text:
Publisher: Springer-Verlag (Total: 2570 journals)

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• Preface: Introducing operations research trends at CLAIO XVIII
• PubDate: 2020-01-31

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762


text:
# Finding explicit formula

Find an explicit formula for $a_k$ if $a_0=0$ and $a_{k+1}=a_k+2^k$ for $k\geq 0$.

Thanks much for the help!

-

HINT: Always begin by collecting some data, unless the computations are impossibly hard. By easy calculation you should find these values:

$$\begin{array}{rcc} k:&0&1&2&3&4&5\\ a_k:&0&1&3&7&15&31 \end{array}$$

The numbers $a_k$ should be very recognizable and should immediately suggest a closed form for $a_k$. Once you have that, prove it by induction on $k$.

-

Hint:

Compute the first few values of $a_k$. Can you guess an explicit formula?

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Write down some of the first elements to get a hint: $0,1,3,7,15,...$

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The first 11 values are

{0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047}

This cries to be $2^k -1$ doesn't it? Now you just need to prove it.

-

So \begin{align} a_k =& a_{k-1} + 2^{k-1} \\ &= a_{k-2} + 2^{k-2}+2^{k-1} \\ &= a_{k-3} +2^{k-3} + 2^{k-2} + 2^{k-1} \\ &= \dots \\ &= a_0 + 2^0 + 2^1 + 2^2 + \dots + 2^{k-2} + 2^{k-1} \\ &= 1 + 2^1 + 2^2 + \dots + 2^{k-2} + 2^{k-1} \end{align} What do you end up with?

-

Let $F(x)=\sum_{k\geq 0}a_k x_k$. Then multiply the recursive formula by $x^{k+1}$ and summing over all $k\in \mathbb{N}$:

$$\sum_{k\geq0}a_{k+1}x^{k+1}=\sum_{k\geq0}a_kx^{k+1}+x\sum_{k\geq0}(2x)^k$$

$$\therefore F(x)=xF(x)+\frac{x}{1-2x}\\F(x)=\frac{x}{(1-x)(1-2x)}=x(1+x+x^2+\cdots+)(1+2x+4x^2+\cdots)$$

$$\therefore a_k=\sum^{k-1}_{i=0}2^i=\boxed{2^k-1}$$

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You gave the complete answer away ... –  Thomas Mar 24 '13 at 20:49
oh derp myself... –  user67258 Mar 24 '13 at 20:52

format: plain


text:
How to entangle two particles?

1. Dec 10, 2014

Breo

I have notions of the mathematics involving (entangled state is one which is not a product state of two qubits, etc) the idea of entanglement. However, still can not figure out how to, let us say, pick a particle (which type of particle we can use as qubit? protons? electrons? ununumbiums!?!?) A in one laboratory of Max Planck and a particle B in Boston and then entangle them. How can we entangle 2 arbitrary particles?

2. Dec 10, 2014

VantagePoint72

Entanglement is what generally happens when things interact with each other in a particular way. So if you want to entangle two particles, you must first bring them together so they can interact.

3. Dec 10, 2014

Breo

How? can you bring me an example?

4. Dec 10, 2014

DrChinese

There are a number of methods, and most use some/all of the following concepts:

a) The particles must be indistinguishable on the basis they are entangled. They can be distinguishable on other bases.
b) There is some conserved quantity or relationship between the particles.
c) The number of objects entangled can be 2 or more, with no theoretical upper limit.

1. Photons are most often entangled via a process called Parametric Down Conversion, in which a single photon is split into 2 entangled ones.
2. A helium atom has 2 electrons, which are entangled when in the same lowest shell together.

5. Dec 10, 2014

Jazzdude

I don't see how that is necessary. First of all, you don't even need several particles for entanglement. A single particle with entangled spin and position is very common. Secondly, I don't see what practically speaks against entangling very different quantities between two different particles.

I'm not sure what you mean with this. Can you give an example of how you think this applies?

With the restriction of the monogamy of entanglement: http://www.quantiki.org/wiki/Monogamy_of_entanglement

Allow me to add that two indistinguishable particles are 1) always entangled if they are fermions 2) entangled if they're in different states and bosons.

Cheers,

Jazz

6. Dec 10, 2014

DrChinese

We are both speaking in general terms. However your objections are themselves objectionable. :-)

1. There is no meaningful way to say a single particle is entangled. Entanglement is represented by a system of 2 or more particles.

2. As to conservation: PDC photons are entangled as to frequency with conserved total momentum (from the input photon) according to a common formula. Spin (polarization) is conserved as a constant in type II PDC.

3. Could you give me an example of this? Specifically, what does the "different states" criteria have to do with anything? And just to be clear about MY objection to your concept about entanglement of fermions versus bosons: there is no requirement that entangled particles be the same kind of particle at all.

7. Dec 10, 2014

VantagePoint72

I agree with @Jazzdude. Entanglement is a general property that states in tensor product spaces may have (with respect to that particular factorization); any collection of independent degrees of freedom can be entangled together. That tensor product can be between any two Hilbert spaces, not just those corresponding to separate particles. Entanglement is just more striking when the factor spaces correspond to separate particles because then you can observe the 'spooky' spatial non-locality for which entanglement is famous. But it's perfectly sensible to talk about, say, the spin of a single particle becoming entangled with its angular momentum in spin-orbit coupling. It's exactly the same phenomenon and that is the language widely used.

Given that, again, there's no need for the things being entangled to even be particles, there's no fundamental relationship between entanglement and conservation. Generally, the physical processes that mediate interactions between different degrees of freedom have certain conservation laws associated with them, but that's completely general and nothing to do with entanglement.

Fermions are subject to the Pauli exclusion principle; so, anti-symmetrizing the state of a collection of fermions necessarily produces an entangled state since you cannot start the tensor product of identical pure states. The state of a collection of bosons needs to be symmetric under particle exchange, so each particle is allowed to be in the same pure state giving an overall product state. On the other hand, if you start out with at least two particles in different pure states and then symmetrize the whole thing you get an entangled state as with the fermions.

Last edited: Dec 10, 2014
8. Dec 10, 2014

Jazzdude

This is not accurate. Entanglement happens between tensor factor spaces, not particles. A single nonrelativistic Schroedinger particle of spin 1/2 lives in a space that factors into a Hilbert space spanned by particle positions and a two dimensional spin space. The particle's state does not necessarily factor into states in those two factor spaces however, and the particle can be entangled. A state like $\left| A \right\rangle \left| \uparrow \right\rangle + \left| B \right\rangle \left| \downarrow \right\rangle$ where $A$ and $B$ are different position states is position-spin entangled.

Obviously, conserved quantities must also be conserved for processes that entangle subsystems. I don't see how the existence of a conserved quantity is required for entanglement though. Of course, in the Hamiltonian formalism energy is always conserved under unitary evolution, but that's surely not what you mean.

No, entangled particles do not have to be of the same kind. However indistinguishability is sufficient for entanglement if the states are different. That follows from the (anti)symmetrisation of multi-particle states. Start with a two particle state $\mathcal{H}^{\otimes 2}$ and the two single particle states $\left| A \right\rangle$ and $\left| B \right\rangle$.

The fermionic state is then $\left| A \right\rangle \left| B \right\rangle - \left| B \right\rangle \left| A \right\rangle$ where the order of the kets indicate the space. This state only exists if $\left| A \right\rangle$ and $\left| B \right\rangle$ are different. But in this case the state does not factor and the combined state is entangled.

The bosonic state is $\left| A \right\rangle \left| B \right\rangle + \left| B \right\rangle \left| A \right\rangle$. This state does exist even if $\left| A \right\rangle$ and $\left| B \right\rangle$ are the same single particle state. If they are the same state the state can trivially be written as a product state. If they're not the same the state does not factor and again the combined state is entangled.

Cheers,

Jazz

9. Dec 10, 2014

DrChinese

I agree with this statement. And I also stand by my post #4 as written.

And again you and I disagree on how to address an answer to the OP. What you and Jazzdude are saying is a deeper level than appropriate, in my opinion (and I realize you are a professor). Any answer given can necessarily be argued with on some level. I didn't shred your answer in post #2 as I could have (as I am sure you know perfectly well that entanglement is possible of particles that never even existed at the same time). But that detail wouldn't be of much help to the OP.

My experience here is that some posters benefit from one person's manner of addressing a question, and some from another's. It is common that different approaches are taken to get to that point. It is more effective to assist the OP that to critique someone else's style or approach in the name of "correctness". For the OP's purposes, I believe my answer is better than what you or Jazzdude have said so far. I would encourage you to provide something more for the OP.

