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# New discrete theory

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They are some common ideas, like using a grid and rules. States are updated by step.

But there's a difference : Edward Fredkin uses a deterministic rules and nokton theory uses some kind of probabilities.

There have been a lot of attempts to develop theories like this (Stephen Wolfram is perhaps the most high profile). As far as I am aware none of them have achieved anything significant.

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I agree with you.

But the common with Wolfram and Fredkin is using non probabilitic rules like cellular automata.

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Can you explain more.

Unfortunately the builders pulled out my line along wiht the phone and internet a couple of days ago.

I have now got a temporary fix in place and I see that things have moved on a bit.

Yes but I need to know where to start.

In other words I don't know your level of mathematical knowledge.

Do you know the difference between a function and an operator or that whilst the solution to a function equation is a value the solution to an operator equation is a function?

If not I will start by explaining this.

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I'm graduated. I know exactly what a function. For operator I have an idea.

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Hi studiot, I see you are connected to forum, can you respond at my last post ?

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Consider this function, which was first enunciated by Gauss well before any quantum or atomic theory.

$f(t) = \frac{1}{{\sqrt {2\pi } }}\frac{1}{\tau }{e^{\left( { - \frac{{{t^2}}}{{2{\tau ^2}}}} \right)}}$

Now take the Fourier transform

$g(w) = \frac{1}{{2\pi \tau }}\int\limits_{ - \infty }^\infty {{e^{\left( { - \frac{{{t^2}}}{{2{\tau ^2}}}} \right)}}} e\left( { - iwt} \right)dt$

Completing the square and executing some algebra leads to

$= \frac{{e\left( { - \frac{{{\tau ^2}{w^2}}}{2}} \right)}}{{\sqrt {2\pi } }}\frac{1}{{\sqrt {2\pi } \tau }}\int\limits_{ - \infty }^\infty {{e^{\left( { - \frac{{{{\left( {t + i{\tau ^2}w} \right)}^2}}}{{2{\tau ^2}}}} \right)}}} dt$

The integral on the right can be shown to equal one by complex integration so

$g(w) = \frac{1}{{\sqrt {2\pi } }}{e^{\left( { - \frac{{{\tau ^2}{w^2}}}{2}} \right)}}$

Which is of the same form ( in w) as the original function in t.

This is, of course, the normal or gaussian distribution in statistics.

The spread or uncertainty for each is

$\Delta t = \tau$ and $\Delta w = 1/\tau$

leading to

$\Delta w\Delta t = 1$

Which is an uncertainty theorem.

A physicist will tell you that if t is time and w is frequency of an electrical impulse than the above pair tells us that the narrower an electrical impulse the greater the spread of the frequency components.

She might also say that in classical wave theory wave number k and position x are similarly related so

$\Delta k\Delta x = 1$

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Thank you, now I understood the origin of Heisenberg inequality.

I see also that Heisenberg inequality is :

My question is : Exists a same kind of this inequality for this "new nokton theory" ?

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Doesn't the very nature of these 'noktons' contradict the uncertainty principle?

In the uncertainty principle the values taken by the two operators are allowed to vary continuously, but the value taken (or assumed) by one affects the value allowable for the other. Since their deltas have an inverse relationship we can say that the large one delta is the smaller the other is, but there is (in mathematical theory) no upper or lower limit to either.

Discretisation of the values (quantisation in physics) but introducing lower (and by implication upper) limitschanges things is is curently the subject of some debate.

Is reality discrete or continuous?

Prefessor Shan Majid of London University has published an interesting book, collecting thoughts from many famous scientists and mathematicians, on this matter.

On Space and Time

Shanh Majid

Cambride University Press

It is interesing reading

Edit a couple of interesting points about the version of the uncertainty theorem in my last post.

The theorem above is purely numerical and has no units, whereas the Heisenberg theorem has units.

My presentation was unusual and constructed to avoid quantum theory for demonstration purposes.

An interpretation of the theorem that is often given is that it applies to processes that it involves the composition/convolution of two operators, say AB.

If the order is important, that is if AB is not equal to BA

Then (AB - BA ) is not zero and the relation can be derived from this.

Physically this means that this allows for the fact that if you first fix the momentum and then measure the position where this occurs you will obtain a different result from if you fix the position and measure the momentum at that position.

This is where the confusion arises leading to the misunderstanding that it is only a measurement issue and not inherent in the theory.

Edited by studiot

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### Home > A2C > Chapter 4 > Lesson 4.2.1 > Problem4-84

4-84.

The slope of $\overline { A B }$is 5, with points $A\left(−3, −1\right)$ and $B\left(2, n\right)$. Find the value of n and the distance between points A and B.

Use the slope and the given point to write an equation.

Substitute into your equation to find n.

Now that you know the coordinates of both points, find the distance between them.

The distance between the points is

$5\sqrt{26}$


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# Properties

 Label 1680.2.f.f Level $1680$ Weight $2$ Character orbit 1680.f Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{3} - q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{3} - q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{21} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{23} + q^{25} + ( -3 - 2 \beta_{2} - \beta_{3} ) q^{27} -2 \beta_{2} q^{29} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( 2 - \beta_{1} + 3 \beta_{3} ) q^{33} + ( -1 + \beta_{1} + \beta_{2} ) q^{35} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{39} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 6 + \beta_{1} - \beta_{3} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{45} + ( 2 + \beta_{1} - \beta_{3} ) q^{47} + ( -3 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{55} + ( -2 + \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{57} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{59} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{61} + ( 3 - \beta_{2} - 5 \beta_{3} ) q^{63} + ( -\beta_{1} - \beta_{3} ) q^{65} + ( -6 - 3 \beta_{1} + 3 \beta_{3} ) q^{67} + ( 6 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{69} + ( -5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{71} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{73} + ( -1 - \beta_{1} ) q^{75} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( -8 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 6 + \beta_{1} - \beta_{3} ) q^{83} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{87} + ( -10 - 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{95} + ( -5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{97} + ( 4 - 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - 4q^{5} + 6q^{7} + O(q^{10})$$ $$4q - 2q^{3} - 4q^{5} + 6q^{7} + 2q^{15} + 4q^{25} - 14q^{27} + 16q^{33} - 6q^{35} + 12q^{39} - 32q^{41} + 20q^{43} + 4q^{47} - 20q^{51} + 2q^{63} - 12q^{67} + 28q^{69} - 2q^{75} - 8q^{77} - 24q^{79} + 4q^{81} + 20q^{83} + 8q^{87} - 32q^{89} + 4q^{91} + 8q^{93} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu + 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{1} - 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.618034i − 0.618034i 1.61803i − 1.61803i
0 −1.61803 0.618034i 0 −1.00000 0 0.381966 2.61803i 0 2.23607 + 2.00000i 0
881.2 0 −1.61803 + 0.618034i 0 −1.00000 0 0.381966 + 2.61803i 0 2.23607 2.00000i 0
881.3 0 0.618034 1.61803i 0 −1.00000 0 2.61803 + 0.381966i 0 −2.23607 2.00000i 0
881.4 0 0.618034 + 1.61803i 0 −1.00000 0 2.61803 0.381966i 0 −2.23607 + 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.f 4
3.b odd 2 1 1680.2.f.j 4
4.b odd 2 1 420.2.d.d yes 4
7.b odd 2 1 1680.2.f.j 4
12.b even 2 1 420.2.d.c 4
20.d odd 2 1 2100.2.d.g 4
20.e even 4 1 2100.2.f.b 4
20.e even 4 1 2100.2.f.h 4
21.c even 2 1 inner 1680.2.f.f 4
28.d even 2 1 420.2.d.c 4
60.h even 2 1 2100.2.d.h 4
60.l odd 4 1 2100.2.f.a 4
60.l odd 4 1 2100.2.f.g 4
84.h odd 2 1 420.2.d.d yes 4
140.c even 2 1 2100.2.d.h 4
140.j odd 4 1 2100.2.f.a 4
140.j odd 4 1 2100.2.f.g 4
420.o odd 2 1 2100.2.d.g 4
420.w even 4 1 2100.2.f.b 4
420.w even 4 1 2100.2.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.c 4 12.b even 2 1
420.2.d.c 4 28.d even 2 1
420.2.d.d yes 4 4.b odd 2 1
420.2.d.d yes 4 84.h odd 2 1
1680.2.f.f 4 1.a even 1 1 trivial
1680.2.f.f 4 21.c even 2 1 inner
1680.2.f.j 4 3.b odd 2 1
1680.2.f.j 4 7.b odd 2 1
2100.2.d.g 4 20.d odd 2 1
2100.2.d.g 4 420.o odd 2 1
2100.2.d.h 4 60.h even 2 1
2100.2.d.h 4 140.c even 2 1
2100.2.f.a 4 60.l odd 4 1
2100.2.f.a 4 140.j odd 4 1
2100.2.f.b 4 20.e even 4 1
2100.2.f.b 4 420.w even 4 1
2100.2.f.g 4 60.l odd 4 1
2100.2.f.g 4 140.j odd 4 1
2100.2.f.h 4 20.e even 4 1
2100.2.f.h 4 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{4} + 28 T_{11}^{2} + 16$$ $$T_{17}^{2} - 20$$ $$T_{41}^{2} + 16 T_{41} + 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$49 - 42 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$16 + 28 T^{2} + T^{4}$$
$13$ $$16 + 12 T^{2} + T^{4}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$400 + 60 T^{2} + T^{4}$$
$23$ $$1936 + 92 T^{2} + T^{4}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$16 + 12 T^{2} + T^{4}$$
$37$ $$( -20 + T^{2} )^{2}$$
$41$ $$( 44 + 16 T + T^{2} )^{2}$$
$43$ $$( 20 - 10 T + T^{2} )^{2}$$
$47$ $$( -4 - 2 T + T^{2} )^{2}$$
$53$ $$1936 + 108 T^{2} + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$4096 + 192 T^{2} + T^{4}$$
$67$ $$( -36 + 6 T + T^{2} )^{2}$$
$71$ $$15376 + 252 T^{2} + T^{4}$$
$73$ $$5776 + 172 T^{2} + T^{4}$$
$79$ $$( 16 + 12 T + T^{2} )^{2}$$
$83$ $$( 20 - 10 T + T^{2} )^{2}$$
$89$ $$( 44 + 16 T + T^{2} )^{2}$$
$97$ $$13456 + 268 T^{2} + T^{4}$$


text:
On the Performance of Selection Cooperation with Imperfect Channel Estimation

