text:
# Math Help - Assume that the time of failure..

1. ## Assume that the time of failure..

Question : Assume that the time of failure, X of a telephone switching system follows exponential distribution with a failure rate $\lambda$.Estimate the maximum likelihood failure rate $\lambda$ from a random sample of n times to failure
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can any one provide me with the maximum failure rate of exponential distribution

2. Originally Posted by zorro
Question : Assume that the time of failure, X of a telephone switching system follows exponential distribution with a failure rate $\lambda$.Estimate the maximum likelihood failure rate $\lambda$ from a random sample of n times to failure
------------------------------------------------------------------------------------------------------------------

can any one provide me with the maximum failure rate of exponential distribution
The question is simply asking you to find the maximum likelihood estimator of the parameter in an exponential distribution. How to do something like this has been explained in other threads started by you. Please use them as a guide to answering this question.

If you get stuck, please show ALL of your working and say where you are stuck.

3. $
\frac{d[ln \ L( \theta)]}{d \theta} = - \frac{n}{\theta}+\frac{1}{\theta^2} \sum_{i=1}^{n} x_i$

But after this i am stuck what should i do now??????

4. Originally Posted by zorro
$
\frac{d[ln \ L( \theta)]}{d \theta} = - \frac{n}{\theta}+\frac{1}{\theta^2} \sum_{i=1}^{n} x_i$

But after this i am stuck what should i do now??????
Do you understand what you're meant to do with the likelihood function? What does post #4 here: http://www.mathhelpforum.com/math-he...ival-rate.html

say you have to do.

You are meant to have done several of these (you have posted several). Go and review this material in your class notes or textbook.

By the way,
please show ALL of your working and say where you are stuck
does not mean show the final line of your working.

5. But that link which u provided is for poisson distribution and here we have to use exponential distribution how are they similar

6. Originally Posted by zorro
But that link which u provided is for poisson distribution and here we have to use exponential distribution how are they similar
The process of finding a maximum likelihood estimator is the same for all distributions. Do you understand this?

You are meant to learn from previous examples (experience) and apply what you have learned from them to the next problem.

7. Originally Posted by zorro
$
\frac{d[ln \ L( \theta)]}{d \theta} = - \frac{n}{\theta}+\frac{1}{\theta^2} \sum_{i=1}^{n} x_i$

But after this i am stuck what should i do now??????
calc 1, set equal to zero, zero

format: plain


text:
# Is it possible to load a TikZ library locally?

Is it possible to load a TikZ library and make it local to a tikzpicture?

I'm writing a long document, and feel like the different TikZ libraries required for different diagrams are "littering" the namespace.

-
You can group it but I don't know if you gain much : \documentclass[tikz]{standalone}\begin{document}\begingroup\usetikzlibrary{calc}\tikz \draw(0,0) -- ($(1,1)+(2,2)$);\endgroup\tikz \draw(0,0) -- ($(1,1)+(2,2)$);\end{document} – percusse Apr 28 '13 at 9:32

Macro \usetikzlibrary is defined at the end of file tikz.code.tex. The macro remembers the state of a loaded module <foo> in the global macro \tikz@library@<foo>@loaded. If a module is loaded locally, all its local definitions and assignments are lost after the group, but tikz still thinks, the module is loaded and ignores further load requests (example taken from percusse's comment):

\documentclass[tikz]{standalone}
\begin{document}
\begingroup
\usetikzlibrary{calc}
\tikz\draw(0,0) -- ($(1,1)+(2,2)$);
\endgroup
\begingroup
\usetikzlibrary{calc}
\tikz\draw(0,0) -- ($(1,1)+(2,2)$);
\endgroup
\end{document}

The result is an error in the second \tikz statement in line 9:

! Package tikz Error: You need to say \usetikzlibrary{calc} for coordinate calculation.

Macro \usetikzlibrary can be patched to make the remembering of the load state local:

\documentclass[tikz]{standalone}

\usepackage{etoolbox}
\makeatletter
\AtBeginDocument{%
\patchcmd{\use@@tikzlibrary}{\global}{}{}{}%
}
\makeatother

\begin{document}
\begingroup
\usetikzlibrary{calc}
\tikz\draw(0,0) -- ($(1,1)+(2,2)$);
\endgroup
\begingroup
\usetikzlibrary{calc}
\tikz\draw(0,0) -- ($(1,1)+(2,2)$);
\endgroup
\end{document}

This works for module calc, because its definitions (tikzlibrarycalc.code.tex) are local. But the global load marker shows that libraries are intended to be loaded at top level. There it does not matter, if a macro is defined locally or globally. Thus there might be modules with relevant global assignments that persist after the group and disturb further tikz code.

-

A way to solve this problem would be to store the TikZ figures as standalone documents:

% file figures/myfigure.tikz
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes,arrows}
\begin{document}
\begin{tikzpicture}
% picture goes here
\end{tikzpicture}
\end{document}

In the main file, include the generated pdf with \includegraphics{figures/myfigure.pdf}.

In order to let latexmk know that it should recompile the PDF files when they are outdated, add this to a .latexmkrc:

add_cus_dep( 'tikz', 'pdf', 0, 'tikz2pdf' );
sub tikz2pdf {
$_[0] =~ m@^(.*/)([^/]+)$@;
my ( $dir,$file ) = ($1,$2);
system("pdflatex -output-directory $dir$_[0].tikz");
}

The latexmk part will only work if you don't include any other PDF files. If you do, you can probably solve it by using another extension for these.

-


text:
## Chapter 13: SIMPLE INTEREST

### Introduction

Simple interest is calculated as
P * R * T / 100
Where P = Principal or the amount borrowed
R = Rate of interest
T = years
Note: The day on which money is deposited isn't counted but date on which money is withdrawn is counted.

• Relation Among Principal, Time, Rate Percent of Interest Per Annum and Total Interest :

• Amount = Principal + Total interest

• A = P + $$\frac{P * t * r}{100}$$

• Time = $$\frac{\text{Total interest}}{\text{Interest on the principal in one year}}$$. Thus, if we have the total interest as  300 and the interest per year is  50, then we can say that the number of years is 300/50 = 6 years.

COMPOUND INTEREST

• Let principal = P, time = n years and rate = r% per annum and let A be the total amount at the end of n years, then A = P[$$1 + \frac{r}{100}$$]n

• When compound interest is reckoned half-yearly. If the annual rate is r% per annum and is to be calculated for n years. Then in this case, rate = (r/2)% half-yearly and time = (2n) half-years then A = P[$$1 + \frac{r/2}{100}$$]2n

• When compound interest is reckoned quarterly. In this case, rate = (r/4)% quarterly and time = (4n) quarter years. \ As before, then A = P[$$1 + \frac{r/4}{100}$$]4n

• The difference between the compound interest and the simple interest over two years is given by $$\frac{P * r^2}{100^2}$$

Ans .

5

1. Explanation :

Let principal = x, time = t years
Then interest = x/4, rate = t%
Now, using the SI formula, we get
Interest = (Principal × Rate × Time)/100
so x/4 = (x × t × t)/100
so t2 = 25
and we get t = 5%

Ans .

912

1. Explanation :

Whenever it is not mentioned whether we have to assume SI or CI we should assume SI.
For any amount, interest for the 1st three years @ 6% SI will be equal to 6 × 3 = 18%
Again, interest for next 4 years will be equal to 7 × 4 = 28%.
And interest for next 4 years (till 11 years) –7.5 × 4 = 30%
So, total interest = 18 + 28 + 30 = 76%
So, total interest earned by him = 76% of the amount = (76*1200 / 100) = 912

Ans .

8.33%

1. Explanation :

Let principal = x, then interest = x, time = 12 years.
Using the formula, Rate = (Interest × 100)/Principal × Time
= (x × 100)/(x × 12) = 8.33%

Ans .

640

1. Explanation :

Let the principal be  x and rate = r%.
Then, difference in between the interest of 5 years and of 2 years equals to
 800 –  704 =  96
So, interest for 3 years =  96
Hence, interest/year =  96/3 =  32
So, interest for 2 years Æ 2 ×  32 =  64
So, the principal =  704 –  64 =  640

Ans .

4000

1. Explanation :

Let the rate be y% and principal be  x and the time be 3 years.
Then according to the question = (x(y + 4) × 3)/100 – (xy × 3)/100 = 480
so xy + 4x – xy = 160 × 100
so x = (160 × 100)/4 =  4000

Ans .

16

1. Explanation :

It trebles itself in 8 years, which makes interest equal to 200% of principal.
So, 200% is added in 8 years.
Hence, 400%, which makes the whole amount equal to five times of the principal, which will be added in
16 years.

Ans .

500

1. Explanation :

Using the formula, amount = Principal (1 + rate/100)time
605 = p(1 + 10/100)2 = p(11/10)2
p = 605(100/121) =  500

Ans .

5000

1. Explanation :

Simple interest and compound interest for the first year on any amount is the same.
Difference in the second year’s interest is due to the fact that compound interest is calculated over the first
year’s interest also.
Hence, we can say that  72 = Interest on first year’s interest Æ 12% on first year’s interest =  72.
Hence, first year’s interest =  600 which should be 12% of the original capital. Hence, original capital =
5000 (this whole process can be done mentally).

Ans .

12,540

1. Explanation :

Population at the end of 1 year will be Æ 10,000 + 10% of 10,000 = 11,000
At the end of second year it will be 11,000 + 20% of 11,000 = 13,200
At the end of third year it will be 13,200-5% of 13,200 = 12,540.

Ans .

6

1. Explanation :

Let the rate of interest be = r%
Then, interest earned from  1200 at the end of year = (1200r)/100 =  12r
Again, interest earned from  1800 at the end of year = (1800/100) × (8/12) × 2r =  24r
So, total interest earned = 36r, which equals 216
fi r = 216/36 = 6%

Ans .

1380

1. Explanation :

The annual interest would be  60. After 3 years the total value would be 1200 + 60 × 3 = 1380

Ans .

10

1. Explanation :

The interest earned per year would be 1500/3=500. This represents a 10% rate of interest

Ans .

2315.25

1. Explanation :

2100 + 5% of 2100 = 2100 + 105 = 2205 (after 1 year). Next year it would become:
2205 + 5% of 2205 = 2205 +110.25 = 2315.25

Ans .

1400, 10%

1. Explanation :

1400 increased by 10% gives 1540 increased by 10% gives  1694.

Ans .

