text:
Conference Paper

Approximation Schemes for Broadcasting in Heterogenous Networks.

DOI: 10.1007/b99805 Conference: Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004, and 8th International Workshop on Randomization and Computation, RANDOM 2004, Cambridge, MA, USA, August 22-24, 2004, Proceedings
Source: DBLP

ABSTRACT

In the Minimum Common String Partition problem (MCSP) we are given two strings on input, and we wish to partition them into the same collection of substrings, minimimizing the number of the substrings in the partition. Even a special case, denoted 2-MCSP, where each letter occurs at most twice in each input string, is NP-hard. We study a greedy algorithm for MCSP that at each step extracts a longest common substring from the given strings. We show that the approximation ratio of this algorithm is between Ω(n 0·43 ) and O(n 0·69 ). In case of 2-MCSP, we show that the approximation ratio is equal to 3. For 4-MCSP, we give a lower bound of Ω(logn).

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Article: A note on the lower bound of centralized radio broadcasting for planar reachability graphs
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ABSTRACT: We consider deterministic radio broadcasting in radio networks whose nodes have full topological information about the network. We focus on radio networks having an underlying reachability graph planar. We show that there are reachability graphs such that it is impossible to complete broadcasting in less than 2.ecc(s,G) rounds, where ecc(s,G) is the eccentricity of a distinguished source ss of a graph GG. They provide a negative answer to the open problem stated by Manne, Wanq, and Xin in [F. Manne, S. Wanq, Q. Xin, Faster radio broadcasting in planar graphs, in: Proceedings of the 4th Annual Conference on Wireless on Demand Network Systems and Services, WONS’2007, IEEE Press, 2007, pp. 9–13] and show that it is impossible to complete the broadcasting task in D+O(1)D+O(1) rounds for each planar reachability graph. Particularly, we propose 3/2.D3/2.D lower bound with respect to the parameter DD — the diameter of a reachability graph. It is a nontrivial lower bound of time of centralized radio broadcasting in the case that an underlying reachability graph is planar. Moreover, we describe a generalized construction which can potentially improve the presented lower bound.
Discrete Applied Mathematics 02/2009; 157(4):853–857. DOI:10.1016/j.dam.2008.09.002 · 0.80 Impact Factor
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Article: Improved lower bound for deterministic broadcasting in radio networks
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ABSTRACT: We consider the problem of deterministic broadcasting in radio networks when the nodes have limited knowledge about the topology of the network. We show that for every deterministic broadcasting protocol there exists a network, of radius 2, for which the protocol takes at least $\Omega(\sqrt{n}) rounds for completing the broadcast. Our argument can be extended to prove a lower bound of Omega(\sqrt{nD}) rounds for broadcasting in radio networks of radius D. This resolves one of the open problems posed in [29], where in the authors proved a lower bound of$\Omega(n^{1/4}) rounds for broadcasting in constant diameter networks. We prove the new lower $\Omega(\sqrt{n})$ bound for a special family of radius 2 networks. Each network of this family consists of O(\sqrt{n}) components which are connected to each other via only the source node. At the heart of the proof is a novel simulation argument, which essentially says that any arbitrarily complicated strategy of the source node can be simulated by the nodes of the networks, if the source node just transmits partial topological knowledge about some component instead of arbitrary complicated messages. To the best of our knowledge this type of simulation argument is novel and may be useful in further improving the lower bound or may find use in other applications. Keywords: radio networks, deterministic broadcast, lower bound, advice string, simulation, selective families, limited topological knowledge.
Theoretical Computer Science 07/2011; 412(29):3568-3578. DOI:10.1016/j.tcs.2011.03.003 · 0.66 Impact Factor

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Browse Questions

# Two samples of lead oxide were separately reduced to metallic lead by heating in a current of hydrogen. The weight of lead from one oxide was half the weight of lead obtained from the other oxide. The data illustrates

$(a)\;\text { Law of reciprocal proportions } \\ (b)\;\text{Law of constant proportions} \\(c)\;\text{Law of multiple proportions} \\(d)\;\text{Law of conservation of mass}$

This shows that the weights of lead combing with field weight of oxygen are in the ratio $1/2 \; : \; 1$ or $1:2$ which is in accordance with law of multiple proportions.
Hence c is the correct answer.

format: plain


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# Exporting a plot without drawing it

I'm trying to create a rather big 3D image and export it. Currently, I create the image then display using Show, then use Export. The problem is that drawing the 3D image as an interactive object is absolutely caning my little laptop, and I don't really need it - I just want a jpeg output. Any way to do this?

What I am doing is basically along the lines of

Show[{object list}, options]
Export["name.jpeg",%,options]

So as an example, if I wanted an image of a Sin[x] graph without Mathematica first "drawing" it, how would I do it?

Thanks

-
I think the following will do the job: plot=Show[...]; Export["name.png",plot] –  Kuba Jul 19 '13 at 10:41
That did the job, thank you. –  Jonathan Thurgood Jul 19 '13 at 11:15
Maybe reading about semicolon will help. –  Jens Jul 19 '13 at 18:11

Export["name.jpeg", Show[{object list}, options], exportoptions]

which returns only the file name of the exported graphics. Another common thing is, as Kuba suggested, to suppress the output with ;
plot = Show[{object list}, options];


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# Solve simultaneous logarithmic equations with different bases? How do I solve these simultaneous equations? 2log_x y+2 log_y x=5 xy=8

Solve simultaneous logarithmic equations with different bases?
How do I solve these simultaneous equations?
$2lo{g}_{x}y+2lo{g}_{y}x=5$
$xy=8$
I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do
$x=8/y$
and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Samantha Braun
Say ${\mathrm{log}}_{x}y=a$
Therefore
$a+\frac{1}{a}=\frac{5}{2}$
$2{a}^{2}-5a+2=0$
$\left(2a-1\right)\left(a-2\right)=0$
$a=2,\frac{1}{2}$
Hence we have
${\mathrm{log}}_{x}y=2,\frac{1}{2}$
Now
$xy=8$
$1+{\mathrm{log}}_{x}y={\mathrm{log}}_{x}8$
Hence
${\mathrm{log}}_{x}8-1=2$
${\mathrm{log}}_{x}8=3$
${\mathrm{log}}_{8}x=\frac{1}{3}$
$x={8}^{\frac{1}{3}}=2$
Similarly we also have
${\mathrm{log}}_{x}8-1=\frac{1}{2}$
${\mathrm{log}}_{x}8=\frac{3}{2}$
$x=4$
The solutions hence follow.


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## WeBWorK Main Forum

### Proctored gateway test

by Lars Jensen -
Number of replies: 5
Does the proctored gateway test work? It doesn't seem to come up for the students, and can't find a place where the instructor can enter a proctor password either. The unproctored version seems to work fine.

We are running webwork 2.3-dev

Thanks,
Lars.

### Re: Proctored gateway test

by Gavin LaRose -
Hi Lars,

I need to change the default on this. In global.conf there is a setting view_proctored_tests in %permissionLevels. The default setting for this is $ta, rather than $student, which is good in cases where one has a proctored lab where a direct link can be set up to get to the proctored test. This avoids students clicking the proctored test and then having to figure out what to do when it demands a proctor login. However, in the absence of that direct link it is a bit confusing when the proctored tests don't show up on the set list page. You may want to change this permission level setting to $student. Anyone who is in the class and has a permissionLevel of 2 ("proctor") or better should be able to authorize proctored tests. A feature that I'm going to look to add is the ability to set some generic per-assignment proctor password, for cases where a class is being administered a test all at once and it's not feasible for a proctor to go around and log in to each students' computer. Please ask if that doesn't address your question. Thanks, Gavin In reply to Gavin LaRose ### Re: Proctored gateway test by Lars Jensen - Hi Gavin, Does this mean that if I want to use the "link method" to assign a proctored gateway quiz, then I need to set up a link to click on for each student? What link do I use? How exactly does the students find his/her proctored quiz? If I use the method setting the permission level to$student will a password from the instructor still be required for the student to take the quiz, or does the $student setting make the quiz unproctored? I'm confused because I don't seem to be able to get a password prompt to pop up for students no matter how I configure the proctored quiz... Lars. In reply to Lars Jensen ### Re: Proctored gateway test by Gavin LaRose - Hi Lars, This slipped past me for longer than I intended; sorry for the delay posting a response. My method of routing students to the correct class/test is by using a select element in a form that uses javascript to then construct the correct URL. It's very simple; c.f. . I can send you the code for the script if you're interested. If you set the "view_proctored_tests" setting to$student, all it changes is whether the test appears on the ProblemSets listing page. All proctoring and other characteristics remain the same.

If you have defined a quiz as a "proctored_gateway_quiz" type of assignment, before the system lets a student start a test it will insert a proctor login requirement. It also requires a proctor authentication to grade the test.

Gavin

### Re: Proctored gateway test

by Lars Jensen -
Hi Gavin,

proctored gateway/quiz doesn't work at all for me. Even when I set "Visible to students" to "Yes," the quiz is not visible. I can't make it show up in the student's list of sets.

gateway/quiz, on the other hand seems to work fine.

