superior processor hardware on my workstation). Comparing CPU-ROCM to GPU-ROCM we see that the GPU actually slows things down- that's not fantastic. For NUTS monte-carlo markov chain sampling, we see that the ROCM stack is MUCH slower than CUDA on the GPU (30x) slower.

## A "Real" Problem

I'm not going to share the code for this one since it is ongoing research. Suffice it to say that the model log-likelihood uses many of the tensorflow operations we explored above (not tf.linalg.inv) and this tensorflow function is used with the tensorflow_probability.mcmc and tensorflow_probability.optimize methods for large tensorflow matrices. Given our findings above, we would expect 1) Log=likelihood evaluations in ROCM to be similar to Cuda (since the function is comprised of tensor operations), and 2) tensorflow_probability calls to be slower than Cuda counterparts.

Model CPU-Cuda GPU-Cuda CPU-ROCM GPU-ROCM
Function and Gradient (ms) 1 0.40 0.50 2.53 1.98
Maximum Likelihood Estimation (ms) 1 110.00 10.00 122.99 138.41
Nuts Sampling (s) 1 36.57 8.12 15.46 31.72
Function and Gradient (ms) 2 30.00 10.00 30.37 12.58
Nuts Sampling (s) 2 36.57 8.12 1546.75 637.78

The real-world example supports what we would expect: the function and gradient calculations are somewhat slower (4x for model 1 and 1.3x in model 2) as these use gather operations on very large matrices, which is where CUDA excels relative to ROCM. These aren't bad results for ROCM particularly for Model 2, which is the most compute intensive of any calculation reported here so far. But when we use either of the tensorflow_probability functions we see severe performance degradation relative to CUDA. Furthermore, we see that the CPU on base tensorflow is faster by some margin (around 20x) than even ROCM GPU. The CUDA GPU stack is orders of magnitude faster than ROCM for sampling. That is very bad news for ROCM for this problem.

## Conclusion

ROCM is being actively developed and I expect these issues will be addressed relatively soon. It seems that tensorflow_probability has not been optimized for ROCM yet and futhermore, ROCM isn't currently optimized for XLA compilation in some instances. Taken together this means that for many "custom" tensorflow applications results will vary and the benchmarks you commonly see on the internet (that I cite in the introduction) are not a reliable indicator for how ROCM will perform for your problem.


text:
# 7.1: Linear Transformations and Matrices

Ordered, finite-dimensional, bases for vector spaces allows us to express linear operators as matrices.

## 7.1.1 Basis Notation

A basis allows us to efficiently label arbitrary vectors in terms of column vectors. Here is an example.

Example 66

Let $$V=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\middle|a,b,c,d\in \mathbb{R}\right\}$$be the vector space of $$2\times 2$$ real matrices, with addition and scalar multiplication defined componentwise. One choice of basis is the ordered set (or list) of matrices

$$B=\left(\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\0&0\end{pmatrix},\begin{pmatrix}0&0\\ 1&0\end{pmatrix},\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\right)=:(e_{1}^{1},e_{2}^{1},e^{2}_{1},e^{2}_{2})\, .\] Given a particular vector and a basis, your job is to write that vector as a sum of multiples of basis elements. Here and arbitrary vector $$v\in V$$ is just a matrix, so we write \begin{eqnarray*} v\ =\ \begin{pmatrix}a&b\\c&d\end{pmatrix}&=&\quad\!\! \begin{pmatrix}a&0\\0&0\end{pmatrix}+\begin{pmatrix}0&b\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\ c&0\end{pmatrix}+\begin{pmatrix}0&0\\0&d\end{pmatrix}\\ &=&a\begin{pmatrix}1&0\\0&0\end{pmatrix}+b\begin{pmatrix}0&1\\0&0\end{pmatrix}+c\begin{pmatrix}0&0\\ 1&0\end{pmatrix} +d\begin{pmatrix}0&0\\0&1\end{pmatrix}\\ &=&a\, e^{1}_{1}+b \, e^{1}_{2}+c \, e^{2}_{1}+d \, e^{2}_{2}\, . \end{eqnarray*} The coefficients $$(a,b,c,d)$$ of the basis vectors $$(e^{1}_{1},e^{1}_{2},e^{2}_{1},e^{2}_{2})$$ encode the information of which matrix the vector $$v$$ is. We store them in column vector by writing$$v= ae_{1}^{1} + be_{2}^{1} + ce_{1}^{2} + de_{2}^{2} =: (e_{1}^{1}, e_{2}^{1}, e_{1}^{2}, e_{2}^{2}) \begin{pmatrix}a\\b\\c\\d\end{pmatrix} =: \begin{pmatrix}a\\b\\c\\d\end{pmatrix}_{B}\]

The column vector $$\begin{pmatrix}a\\b\\c\\d\end{pmatrix}$$ encodes the vector $$v$$ but is NOT equal to it! (After all, $$v$$ is a matrix so could not equal a column vector.) Both notations on the right hand side of the above equation really stand for the vector obtained by multiplying the coefficients stored in the column vector by the corresponding basis element and then summing over them.

Next, lets consider a tautological example showing how to label column vectors in terms of column vectors:

Example 67 (Standard Basis of $$\Re^{2}$$)

The vectors

$e_{1}=\begin{pmatrix}1\\0\end{pmatrix},~~~ e_{2}=\begin{pmatrix}0\\1\end{pmatrix}$

are called the standard basis vectors of $$\Re^{2}=\Re^{\{1,2\}}$$. Their description as functions of $$\{1,2\}$$ are

$e_{1}(k) = \left\{\begin{matrix}1,~~ if ~k = 1\\ 0,~~ if ~k = 2\end{matrix}\right.$

$e_{2}(k) = \left\{\begin{matrix}0,~~ if~ k = 1\\ 1,~~ if ~k = 2\end{matrix}\right.$

It is natural to assign these the order: $$e_{1}$$ is first and $$e_{2}$$ is second. An arbitrary vector $$v$$ of $$\Re^{2}$$ can be written as

$v=\begin{pmatrix}x\\y\end{pmatrix}=x e_{1}+ ye_{2} .$

To emphasize that we are using the standard basis we define the list (or ordered set)

$E=(e_{1},e_{2})\, ,$

and write

$\begin{pmatrix}x\\y\end{pmatrix}_{E}:=(e_{1},e_{2})\begin{pmatrix}x\\y\end{pmatrix}:=x e_{1}+ ye_{2}=v .$

You should read this equation by saying:

"The column vector of the vector $$v$$ in the basis $$E$$ is $$\begin{pmatrix}x\\y\end{pmatrix}$$.''

Again, the first notation of a column vector with a subscript $$E$$ refers to the vector obtained by multiplying each basis vector
by the corresponding scalar listed in the column and then summing these, $$\textit {i.e.}$$ $$xe_{1}+y e_{2}$$. The second notation denotes exactly the same thing but we first list the basis elements and then the column vector; a useful trick because this can be read in the same way as matrix multiplication of a row vector times a column vector--except that the entries of the row vector are themselves vectors!

You should already try to write down the standard basis vectors for $$\Re^{n}$$ for other values of $$n$$ and express an arbitrary vector in $$\Re^{n}$$ in terms of them.

The last example probably seems pedantic because column vectors are already just ordered lists of numbers and the basis notation has simply allowed us to "re-express'' these as lists of numbers. Of course, this objection does not apply to more complicated vector spaces like our first matrix example. Moreover, as we saw earlier, there are infinitely many other pairs of vectors in $$\Re^{2}$$ that form a basis.

Example 68 (A Non-Standard Basis of $$\Re^{2}=\Re^{\{1,2\}}$$)

$b=\begin{pmatrix}1\\1\end{pmatrix}\, ,\quad \beta=\begin{pmatrix}1\\-1\end{pmatrix}.$

As functions of $$\{1,2\}$$ they read

$b(k) = \left\{\begin{matrix}1,~~ if ~k = 1\\ 1,~~ if ~k = 2\end{matrix}\right.$

$\beta(k) = \left\{\begin{matrix}1,~~ if~ k = 1\\ -1,~~ if ~k = 2\end{matrix}\right.$

Notice something important: there is no reason to say that $$\beta$$ comes before $$b$$ or vice versa. That is, there is no $$\textit{a priori}$$ reason to give these basis elements one order or the other. However, it will be necessary to give the basis elements an order if we want to use them to encode other vectors. We choose one arbitrarily; let

$B=(b,\beta)$

be the ordered basis. Note that for an unordered set we use the $$\{\}$$ parentheses while for lists or ordered sets we use $$()$$.