To the OP regarding your original question: there are many ways to entangle particles/systems/properties. As you can see from the responses, the fundamental issues can be perceived in a variety of ways. You might enjoy a few of the following references, in which laboratory entanglement creation is discussed.

http://arxiv.org/abs/quant-ph/0205171
http://www.nature.com/nature/journal/v409/n6822/full/409791a0.html
http://arxiv.org/abs/quant-ph/0303018
http://lanl.arxiv.org/abs/1006.4344[/user]

10. Dec 10, 2014

DrChinese

And again my point, we are mixing the general and the specific. This is actually quite nearly what I said earlier: "The particles must be indistinguishable on the basis they are entangled." You objected to that, saying "I don't see how that is necessary." Almost any experiment with entanglement will mention indistinguishably (as I did) and/or show a related equation with something conserved.

I suspect we will both be happier addressing the OP's question, which I believe you have yet to weigh in on.

PS Not that I am questioning that it exists, but I don't recall any actual experiments involving single particle entanglement. Can you cite a good example for me to add to my collection?

11. Dec 10, 2014

Jazzdude

My reply was intended to address certain, unfortunately quite common, misconceptions about entanglement. They were not specifically directed at the OP but are supposed to stand as a footnote to your contribution. I do agree that answers should be tailored to the level of the original question, but they should also be correct. The fact that you objected to my response clearly shows that the misconceptions are not just simplifications. Also, I've contributed to the OPs question by providing an easy way to "produce" entanglement for identical particles. Frankly, and no offence intended, I think your argument for more suitable answers will appear to others as no more than a distraction from your having been wrong. While that's understandable, a good scientist should stand above this.

Cheers,

Jazz

12. Dec 10, 2014

Jazzdude

There is a difference between sufficient and necessary.

Cheers,

Jazz

13. Dec 10, 2014

Jazzdude

Stern-Gerlach will do nicely.

Cheers,

Jazz

14. Dec 10, 2014

DrChinese

As always, I stand ready to acknowledge the limits of my knowledge. Your argument for "correctness", on the other hand, will appear to some others as a distraction* for the lack of utility to the question at hand. Sorry, a good scientist should also be helpful.

-DrC

* I would call it splitting hairs as I see no meaningful disagreement.

15. Dec 10, 2014

DrChinese

You know, I almost wrote that you would answer with that exact phrase. But I thought at the time that you wouldn't waste time with that. Thank you for pointing out precisely nothing useful in an attempt to trump someone.

16. Dec 10, 2014

DrChinese

I am sure the OP will be delighted this helpful nugget found its way into the conversation.

All this from: "How can we entangle 2 arbitrary particles?"

I am loath to add this definition, but we've already gone so far down this road that the devil in me can't resist. :) I will say that I plan to step out of this conversation until the OP returns, at which time I will comment further if I have anything USEFUL to add.

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently—instead, a quantum state may be given for the system as a whole.

Last edited: Dec 10, 2014
17. Dec 10, 2014

Jazzdude

Yet you don't but rather reiterate something that is wrong:

If you get something obvious wrong, you will have to live with the obvious being pointed out to you. This has nothing to do with trumping you. I had already pointed out how you're wrong. You however tried to restate it in a more obfuscated way hoping I would let it stand and make it appear that you have been right from the beginning. That is trumping. But even worse, if I answer and undermine your plan you suggest that it's me and not you who is not providing anything useful to the discussion.

Together with this:

I cannot think of any other word but "trolling" for what you are doing.

Now hold it for a minute. You specifically asked me for exactly that information. Giving it to you is now dismissed as not being helpful?

How very convenient for you.

Congratulations!

*plonk*

Jazz

Then, clearly, you have not understood the disagreement.

18. Dec 10, 2014

Breo

Thank you all for your responses.

So, could be the entanglement the more nature way to be for quantum particles?

19. Dec 10, 2014

DrChinese

Atoms and molecules naturally exist in configurations in which there is a lot of entanglement. Which is to say there is a lot of superposition of states. Those are not always useful for entanglement experiments, however. The effort to separate the entangled particles can cause them to cease acting as a combined system and instead act as separate ones.

20. Dec 10, 2014

Breo

That should be taught on my quantum mechanics course...

When we use Bell's basis? Which is his usefulness? We use different basis each case of entanglement?

I ask these questions in order to perform a more important one.


text:
# Thread: how to solve this?

1. ## how to solve this?

How to solve this?

"If a farmer buys chicken feed for £1.26 per tonne, pig feed for £2.74 per tonne and cattle feed for 78p per tonne. How much of each would he have if he had 100 tonnes for £100?"

the equations to solve become:

x+y+z = 100
x*1.26+y*2.74+z*0.78 = 100

Now if there three equations for 3 variables then it would be easy to solve this problem using standard equations solving technique such as substitution method or so on.

But with only 2 equations how to solve three variables?

If I start guessing then how to make educated and smart guess so that I don't have to go through the excruciating process of choosing x=1,2,3,... etc? I am hoping there are some techniques such as bisection method, newton-raphson method for solving problems numerically (and iteratively) to apply to this problem.

Some help would be greatly appreciated.

Thanks.

2. Wouldn't know where to start, but I'd have a go at seeing what we've got by attacking it like this.

I would eliminate one of the variables and so get an equation in 2 variables. This will be linear. I would then experiment with plotting it, or something, and see whether there is a constraint that forces a certain solution upon me. Otherwise, you would see there is no unique solution. Unless there is a diophantine equation lurking at the bottom here, in which case I've got this page here:

Solution of Linear Diophantine Equation - ProofWiki

but I don't know whether this will help or if it's way off beam.

I tried to reduce it to one equation getting something like this:

1.48x+1.96z=1.74

then

x = (174-1.96z)/1.48

then I applied bisection method between z = 0 to 88 and finally reaching the value of z=85 giving x = 5.

I am not sure this 'bisection' method of mine is standard way or there is better way (didnt get your plotting technique). But I am not sure how to apply this with a tri-variable single equation like this:

1.26x+2.74y+0.78z = 100

Even if there are multiple solutions for such equation how to enumerate them all? You know for linear algebra problems it happens quite often that some sort of matrix equation like XU = V is given and after applying multiple matrix operations we might reach one or two equations with more than 2 variables and the question asks for finding solutions. If given in a linear algebra test trying to find solutions for such equations starting from x = 1 to 100 (say) exhaustively would not be practical (you would run out entire test time solving one such problem).

4. Originally Posted by amishera
How to solve this?

"If a farmer buys chicken feed for £1.26 per tonne, pig feed for £2.74 per tonne and cattle feed for 78p per tonne. How much of each would he have if he had 100 tonnes for £100?"

the equations to solve become:

x+y+z = 100
x*1.26+y*2.74+z*0.78 = 100

Now if there three equations for 3 variables then it would be easy to solve this problem using standard equations solving technique such as substitution method or so on.

But with only 2 equations how to solve three variables?

If I start guessing then how to make educated and smart guess so that I don't have to go through the excruciating process of choosing x=1,2,3,... etc? I am hoping there are some techniques such as bisection method, newton-raphson method for solving problems numerically (and iteratively) to apply to this problem.

Some help would be greatly appreciated.

Thanks.

Nothing elaborate.
BUT this assumes that the tonnes CANNOT be partial.
That is the sale must be for a whole tonne.
-------------------------------------------
x+y+z=100 [EQUATION 1]

For Eq.2 remove the decimals (multiply by 100)
126x + 274y + 78z = 10000 [EQUATION 2]

from first equation:
z = 100 - x -y
substituting into 2nd equation
126x + 274y + 78(100 -x -y) = 10000
which is
126x + 274y + 7800 -78x -78y = 10000
&
48x + 196y = 2200
divide through by 4
12x + 49y = 550

from this 550/12 = 45.8, therefore x < 46

also

550/49 = 11.22, therefore y < 12

That is where I would start:
y must be in the range 1 to 12.
You have 12 guesses to get the result
x = (550 - 49*y)/12

(550 - 49y) MUST be divisible by 12

550 - 49 = 501 ; 501 is NOT divisible by 12, so y cannot be 1

550 - 49*2 = 452 ; 452 is NOT divisible by 12, so y cannot be 2

etc.

at most ten more guesses to get the answer.

5. Originally Posted by amishera

I tried to reduce it to one equation getting something like this:

1.48x+1.96z=1.74

then

x = (174-1.96z)/1.48

then I applied bisection method between z = 0 to 88 and finally reaching the value of z=85 giving x = 5.