5

1. Explanation :

Simple Interest for 2 years = 100 + 100 = 200.
Compound interest for 2 years: Year 1 = 5% of 2000 = 100.
Year 2: 5% of 2100 = 105 Æ Total compound interest =  205.
Difference between the Simple and Compound interest = 205 – 200 =  5

Ans .

10

1. Explanation :

Interest in 2 years =  240.
Interest per year =  120
Rate of interest = 10%

Ans .

4

1. Explanation :

12500 @ 10% simple interest would give an interest of  1250 per annum. For a total interest of 5000, it would take 4 years.

Ans .

5

1. Explanation :

5% for 3 years (SI) = 15% of the amount; At the same time 4% SI for 4 years means 16% of the amount. The difference between the two is 1% of the amount. 1% of 500 =  5

Ans .

273

1. Explanation :

8% @ 700 =  56 per year for 3 years
7.5% @ 700 =  52.5 per year for 2 years
Total interest = 56 × 3 + 52.5 × 2 = 273

Ans .

cant say

1. Explanation :

8% of 800 for 4 years + 6% of 800 for 4 years = 64 × 4 + 48 × 4 = 256 + 192 = 448. However, we do not know the rate of interest applicable in the 5
th year and hence cannot determine the exact
simple interest for 9 years

Ans .

400

1. Explanation :

Simple interest @ 23% = 4600 × 2 = 9200
Compound interest @ 20%
20000 increase of 20% 24000 increase of 20% 28800
so 8800 compound interest.
Difference = 9200 – 8800 =  400

Ans .

331

1. Explanation :

1000 increase of 10% gives 1100 increase of 10% gives 1210 increase of 10% gives 1331.
Compound interest = 1331 – 1000 =  331

Ans .

2000

1. Explanation :

Solve using options. Thinking about option (a):
2000 gives 2200 (after 1 year) gives 2420 (after 2 years) which gives us an interest of 420 as required
in the problem. Hence, this is the correct answer

Ans .

144

1. Explanation :

P × 7/6 × 7/6 = 196 gives P = (196 × 6 × 6)/7 × 7 = 144

Ans .

9.09

1. Explanation :

1331 × 1.090909 × 1.090909 × 1.090909 = 1331 × 12/11 × 12/11 × 12/11 = 1728. Hence, the rate of compound interest is 9.09%.

Ans .

6

1. Explanation :

Since compounding is half yearly, it is clear that the rate of interest charged for 6 months would be 3%. Thus we get 3300 gives 3% increase and we get 3399

Ans .

144000

1. Explanation :

The value of the van would be 196000 × 6/7 × 6/7 = 144000

Ans .

20

1. Explanation :

Solve through options:
10000 with 20% increase gives 12000 with 20% increase gives 14400 with 20% increase gives 17280.

Ans .

9274.2

1. Explanation :

12% per annum compounded quarterly means that the amount would grow by 3% every 3 months. Thus, 8000 gives 8000 + 3% of 8000 = 8240 after 3 months gives 8240 + 3% of 8240 = 8487.2 after 6
months and so on till five3 month time periods get over. It can be seen that the value would turn out to be 9274.2.

Ans .

5

1. Explanation :

For the last 5 years, the interest earned would be: 30% of 360 = 108. Thus, interest earned in the first 4 years would be  72 and   18 every year on an amount of  360- which means that the rate of
interest is 5%

Ans .

29100

1. Explanation :

He will get 20000 + 45.5% of 20000 = 29100.
[Note: In this case we can take 13% simple interest compounded half yearly to mean 6.5% interest
getting added every 6 months. Thus, in 42 months it would amount to 6.5 × 7 = 45.5%

Ans .

66911.27

1. Explanation :

50000 gives with a 6% interest per annum 53000 gives with a 6% interest per annum 56180 gives with a 6% interest per annum 59550.8 gives with a 6% interest per annum 63123.84 gives with a 6% interest per annum
66911.27

Ans .

126247.69

1. Explanation :

100000 + 6% of 100000 (after the first 6 months) = 106000.
After 1 year: 106000 + 6% of 106000 = 112360
After 1 ½ years: 112360 + 6% of 112360 = 119101.6
After 2 years: 119101.6 + 6% of 119101.6 = 126247.69

Ans .

12.6

1. Explanation :

(73/365) × 0.09 × 700 =  12.6.
(Since the time period is 73 days)

Ans .

1200

1. Explanation :

The average rate of interest he pays is 186 × 100/1500 = 12.4%.
The average rate of interest being 12.4%, it means that the ratio in which the two amounts would
be distributed would be 4:1 (using alligation). Thus, the borrowing at 12% would be  1200

Ans .

9

1. Explanation :

If it doubles in 3 years, it would become 4 times in 6 year and 8 times in 9 years

Ans .

0.5

1. Explanation :

The difference in Simple interest represents 1% of the amount invested. Since this difference has occurred in 2 years, annually the difference would be 0.5%.

Ans .

10

1. Explanation :

It would take another 5 years to double again. Thus, a total of 10 years to become four fold

Ans .

30

1. Explanation :

The sum becomes 4 times Æ the interest earned is 300% of the original amount. In 10 years the interest is 300% means that the yearly interest must be 30%

Ans .

400

1. Explanation :

A × (1.02) + A = 808 × (1.02)
2 Æ A =  400

Ans .

100

1. Explanation :

The value would increase by 4% per year. To go to 5 times it’s original value, it would require an increment of 400%. At 4% SI it would take 100 years

Ans .

5

1. Explanation :

Total effective amount lent for 1 year
=  400 × 2 +  100 × 4 =  1200
Interest being  60, Rate of interest 5%

Ans .

375

1. Explanation :

Solve using options. If we try 500 (option b) for convenience, we can see that the difference between the two is  64 (as the SI would amount to 300 and CI would amount to 100 + 120 + 144
= 364). Since, we need a difference of only  48 we can realize that the value should be 3/4
th of 500. Hence, 375 is correct

Ans .

3725

1. Explanation :

The average Rate of interest is 6.8825%. The ratio of investments would be 1.1175: 1.8825 (@5% is to 8%). The required answer = 10000 × 1.1175/3 = 3725

Ans .

2

1. Explanation :

Interest per year =  25. Thus, an interest of  50 would be earned in 2 years.

Ans .

6 years and 8 months

1. Explanation :

42% on 2500 =  1050. The required answer would be: 1050/157.5 = 6 years and 8 months.

Ans .

700

1. Explanation :

The difference would amount to 8% of the value borrowed. Thus 56 = 0.08 × sum borrowed in each case becomes Sum borrowed =  700.

Ans .

1640

1. Explanation :

882 × (1.05) + 882 = P × (1.05)2
Solve for P to get P = 1640

Ans .

data insufficient

1. Explanation :

The data is insufficient as we do not know the time period involved.

Ans .

1500

1. Explanation :

Based on the information we have, we can say that there would have been  30 extra interest per year. For 2% of the principal to be equal to  30, the principal amount should be  1500`

### Problems

Q. Find the simple interest on Rs. 3000 at rate of interest 6 1/4% for 73 days

A. 73 days = 73/365 = 1/5 yrs.

SI = (3000 * 25/4 * 1/5) / 100

Q. An amount loaned at interest rate 13.5% per annum becomes Rs. 2502.50 after 4 years. Find the sum.

A. P + SI = P + P * 13.5 * 4 / 100 (Adding P on both sides)

2502.5 = P ( 1 + 54/100)

2502.5 = P (1.54)

P = 2502.5 / 1.54 = 1625

Q. A borrowed money at interest rate 6% for first two years, 9% for next three years and 14% for period beyond 5 years. If he pays a total interest of Rs. 11400 after 9 yrs how much did he borrow?

A. 11400 = (P *6*2/100) + (P*9*3/100) + (P*14*4/100)

11400 = (12+27+56)P / 100

11400 * 100 / 95 = P

P = 12000

Q. A certain sum of money amounts to Rs 1008 in 2 years and Rs. 1164 in 3.5 yrs.Find sum and RI

A. SI for 1.5 yrs = 1164 - 1008 = 156

SI for 1 yrs = 156 * 2 / 3 = 104 and two years = 208.

Principal = 1008 - 208 = 800

Use simple interest formula to get RI.

Q. A sum of 1500 is lent in 2 parts where one is at 8% and second is at 6%. If the total annual income is Rs. 106, find money lent at each rate.

A. (x*1*8/100) + ((1500-x) * 1 * 6 / 100 ) = 106

8x/100 + (9000 - 6x)/100 = 106

8x +9000 - 6x = 10600

2x = 1600

x=800

### Compound Interest

When P = principal, n = years and R = rate of interest compounded annually

The Amount = P (1+R/100)^n

When P = principal, R = rate of interest compounded half yearly the
Amount = P ( 1 + (R/2) / 100)^2n

When P = principal, R = rate of interest compounded quarterly the
Amount = P ( 1 + (R/4) / 100)^4n

When P = principal, R = rate of interest compounded annually but the time is in fraction then like 3 2/5 yrs
Amount = P ( 1 + R / 100)^3 * (1 + (2R/5)/100)

When P = principal, R = rate of interest compounded annually but is different for first year R1, second year R2 and third year R3 then
Amount = P ( 1 + R1 / 100) * ( 1 + R2 / 100) * ( 1 + R3 / 100)

Present worth of Rs. x due n years hence is
Present worth = x / (1 + R/100)^n

### Solved Problems

Q. Find CI on Rs.7500, compounded annually at RI of 4% for 2 years.

A. Amount = 7500 * (1 + 4/100)^2

Then amount - 7500 gives CI.

Q. CI is compounded half yearly, principal = Rs. 10000 in rate 4% for 2 years.

A.  Amount = 10000( 1+ 2/100)^4

Amount = 10824.32

CI = Amount - principal = 10824.32 - 10000 = 824.32

Q. Difference between SI and CI accrued on an amount of Rs. 18000 in 2 years is Rs.405. What is the RI.

A. P( 1+ (R/100) )^n - P*R*T/100 = 405

{18000 ( 1 + (R/100))^2 - 18000}  - (18000 * R * 2 / 100) = 405

Solving this you can get R.

Q. Divide 1301 between A and B such that the amount of A after 7 years is equal to amount in B after 9 years. Interest is compounded at 4%.

A. Let the amount be 'x' and 1301 - x.

x(1+4/100)^7 = (1301-x)(1+4/100)^9

Solving this we can get 'x'.