As before we define

$\begin{pmatrix}x\\y\end{pmatrix}_{B} :=(b,\beta)\begin{pmatrix}x \\ y\end{pmatrix}:= xb+ y \beta\, .$

You might think that the numbers $$x$$ and $$y$$ denote exactly the same vector as in the previous example. However, they do not. Inserting the actual vectors that $$b$$ and $$\beta$$ represent we have

$xb+ y \beta=x\begin{pmatrix}1\\1\end{pmatrix}+y\begin{pmatrix}1\\-1\end{pmatrix}=\begin{pmatrix}x+y\\x-y\end{pmatrix}\, .$

Thus, to contrast, we have

$\begin{pmatrix}x\\y\end{pmatrix}_{B}=\begin{pmatrix}x+y\\x-y\end{pmatrix} \mbox{ and } \begin{pmatrix}x\\y\end{pmatrix}_{E}=\begin{pmatrix}x\\y\end{pmatrix}$

Only in the standard basis $$E$$ does the column vector of $$v$$ agree with the column vector that $$v$$ actually is!

Based on the above example, you might think that our aim would be to find the "standard basis'' for any problem. In fact, this is far from the truth. Notice, for example that the vector

$v=\begin{pmatrix}1\\1\end{pmatrix}=e_{1}+e_{2}=b$

written in the standard basis $$E$$ is just

$v=\begin{pmatrix}1\\1\end{pmatrix}_{E}\, ,$

which was easy to calculate. But in the basis $$B$$ we find

$v=\begin{pmatrix}1\\0\end{pmatrix}_{B}\, ,$

which is actually a simpler column vector! The fact that there are many bases for any given vector space allows us to choose a basis in which our computation is easiest. In any case, the standard basis only makes sense for $$\Re^{n}$$. Suppose your vector space was the set of solutions to a differential equation--what would a standard basis then be?

Example 69: A Basis For a Hyperplane

Lets again consider the hyperplane

$V=\left\{ c_{1}\begin{pmatrix}1\\1\\0\end{pmatrix} +c_{2}\begin{pmatrix}0\\1\\1\end{pmatrix} \middle| c_{1},c_{2}\in \Re \right\}$

One possible choice of ordered basis is

$b_{1}=\begin{pmatrix}1\\1\\0\end{pmatrix},~~~b_{2}=\begin{pmatrix}0\\1\\1\end{pmatrix}, ~~~ B=(b_{1},b_{2}).$

With this choice

$\begin{pmatrix}x\\y\end{pmatrix}_{B} := xb_{1}+ y b_{2} =x\begin{pmatrix}1\\1\\0\end{pmatrix}+y \begin{pmatrix}0\\1\\1\end{pmatrix}=\begin{pmatrix}x\\x+y\\y\end{pmatrix}_{E}.$

With the other choice of order $$B'=(b_{2}, b_{1})$$

$\begin{pmatrix}x\\y\end{pmatrix}_{\!B'} := xb_{2}+ y b_{2} =x\begin{pmatrix}0\\1\\1\end{pmatrix}+y \begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}y\\x+y\\x\end{pmatrix}_{E}.$

We see that the order of basis elements matters.

Finding the column vector of a given vector in a given basis usually amounts to a linear systems problem:

Example 70: Pauli Matrices

Let

$V=\left\{\begin{pmatrix}z&u\\v&-z\end{pmatrix}\middle| z,u,v\in\mathbb{C}\right\}$

be the vector space of trace-free complex-valued matrices (over $$\mathbb{C}$$) with basis $$B=(\sigma_{x},\sigma_{y},\sigma_{z})$$, where

$\sigma_{x}=\begin{pmatrix}0&1\\1&0\end{pmatrix}\, ,\quad \sigma_{y}=\begin{pmatrix}0&-i\\i&0\end{pmatrix}\, ,\quad \sigma_{z}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\, .$

These three matrices are the famous $$\textit{Pauli matrices}$$, they are used to describe electrons in quantum theory.

Let
$$v=\begin{pmatrix} -2+i&1+i\\3-i&-2-i \end{pmatrix}\, . \] Find the column vector of $$v$$ in the basis $$B$$. For this we must solve the equation $\begin{pmatrix} -2+i&1+i\\3-i&-2-i \end{pmatrix} =\alpha^{x} \begin{pmatrix}0&1\\1&0\end{pmatrix}+\alpha^{y} \begin{pmatrix}0&-i\\i&0\end{pmatrix}+\alpha^{z} \begin{pmatrix}1&0\\0&-1\end{pmatrix}\, .$ This gives three equations, $$\textit{i.e.}$$ a linear systems problem, for the $$\alpha$$'s $\left\{ \begin{array}{rrrrr} \alpha^{x}&\!\!-\ i\alpha^{y}&&=&1+i\\ \alpha^{x}&\!\!+\ i\alpha^{y}&&=&3-i\\ &&\alpha^{z}&=&-2+i \end{array} \right.$ with solution $\alpha^{x}=2\, ,\quad \alpha^{y}=2-2i\, ,\quad \alpha^{z}=-2+i\, .$ Hence $v=\begin{pmatrix}2\\\ 2-2i\\\!\!-2+i\end{pmatrix}_{B}\, .$ To summarize, the $$\textit{column vector of a vector}$$ $$v$$ in an ordered basis $$B=(b_{1},b_{2},\ldots,b_{n})$$, $\begin{pmatrix}\alpha^{1}\\\alpha^{2}\\\vdots\\\alpha^{n}\end{pmatrix}\, ,$ is defined by solving the linear systems problem $v=\alpha^{1} b_{1} + \alpha^{2}b_{2} +\cdots + \alpha^{n} b_{n} = \sum_{i=1}^{n} \alpha^{i} b_{i}\, . The numbers $$(\alpha^{1},\alpha^{2},\ldots,\alpha^{n})$$ are called the $$\textit{components of the vector}$$ $$v$$. Two useful shorthand notations for this are \[v=\begin{pmatrix}\alpha^{1}\\\alpha^{2}\\\vdots\\\alpha^{n}\end{pmatrix}_{B} = (b_{1},b_{2},\ldots,b_{n})\begin{pmatrix}\alpha^{1}\\\alpha^{2}\\\vdots\\\alpha^{n}\end{pmatrix}\, .$ ## 7.1.2 From Linear Operators to Matrices Chapter 6 showed that linear functions are very special kinds of functions; they are fully specified by their values on any basis for their domain. A matrix records how a linear operator maps an element of the basis to a sum of multiples in the target space basis. More carefully, if $$L$$ is a linear operator from $$V$$ to $$W$$ then the matrix for $$L$$ in the ordered bases $$B=(b_{1},b_{2},\cdots)$$ for $$V$$ and $$B'=(\beta_{1},\beta_{2},\cdots)$$ for $$W$$ is the array of numbers $$m_{i}^{j}$$ specified by$$L(b_{i})= m_{i}^{1}\beta_{1}^{\phantom{1}}+\dots +m_{i}^{j} \beta_{j}^{\phantom{1}}+\cdots\]

Remark

To calculate the matrix of a linear transformation you must compute what the linear transformation does to every input basis vector and then write the answers in terms of the output basis vectors:

\begin{equation*}
\begin{split}
\big((L(b_{1})&,L(b_{2}),\ldots,L(b_{j}),\ldots\big)\\
&
={
\Big((\beta_{1},\beta_{2},\ldots,\beta_{j},\ldots)\begin{pmatrix}m^{1}_{1}\\m^{2}_{2}\\\vdots\\m^{j}_{1}\\\vdots\end{pmatrix},
(\beta_{1},\beta_{2},\ldots,\beta_{j},\ldots)\begin{pmatrix}m^{1}_{2}\\m^{2}_{2}\\\vdots\\m^{j}_{2}\\\vdots\end{pmatrix},\cdots,
(\beta_{1},\beta_{2},\ldots,\beta_{j},\ldots)\begin{pmatrix}m^{1}_{i}\\m^{2}_{i}\\\vdots\\m^{j}_{i}\\\vdots\end{pmatrix},\cdots\Big)}\\&
=(\beta_{1},\beta_{2},\ldots,\beta_{j},\ldots)\begin{pmatrix}
m_{1}^{1} & m_{2}^{1} & \cdots & m_{i}^{1} &\cdots\\
m_{1}^{2} & m_{2}^{2} & \cdots & m_{i}^{2} &\cdots\\
\vdots& \vdots & & \vdots &\\
m_{1}^{j} & m_{2}^{j} & \cdots & m_{i}^{j}& \cdots \\
\vdots&\vdots&&\vdots
\end{pmatrix}
\end{split}
\end{equation*}