I am not sure this 'bisection' method of mine is standard way or there is better way (didnt get your plotting technique). But I am not sure how to apply this with a tri-variable single equation like this:

1.26x+2.74y+0.78z = 100

Even if there are multiple solutions for such equation how to enumerate them all? You know for linear algebra problems it happens quite often that some sort of matrix equation like XU = V is given and after applying multiple matrix operations we might reach one or two equations with more than 2 variables and the question asks for finding solutions. If given in a linear algebra test trying to find solutions for such equations starting from x = 1 to 100 (say) exhaustively would not be practical (you would run out entire test time solving one such problem).
What I meant by "plotting it" was: take the equation 1.48x+1.96z=1.74 (or whatever you've got in 2 dimensions) and using x and z as coordinate axes, and drawing a line corresponding to the points satisfying the above equation. It's just a technique that might help for this sort of exercise.

6. Originally Posted by Matt Westwood
What I meant by "plotting it" was: take the equation 1.48x+1.96z=1.74 (or whatever you've got in 2 dimensions) and using x and z as coordinate axes, and drawing a line corresponding to the points satisfying the above equation. It's just a technique that might help for this sort of exercise.
Hello I'have this solution :

$\begin{array}{l}
12x + 49y = 550 \Leftrightarrow 49y \equiv 550\left[ {\bmod 12} \right] \\
12x + 49y = 550 \Leftrightarrow y \equiv 10\left[ {\bmod 12} \right]....\left( {49 \equiv 1\left[ {\bmod 12} \right]and550 \equiv 10\left[ {\bmod 12} \right]} \right) \\
conclusion:y = 12k + 10....\left( {k \in {\rm N}} \right) \\
\end{array}$

substituting into equation 12x + 49y = 550 : $x = - 49k +5
$

$\left\{ \begin{array}{l}
x \in {\rm N} \\
y \in {\rm N} \\
\end{array} \right. \Leftrightarrow k = 0$

$\left\{ \begin{array}{l}
x = 5 \\
y = 10 \\
z = 85 \\
\end{array} \right.
$


text:
# The power of chi-square test

Under the null hypothesis, if we have $$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$ the test statistic can be construct as: $$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$ And we reject the null hypothesis if $$\hat{\Psi} > \chi^2_{p, 1 - \alpha}$$ under level $$\alpha$$.

Now, if under the alternative hypothesis $$H_1$$, $$\sqrt{n} \left( \vec{x} - \vec{\mu} \right)\, \rightarrow_d \, N(0, \Sigma),$$ I want to know the power of using test statistic $$\hat{\Psi}$$, i.e. $$\mathrm{P} \left( \left. \hat{\Psi} > \chi^2_{p, 1 - \alpha} \right| H_1 \right) = ?$$

I know under $$H_1$$, $$\hat{\Psi}$$ can be decomposed as: $$\hat{\Psi} \, \rightarrow_d \, \sum_{j = 1}^p \xi_j^2, \quad \left( \xi_1, \cdots, \xi_p \right)^{\top} \, \sim \, N(\vec{\mu}, \Sigma),$$ but I don't know how to deal with this square sum. Can anyone help me? Thanks a lot!!

• It seems we cannot use the above method to calculate the power of this test. Since $n^{-1} \hat{\Psi} \, \rightarrow_p \, \vec{\mu}^{\top}\vec{\mu}$, I guess, under $H_1$, $n^{-1/2} \left( \hat{\Psi} - \vec{\mu}^{\top}\vec{\mu} \right)$ is asymptotically normal? So the power goes to 1 when $n \rightarrow \infty$, it that right?
– 香结丁
Jul 29, 2020 at 1:29

If $$\Sigma=I_p$$, then the distribution of $$\sum_{j=1}^p\xi_j^2$$ for $$(\xi_1,\cdots,\xi_p)^\top\sim N(\vec{\mu},\Sigma)$$ is the non-central chi-square distribution with $$p$$ degrees of freedom and non-centrality parameter $$\vec{\mu}^\top\vec{\mu}$$.

If $$\Sigma\ne I_p$$, then the distribution of $$\sum_{j=1}^p\xi_j^2$$ for $$(\xi_1,\cdots,\xi_p)^\top\sim N(\vec{\mu},\Sigma)$$ has no name or closed-form expression of its pdf or cdf -- even when $$\vec{\mu}=\vec0$$.

Responding to a comment by the OP: Let $$x_n:=\vec x$$ and $$\mu:=\vec\mu$$. If $$Z_n:=\sqrt n(x_n-\mu) \to_d Z\sim N(0,\Sigma), \tag{1}$$ then, for any fixed $$\mu\ne0:=\vec0$$, $$nx_n^\top x_n=(\sqrt n\mu+Z_n)^\top(\sqrt n\mu+Z_n) =n\mu_n^\top\mu_n+O_P(\sqrt n),$$ so that $$nx_n^\top x_n$$ converges in probability to $$\infty$$, rather than to a finite random variable.

To get a finite random variable in the limit, you need to consider alternative hypotheses close enough to the null one; that is, in this case, you need to consider nonzero alternative asymptotic mean vectors close enough to $$0$$. In particular, it will make sense to fix some $$\mu\ne0$$ and consider the alternative values $$\mu/\sqrt n$$ of the asymptotic mean.

So, now instead of (1) we are assuming that $$Z_n:=\sqrt n x_n-\mu=\sqrt n(x_n-\mu/\sqrt n) \to_d Z\sim N(0,\Sigma).$$ Then $$nx_n^\top x_n=(\mu+Z_n)^\top(\mu+Z_n)\to_d (\mu+Z)^\top(\mu+Z),$$ and $$\mu+Z\sim N(\mu,\Sigma)$$, so that the limit random variable $$(\mu+Z)^\top(\mu+Z)$$ will have the non-central chi-square distribution with $$p$$ degrees of freedom and non-centrality parameter $$\mu^\top\mu$$.

• Thanks so much. But is there a feasible method to tackle $$\mathrm{P} \left( \sum_{j = 1}^p \xi_j^2 > \chi_{p, 1 - \alpha}^2\right)$$ when $\Sigma \neq I_p$?
– 香结丁
Jul 29, 2020 at 0:38
• @香结丁 : As I said, if $\Sigma\ne I$, then little can be said even for $\vec\mu=\vec0$; see e.g. link.springer.com/article/10.1007%2Fs00440-003-0262-6 for some inequalities. Jul 29, 2020 at 1:09
• @ Iosif Appreciate your helping! It seems we cannot use this method to calculate the power. Since $n^{-1} \hat{\Psi} \, \rightarrow_p \, \vec{\mu}^{\top}\vec{\mu}$, I guess, under $H_1$, $n^{-1/2} \left( \hat{\Psi} - \vec{\mu}^{\top}\vec{\mu} \right)$ is asymptotically normal? It that right?
– 香结丁
Jul 29, 2020 at 1:24
• Thanks for your generous help! Now I know what I should do!
– 香结丁
Jul 30, 2020 at 1:40
• @香结丁 : As a relatively new user, you may want to look at these guidelines Jul 30, 2020 at 15:30


text:
# May, 2016

## A curious identity

There's something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It's one of those that just begs, "prove me!" So, here goes! I'd start with the

## Ask Uncle Colin: Trigonometric craziness

Dear Uncle Colin, My friend claims that $\frac { 2 - \frac{2 \sin(x)}{\cos(x)}}{\sin(x) - \cos(x)} \equiv -2\sec(x)$. I think she's crazy. What do you think? -- I Don't Even Need Trigonometry, I Teach Yoga Hi, IDENTITY -- even yoga teachers need trigonometry, though! Well, there's one way to find out

## How to invert a $3 \times 3$ matrix

So much wasted time. I spent much of my first two years at university cursing the names of Gauss and Jordan, railing at my lecturer (who grim-facedly assured me there were no more useful uses of a student’s thinking time than ham-fistedly rearranging these things), and thinking “there MUST be

## Ask Uncle Colin: I’ve lost my mojo

Dear Uncle Colin, I'm struggling with A-level. I used to love maths when I did [one board] at GCSE and now I'm doing [another board] at A-level, I don't enjoy it any more -- when I see a question, I can't even tell what it is they're asking. My teachers

## Review: The QAMA Calculator

It's billed as the calculator that won't think until you do: if you give it something to evaluate, it will refuse to give you an answer until you give it an acceptable approximation. On the surface, that's a great idea. If I had a coffee for every time I've rolled

## Ask Uncle Colin: Don’t be daunted by D’Hondt?

Dear Uncle Colin, Up here in Scotland, we've got an election tomorrow. It's not as simple as the stupid first past the post elections you have down there in England, but even with our superior Scottish intelligence, some people are still struggling to understand how the system works. Do you

## A frankly ludicrous bit of paper arithmetic

Some time ago, someone asked Uncle Colin what the last two digits of $19^{1000}$ were. That caused few problems. However, Mark came up with a follow-up question: how would you estimate $19^{1000}$? I like this question, and set myself some rules: No calculators (obviously) Only rough memorised numbers (\$e \approx


text:
The "eight queens puzzle" is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other. Thus, a solution requires that no two queens share the same row, column, or diagonal. The eight queens puzzle is an example of the more general N queens problem of placing N non-attacking queens on an N×N chessboard. (From Wikipedia - "Eight queens puzzle".)