# On the Performance of Selection Cooperation with Imperfect Channel Estimation

Mehdi Seyfi,  Sami (Hakam) Muhaidat*,  and Jie Liang,  The work of S. Muhaidat was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant RGPIN372049. The work of Jie Liang was supported in part by NSERC of Canada under grants RGPIN312262-05, EQPEQ330976-2006, and STPGP350416-07. The material in this paper was presented in part at the 2010 IEEE Vehicular Technology Conference, VTC’10, Taipei, Taiwan.M. Seyfi, S. Muhaidat and J. Liang are with the School of Engineering Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada. Phone: 778-782-7376. Fax: 778-782-4951. E-mail: msa119@sfu.ca, muhaidat@ieee.org, jiel@sfu.ca.Corresponding author
###### Abstract

In this paper, we investigate the performance of selection cooperation in the presence of imperfect channel estimation. In particular, we consider a cooperative scenario with multiple relays and amplify-and-forward protocol over frequency flat fading channels. In the selection scheme, only the “best” relay which maximizes the effective signal-to-noise ratio (SNR) at the receiver end is selected. We present lower and upper bounds on the effective SNR and derive closed-form expressions for the average symbol error rate (ASER), outage probability and average capacity per bandwidth of the received signal in the presence of channel estimation errors. A simulation study is presented to corroborate the analytical results and to demonstrate the performance of relay selection with imperfect channel estimation.

{keywords}

Imperfect channel estimation, cooperative communication, relay selection, average symbol error rate, outage probability, average capacity.

## I Introduction

\PARstart

The promise of high spectral efficiency and capability of providing great capacity improvements in a wireless fading environment, as reported by [1] and [2], has led to widespread interest in multi-input multi-output (MIMO) communications. However, due to size, cost, and/or hardware limitations, a wireless device may not be able to support multiple transmit antennas. Cooperative diversity was proposed as an alternative to MIMO systems. It has been demonstrated that cooperative diversity provides an effective way of improving spectral and power efficiency of the wireless networks [3, 4]. The main idea behind cooperative diversity is that in a wireless environment, the signal transmitted by the source () is overheard by other nodes, which can be defined as “relays” [5]. The source and its partners can then jointly process and transmit their information, thereby creating a “virtual antenna array”, although each of them is equipped with only one antenna. It is shown in [6, 7, 8, 9, 10, 11, 12, 13, 14] that cooperative diversity networks can achieve a diversity order equal to the number of paths between the source and the destination, however, the need of transmitting the symbols in a time division multiplexing (TDMA) fashion limits the improvement in capacity. Additionally due to the power allocation constraints, multiple relay deployment is not economic. Relay selection then was proposed to alleviate the loss in spectral efficiency caused by multiple relay schemes and also to moderate the power allocation constraints.

Recently, there have been considerable research efforts on the performance analysis of cooperative diversity including the derivation of closed-form formulas for the average symbol error rate (ASER) [5, 6, 8, 10, 11, 12, 15], the outage probability [7, 9, 10, 13, 16], and the average capacity (ergodic capacity) [7, 10, 11, 17]. Most of the current works, however, assume the availability of perfect channel state information (CSI) at the destination () terminal. Channel estimation for relay communication has also been studied in literature as in[18, 19].

Related work and contributions: The impact of channel estimation error on the ASER performance of distributed space time block coded (DSTBC) systems has been investigated in [20], assuming the amplify-and-forward protocol. Building upon a similar set-up, Gedik and Uysal [21] have extended the work of [20] to a system with relays, assuming the amplify-and-forward protocol. In [22], the symbol error analysis is investigated for the same scenario as in [21]. In this paper, we investigate the effect of channel estimation error on the ASER, outage probability and capacity of a cooperative diversity system with relay selection. We first introduce our system model in Sec. II. In Sec. III, we explain the Gaussian error model introduced by [23] and use it in the selection scenario to obtain the maximum ratio combiner (MRC) output SNR. In Sec. V, we study the ASER performance of our system in the case of imperfect channel estimation. We develop upper and lower bound analysis for the general relay case to formulate the relay selection scheme. We then derive a closed-form formula for the ASER performance with imperfect channel estimation. We also derive closed-form formula for the ASER performance of our system in the high SNR regime as introduced in [5, 24]. We also study the outage probability and average capacity per bandwidth in Sec. VI and Sec. VII, respectively. Diversity analysis is also provided in VIII. Simulation results are presented in Sec. IX, followed by conclusions in Sec. X.

Notation: , and , denote the expected value, the complex conjugate and the estimate of the variable , respectively. and stand for the lower bound and upper bound , respectively. indicates that the relay with SNR is selected. and denote the probability density function (PDF) and the cumulative density function (CDF), respectively.

## Ii system model

In this section we consider a system in which a source node transmits information to a destination node with the help of the best relay node , which is selected under a defined criterion from M available relays. The transmissions are orthogonal, either through time or frequency division.

In the first data sharing time slot, the source node communicates with the destination as well as the relay nodes. In this phase, the signals received by the destination and each relay are

 ysd = √Esdhsdx+nsd, (1) ysi = √Esihsix+nsi, (2)

where , , and denote the transmitted signal with unit energy and the signals received at the destination and the relay node, respectively. and are the channel coefficients of the source-relay and the source-destination channels, which include the effect of fading. and represent the average signal energies received at destination and the relay terminals, respectively, taking into account the path loss and shadowing effects. and are additive white Gaussian noises (AWGN) in the corresponding channels with the same variance , i.e., .

In the next time slot each relay node normalizes its received signal and retransmits it to the destination. For the relay, the normalization factor is and the signal transmitted from the this relay is [7]

 xi=√Esihsix+nsi√Esi|^hsi|2+N0. (3)

Since the relays practically do not have a perfect knowledge about the channel , , the power normalization procedure in each relay is performed based on the estimate of the corresponding channel. We assume in this paper that . Based on (3), the signal received by the destination from the relay node is [7]

 yid = √Eidhidxi+nid, (4) = √EsiEid√Esi|^hsi|2+N0hsihidx+~nid, = αihsihidx+~nid. (5)

where is the channel gain from this node to the destination. is the average signal energy received at the destination via the relay in its corresponding time slot which accounts for the shadowing and path loss effects and . denotes the AWGN of the relay destination channel. is the equivalent noise term in . It can easily be shown that, conditioned on the channel realizations , where

 ω2i=1+Eid|hid|2Esi|^hsi|2+N0. (6)

Supposing only the relay participates in cooperation we look for the condition under which we can select the best relay, , by searching s for .

Since each relay only amplifies the signal from the source, the destination is the only place where an estimate of the information symbol is computed. In order to achieve maximum likelihood performance, the signals from both diversity branches (direct branch and the branch via the relay) are combined using a maximum ratio combiner (MRC). Decoding of is delayed until the relayed signal containing the information symbol is received at the destination. Since the noise power is not the same on the two sub channels, both diversity branches must be weighted by their respective complex fading gain over total noise power on that particular branch before the combiner. Thus we obtain the estimated information symbol as

 ^x = ^h∗sd√EsdN0ysd+α∗i^h∗si^h∗id^ω2iN0yid, (7)

where

 ^ωi= ⎷Esi|^hsi|2+Eid|^hid|2+N0Esi|^hsi|2+N0, (8)

since the destination only uses the estimate of the relay-destination channel.

## Iii Modeling of channel estimation error

Let the true channel gain and its estimate be and , respectively. It is shown in [23, 25] that it is possible to consider the channel estimate as

 ^h=h+e, (9)

where is a zero mean complex Gaussian noise. As and , are jointly Gaussian, the conditional probability density function (PDF), , would also be Gaussian with the mean [26]

 E{h|^h}=σ2hσ2^h^h, (10)

and variance of

 var{h|^h}=(1−σ2hσ2^h)σ2h. (11)

By (10) and (11) and assuming a least squares (LS) estimator, we can consider the channel as

 h=ρ^h+d (12)

where is the correlation coefficient between and and . In this paper, and are the channel estimation error variances and their corresponding factors of the , , and links, respectively.