Lars.

### Re: Proctored gateway test

by Gavin LaRose -
Hi Lars,

Sorry about that; I guess I wasn't clear before. There is a setting in the global.conf file that sets the permission required for a proctored set to appear in a student's list of sets. This is in the %permissionLevels hash, around line 630.

I now also (finally) have some documentation for the GatewayQuiz module on-line in the WeBWorK2 Wiki. That's at http://devel.webwork.rochester.edu/twiki/bin/view/Webwork/GatewayTesting.

Please let me know if this doesn't make sense.

Thanks,
Gavin


text:
# You have 67.0 mL of a 0.400 M stock solution that must be diluted to 0.100 M. Assuming the volumes are additive, how much water should you add?

Dec 30, 2016

$\text{201 mL}$

#### Explanation:

The thing to remember about dilutions is that the ratio that exists between the concentration of the concentrated solution and that of the diluted solution is equal to the ratio that exists between the volume of the diluted solution and that of the concentrated solution.

This ratio is called the dilution factor. You thus have

$\text{DF" = "concentration of the concentrated solution"/"concentration of the diluted solution}$

and

$\text{DF" = "volume of the diluted solution"/"volume of the concentrated solution}$

In your case, the solution must be diluted from an initial concentration of $\text{0.400 M}$ to a final concentration of $\text{0.100 M}$, the equivalent of a dilution factor equal to

"DF" = (0.400 color(red)(cancel(color(black)("M"))))/(0.100color(red)(cancel(color(black)("M")))) = color(blue)(4)

You can now say that the volume of the diluted solution must be $4$ times greater than the volume of the concentrated solution, since

$\text{DF" = "volume of diluted solution"/"67.0 mL}$

$\text{volume of diluted solution" = "DF" xx "67.0 mL}$

and therefore

$\text{volume of diluted solution" = color(blue)(4) xx "67.0 mL" = "268 mL}$

"volume of water added" = "268 mL" - "67.0 mL" = color(darkgreen)(ul(color(black)("201 mL")))
of water to your concentrated solution in order to go from $\text{67.0 mL}$ of $\text{0.400 M}$ solution to $\text{268 mL}$ of $\text{0.100 M}$ solution.


text:
1R21. Thermoelastic Fracture Mechanics using Boundary Elements. Topics in Engineering, Vol 39. - Edited by DN dell’Erba (Wessex Inst of Tech, Southampton, SO40 7AA, UK). WIT Press, Southampton, UK. 2002. 146 pp. ISBN 1-85312-849-X. $109.00. Reviewed by M Bonnet (Lab de Mec des Solides, Ecole Polytechnique, Route de Saclay, Palaiseau Cedex, F-91128, France). This book comprises eight chapters, a bibliographical section, but no index. Its contents may be roughly divided into three parts: introduction and thermoelasticity and fracture mechanics background (Chapters 1 and 2, 22 pages); thermoelastic dual BEM formulation (Chapters 3 and 4, 38 pages); thermoelastic fracture mechanics computations (Chapters 5–7, 66 pages), with conclusions provided in Chapter 8. The goal of the book is to describe the formulation and implementation of the dual BEM (DBEM), which is a variant of the displacement discontinuity method (DDM), in the context of 3D thermoelastic crack problems. The dual BEM for crack problems has been investigated for a long time by the author’s advisor and other collaborators, and the main contribution of the work under review is the treatment of thermoelasticity. Chapter 1 is a general introduction to the work, allowing the author to highlight the rest of the book contents and put it all in perspective, and Chapter 2 reviews the necessary background on thermoelasticity and fracture mechanics. Chapter 3 reviews classical material concerning BEM formulations for thermoelasticity and its standard implementation. Traction boundary integral equations are introduced as well, as they are needed for the dual BEM formulation. The latter is expounded in detail in Chapter 4, together with implementation details such as the handling of the strongly and hypersingular integrals by the direct method of Guiggiani et al. 1 involved in the dual BEM. Special crack front elements and techniques for the numerical evaluation of stress intensity factors (SIFs) are presented in Chapter 5, together with several numerical examples demonstrating the accuracy of SIF evaluation for thermoelastic crack problems by the dual BEM. Chapter 6 is devoted to the formulation and implementation of $J$-integral for 3D thermoelastic crack problems, and the application of a decomposition technique similar to that of Rigby and Aliabadi 2 or Huber, Nickel, and Kuhn 3. Several supporting numerical examples are also presented. Chapter 7 is devoted to the incremental simulation thermal fatigue crack propagation, under Paris law. Several 3D simulations are presented, whose results are physically reasonable, but are not quantitatively compared to other sources. Generally speaking, this book is a clear and readable account of the research undertaken over several years by the author. It is in fact a published version of his doctoral research work, prepared at the Wessex Institute of Technology, to which WIT Press, the publisher of this book, is closely associated. As such, although the research work reported is interesting in itself, the book is very much focused on the particular line of investigation followed and lacks the completeness and perspective which would be expected of a good reference text. The overall presentation is of good quality. The book is hardbound. The typesetting, done with LaTeX, is clear even for the most complex mathematical expressions inherent with the subject matter; the figures are also of good quality. In conclusion, this reviewer expects Theromoelastic Fracture Mechanics using Boundary Elements to be useful mostly to researchers and engineers working in the same general area, and more as a detailed account of recent research than as a reference text. This reviewer strongly objects to the editorial policy behind this book, however, for several reasons: $i)$$109 (US) for 146 pages, advertisements not included, is a much too expensive price, especially since $ii)$ much of the work is available in archival journal publications by the author and his advisor 4,5,6,7; besides $iii)$ the potential buyer is not informed that the book is based on a PhD thesis dissertation, and finally $iv)$ many books in this series, including the one under review, substantially overlap each other simply because they are published versions of several PhD theses on computational fracture mechanics by BEM made in the same research group. Therefore, this reviewer advises against purchase of the book. On the other hand, the work itself is good, and its young author is not responsible for the mercantile editorial policy, so this reviewer instead encourages readers to read the papers 4,5,6,7.

1.
Guiggiani
M
,
Krishnasamy
G
,
Rudolphi
TJ
, and
Rizzo
FJ
(
1992
),
A general algorithm for the numerical solution of hypersingular boundary integral equations
,
ASME J. Appl. Mech.
,
59
,
604
614
.
2.
Huber
O
,
Nickel
J
, and
Kuhn
G
(
1993
),
On the decomposition of the J-integral for 3D problems
,
Int. J. Fract.
,
64
,
339
348
.
3.
Rigby
RH
, and
MH
(
1993
),
Mixed-mode J-integral method for the analysis of 3D fracture problems using BEM
,
Eng. Anal. Boundary Elem.
,
11
,
239
256
.
4.
Dell’Erba
DN
, and
MH
(
2000
),
On the slution of three-dimensional thermoelastic mixed-mode edge crack problems by the dual boundary element method
,
Eng. Fract. Mech.
,
66
,
269
285
.
5.
Dell’Erba
DN
, and
MH
(
2000
),
Three-dimensional thermo-mechanical fatigue crack growth using BEM
,
Int. J. Fatigue
,
22
,
261
273
.
6.
Dell’Erba
DN
, and
MH
(
2001
),
BEM analysis of fracture problems in three-dimensional thermoelasticity using J-integral
,
Int. J. Solids Struct.
,
38
,
4609
4630
.
7.
Dell’Erba
DN
,
MH
, and
Rooke
DP
(
1998
),
Dual boundary element method for three-dimensional thermoelastic crack problems
,
Int. J. Fract.
,
94
,
89
101
.


text:
# Evaluate the iterated integral by converting to polar coordinates. V 81 – x2 sin(x2 6. + y2) dy dx

Question

Evaluate the iterated integral by converting to polar coordinates.

 9 −9
 81 − x2
sin(x2 + y2) dy dx

0


text:
Tag Info

16

You are correct that you can compute Sharpe ratios on portfolios with any return distribution. The issue is comparing Sharpe ratio's of non-normally distributed portfolios (which in reality is almost any portfolio). To take an extreme example. Consider two portfolios, with returns in excess of benchmark. 50% chance of 10% return, 50% chance of a 20% ...

14

The Sharpe ratio $S_i$ of a strategy indexed by $i$ is given by the ratio of the mean excess return $m_i$ to the standard deviation of returns $\sigma_i$, The formula you have quoted is the discrete Kelly criterion. That's not so useful in trading, where the outcomes are continuous. The continuous Kelly criterion states that for every $i$th strategy with ...

13

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...