Here you are NOT asked to solve the puzzles. Instead, you are supposed to judge whether or not a given configuration of the chessboard is a solution. To simplify the representation of a chessboard, let us assume that no two queens will be placed in the same column. Then a configuration can be represented by a simple integer sequence (Q1,Q2,⋯,QN), where Qi is the row number of the queen in the i-th column. For example, Figure 1 can be represented by (4, 6, 8, 2, 7, 1, 3, 5) and it is indeed a solution to the 8 queens puzzle; while Figure 2 can be represented by (4, 6, 7, 2, 8, 1, 9, 5, 3) and is NOT a 9 queens' solution.

Figure 1

Figure 2

## Input Specification:

Each input file contains several test cases. The first line gives an integer K (1<K≤200). Then K lines follow, each gives a configuration in the format "N Q1 Q2 ⋯ QN", where 4≤N≤1000 and it is guaranteed that 1≤Qi≤N for all i=1,⋯,N. The numbers are separated by spaces.

## Output Specification:

For each configuration, if it is a solution to the N queens problem, print YES in a line; or NO if not.

## Sample Input:

4
8 4 6 8 2 7 1 3 5
9 4 6 7 2 8 1 9 5 3
6 1 5 2 6 4 3
5 1 3 5 2 4

YES
NO
NO
YES

## 解答

#define _CRT_SECURE_NO_WARNINGS
#include<iostream>
#include<vector>
#include<cmath>
#include<unordered_map>

using namespace std;

int main(){
int k, n;
scanf("%d", &k);
for (int i = 0; i < k; i++) {
scanf("%d", &n);
vector<int> v(n);
unordered_map<int, int> mp;
bool flag = true;
for (int j = 0; j < n; j++) {
scanf("%d", &v[j]);
mp[v[j]]++;
}
for (int j = 0; j < n; j++) {
if (mp[v[j]] > 1) {
flag = false;
break;
}
for (int u = 0; u < j; u++) {
if (abs(v[j] - v[u]) == abs(j - u)) {
flag = false;
break;
}
}
}
printf("%s\n", flag ? "YES" : "NO");
}
return 0;
}


text:
# Thread: Critical Numbers & Extremas

1. ## Critical Numbers & Extremas

Given the equation y=2x^2-8x+9, find the extreme values of the function and where they occur.

I know I have to find the derivative, then the critical numbers, but when it comes time for the number line, am I plugging numbers into the original equation or the derivative??

Alsooo, what happens after that?

2. you are trying to determine the sign of f'(x) for values between critical numbers ...

3. Originally Posted by skeeter
you are trying to determine the sign of f'(x) for values between critical numbers ...
I don't understand.
See, my teacher told me to first find the derivative, then find the critical numbers, and then make a number line with the critical numbers. So you'd have to pick a number on that line, and plug it in to determine whether you have a max value or min value.
The problem is, Idk if I plug the number from the number line into the derivative of the equation, or the original equation.

Did that confuse you?

4. no, but my response obviously confused you.

once again ... where will you "plug" in those numbers if you have to determine the sign of f'(x) between the critical values?

5. ohh.. xD
now i get it.

ok, how would i find what the minimum value is at x=2 ?

6. minimum values occur when f'(x) changes sign from negative to positive ... why?

7. ok, on the left of the two, i get a neg.. at the right of two, i get a pos.
so is the min at x=2? or do i have to plug 2 into the original equation?
because the back of the book says that the min. is at 1.. so idk..

8. $y=2x^2-8x+9$ has a minimum at $x = 2$

the minimum value is $y = 1$

9. ok, but how did you get that the minimum value is at y=1??

10. plug 2 in for x, calculate the value of y.

11. Originally Posted by skeeter
plug 2 in for x, calculate the value of y.

Thank you very much.
You've been a huge help for the both of us.