Example 71

Consider $$L:V\to \Re^{3}$$ defined by
$$L\begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}0\\1\\0\end{pmatrix}\, ,\quad L\begin{pmatrix}0\\1\\1\end{pmatrix} = \begin{pmatrix}0\\1\\0\end{pmatrix}\, .$$
By linearity this specifies the action of $$L$$ on any vector from $$V$$ as
$$L\left[ c_{1}\begin{pmatrix}1\\1\\0\end{pmatrix} + c_{2} \begin{pmatrix}0\\1\\1\end{pmatrix} \right]= (c_{1}+c_{2})\begin{pmatrix}0\\1\\0\end{pmatrix}.$$
We had trouble expressing this linear operator as a matrix. Lets take input basis
$$B=\left(\begin{pmatrix}1\\1\\0\end{pmatrix} ,\begin{pmatrix}0\\1\\1\end{pmatrix}\right)=:(b_{1},b_{2})\, ,$$
and output basis
$$E=\left(\begin{pmatrix}1\\0\\0\end{pmatrix} ,\begin{pmatrix}0\\1\\0\end{pmatrix} ,\begin{pmatrix}0\\0\\1\end{pmatrix}\right)\, .$$
Then
$$L b_{1} = 0. e_{1} + 1.e_{2}+ 0. e_{3} = L b_{2}\, ,$$
or
$$\big(Lb_{1}, L b_{2}) = \big( (e_{1},e_{2},e_{3})\begin{pmatrix}0\\1\\0\end{pmatrix}, (e_{1},e_{2},e_{3})\begin{pmatrix}0\\1\\0\end{pmatrix}\big)=(e_{1},e_{2},e_{3})\begin{pmatrix}0&0\\1&1\\0&0 \end{pmatrix}\, .$$
The matrix on the right is the matrix of $$L$$ in these bases. More succinctly we could write
$$L\begin{pmatrix}x\\y\end{pmatrix}_{B}=(x+y) \begin{pmatrix}0\\1\\0\end{pmatrix}_{E}$$
and thus see that $$L$$ acts like the matrix
$$\begin{pmatrix} 0&0\\ 1&1\\ 0&0 \end{pmatrix}$$.

Hence

$L\begin{pmatrix}x\\y\end{pmatrix}_{B} =\left( \begin{pmatrix} 0&0\\ 1&1\\ 0&0 \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} \right)_E\, ;$

given input and output bases, the linear operator is now encoded by a matrix.

This is the general rule for this chapter:
\textit{Linear operators become matrices when given ordered input and output bases.}\]

Example 72

Lets compute a matrix for the derivative operator acting on the vector space of polynomials of degree 2 or less:

$V = \{a_{0}1 + a_{1}x + a_{2} x^{2} \,|\, a_{0},a_{1},a_{2} \in \Re \}\, .$

In the ordered basis $$B=( 1,x,x^{2})$$ we write

$\begin{pmatrix}a\\b\\c\end{pmatrix}_{\!B}= a\cdot 1 + b x + c x^{2}$

and

$\frac{d}{dx} \begin{pmatrix}a\\b\\c\end{pmatrix}_{\!B}=b\cdot 1 +2c x +0x^{2}=\begin{pmatrix}b\\2c\\0\end{pmatrix}_B$

In the ordered basis $$B$$ for both domain and range

$\frac{d}{dx} = \begin{pmatrix} 0&1&0\\ 0&0&2\\ 0&0&0 \end{pmatrix}$

Notice this last equation makes no sense without explaining which bases we are using!


text:
# What is the difference of 13.9 - 13.70?

Sep 1, 2016

Another way of thinking about it. Same thing but in disguise!

$0.2$

#### Explanation:

We know that 9 can be split (partitioned) into 7+2 so 0.9 = 0.7+0.2
We also know that 13.70 + 0 = 13.70

So write $13.9 \text{ as } 13.7 + 0.2$

Putting it all together we have

$13.7 + 0.2$
$\underline{13.7 + 0.0} - \leftarrow \text{ Subtract}$
$\text{ } 0 + 0.2$

$\textcolor{red}{\text{Once you more used used to subtraction it becomes a bit easier}}$

Sep 2, 2016

They differ by $0.2$

#### Explanation:

In Maths, the word "DIFFERENCE" implies a subtraction.

It is asking by how much the numbers differ.
We usually take the small one away from the bigger one, which is written on top:
Line up the decimal points under each other and subtract from the right towards the left.

$\textcolor{w h i t e}{\times \times l} 13.90$
$\textcolor{w h i t e}{\times x l} \underline{- 13.70}$
$\textcolor{w h i t e}{\times \times l} \underline{00.20} = 0.2$


text:
Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Category Theory for beginners (4)

Categories
Tsakanikas Nickos
Community Team
Articles: 0
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

### Category Theory for beginners (4)

Note: The terms "monic,epic,section,retraction" used in the post "Category Theory For Beginners - 2" and the terms "monomorphism,epimorphism,split monomorphism,split epimorphism" used in this one are respectively equivalent.

Let $\mathcal{C}$ be a category. Show that

1. An isomorphism is a monomorphism and an epimorphism. Show that the inverse is not true in arbitrary categories. However, show that a morphism is an isomorphism if and only if it is a split monomorphism and a split epimorphism.
2. Let $\displaystyle f : X \longrightarrow Y$ be a morphism in $\mathcal{C}$. If $\displaystyle \left( K, \phi \right)$ is the kernel of $\displaystyle f$ and if $\displaystyle \left( C, \psi \right)$ is the cokernel of $\displaystyle f$ (assuming that both exist in $\mathcal{C}$, show that $\displaystyle \phi$ is a monomorphism and that $\displaystyle \psi$ is an epimorphism.


text:
# Can we make uniformity in marking zero potential? Say it at earths surface or at infinity from earths surface?

The gravitational potential energy at infinity os supposed to be zero. Since body always moves towards lower potential, the gravitational potential is taken as negative so that gravitational potential energy can decrease on moving towards earth.

On the other hand,for near earth surface, potential is taken to be zero at earth surface so that body at height $$h$$ with $$PE= mgh$$ decreases and becomes zero at earth surface.

Can we take potential energy at height $$h$$ to be $$mgh$$ instead of $$mgh$$ and potential at earth surface to be towards negative infinity for sake of uniformity?

Your first mistake is in the first word "the". There is no absolute "gravitational potential energy". Every specific choice of "potential energy" is always relative to some point (in the state space). If you choose that to be the point-at-infinity, then your gravitational potential energy (relative to that point) is negative on earth and increases as you move away from the earth, reaching $$0$$ at infinity. If you choose instead that point to be on the surface of the earth, then your (relative) gravitational potential energy is zero at the surface (by definition).
Your second mistake is in not understanding how the gravitational potential energy based on the assumption of a point mass is derived in Newtonian mechanics. The force on one point mass is $$\frac{G·m_1·m_2}{r^2}$$ towards the other point mass, where $$m_1,m_2$$ are the masses and $$r$$ is the distance between their centres of mass. One antiderivative with respect to $$r$$ is $$-\frac{G·m_1·m_2}{r}$$, but that is clearly not the only antiderivative. You can validly choose to use any antiderivative to define the gravitational potential energy; nothing requires you to make it equal to $$0$$ when $$r = ∞$$. But, it is convenient to make it so just because it is more elegant.
In fact, once there are more than two masses it is not so simplistic anymore. For example, at any point P between stationary point masses X,Y with masses $$m_1,m_2$$, the total gravitational force on an object at P with mass $$m_3$$ is $$\frac{G·m_1·m_3}{r^2}-\frac{G·m_2·m_3}{(d-r)^2}$$ towards X, where $$r$$ is your distance from X and $$d$$ is the distance from X to Y. As before, you can choose any antiderivative with respect to $$r$$ to define the total gravitational potential energy of P, and it is convenient to use $$-\frac{G·m_1·m_3}{r}-\frac{G·m_2·m_3}{d-r}$$ (i.e. no extra constant symbol), but that has nothing to do with being zero at infinity or whatever, since it is negative for every $$r$$ in the interval $$(0,d)$$, and it is maximum at some $$r∈(0,d)$$ such that $$\frac{G·m_1·m_3}{r^2} = \frac{G·m_2·m_3}{(d-r)^2}$$.


text:
# How do you solve the following linear system:  x=1-y , x-y=5 ?