According to (11)

 σ2D = (1−ρ)σ2h (13) = (ρ−ρ2)σ2^h.

Using (12), we can rewrite the first term of (7) as

 Dsd = ^h∗sd√EsdN0(√Esd(ρsd^hsd+dsd)x+nsd) (14) = ρsdEsd|^hsd|2N0x+√Esd^h∗sdN0(√Esddsdx+nsd).

Here, we have inserted (12) in (1) then the resultant expression is substituted into (7). Following [22], the received direct path signal after the MRC can be decomposed into the message part given as and the noise part which is given as . Finally, due to the independence of the , and , the effective SNR due to the direct path is given by

 γeffsd = Ex{|E{Dsd}|2}var{Dsd} (15) = EsdN0ρ2sd|^hsd|21+EsdN0σ2D,sd=ρ2sd^γsd1+ϵsd,

where is the estimated received SNR due to the direct path and .

Following the same approach, we can write the second term of (7) as

 Did =αi^h∗si^h∗id^ω2iN0[αi(ρsi^hsi+dsi)(ρid^hid+did)x+~nid]. (16)

We can extract the message part, error part due to channel estimation error, and the noise part of the received signal as111Conditioned on and the distribution of (III) is approximated as Gaussian. The validity of this approximation has been confirmed by simulation. It should be further noted that the same conclusion has been reported by [22] and the references therein.

 M=α2i^ω2iN0ρsiρid|^hsi|2|^hid|2x, (17) D=α2i^ω2iN0[ρsi|hsi|2h∗iddid+ρid|hid|2h∗sidsi +h∗sih∗iddsidid]x, (18) N=αi^h∗si^h∗id^ω2iN0~nid, (19)

respectively. To obtain the effective SNR expression, we need to find the ratio of the signal power to the overall noise power. Alternatively, the received SNR can be calculated as [27]. Since , , and are zero mean independent processes, we can write the effective SNR of the selected path as

 γeffi=Esiρ2si|^hsi|2Eidρ2id|^hid|2N20(EsiN0|^hsi|2(1+ρ2siEidN0σ2D,id)+EidN0|^hid|2(1+ρ2idEsiN0σ2D,si)+EsiN0σ2D,siEidN0σ2D,id+1). (20)

With a simple manipulation, we may write (20) as follows

 γeffi=ρ2siρ2id^γsi^γid^γsiλsi+^γidλid+ϵsiϵid+1, (21)

where , , , , , and . Following [22], can be ignored 222For a fixed SNR, i.e., dB, we have noticed that the mean square error MSE between in (21) and (22) is percent of for dB.; hence, (22) can be approximated as [6, 22]

 γeffi=ρ2siρ2id^γsi^γid^γsiλsi+^γidλid. (22)

## Iv selection Strategy

In the selection based amplify-and-forward scheme studied in [5, 6, 7, 8, 9, 28, 29] the best relay selected by the destination terminal is the one that leads to the maximum total received SNR, , which is given by

 γr = γeffsd+γeffi⋆, (23)

where

 γeffi⋆=arg maxi{γeffi}, (24)

for . , which is the total received SNR at the destination, is the addition of the SNR received from the direct path, , and the SNR received via the selected relay path . The selection steps are mainly as follows. The source terminal sends a ready to send (RTS) packet to all the relays and the destination. As the relays receive this packet they estimate their corresponding source-to-relay channel . They divide their signal by the factor and wait for the clear to send (CTS) packet from the receiver side. As each relay receives the CTS overhead, it estimates its corresponding relay-to-destination channel and will start a timer which is given by

 τi=1γeffi,

where is given by (21). Each timer counts down to zero right after receiving the CTS packet. The first timer that reaches zero sends a flag to all the relays and the destination, introducing itself as the best relay. Then all the relays would remain silent, and the selected relay forwards its signal to the destination, with an overhead package containing information about its source-to-relay link and the relay-to-destination channel . It is also possible for the relays to use a separate bit feedback link to send the CSI to the destination. Further details on the selection procedures can be found in [16, 14].

## V ASER analysis

In this section, we derive an ASER expression for the relay selection scheme. We first introduce the following theorems [30]

• Theorem 1: If and are two random variables with , then

 fY(γ)=1|m|fX(γm).
• Theorem 2: For two independent random variables and if , then

 fZ(γ)=fX(γ)+fY(γ)−fX(γ)FY(γ)−fY(γ)FX(γ).
• Theorem 3: Let , , where s are independent random variables, then

 FY(γ)=M∏i=1FXi(γ).

It is shown in [5, 6, 8, 10, 8, 11, 12, 15] that the symbol error rate (SER) in a cooperative scenario is given by , where

 Q(x)=1√2π∫∞xexp(−t22) dt,

and depends on the modulation scheme.333For instance for BPSK modulation . The analysis can be generalized to QPSK or QPAM modulation as well[5].

To find the average symbol error rate, we need to integrate over the probability density function (PDF) of , i.e.,

 ¯Pe=∫∞0Q(√kγr) fγr(γr) dγr. (25)

Finding the exact expression for can be quite cumbersome, therefore, in the following and to simplify analysis, we develop lower and upper bounds on . It is straightforward to show that for two arbitrary independent random variables, the following inequality holds [6]

 12min(X,Y)≤XYX+Y≤min(X,Y). (26)

Applying (26) to in (22) yields

 γi,lb≤γeffi≤γi,ub, (27)

where

 γi,ub = ρ2siρ2idλsiλidmin(^γsiλsi,^γidλid), (28) γi,lb = 12ρ2siρ2idλsiλidmin(^γsiλsi,^γidλid), (29)

Finally, , can be bounded as

 γlb≤γr≤γub, (30)

where and are the lower bound and the upper bound SNR values defined as

 γub = ρ2sd^γsd1+ϵsd+γi⋆,ub, (31) γlb = ρ2sd^γsd1+ϵsd+γi⋆,lb, (32)

respectively, where

 γi⋆,ub \lx@stackrelΔ= maxi{γi,ub}, (33) γi⋆,lb \lx@stackrelΔ= maxi{γi,lb}. (34)

Using (30), (31) and (32), (25) can be bounded as

 ¯Plb≤¯Pe≤¯Pub, (35)

where and , are the lower and the upper bounds for ASER’s. In this paper, we limit our discussion to the lower bound analysis. The upper bound analysis can be obtained similarly. Since the ASER is a decreasing function of SNR, a lower bound for would simply be .

Since our final goal is to obtain the PDF of we need to find the PDF of , for , first. Note that the PDF of can be expressed in terms of the average SNR as . Similar expressions can be found for the PDF of and as well. Thus, using Theorem 1, the PDF of can be written as 444Here we have to note that s are not i.i.d random variables, since s are not analogous for different relay paths.

 fγi,ub(γ)=1βiexp(−γβi), (36)

where . Using Theorem 2 and noting that , we obtain

 fγi⋆,ub(γ) = ∂Fγi⋆,ub(γ)∂γ (37) = ∂∏Mi=1Fγi,ub(γ)∂γ (38) = M∑l=1fγl,ub(γ)M∏i=1i≠lFγi,ub(γ). (39)

The closed-form formulation for would then be (see Appendix I)

 fγub(γ)=M∑p=1(Mp)∑m=1(−1)(p+1)[Ψpb]m¯μ[Ψpb]m−1[exp(−γ¯μ)−exp(−γ[Ψpb]m)], (40)

where and is a binary permutation matrix with dimension of which is defined in the Appendix I and

 b\lx@stackrelΔ=[1β1 1β2 … 1βM]T, (41)
 Q(γ)=12−12Erf(γ√2). (42)

Having noted that

 ¯Plb=∫∞0[12−12Erf(√k2γub)]fγub(γub)dγub (43)

by substituting (40) into (43) and setting , we can write as

 ¸¯Plb=12−12M∑p=1(Mp)∑m=1(−1)(p+1)[Ψpb]m¯μ[Ψpb]m−1[¯μ√k/2¯μ−1+k/2−[Ψpb]−1m√k/2[Ψpb]m+k/2]. (44)

Here, we have used the fact [31]

 ∫∞0γ Erf(aγ)exp(−b2γ2)dγ=a2b2√b2+a2    b≠0. (45)

Note that the upper bound analysis can be derived similarly.

To gain further insight into the performance of the selection scheme under consideration, we resort to the high SNR analysis. Defining , where is the average SNR, and following [24, 5], we can approximate the asymptotic behavior of in the high SNR regime with its Mclaurin series expansion, i.e.,

 f^γub=a^γnub+o(^γub), (46)

where , if the derivatives of up to order are null and

 o(^γub)=∞∑t=n+11t!∂tf^γub∂^γtub(0)^γtub.

It can be easily shown that by replacing (46) into (43), the asymptotic behavior of ASER is given by [5]

 ¯Plb→∏n+1m=1(2m−1)2(n+1)k(n+1)1n!∂nfγub∂γnub(0). (47)

where we have used the fact that .