text:
01 - 02 - SE - 03 - 04 - 05 - 06 - 07 - 08 - 09 - 10 - 11 - 12 - 13 - 14
 Alive 8 ALT V PART 2 (18.1.04) Okay, so this is a lot later, and not at all realtime. I had planned to do some more realtime text-tapping when I was back at Q-Funk's place, but limited opportunity, inclination, and space to set up in, meant that I didn't. So anything else written from here on, is going to be in a past tense. One of the corporately sponsored competitions was the mobile phone camera picture competition. This was laid on by Nokia, generously providing the facilities, and also a prize for this, namely one of the top of the range mobile camera phones in question. Entrants to this, had a short time with one of these cameraphones to rush around Gloria and the surrounding wilderness, to take pictures of whatever they thought would win a prize. This one goes to a judging panel, rather than a vote, I am surprised to find myself part of that panel, but the bulk of the decision-making is left to still photography fan Q-Funk, and probably a good job too. In the meantime, I am distracted by another competition being outputted to the big screen. This is the Text Mode Demo Competition, organised by Finnish sceners tAAt, with the entries for this one gathered in earlier, and screened here at the Alt Party. This uses a hairy combination of PeeCee demo coding, and the ancient ANSI text mode, beloved of a thousand and one bulletin board logo's c.1991. Apart from the blockiness evident in using such a screen mode, the effects being shown bear every resemblance to contemporary demo stuff. It looks like there are more colours used, than for an old-style CGA/EGA PeeCee, and the level of sound support is 'improvised' in places. The overall standard is very high, and all the demos seem to run from within a smart 'shell' which details the ranking of each demo, counting back to the number #1 entry. In the end, a cool Amiga style oldschool demo, pips a more recognisable Newschool effort to the winning post... With all thoughts of going back to Q-Funk's place before the prizegiving now gone, we decide to repair a food-related deficiency at that purveyor of near-substitute foodlike substances called Madonalds. They do their thing rather smartly, and we return to Gloria, grab a vacant table upstairs, and pig out on the food-substitutes. Soon, the end of Alt Party V is near, only the prizegiving and closing ceremonies are left.... Here is the official results file, with comments added by myself at the end of each category, where it might hopefully help to explain things better. ALTERNATIVE PARTY V - Results Compo: Realtime Mobile Gaming 1) Rasak That one slid right past me... Compo: Realtime Mobile Photography 1) Hydro/Gluterol The event which was judged rather than voted on. There were quite a lot of cool entries for this one. Only one prize winner though! Compo: Two-colour Graphics 1) Is It True? by Ansichrist 85 2) Sad Creature by Ravel/Sleber Eid 84 3) Apt Pupil by DiamondDie/foobug^Numedia Cyclops 53 4) This Forest of Love by Nosfe 47 5) XTC Legalized in Finland Only Minutes Ago by Geel/Faktory 46 6) Bouh! by Baggio/Evulution 45 (*) 7) Really by Serialthriller/Moonhazard 38 8) Paeaestaekaeae minut pois kalliosta by Scoopr 36 9) Uncle-X:n iloksi by 333elite/ISO 27 10) Turin by Visy/Tukka 6 DISQUALIFIED: Sadedelfi by DiamonDienpikkusisko/Numedia Cyclops (reason: author not present at the party) Mun bestiksen penis by 18v jatkaeae jonka mutsi jne of Jumalauta (reason: wrong file format, should have been png) (*) Not such bad going from our Swedish hero. A few more votes would have put him in the prizes. Compo: Realtime Short Story 1) 6 by DiamonDie 43 2) 5 by Elfie 37 2) 2 by Nosfe 37 4) 4 by MegaMies 35 5) 1 by Sauli/JML 20 6) 3 by Molox 19 7) 7 by jobe 15 Compo: Elite Ship Building 1) Battle-damaged Viper by Depressio/Naemok 84 2) Shadow by Spiikki/Dekadence 60 3) Kukkia ja mehiläisiä by Tazze 30 4) Nosfe by Nosfe 30 5) Kirkkovenealus by Britelite/Dekadence 29 6) Afrikka by Ansichrist 28 7) Sidewinder by Ravel/Sleber Eid 26 8) j1g54w by DiamonDie/Foobug^Numedia Cyclops 25 9) Cobra MK-1 Backwards Edition by Scoopr 10 10) Cobra MkII (?) by Tuoppi/Sleber Eid 9 This was a cool competition. Most people contented themselves with colouring in advanced origami recreations of classic Elite spaceships, but the first prize winner was a decent attempt at model-making, including authentic battle damaged laser burning and scarring near the rear of the craft. Now if it could have been flown?! Compo: Sewing 1) Pupu by ISO 44 2) "Dragon" Lohikäärme by Elfie 41 3) Freedom Alt 2004 SEW by Entil / Fractal Kru 31 4) Sunflower by Spiikki/Dekadence 26 5) Alternative Pacman by DiamonDie/Numedia Cyclops 25 6) Squat the Planet by Nosfe 18 6) A Stupid Dog on a Surfboard by MegaMies 18 8) ? by Karvaisten rkkien kasvatusyhdistys 17 9) Growth by Stibe 16 10) Old School by Kristoffer Lawson 14 11) Kala-allas by Molox 12 11) "Die Rebels" by Tuoppi 12 13) How to Get AMIGA by DeLada 10 13) Porkkana by Maarit "Tazze" Klami 10 15) Bernhardilainen by Rasputina 8 15) Squiggly by Liberty 8 17) QBE by Scoopr 7 18) OM by Atte Joutsen 5 Yes, there was such a compo, and the idea blows my brain as well. Compo: General MIDI Music 1) Gone Laserdancing by Little Bitchard/MFX^Kwl 137 2) Blue Sparks by Pinza/Unique 90 3) We Operate by Jobe/Matt Current 76 4) BaBiBu by Paaris/Tukka 48 5) Sunny Games by Sauli 44 6) Rue Tama by Tonic 32 6) Mun Biiisi by Nunna/ISO 32 8) Hiippailua by Waffle 19 Missed this one, sorry, food was more important. Compo: Dynamic Demo 1) Juustomaariageneraattori by Exile Studios 194 (Windows) 2) Micery by Bandwagon & Fit 141 (Mac) 3) L'art_pretentiux by MFX 61 (Windows) 4) Noir by Mäyrä 32 (Windows) 5) Tukkage! by Visy/Tukka 17 (Vic-20) 6) TI-557 by Paaris/Tukka 12 (TI-86) DISQUALIFIED: Hilary Rozen by Donkeysex.com (Atari Falcon) (*) (reason: demo not dynamic) (*) That result wasn't unexpected, but we could have got around it, if we knew in advance the calibre of some of the other entries. (Falling far short of the competition rules by going for some lame randomisation, in my view!) Still, it got shown, and got the most laughs of the evening, which is what counted! The first place entry was some kind of cartoon with a random conversation in Finnish going on between two characters, apparently using incomplete fragments of sentences, and very funny for that part of the audience. The most impressive entry, technically, was the Mac-based 'Micery' demo, which seemed to be more of a simulation with packs of minature clockwork mice roaming around a maze, being chased by different kinds of bouncing balls. Third prize went to Nosfe, for something that was a little bit too messed up through the projector, which had problems handling heavy screenmode switching? The fourth place 'Noir' was a black and white photo slideshow, with a film noir soundtrack, both parts playing randomly. Tukkage, the other oldschool entry, was a fairly simple Vic 20 demo, which managed to randomise colours, and the TI-86 demo did something fairly dull with sine waves. Compo: Software Sound Synthesis 1) Glowing by ld0d 165 (Windows; c; executable size 20480 bytes) 2) Trollenscheisse by Viznut/pWp 140 (GNU/Linux, c & assembly, executable size 2100 bytes) 3) Llamazonia by Gasman 95 (GNU/Linux, perl; source size 1259 bytes) 4) Wavebuilder by Bass Cadet 65 (GNU/Linux, c; executable size 8231 bytes, patch file 2186 bytes) DISQUALIFIED: Jormation by Vesuri/da Jormas (AmigaOS, assembly; 7 executables, ~6370 bytes each) (reason: author not present at the party) Missed that one too, (See MIDI music compo). It was a really good curry though! Compo: Poetry Recital 1) Poetry Resist by ISO 69 2) Fire in Our Hearts by Nosfe 56 3) Elämä by Ravel 55 4) Poetry Recital by Abho 54 5) Ode to the RIAA by Felice/Maggie Team 50 6) Aamukuva by DiamonDie/foobug^Numedia Cyclops 43 7) SMS by Lobotomy 21 A closely packed top order there, only a handful of votes separating 2nd from 5th place, Felice was unlucky there! Felice was also the only english language entry, as far as I remember? Compo: Silent Movie 1) Vihannexia - Vegetables by MegaMies 125 2) Harmu 19,417 by Kameli 59 3) Paranoid by Numedia Cyclops 45 4) Agnosia Live by Agnosia 33 5) Virne voittaa Nosfen kompossa by Virne 30 6) The Last Best Hope by Ravel 25 7) Infinity^2 by Nosfe 22 8) Untitled by Stibe 20 9) Aropupu by ISO 19 10) Mun on niin paha olla by Perselintu! 17 11) Kohtalon tanssi by Tukka 13 One of the more memorable competitions, apart from the fact that all the entries blur together in a series of half-recalled impressions and images, so the war grave movie blends into the silent band playing a gig, closely followed by a couple of cartoon detectives blowing up a petrol station! Bass Cadet on piano, with his classic 'silent movie soundtrack' style, proved to be an effective accompaniment to these films. Compo: Realtime Comic-drawing 1) 11 by Ansichrist 51 2) 5 by Britelite/Dekadence 39 3) 1 by J. Pupu 26 4) 3 by DiamonDie 25 5) 7 by Nosfe 21 6) 6 by Jobe/MC 20 7) 16 by Spiikki 19 8) 13 by Modust/Exile 18 9) 10 by Gasman/Hooy Program 16 10) 4 by Uhef/Exile 14 11) 12 by Kanttu 11 12) 2 by cih (*) 10 12) 14 by Felor 10 14) 15 by Antti Mikkinen 9 14) 9 by Paaris/Tukka 9 16) 8 by Vizy/Tukka 3 (*) Yes I did have a go at that. Twelth place reflects the fact that from a thirty minute competition, I spent ten minutes on my entry! The funniest entry was a satire of 'Clippy', the Microsoft annoying desktop widget, given 'alternative' work as a stethoscope. A: You are feeling a funny pain in your left arm. B: You are about to die! C: Why don't you reboot windows.. D: Please buy this expensive health insurance! Or something like that, anyway, I hope it won. Compo: BASIC Demo 1) Excellence[tm] by Ananasmurska 90 (Excel) 2) Solutus by ISO 71 (QBasic) 3) Hikiliikkuja - Perspiratual Motion Machine by tAAt 2004 61 (GW-BASIC) 4) Sieni pakottaa by Nyyrikki 49 (MSX Basic 4 & X-Basic) 5) Threederles by Spiikki/Nalleperhe^DKD 40 (QuickBasic) 6) Oksennus 2 by Megamies/Gluterol 34 (QuickBasic) 7) One Last Time by Lucid 16 (AmigaBasic) 8) Sikakreisi by Pahamoka/Naemok 14 (QuickBasic) 9) Mutsiis kans by Visy/Tukka 12 (QBasic) 10) 288 Lines of QBasic Power! by RV-1991 11 (QBasic) 11) Weeden by Depili 4 (Casio FP-200) Cool! The Excel demo won! The overall standard of this compo was pretty good. Compo: World Domination 1) IRC-Galleria This was the obligatory Setok fit of self-indulgence competition. The general aim of this competition was to achieve 'World Domination', or a state as close as possible, and to have this process documented by the media. This could have meant anything, from Finnish demo collective advances world revolution agenda with excessively preachy manifesto, to finnish demo collective swallows and destroys Microsoft through cunning use of global capital etc... Compo: Most Obscure Computer 1) Bethvoid/Bleubtreu (Texas Instruments some model) Do you get the feeling that the original interest in older hardware is slipping away rather too quickly to be decent?! ------------------------ A:\ Run exit procedures -------------------------- The prize giving ceremony goes on for a while, with all the different competitions. The lucky (male) winners have the added bonus of being greeted by the lovely Alt Party mascot 'MaryCloud', and more of her later, yes yes, you've said that before! From there, dissolution is remarkably quick, and the end is near. it is at that point, that we get to meet some last minute people, in the form of a couple of UK'ers who we managed to miss seeing for the entire duration of the party up to now! These are a couple of Welsh females, with the look of a mother and daughter set-up. I didn't get enough information about the precise nature of their relationship to each other? They are connected with the inner workings of Llamasoft and the sheep cult surrounding Jeff "otherwise quite sane" Minter. We get on like the proverbial house-inferno, they mention something about a Llamasoftie camping trip that might well come about in the summer. With that news, Alt Partygoers are flying in all directions, and soon, it is our turn to leave the scene of the crime too. Q-Funk has managed to book a massively luxurious taxi-like minibus. A vehicle which can carry seven people and luggage in comfort has no problem with us three, and our slender baggage requirements, and pretty soon, we are at the new Q-Funklie GHQ, somewhere in the outer reaches of Helsinki. Whilst we have been running in a self-contained manner in the party environment, outside, some fresh falls of snow have been making life interesting, as we slather the last couple of yards back to Q-Funks apartment. We soon settle in, a large degree of tiredness