Q. A sum of money amounts to Rs.6690 after 3 years and Rs. 10035 after 6 years on CI. Find the sum.

A. P(1+R/100)^3 = 6690 ; P(1+R/100)^6 = 10035

Dividing first eqn by second eqn we get (1+R/100)^3 = 10035/6690 = 3/2

Substituting this value in first equation we get P = 6690 * 2 / 3 = 4460.

Q. A sum doubles itself in 9 years how many will it take to become 8 times.

A. P(1+R/100)^9 = 2P

(1+R/100)^9 = 2

Now finding P(1+R/100)^n = 8P

we need to get (1 + R/100)^n = 8 but 8 = 2^3

(1+R/100)^n = (1 + R/100)^ 9 ^3

we know that A^b^c = A^b*c

so (1+R/100)^n = (1 + R/100)^ 9*3 = (1 + R/100)^27

n = 27

### CAT Problems

Q.A sum of money compounded annually becomes Rs.625 in two years and Rs.675 in three years. The rate of interest per annum is

1. 7
2. 8
3. 6
4. 5

Ans.b

### Quiz

Score more than 80% marks and move ahead else stay back and read again!


text:
### Home > APCALC > Chapter 11 > Lesson 11.2.1 > Problem11-46

11-46.

What percent of the area inside the limaçon $r = 3 + 6\cos\left(θ\right)$ lies within the inner loop? .

Determine where the graph intersects the pole (origin):

$0=3+6\cos(\theta)$

$\cos(\theta)=-\frac{1}{2}$

$\theta=\pm\frac{2}{3}$

The full area is:

$2\int_0^{\frac{2\pi}{3}}\frac{1}{2}(3+6\cos(\theta))^2d\theta$

The area of the inner loop is:

$2\int_{\frac{2\pi}{3}}^\pi\frac{1}{2}(3+6\cos(\theta))^2d\theta$

Determine the percent of the area that is inside the limacon.


text:
# Finitism and .999....=1 question

• B
huginn
I had a discussion with someone about does .999...=1. She used finitism and I don't know enough about it to reply. I read a little about finitism but I don't know what to make of it, it just seems wrong
Can someone recommend a site with a good rebuttal

Mentor
2021 Award
The shortest version: ##0,999\ldots = 1##.
The short version: The left hand side is a different representation than on the right. So first of all, the left hand side has to be explained. If it is a limit, then it equals ##1##. If it is something else, then what?

Gold Member
##0.99999... = 0.\bar9= 3 \times \frac{1}{3} = \frac{3}{3} = 1##

I don't know anything about finitism. And I also don't know why one needs to be all philosophical about the question.

As I see it the problem just lies in the way numbers are represented in the decimal number system. Nobody would ever argue that ##\frac{3}{3} = 1##. If we used a number system base 60 like on our clocks where ##\frac{1}{3}## would be something like 20 minutes, the question whether 60 minutes is one hour wouldn't come up either.

Last edited:
Gold Member
Just to pile on, apparently Finitism sees some academic discussion occasionally, but unless your friend is a research logician, she probably doesn't know as much about it as she thinks she does. The type of finitism that asserts ##0.999... \neq 1## also must assert some weird propositions, like ##N+1\ngtr N## for sufficiently large ##N##.

SSequence
I suppose one particular way to see this equality can be to see it as a geometric series (probably already mentioned in thread linked above). For example, set:
a = initial term = 9/10
r = multiplication factor = 1/10

Because r<1 we know that the following sum converges:
a+ar+ar2+ar3+ar4...

S=a/(1-r)=1

Interestingly, a similar equality will seemingly follow no matter number-base system we use.

What is more interesting would be to prove the converse: That is, the only source of ambiguity in decimal representation can occur between:
---- decimal representations with infinite strings of 9's
---- decimal representations with infinite strings of 0's
I think it might be a bit tedious to prove but it could be done by a more careful analysis.

===============

Also I think probably the term ultrafinitism might be more in line with regards to question in OP.

Last edited:
Mentor
What is more interesting would be to prove the converse: That is, the only source of ambiguity in decimal representation can occur between:
---- decimal representations with infinite strings of 9's
---- decimal representations with infinite strings of 0's
I think it might be a bit tedious to prove but it could be done by a more careful analysis.
Actually, I don't think it would be hard to prove at all. Any decimal fraction (i.e., base-10) that has an expression of the form ##.d_1d_2 \dots n999\dots## can also be represented as ##.d_1d_2 \dots m000\dots##, where m = n + 1. Of course, I'm not proving anything here, but for any given decimal fraction in the first form, you could prove that it is equal to the second form, using the same geometric series argument that you gave.

The ambiguity of these two decimal fractions has to do with 9 being the largest decimal digit. If we were dealing with base-2 fractions, 1 is the largest binary digit, so a binary fraction of the form ##.d_1d_2\dots n111\dots## would be the same as ##d_1d_2\dots m000\dots##, again with m = n + 1, and this could be proved in the same manner. In base-3, we have the ambiguity in trinary fractions of the form ##.d_1d_2 \dots n222\dots##, as 2 is the largest base-3 digit.

SSequence
SSequence
Yeah, basically I think the most illustrative case would be of:
1=0.999...
I think apart from the equality of these (as is mentioned frequently), it is also important to show that 0.9999... is the only other decimal representation for 1**. I don't think it should be quite difficult (but certainly illustrative). Once one shows that it would be reasonably clear that it can be extended as a more general result.

Furthermore, one would also want to show that apart from certain kind of decimal representations(terminating ones and ones with infinite string of 9's), all other decimal representations uniquely describe a real number.

** Obviously assuming the convention that unnecessary 0's towards the end are cut-off. For example, 1.000 is counted as same decimal representation as 1.

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Well, how does she define 0,999... in her view?

fresh_42


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# How to make this equations aligned

I have an equation as below, I want text1 and text2, text3 are aligned as well.

\begin{aligned} A &= B \quad \text{text1}\\ C &= \begin{cases} 1 & \text{text2} \\ 0 & \text{text3} \end{cases} & \end{aligned}

You need to insert the same spacing as for the cases (only the first column), and correct for a "null delimiter" on the right:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\begin{align*}
A &= \makebox[0pt][l]{$B$}
\hphantom{\begin{cases} 1\\0\end{cases}}\kern-\nulldelimiterspace
\text{text1} \\
C &=
\begin{cases}
1 & \text{text2} \\
0 & \text{text3}
\end{cases}
\end{align*}
\end{document}

The element B is inserted with a zero-width right overlap (equivalent to \rlap) to not interfere with the placement of text1.

An alternative where array environment with @{quad} alignment and {... \right is used.

\documentclass[]{article}
\usepackage{amsmath}
\begin{document}
\begin{aligned} A &= B \quad\quad \text{text1}\\ C &= \left \{ \begin{array}{r@{\quad}l} 1 & \text{text2} \\ 0 & \text{text3} \end{array} \right . & \end{aligned}
\end{document}


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## Real zeros of algebraic polynomials with nonidentical dependent random coefficients

16 Oct 2019  ·  Sahoo Sabita, Maharana Partiswari ·

The expected number of real zeros of an algebraic polynomial $a_0+a_1x+a_2x^2+a_3x^3+....+a_{n-1}x^{n-1}$ depends on the types of random coefficients, with large $n.$ In this article, we show that when the random coefficients $\{a_i\}_{i=1}^{n-1}$ are assumed to be negatively dependent with $var(a_i)=\sigma^{2i}$ and correlation between any two coefficients for $i\neq j,$ assumed to be $\rho_{ij}=-\rho^{|i-j|},$ where $0<\rho<\frac{1}{3}$, then the expected number of real zeros is asymptotically equal to $\frac{2}{\pi \sigma}logn.$..

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# Categories

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# Sunday Times Teaser – Field for Thought

### by Peter Good

#### Published Sunday August 16 2020 (link)

Farmer Giles had a rectangular field bordered by four fences that was 55 hectares in size. He divided the field into three by planting two hedges, from the mid-point of two fences to two corners of the field. He then planted two more hedges, from the mid-point of two fences to two corners of the field. All four hedges were straight, each starting at a different fence and finishing at a different corner.

What was the area of the largest field when the four hedges had been planted?

The layout used is shown above.  Since the answer doesn’t depend on shape of the rectangular field, we can derive the answer by considering it to be a square.  All triangles are similar with non-hypotenuse side ratios of 2:1. Using BC as a unit of length, triangles ABC and ADE are similar with ADE being double the size of ABC. So the side CE of the central square field is 2 and the whole field has a side length AD of $$2\sqrt{5}$$. Hence AD/BC = $$1/\sqrt{5}$$, which means that the area ratio of the inner and outer squares is $$1/5$$ making the central field 11 hectares. The red figures show the relative areas of all the sub-fields.


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constants¶

Contains a collection of physical constants for the atmosphere and ocean.

import numpy as np

a = 6.373E6      # Radius of Earth (m)
Lhvap = 2.5E6    # Latent heat of vaporization (J / kg)
Lhsub = 2.834E6   # Latent heat of sublimation (J / kg)
Lhfus = Lhsub - Lhvap  # Latent heat of fusion (J / kg)
cp = 1004.     # specific heat at constant pressure for dry air (J / kg / K)
Rd = 287.         # gas constant for dry air (J / kg / K)
kappa = Rd / cp
Rv = 461.5       # gas constant for water vapor (J / kg / K)
cpv = 1875.   # specific heat at constant pressure for water vapor (J / kg / K)
Omega = 2 * np.math.pi / 24. / 3600.  # Earth's rotation rate, (s**(-1))
g = 9.8          # gravitational acceleration (m / s**2)
kBoltzmann = 1.3806488E-23  # the Boltzmann constant (J / K)
c_light = 2.99792458E8   # speed of light (m/s)
hPlanck = 6.62606957E-34  # Planck's constant (J s)
# sigma = 5.67E-8  # Stefan-Boltzmann constant (W / m**2 / K**4)
#  sigma derived from fundamental constants
sigma = (2*np.pi**5 * kBoltzmann**4) / (15 * c_light**2 * hPlanck**3)

S0 = 1365.2       # solar constant (W / m**2)
# value is consistent with Trenberth and Fasullo, Surveys of Geophysics 2012

ps = 1000.       # approximate surface pressure (mb or hPa)

rho_w = 1000.    # density of water (kg / m**3)
cw = 4181.3      # specific heat of liquid water (J / kg / K)

tempCtoK = 273.15   # 0degC in Kelvin
tempKtoC = -tempCtoK  # 0 K in degC
mb_to_Pa = 100.  # conversion factor from mb to Pa

#  Some useful time conversion factors
seconds_per_minute = 60.
minutes_per_hour = 60.
hours_per_day = 24.

# the length of the "tropical year" -- time between vernal equinoxes
# This value is consistent with Berger (1978)
# "Long-Term Variations of Daily Insolation and Quaternary Climatic Changes"
days_per_year = 365.2422
seconds_per_hour = minutes_per_hour * seconds_per_minute
minutes_per_day = hours_per_day * minutes_per_hour
seconds_per_day = hours_per_day * seconds_per_hour
seconds_per_year = seconds_per_day * days_per_year
minutes_per_year = seconds_per_year / seconds_per_minute
hours_per_year = seconds_per_year / seconds_per_hour
#  average lenghts of months based on dividing the year into 12 equal parts
months_per_year = 12.
seconds_per_month = seconds_per_year / months_per_year
minutes_per_month = minutes_per_year / months_per_year
hours_per_month = hours_per_year / months_per_year
days_per_month = days_per_year / months_per_year

area_earth = 4 * np.math.pi * a**2

# present-day orbital parameters, in the same format generated by orbital.py
orb_present = {'ecc': 0.017236, 'long_peri': 281.37, 'obliquity': 23.446}


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# Logistic Regression with Multiple Independent Variables vs One Independent Variable

Will the estimates and odds ratios change for an independent variable if it is by itself vs if there are other independent variables? I would think that it would change since thinking of it as an equation $y=x$ (one variable) is different than $y=x+z+q$ (3 variables).