Similarly, the asymptotic upper bound ASER can be found as

 ¯Pub→∏n+1m=1(2m−1)2(n+1)k(n+1)1n!∂nfγlb∂γnlb(0). (48)

Theorem 4: For all , the values of and are zero and for

 ¯Plb→∏M+1m=1(2m−1)2(M+1)k(M+1)1μM∏i=11βi, (49) ¯Pub→∏M+1m=1(2m−1)(M+1)k(M+1)1μM∏i=11βi, (50)

where .

Proof of Theorem 4: (see Appendix II)

## Vi Outage probability Analysis

The outage probability , which is a valid measure of performance in slowly fading channels, is defined as the probability that the source-destination mutual information falls below the transmission rate . Defining

 Isd\lx@stackrelΔ=12log2(1+γeffsd+γeffi⋆)≤R, (51)

the outage probability is given by

 Pout = (52) =

Substituting the introduced upper bound for leads to adopt an upper bound on the outage probability, i.e.,

 Pout≤Pubout = (53) = Fγub(22R−1).

We can simply find by integrating (40) with respect to the result of which is given by

 ¸Fγub(γ) = M∑p=1(


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# Batch Processing¶

## Algorithm Parameters¶

The DBSCAN clustering algorithm has the following parameters:

Algorithm Parameters for DBSCAN (Batch Processing)

Parameter

Default Valude

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Available methods for computation of DBSCAN algorithm:

• defaultDense – uses brute-force for neighborhood computation

epsilon

Not applicable

The maximum distance between observations lying in the same neighborhood.

minObservations

Not applicable

The number of observations in a neighborhood for an observation to be considered as a core one.

memorySavingMode

false

If flag is set to false, all neighborhoods will be computed and stored prior to clustering. It will require up to $$O(|\text{sum of sizes of all observations' neighborhoods}|)$$ of additional memory, which in worst case can be $$O(|\text{number of observations}|^2)$$. However, in general, performance may be better.

Note

On GPU, the memorySavingMode flag can only be set to true. You will get an error if the flag is set to false.

resultsToCompute

$$0$$

The 64-bit integer flag that specifies which extra characteristics of the DBSCAN algorithm to compute.

Provide one of the following values to request a single characteristic or use bitwise OR to request a combination of the characteristics:

• computeCoreIndices for indices of core observations

• computeCoreObservations for core observations

## Algorithm Input¶

The DBSCAN algorithm accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Algorithm Input for DBSCAN (Batch Processing)

Input ID

Input

data

Pointer to the $$n \times p$$ numeric table with the data to be clustered.

Note

The input can be an object of any class derived from NumericTable.

weights

Optional input. Pointer to the $$n \times 1$$ numeric table with weights of observations.

Note

The input can be an object of any class derived from NumericTable except PackedTriangularMatrix, PackedSymmetricMatrix.

By default all weights are equal to $$1$$.

Note

This parameter is ignored on GPU.

## Algorithm Output¶

The DBSCAN algorithms calculates the results described below. Pass the Result ID as a parameter to the methods that access the result of your algorithm. For more details, see Algorithms.

Algorithm Output for DBSCAN (Batch Processing)

Result ID

Result

assignments

Pointer to the $$n \times 1$$ numeric table with assignments of cluster indices to observations in the input data.

Noise observations have the assignment equal to $$-1$$.

nClusters

Pointer to the $$1 \times 1$$ numeric table with the total number of clusters found by the algorithm.

coreIndices

Pointer to the numeric table with $$1$$ column and arbitrary number of rows, containing indices of core observations.

coreObservations

Pointer to the numeric table with $$p$$ columns and arbitrary number of rows, containing core observations.

Note

By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.


text:
Explore BrainMass

# math

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Let K be a field. (All of this works just as well for a division ring.) Let V be an n-dimensional vector space over K and let A = EndK(V), the ring of n n matrices over K. The ring A is clearly Artinian since it has finite dimension over K. We need to see that the 2-sided ideals are either (0) or A.

1. Let W be a subspace of V. Then LW = { f  A | f (W) = 0 } is clearly a left ideal of A.
Prove that every left ideal has this form.

Here is an outline of one method. Let I be a left ideal, W = { x  V | f (x) = 0 f  I },
so I  LW. Let e1, . . . , en be a basis of V with e1, . . . , er a basis of W.

a. Show that if f (x) has zero e1-coordinate for all f  I, then x  W. [Note that if π permutes the ei, then π ◦ f  I.]

b. Note that H = {e1-coordinate function of f | f  I } is a subspace of V * = hom(V, K).
Use duality and (a) to conclude that H is the annihilator of W, and so any linear function
V → K vanishing on W is in H.

c. Let Eij be the function taking ej to ei and vanishing on ek for k ≠ j. Using (b) and a projection onto e1, prove that E1j  I for j > r.

d. Permute the ei to show all Eij  I for j > r.

e. Show these elements span LW.

2. Prove that (LW)T = LT - 1W (L sub T-1W) for any T  A. Deduce that (0) and A are the only 2-sided ideals of A, so that matrix rings are simple.

https://brainmass.com/math/basic-algebra/few-quest-functions-math-237009

#### Solution Preview

1. Let be a subspace of . Then is clearly a left ideal of . Prove that every left ideal has this form.
Proof:
Suppose is a left ideal of , . Then for any , we have . So . Let be a basis of with a basis of .
a. I show that if has zero -coordinate for all , then .
Consider any , since is a left ideal, then for any permutation matrix , we have .
Now has zero -coordinate for all , I claim that . If not, then we can find some , such that . Then must have some non-zero coordinate, say ...

#### Solution Summary

This solution is comprised of a detailed explanation to answer math questions.

\$2.49


text:
# Flatbed truck going round a circular road on an incline

1. Sep 23, 2014

### whdahl

1. The problem statement, all variables and given/known data
The problem statement is in the attachments.

2. Relevant equations
F=ma

3. The attempt at a solution

I uploaded the solution to the problem. I have had no trouble deriving the first two equations for the summation of forces in the x, n and t directions, however I am having trouble understanding the algebra involved in the solution, specifically where it says, "Solve the first two equations for N and Fn to obtain:", then it lists the equations for N and Fn. I can't figure out how they derived those two equations for N and Fn from the first two equations.

2. Sep 23, 2014

### Simon Bridge

... they did exactly what was described - they too the first two equations... these were:
$$N\cos(10^\circ)-F_{n'}\sin(10^\circ) - mg = 0\\ F_{n'}\cos(10^\circ) + N\sin(10^\circ) = m\frac{2t^2}{30}$$ ... do you not know how to solve simultaneous equations?

3. Sep 23, 2014

### whdahl

I know how to solve simultaneous equations, but I am not getting the solution they show in this case. Solving one of the equations for Fn or N and substituting that in the second equation will allow you to eliminate one of the variables. However I am not getting the solutions they provide, I am getting jumbles of sin squared's over cosines and tangents...

Last edited: Sep 23, 2014
4. Sep 23, 2014

### Simon Bridge

5. Sep 23, 2014

### whdahl

I don't see where my mistake is, but I've ended up with an extra $$cos(10)$$ in the final answer.

#### Attached Files:

• ###### photo(1).JPG
File size:
37 KB
Views:
59
6. Sep 23, 2014

### whdahl

Literally the moment I sent that, I found my mistake. I did not carry the $$\frac{mgsin10}{cos10}$$ over correctly.

7. Sep 24, 2014

### Simon Bridge

Well done.
Note: if you put a backslash before the name of the trig function in LaTeX, it will typeset properly.
$$\frac{mg\sin\theta}{\cos\theta}=mg\tan\theta$$


text:
# Conditions for equilibrium

Second condition of equilibrium an object in equilibrium does not move along a straight line -- it does not translate -- that means the sum of all the forces on it is zero that was the first condition of equilibrium but an object in equilibrium also does not rotate that means the sum of all the rotational forces on it is also zero. Torque the second condition necessary to achieve equilibrium involves avoiding accelerated rotation (maintaining a constant angular velocity a rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it to understand what factors affect rotation, let us. A solid body submitted to three forces whose lines of action are not parallel is in equilibrium if the three following conditions apply : the lines of action are coplanar (in the same plane) the lines of action are convergent (they cross at the same point) the vector sum of these forces is equal to the zero vector in the animation. There are two conditions of equilibriumaccording to first condition of equilibrium a body at rest or moving with uniform velocity has zero accelerationfrom newton's first law of motion the vector sum of all forces acting on it must be zero in second condition vector sum of all torques acting on body is zero. During the emerging period it is doubtful if anchor ice could smother trout eggs because this ice condition seldom persisted over i2 hours and because the ice had a porous texture also, atmospheric conditions which were apparently necessary for the formation of anchor ice did not occur often on the pigeon river in many. The four forces acting on an aircraft fig_28-05gif source: fas an aircraft in powered flight can be said to be under the influence of the four main forces: lift (l ) lift of main planes, acting vertically upwards through the centre of pressure the main purpose of lift is to keep the aircraft airborne lift is a.

The following conditions of equilibrium are regarded as satisfactory: (a) downflooding the final waterline after flooding, taking into account sinkage, heel , and trim, is below the lower edge of any opening through which progressive flooding can take place such openings include air pipes, ventilators, and. Let us consider a heterogeneous system of volume v with several homogeneous phases (φ = a, b,c r) existing in equilibrium let us further suppose that each phase consists of i(i=1,2,3 ,c) constituents further, number of constituents in any phase is different from others with each phase, a change in internal energy. Covers the linear (force) and rotational (torque) conditions for equilibrium. This text is meant to accompany class discussions it is not everything there is to know about the basics of torque and the two conditions of equilibrium it is meant as a prep for class more detailed notes and examples are given in the class notes, presentations, and demonstrations (click here).