takes over for the next couple of hours or so, as Martin's bed is so very comfortable, and he catches up with various domestic items and IRC'ing, and Felice makes various people reassuring phone calls to distant parts of the world. Thoughts eventually turn to getting an evening meal. This is within a short walking distance at the local Chinese restaurant, which is having a quiet post-Xmas Sunday. Whilst waiting for the three-course meal that we ordered, Q-Funk explains that the Chinese restaurant trade around these parts is very helpful, as they are the only places open on Xmas day! Fine if you can't be too bothered yourself. Whilst we are eating our meal, Q-Funk starts to worry about a French guy called 'Nystep' who turned up to Alt, who is staying at some seedy dive in the downtown part of Helsinki. Nystep latched onto Marty as a friendly French-speaking source of help, in an otherwise strange country. Nystep is experiencing hassles with the owners of said seedy dive, and having difficulties in drawing cash out of the hole in the wall ATM. (Both of these problems proved to be entirely solveable upon closer examination.) Anyway, there is a pre-planned pub crawl taking place from 20.00hrs on, in that same downtown area. So it is, we are headed there just a bit later on... The journey to downtown takes in quite a lot of public transport, both above ground, and the subway and also about half a mile's worth of sliding across snow-covered pavements. Setok is tantalisingly imprecise about where he and the other organisers have located themselve to, but a phone call later finds us in the right place. We gratefully enter the bar, and the bartender seems to be glad to see us, to the point of shouting greetings.... "Hey hey!" he shouts, we think, "Great, he knows us!", but his remarks are addressed to the two drunks nearby, who are busily twatting each other in the near corner. Well it was mainly one guy hitting the other, who seemed to be rolling rather well with the incoming punches. We conclude that it would perhaps be a bit premature to start ordering drinks at that moment?! The warring parties are disentangled and evicted, with a cheery "Fuck you!" issuing from the more insulted and aggressive party, and we are able to settle down for the rest of the evening, robbing the chairs that have just been forcibly cleared for our own use! Apart from Setok, several other organisers and their guests are present as well, including Baggio, Partycle, Wiztom, and several females seen in various places around the party. Q-Funk goes to talk to Nystep, to sort out his situation, even leaving the bar with him at some point, to sort out the hotel hassles. The rest of the evening passes very pleasantly, but we gradually become aware of another fuckwit starting to intrude on our enjoyment horizon. This is a middle-aged drunk, complete with a Saddam Hussein style untidy beard, whose idea of haute couture is to wear clothing with the beer company logo printed on it. He is not unpleasant at first, even speaking some drunkard French dialect to the unfortunate Nystep. He, without his prior consent, has been previously appointed this evening's "Loony magnet", thus freeing the rest of us from such drunken menaces! By the time people are starting to leave, it is almost the midnight hour, and this guy is offering to fight people in English. We don't get to witness the aftermath, but gather that a vigilant bar tender managed to sort out this person, and send him on his way too. Back across town, and a dash for a late bus home. Miss late bus, pass a few not unpleasant minutes waiting for the next one, and reflecting on the fun and games of the party thus far. We get back home soon after, and put an end to Sunday, a general collapse into various forms of sleeping equipment follows! READ B:\monday\big_sleep_in\dontwake.prg Ah, pleasant blackness, soft but unyielding. Well almost, as when it is just starting to get light, a dreamstorm across my cerebral cortex blasts me to a brief wakefulness. A quick bathroom pitstop helps greatly, and the next bout of sleep is a much more relaxing affair. Eventually, at some point very late in the morning, or early in the afternoon, we all manage to stir ourselves very reluctantly, eventually doing the getting up thing. I finally get to meet Q-Funks' new room-mate, who came with the new apartment. He is a big improvement on the mad drunk from the two previous Alt sleepovers. His only fault is to look something like a Finnish edition of Yahoo Serious! In the adjoining room, his very nice cutie blonde girlfriend can be seen playing something with a shooty- death mayhem theme on their PeeCee.. We are only able to rouse ourselves very slowly, in the post party aftermath. It is getting into the late afternoon period by the time we manage to drag ourselves off to town. We make sure that our after-party gear is taken with us, as we are planning to make our way straight there in the evening, without coming back to Martin's place. Outside, it has been snowing somewhat, not too heavily, but enough to provide something of fresh interest to walk on. A bus looms out of the grey and blizzardy conditions and we get on gratefully. Then a very lengthy episode of bus and local railway gets us to the residence of Q-Funk's current squeeze, whom we met briefly at the party. She is at work, but Martin catches up on his sanitary refreshment there, and that apartment is very comfortable, too much so, when we have to venture forth into the snowfields once more.. We go back into the city centre, this time on a search for pizza. This is duly found, in a neat bar-cafe not too far from Rax's, the domain of 'Eat until you explode'. Q-Funk is a regular there, and the bar person seems to know him well. What is given to us foodwise, is plenty to be going on with. Whilst we are contentedly pigging away, Nystep turns up, his hotel and monetary hassles solved, and he is seemingly pleased to see us. The next vital and necessary step, is to secure some inportant alcohol supplies for the after-party, and it is to the 'Alko' booze shoppe, that we adjourn to next. We spend an enjoyable fifteen or so minutes mulling the many choices over. Felice opts for a bottle of fine old red wine, I am caught in a dilemma over several different varieties of flavoured vodka, choosing a big bottle for public enjoyment, and a smaller bottle for my own abuse! We pay for our liquor, carefully averting the distainful gaze of the cashier, who no doubt has had state-sponsored training, with periodic refresher courses, in the art of looking down on people who choose intoxicating liquids for pleasure. Time is getting on by then, the appointed hour of sauna is near! Thence we goto the mighty After-party! A bus ride gets us to the spot where we came upon Jeff Minter and visiting party last year, and we head, bottles clinking, firstly to the place where the Reservoir Gods refused to bow down and worship the sauna after the third Alt Party. This turns out to be the wrong venue, so we head on over to the scene of last year's Yak-baptism of steam, which happens to be the right location for this year's shindig. RUN C:\afterprt.prg Now this afterparty left lots and lots of different impressions. It might take me a while to gather in all the juicy ones, so bear with me a bit. Anyway, I can tell you straight off, that we were there nice and early, before most other people had arrived. We donate our booze to the kitchen supplies, and start to discreetly crack open the odd bottle or two ourselves. Felice develops a taste for the vodka that I brought. Meanwhile, he has unpacked his laptop, having first remembered to bring it with him, and we are initially absorbed in browsing off the 'No Fragments' ST demo collection CDR, via the STeem emulator. Other people gradually start to arrive, the sauna is warming up, a big log fire crackles invitingly in the fireplace nearby. Another stout Atarian, Baggio, has opted to stay around for this afterparty, which was a good decision on his part. Setok turns up briefly, and disappears again for a time, the better to catch some more sleep, other recognisable faces, such as Twilighte, Partycle, and Wiztom all turn up before too long too. The expensive Epsom projector, which made the Alt Party bigscreen so viewable, is back with us again. The sponsors would no doubt be horrified to see it put to work in such a drunken and bohemian environment! There is a muchly varied menu of things shown on the big screen. A strange game by Finnish demo collective 'tAAt' seems to be very popular, which is a person- mashing game at the hands, or wheels of a big truck. You take turns to crash this person and truck into a wall, tweaking settings for position and angles of ramps, and even where to have this person sitting, inside, outside, or right in front of the truck! Damage points are built up, according to how catastrophic your "solution" is! Nobody we saw got into the really big points shown on the highscore, but we did have some amusing moments on the way, such as the run which catapaulted the player perfectly onto the top of the wall, or the crash which suceeded in lobbing them over the wall, but the camera panned around just in time to catch the last limb-twitch of the newly created corpse! Finnish sceners, criminally insane? Naah! Just "differently moralled"! There are lots of different PeeCee things. A wide selection of demo material over the course of the evening, fortunately without the Amiga gayscene material which blighted last year. We even have a little bit of a go ourselves, as Felice manages to hook up his laptop, with the No Fragments disk, and bash the population with a selection of the ST's greatest hits, he chooses oldschool stuff such as the evergreen Mindbomb, where as I opt for some newer smash hits, such as EIL 3 hit 'Darkness'. Of course, the still mighty 'Brain-Damage' got a choice spot, out of respect to the members of Aggression who attended the party. Wiztom delights in telling us which bits of that demo, were cunning coding cheats! And did we manage to get away without playing "H-Demo IV"? Of course not! It is made even better when we are able to point out one of the participants in that notorious video capture, namely Baggio, who tried to accentuate his lack of hair and growth of beard compared with back then, but without success! It may have been a couple of hours, or possibly even more in, by the time the sauna kicked off? The arrival of a still uncharacteristically quiet Nosfe changed the mood, as he was in there first. Soon, a stream of people were making their way to the changing rooms, for the latest instalment of naked steam powered action! This starts off in the usual manner, familiar to readers of past Alt Party reports. But after a while, certain format changes become apparent. A number of females conncted with the Alt Party have also turned up. This injection of the fairer sex includes the fragrant presence of the delicious Marycloud!  Marycloud Satan, erm Setok! It soon becomes apparent that these female members are not waiting for the creation of a segregated sauna area, as they strip off and join right in! Suddenly, those people not already committed to the Sauna, become more than usually interested! Without going into details, and looking like a bit of a pervert in the course of writing this report, I might take this opportunity to pass my compliments on to all the women who entered the sauna with us, on the night of that afterparty! You were all very nice indeed! Notwithstanding the interesting fact that a mixed sauna is not so common in Finland, most people take matters in their stride. A hotbox with added steam isn't the sort of place where you get too excited. However, others weren't so easy with the concept, as poor old Nystep managed to get a stiffy upon sighting the fair and unencumbered female flesh. (I'm not taking bets, but I've got a feeling it was MaryCloud 'wot triggered that reaction!?) When this was mentioned to her later, she took it as a compliment! The whole sauna thing goes on for quite a while, a lot longer even than at the previous after-parties. The action transfers to the smaller wood burning sauna. I was one of the first in there, whilst it was still getting to full temperature. It was a very nice sauna, especially when the steam was up. According to sauna expert Setok, a lot less harsh and easier to tolerate for longer periods than the big electric one. Some people went outside briefly, Nystep with the ladies rolling in the snow, possibly to reduce his groinal swelling?! I went outside for a while with Twilighte and others, which was a tolerable feeling, apart from everyone's feet getting very cold. The sauna action climaxes (ahem!) with a record of sorts being established for the most people in a confined space. This would be two women, and a dozen blokes in a space which would fit nine people comfortably! And I was in there, at one stage, sitting next to MaryCloud, who was the 'middle layer' of a lap-sitting 'sandwich!' She was clearly enjoying the situation, and the guy underneath had never suffocated more enjoyably! The whole sauna experience, naked chicks and all lasted a long time, I reckon past midnight. In between skinny-dipping sessions in steam, Marycloud was indulging in a series of make-up experiments on various people, who were prepared to put up with the minor inconvenience of her wrigging on them! Twilighte and Partycle were painted up to look like Rocky Horror Show extras.  