• Yes, it would. You can verify yourself with any example. – Gregor May 8 '17 at 22:26
• It'd most likely be different due to other variables. – SmallChess May 9 '17 at 2:08
• Yes, it will change because more of the variation in the dependent variable y is now being attributed to x. – Michael Grogan May 9 '17 at 2:46

The estimate of the effect of $x$ will certainly change if $z$ or $q$ (or both) have an effect on $y$ net of $x$. It will change even if $z$ and $q$ are orthogonal to $x$ as long as $z$ and $q$ explain any portion of $y$. This happens because adding new variables changes the scale in which the entire model is expressed. The logistic model given by:

(1)$$\ln\bigg(\frac{1}{1-p_i}\bigg) = \beta_{0} + \beta_{1}x_{1i}$$

Is expressed in the latent variable formulation as:

(2)$$y^* =\alpha_{0} + \alpha_{1}x_{1i} + \sigma \varepsilon$$

The total variance in the model is made of explained (modelled) and unexplained (residual/errors) variance. The logistic model forces the errors to have a fixed variance of 3.29 (and a logistical distribution). Hence, any changes in the amount of explained variance will force the total variance of $y*$ to change, causing its scale to change because the variance of the errors is fixed and cannot change. This affects the size of the coefficients because they now explain change in $y*$ in a different scale. The scaling factor is given by $\sigma$ in (2). The alphas in (2) are related to the betas in (1) by:

(3)$$\beta_{j} = \frac{\alpha_{j}}{\sigma}\;\;j=1,...,J.$$

Adding more covariates that explain any portion of $y$ will reduce the amount of unexplained heterogeneity and will consequentially change the other coefficients in the model. You can refer to my answer here or to the following literature:

References

• Thanks for your information! Does that mean the more independent variables you include the better your model will be, thus, the better your output would be even if it changes some of the odds ratios a bit? – Mike May 25 '17 at 19:12
• Not necessarily. Usually it is better to have more of the relevant variables, but there are cases in which controlling for a variable will lead to biased estimates of other coefficients. One example of these is what is called collider effects in some fields, and is probably a matter for a different question here in this site (if there is not any already). – Kenji May 26 '17 at 21:11


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Aug 06

### Building a Better Robo Advisor

The more we learned about the current crop of robo advisory firms, the more we realized we could do better. This brief blog post hits the high points of that thinking.

### Not Just the Same Robo Advisory Technology

It appears that all major robo advisory companies use 50+ year-old MPT (modern portfolio theory). At Sigma1 we use so-called post-modern portfolio theory (PMPT) that is much more current. At the heart of PMPT is optimizing return versus semivariance. The details are not important to most people, but the takeaway is the PMPT, in theory, allows greater downside risk mitigation and does not penalize portfolios that have sharp upward jumps.

Robo advisors, we infer, must use some sort of Monte Carlo analysis to estimate “poor market condition” returns. We believe we have superior technology in this area too.

Finally, while most robo advisory firms offer tax loss harvesting, we believe we can 1) set up portfolios that do it better, 2) go beyond just tax loss harvesting to achieve greater portfolio tax efficiency.

Aug 01

## I Robo: The Rise of the Robo Advisor

Financial innovation is here and it is here to stay.  Financial advisors, broker/dealers, hybrids, and even financial planners should be thinking about how to adapt to inevitable changes launched by disruptive investing technologies.

### Robo Design — Chip designers have been using it for decades

I have an unique perspective on technological disruption.  For over ten years, my job was to develop software to make microchip designers more productive. Another way of describing my work was to replace microchip design tasks done by humans with software. In essence, my job was to put some chip designers out of work. My role was called (digital circuit) design automation, or DA.

In reality my work and the work of software design automation engineers like myself resulted in making designers faster and more productive — able to develop larger chips with roughly the same number of design engineers.