Nov 13, 2015

$\textcolor{b l u e}{x = 3}$
A large amount of detail given for the method on purpose!
Using my methods you should be able to find the value of y$\to \textcolor{b r o w n}{\text{Practice makes it much faster}}$

#### Explanation:

$\textcolor{b r o w n}{\text{You will get used to this and be able to do simple ones in your head}}$

Given:

$x = 1 - y \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)$
$x - y = 5. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(2\right)$

In each case make y the dependant variable (answer) then equate to each other through y to find x

$\textcolor{b l u e}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$
$\textcolor{red}{\text{For (1)}}$

Add y to both sides giving:

$x + y = 1 - y + y \textcolor{w h i t e}{\times x} \text{Note that" -y +y=0 " so disappears!}$
$x + y = 1$

Subtract $x$ from both sides
$x - x + y = 1 - x$
$y = 1 - x \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left({1}_{a}\right)$

$\textcolor{b l u e}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$
$\textcolor{red}{\text{For (2)}}$

Add y to both sides giving:
$x - y + y = 5 + y$
$x = 5 + y$

Subtract 5 from bot sides giving:
$x - 5 = 5 - 5 + y$
$x - 5 = y$

Reverse so that y is on the left
$y = x - 5. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({2}_{a}\right)$

$\textcolor{b l u e}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$
$\text{Equation "(1_a) = color(blue)(y) = "Equation } \left({2}_{a}\right)$

$\textcolor{b r o w n}{1 - x} = \textcolor{b l u e}{y} = \textcolor{b r o w n}{x - 5}$
$\textcolor{b r o w n}{1 - x} = \textcolor{b r o w n}{x - 5}$

Add $x$ to both sides

$1 - x + x = x + x - 5$
$1 = 2 x - 5$

$1 + 5 = 2 x - 5 + 5$

$2 x = 6$

$\textcolor{b r o w n}{\underline{\text{DIVIDE}}}$ BOTH SIDES BY 2

$\frac{2}{2} x = \frac{6}{2}$

## $\textcolor{b l u e}{x = 3}$

$\textcolor{b l u e}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

Using the methods I have shown you first substitute $x = 3$ in equations #(1_a) "or " (2_a) then manipulate to find y


text:
Transition amplitude of n-vacuua in QCD

The vacuum state of QCD is a superposition of different ground states with non-tirvial topological charge $\propto n\in \mathbb{Z}$. We lable this vacuum configurations with $|{n}\rangle$. The transition amplitude is given by:

$$\langle n|m\rangle = \int \mathcal{DA}_{n-m}\ \ \text{exp} (-iS)$$

where $\mathcal{A}_{n-m}$ denots the gauge field configuration to the topological charge $n-m$.

Maybe I missed something but I don't understand why this transition amplitude depends only on the difference $n-m$? Shouldn't one calculate the the vacuum-to-vacuum transition amplitude by the generating functional (including all configurations)?

• $DA_{n-m}$ is funny (and unclear) notation, but it is clear that the transition amplitude only depends on $n-m$, not on $n,m$ separately. Of course you have to integrate over all configurations that start in $n$ and end in $m$. – Thomas Dec 17 '17 at 0:10

The path integral starts from a configuration $A^{(m)}$ of a topological charge $m$ and ends in a configuration $A^{(n)}$ of topological charge $n$.
The reason that it depends only on the difference is that both the QCD Lagrangian and the path integral measure are invariant under large gauge transformation, i.e., gauge transformation with nonvanishing winding number: $$Q = \frac{1}{24 \pi^2} \int \mathrm{Tr}\left((g^{-1}dg)^3\right)$$ If we change the integration variables by performing a gauge transformation $$g\cdot A = g^{-1}Ag + g^{-1} dg$$ The Lagrangian will not change, while the boundary configurations will become of topological charges $m+Q$ and $n+Q$, respectively. Since a transformation of the integration variables does not change the result. The final value should depend only on the difference $m-n$.


text:
## Quantization and representations of solvable Lie groups.(English)Zbl 0203.03302

Let $$G$$ be a connected, simply connected, solvable Lie group. Theorem 1 provides a necessary and sufficient condition for $$G$$ to be of Type I. Assuming that $$G$$ is of Type I, Theorem 2 provides a description of the irreducible representations of $$G$$ and Theorem 3 provides a construction of such representations (to within equivalence). The article is concerned solely with discussing and setting forth these theorems, and contains no proofs.
Reviewer: J. W. Baker

### MSC:

 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Full Text:

### References:

 [1] Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. No. 62 (1966), 199. · Zbl 0204.14202 [2] P. Bernat, Sur les représentations unitaires des groups de Lie résolubles, Ann. Sci. École Norm. Sup. (3) 82 (1965), 37 – 99 (French). · Zbl 0138.07302 [3] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57 – 110 (Russian). · Zbl 0090.09802 [4] Lajos Pukánszky, On the theory of exponential groups, Trans. Amer. Math. Soc. 126 (1967), 487 – 507. · Zbl 0207.33605
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
(This article was first published on Thinking inside the box , and kindly contributed to R-bloggers)

A new release 4.300 of Armadillo, the templated C++ library for linear algebra, was released by Conrad a few days ago.

As usual, this was rolled up in a new RcppArmadillo release 0.4.300.0 which is now on CRAN and in Debian.

This release brings nice new upstream changes such as faster find() as well as corresponding functions to find elements which are either finite or non-finite, based on a suggestion we made. This may need an RcppArmadillo specific extension as R extends the IEEE logic to supported NA, NaN and Inf on floating point variables whereas standard C++ (and hence Armadillo) only supports the latter two. We hope to work on this shortly.

Kevin also put in some code to make sure unsigned int vectors and matrices are converted correctly, and I updated the short configure script checking for complete LAPACK support to account for the R 3.0.3 or later case.

#### Changes in RcppArmadillo version 0.4.300.0 (2014-05-04)

• faster find()

• added find_finite() and find_nonfinite() for finding indices of finite and non-finite elements

• expressions X=inv(A)*B*C and X=A.i()*B*C are automatically converted to X=solve(A,B*C)

• Corrected conversion to unsigned int vectors and matrices

• Configure script now checks for R version 3.0.3 or newer to enable complex divide-and-conquer SVD in case of R-supplied LAPACK

Courtesy of CRANberries, there is also a diffstat report for the most recent release As always, more detailed information is on the RcppArmadillo page. Questions, comments etc should go to the rcpp-devel mailing list off the R-Forge page.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.


text:
# How do you calculate log_2 (3.16)?

May 5, 2016

${\log}_{2} \left(3.16\right) = \log \frac{3.16}{\log} \left(2\right) \approx 1.66$

#### Explanation:

Using the change of base formula, we have:

${\log}_{2} \left(3.16\right) = \log \frac{3.16}{\log} \left(2\right)$

One way of finding an approximation for this is to note that:

$\sqrt{10} \approx 3.16227766 \approx 3.16$

So:

$\log \left(3.16\right) \approx \log \left({10}^{\frac{1}{2}}\right) = \frac{1}{2}$

Also use:

$\log \left(2\right) \approx 0.30103$

Then:

${\log}_{2} \left(3.16\right) = \log \frac{3.16}{\log} \left(2\right) \approx \frac{1}{2 \cdot 0.30103} = \frac{1}{0.60206} \approx 1.66$


text:
Question

A commercial airplane has an air speed of 280 m/s due east and flies with a strong tailwind. It travels 3000 km in a direction $5^\circ$ south of east in 1.50 h. (a) What was the velocity of the plane relative to the ground? (b) Calculate the magnitude and direction of the tailwind's velocity. (c) What is unreasonable about both of these velocities? (d) Which premise is unreasonable?

a) $556 \textrm{ m/s}$

b) $278 \textrm{ m/s, } 10.0^\circ \textrm{ S of E}$

c) & d) see video for discussion.

Solution Video


text:
# Define the relaxation time of the free electrons drifting in a conductor: How is it related to the drift velocity of free electrons? Use this relation to deduce the expression for the electrical resistivity of the material.