10

If we assume that by ensemble you mean an equally weighted portfolio of the two. We can express that portfolio as $$P = \frac{1}{2}x + \frac{1}{2}y$$ and the sharpe ratio of $P$, $S(P)$, will be $$\frac{\frac{1}{2}\mu_x + \frac{1}{2}\mu_y - r_f}{\sigma_{\frac{1}{2}x + \frac{1}{2}y}}$$ becuase $x$ and $y$ are uncorellated, this reduces to $$\frac{\mu_x + \... 9 I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ... 9 The HJ bounds state that$$ \frac{\sigma(m)}{\mathbb{E}[m]} \geq \frac{|\mathbb{E}[R^e]|}{\sigma(R^e)} $$where R^e is the excess return of an asset or portfolio, \sigma denotes standard deviation, \mathbb{E} denotes expectation w.r.t. the statistical measure, and m is a stochastic discount factor (or state-price density/kernel, etc.) that prices the ... 9 If Q is your covariance matrix, and r is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program.$${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$subject to$$ 1^t x = m x \in \{0,1\}^n$$Where x is a vector of indicators of which of the n assets are part of the m selected assets. While the ... 9 A Sharpe ratio of at least 1 in backtesting is a promising start, but that is just one of many statistics of interest. The Sharpe ratio measures return per unit volatility, i.e., return per unit risk. Some other important Sharpe-like measures with different definitions of risk include: Return per unit turnover (aka yield): A high yielding strategy is more ... 8 To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD. How much weight you should assign to each strategy depends on your goal. You might want to maximize the daily profit, minimize the volatility, or maximize the ... 8 Let's say your cumulative return series is \{R_i \mid i=0,1,...,N-1\} of length N days. There's 3 conventional ways to do this at this stage. You may convert the cumulative dollar return curve into arithmetic returns: \displaystyle{r_i}= \dfrac{R_i-R_{i-1}}{R_{i-1}} Or dollar returns: \displaystyle{r_i=R_i-R_{i-1}} Then take the ratio: \... 7 Your approach of computation is not very standard. Specifically, you do not need to compute the annualized monthly return. One can compute the annualized Sharpe ratio from return sampled at any frequency using the following Generalized formula:$$ Sharpe = \frac{E|R_p - R_{rf}|}{\sqrt{var(R_p - R_{rf})}} * \sqrt{N}$$where R_{rf} is the benchmark/ risk-... 7 Let R be a random vector of risky returns and let r_f denote the risk free rate. Let vector of expected returns \boldsymbol{\mu} = \operatorname{E}[R] and covariance matrix \Sigma = \operatorname{Cov}(R). The maximum Sharpe ratio portfolio among risky assets is called the tangency portfolio. Quick method to tangency portfolio Let's find the ... 6 There are sufficiently different ways to calculate the Sharpe ratio that the best advice I can give is to do whatever your boss wants. Also, if it is for a paper or research document, just make clear you document your method. My approach is usually to calculate the highest frequency Sharpe ratio I can based on the data. The higher frequency choice is to get ... 6 There is no way to calculate returns here. Let me stop you right there. You didn't open a brokerage account with zero dollars. The money you put-up for margin is your starting position. After a year of trading, you have a stopping position represented by a different amount of money in your account. The change from your starting position to your stopping is ... 6 Edited Comments: Sharpe Ratio covers both future and historical time frames (as @Freddy points out). Referencing the "Geometric Return and Portoflio Analysis", for the historical calculation, you want to make as few assumptions as possible (in my opinion). Let m_i \triangleq the monthly return for period i and r_t \triangleq annual return, for i\... 6 There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system. With riskless lending and borrowing The existence of a riskless lending and borrowing rate r_f implies that there is a single portfolio of risky assets, that is preferred to all other ... 6 To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change r to something like r-r_f, but this doesn't affect the algebraic structure). Our goal is to find the maximiser for the problem$$\max_{w}f(w):=\frac{w^T r}{(w^T\Sigma w)^{1/2}}.$$Let$$\phi: w\...

5

I would not put too much weight on any relationship between Sharpe ratio and Kelly criterion. The two are simply not logically related other than they both share common inputs. Kelly relates to sizing your position while Sharpe ratios relate your excess returns to the volatility of those. As long as you find common inputs you can always setup a ...

5

Sharpe's 1966 equation had $R_b$ defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent. If we refer to the original definitions, then that is the primary difference - Sharpe's ratio looks at reward/risk of the excess return for an asset over the risk-free rate while the ...

5

If you're annualising your data with T it should always be the same, not changing with the length of your data. To demonstrate, annualising monthly returns, the Sharpe ratios turn out fairly similar:- Note The reason for multiplying by root 12 is that the mean return is annualised by multiplying by 12 and volatility is annualised by m = 12. 12 on the ...

5

Probably missing something here but if $X$ has $E(X) = \mu$ and $variance(X) = \sigma^2$ then $2X$ has $E(2X) = 2 \mu, variance(2X) = 4\sigma^2$. Thus the sharp ratio defined as $\frac{\mu}{\sigma}$ stays the same for the 2x leveraged and the regular index.

5

You seem to use the term "volatility" to describe two very different quantities: (1) the diffusion coefficient of your SDE and (2) the standard deviation of the log-returns under your modelling assumptions. While the first may be negative, the second may not. [Interpretation 1] Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a standard ...

5

For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference.

5

try: library(PerformanceAnalytics) SharpeRatio.annualized(Returns, Rf = 0.05, scale = 252, geometric = TRUE)

5

Another intuitive interpretation of the Sharpe ratio is as a signal-to-noise ratio: $$\frac{\mu}{\sigma}$$ where you compare the strength of the signal (= return) to the level of noise (= risk). The bigger this ratio is the better: either you have more return (= signal) or you have less risk (= noise).

5

To give you an idea of industry standards for funds (although not hedge-fund specific), Morningstar and Trustnet both use monthly returns and annualize their data. See, for an example plucked at random, https://www.trustnet.com/factsheets/o/gnol/aberdeen-asia-pacific--japan-equity-i-acc. Monthly returns remain the standard because some funds only publish ...

5

Use daily P&L rather than return rate1. $$Sharpe = \frac{\mu}{\sigma}$$ To annualize, multiply by the square root of the number of trading days in the year. For US equities, that would be 252. $$Annualized\ Sharpe = \frac{\mu}{\sigma} \times \sqrt{252}$$ As for what kind of Sharpe you should target, the lowest I've seen is 5 in practice. A good ...

5

Portfolio risk metrics matter a lot for all fund managers. Though certain type fund vehicles can have completely different sets of performance metrics. It's hard to imagine a Venture Fund analyzing their portfolio using Sharpe and Sortino. Passive ETF funds probably care about Asset Under Management (AUM), inflow/outflow, and top allocation the most. Since ...

5

The Sharpe ratio and the Sortino ratio are not under the control of the ETF managers, they will be equal (or very close) to the ratios for the Index that the ETF tracks. There is not much room for differentiation here. To make their ETF attractive to customers, fund managers care about the tracking error between their fund and the index. They would like ...

4

I would absolutely use a mark-to-market value in your daily pnl for the purposes of evaluating performance (e.g. Sharpe). So, yes, that would include the value of open positions in addition to your cash balance. If you hold something for a year, that performance was earned one day at a time, not all at once. If you only look at cash, you will have a large ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Working with symmetric polynomials

I have a question related to working with symmetric polynomials in some variables. Let us say, I have an expression

expr = x^2 + y^2 + 2*x^3 + 2*y^3 + 3 x^2 y + 3 y^2 x

this is symmetric as x <-> y

I want to express this as expr1 + expr2 where

expr1 = x^2 + 2 x^3 + 3 x^2 y

and

expr2 = expr1/.{x -> y, y-> x}

How to use Mathematica to achieve this?

• Perhaps I'm missing something in the question, but Together[expr] gives what you want. – ciao Mar 4 '14 at 8:07
• What distinguishes your target expressions from, say, expr1=expr2=expr/2? The two add up to your original expression and can be obtained from each other by doing x$\leftrightarrow$y. The class of functions that does that is rather broad! How do you pick your target from within that? – Emilio Pisanty Mar 4 '14 at 12:51

The original question is not clear. There is a wide range of functionality to work with polynomials symbolically e.g. PolynomialReduce, SymmetricReduction, etc. however it is not clear why one just assumes that expr1 and expr2 are as given above and if they are unique representants of possible reductions in terms of third order polynomials of two variables, most likely they are not.
So I find that this question is asking: how can I find appropriate coefficients {a, b, c} such that expr == p[x,y] + p[y,x] where:

p[x_, y_] := a x^3 + b x^2 + c x y^2

There is SolveAlways which helps in such cases:

{p[x, y], p[y, x]} /. SolveAlways[ expr == p[x, y] + p[y, x], {x, y}]

 {{x^2 + 2 x^3 + 3 x y^2, 3 x^2 y + y^2 + 2 y^3}}

PolynomialReduce demonstrates that expr1 and expr2 can express the original polynomial expr

PolynomialReduce[ expr, {expr1, expr2}, {x, y}]

{{1, 1}, 0}

SymmetricReduction serves a different purpose, see e.g.