“The robos are coming, the robos are coming!” It’s true. Data though the end of 2014 shows that robo advisors managed $19 billion in assets with a 65% growth rate in just eight short months. This is essentially triple-digit growth, annual doubling.$19 billion (likely $30 billion now), is just a drop in the bucket now… but with firms like Vanguard and Schwab already developing and rolling out robo advising option of their own these crazy growth rates are sustainable for a while. With total US assets under management (AUM) exceeding$34 trillion, an estimated $30 billion for robo advisors represents less than 0.1% of managed assets. If, however, robo advisors grow double their managed assets annually for the next five years that amounts to about 3% of total AUM management by robo advisors. If in the second five years the robo advisory annual grow rate slows to 50% that still mean that robo advisors will control in the neighborhood of 20% of managed assets by 2025. ### “Robo-Shields” and Robo Friends Deborah Fox was clever enough to coin and trademark the term “robo-shield.” The basic idea is for traditional (human) investment advisors to protect their business by offering robo-like services ranging from client access to their online data to tax harvesting. I call this the half-robo defense Another route to explore is the “robo friends”, or “full robo-hybrid” approach. This is partnering with an internal or external robo advisor. As an investment advisor, the robo advisor is subservient to you, and provides portfolio allocation and tax-loss harvesting, while you focus on the client relationship. I believe that the “robo friends” model will win over the pure robo advising model — most people prefer to have someone to call when they have investment questions or concerns, and they like to have relationships with their human advisors. We shall see. What matters most is staying abreast of the robo advisor revolution and having a plan for finding a place in the brave new world of robo advising. Jul 31 ## Semivariance Excel Example The most in-demand topic on this blog is for an Excel semivariance example. I have posted mathematical semivariance formulas before, but now I am providing a description of exactly how to compute semivariance in “vanilla” Excel… no VBA required. The starting point is row D. Cell D$2 contains average returns of over the past 36 months. The range D31:D66 contains those returns.  Thus the contents of D$2 are simply: =AVERAGE(D31:D66) This leads us to the semivariance formula: {=SQRT(12)*(SQRT(SUM(IF((D31:D66-D$2)<0,(D31:D66-D$2)^2,0))/(COUNT(D31:D66-1))))} We will now examine each building block of this formula starting with IF((D31:D66-D$2)<0,(D31:D66-D$2)^2,0) We only want to measure “dips” below the mean return. For all the observations that “dip” below the mean we take the square of the dip, otherwise we return zero. Obviously this is a vector operation, the IF function returns a vector of values. Next we divide the resulting vector by the number of observations (months) minus 1. We can simply COUNT the number of observations with COUNT(D31:D66-1). [NOTE 1: The minus 1 means we are taking the semivariance of a sample, not a population. NOTE 2: We could just as easily taken the division “outside” the SUM — the result is the same either way.] Next is the SUM. The following formula is the monthly semivariance of our returns in row D: {=SUM(IF((D31:D66-D$2)<0,(D31:D66-D$2)^2,0))/(COUNT(D31:D66-1))} You’ll notice the added curly braces around this formula. This specifies that this formula should be treated as a vector (matrix) operation. The curly braces allow this formula to stand alone. The way the curly braces are applied to a vector (or matrix) formula is to hit <CTRL><SHIFT><ENTER> rather than just <ENTER>. Hitting <CTRL><SHIFT><ENTER> is required after every edit. We now have monthly semivariance. If we wanted annual semivariance we could simply multiply by 12. Often, however, we ultimately want annual semi-deviation (also called semi-standard deviation) for computing things like Sortino ratios, etc. Going up one more layer in the call stack brings us to the SQRT operation, specifically: {=SQRT(SUM(IF((D31:D66-D$2)<0,(D31:D66-D$2)^2,0))/(COUNT(D31:D66-1)))} This is monthly (downside) semi-deviation. We are just one step away from computing annual semi-deviation. That step is multiplying by SQRT(12), which brings us back to the big full formula. There it is in a nutshell. You now have the formulas to compute semivariance and semi-deviation in Excel. Jun 28 ## Quant Cross-Training A very astute professor of finance told our graduate finance class that the best way to become a bona fide quant is NOT to get a Ph.D. in Finance! It is better, he said, to get a Ph.D. in statistics, applied mathematics, or even physics. Why? Because a Ph.D in Finance is generally not sufficiently quantitative. A quant needs a strong background in Stochastic Calculus. ### “Quants for Hire?” Our company has been described as a “quants for hire” firm. That is flattering. While we currently have 4 folks with master’s of science degrees (and one close to finishing a master’s) what we do is probably more accurately described as “quant-like” or “quant-lite” software and services. However “Quants for Hire” definitely has a nice succinct ring to it. ### Quant-like Tangents to Financial Learning Most of our quant-like work has been fairly vanilla — back testing trading strategies in Excel, Monte Carlo simulations (also in Excel), factor analysis, options strategy analysis. So far our clients like Excel and are not very interested in R. The main application of R has been to double-check our Excel back tests! We have attracted fairly sophisticated clients. They seem reasonably comfortable about talking about viewing portfolios as unit vectors that can be linearly combined. They tend to understand correlation matrices, Sortino ratios, and in some cases even relate to partial derivatives and gradients. But they tend to push back on explanations involving geometric Brownian motion, Ito’s lemma, and the finer points of Black-Scholes-Merton. They do, however, appear to appreciate that we “know our stuff.” I’ve got a decent set of R skills, but I’m looking to take them to the next level. I’m taking a page from my professor in tackling non-financial quantitative problems. My current problem du jour image compression. I came up with an R script that achieves very high compression levels for lossy compression. It is shorter than 200 lines commented and shorter than 100 lines when stripped of comments and blank (formatting) lines. It can easily achieve 20X or greater compression, albeit with a loss in quality. In my initial tests my R algorithm (IC_DXB1.1) was somewhat comparable to JPEG (GIMP 2.8) at 20X compression, though I the JPEG clearly looks better in general. I also found an elegant R compressor that is extremely compact R code… the kernel is about 5 lines! Let’s call this SVD (singular value decomposition) for reference. So here’s the bake off results (all ~20X compressed to ~1.5KB): JPEG: IC_DXB1.1: JPEG 20X Compressed IC_DXB1.1 20X Compressed SVD 20X Compressed What’s interesting to me is that each algorithm uses radically different approaches. JPEG uses DCT (discrete cosine transform) plus a frequency “mask” or filter that reduces more and more high-frequency components to achieve compression. My ic_dxb1.1 algorithm uses a variant of B-splines. The SVD approach uses singular value decomposition from linear algebra. Obviously tens of thousands of hours have been invested in JPEG encoding. And, unfortunately, 99%+ of JPEG images are not as compact as they could be due to a series of patent disputes around arithmetic coding. Even thought the patents have all (to the best of my knowledge) expired, there is simply too much inertia behind the alternative Huffman coding at the present. It is worth noting that my analysis of all 3 algorithms is based on Huffman coding for consistency. All three approaches could ultimately use either Huffman or arithmetic coding. ### So this Image Stuff Relates to Finance How? Another of my professors explained that, fundamentally, finance is about information. One set of financial interview questions start with the premise that you have immediate (light-speed, real-time) access to all public information. Generally how would you make use of this information to make money trading? Alternatively you are to assume (correctly) that information costs money… how would your prioritize your firm’s information access? How important is frequency and latency? Having boat loads of real-time data and knowing what to do with it are two different things. I use R to back test strategies, because it easy to write readable R code with a low bug rate. If I had to implement those strategies in a high-frequency trading environment, I would not use R, I would likely use C or C++. R is fast compared to Excel (maybe 5X faster), but is slow compared to good C/C++ implementations (often 100X slower). My thinking is that while knowledge is important, so is creativity. By dabbling in areas outside of my “realm of expertise”, I improve my knowledge while simultaneously exercising my creativity. Both signal processing and quant finance can reasonably be viewed as signal processing problems. Signal processing and information theory are closely related. So I would argue that developing skills in one area is cross-training skills in the other… and with greater opportunity for developing creativity. Finance is inextricably linked to information. ### The Future of Finance Requires Disruptive (Software) Technology It aint gonna be pretty for traditional financial advisors, hybrid advisors, broker/dealers, etc. Not with the rapid market acceptance of robo advisors. Robo advising will have at least three important disruptive impacts: 1. Accelerating downward pressure on advisory fees 2. Taking of market share and AUM 3. Increasing market demand for investment tax management services such as tax-loss harvesting Are you ready for the rise of the bots? We at Sigma1 are, and we are looking forward to it. That is because we believe we have the software and skills to make robo advisors work better. And we are not resting on our laurels — we are focusing our professional development on software, computer science, advanced mathematics, information theory, and the like. Jun 18 ## Dividends and Tax-Optimal Investing The previous post showed after-tax results of a hypothetical 8% return portfolio. The primary weakness in this analysis was a missing bifurcation of return: dividends versus capital gains. The analysis in this post adds the missing bifurcation. It is instructive to compare the two results. This new analysis accounts for the qualified dividends and assumes that these dividends are reinvested. It is an easy mistake to assume that since the qualified dividend rate is identical to the capital gains rate, that dividends are equivalent to capital gains on a post-tax basis. This assumption is demonstrably false. Tax Efficiency with dividends. Though both scenarios model a net 8% annual pre-tax return, the “6+2” model (6% capital appreciation, 2% dividend) shows a lower 6.98% after-tax return for the most tax-efficient scenario versus a 7.20% after-tax return for the capital-appreciation-only model. (The “6+2” model assumes that all dividends are re-invested post-tax.) This insight suggests an interesting strategy to potentially boost total after-tax returns. We can assume that our “6+2” model represents the expected 30-year average returns for a total US stock market index ETF like VTI, We can deconstruct VTI into a value half and a growth half. We then put the higher-dividend value half in a tax-sheltered account such as an IRA, while we leave the lower-dividend growth half in a taxable account. This value/growth split only produces about 3% more return over 30 years, an additional future value of$2422 per $10,000 invested in this way. While this value/growth split works, I suspect most investors would not find it to be worth the extra effort. The analysis above assumes that the growth half is “7+1” model. In reality the split costs about 4 extra basis points of expense ratio — VTI has a 5 bps expense ratio, while the growth and value ETFs all have 9 bps expense ratios. This cuts the 10 bps per year after-tax boost to only 6 bps. Definitely not worth the hassle. Now, consider the ETF Global X SuperDividend ETF (SDIV) which has a dividend yield of about 5.93%. Even if all of the dividends from this ETF receive qualified-dividend tax treatment, it is probably better to hold this ETF in a tax-sheltered account. All things equal it is better to hold higher yielding assets in a tax-sheltered account when possible. Perhaps more important is to hold assets that you are likely to trade more frequently in a tax-sheltered account and assets that you are less likely to trade in a taxable account. The trick then is to be highly disciplined to not trade taxable assets that have appreciated (it is okay to sell taxable assets that have declined in value — tax loss harvesting). The graph shows the benefits of long-term discipline on after-tax return, and the potential costs of a lack of trading discipline. Of course this whole analysis changes if capital gains tax rates are increased in the future — one hopes one will have sufficient advanced notice to take “evasive” action. It is also possible that one could be blindsided by tax raising surprises that give no advanced notice or are even retroactive! Unfortunately there are many forms of tax risk including the very real possibility of future tax increases. Jun 15 ## Investment Tax Management Boosts Returns in Surprising Ways Figure 1. Tax Deferral benefits both parties. ### A Common Tax Misconception When asked to consider tax deferral investment strategies, many people instinctively conclude that tax deferral benefits the investor at the expense of the government. Such a belief is half-right. Tax deferral ultimately benefits both the investor and the government’s tax revenues. While there are exceptions involving inheritance, in most other cases both parties benefit. Figure 1 summarizes the relationship between higher after-tax returns and higher nominal net cash flows to the government. The reason I lead with the government’s side of the tax equation is for tax policy wonks in Washington D.C. I suspect many of them already know this information, and this is simply another data point to add to their arsenal of tax facts. For the others, I hope this a wake-up call. The message: When investors, investment advisors, and fund managers successfully defer long-term capital gains, investors and governments win in the long run. The phrase “in the long run” is important. When taxes are deferred, the government’s share grows along with the investor’s. In the short term, taxes are reduced; in the long run taxes are increased. For the investor this long-run tax increase is more than offset by increased compounding of return. Please note that all of these win/win outcomes occur under a assumption of fixed tax rates — which is 20% in this example. It is also worth noting that these outcomes occur for funds that are spent at any point in the investor’s lifetime. This analysis does not necessarily apply to taxable assets that are passed on via inheritance. Critical observers may acknowledge the government tax “win” holds for nominal tax dollars, but wonder whether it still holds in inflation-adjusted terms. The answer is “yes” so long as the the investor’s (long-run) pre-tax returns exceed the (long run) rate of inflation. In other words so long as g > i (g is pre-tax return, i is inflation), the yellow line will be upward sloping; More effective tax-deferral strategies, with higher post-tax returns, will benefit both parties. As inflation increases the slope of the yellow line gets flatter, but it retains an upward slope so long as pre-tax return is greater than inflation. ### Tax Advantages for Investors Responsible investors face many challenges when trying to preserve and grow wealth. Among these challenges are taxes and inflation. I will start by addressing two important maxims in managing investment taxes: 1. Avoid net realized short-term (ST) gains 2. Defer net long-term gains as long as possible It is okay to realize some ST gains, however it is important to offset those gains with capital losses. The simplest way of achieving this offset is to realize an equal or greater amount of ST capital losses within the same tax year. ST capital losses directly offset ST capital gains. A workable, but more complex way of offsetting ST gains is with net LT capital losses.The term net is crucial here, as LT capital losses can only be used to offset ST capital gain once they have been first used to offset LT capital gains. It is only LT capital losses in excess of LT capital gains that offset ST gains. If the above explanation makes your head spin, you are not alone. Managing capital gains is really an exercise in linear programming. In order to make this tax exercise less (mentally) taxing, here are some simple concepts to help: • ST capital losses are better than LT capital losses • ST capital gains are worse than LT capital gains • When possible offset ST losses with ST gains Because ST capital losses are better than LT, it often makes sense to see how long you have held assets that have larger paper (unrealized) losses. All things equal it is better to “harvest” the losses from the ST losers than from the LT losers. Managing net ST capital gains can potentially save you a large amount of taxes, resulting in higher post-tax returns. ### Tax Advantages for the Patient Investor Deferring LT capital gains requires patience and discipline. Motivation can help reinforce patience. For motivation we go back to the example used to create Figure 1. The example starts today with$10,000 investment in a taxable account and a 30-year time horizon. The example assumes a starting cost basis of zero and an annual return of 8%.

This example was set up to help answer the question: “What is the impact of ‘tax events’ on after-tax returns?” To keep things as simple as possible a “tax event” is an event that triggers a long-term capital gains tax realization in a tax year. Also, in all cases, the investor liquidates the account at the start of year 31. (This year-31 sale is not counted in the tax event count.)

It turns out that it not just the number of tax events that matters — it is also the timing. To capture some of this timing-dependent behavior, I set up my spreadsheets to model two different timing modes. The first is called “stacked” and it simply stacks all tax events in back-to-back years. These second mode is called “spaced” because the tax events are spaced uniformly.  Thus 2 stacked tax events occur in years 1, 2, while 2 spaced tax events occur in years 10 and 20. The results are interesting:

Tax “event” impact on after-tax returns.