Verified
147k+ views
Hint: The relaxation time of the free electrons drifting in a conductor is given by the formula, $T=v\dfrac{m}{e}E$
The expression of electrical resistivity of the material is obtained by modifying the expression,
Electric field is equal to E and voltage is equal to V
\begin{align} & E=\dfrac{V}{L} \\ & V=IR \\ \end{align}

Complete step-by-step solution:
First of all, let us check what relaxation time actually means.
Relaxation time is the time needed for the exponential decrease of the variable from its initial value to the $\dfrac{1}{{e}^{th}}$ value or 0.368 of that value. The relaxation time of an electron is the time gap between the successive collision of electrons in a conductor.
Now let us check the question,
Let I is the current that flows through the conductor, n is the number of electrons in that conductor, A is the area of the conductor, v is the drift velocity of the electron, e is the charge of the electron in that conductor, E is the electric field in that region, m is the mass of electron and T is the relaxation time period.
Then, $V=IR$
Rearranging this equation will give, $T=v\dfrac{m}{e}E$
The current flowing in the conductor I is
$I=-neA{{v}_{d}}=neA\left( e\dfrac{E}{m} \right)TI=\dfrac{n{{e}^{2}}EA}{m}T$
Also we know that electric field is given by the equation,
$E=\dfrac{V}{L}$
Substituting this in equation of current will give,
$I=\dfrac{n{{e}^{2}}VA}{mL}T\dfrac{V}{I}=\dfrac{mL}{n{{e}^{2}}TA}$
According to ohm's law,
$V=IR$
$R=\dfrac{V}{I}$
Therefore the equation becomes
$R=\left( \dfrac{m}{n{{e}^{2}}T} \right)\dfrac{L}{A}; R=\rho \dfrac{L}{A}$
$R=\rho \dfrac{L}{A}$
is the required solution.

Note: As the temperature of a conductor decreases, the number of collisions of electrons in the conductor also decreases. Therefore the free travel time increases or relaxation time increases. This is the temperature dependence of relaxation time.


text:
# Quote to show students don't have to fear making mistakes

I have some high school students which seem to be afraid of making mistakes. They are hesitant to make exercises in class because they want their course notes to be super clean, without any mistakes. The following has often happened in my class. A student writes (when $$a$$ is a positive real) $$\sqrt{16a^2+9a^2} = 4a+3a = 7a$$. When the correct solution is shown and the mistake the student made is discussed, the student erases his mistake and writes down the correct solution. However, when studying the contents, he is not reminded of the mistake. (Which would be very valuable)

Does anyone know of a nice quote which shows that there is great learning potential in making mistakes (and figuring out why!). (Especially in math)

I know of the following quote by prof. Francis Su. A nice quote, but it's more about the value of persistence.

Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.

• When the correct solution is shown and the mistake the student made is discussed, the student erases his mistake and writes down the correct solution. --- Perhaps have a policy of NOT erasing mistakes, but rather crossing them out with a big 'X'. You could also incorporate the idea of learning from mistakes by including questions/problems (verbal in-class, as well as written homework) that consist of solutions designed with specific intended mistakes that students are to identify, explain, and correct. – Dave L Renfro Sep 26 '19 at 7:10
• Grothendieck was used $57$ as an example of a prime number, so $57$ is now the Grothendieck prime. The effectiveness of that anecdote will depend on the level of the students, but I think it shows that even the very best mathematicians make silly mistakes. (Comment because it doesn't directly answer the question.) – user12756 Sep 26 '19 at 7:33
• Two Books: A great idea my maths teacher had was to have 2 writing books. One is the normal one where you put all the examples you try yourself. Then the other book is the special book which the teacher writes the notes for the subject on the board which we copy. He would then then go through some worked examples he had made up to highlight many of the issue you would find in the subject. We would copy those into the special book as well. It was really great. This meant you had a perfect writing-book to revise from with absolutely perfect examples. – Rewind Sep 27 '19 at 16:41
• There is a problem when a student has a high error rate and also lacks the skills to check their work or detect the mistakes. The ethos I try to instill in my students is that everyone makes mistakes, and it's OK to make them, but it's not OK not to catch them. Many students are extremely reluctant to check their own work, even when there are easy ways to do so, such as checking an integral by differentiating, or checking a factorization by multiplying it out. For a student who will be a pharmacist, it is not OK to make a mistake in converting grams to milligrams -- without catching it. – Ben Crowell Sep 27 '19 at 21:41

• Johann Wolfgang von Goethe: "By seeking and blundering we learn." Original German, 1825.
• Albert Einstein: "Anyone who has never made a mistake has never tried anything new." (However, attribution to Einstein is weak. See quoteinvestigator.com.)
• Jo Boaler: "When I have tutored people in math, I've always started by saying, 'By the way, I just want you to know that I love mistakes the most. They are the time that your brain grows, when you really learn, so it's really great to make mistakes.' [...] people immediately relax and breathe a sigh of relief and are much more willing to jump in to problems and persist longer." Cited here.
• Keith Devlin: "Learning occurs when we get something wrong and have to correct it." From Devlin's Angle, Oct 1st 2019.
• I didn't know of Jo Boaler, looks like she has written very interesting stuff on mathematics education. – dietervdf Sep 26 '19 at 9:03
• The Jo Boaler quote - When I start with a student studying for the SAT, I ask them to take a sample test. Not interested in the score. Just which questions they got wrong. I then focus on the source of those mistakes to identify any potential weakness in their skills. – JTP - Apologise to Monica Sep 26 '19 at 9:15

"I have not failed. I've just found 10,000 ways that won't work."

"Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time."

"Many of life's failures are people who did not realize how close they were to success when they gave up."

Thomas A. Edison

• It seems that the 10,000 is not exaggerated. The number of designs tried by Edison before a workable electric light was found. – Gerald Edgar Sep 27 '19 at 11:01
• Should come with an obligatory note that Edison had an army of assistants who did the dirty work for him, and from whose ideas he profited. Tesla is more sympathetic to me than Edison. – Rusty Core Oct 1 '19 at 0:24
• @Rusty Core, I don't think we have precise information about any idea in history to judge such issues, but myself I don't think that Edison would have been famous and well known for his inventions if he just stole ideas from others, he must have been pretty smart and ofcourse he did most of the work himself using his own ideas and not only the ideas of others. Also I just found this article edison.rutgers.edu/tesla.htm, and I see that it is objective. – Fareed Abi Farraj Oct 1 '19 at 6:15
• @FareedAF I don't agree with your logic. There are a number of named maths theorems that are not the work of the people they are named after. – Jessica B Oct 22 '19 at 6:07
• @FareedAbiFarraj, those who are culturally in power often get the credit for the work of the less powerful working with them. Check out Rosalind Franklin's work on DNA, for one example. Many women and POC have not been given the credit they are due. Many white men have gotten credit where it was not due. – Sue VanHattum May 26 at 19:14

“There is no man,” he began, “however wise, who has not at some period of his youth said things, or lived in a way the consciousness of which is so unpleasant to him in later life that he would gladly, if he could, expunge it from his memory. And yet he ought not entirely to regret it, because he cannot be certain that he has indeed become a wise man — so far as it is possible for any of us to be wise — unless he has passed through all the fatuous or unwholesome incarnations by which that ultimate stage must be preceded. I know that there are young fellows, the sons and grandsons of famous men, whose masters have instilled into them nobility of mind and moral refinement in their schooldays. They have, perhaps, when they look back upon their past lives, nothing to retract; they can, if they choose, publish a signed account of everything they have ever said or done; but they are poor creatures, feeble descendants of doctrinaires, and their wisdom is negative and sterile. We are not provided with wisdom, we must discover it for ourselves, after a journey through the wilderness which no one else can take for us, an effort which no one can spare us, for our wisdom is the point of view from which we come at last to regard the world. The lives that you admire, the attitudes that seem noble to you are not the result of training at home, by a father, or by masters at school, they have sprung from beginnings of a very different order, by reaction from the influence of everything evil or commonplace that prevailed round about them. They represent a struggle and a victory. I can see that the picture of what we once were, in early youth, may not be recognisable and cannot, certainly, be pleasing to contemplate in later life. But we must not deny the truth of it, for it is evidence that we have really lived, that it is in accordance with the laws of life and of the mind that we have, from the common elements of life, of the life of studios, of artistic groups — assuming that one is a painter — extracted something that goes beyond them.”

It is about a wise man who has committed foolish acts in his youth. Perhaps a little bit wordish (hey, it's Proust), but you might extract something like the part I have highlighted.

• Tapestry, TNG S6e15. "There are many parts of my youth that I'm not proud of… there were loose threads… untidy parts of me that I would like to remove. But when I pulled on one of those threads… it unraveled the tapestry of my life." – J... Sep 27 '19 at 17:29
• Thank you for the quote, great life lesson from Proust! – dietervdf Oct 23 '19 at 19:38

There was a post over at academia where somebody essentially said that after starting by investigating other people's mistakes:

Eventually I got better — I started making my own mistakes.

As I said over there, I liked that a lot and it stuck with me. Making your own mistakes is a sign of growing up.

Mistakes Allow Thinking to Happen

"I hope that in this year to come, you make mistakes. Because if you are making mistakes, then you are making new things, trying new things, learning, living, pushing yourself, changing yourself, changing your world. You're doing things you've never done before, and more importantly, you're doing something."