SymmetricReduction[expr, {x, y}, {expr1, expr2}]

The usual way is to make a Groebner basis from the polynomials that define the relationships you intend to apply ("reduce with", if you will), and then do that with PolynomialReduce.

poly = x^2 + 2 x^3 + 3 x^2 y;
polys = {poly - e1, (poly /. {x -> y, y -> x}) - e2};
gb = GroebnerBasis[polys, {x, y}];

Now for the example in question.

expr = x^2 + y^2 + 2*x^3 + 2*y^3 + 3 x^2 y + 3 y^2 x;
PolynomialReduce[expr, gb, {x, y}][[2]]

(* e1 + e2 *)

Using knowledge from CoefficientList docs (Properties and Relations section) of how to recreate multivariate polynomial from coefficient list:

coe = CoefficientList[expr, {x, y}][[ All , ;; 2]];

expr1 = Fold[FromDigits[Reverse[#1], #2] &, coe, {x, y}] // Expand

expr2 = expr1 /. {x -> y, y -> x}

x^2 + 2 x^3 + 3 x^2 y
y^2 + 3 x y^2 + 2 y^3

One way to do that is :

expr = x^2 + y^2 + 2*x^3 + 2*y^3 + 3 x^2 y + 3 y^2 x;

expr1 = x^2 + 2 x^3 + 3 x^2 y;

expr2 = expr1 /. {x -> y, y -> x}

expr1 + expr2 == expr

Another way is to define functions :

expr1[x_,y_]:=x^2 + 2 x^3 + 3 x^2 y;
expr2[x_,y_]:=expr1[y,x];


text:
# ${{\boldsymbol X}^{0}}$ (Heavy Boson) Searches in ${{\boldsymbol Z}}$ Decays INSPIRE search

Searches for radiative transition of ${{\mathit Z}}$ to a lighter spin-0 state ${{\mathit X}^{0}}$ decaying to hadrons, a lepton pair, a photon pair, or invisible particles as shown in the comments. The limits are for the product of branching ratios.
VALUE CL% DOCUMENT ID TECN  COMMENT
• • • We do not use the following data for averages, fits, limits, etc. • • •
1
 2019
RVUE ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$
2
 2019 AZ
CMS ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
3
 1998 U
ALEP ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \ell}}{{\overline{\mathit \ell}}}$ , ${{\mathit q}}{{\overline{\mathit q}}}$ , ${{\mathit g}}{{\mathit g}}$ , ${{\mathit \gamma}}{{\mathit \gamma}}$ , ${{\mathit \nu}}{{\overline{\mathit \nu}}}$
4
 1997 Q
L3 ${{\mathit X}^{0}}$ $\rightarrow$ invisible particle(s)
5
 1993 E
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$
6
 1992 D
DLPH ${{\mathit X}^{0}}$ $\rightarrow$ hadrons
7
 1992 F
L3 ${{\mathit X}^{0}}$ $\rightarrow$ hadrons
8
 1991
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ anything
$<1.1 \times 10^{-4}$ 95 9
 1991 B
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$
$<9 \times 10^{-5}$ 95 9
 1991 B
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
$<1.1 \times 10^{-4}$ 95 9
 1991 B
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$
$<2.8 \times 10^{-4}$ 95 10
 1991 D
L3 ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$
$<2.3 \times 10^{-4}$ 95 10
 1991 D
L3 ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$
$<4.7 \times 10^{-4}$ 95 11
 1991 D
L3 ${{\mathit X}^{0}}$ $\rightarrow$ hadrons
$<8 \times 10^{-4}$ 95 12
 1990 J
OPAL ${{\mathit X}^{0}}$ $\rightarrow$ hadrons
1  RAINBOLT 2019 limits are from B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \ell}^{-}}{{\mathit \ell}^{+}}{{\mathit \ell}^{-}}$ ). See their Figs. 5 and 6 for limits in mass-coupling plane.
2  SIRUNYAN 2019AZ search for ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit Z}}$ $\rightarrow$ ${{\mathit X}^{0}}{{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}{{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ events in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. See their Fig. 5 for limits on ${\mathit \sigma (}$ ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit X}^{0}}{{\mathit \mu}^{+}}{{\mathit \mu}^{-}}{)}\cdot{}$B( ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ ).
3  BARATE 1998U obtain limits on B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}^{0}}$ )B( ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \ell}}{{\overline{\mathit \ell}}}$ , ${{\mathit q}}{{\overline{\mathit q}}}$ , ${{\mathit g}}{{\mathit g}}$ , ${{\mathit \gamma}}{{\mathit \gamma}}$ , ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ ). See their Fig.$~$17.
4  See Fig.$~$4 of ACCIARRI 1997Q for the upper limit on B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}^{0}}$ ; $\mathit E_{{{\mathit \gamma}}}>\mathit E_{{\mathrm {min}}}$) as a function of $\mathit E_{{\mathrm {min}}}$.
5  ACTON 1993E give $\sigma\mathrm {( {{\mathit e}^{+}} {{\mathit e}^{-}} \rightarrow {{\mathit X}^{0}} {{\mathit \gamma}} )}\cdot{}$B( ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ )$<0.4~$pb (95$\%$CL) for ${\mathit m}_{{{\mathit X}^{0}}}=60$ $\pm2.5$ GeV. If the process occurs via $\mathit s$-channel ${{\mathit \gamma}}$ exchange, the limit translates to $\Gamma\mathrm {({{\mathit X}^{0}})}\cdot{}$B( ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ )${}^{2}<$20 MeV for ${\mathit m}_{{{\mathit X}^{0}}}$ = $60$ $\pm1$ GeV.
6  ABREU 1992D give ${{\mathit \sigma}_{{Z}}}$ $\cdot{}$ B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}^{0}}$ ) $\cdot{}$ B( ${{\mathit X}^{0}}$ $\rightarrow$ hadrons) $<(3 - 10)~$pb for ${\mathit m}_{{{\mathit X}^{0}}}$ = $10 - 78$ GeV. A very similar limit is obtained for spin-1 ${{\mathit X}^{0}}$.
7  ADRIANI 1992F search for isolated ${{\mathit \gamma}}$ in hadronic ${{\mathit Z}}$ decays. The limit ${{\mathit \sigma}_{{Z}}}$ $\cdot{}$ B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}^{0}}$ ) $\cdot{}$ B( ${{\mathit X}^{0}}$ $\rightarrow$ hadrons) $<(2 - 10)~$pb (95$\%$CL) is given for ${\mathit m}_{{{\mathit X}^{0}}}$ = $25 - 85$ GeV.
8  ACTON 1991 searches for ${{\mathit Z}}$ $\rightarrow$ ${{\mathit Z}^{*}}{{\mathit X}^{0}}$ , ${{\mathit Z}^{*}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , or ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ . Excludes any new scalar ${{\mathit X}^{0}}$ with ${\mathit m}_{{{\mathit X}^{0}}}<9.5$ GeV/$\mathit c$ if it has the same coupling to ${{\mathit Z}}{{\mathit Z}^{*}}$ as the MSM Higgs boson.
9  ACTON 1991B limits are for ${\mathit m}_{{{\mathit X}^{0}}}$ = $60 - 85$ GeV.
10  ADEVA 1991D limits are for ${\mathit m}_{{{\mathit X}^{0}}}$ = $30 - 89$ GeV.
11  ADEVA 1991D limits are for ${\mathit m}_{{{\mathit X}^{0}}}$ = $30 - 86$ GeV.
12  AKRAWY 1990J give $\Gamma\mathrm {( {{\mathit Z}} \rightarrow {{\mathit \gamma}} {{\mathit X}^{0}} )}\cdot{}$B( ${{\mathit X}^{0}}$ $\rightarrow$ hadrons) $<~1.9$ MeV (95$\%$CL) for ${\mathit m}_{{{\mathit X}^{0}}}$ = $32 - 80$ GeV. We divide by $\Gamma\mathrm {({{\mathit Z}})}$ = $2.5$ GeV to get product of branching ratios. For nonresonant transitions, the limit is B( ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit q}}{{\overline{\mathit q}}}$ ) $<~8.2$ MeV assuming three-body phase space distribution.
References:
 RAINBOLT 2019
PR D99 013004 Branching fraction for $Z$ decays to four leptons and constraints on new physics
 SIRUNYAN 2019AZ
PL B792 345 Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\to4\mu$ events in proton-proton collisions at $\sqrt{s} =$ 13 TeV
 BARATE 1998U
EPJ C4 571 Search for Evidence of Compositeness at LEP I
 ACCIARRI 1997Q
PL B412 201 Search for New Physics in Energetic Single Photon Production in ${{\mathit e}^{+}}{{\mathit e}^{-}}$ Annihilation at the ${{\mathit Z}}$ Resonance
 ACTON 1993E
PL B311 391 Search for Anomalous Production of High Mass Photon Pairs in ${{\mathit e}^{+}}{{\mathit e}^{-}}$ Collisions at LEP
 ABREU 1992D
ZPHY C53 555 Study of Final State Photons in Hadronic ${{\mathit Z}^{0}}$ Decay and Limits on New Phenomena
PL B292 472 Isolated Hard Photon Emission in Hadronic ${{\mathit Z}^{0}}$ Decays
PL B273 338 A Measurement of Photon Radiation in Lepton Pair Events from ${{\mathit Z}^{0}}$ Decays
PL B262 155 Search for Narrow High Mass Resonances in Radiative Decays of the ${{\mathit Z}^{0}}$
PL B246 285 Evidence for Final State Photons in Multihadronic Decays of the ${{\mathit Z}^{0}}$


text:
The Universe of Discourse

Wed, 25 Sep 2013

In which I revisit the pastimes of my misspent youth
Last weekend I was at a flea market and saw an HP-15C calculator for \$10. The HP-15C was the last pocket calculator I owned, some time before pocket calculators became ridiculous. It was a really nice calculator when I got it in 1986, one of my most prized possessions.