The most important thing to notice is that if an investor can completely avoid all “tax events” for 30 years the (compound) after-tax return is 7.2% per year, but if the investor triggers just one taxable event the after tax return is significantly reduced. A single “stacked” tax event in year 1 reduces after tax returns to 6.49% while a single “spaced” tax event in year 15 reduces returns to 6.67%. Thus for a single event the spaced tax event curve is higher, while for all other numbers of tax events (except 30 where they are identical) it is lower than the stacked-event curve.

The main take-away from this graph is that tax deferral discipline matters. The difference between 7.2% and 6.67% after-tax return, over thirty years is huge when framed in dollar terms. With zero (excess) tax events the after-tax result in year 31 is $80,501. With one excess tax event (with the least harmful timing!) that sum drops to$69,476.

In the worst case the future value drops to 51,444 with an annual compound after-tax return of only 5.61%. ### Tax Complexity, Tax Modeling Complexity, and Other Factors One of the challenges faced when bringing fresh perspectives to the tax-plus-investing dialog is in providing examples that paint the broad portfolio tax management themes in a concise way. The first challenge is that the tax code is constantly changing, so predicting future tax rates and tax rules is an imprecise game at best. The second challenge is that the tax code is so complex that any generalization will mostly likely have a counterexample buried somewhere in the tax code. The third complication is that baring significant future tax code changes and obscure tax code counterexamples, creating a one-size-fits-all model for investors results in large oversimplifications. I believe that tax indifference is the wrong answer to the question of portfolio tax optimization. The right answer is more closely aligned with the maxim: All models are wrong. Some are useful. This common saying in statistics gets to the heart of the problem and the opportunity of investment tax management. It is better to build a model that gives deeper insight into opportunities that exist in reconciling prudent tax planning with prudent investment management, than to build no model at all. The simple tax model used in this blog post makes some broad assumptions. Among these is that the long-term capital gains rate will be the same for 30 years and that the investor will occupy the same tax bracket for 30 years. The pre-tax return model is also very simple: 8% pre-tax return each and every year. I argue that models as simple as this are still useful. They illustrate investment tax-management tax principles in a manner that is clear and draws the same conclusions as analysis using more complex tax modelling. (Complex models also have their place.) I would like to highlight the oversimplification I think is most problematic from a tax perspective. The model assumes all the returns (8% per year) are in the form of capital appreciation. A better “8%” model would be to assume a 2% dividend and 6% capital appreciation. Dividends, even though receiving qualified-dividend tax treatment, would bring down the after-tax returns, especially on the left side of the curve. I will likely remedy that oversimplification in a future blog post. ### Investment Tax Management Summary 1. Tax deferral does not hurt government revenues; it helps in the long run. 2. Realized net short-term capital gains can crater post-tax investment returns and should be avoided. 3. Deferral of (net) long-term capital gains can dramatically improve after-tax returns. 4. Tax deferral strategies require serious investment discipline to achieve maximum benefit. 5. Even simple tax modelling is far better than no tax modelling at all. Simple tax models can be useful and powerful. Nonetheless, investment tax models can and should be improved over time. Jun 03 ## Managing Portfolio Taxes and Tax Risk ### Portfolio Tax-Management is Key to Attracting and Keeping High-Net-Worth Investors With the right tools, the most reliable way to generate portfolio alpha is via tax alpha. One way to look at traditional alpha is as a zero-sum game (after subtracting risk-adjusted market returns). Very smart people with substantial financial backing are searching for tradition alpha every day. The competition for alpha is intense, and tautologically when someone generates positive alpha, elsewhere in the market others are offsetting with negative alpha. Producing tax alpha, however, can be a viewed as win/win cooperative “game” involving the investment advisor representative (IAR) and client. Among the IAR’s many tasks in creating positive tax alpha is asking for regular (quarterly or twice-per-year) updates on the client’s tax situation and expected tax bracket. The client’s responsibility is to monitor and know their tax situation and to provide timely and accurate updates when their expected tax situation has significant changes. Net-net worth investors tend to be both more tax savvy and tax aware than other investors. Whereas5M in liquid assets used to be the threshold where investors expect and demand portfolio tax management, investors at the $2M and even$1M level are increasingly demanding portfolio tax management.

Firms that fail to offer effective portfolio tax management services are at risk of losing their most revenue-producing clients. Conversely, firms that offer and effectively communicate quality investment tax-management services stand to win highly sought-after high-net-worth advisory clients.

### Developing a Tax-Aware Portfolio Mindset

Effective investment portfolio tax management begins with awareness of the tax implications of investment actions. As a rule of thumb, the two most tax important considerations are:

1. Awareness of whether asset sale will: A) Generate a large capital gain or loss. B) Result in a short-term (ST) or long-term (LT) gain or loss.
2. Awareness of the client’s: A) Tax situation, B) Tax preferences.

After choosing to increase one’s own investment tax proficiency, we advise clients to begin slowly and focus primarily on tax awareness for the first month or two. There are many reasons we favor this approach. First there is a tendency to initially swing to a tax-first mentality. We believe that it is best to maintain an investment-first mentality, with tax-awareness and tax-management playing an important secondary role in most cases. Developing a tax-aware mentality requires much more mental energy, especially in the beginning. Much like an amateur juggler adept at juggling three balls, adding a forth is very difficult at first, and it dropping one (or more) balls is increasingly likely. If overwhelmed it is best to drop the “tax ball” first, and pick it up later when circumstances permit.

Our view is that it takes 2-3 months of effort to reach a point where portfolio tax awareness becomes largely second nature. It is only after tax consciousness becomes second nature that the foundation for investment tax mastery is ready. Up until this point, and advisor can learn and become modestly proficient with a small number of basic tax techniques. The long-term goal of developing portfolio tax mastery is an art that requires attention, knowledge, creativity, and perseverance. The benefits to both the client and the practitioner are well worth the effort.

### Portfolio Tax-Management Corporate Training

We at Sigma1 are working diligently to develop a set of portfolio tax-management training modules ranging from basic to advanced. We have begun offering limited free “beta” courses in investment management tax basic and intermediate level techniques to investment professionals in the the Northern Colorado region. All we ask in exchange is for participants to agree to fill out a brief survey on the day of our presentation and a followup survey in 6-8 weeks to see if and how they are using the training in their investment management process. If you or your firm is interested in participating in this free beta training, feel free to contact us.

Dec 20

## How to Write a Mean-Variance Optimizer (Part III)… In R

Parts 1 and 2 left a trail of breadcrumbs to follow.  Now I provide a full-color map, a GPS, and local guide.  In other words the complete solution in the R statistical language.

Recall that the fast way to compute portfolio variance is:

$\inline&space;\dpi{300}&space;\large&space;\sigma_{p}^{2}=&space;\mathbf{w}^\top\mathbf{V}\mathbf{w}$

The companion equation is rp = wTrtn, where rtn is a column vector of expected returns (or historic returns) for each asset.  The first goal is to find find w0 and wn. w0 minimizes variance regardless of return, while wn maximizes return regardless of variance.  The goal is to then create the set of vectors {w0,w1,…wn} that minimizes variance for a given level of expected return.

I just discovered that someone already wrote an excellent post that shows exactly how to write an MVO optimizer completely in R. Very convenient!  Enjoy…

http://economistatlarge.com/portfolio-theory/r-optimized-portfolio

Sep 19

## The Equation Everyone in Finance Should Know (MV Optimization: How To, Part 2)

As the previous post shows, it all starts with…

$\inline&space;\dpi{300}&space;\large&space;\sigma_{p}^{2}=&space;\mathbf{w}^\top\mathbf{V}\mathbf{w}$

In order get close to bare-metal access to your compute hardware, use C.  In order to utilize powerful, tested, convex optimization methods use CVXGEN.  You can start with this CVXGEN code, but you’ll have to retool it…

• Discard the (m,m) matrix for an (n,n) matrix. I prefer to still call it V, but Sigma is fine too.  Just note that there is a major difference between Sigma (the covariance-variance matrix) and sigma (individual asset-return variances matrix; the diagonal of Sigma).
• Go meta for the efficient frontier (EF).  We’re going to iteratively generate/call CVXGEN with multiple scripts. The differences will be w.r.t the E(Rp).
• Computing Max: E(Rp)  is easy, given α.  [I’d strongly recommend renaming this to something like expect_ret comprised of (r1, r2, … rn). Alpha has too much overloaded meaning in finance].
• [Rmax] The first computation is simple.  Maximize E(Rp) s.t constraints.  This is trivial and can be done w/o CVXGEN.
• [Rmin] The first CVXGEN call is the simplest.  Minimize σp2 s.t. constraints, but ignoring E(Rp)
• Using Rmin and Rmax, iteratively call CVXGEN q times (i=1 to q) using the additional constraint s.t. Rp_i= Rmin + (i/(q+1)*(Rmax-Rmin). This will produce q+2 portfolios on the EF [including Rmin and Rmax].  [Think of each step (1/(q+1))*(Rmax-Rmin) as a quantization of intermediate returns.]
• Present, as you see fit, the following data…
• (w0, w1, …wq+1)
• [ E(Rp_0), …E(Rp_(q+1)) ]
• [ σ(Rp_0), …σ(Rp_(q+1)) ]

My point is that —  in two short blog posts — I’ve hopefully shown how easily-accessible advanced MVO portfolio optimization has become.  In essence, you can do it for “free”… and stop paying for simple MVO optimization… so long as you “roll your own” in house.

I do this for the following reasons:

• To spread MVO to the “masses”
• To highlight that if “anyone with a master’s in finance and computer can do MVO for free” to consider their quantitative portfolio-optimization differentiation (AKA portfolio risk management differentiation), if any
• To emphasize that this and the previous blog will not greatly help with semi-variance portfolio optimization

I ask you to consider that you, as one of the few that read this blog, have a potential advantage.  You know who to contact for advanced, relatively-inexpensive SVO software. Will you use that advantage?

Sep 05

## How to Write a Mean-Variance Optimizer: Part 1

### The Equation Everyone in Finance Show Know, but Many Probably Don’t!

… With thanks to codecogs.com which makes it really easy to write equations for the web.

This simple matrix equation is extremely powerful.  This is really two equations.  The first is all you really need.  The second is just merely there for illustrative purposes.

This formula says how the variance of a portfolio can be computed from the position weights wT = [w1 w2 … wn] and the covariance matrix V.

• σii ≡ σi2 = Var(Ri)
• σij ≡ Cov(Ri, Rj) for i ≠ j

The second equation is actually rather limiting.  It represents the smallest possible example to clarify the first equation — a two-asset portfolio.  Once you understand it for 2 assets, it is relatively easy to extrapolate to 3-asset portfolios, 4-asset portfolios, and before you know it, n-asset portfolios.