Neil Gaiman

"Failure is simply the opportunity to begin again, this time more intelligently."

Henry Ford

Is a quote about failing in topic? I think so. Here we go.

“Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.”

Samuel Beckett

The one I'm using is:

An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field.

Attributed to Niels Bohr, quoted by Edward Teller, in Dr. Edward Teller's Magnificent Obsession by Robert Coughlan, in LIFE magazine (6 September 1954), p. 62.

If you know what you are doing, then you are wasting your time.

Anonymous

• I can't find this one online – aloisdg moving to codidact.com Sep 26 '19 at 13:42
• @aloisdg, true, but now you can!--Anonymous – user52817 Sep 27 '19 at 3:46

"Human is the only animal that trips twice over the same stone" - Anonymous

"Next time you trip over a stone, instead of stepping over it, place a big flashing sign that will remind you where you fell to the ground. The next time you travel the same path you will remember your past mistake and be able to avoid the hazard" - Me

This is trying to be funny to convince them to mark their own mistakes so they can learn from them, instead of just getting over them. Please adjust to proper English where needed.

"In the remaining sections of this paper we briefly discuss various occurrences of the stability and pinching phenomena in differential geometry. The results we present are, for the most part, not new and we do not provide the detailed proofs. (These can be found in the papers cited in our list of references). What may be new and interesting for non-experts is an exposition of the stability/pinching philosophy which lies behind the basic results and methods in the field and which is very rarely (if ever) presented in print. (This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which [the] idea appears non-obvious.)" --Mikhail Gromov, "Stability and Pinching" pp. 64-65. Bold emphasis added.

This quote (from an absolutely towering figure of modern mathematics) pretty much guides my entire pedagogical philosophy. In particular, we must make a conscientious effort as both instructors and students of mathematics to combat this binary nature of mathematical perception. As instructors, we need the empathy and humility to remember that a topic was not always blindingly obvious and that we too once struggled with it. As students, we need the perseverance to push through until the current struggle becomes blindingly obvious in hindsight.

I do what I can to encourage this metacognition. Much of mathematics education can feel like an endless treadmill, of being perpetually stuck in the dark of new ideas and concepts. Seek out time to reinforce and reflect upon topics that once seemed impossible: "Remember when multiplication tables / equations of parabolas and lines / the product rule / linear differential equations / ... seemed hard? This too shall pass...". As grating as we might find it when students write $$\sqrt{a+b} = \sqrt{a} + \sqrt{b}$$, who among us didn't make this mistake at least once upon our own journeys? While we are tasked with correcting this mistake, don't fall into the trap of believing it to be trivial or the sign of an inferior intelligence. Embrace and encourage acknowledgement of mistakes.

That even an Abel prize winning mathematician can feel this way about mathematical understanding only further hammers this point home. If even Gromov can admit being embarassed about not grasping a subject at first pass, then what shame should the rest of us possibly bear? The only crimes would be to either lie to ourselves about having ever struggled or to write off those who struggle with it.

### Edit: Another quote

"Understanding what and why did not work may be more instructive than celebrating our successes." ~ Mikhail Gromov

I am having trouble finding a rigorous citation for this quote, but I will edit if I can track it down.

The road to wisdom? — Well, it's plain
and simple to express:
Err
and err
and err again
but less
and less
and less.”

• Please don't use codeblocks for anything that is not actual code. Quoteblock exists specifically for quotes. – Nij Sep 28 '19 at 4:24

I appreciate Ben Orlin's "Math with Bad Drawings" post about Andrew Wiles, for which the theme is not mistakes but rather being stuck. I have the sense that it contains some relevant content for the question that is asked here; for example, consider the following excerpt (emphasis added by me):

For example, take Wiles’ musings on the value of forgetfulness. “I think it’s bad to have too good a memory if you want to be a mathematician,” Wiles said. “You need to forget the way you approached [the problem] the previous time.”

It goes like this. You try one strategy on a problem. It fails. You retreat, dispirited. Later, having forgotten your bitter defeat, you try the same strategy again. Perhaps the process repeats. But eventually—again, thanks to your forgetfulness—you commit a slight error, a tiny deviation from the path you’ve tried several times. And suddenly, you succeed.

Wiles has a nifty analogy for this: it’s like a chance mutation in a strand of DNA that yields surprising evolutionary success.

“If you remember all the false, failed attempts before,” said Wiles, “you wouldn’t try. But because I have a slightly bad memory, I’ll try essentially the same thing again, and then I’ll realize I was just missing this one little thing.”

• I like the "slightly bad memory" remark. I think we (humans in general) tend to underestimate and undervalue the brain's capacity to forget, which is very important in learning mathematics. Here forgetfulness is shown as a positive virtue. There are other positive aspects of forgetfulness, but generally we are taught that forgetfulness is a negative defect. Indeed, there are things to remember in order to be able to solve mathematics problems. Given the brain's tendency to forget, being able to remember them takes preparation and work. – user1027 Oct 13 '19 at 15:41

There are plenty of quotes here...

But if you're trying to increase willingness to make and learn mistakes, I doubt that quotes will change a lot of minds, because a quote will be a tiny trickle of words pushing in the direction you want versus a raging river pushing the other way. Even Michael Jordan's quote about being cut from his high school basketball team or a Jo Boaler video just isn't going to make a lick of difference against the forces pushing against you.

A few considerations:

• How much of their letter grades for the course are explicitly about making mistakes and learning from them? E.g. "Exams are 50% of your mark, quizzes are 25% of your mark, and the other 25% of your mark will be how you recorded and learned from your mistakes." Most teachers don't do this. Every student sees the gap between what teachers say and what they incentivize. The normal reaction to this is to ignore the words and internalize the "obvious" message from the incentives: "You'll get high marks for correct answers and low marks for mistakes so don't make mistakes!" I am aware that many teachers are not allowed to have grading schemes that incorporate work habits. And I am also aware that the message you want to get across is to make the mistakes before the stakes are high, but...
• A huge share of math students, in the short-term, can't really learn from their mistakes because they lack the background knowledge to do so. If you're highly confused by the difference between $$x=2$$ and $$x+2$$, then how do you learn from your mistakes in a 9-step algebra problem? You can't. But you've probably been told to learn from such mistakes a thousand times, all to no avail. The normal reaction to this is to believe that most attempts to "learn from mistakes" in math class are a total waste of time and to be annoyed by your teacher who tells you to waste a huge amount of your own time for the thousandth time. Frustrated students will just say "I don't care what this means. Just show me the one right way to do it." [This problem is obviously solvable by backfilling knowledge gaps, but that's an enormous effort in and of itself.]
• What happens to the social status of students who make mistakes publicly?
• Do you exemplify learning from mistakes as a teacher? For example, do you video record what happens in your classroom, then share that with students, parents, colleagues, administrators, etc. all in order to find out how to improve? Do you try to make sure lots of people find all of your mistakes as a teacher? Would you focus your recording efforts on your least favorite lessons and topics to teach? Most teachers don't do this.

Overall, if it's a quote vs incentives, weak prior knowledge, countless bad experiences, lessened social status, and "do as I say not as I do", it's pretty obvious which side will win out.

• Thank you for the answer. (This question got a lot more useful/interesting comments then I anticipated, love the community!) The quotes I was looking for aren't going change the pupils mind. But they are a nice introduction into the theme. Also to clarify, I don't want students to make mistakes, but they shouldn't erase them from their notes when they do. I have the feeling that some (or most) don't realize the learning potential in leaving (and marking) the mistake in their notes. – dietervdf Oct 23 '19 at 19:36
• @dietervdf But you should want them to make mistakes! That's one of the themes of all these quotes. We need to dare to make mistakes, if we are to freely explore. – Sue VanHattum May 26 at 19:20
• @WeCanLearnAnything I am inspired by your first point. Perhaps I will change my grading scheme somehow to reflect this idea! – Sue VanHattum May 26 at 19:22
• @SueVanHattum - Awesome! I've failed to think clearly about how students perceive incentives many times. I would love to hear about how grading scheme evolves. :) – WeCanLearnAnything May 27 at 19:40

Good decisions come from experience, and experience come from bad decisions.