I lost my original one somewhere along the way, and also the spare I had bought from a friend against the day when I lost the original, and I was glad to get another one, even though I didn't have any idea what I was going to do with it. My phone has a perfectly serviceable scientific calculator in it, a very HP-ish one called RealCalc. (It's nice, you should check it out.) The 15C was sufficiently popular that someone actually brought it back a couple of years ago, in a new and improved version, with the same interface but 21st-century technology, and I thought hard about getting one, but decided I couldn't justify spending that much money on something so useless, even if it was charming. Finding a cheap replacement was a delightful surprise.

Then on Friday night I was sitting around thinking about which numbers n are such that !!10n^2+9!! a perfect square, and I couldn't think of any examples except for 0, 2, and 4. Normally I would just run and ask the computer, which would take about two minutes to write the program and one second to run it. But I was out in the courtyard, it was a really nice evening, my favorite time of the year, the fading light was beautiful, and I wasn't going to squander it by going inside to brute-force some number problem.

But I did have the HP-15C in my pocket, and the HP-15C is programmable, by mid-1980s programmable calculator standards. That is to say, it is just barely programmable, but just barely is all you need to implement linear search for solutions of !!10n^2+9 = m^2!!. So I wrote the program and discovered, to my surprise, that I still remember many of the fussy details of how to program an HP-15C. For example, the SST button single-steps through the listing, in program mode, but single-steps the execution in run mode. And instead of using the special test 5 to see if the x and y registers are equal you might as well subtract them and use the x=0 test; it uses the same amount of program memory and you won't have to flip the calculator over to remember what test 5 is. And the x2 and INT() operations are on the blue shift key.

Here's the program:

        001 - 42,21,11
002 -    43 11
003 -        1
004 -        0
005 -       20
006 -        9
007 -       40
008 -       36
009 -       11
010 -       36
011 -    43 44
012 -       30
013 -    43 20
014 -       31
015 -    43 32
016 - 42,21,12
017 -       40
018 -    45  0
019 -    32 11
020 -        2
021 - 44,40, 0
022 -    22 12

I see now that when I tested !!\sqrt{10n^2+9}!! for integrality, I did it the wrong way. My method used four steps:
        010 -       36   -- push stack
011 -    43 44   -- x ← INT(x)
012 -       30   -- subtract
013 -    43 20   -- test x=0 ?

but it would have been better to just test the fractional part of the value for zeroness:
                 42 44   -- x ← FRAC(x)
43 20   -- test x=0 ?

Saving two instructions might not seem like a big deal, but it takes the calculator a significant amount of time to execute two instructions. The original program takes 55.2 seconds to find n=80; with the shorter code, it takes only 49.2 seconds, a 10% improvement. And when your debugging tool can only display a single line of numeric operation codes, you really want to keep the program as simple as you can.

Besides, stuff should be done right. That's why it's called "right".

But I kind of wish I had that part of my brain back. Who knows what useful thing I would be able to remember if I wasn't wasting my precious few brain cells remembering that the back-step key ("BST") is on the blue shift, and that "42,21,12" is the code for "subroutine B starts here".

Anyway, the program worked, once I had debugged it, and in short order (by 1986 standards) produced the solutions n=18, 80, 154, which was enough to get my phone to search the OEIS and find the rest of the sequence. The OEIS entry mentioned that the solutions have the generating function

$$\frac{2x^2(1+2x+9x^2+2x^3+x^4)}{1-38x^3+x^6}$$

and when I saw that !!38x^3!! in the denominator, I laughed, really loudly. My new neighbor was in her back yard, which adjoins the courtyard, and heard me, and said that if I was going to laugh like that I had to explain what was so funny. I said “Do you really want to know?” and she said yes, but I think she was mistaken.

Sat, 18 Sep 2010

The coming singularity
A Wikipedia user asked on the Wikipedia reference desk page:

You know the idea that eventually we'll be able to download our brains/personalities to computer, to achieve physical immortality?
There is a big trouble with this version of the immortality thing that people rarely mention. You go to the scanning and uploading center one day and write them a check. They scan and upload your brain, and say "All done, time to go home!"

"That's it?" you say. "I don't feel any different."

"Well, of course not. You're no different. But the uploaded version is immortal."

Then you go home and grow old, and every once in a while you get an email:

	From: Mark Dominus (Immortal version) <mjd@forever.org>
Subject: Wish you were here

Having a great time here in paradise! Haven't aged a day.
Thanks!

Or maybe one like this:

	From: Mark Dominus (Immortal version) <mjd@forever.org>
Subject: Guess what I just did?

Today I had sex with Arthur C. Clarke while swimming in a
giant hot-fudge sundae. It was totally awesome! Too bad you
couldn't be here. I know exactly how much you would
have enjoyed it.

Sorry to hear you're sick.  Hope the chemo works out.

Then you die, and some computer program somewhere simulates the immortal version of you having a little ritual observance to mark your passing.

Sometimes the proponents of this scheme try to conceal this enormous drawback by suggesting that they obliterate the original version of you immediately after the upload. Consider carefully whether you believe this will improve the outcome.

Thu, 08 May 2008
 Order The Origin of Consciousness in the Breakdown of the Bicameral Mind with kickback no kickback
One of my favorite books is The Origin of Consciousness in the Breakdown of the Bicameral Mind, by Julian Jaynes, a psychologist at Princeton University. Nearly everyone seems to agree that this is either a work of profound genius, or of profound crackpottery, and also that they aren't sure which it is. Jaynes' theory, as nearly as I can summarize the book, is something like this:

Human consciousness (which Jaynes describes and defines in considerable detail) is a relatively recent development, dating back at most only about 3,000 years or so.

That is the shocking part of the theory. Most people probably imagine consciousness arising much, much earlier, perhaps before language. Jaynes disagrees. In his theory, language, and in particular its mediation of thought through the use of metaphors, is an essential prerequisite for consciousness. And his date for the development of consciousness means that human consciousness would postdate several other important developments, such as metalworking, large-scale agriculture, complex hierarchical social structures, and even writing. Jaynes thinks that the development of consciousness is a historical event and is attested to by written history. He tries to examine the historical record to find evidence not only of preconscious culture, but of the tremendous upheavals that both caused and were the result of the arrival of consciousness.

If preconscious humans farmed, built temples and granaries, and kept records, they must have had some sort of organizing behavior that sufficed in place of consciousness. Jaynes believes that prior to the development of consciousness, humans had a very different mentality. When you or I need to make a decision, we construct a mental narrative, in which we imagine ourselves trying several courses of action, and attempt to predict the possible consequences. Jaynes claims that Bronze Age humans did not do this. What then?

Instead, says Jaynes, the two halves of the brain were less well-integrated in preconscious humans than they are today. The preconscious mentality was "bicameral", with the two halves of the brain operating more independently, and sometimes at odds with each other. The left hemisphere, as today, was usually dominant. Faced with a difficult decision, preconscious human would wait, possibly undergoing (and perhaps even encouraging) an increasingly agitated physical state, until they heard the voice of a god directing them what to do. These hallucinated voices were generated by the right hemisphere of the brain, and projected internally into the left hemisphere.

For example, when the Iliad says that the goddess Athena spoke to Achilles, and commanded and physically restrained him from killing Agamemnon, it is not fabulating: Achilles' right brain hallucinated the voice of the Goddess and restrained him.

In Jaynes' view, there is a large amount of varied literary, anthropological, and neurological evidence supporting this admittedly bizarre hypothesis. For example, he compares the language used in the Biblical Book of Amos (bicameral) with that in Ecclesiastes (conscious). He finds many examples of records from the right period of history bewailing the loss of the guidance of the gods, the stilling of their voices, and the measures that people took, involving seers and prophets, to try to bring the guiding voices back.

Jaynes speculates that mental states such as schizophrenia, which are frequently accompanied by irresistible auditorily hallucinated commands, may be throwbacks to the older, "bicameral" mental state.

Whether you find the theory amazingly brilliant or amazingly stupid, I urge to to withhold judgment until you have read the book. It is a fat book, and there is a mass of fascinating detail. As I implied, it's either a work of profound genius or of profound crackpottery, and I'm not sure which. (Yaakov Sloman tells me that the response to Wittgenstein's Tractatus Logico-Philosophicus was similarly ambivalent when it was new. I think the consensus is now on the genius side.) Either way, it is quite fascinating. There needs to be some theory to account for the historical development of consciousness, and as far as I know, this is the only one on offer.