Now I show the truly powerful “naked” general form equation:
$\inline&space;\dpi{300}&space;\large&space;\sigma_{p}^{2}=&space;\mathbf{w}^\top\mathbf{V}\mathbf{w}$
This is really all you need to know!  It works for 50-asset portfolios. For 100 assets. For 1000.  You get the point. It works in general. And it is exact. It is the E = mc2 of Modern Portfolio Theory (MPT).  It at least about 55 years old (2014 – 1959), while E = mc2 is about 99 years old (2014 – 1915).  Harry Markowitz, the Father of (M)PT simply called it “Portfolio Theory” because:

Yes, I’m calling Markowitz the Einstein of Portfolio Theory AND of finance!  (Now there are several other “post”-Einstein geniuses… Bohr, Heisenberg, Feynman… just as there are Sharpe, Scholes, Black, Merton, Fama, French, Shiller, [Graham?, Buffet?]…)   I’m saying that a physicist who doesn’t know E = mc2 is not much of a physicist. You can read between the lines for what I’m saying about those that dabble in portfolio theory… with other people’s money… without really knowing (or using) the financial analog.

### Why Markowitz is Still “The Einstein” of Finance (Even if He was “Wrong”)

Markowitz said that “downside semi-variance” would be better.  Sharpe said “In light of the formidable
computational problems…[he] bases his analysis on the variance and standard deviation.”

Today we have no such excuse.  We have more than sufficient computational power on our laptops to optimize for downside semi-variance, σd. There is no such tidy, efficient equation for downside semi-variance.  (At least not that anyone can agree on… and none that that is exact in any sense of any reasonable mathematical definition of the word ‘exact’.)

Fama and French improve upon Markowitz (M)PT [I say that if M is used in MPT, it should mean “Markowitz,” not “modern”, but I digress.] Shiller, however, decimates it.  As does Buffet, in his own applied way.  I use the word decimate in its strict sense… killing one in ten.  (M)PT is not dead; it is still useful.  Diversification still works; rational investors are still risk-averse; and certain low-beta investments (bonds, gold, commodities…) are still poor very-long-term (20+ year) investments in isolation and relative to stocks, though they still can serve a role as Markowitz Portfolio Theory suggests.

### Wanna Build your Own Optimizer (for Mean-Return Variance)?

This blog post tells you most of the important bits.  I don’t really need to write part 2, do I?   Not if you can answer these relatively easy questions…

• What is the matrix expression for computing E(Rp) based on w?
• What simple constraint is w subject to?
• How does the general σp2 equation relate to the efficient frontier?
• How might you adapt the general equation to efficiently compute the effects of a Δw event where wi increases and wj decreases?  (Hint “cache” the wx terms that don’t change,)
• What other constraints may be imposed on w or subsets (asset categories within w)?  How will you efficiently deal with these constraints?
• Is short-selling allowed?  What if it is?
• OK… this one’s a bit tricky:  How can convex optimization methods be applied?

If you can answer these questions, a Part 2 really isn’t necessary is it?

Jul 16

## Binary Options and Test Taking

Most of the important financial industry tests (Series 6, 7, 24, 26, CFA I, CFA II, CFA III, etc) only have two possible binary outcomes: PASS or FAIL. Failure is a waste of time and money. Over-studying, however, can also waste time. (Studying for a PASS/FAIL test is investing in a binary “real option.”)

All of the material is worth knowing for someone, but some information is simply not relevant to everyone. For example, investment advisor reps don’t necessarily need to know all of the rules for broker dealer agents (and vise versa). Knowing the stuff that is relevant to you is more valuable than simply allowing you to pass a test.

That said, the goal is to PASS. And you’ve got a million other things to do. So what’s a quant to do? Get quantitative of course!

### Quantitative Test Prep

Step 1: Find representative sample tests. All else hinges on this. Obtaining sample tests from multiple independent sources may help.

Step 2: Determine your average score on practice tests.

Step 3: Determine the standard deviation of your scores.

Step 4: Calculate the probability of achieving a passing score given your mean score and standard deviation.

Step 5: Decide the risk/reward and whether more study provides sufficient ROI.

Assuming normal distributions, I use the 68/95/99.7 rule. Regardless of the standard deviation, if your practice average is the same as the minimum score your chance of success is only 50%. Naturally, if your mean practice score is 1-sigma above the threshold for passing, your chance on the real test is 84% [1-(1-0.68)/2]. If your mean score is plus 2 sigma, your chance of passing is almost 98% [1-(1-0.95)/2].

This little exercise shows two possible ways to improve your expected pass rate. The obvious way is getting better with the material. The less obvious way is reducing your standard deviation. Can this second way be achieved? If so how?

Keeping in mind the four-answer multiple-choice format, the mean deviation is:

MD = 2*p*(1-p)

Where p is the probability of answering a particular question correctly. Per-question deviation (PQD) is highest for p=0.5 at 0.5. PQD is lowest when p=1 at 0. For random guessing, PQD is 0.375.

Increasing your pq to from random guess 0.25 to 0.5 for a given category q will increase your expected score, but will also increase sigma. Taking the first derivative of MD with respect to p gives: 2-4p. Because the range of p is [0,1] (arguably [0.25,1)) the best incremental decrease in MD is greatest near p=1.

Now, the test candidate must decide what the the d/dt(pqc(t)) is for each question category (where t is time spent studying that category).  Studying the categories (qc) with the highest d/dt(pqc(t)) will most efficiently improve the expected score. Further studying the categories with the maximum d/dt(pqc(t))*(4p-2) will reduce PQD and hence reduce test standard deviation.

### Deeper Analysis of the Meta Problem

Naturally, this analysis only scratches the tactical surface of the “binary-test optimization meta problem.” [The test itself is the problem, the tactics are part of the meta-problem of optimizing generalized multiple-choice test prep]. Improving from p=0.8 to p=0.9 is clearly better than improving from p=0.4 to p=0.5 in terms of PQD reduction, and equal in terms of increase of expected score.

Also of relevance is PQD (modified) downside semi-deviation, which I will call PQDd. I’ll spare you the derivation; it turns out that:

PQDd = p*sqrt(2*(1-p))

This value peaks at p=2/3 with a value of 0.5443. PQDd slowly ascends as p goes from 0.25 up to 0.667, then falls pretty rapidly for values of p>0.8.

We care about the random variable S which represents the actual test score. S is a function of the mean expected score μ and standard deviation σ… in a normal distribution. What we really care about is Pr(S>=Threshold), the probability that our score meets or exceeds the minimum passing score.

PQD = PQDd only when p = 0, 0.5, or 1.  For p in (0,0.5) PQDd<PQD and for p in (0.5,1) PDQd>PQD. Even though it seems a bit strange for discrete binary distribution, p in (0,0.5) has positive skewness and p in (0.5,1) negative skewness.

In the “final” analysis the chance of passing, pr(S=>Threshold), depends on score mean, μ, and downside deviation, σd.  In turn σd depends on PQD and PQDd.

### Summary and Conclusions

Theoretically, one’s best course of action is to 1) increase the average expected score and 2) reduce σd. If practical, the best and most efficient way to achieve both objectives simultaneously is to improve areas that are in the 60-75% range (p=0.6 to 0.75) to the mid to high 90% range (p>=0.95).  This may seem counter-intuitive, but the math is solid.

Caveats: This analysis is mostly an exercise in showing the value of statics, variance, and downside variance in an area outside of finance.  It shows that there is more than one way to approach to a goal; in this case passing a standardized test.

Jun 29

## Clover Patterns Show How Portfolios Manage Risk

Illustration of Classic Covariance.

The red and green “clover” pattern illustrates how traditional risk can be modeled.  The red “leaves” are triggered when both the portfolio and the “other asset” move together in concert.  The green leaves are triggered when the portfolio and asset move in opposite directions.

Each event represents a moment in time, say the closing price for each asset (the portfolio or the new asset).  A common time period is 3-years of total-return data [37 months of price and dividend data reduced to 36 monthly returns.]

### Plain English

When a portfolio manager considers adding a new asset to an existing portfolio, she may wish to see how that asset’s returns would have interacted with the rest of the portfolio.  Would this new asset have made the portfolio more or less volatile?  Risk can be measured by looking at the time-series return data.  Each time the asset and the portfolio are in the red, risk is added. Each time they are in the green, risk is subtracted.  When all the reds and greens are summed up there is a “mathy” term for this sum: covariance.  “Variance” as in change, and “co” as in together. Covariance means the degree to which two items move together.

If there are mostly red events, the two assets move together most of the time.  Another way of saying this is that the assets are highly correlated. Again, that is “co” as in together and “related” as in relationship between their movements. If, however, the portfolio and asset move in opposite directions most of the time, the green areas, then the covariance is lower, and can even be negative.

### Covariance Details

It is not only the whether the two assets move together or apart; it is also the degree to which they move.  Larger movements in the red region result in larger covariance than smaller movements.  Similarly, larger movements in the green region reduce covariance.  In fact it is the product of movements that affects how much the sum of covariance is moved up and down.  Notice how the clover-leaf leaves move to the center, (0,0) if either the asset or the portfolio doesn’t move at all.  This is because the product of zero times anything must be zero.

Getting Technical: The clover-leaf pattern relates to the angle between each pair of asset movements.  It does not show the affect of the magnitude of their positions.

If the incremental covariance of the asset to the portfolio is less than the variance of the portfolio, a portfolio that adds the asset would have had lower overall variance (historically).  Since there is a tenancy (but no guarantee!) for asset’s correlations to remain somewhat similar over time, the portfolio manager might use the covariance analysis to decide whether or not to add the new asset to the portfolio.

### Semi-Variance: Another Way to Measure Risk

Semi-variance Visualization

After staring at the covariance visualization, something may strike you as odd — The fact that when the portfolio and the asset move UP together this increases the variance. Since variance is used as a measure of risk, that’s like saying the risk of positive returns.

Most ordinary investors would not consider the two assets going up together to be a bad thing.  In general they would consider this to be a good thing.

So why do many (most?) risk measures use a risk model that resembles the red and green cloverleaf?  Two reasons: 1) It makes the math easier, 2) history and inertia. Many (most?) textbooks today still define risk in terms of variance, or its related cousin standard deviation.