Partycle walks like an Egyptian, Wiztom and Felice reflect on a lucky escape! In the meantime, our old friend, and the refuge of the labouring classes, DRINK had not been forgotten. There was plenty of it brought into the party. I soon managed to finish off my own personal supplies, with the enthusiastic help of Felice, who turned into a little bit of a vodka fan that night! Then it was time to tuck into the substances that others had brought with them! Quite a bit later, some people fell off the pace. Twilighte was one such, as he crashed for a couple of hours near the end of the party. Some people resorted to a Playstation dance-mat game for crazed entertainment. The log fire roared on, the drink was gradually searched out and drunk. I seem to remember carrying out a search of the kitchen area with someone, possibly Wiztom, and carrying on a loud conversation like something out of the Viz Comic "Drunk Bakers" strip. "Looks like all the vodka's gone... Good stuff that is.." "Here, have a shot at this Pastis that Nystep brought, it's strong stuff on its own, but better with water, turns all milky..." "Mmmm, not sure about that one, tastes funny, like aniseed.." A bit later still and some people get an attack of the munchies. This entails eating anything to hand... There was a nice open fire, which some people were in danger of falling into, but there was no reindeer stew, or barbeque sausages organised. The snacks and crisps supplied had long gone by then. I do remember getting my hands on a small microwave pizza, which tasted fantastic in the booze-fuddled and hunger-sharpened conditions. Eventually, at around 06.00, the party is beginning to run down. Everything fun or useful has been done, and everything consumable has run dry. I am starting to feel more than a little sleepy myself. There is something of an en-masse departure for early buses, as the bleary-eyed crowd staggers into the new day.. Our bus takes us most of the way home. Not quite all, as we walk the last half a mile partway across the bridge back to the city centre, which was alarmingly described as 'two kilometres' by the Q of Funk. So either he is a very poor judge of distance, (See Altparty III report!) Or a particularly miraculous "beer taxi"(*) was operating that morning! (*) Definition time, ho hum, "Beer Taxi", the means whereby you somehow get home when you are very drunk, without being able to recall too much of the journey back! So about ten minutes of walking later, I recognise that I am back in Q-Fey's neighbourhood, and the cheer in my heart substains me the last hundred metres or so to my waiting bed and SLEEP! FIND D:\wake_up_dammit\tuesday.prg We struggled awake sometime around midday, maybe a bit later. There was a simple plan for our last day, which was to meet up with the remaining international visitors, at the premises of 'Molly Malones for one last session. It is late afternoon before we have pulled ourselves together properly, to be able to get away to town. Q-Funk's room-mate is very helpful with a big supply of fresh hot coffee on tap, but he has to leave as well. We meet with Twilighte in the rather congenial 'Pullman Cafe', which is on the top floor of the Central Railway Station. This has a limited but tasty menu, so we order food, and linger for a fair while there. We do get talking about various ALT party issues, and dishing out the gossip on some of the people at Alt. We also get to hear a bit more of the inside story of 'Alt- TV', and the people doing silly crazy things with it, early in the morning! (Hi Setok!) We get to Molly's first, order drinks, and find out that a lot of our first choices aren't actually available, for one reason or another, try again, and again, until we all have something we're reasonably happy with, then proceed to find a spot in the far corner that can accomodate us, and the rest when they turn up. Baggio, Partycle, Wiztom all arrive soon after. We spend a considerable dollop of time at Mollys, at least getting another drink or two there. Some kind of live music act is setting up on the small stage nearby, so most of the visiting party leaves, apparently going on to some snooker place. But Q-Funk, Felice and I stay where we are, as Q-Funk's girlfriend, and her best friend are coming to meet us at Molly's. The band start to play, and it turns out to be an undemanding rock outfit, playing cover versions of middle of the road stuff, at a not too harsh volume. Clearly the exposure to the many and varied live acts at Alt, has raised our game for musical endurance! We even get talking to some of the local females who are watching the band next to us. I end up chatting to a redheaded girl called Miha. (excuse spelling of the name?) She had been to the UK, and had gained some interestingly hairy work experience over there. She was good company, a good laugh even, and she revealed that she had just finished with her significant other, and was going out with some friends to cheer herself up. It seemed to be working... I get the feeling that I could have been onto a good thing if the situation had been slightly different? But then again, I could be kidding myself ;-) As the midnight hour approaches, Mollys is getting very crowded, awaiting the climax of the live music, but we bid farewell to our new friends, and start back home, as we have a bit of an early start for the flight home. RUN E:\oh_fuck_is_it_that_early\time_to_go_home\exit.prg Wednesday, 06.30hrs.. Wake up, stagger in darkness, lights go on, bathroom, get dressed, pack everything, crush last gasp of air from air mattress. Between the three of us, this final act is carried out awesomely quickly, to the point where we are ready to set off by 07.00. So we do.. There is not a lot more to add, we are well-served by the various buses going from Q-Funk's to the airport, the latter service being the luxurious direct shuttlebus connection. It is still very dark out, as we glimpse the rush hour traffic building up but going the other way into town. The flight back is a weary action-replay of the flight over, and all too soon, we are back in time-adjusted to GMT Britain. Stanstead proved to be easy to negotiate, and we reacquaint ourselves with Felice's car, driving on the correct side of the road, and so on. One little surprise was brought back with us, no, not Nosfe springing out from the aircraft undercarriage well screaming "Fire in your heart!" in Finnish, but an increasingly Suomi-style snowfall, which got more intense as the journey back to Felice's wore on. I pause for a short time at his place, in order to deposit the Alt-travelled Falcon with him, and take up my car keys for part II of the journey back. Now the snowfall is really looking serious, and by the time I get back to the Northampton traffic mix, I have more to fear from the stumbling idiots failing to cope with the wet and slippery conditions than I ever did in Helsinki. I do get home in one piece though, and gratefully collapse, before logging on to check out all the lovely spam that has silted up my mailbox while I've been away. Another Alt Adventure is over! PAUSE A:\rounding_up\summing-up\endbit.prg Okay, so what have we learned from our latest soujourn to Finland, this time around? This fifth edition was never going to be quite able to match up to last years Minterquake. This years guest, a Finnish geek turned MP was most notable for keeping his appearance brief and to the point, but then again, where were they ever going to get another guest of Mr Minter's calibre? The view in some quarters was, that perhaps some of the competitions were perhaps a bit too alternate? The main demo competition suffered from an interesting idea, which turned out to be not so great when people attempted to put it into practice. We also have to sadly note a certain number of missing people from this year's party. The big man Teque briefly arrived at the party, but wasn't seen by many people. Pahartik didn't make it at all, and also on the list of the missing, was Oric killer Dbug, who couldn't make it either. The party atmosphere felt a bit more self-contained and parochial in places this time around. There seems to be a confusion over wanting to attract more international visitors, but there was a feeling that Alt could equally turn inward into a very local event, with little to appeal to outsiders. There was a tendency for the organisers and their friends to be a little bit cliquey and self-indulgent in some ways. Is it possible we could all be running out of sharpness and excitement and stumbling into a well worn rut of routine now? Maybe we were feeling the acute lack of something Minteresque, to act as a sharp poke in the eye of complacency? A lot of the competition entries and general entertaiment was in the Finnish language, and much of it tended to run on in-jokes made for the benefit of a few select people. This is fair enough, as it was their party, to run as you wish, but it got a bit teeth-grating to be told that something like that is the "funniest" thing on this planet, when it isn't to your ears, which happened to me more than once! The 'oldschool' computer theme staged a brave rearguard action against an increasing marginalisation, aided and abetted by a certain Falcon '030, against the waves of relentless laptops. There was even a semi-respectable amount of entries from Amiga, MSX, and Vic 20 in the competitions this time around. With regard to the usual rowdy crowd, they were there as well, very much in a nocturnal sense. I thought of making some kind of simulation, computer wargaming Drunk Polski vs Suomi sceners! It would be too close to tell which would out-drink the other?! These seem to be a well-established ALT tradition by now. It was great to see many familiar people again. The usual suspects like Marty (Q-Funk), Setok, Partycle, Wiztom, and Ravel all came forward. Lobotomy, another of the organisers, and the evil arch-compo entry disqualifier, became better known to me this time. Of newer people, I got talking to Diamondie, sometime diskmag queen (writes for Hugi) and we met some new people such as Nystep and the welshies for the first time. There was also the welcome presence of fellow UK'ers like Ian 'Jagfest' Smith, and the return of Twilighte. He now seems to be more involved in the Alt-TV side of things, and attended last year's Assembly Party in the summer. Also Baggio made up for the rest of the Swedes not being there, by sticking around for the afterparty, a good move, even when H-demo IV was being played! I seem to remember that there was at least one other UK person there, an individual going under the nick of "Gasman", but we never got around to spotting him. The afterparty rocked! It gets better each year! No other words needed really... Well, just one more thing. I send out greetings to Marycloud, Osyn and the other females who joined in the sauna with us. You were all fantastically cool! (In a very steam-heated way!) On a personal note, it was great to get the second Dildo Fatwa production out of the door, and onto the big screen. Even if it didn't get anywhere in the competition, we did seem to wake up those parts of the audience sent comatose by some of the previous entries. Maybe some people capable of making proper demos on Atari might be more motivated to challenge at the next Alt Party? I also give a lot of thanks to Martin (Q-Funky person) for putting up with us as always. We hope that this year goes a lot better for you than this last one. Thanks again pal! To conclude, I ejoyed this Alt edition a hell of a lot, I'm looking forward to the next one, but I won't get too wound up if it is rested for a little while, and allowed to freshen up a bit. CiH, for Alive Mag,Jan '04. Photographic credits on those pictures I 'borrowed' for use in this report go as follows to Timo Rossi for the Battle Damaged Viper, and those stunning After-Party pics. MSK takes the rap for that suggestive piccy of Marycloud doing "things" with my Falcon!  Alive 8