Anyway, I did not mean to get into this in so much detail. The reason I brought this up is that because of my continuing interest in Jaynes' theory, and how it is viewed by later scholars, I am reading Muses, Madmen, and Prophets: Rethinking the History, Science, and Meaning of Auditory Hallucination by Daniel B. Smith. I am not very far into it yet, but Smith has many interesting things to say about auditory hallucinations, their relationship to obsessive-compulsive disorder, and other matters.

On page 37 Smith mentions a paper, which as he says, has a wonderful title: "Involuntary Masturbation as a Manifestation of Stroke-Related Alien Hand Syndrome". Isn't that just awesome? It gets you coming and going, like a one-two punch. First there's the involuntary masturbation, and while you're still reeling from that it follows up with "alien hand syndrome".

To save you the trouble of reading the paper, I will summarize. The patient is a 72-year-old male. He has lesions in his right frontal lobe. He is experiencing "alien hand syndrome", where his hand seems to be under someone else's control, grabbing objects, like the TV remote control, or grabbing pieces of chicken off his plate and feeding them to him, when what he wanted to do was feed himself with the fork in his right hand. "During his hospital stay, the patient expressed frustration and dismay when he realized that he was masturbating publicly and with his inability to voluntarily release his grasp of objects in the left hand."

Reaction time tests of his hands revealed that when the left hand was under his conscious control, it suffered from a reaction time delay, but when it was under the alien's control, it didn't.

Whee, freaky.

Sun, 24 Jun 2007

Do you dream in color?
People have occasionally asked me whether I dreamt in color or on black-and-white, by which I suppose they meant grayscale. This question was strange to me the first time I heard it, because up to then it had not occurred to me that anyone did not dream in color. I still find it strange, and I had to do a Google search to verify that there really are people who claim not to dream in color.

One time, when I replied that I did dream in color, my interlocutor asked me if I was sure: perhaps I dreamt in black and white, but only remembered it as being in color later.

I am sure I dream in color, because on more than one occasion I have had discussions in dreams about colors of objects. I can't remember any examples right now, but it was something like this: "Give me the red apple." "Okay, here." "That is not the red apple, that is the green apple!" And then I looked and saw that the apple I had thought was red was really green.

One could still argue that I wasn't really dreaming in color, that it only seemed like that, or something. It's a delicate philosophical point. One could also argue that I didn't have any dream at all, I only thought I did after I woke up. I suppose the only refutations of such an argument either appeal to neurology or involve a swift kick in the pants.

And then suppose I have a dream in which I take LSD and have marvelous hallucinations. Did I really have hallucinations? Or did I only dream them? If I dream that I kill someone, we agree that it wasn't real, that a dream murder is not a real murder; it is only in your head. But hallucinations, by definition, are only in your head even when they are real, so don't dream hallucinations have as much claim to reality as waking hallucinations?

One might argue that dreamt LSD hallucinations are likely to be qualitatively very different from real LSD hallucinations—less like real LSD hallucinations, say, and more like, well, dreams. But this only refutes the claim that the dream hallucinations were LSD hallucinations. And nobody was going to claim that they were LSD hallucinations anyway, since no actual LSD was involved. So this doesn't address the right question.

Stickier versions of the same problem are possible. For example, suppose I give Bill a little piece of paper and tell him it is impregnated with LSD. It is not, but because of the placebo effect, Bill believes himself to be having an LSD trip and reports hallucinations. There was no LSD involved, so the hallucinations were only imaginary. But even real hallucinations are only imaginary. Are we really justified in saying that Bill is mistaken, that he did not actually hallucinate, but only imagined that he did? That seems like a very difficult position to defend.

I seem to have wandered from the main point, which is that I had another dream last night that supports my contention that I dream in color. I was showing my friend Peter some little homunculi that had been made long ago from colored pipe cleaners, shiny paper, and sequins by my grandmother's friend Kay Seiler. Originally there had been ten of these, but in the dream I had only five. When my grandmother had died, my sister and I had split the set, taking five each. In place of the five originals I was missing, I had five copies, which were identifiable as such because they were in grayscale. Presumably my sister had grayscale copies of the originals I retained. I explained this to Peter, drawing his attention to the five full-color homunculi and the five grayscale ones.

So yes, barring philosophical arguments that I think deserve a kick in the pants, I am sure that I dream in color.


text:
# Linear dependence

Let $X$ be a Hilbert space and let $f:X\to\mathbb{R}$.

Let $M=\{x\in X:f(x)=0\}$ be the nullspace of $f$.

Let $M^\perp=\{x\in X:(x,y)=0\text{ for all }y\in M\}$ be the orthogonal complement of $M$.

Show that $M^\perp$ is at most one-dimensional, i.e. that if $x$ and $y$ are members of $M^\perp$ then there exist scalars $a$ and $b$, not both zero, such that $ax+by=0$.

-
I take it $f$ is assumed linear. – Gerry Myerson Mar 7 '12 at 5:52

(Assuming $f$ is linear, of course...)
If $M=X$, then $M^{\perp}=\{\mathbf{0}\}$ and we are done.
Assume then that $M\neq X$. Let $x,y\in M^{\perp}$. If $x=\mathbf{0}$, then $1x+0y = \mathbf{0}$ and we are done. If $y=\mathbf{0}$, then $0x+1y=\mathbf{0}$ and we are done. So we may assume that $x\neq\mathbf{0}$ and $y\neq\mathbf{0}$.
Then $x\notin M$, and $y\notin M$. Hence $f(x)=\alpha\in\mathbb{R}$, $f(y)=\beta\in\mathbb{R}$, and $\alpha\beta\neq 0$.
What can you say about $\beta x-\alpha y$?