There is an alternative risk measure: semi-variance. The multi-colored cloverleaf, which I will call the yellow-grey cloverleaf, is a visualization of how semi-variance is computed. The grey leaf indicates that events that occur in that quadrant are ignored (multiplied by zero).  So far this is where most academics agree on how to measure semi-variance.

### Variants on the Semi-Variance Theme

However differences exist on how to weight the other three clover leaves.  It is well-known that for measuring covariance each leaf is weighted equally, with a weight of 1. When it comes to quantifying semi-covariance, methods and opinions differ. Some favor a (0, 0.5, 0.5, 1) weighting scheme where the order is weights for quadrants 1, 2, 3, and 4 respectively. [As a decoder ring Q1 = grey leaf, Q2 = green leaf, Q3 = red leaf, Q4 = yellow leaf].

Personally, I favor weights (0, 3, 2, -1) for the asset versus portfolio semi-covariance calculation.  For asset vs asset semi-covariance matrices, I favor a (0, 1, 2, 1) weighting.  Notice that in both cases my weighting scheme results in an average weight per quadrant of 1.0, just like for regular covariance calculations.

### Financial Industry Moving toward Semi-Variance (Gradually)

Semi-variance more closely resembles how ordinary investors view risk. Moreover it also mirrors a concept economists call “utility.” In general, losing $10,000 is more painful than gaining$10,000 is pleasurable. Additionally, losing $10,000 is more likely to adversely affect a person’s lifestyle than gaining$10,000 is to help improve it.  This is the concept of utility in a nutshell: losses and gains have an asymmetrical impact on investors. Losses have a bigger impact than gains of the same size.

Semi-variance optimization software is generally much more expensive than variance-based (MVO mean-variance optimization) software.  This creates an environment where larger investment companies are better equipped to afford and use semi-variance optimization for their investment portfolios.  This too is gradually changing as more competition enters the semi-variance optimization space.  My guestimate is that currently about 20% of professionally-managed U.S. portfolios (as measured by total assets under management, AUM) are using some form of semi-variance in their risk management process.  I predict that that percentage will exceed 50% by 2018.


text:
# Symplectic boundary

Is there any work on manifolds (perhaps contact) with symplectic boundary (not asking about the boundary of a symplectic manifold)?

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Perhaps it would be good to make your question more precise : are you asking about manifolds whose boudary can be endowed with a symplectic structure ? – Samuel Tinguely May 8 '12 at 14:54
One way to start thinking about this would be to understand what type of structure $M\times\mathbb{R}$ can have if $M$ is equipped with a symplectic form. If $\omega=-d\theta$ is exact then $dt+\theta$ is a contact form on $M\times\mathbb{R}$, however this exclude compact $M$'s. On the other side, the good notion of boundary of contact manifolds seems to be that of convex hypersurfaces, which outside a dividing set are exact symplectic. – Noz May 8 '12 at 18:14
For closed symplectic manifold, here springerlink.com/content/m080126681712458 some symplectic manifold arises as boundary of quasi-symplectic manifold which are manifold whith a closed $2$-form with one dimensionnal kernel (in the case of $M\times\mathbb{R}$ it is simply $\omega$. I don't know whether or not those type of manifold have been extensively studied. – Noz May 8 '12 at 18:15
Maybe one could make a definition like this. A filling of symplectic manifold $(M,\omega)$ is a $2n+1$ manifold with boundary $M$ and with stable Hamiltonian structure $(\Omega,\lambda)$ such that $\Omega\vert_M=\omega$. – Noz May 8 '12 at 18:27

## 2 Answers

There is a book by Guillemin, Gizburg and Karshon: Moment Maps, Cobordisms, and Hamiltonian Group Actions.

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For contact manifolds there is the notion of a convex hypersurface due to Giroux, ie, a hypersurface that is transverse to a contact vector field. A convex hypersurface is composed by two exact symplectic domains (not necessarily connected) glued together along the dividing set (the dividing set is a hypersurface in the hypersurface) that carries a natural contact structure (see also this question).

I don't know if this helps, since the boundary is not a closed symplectic manifold, but it is a very natural object to consider in contact topology.

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text:
## On the work performed by a transformation semigroup

### James East and Peter McNamara

#### Abstract

A (partial) transformation $$\alpha$$ on the finite set $$\{1,\ldots,n\}$$ moves an element $$i$$ of its domain a distance of $$|i-i\alpha|$$ units. The work $$w(\alpha)$$ performed by $$\alpha$$ is the sum of all of these distances. We derive formulae for the total work $$w(S)=\sum_{\alpha\in S}w(\alpha)$$ performed by various semigroups $$S$$ of (partial) transformations. One of our main results is the proof of a conjecture of Tim Lavers which states that the total work performed by the semigroup of all order-preserving functions on an $$n$$-element chain is equal to $$(n-1)2^{2n-3}$$.

Keywords: Transformation semigroup, work.

: Primary 20M20; Secondary 05A10.

This paper is available as a pdf (160kB) file.

 Thursday, January 14, 2010


text:
## Converting 1/3 to binary

I don't understand the first line of this conversion.  Isthis a typo?  How does 1/3 * 2 produce 1.3332?
Thanks,
Mark


text:
discussion post by Stuart Armstrong 282 days ago | 2 comments

A putative new idea for AI control; index here.

A reminder that we won’t be looking at any “utilities I might have had before I knew who I was” scenarios.

This post is for illustrating a point about acausal trade: weakening the acausal trade network(s) in any way tends to reduce acausal trade more than linearly, as the traders cut back further on their trading. And the converse for strengthening the acausal trade network(s).

# How to weaken the network

How could the acausal trade network be weakened? In potentially many ways. Greater uncertainty about the existence or the utilities of other agents, for instance. More agents who might defect from the trade, not have right utility function, or with who you can’t reach a deal because of negotiation breakdown.

Basically anything that lowers the expected number of agents acausally trading with you - and also causes those agents to similarly have a lower expectation on the number of agents trading with you.

# Illustration

Take the case where $$N=2$$, so there are only two possible agents, you ($$A_1$$) and one other ($$A_2$$), with utilities $$u_1$$ and $$u_2$$ respectively. Both agents are sure to exist, so $$q_1=q_2=1$$.

Trade can’t happen unless there is some gain from trade - if it costs you more (in terms of $$u_1$$) to increase $$u_2$$, than the gain in $$u_1$$ that the other agent is capable of giving you in exchange, then there is no trade that can happen.

So suppose you can increase $$u_2$$ quite easily initially, but it gets harder and harder as you increase it more. Specifically, if you’ve already increased $$u_2$$ by $$x$$, then it costs you, marginally, $$x$$ to increase $$u_2$$ further.

So the marginal cost is linear in $$x$$; cost, here, always refers to the decrease in $$u_1$$ needed to pay for the increase in $$u_2$$.

Assume the other agent is in exactly the same situation, mirrored.

Then, since we’re assuming that the negotiations divide the gains from trade equally, then if you and the other agent have full knowledge of these facts and use a functional decision theory.

Then the rational acausal decision is for both of you to increase the utility of the other agent by $$1$$, paying $$\int_0^1 x dx = 1/2$$ utility each, and hence each gaining $$1/2$$ utility total.

But now imagine that the probability of each agent existing is $$q=q_1=q_2$$, and that $$q$$ is not necessarily $$1$$. You know you yourself exist, so put the probability of the other agent existing at $$q$$ (note that this argument is robust to different types of anthropic reasoning, as it’s the change that happens when $$q$$ varies that’s important).

Then the rational thing for both of you is to increase the other utility until the marginal cost of doing so reaches $$q$$. Thus each agent increases the utility by $$q$$, at a cost of $$\int_0^q xdx = q^2/2$$. With probability $$q$$, the other agent exists and will thus give you $$q$$ utility. Thus the expected gain for each of you is $$q(q)-q^2/2=q^2/2$$.

The fact that this is quadratic in $$q$$ rather than linear is the “double decrease” effect: as the expected size of the network goes down, the expected return for participation goes down as well, causing those in it to decrease their own participation, until an equilibrium is reached at a lower level.

 by Owen Cotton-Barratt 277 days ago | link I think the double decrease effect kicks in with uncertainty, but not with confident expectation of a smaller network. reply
 by Stuart Armstrong 274 days ago | link I think it does do the double decrease for the known smaller network. Take three agent $$A_1$$, $$A_2$$, and $$A_3$$, with utilities $$u_1$$, $$u_2$$, and $$u_3$$. Assume the indexes $$i$$, $$j$$, and $$k$$ are always distinct. For each $$A_i$$, they can boost $$u_j$$ at the cost described above in terms of $$u_i$$. What I haven’t really specified is the three-way synergy - can $$A_i$$ boost $$u_j+u_k$$ more efficiently that simply boosting $$u_j$$ and $$u_k$$ independently? In general yes (the two utilities $$u_j$$ and $$u_k$$ are synergistic with each other, after all), but let’s first assume there is zero three-way synergy. Then each agent $$A_i$$ will sacrifice $$1/2+1/2=1$$ in $$u_i$$ to boost $$u_j$$ and $$u_k$$ each by $$1$$. Overall, each utility function goes up by $$1+1-1=1$$. This scales linearly with the size of the trade network each agent sees (excluding themselves): if there were two agents total, each utility would go up by $$1/2$$, as in the top post example. And if there were $$n+1$$ agents, each utility would go up by $$n/2$$. However, if there are any three-way, four-way,…, or $$n$$-way synergies, then the trade network is more efficient than that. So there is a double decrease (or double increase, from the other perspective), as long as there are higher-order synergies between the utilities. reply

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text:
# zbMATH — the first resource for mathematics

Quantum nonlinear Schrödinger equation. I: Intertwining operators. (English) Zbl 0614.35086
The quantum nonlinear Schrödinger equation is studied as a model of the quantum (nonrelativistic) field theory in $$1+1$$ dimensions. In § 1 the calculus of intertwining operators $$P_ N$$, $$P^*_ N$$, $$P_ N^{*- 1}$$, $$P_ N^{-1}$$ on $${\mathcal H}_ N$$ is developed which produce the equivalence of the N-particle Hamiltonian $$H_ N$$ and the free Hamiltonian $$-\Delta_ N$$. In § 2 the intertwining operators on the Fock space are studied, which are the direct sums of the corresponding operators on $${\mathcal H}_ N$$. This calculus is supposed to be the basis for subsequent publications on the explicit solution of an initial value problem for the nonlinear Schrödinger equation.
Reviewer: J.Weidmann

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35G20 Nonlinear higher-order PDEs 81T08 Constructive quantum field theory
Full Text:
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