An analytical criterion is given for the choice of independent components an inclusive discussion of equilibrium in systems of many constituents is thereby achieved by reason of symmetry, the resulting conditions are particularly suited to form the basis of a calculation of the equilibrium composition of the system. The first condition for equilibrium the second condition for equilibrium stability applications of statics, including problem-solving strategies simple machines forces and torques in muscles and joints rotational motion and angular momentum angular acceleration kinematics of rotational motion dynamics of.

Equilibrium constants and le chatelier's principle this page looks at the relationship between equilibrium constants and le chatelier's principle students often get confused about how it is possible for the position of equilibrium to change as you change the conditions of a reaction, although the equilibrium. You need to check that two conditions are satisfied before you can say that something is in equilibrium the sum of the forces in any direction = 0 if this is satisfied, the object will have no linear acceleration (for instance, it won't accelerate in any direction) the sum of the moments about any point (not just the pivot point) = 0. The equilibrium state can only be reached if the chemical reaction takes place in a closed system otherwise, some of the products may escape, leading to the absence of a reverse reaction (note that in the diagrams under characteristics of chemical equilibrium, all reactions are. It is the (very important) special case where the center of mass of the object has no motion and the object is not rotating 311 conditions for equilibrium of a rigid object for a rigid object which is not moving at all we have the following conditions: • the (vector) sum of the external forces on the rigid object must equal zero.

Here, we investigate the generality of these equilibrium solid-stabilized emulsions with respect to the basic constituents: aqueous phase with ions, oil, and stabilizing particles from systematic variations of these constituents, we identify general conditions for the spontaneous formation of monodisperse. Equilibrium of force systems the necessary and sufficient conditions for equilibrium of a force system are that the resultant force $\bf f_{\rm r}^{}$ and the resultant couple $\bf c_{\rm r}^{}$ be zero vectors:. Conditions of equilibrium thus two conditions must be satisfied for a consumer tq attain an equilibrium: (i) the price line should be tangent to an indifference curve or mrs of one c commodity for another should be equal to their relative pries this is no doubt a necessary condition but not a sufficient condition of.

## Conditions for equilibrium

An object is in equilibrium if the resultant force acting on the object is zero. A body acted upon by several forces said to be in equilibrium if it does not move or rotate under this equilibrium condition, the sum of the forces acting in one direction (eg upwards) must be equal to the sum of the forces acting in opposite direction acting downwards must balance the total prices acting. Let's say the forces are acting in the xy plane therefore, for equilibrium, summation of all forces in x-direction should be equal to zero summation of all forces in y-direction should be equal to zero these are the only conditions that nee.

• The conditions for equilibrium are basic to the design of any load-bearing structure such as a bridge or a building since such structures must be able to maintain equilibrium under load they are also important for the study of machines, since one must first establish equilibrium and then apply extra force or torque to produce.
• Hart (j econ theory 9 (1974), 293–311) gave conditions for equilibrium to exist in a securities model where each agent undertakes asset transactions to maximize expected utility of wealth these conditions rule out agents wanting to undertake unbounded balanced transactions to reach a pareto superior allocation given.

What the book meant by two dimensions is actually a system whose forces acting on are lying in a plane, ie, they are coplanar forces since the torque m → = r → × f → is a vector product involving forces, it points in a direction perpendicular to that plane if the forces are in the x y plane, then the torque is in the z. A three minute video introducing the two conditions for equilibrium relates to young and freedman's university physics. A system is said to be at equilibrium if the properties of the system: 1 are uniform throughout the system 2 they do not vary with time if both the conditions are satisfied then it is said to be in thermodynamic equilibrium the thermodynamic e. 111 the first condition of equilibrium the first condition of equilibrium deals with the forces that cause possible translations of a body the simplest way to define the translational equilibrium of a body is to say that a body is in translational equilibrium if it has no translational acceleration1 that is, if the acceleration of a.

Conditions for equilibrium
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2018.


text:
$\Delta U=q+w$

Hena Sihota 1L
Posts: 51
Joined: Fri Sep 29, 2017 7:07 am
Been upvoted: 2 times

The following equation was given in lecture today: DeltaU=energy transferred by heating (+q) + energy transferred by compression (+w)
If the system was cooled instead of heated you would just replace "heating (+q)" with "cooling (-q)" and if the system was expanded instead of compressed you would just replace "compression (+w)" with "expansion (-w)". Is that correct?

Jeremiah Samaniego 2C
Posts: 33
Joined: Fri Sep 29, 2017 7:05 am

Re: Clarification About DeltaU=q+w

I believe so. If a system is being cooled, then heat would be leaving the system (-q) and if the system is expanding, then work is being done by the system (-w).

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text:
# Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the law of $S_t$ knowing the value at $0$ and at $T$, for $t \in [0,T]$.

Thank you very much.

• the solution is known for sigma constant but I didnt find anything about this case. – mawchne Sep 15 '18 at 18:24
• The variance between $t_1$ and $t_2$ is going to be the average $\sigma_t^2$ in this interval i.e. $V(t_1,t_2)=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\sigma^2_t dt$. You could try to work from this, I haven't tried but I have a feeling things may get extremely complicated, especially for a completely general $\sigma_t$. Do we know anthing about how $\sigma_t$ varies with $t$? – Alex C Sep 15 '18 at 21:18
• Hi Alex, let's say it is constant wise, if that may simplify things – mawchne Sep 16 '18 at 7:36


text:
# Simulated MBE Wrong Answers - TORTS AND CON LAW

TORTS: As a general rule, one may use reasonable force to prevent the commission of a tort against one's property; HOWEVER, may NOT use force...
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text:
# CSE 643: Computer Security (Fall 2019)

## Basic Information

• Instructor: Prof. W. Du
• Classroom :
• 020 Dineen Hall (08/26/2019 - 09/11/2019)
• Lyman Hall 132 (from 09/16/2019 on)
• Time: Monday, Wednesday 14:15-15:35
• Class website:
• Textbook:
• Computer & Internet Security: A Hands-on Approach (2nd Edition) [amazon]
• Computer Security: A Hands-on Approach 2nd Edition (2nd Edition) [amazon]

The second book is a subset of the first one. However, this class will only cover the content of the second book.

• Office hours:
• Prof. W. Du: Tuesday 14:00-15:00 @ CST 4-285
• Misc:
• The lab environment can be acquired on this page. It runs on VirtualBox VM. Only version 6.0.4 is advised, using other ones may cause compatibility issues.

## Lab 1: Environment Variables and Set-UID Programs

### User id & privilege

In Linux, each user is associated with a unique user id. The root user (who has the greatest privilege) has user id 0. The permission granting mechanism verify a user’s id, instead of name when giving out permissions. Therefore, if one can somehow change his/her user id to 0, he/she can execute the rights of the root.

In Linux, each running process has three types of user ids, which is introduced here in detail. We are only interested in real user id and effective user id. Real user id is the id of owner of this process; effective user id is the id of the user whose rights will be given to the process. Usually, real user id is equal to effective user id.

One can finds out his/her own user id by typing id in the terminal.

[09/02/19]seed@VM:~$id uid=1000(seed) gid=1000(seed) groups=1000(seed),4(adm),24(cdrom),27(sudo),30(dip),46(plugdev),113(lpadmin),128(sambashare)  ### Linux file permission Linux uses 10 bits to describe file permissions. If we type command ls -l in terminal, we can get the permission information of the files in current folder. [09/02/19]seed@VM:~$ ls -l
total 1688
drwxrwxr-x 4 seed seed    4096 May  1  2018 android
drwxrwxr-x 2 seed seed    4096 Jan 14  2018 bin
drwxrwxr-x 2 seed seed    4096 Jan 14  2018 Customization
drwxr-xr-x 3 seed seed    4096 Aug 31 10:37 Desktop
drwxr-xr-x 2 seed seed    4096 Jul 25  2017 Documents
-rw-r--r-- 1 seed seed    8980 Jul 25  2017 examples.desktop
-rw-rw-r-- 1 seed seed 1661676 Jan  2  2019 get-pip.py
drwxrwxr-x 3 seed seed    4096 May  9  2018 lib
drwxr-xr-x 2 seed seed    4096 Jul 25  2017 Music
drwxr-xr-x 3 seed seed    4096 Jan 14  2018 Pictures
drwxr-xr-x 2 seed seed    4096 Jul 25  2017 Public
drwxrwxr-x 4 seed seed    4096 May  9  2018 source
drwxr-xr-x 2 seed seed    4096 Jul 25  2017 Templates
drwxr-xr-x 2 seed seed    4096 Jul 25  2017 Videos

In the output:

• 1st column will give detailed information regarding file permission,
• 2nd column will tell you about the number of links to the file,
• 3rd and 4th columns are associated with owner and group of the file,
• 5th column will be displaying the size of the file in bytes,
• 6th column will display the recent time and date at which the file was modified, and the last and 7th column is the actual file/directory name.