text:
# zbMATH — the first resource for mathematics

A large deformation, rotation-free, isogeometric shell. (English) Zbl 1228.74077
Summary: Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner-Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using $$C^{0}$$ basis functions are possible, they are complicated in comparison to their $$C^{1}$$ counterparts. A $$C^{k}$$-continuous, $$k\geq 1$$, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74K25 Shells 65D17 Computer-aided design (modeling of curves and surfaces)
##### Keywords:
isogeometric analysis; NURBS; shells; rotation-free; metal stamping
LS-DYNA; ISOGAT
Full Text:
##### References:
 [1] Akkerman, I.; Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Computational mechanics, 41, 371-378, (2008) · Zbl 1162.76355 [2] Auricchio, F.; Beirao da Veiga, L.; Buffa, A.; Lovadina, C.; Reali, A.; Sangalli, G., A fully locking-free isogeometric approach for plane linear elasticity problems: a stream function formulation, Computer methods in applied mechanics and engineering, 197, 160-172, (2007) · Zbl 1169.74643 [3] Auricchio, F.; Da Veiga, L.B.; Hughes, T.J.R.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Mathematical models and methods in applied sciences, 20, 11, 2075-2107, (2010) · Zbl 1226.65091 [4] Bazilevs, Y.; Calo, V.M.; Cottrell, J.A.; Evans, J.A.; Hughes, T.J.R.; Lipton, S.; Scott, M.A.; Sederberg, T.W., Isogeometric analysis using T-splines, Computer methods in applied mechanics and engineering, 199, 5-8, 229, (2010) · Zbl 1227.74123 [5] Bazilevs, Y.; Calo, V.M.; Cottrell, J.A.; Hughes, T.J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Computer methods in applied mechanics and engineering, 197, 173-201, (2007) · Zbl 1169.76352 [6] Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R.; Zhang, Y., Isogeometric fluid – structure interaction: theory, algorithms, and computations, Computational mechanics, 43, 3-37, (2008) · Zbl 1169.74015 [7] Bazilevs, Y.; Calo, V.M.; Zhang, Y.; Hughes, T.J.R., Isogeometric fluid – structure interaction analysis with applications to arterial blood flow, Computational mechanics, 38, 310-322, (2006) · Zbl 1161.74020 [8] Bazilevs, Y.; Hughes, T.J.R., NURBS-based isogeometric analysis for the computation of flows about rotating components, Computational mechanics, 43, 143-150, (2008) · Zbl 1171.76043 [9] Belytschko, T.; Lin, J.I.; Tsay, C.S., Explicit algorithms for the nonlinear dynamics of shells, Computer methods in applied mechanics and engineering, 42, 225-251, (1984) · Zbl 0512.73073 [10] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Computer methods in applied mechanics and engineering, 99, 2356-2394, (1992) · Zbl 0763.73052 [11] Benson, D.J., Stable time step estimation for multi-material Eulerian hydrocodes, Computer methods in applied mechanics and engineering, 167, 191-205, (1998) · Zbl 0945.74064 [12] D.J. Benson, Y. Bazilevs, E. De Luycker, M.-C. Hsu, M. Scott, T.J.R. Hughes, T. Belytschko, A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM, International Journal for Numerical Methods in Engineering, in press. · Zbl 1197.74177 [13] D.J. Benson, Y. Bazilevs, M.C. Hsu, T.J.R. 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Dörfel, B. Jüttler, B. Simeon, Adaptive isogeometric analysis by local h-refinement with T-splines, Computer Methods in Applied Mechanics and Engineering (2009). doi:10.1016/j.cma.2008.07.012. [20] R. Dick, Personal communication, Alcoa, 2009. [21] Elguedj, T.; Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R., $$\overline{B}$$ and $$\overline{F}$$ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements, Computer methods in applied mechanics and engineering, 197, 33-40, (2008) · Zbl 1194.74518 [22] Flanagan, D.P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, International journal for numerical methods in engineering, 17, 679-706, (1981) · Zbl 0478.73049 [23] Gresho, P.M.; Sani, R.L., Incompressible flow and the finite element method, (1998), Wiley New York, NY · Zbl 0941.76002 [24] J.O. Hallquist, LS-DYNA Theoretical Manual, Technical Report, Livermore Software Technology Corporation, 1998. [25] Hill, R., A theory of yielding and plastic flow of anisotropic metals, Proceedings of the royal society of London, series A, 193, 281-297, (1948) · Zbl 0032.08805 [26] Hughes, T.J.R., The finite element method, linear static and dynamic finite element analysis, (2000), Dover [27] Hughes, T.J.R.; Liu, W.K., Nonlinear finite element analysis of shells: part I. three-dimensional shells, Computer methods in applied mechanics and engineering, 26, 331-362, (1981) · Zbl 0461.73061 [28] Hughes, T.J.R.; Pister, K.S.; Taylor, R.L., Implicit – explicit finite elements in nonlinear transient analysis, Computer methods in applied mechanics and engineering, 17/18, 159-182, (1979) · Zbl 0413.73074 [29] Hughes, T.J.R.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis, Computer methods in applied mechanics and engineering, 199, 5-8, 301-313, (2010) · Zbl 1227.65029 [30] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer methods in applied mechanics and engineering, 194, 4135-4195, (2005) · Zbl 1151.74419 [31] Hughes, T.J.R.; Reali, A.; Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS, Computer methods in applied mechanics and engineering, 197, 4104-4124, (2008) · Zbl 1194.74114 [32] B.M. Irons, Applications of a Theorem on Eigenvalues to Finite Element Problems, Technical Report CR/132/70, University of Wales, Department of Civil Engineering, Swansea, UK, 1970. [33] Kiendl, J.; Bazilevs, Y.; Hsu, M.-C.; Wnchner, R.; Bletzinger, K.-U., The bending strip method for isogeometric analysis of kirchhoff – love shell structures comprised of multiple patches, Computer methods in applied mechanics and engineering, 199, 37-40, 2403-2416, (2010) · Zbl 1231.74482 [34] J.K. Lee, G.L. Kinzel, R.H. Wagoner, in: NUMISHEET 96 Third International Conference: Numerical Simulation of 3-D Sheet Metal Forming Processes Verification of Simulations and Experiments, Ohio State University, 1996. [35] Morino, L.; Leech, J.W.; Wittmer, E.A., An improved numerical calculation technique for large elastic – plastic transient deformations of thin shells: part 2 - evaluation and applications, Journal of applied mechanics, 429-435, (1971) [36] Piegl, L.; Tiller, W., The NURBS book (monographs in visual communication), (1997), Springer-Verlag New York [37] Schoenfeld, S.; Benson, D.J., Quickly convergent integration methods for plane stress plasticity, Communications in applied numerical methods, 9, 293-305, (1993) · Zbl 0777.73071 [38] Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (2000), Springer New York · Zbl 0934.74003 [39] Wall, W.A.; Frenzel, M.A.; Cyron, C., Isogeometric structural shape optimization, Computer methods in applied mechanics and engineering, 197, 2976-2988, (2008) · Zbl 1194.74263 [40] Wikipedia.org. . [41] Wilkinson, J.H., The algebraic eigenvalue problem, (1965), Oxford University Press · Zbl 0258.65037
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# Another way to solve $\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$ without the substitution $y=\tan\left(\frac{x}{2}\right)$?

Is there another way to solve an integral $$\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$$ without the substitution $$y=\tan\left(\frac{x}{2}\right)$$?

$$\large \int \frac{\sin^3(x)}{1+\cos^2(x)}\ dx$$ is easily solved using the substitution $$y=\cos(x)$$. What if the power of sine is even?

• $y = \tan x$ and $y = \cot x$ should also work. – an4s Jul 9 at 19:53
• You may solve it numerically. But i am sure you want an analytic solution. – ConvexHull Jul 9 at 19:55
• @ConvexHull: Huh? How can you solve that indefinite integral "numerically"?!? – David G. Stork Jul 9 at 20:02
• You may always find an anti-derivative (or indefinite integral) approximately if you replace the integration kernel with a simpler expression, e.g. with a polynomial expansion of arbitrary order, before you apply the integration. This is often done with analytical !not! integrate able functions, e.g. the error function. – ConvexHull Jul 9 at 20:08
• @DavidG.Stork Simply spoken, approximate $\frac{\sin^4(x)}{1+\cos^2(x)} \approx \sum_i^N a_i \Phi_i(x)$. With high $N$ you will get a quasi exact solution up to round of errors. – ConvexHull Jul 9 at 21:27

Hint:

Bioche's rules suggest to use the substitution $$t=\tan x,\quad \mathrm d x=\frac{\mathrm dt}{1+t^2}.$$ Indeed, as $$\cos^2x=\frac 1{1+t^2}$$, $$\:\sin ^2x=\frac{t^2}{1+t^2}$$, one obtains $$\int\frac{\sin^4(x)}{1+\cos^2(x)}\,\mathrm dx=\int\frac{\frac{t^4}{(1+t^2)^2}}{1+\frac1{1+t^2}}\,\frac{\mathrm dt}{1+t^2}= \int\frac{t^4\,\mathrm dt}{(1+t^2)^2(2+t^2)}$$ which is comptuted with a decomposition into partial fractions: $$\frac{t^4}{(1+t^2)^2(2+t^2)}=\frac{At+B}{1+t^2}+\frac{Ct+D}{(1+t^2)^2}+\frac{Et+F}{2+t^2}.$$

HINT: $$\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx=\int \frac{(1-\cos^2(x))^2}{1+\cos^2(x)}\ dx$$ $$=\int \frac{(1+\cos^2(x))^2-4(1+\cos^2(x))+4}{1+\cos^2(x)}\ dx$$ $$=\int \left(1+\cos^2(x)-4+\frac{4}{1+\cos^2(x)}\right)\ dx$$ $$=\int \left(\frac{1+\cos(2x)}{2}-3+\frac{4\sec^2x}{\tan^2(x)+2}\right)\ dx$$

• Will this work for every even power of sine? – Andrew Fount Jul 9 at 20:06
• for higher even powers, the conversion $\sin^2x\to \cos^2x$ is possible but becomes much lengthier and complicated – Harish Chandra Rajpoot Jul 9 at 20:10
• For higher even powers of sine, you can use a recurrence relation combined with Harish' answer, as you can see below @AndrewFount – Riemann'sPointyNose Jul 9 at 20:36

Denote

$${I_n = \int\frac{\sin^{2n}(x)}{1 + \cos^2(x)}dx}$$

Then

$${I_{n}=\int\sin^2(x)\frac{\sin^{2n-2}(x)}{1+\cos^2(x)}dx=\int(1-\cos^2(x))\frac{\sin^{2n-2}(x)}{1+\cos^2(x)}dx}$$

If you expand this, you get

$${=I_{n-1} - \int \cos^2(x)\frac{\sin^{2n-2}(x)}{1+\cos^2(x)}dx=I_{n-1}-\int \sin^{2n-2}(x) - \frac{\sin^{2n-2}(x)}{1+\cos^2(x)}dx}$$

Hence

$${I_{n}=2I_{n-1} - \int \sin^{2n-2}(x)dx}$$

Now define $${S_n = \int\sin^{2n}(x)dx}$$. Then

$${S_{n}=\int \sin^2(x)\sin^{2n-2}(x)dx=S_{n-1}-\int \cos^2(x)\sin^{2n-2}(x)dx}$$

On the rightmost integral, using integration by parts yields

$${\int\cos^2(x) \sin^{2n-2}(x)dx=\frac{\cos(x)\sin^{2n-1}(x)}{2n-1}+\frac{1}{2n-1}\int \sin^{2n}(x)dx}$$

So overall

$${\Rightarrow S_n = S_{n-1}-\frac{\cos(x)\sin^{2n-1}(x)}{2n-1} - \frac{1}{2n-1}S_n}$$

And so

$${\left(\frac{2n}{2n-1}\right)S_n = S_{n-1} - \frac{\cos(x)\sin^{2n-1}(x)}{2n-1}}$$

$${\Rightarrow S_n = \frac{(2n-1)S_{n-1}}{2n} - \frac{\cos(x)\sin^{2n-1}(x)}{2n}}$$