Unknown

Here are a few of my favorites:

Sometimes life's going to hit you in the head with a brick. Don't lose faith. - Steve Jobs

• I don't see how this is related to making mistakes. – Dirk Sep 28 '19 at 15:18

The But They Did Not Give Up site contains a large number of quotes and anecdotes regarding successful people from many different walks of life who achieved success only after many repeated failures. For example:

Albert Einstein did not speak until he was 4-years-old and did not read until he was 7. His parents thought he was "sub-normal," and one of his teachers described him as "mentally slow, unsociable, and adrift forever in foolish dreams." He was expelled from school and was refused admittance to the Zurich Polytechnic School. He did eventually learn to speak and read. Even to do a little math.

and:

Thomas Edison's teachers said he was "too stupid to learn anything." He was fired from his first two jobs for being "non-productive." As an inventor, Edison made 1,000 unsuccessful attempts at inventing the light bulb. When a reporter asked, "How did it feel to fail 1,000 times?" Edison replied, "I didn’t fail 1,000 times. The light bulb was an invention with 1,000 steps."

• Thank you for adding some details to your answer. It is much improved now. – Xander Henderson Sep 27 '19 at 16:16
• The quote about Einstein is absurdly misleading (or even false, although I can't check the claim about speaking and reading). Einstein was an excellent mathematician, even from a very young age. Although he was refused admittance to Zürich Polytechnic initially (at age 16), it was because of his scores on other subjects than physics and mathematics, and he did "eventually" get in at age 17. – Mees de Vries Sep 28 '19 at 7:21

Meyer began his career as a high school math teacher, an experience for which he felt ill-suited but which nonetheless, he said in the interview, “shaped [his] entire life.” He “felt guilty to be the only one who is always right while kids are wrong most of the time,” he explained, adding that he preferred the role of student: “To do research means to be ignorant most of the time and to often make the many mistakes I criticized in correcting my students’ homework.”

Article in Quanta about Yves Meyer, 2017 Abel Prize Winner

I'm not sure if I understand your point here: your students want to keep their course notes clean and don't want to mix correct calculations with wrong ones. You, however, want to add wrong calculations, in order for them to learn from those mistakes, and you are looking for quotes to support your point.

I, however, see that your students have a valid point. You do too. So, instead of forcing one point of view, I'll try to mix both:

• You can tell your students to write wrong calculations in their course, but they need to do this in another colour (green, for instance). That way, their course (written in blue or black) will still be completely correct, taking into account that the green text needs to be understood in another way (as the way NOT to calculate something).
• You can tell your students to flip their course notes, and write the wrong calculations in the backside of their notes. Like this, the wrong calculations will still be in their notes but in such a way that it does not disturb the rest of their notes.
• Thank you for the answer. I like your suggestions too. Just to clarify, I don't want pupils to mix wrong calculations with the correct ones. The problem I am/was facing, was that pupils are working on problems and when the correct answer was being worked out on the blackboard, a lot of pupils who made mistakes just erased their answer and wrote the correct, blackboard version down. Instead of tracing down the error in their own solution, marking it, and then writing the right solution next to it. They didn't see the learning potential of their wrong answers. – dietervdf Oct 23 '19 at 19:44

“Success is not final, failure is not fatal: it is the courage to continue that counts.” ― Winston S. Churchill

“There is only one thing that makes a dream impossible to achieve: the fear of failure.” ― Paulo Coelho, The Alchemist

“Failure is the condiment that gives success its flavor.” ― Truman Capote

Although (as with several other answers) this one is more about persistence, I think it's relevant enough ("depressingly slowly" sure implies intermittent failure to me) to post.

This is a poem by Piet Hein.

T.T.T.

Put up in a place
where it's easy to see
T.T.T.

When you feel how depressingly
slowly you climb,
it's well to remember that
Things Take Time.

(Amazingly, you can even buy plaques with a shorter version of the Danish original (?) of this on Etsy, with the famous picture that goes with it.)

• Seems more like a life lesson, but I love it. Thanks! – dietervdf Oct 23 '19 at 19:32

I had a poster of Jake the Dog (from Adventure Time) in my high school geometry classroom, with this phrase:

I also gave frequent closed-notes quizzes, and then made my students correct their quizzes on an error analysis sheet like this one (source):

I had a poster with a description of different types of errors, in order to encourage them to analyze why they made the mistake they made and how to avoid it in the future. I didn't use the same types of errors as the above blog, instead I used the following:


text:
# Waves frequency

thereddevils

## Homework Statement

One end of a string of mass 1.45 g and length 50.0 cm is attached to a frequency generator and the other end to a weight holder which hangs over a fixed pulley . The part of the string between the generator and the pulley is horizontal . THe string is set to vibrate . Several weights are added until a fundamental frequency of 120 Hz is achieved . Calculate the tension in the string when the fundamental frequency is achieved .

## The Attempt at a Solution

f=v/2L

120=v/(2 x0.05)

v=120 m/s

not really sure whether this is even an attempt . I do not know how to link the v i got here to tension in the string .

Homework Helper
Try to find the relation between tension, mass per unit length and the velocity of the wave in the stretched string.

thereddevils
Try to find the relation between tension, mass per unit length and the velocity of the wave in the stretched string.

i am not sure how to relate them since the unit of tension is kgm/s^2 , mass per unit length is kg/m and velocity , m/s

how can i combine m/s and kg/m to get kgm/s^2 ?

Homework Helper
i am not sure how to relate them since the unit of tension is kgm/s^2 , mass per unit length is kg/m and velocity , m/s

how can i combine m/s and kg/m to get kgm/s^2 ?

What is the value of
$$\sqrt{\frac{kgm/s^2}{kg/m} = ?}$$


text:
"Mini" fusion categories via 6j symbols - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:18:58Z http://mathoverflow.net/feeds/question/105151 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105151/mini-fusion-categories-via-6j-symbols "Mini" fusion categories via 6j symbols Hauke Reddmann 2012-08-21T11:31:27Z 2012-08-21T21:43:22Z
Just for fun, I set up the following scheme:
 - A 6j symbol is everything that fulfils Biedenharn-Elliott. (Plus symmetry, orthogonality etc. if that doesn't follow from it anyway.)
 - There are only a finite number $i$ of irreps (including $1$, the 1-dim "neutral"). Then I tried out which of the many possible Clebsch-Gordan expansions are mutally consistent with all the 6j equations via Biedenharn-Elliott. For two irreps I got only this: $1\bigotimes{X}=X (X=1,2), 2\bigotimes{2}=1+2.$
 - Ha!
 - Even I did see this before, it's the smallest Fibonacci fusion category. Now a quick research gave me a load of adjectives that come with fusion categories. Can you tell me which of them apply to my scheme?
 - $i=3$ produces 3 "minis", with $i=4$ I get 6 (and I stopped here because of the dreaded combinatiorial explosion - are these already classified somewhere?).
 - My main question, though, is whether you get a free parameter when you choose $i$ large enough. All my "minis" give only a set of fixed complex numbers for the values of the 6j symbols, the quantum dimension and the writhe normalizer (the minis all seem to be knot-theory compatible). The latter is a root of unity so I can speculate all the minis correspond to special values of the Jones polynomial. Or suchlike.
 http://mathoverflow.net/questions/105151/mini-fusion-categories-via-6j-symbols/105162#105162 Answer by Bruce Westbury for "Mini" fusion categories via 6j symbols Bruce Westbury 2012-08-21T15:00:15Z 2012-08-21T15:00:15Z
Fusion categories are discrete (like finite groups) and you never have a "free parameter". This is an observation by Ocneanu.
 http://mathoverflow.net/questions/105151/mini-fusion-categories-via-6j-symbols/105186#105186 Answer by Noah Snyder for "Mini" fusion categories via 6j symbols Noah Snyder 2012-08-21T18:57:16Z 2012-08-21T21:43:22Z
The number of non-isomorphic simple objects is called the rank of the fusion category.

You've made an error somewhere, as in rank 2 you should also get an example where the nontrivial simple object squares to $1$. (In your notation $2 \otimes 2 \cong 1$.)

For rank 2, Ostrik gave a complete classification (http://arxiv.org/abs/math/0203255). Ostrik's argument is somewhat indirect (via the Drinfel'd center). As far as I know, no one has given a direct classification in rank 2 via 6j symbols. The hard part is figuring out why there's no fusion categories with the fusion rule $X^2 = 1 + n X$ for $n>1$.

For rank 3 if you only look at fusion categories that give knot invariants, then Ostrik has a complete classification (http://arxiv.org/abs/math/0503564).

A great source of all information about "minis" that give knot invariants is Section 5.3 of Rowell-Strong-Wang (http://arxiv.org/abs/0712.1377). "Modular" means that you get knot invariants plus you have a certain "non-degeneracy" condition that's a bit hard to explain.


text:
## Free Speech Slip and Slide

In the past I’ve written at length about my concern that the newly invigorated attitude that we must outlaw, or at least severely socially punish the speakers, racist/sexist/etc.. speech is a mistake. I have doubts about the efficacy of such punishments and believe that pushing racism adjacent views into a hidden underground where they fester and mutate1 creates more hate. However, the primary thrust of my concern was the usual slippery slope argument (importantly serious harms arise as soon as well-intentioned people start to fear that an epistemic mistake could land them in trouble). Unfortunately, evidence for a steep slippery plastic slope with extra soap arrived all too quickly.