The first column indicates permission. We can split them into four segments as follows.

d rwx rwx rwx
if directory permission of owner permission of owner’s group permission of others

Notice that the meaning of each bit is different for files and directories. For normal files, “r” indicates the right to read; “w” indicates the right to write and “x” indicates the right to execute a file. For directories, “r” indicates the right to list its contents; “w” indicates the right to modify the content inside and “x” indicates the right to cd into the directory. We can compress the last nine file permission bits into three decimal digits. The way of doing it is to treat “rwx” as a three-bit binary representation of a decimal number. For example, “rwxrwxr-x” can be expressed as “775”.

There are also three other special permission bits that are not displayed by default. This page describes all three of them in detail. Here, we are only interested in the Set-UID bit. For a program, if this bit is set, then when someone runs the program, its effective user id would equal to its owner’s, instead of its runner’s (i.e. the effective user id is set to be the owner’s user id, instead of assigning real user id).

In order to make a program Set-UID program, we can use the following command:

[09/02/19]seed@VM:~$chmod 4755 progname  The “4” preceding normal file permissions indicates toggling the Set-UID bit only (in this demonstration, the command also changes the file permission to 755). When we run ls in a folder, Set-UID programs will be highlighted with red background. For example: [09/02/19]seed@VM:~$ cd /bin
[09/02/19]seed@VM:/bin$ls bash date hostname mountpoint ntfswipe sleep uname bash_shellshock dd ip mt open ss uncompress bunzip2 df journalctl mt-gnu openvt static-sh unicode_start busybox dir kbd_mode mv pidof stty vdir bzcat dmesg kill nano ping su wdctl bzcmp dnsdomainname kmod nc ping6 sync which bzdiff domainname less nc.openbsd plymouth systemctl whiptail bzegrep dumpkeys lessecho netcat ps systemd ypdomainname bzexe echo lessfile netstat pwd systemd-ask-password zcat bzfgrep ed lesskey networkctl rbash systemd-escape zcmp bzgrep efibootmgr lesspipe nisdomainname readlink systemd-hwdb zdiff bzip2 egrep ln ntfs-3g red systemd-inhibit zegrep bzip2recover false loadkeys ntfs-3g.probe rm systemd-machine-id-setup zfgrep bzless fgconsole login ntfs-3g.secaudit rmdir systemd-notify zforce bzmore fgrep loginctl ntfs-3g.usermap rnano systemd-tmpfiles zgrep cat findmnt lowntfs-3g ntfscat run-parts systemd-tty-ask-password-agent zless chacl fuser ls ntfscluster rzsh tailf zmore chgrp fusermount lsblk ntfscmp sed tar znew chmod getfacl lsmod ntfsfallocate setfacl tempfile zsh chown grep mkdir ntfsfix setfont touch zsh5 chvt gunzip mknod ntfsinfo setupcon true cp gzexe mktemp ntfsls sh udevadm cpio gzip more ntfsmove sh_backup ulockmgr_server dash hciconfig mount ntfstruncate sh.distrib umount  When the file permission details are displayed, it can be seen that an “s” bit is present in place of the usual “x” bit. [09/02/19]seed@VM:/bin$ ls -l mount
-rwsr-xr-x 1 root root 34812 Dec 16  2016 mount

### Password dilemma & Set-UID programs

In Linux, the password entries of each user are stored in /etc/shadow, which is owned by root user with permission 640. That is to say, normal users cannot read or modify this file. However, normal users should be able to change their own passwords freely. But if we allow them to access /etc/shadow, they may break the security of the system by changing other user’s password entry. This is the password dilemma: if we enable users to access the password file, they may compromise system security; but if no access is allowed, they are unable to change their own passwords.

Linux resolves this dilemma by implementing Set-UID mechanism, which was initially invented by Dennis Ritchie at Bell Labs [wiki]. As it is mentioned above, Set-UID programs would have their owner’s privilege. In Linux, the password utility passwd is a Set-UID program owned by root. That is, when a normal user run the program, the program itself would have root privilege, which means it can modify the password file (also any other file on the system). But (ideally) this does not bring any security hazard because the passwd program itself is programmed to handle the password file with grace and the password file only. That is, it only updates the password file when it finds out the the user provides the correct old password, and it would only change the user’s entry, leaving other ones untouched. It will never read or modify any other unrelated files.

### Potential attacks on Set-UID programs

Because Set-UID programs allows a normal user to gain (limited) root privilege, attacks on them become very appealing. We can first analyze the attack surface of Set-UID programs. The attack surface of a software environment is the sum of the different points (the “attack vectors”) where an unauthorized user (the “attacker”) can try to enter data to or extract data from an environment. For Set-UID programs, the attack surface is the sum of places where the program gets its inputs.

#### User inputs

If a Set-UID program fails to sanitize inputs from a user, it may create a security loophole. In Linux, the user information is stored in /etc/passwd, where each line represents one user. Two sample lines are shown as below. There are multiple fields in each line, which are separated by colons. The detailed explanation of each filed can be seen here. Now we are only interesed in the last field, which specifies the default shell program of the user, i.e. the first program that would be run after logging in.