Now you have two recursive relations that will help you compute the integral for higher even powers of $${\sin(x)}$$:

$${I_n = 2I_{n-1} - S_{n-1}}$$

$${S_{n} = \frac{(2n-1)S_{n-1}}{2n} - \frac{\cos(x)\sin^{2n-1}(x)}{2n}}$$

HINT

You can use $$\sin(x) = \tan(x) / \sec(x)$$ and $$\cos(x) = 1/\sec(x)$$.

and then you it will bring you to Partial Fractions Decomposition.

$$\sin^4x=(\sin^2x+1)(\sin x+\sqrt{2})(\sin x-\sqrt{2})+(2+\sin^2x)$$ $$1+\cos^2x=(\sqrt{2}+\sin x)(\sqrt{2}-\sin x)$$ and so your integral becomes: $$\int\frac{\sin^4x}{1+\cos^2x}dx=\int\frac{(\sin^2x+1)(\sin x+\sqrt{2})(\sin x-\sqrt{2})+(2+\sin^2x)}{(\sqrt{2}+\sin x)(\sqrt{2}-\sin x)}dx=-\int1+\sin^2xdx+\int\frac{2+\sin^2x}{2-\sin^2x}dx$$

If you enjoy special functions, using $$t=\tan(x)$$ $$I_n=\int \frac{\sin^n(x)}{1+\cos^2(x)}\ dx=\int \left(\frac{t}{\sqrt{t^2+1}}\right)^n\frac{dt}{t^2+2}$$

$$I_n=\frac {t^{n+1}}{2(n+1)}\,F_1\left(\frac{n+1}{2};\frac{n}{2},1;\frac{n+3}{2};-t^2,-\frac{t^2}{2}\right)$$ where appears the Appell hypergeometric function of two variables.


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# "I need some guidance, please" - " i want to power six 10 W led-s."

#### Rizalex

Joined Mar 25, 2016
6
Hi. Im not that new to electronics but Im having some problems figuring this out.
The basic idea is that i want to power six 10 W led-s. They have a Vf of 9-12V and
a current draw of 1050 mA (china). I also want to buy a 12 V power supply 16.5 amps
or something like that. In the past i have used 12 V led-s and connected them directly to the
power supply and they worked fine. Is is true that the current draw from the source was getting
higher and higher as the led-s temperature climbed. This time around i want to limit the current that
the led-s take in. For that i want to use the LM317. Now, with the 317 i want to use a 2W 1.5 ohm resistor.
I want to limit the current draw to about 800 mA. I dont know for sure that the 12 volts from the power supply
will be enough to power the led-s. So could anyone give me some advice on this matter.
Tanks a lot!

#### wayneh

Joined Sep 9, 2010
17,167
The normal application of the 317 is a voltage regulator and in that circuit it has too large a dropout voltage for your application. It also can not handle a 1A load without a big heat sink.

You need a current regulator. The 317 can be used for that but by the time you add an external transistor, you might as well get a DC-DC converter rated for your load.

#### Rizalex

Joined Mar 25, 2016
6
Yes. Probably the use of the 317 is not the best one in my case because of the big heat sink and also i have to make one circuit for each of the led-s. What is the best way to power the leds? Lets say i want to power just one. Can i use a 12 volts 10 amps power supply designed for led strips? OR this will damage the led unless i have something that will limit the current that can be used by the led?

#### #12

Joined Nov 30, 2010
18,223
Not being one to tell you how to do your business, I just show you how.
The TO-220 will have 108 C die temperature if you apply an infinite heat sink. I think a TO-3 metal can is the only answer for a chip like this in a job like this.

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#### Rizalex

Joined Mar 25, 2016
6
But still. Can i power the led straight from the led strip power supply?

#### wayneh

Joined Sep 9, 2010
17,167
Lets say i want to power just one. Can i use a 12 volts 10 amps power supply designed for led strips? OR this will damage the led unless i have something that will limit the current that can be used by the led?
That won't work, since the 10A control would try to drive 10A through the single strip. You need a current control for
each strip. If you use a higher voltage, you can put strips in series. A single controller can control multiple strips in series.

#### mcgyvr

Joined Oct 15, 2009
5,394
I highly suggest meanwell LDD drivers.. They are around \$5 each and you would use 1 for each LED and need a 24V power supply (of sufficient wattage)..
12V will not work as the Vf of the LED's could be 12V already and you need about a 2V overhead for the LDD's.. (next logical power supply is 24V.. or a 19V laptop supply)
Each (LED + Meanwell) is simply wired in series with the power supply..
They also make 5-up,etc.. PC boards for the LDD's with the through hole pins

Now there are many other ways.. But I always recommend meanwell drivers with ANY high current LED application vs cheap chinese drivers commonly found on ebay or attempting to roll your own for only 6 LEDs..

This will work great for you and run them at 700mA..
and IF you ever want dimming they accept a 5V PWM signal (right from an Arduino PWM pin or similar)

Use this one if you want 1000mA

IMO you will spend more in time/money trying to roll your own or have to use resistors that just give off heat and don't regulate the current in the event of changing Vf or Vin

#### Rizalex

Joined Mar 25, 2016
6
Thank you very much for the answers.
As this project is meant for an aquarium lightning, i decided to take another route.
Since using 10W led-s imply using more then a 12 v power supply and the actual leds
i started to look for other ways to power the light for my aquarium.
So i decided to go for the 5050 led strips. Now, since i want to grow my plants with it i
want a little red and a little blue in the lights. I have to mention right now that i dont want
RGB led strip. I want 6000-6500k white and maybe half a dozen of the red and blue led.
Does it make any difference if i connect a 3 led red strip in series to my big 5 meter led strip?
Since it is powered by 12 volts im thinking that the should not be any problems.

And another question. I saw on the internet that you could buy 110/220 AC led strips.
So instead of giving the strip 12 volts, you buy a little adapter that plugs into the outlet which has,
what i believe it is, a ac to dc rectifier and in the end the strip still gets 12 volts.
Am i right or am i wrong? It would be way easier to bypass the power source all together and would
make this project just a little bit easier. Thank you very much.

#### mcgyvr

Joined Oct 15, 2009
5,394
We don't like to guess..
Typically those 5050 strips use a regular ac/dc adapter anyways so I'm not sure whats the difference between those and the 110/220 AC led strips you talk about (could be nothing is different but without links we don't know)

And...Use caution with AC voltages around water..

#### Rizalex

Joined Mar 25, 2016
6
http://www.aliexpress.com/item/AC-2...3856713.html?spm=2114.40010508.4.2.BQa2ge&s=p

This is a comparable product and i believe that 220 AC is not the best way to go with my project.
It seems that the input is 220 and also the output is 220 AC. And it is to dangerous have 220 that close
to the water.

In the end i believe that the best way is to buy a 12 volts power supply and stick with the 12 volts led strips.

So in regards with the combinations of colors not using the RGB strips, can i pair sections of any color with
the white one, am i right? If i want to replace a single white led with a red led with its respective
resistor will this work? I mean, in a section of tree white led-s (white leds), if i want to replace a white one
with a red one (and the resistor - i say with the resistor because i know that the are different values for white,
red, blue led and so on) will i get who white leds and a red one all glowing at optimal brightness?

Thanks guys!


text:
# Finding the number of analytic functions which vanish only on a given set.

Let $S = \{0\}\cup \{\frac{1}{4n+7} : n =1,2\ldots\}$. How to find the number of analytic functions which vanish only on $S$?

Options are

a: $\infty$

b: $0$

c: $1$

d: $2$

-
Hint: the zeroes of an analytic function can't be arbitrarily close together. –  Pedro May 13 '12 at 7:09
@Pedro Then what would be conclusion sir? –  srijan May 13 '12 at 7:16
@pedro Can you write the answer with little explanation please? I would be very much thankful to you. –  srijan May 13 '12 at 7:32
Actually the zeros of an analytic function can be arbitrarily close together, e.g. $\sin(z^2)$. But the zero set an analytic function whose domain is a connected open subset of the plane cannot have an accumulation point in the domain, unless the function is identically zero. –  Jonas Meyer May 13 '12 at 7:49
@srijan: Have you seen the "identity theorem"? (If you scroll down to "an improvement" you will find stuff about accumulation points.) That article refers to $2$ analytic functions $f$ and $g$, but the version Pedro and I are alluding to is the special case where $g=0$. So suppose that you have an analytic function $f$ that is zero on $S$. Can you find a way to apply the identity theorem to make a conclusion about $f$? By the way, are your analytic functions assumed to be defined on the entire plane? –  Jonas Meyer May 13 '12 at 8:07

First I would like to say that zeroes of analytic function are isolated point.Identity Theorem or In some books uniqueness theorem says that: $f$ be analytic in a domain $D$, If the set of zeroes has a limit point in the domain $D$ then $f\equiv 0$. In your case $D=\mathbb{C}$ and set of zeroes=$S$(as you have already defined in your question),Notice that $S$ has a limit point namely $0\in S$, so Uniqueness theorem says that only all the analytic function that has zero set as $S$ must be $\equiv 0$ function