#### Superiority of Western Culture

First we had this really stupid opinion piece that I would have guessed was written by a machine learning algorithm trained on 1980s era conservative values pieces if it had only mentioned crack (still managed a shout out to the pill for destroying our perfect 1950s society). Personally, I thought it was just as stupid this time around as I did in the late 80s and early 90s except these authors should have seen how that went and known better. However, as far as offensiveness goes it rates as a “kids these days…have no … always on their..” but somehow it has become the subject of accusations of racism and the subject of serious controversy (yes, that last article is written by a friend of the original author so take its slant with a grain of salt).

True, there is no credible effort to have the author fired from her position in the law school but it has generated enough outrage for students to get up in time to picket Wax’s class as racist and its not just some hasty people with signs. At least a non-trivial segment of the Penn campus left is willing to call this piece racist, sexist or otherwise suggest it isn’t just dumb and wrong but deserving of open moral scorn.

While one might try and charitably reconstruct some argument based on the text of the oped2 what is going on is what is always going on with accusations of racism/sexism/islamophobia etc.. Rather than parsing the literal content of a piece and asserting those claims amount to racism (or providing evidence that the author was being disingenuous) people decide to call something racist if it feels like the things racists would say. In this case there is no doubt this oped has that feel. Indeed, it hits many of the points that one would expect from a racist dog-whistle: glorification of European/western culture, suggestion that something associated with whites is superior, a nostalgic comparison to the 1950s, reference to some aspect of black culture the author disapproves of (“anti-“acting white” rap culture of inner-city blacks”) and even the obligatory focus on whites that have the traits you are criticizing.

The problem with taking this as grounds for accusations of racism is that it confuses being the sort of person whose strong affinity for traditionalism and reverence for long lived institutions and practices may make needed reform more difficult with actual racism. However, we are generally quite willing to let the earnest man who is such a strong believer in feminism that he frequently gives a piece of his mind to men who he views as pushing an aggressive male-centric approach on women and thereby does more to perpetuate the stereotype of women as unable to handle these situations than anyone he criticizes. This case is only different in that it is harder to imagine genuinely feeling that these old school conservative values are the secret to a better life and wanting to help minorities by sharing. Also in that often people who feel this way about morals and newfangled social innovations also feel this way about minorities but that’s just a stereotype.

Most importantly, it renders the standard for racism uselessly subjective. If it is no longer necessary to have overt animus or believe in some particular stereotype then it is insanely easy to apply the term to virtually anyone you want. Especially given that as the sphere of things that have been labeled racist expands fewer and fewer non-racists say anything in that sphere so just imagine the same dialog in 20 years about pieces supporting free speech. It would be something mostly racists talk about as a cover, anyone like me writing about it would explain that we believed in it for everyone (while detractors would point out that we kept focusing on the free speech of the racists as they don’t see it from the context in which that is the right place to make one’s stand), one could raise analogies to the contract rights arguments offered in the civil rights movement (yes its bad but the constitution…we just can’t do anything). The only thing this lacks is the subjective feel that comes from hearing lots of racists say something that sounds similar but we can’t cede to racists the power to decide what is and isn’t considered.

Also, as a practical matter this kind of use of the accusation of racism isn’t productive. The reason to use the term at all is to invoke our shared disapprobation of certain behaviors to change people’s behavior. Telling someone ‘suggesting that blacks only eat fried Chicken or look like Gorillas’ is racist usually results in an immediate change and the world is a better place but when you say that some vague thing about the gestalt I get from your article is racist doesn’t. If I were the author and was willing to sell out my views so I wouldn’t be racist how would I even know where to start?

Call these ideas out as stupid or even the kind of progress phobic thinking that perpetuates racism that’s great but its just not racism.

#### University of Tampa’s Impolitic Twitter Firing

Also, we have the University of Tampa firing a visiting professor for the following poorly considered and bumblinging inappropriate tweet

I dont believe in instant karma but this kinda feels like it for Texas. Hopefully this will help them realize the GOP doesnt care about them.

This is obviously just a case of someone not realizing how what he said would be taken in context. When he did he apologized. That should have been the end of it.

While at first glance one might feel that this isn’t really relevant to the broader picture at the moment. However, while it wasn’t exactly an academic paper this tweet is fundamentally nothing but an expression of a political sentiment. Indeed, suppose the author really believed this was some kind of divine vengeance on Texas for voting GOP. Surely that is core political-religious speech if anything is so its hard to see how this is anything but a direct attack on the idea that Professors get to comment on current events and broader social issues without fear of being fired for controversial views (assuming they don’t bear on their academic qualifications…mathematicians probably shouldn’t say $\omega$ and $2^\omega$ have the same cardinality).

#### Mistakes

We need room for people to make mistakes! Even mistakes about what to believe on controversial issues because only when people feel they won’t lose their jobs or be shunned if they get it wrong can they allow themselves to explore the issue and reach the right conclusions.

I know its really hard in these discussions to imagine any other perspective than your own but rarely is it the case that someone just wakes up out of the blue filled with hate and the desire to see another race suffer. Sure, sometimes the reasons are just visceral (your gang is white they are black) but in most cases there is some chain of thought and emotion that made every step they took seem reasonable so if you suspect the target of your criticism of simply reasonless hate you should probably reevaluate that view.

However, that is what makes the situation so dangerous as well. Given that even racists think they have good and sound justifications for their beliefs an atmosphere which imposes severe penalties for even minor infractions allows only one safe response: parrot back the official dogma.

But, if we are going to fix the remaining barriers and harms inflicted by problematic stereotypes and structural racism/sexism we need to find them in non-obvious places and that takes open speculation. We’ve picked all the low hanging fruit so more looking for white or male ‘perpetrators’ (if it could have been fixed easily that way we would have) we instead need to look at the less examined reservoirs of stereotypes such as members of the group themselves or the well-intentioned helper3. That means we need to walk on the edge and consider possibly offensive or unpleasant possibilities if we are going to figure out what is really going on so we can do something to fix things.

1. I’ve seen any number of scenarios in which the perception that certain topics can’t even be discussed doesn’t erase those ideas from people’s minds. Rather, it pushes them to form groups (the ones that go silent when a woman or minority comes by and we work so hard to eliminate) in which they feel they can comfortably express views they are sympathetic to but are too controversial for general consumption. Unfortunately, when people gather together for the purpose of feeling safe sharing controversial views creates a strong social pressure not to call anyone else’s views in that group out for sexism/racism/etc.. even in a polite friendly way. I’m constantly amazed at how quickly both such groups form and how quickly they descend to the lowest common denominator and serve as a breeding ground where hateful ideas can infect good people because there is no opportunity to apply the corrective of a good counterargument and criticism.
2. Taking their complaints at face value would seem to suggest the problem is that suggesting WASP culture (not so named) is superior is racist or at least unacceptable and bad. While those of us immersed in liberal sensibilities naturally flinch a bit when the suggestion is made that one culture is superior to another that doesn’t make the claim wrong or racist. Indeed, we all believe that, at least in the modern context, modern western culture is superior to the violent revenge culture in some New Guinean tribes all things considered (of course cultures have so many traits surely we could cherry pick a few improvements but the original piece doesn’t deny this). Hell, the very idea of tolerance and equality that those on the left are fighting for is a rare value for a culture to have and we are right to identify it as something good and important. But I think this “can’t say one culture is better than another” line isn’t a very charitable interpretation.
3. Everyone knows that a great deal of slut-shaming and outfit policing is done to women by women and we’ve learned recently that it is other women who do the majority of interrupting women and may very well be the ones preventing more competitive female involvement. This matches both my experience at caltech (women who had few if any female friends their whole lives were way more likely to just blunder in and shot their load on the conversation or dismiss someone else’s contribution as stupid) and what evolutionary psychology would suggest (men have little interest in policing women but each gender needs to police rivals). Of course, men aren’t on the hook they are just on the hook for something else perpetuating harmful male stereotypes which can harm women as much as they do men (say by men not being willing to become primary caregivers).


text:
Hongyi Sheng

UC Irvine

## Time:

Tuesday, November 30, 2021 - 4:00pm

## Location:

NS2 1201
In Riemannian manifold $(M^n, g)$, it is well-known that its minimizing hypersurface is smooth when $n\leq 7$, and singular when $n\geq 8$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.