[09/02/19]seed@VM:~sudo cat /etc/passwd root:x:0:0:root:/root:/bin/bash seed:x:1000:1000:seed,,,:/home/seed:/bin/bash  The Linux provides a utility called chsh to modify the default shell program. If it fails to sanitize user input properly, a user can provide a new shell program string with a line break character inside, effectively creating a new user in the system. By assigning 0 to user id, the normal user can even plat a new root account in the system. #### System inputs Sometimes even system inputs can be the point of attack on Set-UID programs. Suppose a privileged Set-UID program needs to write some file in /tmp/abc. Because the /tmp folder is globally accessible, one can create a symlink in /tmp with the name abc, but actually points to /etc/shadow. When the Set-UID program writes its output, it actually destroys the password file. #### Environment variables If a Set-UID program needs to read values of environment variables directly, it is very import to sanitize the values. However, threats from environment variables can occur even when the Set-UID program is not reading them directly. 1. When invoking other programs. Problems can occur when a Set-UID program is calling external procedures with system(). For example, a program can call system(date) to show the date. What this command does is to tell the bash program to search for a program named date in directories specified by PATH environment variable. However, if the user changes the PATH variable to a folder that includes his/her own version of date, the wrong version of date gets invoked instead. If the program happends to be a privileged Set-UID program, then the wrong date program gets executed with root privilege. 2. During dynamic linking. In Linux, the dynamic linker finds dynamic libraries in folders specified by LD_LIBRARY_PATH and LD_PRELOAD. If one changes these two environment variables to directories containing their own malicious versions of dynamic libraries, they can change the behavior of the program. #### Capability leaking It is very common for a privileged program to relinquish the privileges after completing certain operations. A common mistake made by programmers is capability leaking. It happens when the programmer forgets to release some resource gained when the program still owns the privileges, even tough he/she had explicitly downgraded the program’s rights. For example, consider the pseudocode below. At the first glimpse, everything looks fine. However, that is not the case. When calling fork, the child process gets a copy of all parent process’s file handles, which includes the one that has access to the important file. Although the programmer has explicitly relinquished the privileges before forking, indicating that he/she does not want the child process to access the file, the attempt obviously failed. Therefore, it is important for programmers to relinquish privileged resource as soon as possible, i.e. in this case, to close the file before forking. #### Invoking other programs We have discussed using system() to call an external procedure above. Now this concept is to be elaborated. When calling an external procedure with system(command), it actually calls /bin/sh -c command. For a privileged program, this can be very sloppy. Consider a program that scans virus on any given file/directory on the system. Because it has to be able to open any file on the system, it is a privileged Set-UID program. Suppose this program takes path as input from users, and calls system("ls " + path) to check the content of the folder, a normal user can easily run a root shell by passing "/abc;/bin/sh" as the path. Here, /abc is just a path name that is not relevant. What matters is the semicolon, which acts as a separator between two commands in shell syntax, and /bin/sh, which invokes the shell. Together with the environment variable vulnerability introduced above, it should be noted that system() is a very dangerous way of invoking other programs. A safer approach is to use execve function. It takes three arguments: the filename of the executable; the arguments passed to the executable and the environment variables defined during this call. Instead of using /bin/sh -c command, this function uses an internal system call to invoke the program, which eliminates many potential safety hazards. In this case, "ls" would be passed as the first argument, and the path would be passed as the second argument. If one tries to pass "/abc;/bin/sh" to the virus scanner program, this entire string will be recognized as a path argument, which stops the attack. The ability to specify environment variables also provides stronger security. ### The principle of isolation The difference between system() and execve() reflects the principle of isolation in computer security, which states that data should be clearly isolated from code. In the virus scanner example above, "ls" is code, which specifies what program we would like to call and should not be changed; path is data, which determines which folder to scan. The system() approach violates this principle by blending data the code together into one string, which introduces many security problems. ### The principle of least privilege The principle of least privilege is introduced by J.H. Saltzer and M.D. Schroeder in 1975. It states that every program and every privileged user of the system should operate using the least amount of privileges necessary to complete the job. In Linux, privileged Set-UID programs violates this principle because they have the power of the root user, which has every possible privilege. Modern operating systems like Android provide find-grained privileges. When we open the application settings page on Android phones, we can set the privilege for a program to access location, camera, microphone, etc. Another implication derived from this principle is that if a privileged program does not need some privileges for part of its execution, it should disable the privileges either temporarily or permanently, depending on whether this privilege needs to be reused later on. For example, if we close the privileged file handle as soon as possible in the capability leaking case, then this problem will not occur. ## Lab 2: Buffer-Overflow Vulnerability ### Process memory layout Thanks to virtual memory, each Linux process has its own address space. The layout of the address space can be seen as below. There are mainly five segments: 1. Text segment: stores the executable code of the program. This segment is usually read-only. 2. Data segment: stores (initialized) global and static variables. 3. BSS segment: stores uninitialized global and static variables. 4. Heap: provides space for dynamic memory allocation. The heap grows from low address to high address. 5. Stack: stores local variables and maintains the structure of the program. The stack grows from high address to low address. #### x86 stack layout Each function has its corresponding stack frame, which stores its local variables and other important status. When a new function is called, a new stack frame is created, and the stack grows (from high address to low address). void bar(){ // <--here } void foo(){ bar(); } int main(){ foo(); return 0; }  For the following above, when the execution goes inside bar(), the stack frames can be illustrated below. Inside the stack frame are arguments, local variables and other values that keep the program running correctly. int func(int a, int b){ int x, y; }  For the func(int, int) above, its stack frame is shown as below. The arguments of this function are at the top of the stack. The return address stores where the execution should continue after func() returns. The %ebp register (aka “stack base pointer”) points to the base of the stack frame, allowing the program to access saved %ebp, return address and its arguments by a fixed offset. The saved %ebp value stores the former %ebp of func()’s caller. When func() returns, the %ebp value will be reverted to the saved value. Below %ebp are the local variables of func(). Its layout is completely determined by the compiler, so we do not know where x and y are exactly. We only know that they are roughly in that region. There is also another “stack pointer” that points to the top of the stack, which is stored in register %esp. ### Buffer overflow attack #### Copying data to buffer Usually, we use the strcpy(char *dest, const char *src) function to copy strings from source to destination. We do not need to specify how long src is, since the copying is automatically terminated when a '\0' is encountered in src. #### Buffer overflow Since the strcpy function does not compare the length of src and dest, it may be the case that the length of src exceeds the maximum length of dest. This is called buffer overflow. This is dangerous, because depending on the layout, other local variables can be overwritten. In the worst case, even the saved %ebp and return address will be modified. When the return address is changed, the program will almost always malfunction since the execution flow is broken. Usually, there are four consequences: 1. Jumping to a invalid address: the memory protection mechanism of OS prevents the program from accessing unallocated memory space. 2. Jumping to a protected address: the target memory space is allocated, but it is protected (e.g. it is reserved for the kernel). Usually these memory access violations will causes segmentation faults. 3. Invalid instruction: the target memory space is allocated, and it is accessible. However, the data there is not a valid machine instruction. 4. Normal execution: the target memory space is allocated, and it is accessible. The data there happens to be valid machine instructions. By making use of buffer overflow carefully, an attacker can deliberately change the return address so that it points to some malicious code in another memory location, thus compromising the logic of the program. #### The vulnerable program Since analyzing complicated programs for buffer overflow vulnerabilities is difficult, we would try our attacks with the following program. The program has a buffer overflow problem, since we are copying str, a string of maximun length of 517 into buffer, whose maximum length is 24. /* stack.c */ /* This program has a buffer overflow vulnerability. */ /* Our task is to exploit this vulnerability */ #include <stdlib.h> #include <stdio.h> #include <string.h> int bof(char *str) { char buffer[24]; /* The following statement has a buffer overflow problem */ strcpy(buffer, str); return 1; } int main(int argc, char **argv) { char str[517]; FILE *badfile; badfile = fopen("badfile", "r"); fread(str, sizeof(char), 517, badfile); bof(str); printf("Returned Properly\n"); return 1; }  Now we analyze how to exploit the buffer overflow vulnerability of this problem. The bof function’s stack frame is shown in the figure below. When we call the strcpy function, it copies data from str[0] to buffer[0], str[1] to buffer[1], etc. If the length of str is longer than 24, it can be seen that we will tamper the variables above buffer, namely saved ebp, return address and arguments. We may also modify the main function’s stack frame. Our goal is to modify the return address slot so that when the execution of bof finishes, we can mislead the program to “return” to our own malicious code segment. It is worth noticing that our malicious code segment is transferred into the victim computer using the input buffer (in this case, a file named badfile; but it could be input acquired from standard input). This immediately brings us the two major challenges of a buffer overflow attack: 1. How to determine the offset of the return address? Since we need to overwrite the old return address with our own desired one, we need to guess where to place the return address value in our input buffer. If there is a misalignment, the attack will not succeed. 2. How to determine which address to return to? The return address will point to the absolute location of the code that will be executed next. In order to trick the program into running our own code, we have to know the absolute location of where our input will be put into the memory. It is very difficult to meet these two requirements when we do not have access to the vulnerable program’s source code. However, as we can see that there are ways to make these two conditions “fuzzy”, i.e. sometimes we can launch the attack successfully without knowing the exact address. #### Conducting a buffer overflow attack To conduct a (toy-level) buffer overflow attack, we need to follow these steps: 1. Turn off countermeasures There are a lot of countermeasures that are already implemented in the Linux system to defend against buffer overflow attack. We need to disable all of them to make our life easier. • Turn off address space randomization Address space randomization will shuffle the address space, which makes it more difficult to guess the correct value of return address. It can be turned off using the following command: [09/13/19]seed@VM:~ sudo sysctl -w kernel.randomize_va_space=0
kernel.randomize_va_space = 0

• Turn off compiler’s protections

We need to compile stack.c with the following command:

[09/13/19]seed@VM:~/.../lab2\$ gcc -z execstack -fno-stack-protector -o stack stack.c

The -z execstack flag disables non-executable stack protection. Because our malicious code is stored inside the stack frames, this mechanism will stop the code from executing.

The -fno-stack-protector flag disables stack protector, which is a special code segment added to the program to detect if there is a stack overflow. Again, if it is not canceled, our attack will not succeed.


text:
Global well-posedness of a system from quantum hydrodynamics for small data
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 7-16.

This article describes a joint work of the author with B.Haspot on the existence and uniqueness of global solutions for the Euler-Korteweg equations in the special case of quantum hydrodynamics. Our aim here is to sketch how one can construct global small solutions of the Gross-Pitaevskii equation and use the so-called Madelung transform to convert these into solutions without vacuum of the quantum hydrodynamics. A key point is to bound the the solution of the Gross-Pitaevskii equation away from $0$, this condition is fullfilled thanks to recent scattering results.

DOI : https://doi.org/10.5802/cml.21
Classification : 35A01,  35Q31,  35Q55,  76D45
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author = {Audiard, Corentin},
title = {Global well-posedness of a system from quantum hydrodynamics for small data},
journal = {Confluentes Mathematici},
pages = {7--16},
publisher = {Institut Camille Jordan},
volume = {7},
number = {2},
year = {2015},
doi = {10.5802/cml.21},
language = {en},
url = {http://www.numdam.org/articles/10.5802/cml.21/}
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Audiard, Corentin. Global well-posedness of a system from quantum hydrodynamics for small data. Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 7-16. doi : 10.5802/cml.21. http://www.numdam.org/articles/10.5802/cml.21/

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[2] Audiard, Corentin; Haspot, Boris From Gross-Pitaevskii equation to Euler-Korteweg system, existence of global strong solutions with small irrotational initial data, preprint

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[4] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations (2006), pp. No. 59, 35 pp. (electronic) | MR 2226932 | Zbl 1114.76058

[5] Bona, J.; Ponce, G.; Saut, J.C.; Sparber, C. Dispersive blow up for nonlinear Schrödinger equations revisited, preprint

[6] Bourgain, J. Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., Volume 12 (1999) no. 1, pp. 145-171 | Article | MR 1626257 | Zbl 0958.35126

[7] Carles, Rémi; Danchin, Raphaël; Saut, Jean-Claude Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, Volume 25 (2012) no. 10, pp. 2843-2873 | Article | MR 2979973 | Zbl 1251.35142

[8] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${ℝ}^{3}$, Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 767-865 | Article | MR 2415387 | Zbl 1178.35345

[9] Gérard, P. The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 23 (2006) no. 5, pp. 765-779 | Article | MR 2259616 | Zbl 1122.35133

[10] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), Volume 175 (2012) no. 2, pp. 691-754 | Article | MR 2993751 | Zbl 1241.35003

[11] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN (2009) no. 3, pp. 414-432 | Article | MR 2482120 | Zbl 1156.35087

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[13] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng Scattering for the Gross-Pitaevskii equation, Math. Res. Lett., Volume 13 (2006) no. 2-3, pp. 273-285 | Article | MR 2231117 | Zbl 1119.35084

[14] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, Volume 8 (2007) no. 7, pp. 1303-1331 | Article | MR 2360438

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# How to find the wave function using WKB?

1. Dec 15, 2013

### a1111

I would like to understand how to find wave functions using WKB.