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Close - Maple Help

# Online Help

###### All Products    Maple    MapleSim

Connection

 Close
 close the Connection Module

 Calling Sequence connection:-Close( )

Parameters

 connection - Connection module

Description

 • Close frees the resources associated with connection. This happens automatically when connection is garbage collected; however, you can call Close to release the resources immediately.
 • Any descendant modules of connection are closed when connection is closed.  (A module is a descendant of a parent module if it is returned by one of the parent module's exports or if it is a descendant of one of the parent module's descendants.)

Examples

 > $\mathrm{driver}≔\mathrm{Database}\left[\mathrm{LoadDriver}\right]\left(\right):$
 > $\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):$

Create a descendant of conn.

 > $\mathrm{res}≔\mathrm{conn}:-\mathrm{ExecuteQuery}\left("SELECT * FROM animals"\right):$

Close conn.

 > $\mathrm{conn}:-\mathrm{Close}\left(\right)$

Try to use conn.

 > $\mathrm{conn}:-\mathrm{ExecuteQuery}\left("SELECT * FROM animals"\right):$

 Error, (in ExecuteQuery) this object has been closed, no further operations are possible

Try using conn's descendant.

 > $\mathrm{res}:-\mathrm{Next}\left(\right)$

 Error, (in ExecuteQuery) this object has been closed, no further operations are possible


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Browse Questions

# Using matrices, solve the following system of linear equations: $$x + y + z = 4$$ $$2x - y + z = -1$$ $$2x + y - 3z = -9$$

Toolbox:
• If the value of determinant of a $3 \times 3$ square matrix is not equal to zero, then it is a non-singular matrix
• If it is a nonsingular matrix, then inverse exists
• $A^{-1}=\frac{1}{|A|} adj (A)$
• $A^{-1}B=X$
Step 1:
Given $x+y+z=4,2x-y+z=-1,2x+y-3z=-9$
The above system of the equation is of the form $AX=B$
$(ie)\begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ 2 & 1 & -3 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} 4 \\ -1 \\ -9 \end{bmatrix}$
where $A=\begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & -1 \\ 2 & 1 & -3 \end{bmatrix}\qquad X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}\;and \; B=\begin{bmatrix} 4 \\ -1 \\ -9 \end{bmatrix}$
Let us first find the determinant of A
$|A|=1(-1 \times -3 - 1 \times 1)-1(2 \times -3-2 \times 1)+1(2 \times 1- 2 \times 1)$
$=2+8+4$
$=14 \neq 0$
Hence this is a non singular matrix and $A^{-1}$ exists
Step 2:
let us now find the adjoint of A
$A_{11}=(-1)^{1+1} \begin{vmatrix} -1 & 1 \\ 1 & -3 \end{vmatrix}=3-1=2$
$A_{12}=(-1)^{1+2} \begin{vmatrix} 2 & 1 \\ 2 & -3 \end{vmatrix}=-(-6-2)=8$
$A_{13}=(-1)^{1+3} \begin{vmatrix} 2 & -1 \\ 2 & 1 \end{vmatrix}=2+2=4$
$A_{21}=(-1)^{2+1} \begin{vmatrix} 1 & 1 \\ 1 & -3 \end{vmatrix}=-(-3-1)=4$
$A_{22}=(-1)^{2+2} \begin{vmatrix} 1 & 1 \\ 2 & -3 \end{vmatrix}=-3-2=-5$
$A_{23}=(-1)^{2+3} \begin{vmatrix} 1 & 1 \\ -1 & 1 \end{vmatrix}=-(1-2)=1$
$A_{31}=(-1)^{3+1} \begin{vmatrix} 1 & 1 \\ -1 & 1 \end{vmatrix}=1+1=2$
$A_{32}=(-1)^{3+2} \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}=-(1-2)=1$
$A_{33}=(-1)^{3+3} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix}=-1-2=-3$
Adj $A=\begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}$
$=\begin{bmatrix} 2 & 4 & 2 \\ 8 & -1 & 1 \\ 4 & 1 & -3 \end{bmatrix}$
step 3:
Therefore $AX=B => X=A^{-1}B$
$A^{-1}=\frac{1}{|A|} adj (A),$ we know $|A|=12$
Hence $A^{-1}=\frac{1}{12}\begin{bmatrix} 2 & 4 & 2 \\ 8 & -1 & 1 \\ 4 & 1 & -3 \end{bmatrix}$
$AX=B \qquad => X=A^{-1}B$
Now Substituting for $X, A^{-1}$ and B we get
$\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\frac{1}{14}\begin{bmatrix} 2 & 4 & 2 \\ 8 & -5 & 1 \\ 4 & 1 & -3 \end{bmatrix} \begin{bmatrix} 4 \\ -1 \\ -9 \end{bmatrix}$
$\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\frac{1}{14}\begin{bmatrix} 8-4-18 \\ 32+5-9 \\ 16-1+27 \end{bmatrix}=\begin{bmatrix} -\frac{14}{14} \\ \frac{28}{14} \\ \frac{42}{14} \end{bmatrix}$
$\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$
Therefore $x=-1,y=2,z=3$