1. The problem statement, all variables and given/known data

Given an electron, say, in the nuclear potential

$$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0} \end{cases}$$

With the barrier region given by:

$$r_{0} < r < k/E$$

What is the wave function inside the barrier region?

2. Relevant equations

See below.

3. The attempt at a solution

For E > U(r):

$$\psi(r)=\frac{Ae^{i\phi(x)}}{\sqrt{2m(E-U(r)}}+\frac{Be^{-\phi(x)}}{\sqrt{2m(E-U(r)}}$$

$$\phi(r)=\frac{1}{\hbar}\int^{r}_{0}\sqrt{2m(E-U(r')}=\frac{1}{\hbar}\int^{r}_{0}\sqrt{2m(E+U_{0})}dr'=\frac{r\sqrt{2m(E+U_{0})}}{\hbar}$$

Is the above integral correct? Is that simply all that I needed to do?

Last edited: Dec 15, 2013


text:
# werkzeug.BaseRequest.path¶

BaseRequest.path[source]

Requested path as unicode. This works a bit like the regular path info in the WSGI environment but will always include a leading slash, even if the URL root is accessed.


text:
Znaj siebie.

## Trygonometryczne wzory redukcyjne

Korzystając z wzorów redukcyjnych możemy wartość funkcji trygonometrycznych dowolnego kąta zapisać w postaci funkcji trygonometrycznej kąta ostrego, co ułatwi nam obliczenie wartości jeśli nie posiadamy przy sobie kalkulatora. Takie uproszczenie pozwoli nam wykonywać działania np. mając $$cos \: 135^o$$ możemy go zapisać w postaci $$cos (180^o - 45^o)$$, co po uproszczeniu daje nam $$-cos \: 45^o$$ co jest równe $$\dfrac{\sqrt{2}}{2}$$.

 $$\varphi$$ $$sin \: \varphi$$ $$cos \:\varphi$$ $$tg \: \varphi$$ $$ctg \: \varphi$$ $$- \alpha$$ $$- \alpha$$ $$- sin \: \alpha$$ $$cos \: \alpha$$ $$- tg \: \alpha$$ $$- ctg \: \alpha$$ $$90^o + \alpha$$ $$\dfrac{\pi}{2} + \alpha$$ $$cos \: \alpha$$ $$- sin \: \alpha$$ $$- ctg \: \alpha$$ $$- tg \: \alpha$$ $$90^o - \alpha$$ $$\dfrac{\pi}{2} - \alpha$$ $$cos \: \alpha$$ $$sin \: \alpha$$ $$ctg \: \alpha$$ $$tg \: \alpha$$ $$180^o + \alpha$$ $$\pi +\alpha$$ $$- sin \: \alpha$$ $$- cos \: \alpha$$ $$tg \: \alpha$$ $$ctg \: \alpha$$ $$180^o - \alpha$$ $$\pi -\alpha$$ $$sin \: \alpha$$ $$- cos \: \alpha$$ $$- tg \: \alpha$$ $$- ctg \: \alpha$$ $$270^o + \alpha$$ $$\dfrac{3}{2} \pi + \alpha$$ $$- cos \: \alpha$$ $$sin \: \alpha$$ $$- ctg \: \alpha$$ $$-tg \: \alpha$$ $$270^o - \alpha$$ $$\dfrac{3}{2} \pi + \alpha$$ $$- cos \: \alpha$$ $$- sin \: \alpha$$ $$ctg \: \alpha$$ $$tg \: \alpha$$ $$360^o + \alpha$$ $$2\pi + \alpha$$ $$sin \: \alpha$$ $$cos \: \alpha$$ $$tg \: \alpha$$ $$ctg \: \alpha$$ $$360^o - \alpha$$ $$2\pi - \alpha$$ $$- sin \: \alpha$$ $$cos \: \alpha$$ $$- tg \: \alpha$$ $$- tg \: \alpha$$

$$sin \:150^o = sin(90^o + 60^o) = cos \: 60^o = \dfrac{1}{2}$$
$$cos \: 120^o = cos(180^o - 60^o) = -cos \: 60^o = -\dfrac{1}{2}$$
$$tg \: (240^o) = tg(270^o - 30^o) = ctg \: 30^o = \sqrt{3}$$

Wartości funkcji trygonometrycznych wybranych kątów

$$\dfrac{\sqrt{2}}{2}$$

 $$\varphi$$ $$sin \: \varphi$$ $$cos \:\varphi$$ $$tg \: \varphi$$ $$ctg \: \varphi$$ $$- \alpha$$ $$- \alpha$$ $$- sin \: \alpha$$ $$cos \: \alpha$$ $$- tg \: \alpha$$ $$- ctg \: \alpha$$ $$90^o + \alpha$$ $$\dfrac{\pi}{2} + \alpha$$ $$cos \: \alpha$$ $$- sin \: \alpha$$ $$- ctg \: \alpha$$ $$- tg \: \alpha$$ $$90^o - \alpha$$ $$\dfrac{\pi}{2} - \alpha$$ $$cos \: \alpha$$ $$sin \: \alpha$$ $$ctg \: \alpha$$ $$tg \: \alpha$$ $$180^o + \alpha$$ $$\pi +\alpha$$ $$- sin \: \alpha$$ $$- cos \: \alpha$$ $$tg \: \alpha$$ $$ctg \: \alpha$$ $$180^o - \alpha$$ $$\pi -\alpha$$ $$sin \: \alpha$$ $$- cos \: \alpha$$ $$- tg \: \alpha$$ $$- ctg \: \alpha$$ $$270^o + \alpha$$ $$\dfrac{3}{2} \pi + \alpha$$ $$- cos \: \alpha$$ $$sin \: \alpha$$ $$- ctg \: \alpha$$ $$-tg \: \alpha$$ $$270^o - \alpha$$ $$\dfrac{3}{2} \pi + \alpha$$ $$- cos \: \alpha$$ $$- sin \: \alpha$$ $$ctg \: \alpha$$ $$tg \: \alpha$$ $$360^o + \alpha$$ $$2\pi + \alpha$$ $$sin \: \alpha$$ $$cos \: \alpha$$ $$tg \: \alpha$$ $$ctg \: \alpha$$ $$360^o - \alpha$$ $$2\pi - \alpha$$ $$- sin \: \alpha$$ $$cos \: \alpha$$ $$- tg \: \alpha$$ $$- tg \: \alpha$$
cos \: 60^o = \dfrac{1}{2}\)

Potrzebujesz pomocy? Przejrzyj ogłoszenia korepetytorów z matematyki na http://e-korepetycje.net/


text:
Buckling load will be maximum for a column if its

This question was previously asked in
UPPSC AE Mechanical 2022 Official Paper I (Held on 29 May 2022)
View all UPPSC AE Papers >
1. One end is fixed and other end is free
2. One end of the column is hinged and other end is free
3. Both ends of the column are clamped
4. Both ends of the column are hinged

Option 3 : Both ends of the column are clamped
Free
CT 1: Indian History
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Detailed Solution

Explanation:

$${P_b} = \frac{{{\pi ^2}EI}}{{L_e^2}}$$
 End conditions Le Buckling load Both ends hinged Le = L $${P_b} = \frac{{{\pi ^2}EI}}{{L_e^2}}$$ Both ends fixed Le = L/2 $${P_b} = \frac{{{4\pi ^2}EI}}{{L_e^2}}$$ One end fixed and another end is free Le = 2L $${P_b} = \frac{{{\pi ^2}EI}}{{4L_e^2}}$$ One end fixed and another end is hinged $${L_e} = \frac{L}{{\sqrt 2 }}$$ $${P_b} = \frac{{{2\pi ^2}EI}}{{L_e^2}}$$


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# On the Complexity of Bandwidth Allocation in Radio Networks with Steady Traffic Demands

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In this paper we define and study a call scheduling problem that is motivated by radio networks. In such networks the physical space is a common resource that nodes have to share, since concurrent transmissions cannot be interfering. We study how one can satisfy steady bandwidth demands according to this constraint. This leads to the definition of a call scheduling problem. We show that it can be relaxed into a simpler problem: The call weighting problem, which is almost a usual multi-commodity flow problem, but the capacity constraints are replaced by the much more complex notion of non interference. Not surprisingly this notion involve independent sets, and we prove that the complexity of the call weighting problem is strongly related to the one of the independent set problem and its variants (max-weight, coloring, fractional coloring). The hardness of approximation follows when the interferences are described by an arbitrary graph. We refine our study by considering some particular cases for which efficient polynomial algorithm can be provided: the Gathering in which all the demand are directed toward the same sink, and specific interference relations: namely those induced by the dimension $1$ and $2$ Euclidean space, those cases are likely to be the practical ones.
Keywords :
Document type :
Reports
Domain :

Cited literature [22 references]

https://hal.inria.fr/inria-00070575
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 8:55:28 PM
Last modification on : Monday, October 12, 2020 - 10:30:21 AM
Long-term archiving on: : Sunday, April 4, 2010 - 9:30:16 PM

### Identifiers

• HAL Id : inria-00070575, version 1

### Citation

Ralf Klasing, Nelson Morales, Stéphane Pérennes. On the Complexity of Bandwidth Allocation in Radio Networks with Steady Traffic Demands. [Research Report] RR-5432, INRIA. 2004, pp.27. ⟨inria-00070575⟩

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